Lee - The Design of CMOS RF Integrated Circuits

This co mprehensive and insig htful book sets ou t in det ail how to design gigahertzspeed radio-frequency integrated circuits in C MOS tec hnology. Slafti ng ,,"'ilh d hi!olury of ,ihliu to c~wblbb OJ fou lk.lOJtiun
THE DESIGN OF CMOS RADIO -FREQUENCY INTEGRATED CIRCUITS

THOMAS H. LEE Sial/ford University

U

V

C AMBR ID G E UN IVERS ITY PR ESS

PU BLIS HED BY TH E PR ESS SYNDICATE OF TilE UNIVERS ITY OF CAMBRIDGE The Pitt Building, Trump ington Street. Cam bridge, United Kingdom CA MBRIDG E UN IVERS ITY PRESS The Edinbu rgh Building. Cambridge C B2 2RU. UK 40 Wesl 20th Street , Ne w York . N Y 100 11-421 1. USA 10 Stamford Road. Oak leig h, VIC 3 166, Australia Ruiz de Alarcon 13. 28 0 14 Madrid, Spain Dock House. Th e Wate rfro nt. Cape Town 800 1, South Afri ca http://www.cambridge.org

o Cam bridge

Universi ty Pre ss 1998

This boo k is in copyrig ht. Subjcct lo statutory exception and to the provisions of rele vant co llect ive licen sing agree ments. no reprcdccncn of any pan may take place wi thou t the written permission of Cam bridge Un iversity Press. First pu blished 1998

Reprinted with co rrec tion s 1998 Reprin ted 1999,2000,200 1 Printed in the: Unued Stares of America Type set in Tim es using AMS·TEX

Libra ry of Congress Ca taloging-in-Publication Dtlla

Lee, Thom as H.• 1959The de sign of C MOS radio-freque"')' inleg raled circu its I Thomas U. Lee. p.

em.

ISBN 0-52 1-6306 1-4 ( hb) - ISBN 0-52 1·63922-0 l pb) I. Metal ox ide semico nduc tors , Compleme ntary - Design and conwccuon . 2. Integrated circuits - Design and co nstruction . J . Radio - Traesrmuer-recetvers. 1. Tille . TK787 1.99.M44L44 1998 621.39'732 - !X2 1 97-34 158

CIP A catalog rrcord for this book is m'aiiab/e fro m the British Library ISBN 0 521 6306 1 4 hardback ISBN 0521639220 paperback

To my parents. who had no idea what they were starting when they bought me a pair of walkie-talkies for my sixth birthday

•

CONTENTS

Pref ace

p ag"

XIII

A NONLINEAR HISTORY OF RADIO 1. Introduction 2. Maxwell and Hertz

3. 4. 5. 6. 7. 8. 9. 10.

Pre-Vacuum Tube Electron ics Hirth of the Vacuum Tube Armstrong and the Regenerative Amplifer/ Detecror/Oscillaror Other Radio Circuits Armstrong and the Supe rregenerat or Olcg Losev and the First So lid -Slate Amplifi er

Epilog A ppendix: A Vacuum Tube Primer

2 CHARACTERISTICS OF PASSIVE IC CO M PO N EN TS I. Introdu ction 2. Resistors

)4 J4 34

3. Capacitors

37

4. Inductors

47

5. Summary 6. Appendi x: Summary of Ca paci tance Equat ions Problem Set

57 57 58

62 62 62

3 A REVIEW OF MOS DEVICE PHYSICS

I. lntrodur.-tion 2. A lill ie History

•

2 9 13 15 18 20 21 22

3. FETs: The Shon Sto ry 4. MOSF ET Physics: The Long-Chan nel Approxi mation 5. Operatio n in Weak Inversion (Subthreshold)

6) 64

n

vii

THE LIBRA RY UNIVE RSITY OF WE ST FLORIDA

CO NTE NTS

\/iii

6 . MOS Device Phys ics in the Short -Ch a nne l Regime 7. O ther Effects 8. Tran sit Tim e Effects 9. Summary 10. A ppe ndix: O.5·,Hl 1 Level-S SP ICE Models Problem Sc i

A PASSIVE RIC NETWORKS I. lr ur oductic n 2. Parallel R LC Ta nk 3. Se ries RL C Networks 4. Ot her Resona nt RL C Ne twor ks 5. R t. C Ne tworks as Impedance Transform e rs 6. Exam ples Problem Sci

5 DI STRIBUTED SYSTEMS I. Introd uction 2. Link Bet ween Lumped and Distrib uted Regimes 3. Driving-Point Impeda nce of Itera ted St ruct ures 4 . Tra nsmission Lines in More De ta il 5. Behavior uf Finite- Len gth Tra nsmiss io n Lines 6. Sum ma ry of Transmi ssio n- Line Equatio ns 7. Artificia l Lines 8. Summary Proble m Sci

6 THE SMITH CHART AND S-PARAMETERS I. Introd uc tio n 2. Th e S mith C hart 3. S-Paralllcler s 4 . Appen dix : A Sho rt Note on Units 5. A ppe ndix: Why 50 (or 75) Q? Problem Set

7 BANDWIDTH ESTIMATION TECHN IQUES I. Introduction 2. Th e Method of Open-Circuit T ime Constant s 3. Th e Method of Short -Circuit Time Constant s 4. Further Read ing 5. Risctime. Del ay. a nd Bandwidth 6. Sum mary Prob lem Sc i

73 77 7~

XO XO XI X6 X6 86 ~I ~I

~3

104 106

114 11 4 11 6 117 11 9 124 127 127 131 13 1 134 134 134 I3 X 140 141 144 146 146 147 16 1 166 167

174 174

;,

CO NTENTS

8 HIGH ·fREQUENCY AMPLIfiER DESIGN I. 2. 3. 4.

Intr oduc tio n Zeros as Band widt h Enha ncers T he Sh unt-Series A mp lifie r Band wid th Enha nce ment with Ir Dou ble rs 5. TImcd Am pl ifiers 6. Ne utrali zation a nd Unil areralizarion 7. Ca sca dc-d Amplifiers 8. Su mmary Problem Se t

9 VOLTAGE REFERENCES AND BIASING I . Introd uctio n

2. Review of Diod e Beh avior 3. 4. 5. 6. 7.

Diodes and Bipol ar Tran sistors in C MOS Tec hnol og y Supply-Independent Bias C ircuits Bandgap Voltage Reference Co nstant-g; Bias Su mmary Problem Sc i

10 NOISE l . Introd uctio n 2. T hermal No ise .1 . 4. 5. 6.

II

17M 178 179 191 197 199 203 206

216 2 17

223 223 22.1 225

225

227 235

237 237 243 243 24.1 250 252 255 256

Sho t No ise Flicker No ise Popcorn Noise C lassica l Two -Port Noise T heory 7. Exam ples of Noise Calculations l'I . A Hand y Rule of Thumb 9. Typical Noise Perfo rmance 10. Appendix: Noise Models Pro blem Set

265 266

LNA DESIGN

272

J. Introd uction 2. Der ivatio n of MOS F ET Two -Po rt Noi se Parameters 3. 4. 5. 6. 7.

LNA To po logies: Pow er Matc h versus Noi se Milich Po wer-Con strained Noi se O ptimiza tio n Design Exa mples Li nea rity and Large-Sig nal Perform ance Spurio us-f ree Dyna mic Ra nge

260 263 2M

272

273 277 284 2MM 295 302

CO NTENTS

•

12

13

8. Summary Problem Sci

303 304

MI XERS I. Introd uction 2. Mixer Fundamentals 3. Nonlinear Systems as Linear Mixers 4 . Multi plier -Based Mixers

308 308

335 337

RF PO W ER AMPLIF I ERS I. Introd uction 2. Gen eral Conside ration s 3. Cl ass A. AB. B. and C Power Amplifi ers

J.I4 J.I4 344 345 355 357 359 362 364 365

5. Class E Ampli fiers 6. Class F Amp lifiers 7. M odul ation of Power A mplifiers 8. Sum mary of PA C haracteristics 9. RF I'A Design Examp les 10 . Additional Design Considerations 11. Design Summary Problem Set FEE DBAC K SYSTEM S I. Introd uction 2. A Brief History of Mod em Feedback

3. 4. 5. 6. 7. 8. 9. 10 . I I. 12. 13. 14 . 15.

I

~

3 14 3 18

5. Subsampling Mi xers 6. Appendi x: Diode -R ing Mixers Problem SCi

-I. C1u!Os [) Amplifiers

14

309

A Puzzle Desen sitivity of Negative Feedb ack Sy stems Stabili ty of Feedb ack Sy stems Gain and Phase Margin as Stability Measures Root-Locus Techniq ues Summary of Stability Cr iteria Modeling Feed back Systems Errors in Feedb ack Systems Frequency- and Tim e- Domai n Cha racteris tics of First- and Second-Orde r Syste ms Useful Rules of Thumb Root-Locus Example s and Compe nsation Sum mary of Root -Loc us Techniques Co mpen sation

340

372

379 380

385 385 385 390 391 395 396 398 404 404

408 4 10 4 14

41 5 4 22 4 23

CO NTEN TS

15

,i

16. Compe nsa tio n th rou gh Gain Red uction 17. Lag Co mpe nsa tion 18. Lead Compensation 19. Summary of Compe nsalion Probl em Set

423 426 -l29 -l32

PHA SE-LOCK ED LOOP S

438 43H 43H 44 1 447 450 455 46 3 4711 47H 47H

I. Introducti on 2. A Short History o f I'LL~ 3. Lineari zed PLL Model s -I. Some No ise Propertie s of PLL s 5. Phase Detectors 6. Sequen tial Phase Detectors 7. Loop Filters and Cha rge Pumps R. I'LL Design Examples 9. Summary Proh le m Ser

16 OSCILLATOR S AND SYNTHES IZE RS I. 2. 3. -I. 5. 6. 7. 8.

Introdu ction Th e Problem w ith Pur ely Linear O sc illat ors Describing Fu nctions Reso nato rs A Catalog of Tuned O scill ators Nega tive Resistance O sci llato rs Freque ncy Sy nthesis Su mm ary Problem Set

17 PHA SE NOI SE I . Introd uction 2. General Co nsidera tions 3. Detai led Conside rations: Phase No ise 4. The Haj imiri Model : A Time-Varying Phase No ise Th eory 5. Summary Pro hle m Set

18 ARCHITE CTURE S I. 2. 3. -I. 5.

Introd uction Dyn amic Range Subsa mpling Tn•ansmi tter Archi tectu res O scill ator Stabi lity

-132

484 4H4 484 485 5(X)

5().j

5"" 5 1-1 524 525 5311 5311 530 532 536 544

5-15 5511 550

551 565 566 567

CO NTE NTS

xii

19

6. Summary Problem Sci

568 568

RF CIRCU ITS THROUGH THE AGES

57\ 571 57\ 575 578 580

I. 2. 3. 4. 5.

Intr oduction Armstro ng The "All-American" 5-Tube Superhet The Regency TR -I Transistor Radio Three -Transistor Toy CB Walkie -Talkie

PREFACE

The field of radio frequency (RF) circuit design is currently enjoying a renaissance. driven in particular by the rece nt. and largely unanticipated. explosive growth in wireless telecommunications, Because this resurgence of interest in Rf ca ught industry and academ ia by surprise. there has been a mad scramble to educate a new generation of RF engineers. However. in trying to synthesize the two traditi ons of "conventional" RF and lower-freq uency IC design. one enco unters a problem : "Traditional" RF engineers and analog IC designers often find communication with each other difficult because of their diverse backgrounds and the d iffere nces in the media in which they realize their circuits. Radio -frequ ency IC design . particularly in C ~f OS . is a different activity alto gether from discrete RF design. T his hook is intended as both a link to the past and a pointe r to the future. The contents of this boo k derive from a set of notes used to teach a one -term ndvanccd graduate course on RF IC design at Stan ford University. The course was a follow-up to a low-frequency ana log Ie design cla ss. and this hook therefore assumes that the reader is intimately famil iar with that subjec t. described in standard texts such as Al/aly.\I \' and lk.\';RII /d Analog i ntegrated Circu its by P. R. Gray and R. G. Meyer (Wiley. 19 lJ) ). Some review materia l is provided. so that the practicin g engineer with a few neurons surviving from undergraduate education will be able to dive in without too much disorientation. The amount (If materia l here is significantly beyond what students can comfo rtubly assimilat e in one quarter or semester. and instructors arc invited to pick and choose topics to suit their tastes. the length of the academic term. and the background level of the students, In the cha pter descript ions that follow are included some hints about what chapters may he comfortably omit ted or deferred . Chapter I presen ts an erratic history of rad io. T his material is presented largely for cultural reasons. T he author recog nizes that not everyone finds hislory interesting. so the impatie nt reader is invited to skip ahead to the more tech nical chapters. Chapter 2 surveys the passive compo nents norm ally available in standard C ~1 0S processes. There is a focus on inductors bec ause of their pro minent ro le in kF ...iii

•

••

PREFACE

circuits. and also because material o n this subject i.. . scattered in the currentliterature (although. hap pily, this situation is rapidly changing). Chapter 3 provides a quick review of MOS device physics and modeli ng. Since deep submicron technology is now co mmonplace, there is a focus on approximate annlyrical models that account for short-channel effects. This chapter is necessarily brief. and is intended only as a supplement 10 more detailed treatments available elsewhere. Chapter 4 exa mines the properties of lumped. passive RLC networks. For advanced students. this chapter may he a review ami may be skipped if desired . In the author's experience . most undergraduate curricula essentially abandoned the teaching (I f inductors long ago. so this chapter spends a fair amount of time examining the issues of reso nance. Q. and impedance matching. Chapter 5 extends into the dist ributed realm many of the concepts introduced in the context of lumped networks. Transmission lines are introd uced in a somewhat unusual way. with the treatment a..o iding altogether the derivat ion o f the telegrapher 's equation with its attendant wave solutions. The charac teristic impedance and propagation constant of a uniform line are derived entirely from simple extensions of lumped ideas. Although distributed networks play hut a minor role in the current generation of silico n IC technology. that slate of affairs will be temporary. given that device speeds are doubling abo ut every three years. Chapter 6 provides an import ant bridge between the traditional " microwave plumber 's" mind-set and the IC designer 's world view by prese nting a simple derivation of the Smith chart, explaining what Svpara me ters are and why they are useful. Even though the rypicul IC engineer will almo st certainly not design circuits using these tools. much instrumentation present s data in Smith-chart and Scparamctcr form. so modern engineers still need to be conversant with them . Chapter 7 presents numerous simple methods for estimating the band width of highorder systems from a series of first-order calculations or from simple measurements. The former set of techniq ues. called the method o f ope n-circuit (o r zero -value) time constants. allows one to identify bandwidth-lim iting parts of a circuit w hile providing a typically conserva tive bandwidth estimate. Relationshi ps among bandwidth . delay. ami risetimc allow us to identify important degrees of freedo m in trading off various parameters. In particular. gain and bandwidth are shown not to trade off with one another in any fundamental way. contrary to the beliefs of many (if not most) engineers. Rather. gain ami delay are shown to be more tightly coupled. opening signifi cant loopholes that point the way 10 amplifier architectures which effect thai tradeo ff and leave bandwidth largely untouched. Chapter 8 take!"> a detailed look at the problem of designing extremely highfrequency amplifiers. both broad- and narrowband. with many " tricks" evolving from a purposeful violation of the assumptions underlying the method o f ope n-ci rcuit time constants. Chapter 9 surveys a number of biasing methods. Although intended mainly as a review, the problems o f implementing good references in standard C MOS are

.,. PR EFACE

large enough to risk so me repetition. In particu lar. the design of CM OS -co mpatible bandgap voltage references and co nsrant-transcon ductunce bias ci rcuits are empha sized here. perhaps a little more so Ihan in most standard analog texis. Chapter 10 stud ies the all-important issue of noise . Simply obtaining sufficie nt gain over some acceptable bandwidth is frequently insuff icient . In many wireless applications, the rece ived signal amplitude is in the microvolt range . The need 10 amplify such minu te signals as no iselessly as possible is selt-evide nt. and this chapter provides the nec essary founda tion foridentifyi ng co nditions for achieving the best possible noise performance from a given technology, Chapter II follow s up on the previou s two or three chapters to identify low -noi se amplifier ( LNA) architectu res and the specific cond itions that lead 10 the best possible noise performa nce . given an ex plicit co nstrai nt on po wer consumption. Thi s power-con strai ned approac h differs consider ably from standard discrete -oriented method s. and explo its the freedom enjoyed by Ie designers to tail or device sizes to achieve a particu lar optim um. The important issue of dy nam ic range is also examined. and a simple analytical method for estimating a large-signal linearit y limit is presented . Chapter 12 introduce s the first intentionally no nlinear element. and the heart of all modem transceivers : the mixer. After identifyin g key mixer performa nce parameters. numerous mixer topologies are examined. As with the LNA . the issue of dynamic range is kept in focu s the entire time. Chapter 13 presents numero us topo logies for bui lding RF power amplifiers. The serious and often unsatisfactory tradeo ff's among ga in. efficicncy. Iincarity, and output power lead to a family of topo logies. eac h with its particular do ma in of application. The cha pter clo ses with an exa mination of load-pull ex perimental characterizations of real power ampli fiers. Chapter 14 pro vides a review of cla ssical feedback concepts. mainly in prep aration for the following chapter on phase -locked loops. Reader s with a so lid back groun d in feedback may wish to skim it . or even skip it entirely. Chapter 15 surveys a number of phase-locked loop circuits after presenting basic operating theory of both first- and second-order loo ps. Loo p stability is examin ed in detail. and ,I simple criterion fo r assessing a PLL' s sensitivity to power supply and substrate noise is offered. Chapter 16 exam ines in detail the issue o f oscillators and frequency synthes izers. Both relaxation and tuned o..cillators are considered. with the Jailer category further subdivided into LC and crystal-contro lled oscillators. Both fixed and cor urolls ble oscillator s are presented . Prediction of oscillation amp litude. crit eria for start-up. and device sizing are all studied . Chapter 17 extends to oscillators the earlier work on noise. After elucidating some general criteria for optimizing the no ise performa nce of oscillators. a powerfu l theory of phase noi se based on a linear. time -vary ing mood is presented . The model makes some surprisingly optimistic (and ex perimentally verified ) predictions abou t what

PR EfACE

one may do to red uce the phase noise o f osc illators buill with such infamously noisy devices as MO SFETs, Chapter 18 ties all the previous chapters together and surveys arc hitectures of rece ivers and transmit ters. Rules are derived for computing the intercept and noise figure of a ca...cadc of subsys tems. Trad itiunal superheterody ne architect ures are exumincd. along with low- IF image-reject and direct-conversion receivers. The relative merits and disad vantages of each of these is studied in detail. Finally, Chapter 19 clo ses the boo k the way it began: with some history. A nonuniform sampling of c lassical (and di stinctly non-CM OS ) RF ci rcuits takes a look at Armstrong's earliest inven tions. the "All-A mcrican Five" vac uum tube table md lo. the first transistor radio. and the first toy walkie-tal kie. As with the first chapter . this one is present ed purely for enjoyme nt. so those who do not find history lessons enjoyable or worthw hile arc invited to close the book and revel in having made it through the whole thing. A book o f this length could not have bee n com ple ted in the given time were it not for the ge nerous and co mpe tent help of colleag ues and stude nts. My wonde rful administrative assistant. Ann G uerra. magically created time by handli ng everything with her remarkable good cheer and efficiency, Also . the followi ng Ph .D. stude nts went far beyond the call of duty in proofreadi ng the manuscript and suggesting or ge nerating examples and many o f the problem-set questions: Tamara Ahren s, Rafael Hetancourt -Zarnoru. David Collera n, Ramin Farjad-Rad, Mar Hershen son . Joe Ingino. Adrian O ng. Hamid Raregh . Hirad Sa mavati. Brian Se ne rbc rg, Arv in Sbaha ni. and Kevin Yu. Ali Haj imiri , Su nderarajan S. Mohan . and Derek Shaeffer merit special men tion for thei r co nspicuous contribut ions. Without the ir help. given in the eleven th hour. this txXIK would still be awaiting comple tion, The autho r is also ex tremely grateful to the text's reviewers. both known and anonymous, who all had excellent. thou ghtful sugges tions. Of the former gro up, Mr. Howard Swain (formerly of Hewlett -Pack ard ), Dr. Gilly Nusserbakfu of Texas Instrumc nts. and Professors James Robe rge of the Massachusetts Institute of Technol ogy and Kurtikeya Maya nnn of Washin gton State Un iversity deserve speci al thanks fo r spotting typo graphical and gruphicnl errors, and also for thei r valuable edito rial suggestions. Malt and Vickie Darn e ll of Four-Hand Book Pnckuging did a fantastic j ob of co pyedui ng and type setting. Their valiant efforts to convert my "sow ' s car" of a manuscript into the proverbial silk pu rse were noth ing short of supe rhuman. And Dr. Phi lip Meyler of C ambridge University Press started this whole thing by urging me to write this book in the first place. so he 's the one to blam e . Despite the deli ght taken by students in findin g mistakes in the professor 's notes. some errors have managed to slip through the sieve. eve n after three years of filtering. Sadly. this suggests that more await discovery by you. I suppose that is whal second edit ions are for ,

i!2

CHAPTER ONE

A NONLINEAR HISTORY OF RADIO

1.1

INT RODUCTION

Integrated circuit engineers have the luxury of lak ing for granted thai the incremen tal cost of a tran sistor is essentia lly zero, a nd this has led ( 0 the high-device-co unt circuits that are common toda y. Of course. this situation is a relatively rece nt develop ment: during most of the history of electronics. the economics of circu it de sign were the inverse of wha t they are tod ay. II rea lly wasn't all thai long ago when an engi neer was forced by the relatively high cos t of ac tive devices 10 try to get blood (or atlea..1 rectification) from a stone . And it is indeed remark able j ust how much performance radio pionee rs were ah le to squee ze out of just a handful (If compo nents. For exa mple. we' Il SI."e how American radio ge nius Ed win Armstrong devised circuits in the early 1920 s that trade log of gain for ban dwidth . contrary 10 the conventional wisdom that gain and bandwidth should trade o ff mo re or less directl y. And we ' ll SCI.' that at the same time Armstron g was developing those circuits, self-taught So viet radio engineer Oleg Loscv was experime nting with blu e LEDs and constructing com pletely solid -state radios that func tio ned up to 5 MHz , a qua rter ce ntury before the transistor was invented . These fascinati ng stories are rare ly told because they tend to fall into the cracks between history and engineering curricula. Somebody ought to tell these stories, though, since in so doin g, man y commonly asked question s (" why don' t they do it this way?") arc answered (vth cy used [0 , but it caused key body pa n s to fall off") . This highly non linear history of radio touches briefly on j ust so me of the mai n stories, and provides poin ters to the literature for those who want 10 probe further .

1.2 MAXWELL AND HERTZ Every elec trical engineer knows at least a bit about James C lerk ( pronounced "clark") Maxwell; he wrote tho se equat ions rhar made life extra busy back in sophomore yea r

2

CHAPTER 1 A NONLINEAR HISTOR Y O f RADIO

or thereabouts. Nor only d id he write the electrodynamic eq uations! that bear his name. he also published the first mathematical treatm ent of stability in feedback systems ("On Governors: ' which explained w hy speed controllers for "team engines could sometimes he unstab tc-) . Maxwell collec ted all that was then know n abo ut electromagnetic phenomena and . in a mysterious) ami brilli ant stroke. inven ted the displacemen t (capac itive) current term that allowed him to derive an equation thai led 10 the prediction of electromagnetic wave propagation . Then came Heinrich Hertz. who was the first to verify experimentally Maxwell's prediction thai electromagnetic waves exist and propagate with a finite velocity. His "t ransmitters" worked on this simple idea : discha rge a coil acro ss a spark gap and hook up some kind of an antenna to launch a wavet uninrentionally) rich in harmonics. His setup naturally provided only the must rudimentary fi ltering of this dirt y signal. so it took extraordinary care and persistence to verify the existence o f (and to qua ntify) the interference nulls and peaks that are the earmarks of wave phenomena . He also managed to demon strate such qu intessential wave behavior as refraction and polarization . And you may be surprised that the fundamental frequencies he worked with were bctween Sn and S{XJ Mi ll . He was actually forc ed to these frequencie s because his laboratory was simply too small 10 enclose several wavelengths of anything lower in frequency. Because Hertz's sensor was ano ther spark gap (integra l with a loop resonator). the received signal had to be large enough to induce a visible spark . Although adequate for verifying the validity of Maxwell' s equa tions. you can apprec iate the difficulties of trying to use this apparatus for wireless communication. After all. if the reo ceived signal has to be strong enough to generate a visible spark. scaling up to global proportions has rather unpleasant implications for those of us with metal dental work. And then Hertz died - young, Enter Marconi.

1.3

PRE-VACUUM TUBE ElECTRONICS

For his rad io experiments Marconi simply copied Hertz's transmitter and tinkered like crazy with the so le intent to usc the system for wireless communication (and not

I Actually. O liver Hcavisid c was the one who llrst used the nota tional conventions of vec tor calcu lus 10 ca'" Maxwell's equeuon.. in me form familia r 10 mo"l cl1~ il1«~ totl" y. 2 Proc. Ro)'. Sf~.. 18tl11. J Many electncuy and magnen sm ( E&M I lexts o ffer the logical . bul hi"lorically ....ro ng. explana tion that Max well Inven ted lhe disp lace ment current tenn a fter rea lizing that there was an inconsistency between the known taws of E& M and the continuity equation for current. The lruth is that Maxwell was a genius. and the inspiration s o f a genius o ften have elusive or igins. This is one of those cases.

1.3

PRE - VACU UM TUBE ELE CTRO NI CS

3

[A,---O-,j _ _Lr------' ° RJ FIGURE 1.1. Branly's cohere r.

•

0---0 Re lay/Paper Tape lnker (As\umcd to have high RF impedance ]

•

FIGURE 1. 2 . Typical rece iver with coherer.

incidentall y to make a lot of mon ey in the process). Recognizing the inhere ntlir nitarions of Herta' s spark-gap defector. he instead used a bizarre creat ion that had been developed by Edouard Branl y in 1890 . As seen in Figure 1. 1. the device < d ubbed the "coherer" by Sir Oliver Lodge - co nsisted of a glass enclos ure filled with a loo sely packed. perhaps slightly ox idized . metall ic powd er whose resistance turn ed out to have interesting hysteretic behav ior. No w. it must be emphasized that the detailed principles underlying the oper ation of coherers have never been satisfactorily clucid
~

Under lar~e - s ignal e xcitation, the tilings could he seen to snc k together ! hence the name "coherer" ]. and it's , nol hard to understand the drop in resistance ill that case . However. apparently unknown 10most aUlhuT'. the coberer also worked with input energies St l small that Il U such "cohere nce" is observed. so I assert thatlhe detailed principles of operation remain unknown.

,

CHAPTER I A N ONLINE AR HISTQRY O F RADIO

-

--e:::::::j FIGURE

1.3 . Marconi's coherer .

As can he seen. the co hcre r acti vated a relay (for a udi ble click s ) or paper tape inker (fo r a perma ne nt reco rd ) when a received s ig nal triggered the transitio n to a low-resista nce sta te. II is ev ide nt thai the cohe rer was ba sicall y a digita l device. a nd ther efore unsuita ble fo r uses ot her th an rad iote legra phy. Marcon i spe nt a grea t deal of time improvin g what wus inheren tly u terrible dct cclor and finally se ttled o n the co nfiguratio n sho wn in fi gu re 1.3. lie great ly red uced the spaci ng be twee n the end plugs (to a minimum of 2 nun). filled the intervening space with a particular mi xtur e of nic kel and silver filings (i n 19 : 1 ratio ) of carefully selected size. and sealed the en tire assembly in a partiall y evacuated tube . As a n additional refinem cm in the receiver. a so lenoid pro vided a n a udible indication in the process o f auto ma ticully whacking the det ector bac k into its ini tia l state after e ac h rec ei ved pu l ~ .' As yo u can imagin e. man y Et\.f even ts other than the desired sig na l co uld trigger a co herer. re sulti ng in some difficult-to-read messages. Eve n so. Marconi wa s able to refin e his apparatus to the po int of achieving transatlan tic wireless co mmunica tio ns by 190 1, w ith much o f his success attributa ble to mo re powerful trunsrmuers and large. e levated ante nnas that used the e arth as o ne te rm ina l (a s d id his tran smitter ). as we ll a~ to hi s imp roved coherer. It sho uldn't surp rise you. though . thai the coherer. eve n at its be st. pe rformed q uite poorly. Frustratio n w ith the co herer's e rra tic nat ure im pelled a n aggressive search fo r bett e r detec to rs. Witho ut a suita ble theoreti ca l fra mewo rk as a guide , however, thi s sea rch som etimes took macabre turns. In o ne c ase. eve n a human brain from a fre sh cadave r wa s used as a cohe rcr. with the expe rime nter claiming remarkable se llsitivity fo r hi s apparatus." Let us a ll be thankful that this particul a r type o f cohcrer neve r qui te ca ught o n. Most rese a rch wa s g uided by the vagu e intuiti ve not ion thai the co here r's operalion depended o n so me mysterious pro perty of imperfect contacts, and a variety of experime nters stu mb led. virtually sir nulra neo usly, o n the point-cont act crys tal dcrccto r t f'ig urc I A), Th e first pat en t fo r such a device wa .. awa rded in 190-1 (filed in 19( 1)

~ The cohcre r was most recently used in a radio -contro lled toy truck in the late 1 'J.'i O ~ , f>

A . F, Cotbns, Electr ical World and Enginrrr. and worked his way up In humans,

\I,

:\9 , 1902 : he started out with bruins of othe r spcde~

1.3

PRE -VACUUM TUBE ELECTRONI C S

5

o

•

FIGURE 1.4 . Typical crysk:ll detector.

J. C. Bo'-C for a detector thai used galena (l ead sulfide j.? Th is appears 10 he the first patent awarded for a semiconductor det ector . although it was not reco gni zed as

10

such ( Indeed. the word "semicond uctor" had not yet been co ined ). Work along these lines continued. and Genera l Henry Harrison Chase Dunwood y receiv ed a patent in late 1906 for a detector using ca rborundu m (silicon carbide) . followed in early 1'X)7 by a patent to Greenlea f Whini er Pick ard (an M IT graduate whose great-uncle was the poet Joh n Greenleaf Wh inier) for a silico n (!) detector . As show n in the fi gure. one co nnection to thi... type o f detec tor co nsisted of a small wire (whimsically known as a cutwhisker} that made a point contact to the crystal surface . The other conncc tion was a large area co ntact typica lly formed by a low-melting-poi nt alloy (usually a mixture of lead. tin, bismuth and cad mium. known ;1.'-' wood's metal . with a melt ing temperatu re o f under 80' C) that surrounded the crystal. One might cal l a device made in this way a po int-co ntact Sc hottky diode. although mea.surerncr us 'Ire not always easily reco nciled with such a description, In any eve nt. we ca ll sec how the modern symbol for the d iode evolved from a depiction of this physical arrange ment, with the arrow representing the ca twbiskcr point contact. us seen in the figure, Figure 1.5 shows a si mple crystal" radio made with these devices." An L C circuit tunes the desired signal. which the crys tal then rectifies, leaving the demodul ated

' 1. C, Bose, U,S.I'atellt #755,840, granted 19 March 1904. Actually, Ferdinand Braun had reported asynuncrrica l co nduction ill gale na and copper pyrite s (among others ) hack in 11174. in "Ueber die Suornlcinmg durch Schwefclmctallc ("On Current Fluw through Metallic Sulfides"], /'0I:IW/l o d0rff',f Antlllit'/! dc r /'h)'.~i1( und Chemie , v, 153, pp. 556- 63. The large -area conta ct was made through partial immersion in merc ury. and the other with copper, platinum, and silver wires, None of the !kImples ~hnwed more than a 2: I for ward / reverse current ratio. Braun later shared a Nnhd Prize with Marcon i for co nmbu nons 10 the radio art. , In modem electronics. "crystal" usually refers to quartz reson ators used . for example, as fre q uency determini ng elements in lI<;cilJators; these hear ahsolUlely no relat ionship 111 the cry stals used in crystal radios . • A INJ4A germanium d iode worts fine and is more read ily available, burlacks the charm o f galen a . Wnud 's me tal, and a cat....biskcr to fiddle with.

,

CHAPTER. 1 A N ONLI NE AR HISTORY O F RAD IO

.. -

Long ...in: for A M !>and

L : Appru x . 25( ~H C:

L

Headphone s

•C .. _ _

Af'l'"'~ . JlJ-Jmrf.o ~

IUnc A M ...OO

to be h lJ h.Z (> I fe... kU I

l'Ottd.. good conlle<1i...n 10 0:.,,11 Jf' .........fllr "".. fnu hs (TIw ground ~"mnal of an AC POW"oullrf ofkn wor\s " kay . bul be ~ 10 hoc.. Ihinp up " ght ,.. the", could be ..lInC' ntl''''-' ul'l' of un pk~l llI:>"I)

FI GURE 1.5 . Sim~c~101 radio.

audio 10 drive the headphones. A bias so urce is nor need ed wi th some detect ors (such as ga lena) , so it is possibl e to make a "tree-energy " rad io ! 10 Pickard worked harder than anyone else 10 develop crys tal detectors. eve ntually trying ove r 30.()(x)combinations of wires and crystals. Among these were iron pyrite s (foo l' s gold ) and rusty scissors, in addi tio n to silicon. Gal ena detect ors bec ame qu ite popular becau se they were inexpen sive and need ed no bias. Unfortunately. prope r adju stment of me catwhisker wire contac t was difficult to maintain beca use anything ot he r tha n the lightest pressure on galena destroyed the rec tifica tio n. Plus. you had to hu nt around the crystal surface for a se ns itive spo t in the first place. On the ot her hand. although carborundum detectors needed a bias of a co uple o f volts. they were mo re mech anically stable (a relat ively high con tact pressure was .111 right). and found wide use on ships as a conseq uence.'! At abo ut the same time that these crud e semicon ductors were first coming into usc. radio enginee rs began to struggle with a problem that was assuming greater and greater prominence: interfere nce . The broa d spectrum of a spark signal made it imprac tical to attempt much other than Morse -code types of tran smissions (a lthough some in trepid engi neers did attempt A M tra nsmissio ns with spark ga p equipment. with little success ). T his broadhand nat ure fit we ll with cchercr technology. since the vary ing impedan ce of the latter mad e it difficult to realize tuned circuits anyhow. However. the inability to provide any useful degree of selec tivity became increasingly vexing as the num ber of tra nsmitters multiplied .

III Perhaps we shou ld give a little credit to the human aud illlry system: the thresho ld of hearing (:lIITe\ ptmds 10 an eardrum displacement o f ahout the diameter IIf a hydroge n ato m! II Carbo rundum detectors were typica lly packaged in cartridges and o ften adjusted throu gh the de lkate procedu re of slamming the m against a hard surface .

I

I

j

1.3 PRE- VACUUM

ruee El ECTRO N ICS

7

Marconi had made head lines in 1899 by contrac ting with the Ne w York Herald and the Evenin g Telegram to provide up-to -the-m inute coverage of the America 's Cup yacht race, and was so successful that two add itional groups were enco uraged to try the same thin g in 190) . One o f these was led by Lee de Forest. whom we ' ll meet later. and the other by an unexpected interloper (who turned ou t to be none othe r Ibm Pickard ) from American Wireless Telephone and Telegraph . Unfortunately with th rr t: grou ps s im u lt a n e o usly s p ark in g a way lhal y e ar. " 0 o m ' was able 10 receive intelligible signals. and race results had to be repo ned the o ld way, by semaphore. A thoroughly disgusted de Forest threw his tran smitt er overboa rd, and news-starved relay stations on shore resorted 10 making up much of what they repo n ed. This failure was all the more discou raging because Mar coni. Lod ge. and that errat ic genius Nikola Tesla had actuall y alread y patented ci rcuits for luning. and Marcon i' s apparatus had employed bandpa ss filters to reduce the possibility of interference. 12 The prob lem was that . even though addin g tun ed circuits to spark transmitters and receivers certainly helped to filter the signal. no practical amou nt o f filterin g could ever really convert a spark train into a sine wave. Recognizing this fundam ent al truth. a number of engineers sought ways of generating co ntinuous sine waves at radio frequencies. One gro up. which included Dani sh engineer Valdemar Poul sen' ! (w ho had also invented a crude magnetic recording device called the telegra phone ) and Australian-American engi nee r (and Stanford graduate) Cyri l Elwell. used the negative resistance associated with a glowing OC arc to keep an L C ci rcuit in constant osc illation 14 to prov ide a sine -wave radio -frequ ency ( RF) carrie r. Enginee rs quickly discovered that this approach co uld be scaled up to impressive power levels: an arc tran smitter of ove r I lrIegllwatt was in use shortly after WWI! Pursuing a somewhat differe nt approach , Ernst F. W. Alcxanderson of Genera l Electric (GE) acted on Reginald Fessenden's request to produ ce RF sine waves at large power le....els with huge alternators (realty big. high -speed versions of the thing that charges your ca r battery as yo u d rive). Th is dead -en d tec hno logy culminated in the construction of an alterna tor that put out 200 kw at 100 kl tz! It was completed just as WW I ended. and was already obso lete by the time it becam e ope rational.'?

12 Marcon i was the only one hacked by str ong financial interests (essentiullythe Briti sh gove rnme nt). and his Bnush palenl (#7777. lhe famou s "four sevens" pate nt, granted 26 April I9
a

CHAPTER. 1 A N ON LINEAR HISTORY O F RAD IO

nirn c acid --

_

FIGURE 1.6 . Feuendeo's liquid bcr retter.

Th e superiority of the continuous wave over spark signals was immedia tely evident . and spurred the develop ment of bet ter receiving equipment. Thankfully. the cohercr was grad ually supplanted by a number o f improved devices, including the semicond uctor devices described earlier. and was well on its way to exti nction by 19 10 (alt hough as late as the 1950 s there was atleast thai one radio -co ntrolled toy that used a cohcrer). One such improvement . invented by Fessenden. was the " liquid barretter " shown in Figure 1.6. Thi s detector consisted of a th in. silver-coa led pla tinum wire (a "Wollaston wire" ) encased in a gla ss rOO . A tiny bit of the wire protruded from the rod and made contact with a small poo l of nitric acid. Thi s arrangeme nt had a quasiquadratic V- I characte ristic near the origin and therefore co uld actually demod ulate RF signals. T he barreuer was widely used in a number of incarnatio ns since it was a "self-res toring' device (unlike typical co herers ) and requi red no adjustments (unlike crystal detectors) . Except for the hazards assoc iated with the acid , the barretter was apparently a satisfactory detector. judging from the many infringem ents ( including an infamous one by de Forest) of Fessenden ' s pate nt. Enough rectifyin g detectors were in usc by late 1906 to allow shipboard ope rators on the east coast of the United States to hear, much to their am azement (despite a forewarning by radiotelegra ph three days before ). the fi rst AM broadcast by f essenden himself 011 Christ mas Eve.lf> Delight edlisteners were treated to a program of poet ry, Fessende n' s violin playing of Christma s carols. and some singing. li e used a water-coo led ca rbon microphone i ll series witli the ml1 n /1/tl to modu late a 5-kW (approximately), 50 -kHz (also appro ximate) carrier generated by a prototype Alexanderson alternator loc ated at Brant Rock , Massnch usens. Those unfortun ate enough 10 use cohcrers missed out on the historic event, since coherers a." typicall y used are comple tely unsuited to AM demodulation. Fessenden repeated his feat a week later. on New Year ' s Eve. to give more people a chance to ge l in on the fun .

•" Auken [see Sec non 1.9 1erroneous ly !lives the dale as Christmas Day.

1.4

BIRTH O F THE VACUUM TUBE

9

The next yea r. 1907. was a significant one for ele c tro nics. Aside front Iollowing on the heel s o f Ihe first AM broad cast (w hich ma rked the rrunsiti on from radi otel egraphy to radiot elephony ). it sa w the e me rge nce of im porta nt se mic onductors. In ad dition to the pa te nti ng of the silicon detector. the LED was also discovered that ye ar! In a brief art icle in Wi l'l'les.f UiJrltl titled "A No te on Carborundum." Il en ry J. Round of Great Britain report ed the pu zzling e mission of a co ld. blue l7 1ighl from ca rboru ndum detectors und er ce rtai n cond itions (usually when the ca twtuc ker porennal was very negativ e relative to thai o f the crys tal). Th e e ffec t was largely ignored and ultimately forgott en , as there were just so man y more pressing problems in rad io at the lime. Toda y, howe ver. carbo rundum is in fact used in blue LEDs,18 a nd has been investigated by so me 10 make transistors that ca n operate a t elevated tem peratures. And as for silicon. wel l. we all know how that turned out.

1.4

BIRTH OF THE VACUUM TUBE

The year 1907 also saw the pa tenting, by Lee de Forest , of the first electronic device capable of am pli f ica tion: the triode vacuum lube . Unfo rtunately, de Forest didn 't understand how his inve ntion ac tual ly worked . ha ving stum bled upon it by way of a circuitous (a nd occasionally unethical) route. The vacuum lube ac tually traces its a nce stry to the low ly incandesce nt lig ht bu lb of Thomas Edison. Edison's bulb s had a problem wit h progress ive dark en ing ca used by the accumu lation of '10 0 1 (given off by the carbon filame nts) on the in ner surface of the hulb. In a n atrernpt rc c ure the problem, he insert ed a met al elec trode. hopin g somehow to attract the '10 0 1 to th is pla te ra ther than to the glass. Ever the ex pcrimentalist . he applied both positive and negati ve vo ltage s (relative to one of the filament connections ) to thi s plat e, and noted in 1883 that a c urre nt m yste rio usly Ilowed when the plate was po sitive, but none flowed whe n the pla te was negative, Furthermore, the current that flowed depended on how hot he made the filamen t. li e had no theory to explain these ob ser vations (rem ember, the word "electron" wasn 't even coined unti l IXlJ 1, and the panicle itself wasn' t unambi guously identified until J. J, Thomson' s experimen ts of IXlJ7 ), but Ediso n went a head and paten ted in IXX4 the fi rst electronic (as opposed to electrical) devic e. one that ex ploi ted the dependence of plate cUITCnl on filament temperatu re 10 measure line vo ltage indirectly. This Rube Goldberg instrume nt ne ver mad e it into production since it was in fe rior to a sta ndard voltmete r: Edison j ust wanted a nother patent . that ' s all (tha t's on e way he ended up with over a thou sand of the m).

17 1 1e ~ ..... IIrange and yellow, too. He may have been drinking. 18 II stMluld he me ntioned lhal Ga N-N.<;('d LEDs olTer much highe r efficiency. hul il was on ly very recently Ihat people ngured oe r hnw to dope the stuff without inlrodu cing M' rinufo defects . GaN blue l.Eo.. are much mo re e fficient man SiC ones.

10

CHAPTER 1 A N ONLINE AR HISTORY O F RADIO

~

cy lindrical plate -

f

,

'

4+

f-;-\

9.-

Plate con recnon

FIGURE 1.7. Fleming

Filament connecnons

V'CIIve.

Th e fun ny thing about this ep isode is that Ed iso n arg uably had ne ver inve nted anything in the fundamental se nse of the term. and here he had stumbled across an electron ic rectifier but neverth eless failed to recogn ize the impl ication s of what he had found. Part of this blindness was no doubt related to his emotio nal (and financial) fixation on the DC tran smi ssion of power. whe re a rectifi er had no role. At about this time. a consultant to the British Edison Com pany named John Ambro se Fleming hap pen ed to atte nd a conferen ce in Ca nada . He drop ped down to the United Slate s to visi t his brother in New Jerse y a nd also stopped by Ed iso n's lab. He was greatly intri gued by the " Ed ison effec t" (much more liO than Edison. who fou nd it difficult to underst and Fleming's exciteme nt ove r something that had no obvio us pro mise of practical app lica tio n). a nd eventually publi shed papers on the Ed ison effect from 1890 to 1896 . A lthou gh his expe rime nts cre ated a n init ial stir. Ront ge n's announce me nt in Ja nuary 1896 of the d iscov ery o f Xvrays - as well as the discovery o f natu ral radioact ivity later that same yea r - soo n do min a ted the interest of the ph ysics co mm unity. a nd the Ed ison effec t qu ick ly lapsed into obscurity, Se veral years later . though . Flemi ng bec ame a consultant to British Marconi and joined in the search for improved detectors. Recalling the Ediso n effect . he tested some bulbs, fo und out tha t they worked all righ t as RF rect ifiers. a nd patent ed the Fleming valve (vac uum tube s are thus still known as valves in the Un ited Kingdom ) in IljOj ( Figure 1.7). Th e nearly deaf Fleming used a mirror galvanome ter to prov ide a vis ual ind ication of the rece ived signal. and included this featu re as part of his pate nt. Alt hough not partic ular ly se nsi tive. the Fle min g valve wax at least continually responsive and req uired no mec hanica l adjustm ent s. various Marco ni installations used the m (largely out of contractua l obli gations) . but the Fleming valve was never popular (contra ry 10 the asse rtio ns of some poorly resea rched histories) - it needed too much pow er, filament life was poor. the thi ng was ex pens ive, and il was a rema rkab ly insens itive de tec tor com pared to. say. Fesse nden 's barretter or well-made cry sta l dete c tors. De Forest . mean whil e. was busy in A mer ica se tting up shady wireless companies who se so le purpose was to ea rn mone y via the sale of stock . "Soo n. we belie ve.

1.4 BIRTH O F THE VACUUM TUBE

\I

the sucke rs will begin to bite," he wro te in his journal in early 1l)()2. As soo n as the stock in one wi reless installation was sold, he and his cronies pic ked up Slakes (.... hether or nOI the station was actually completed) and moved un to the next town. In another demonstration o f his sterling character. he outright sto le Fessenden' s barretrer (simply reform ing the Wollaston wire into the shape of a spade ) after visiting Fessenden 's laboratory, and even had the audaci ty to claim a prize for its invention. In this case, however. ju stice did prevail. and Fessende n won an infrin gement suit against de Forest. Fortunately for de Forest. Dunwood y invent ed the carbo rundum detector just in time to save him fro m ba nkrupt cy. NOI content 10 develop this legitimate invention . 19 though. de Forest proceeded to steal F leming's vacuum tube d iode and actually received a patent for it in 1905. He simply rep laced the mirror gal vanometer with a headphone and added a huge forwa rd bias (thus red ucing the sensitivity of an already insensitive detector). De Forest repe atedl y and unconvincingly denied throughou t his life that he was aware of Fle ming 's prior work (even though Fleming publi shed in professional jo urnals tha t de forest habi tuall y and assiduously scanned ): to 00 1· srer his claims. de Forest pointed 10 his usc of bias where Flemin g had used none.P' Conclusive evide nce that de Forest had lied o utright fina lly came to ligh t when historian Gerald Tyne obtained the business record" of W. Mccand tess. the man who made all of de Forest' s first vacu um lubes (de Forest called them "audions"). The records clearl y show that de Forest had asked McCandless 10 dupl icate some Flem ing valves months before he filed his patent. There is th us no mom for a charitable interpretation tha t de Forest independently invented the vac uum tube diode. lI is crowning achieve ment ca me soo n after. however. De Forest add ed a zigzag wire elec tr ode. which he called the grid. between the filament and wing electrode (later known as the plate }, and thus the triode was born (see Figure 1.8). This three element audion was ca pable of amplifica tion, but de Forest d id nol reali ze this fact until years later . In fact. hi!'> patent application described the trio d e audio n on ly as a detector. not as an amplitier.U Motivation for the addit ion o f the grid is thus still curiously unclear. He certainly d id no t add the grid as the co nsequence (If careful rea soning. as some histories claim. The filet is that he added electrodes all over the place. Ile even tried "control electrodes" o utside of the plate ! We must therefo re regard his addition of the grid as merely the result of haphazard bUI persistent tinkering in his

I ~ Dun.....o ody had performed l h i ~ Iol.ur l as a consulta nt to de F(lrt"~t. lie .....as unsecce ss nr l in his

rtTorts to get de Fo resl to pa y him fo r it . :to In his efforts 10 establish that he had Iol. ur lcd inde pe nde nt ly of Fleming. de Forest re peatedly and

21

!Mide nlly suned Ihat il was hi!> rese arcbes Into the conductivity properties of lIalfl('s that informed hi ~ ..... od in vacu um 1Uhe!>. arg uing thai io nic conduct ion wa~ the l ey 10 the ir operation. As a consequence. he b oxed him self into a comer thaI he found difficult to escape later . afte r oebers deve loped the superior hig h-vac uum robe s th at were e~ntia lly free of iun ~ . Curiou~ly enough . thoug h, his patent fUf the: two -ele me nt au dion JoC's Ifl('ntion am phficat ion .

12

CHAPTER 1 A N O NliNEAR HISTORY O F RADIO

Grid connection

Plate

Plate (Mwing" ) con nectio n

Grid

(new)

Wing

G rid

-

f--/\

(o ld)

Filament FIGU RE 1. 8 . De Fore st triode c udicn and syrnbo15.

search for a detector to call his own . It wou ld not be inaccurate to say that he srumbled onto the triode. and it is ce rtainly true that others had to explain its operation to him.n From the available evidence. neither de Forest nor anyone else thought much of the audion for a number of years ( 1906- I 9{)t) saw essentially no activi ty on the audion ). In fact, when de Forest barely esca ped conviction and ajail sentence for stock fraud after the collapse of one of his companies. he had to relinquish interest in all of hi!'> inventions as a co ndition of the subsequent reorganization of his companies. with just one exce ption: the lawyers let him keep the patent for the audion. thinking it worthless.P He intermittently puttered around with the nudion and eventually di scovered its amplifying potential. as did others almost simultaneously (including rocket pioneer Robert God dard).24 He managed to sell the device to AT&T in 191 2 as a telep hone repea ter amplifier. but initially had a tough time because of the erratic behavior of the audicn. Rep roducibi lity of device characteristics was rather ptxlr and the tube had a limited dynamic range. It functioned well for small signals. but behaved badly upon overload (the residual gas in the tube would ionize. resu lting in a blue glow and a frying noise in the output signal). To top things off the audion filaments (made II Auken (see Section 1.9) argues that de Ft,m,t ha., been unfairly acc used of not unde~tand ing h i ~ own invennon . However. the bulk o f the evidence contradicb Aitken ' ~ ge ocnJus view. 13 The rece ntly unem ployed de Forest the n wem te work fur Elwell at Federal Telephone and Tele graph in Palo Atto. N His U.S_Patent III, 159,209 . filetl l AugUSl IIJI 2 and granted 2 November 191S. describes an audion uscillaillf and Ihu~ actually predates even Aml~tn," g 's oocu rrenred won .

1. 5 TH E REG ENERATIV E AMPli fiE R/ DETECTOR/ O SCIL LATOR

13

B+

c •

Ond-leak bias resistor

Til Fil. Supply

Tbc: ~m"unl of P'" iliw fee'dbocL i, ~olll ll ll kd by 1'-' coupling bct"' ....m U . 00 L.2 .

FIGURE 1.9. A rmstrong regenerative receiver.

of tantalum) had a life of only about 100 - 200 hours. It would he a while befor e the vacuum tube could ta ke over the wor ld .

1.5 ARMSTRONG AND THE REG ENERATIVE AMP 1IFIE R/ DHE CTO R/ O SC II I ATOR Fortunately, some gi fted people fina lly became interested in the audion. Irving La ngmuir at GE Lab s in Sc henectady wor ked to ach ie ve high vacua. thu s eliminating the erratic behavior ca used by the prese nce of (easily ioni zed ) res idual gases. De Forest nev er thought to do thi s (a nd in fact wa rned aga inst it. be liev ing that it wo uld redu ce the sensitivity ) beca use he never rea lly be lieved in thermionic e mission of elec trons (indeed. it isn't clear he e ven believed in elec tro ns at the time ), asse rting instead that theaudion depended fundam e ntally on ioni zed gas for its operat ion. Afler Lan gmui r' s ac hie veme nt , the way was paved for a bright e ngi neer to de vise useful circuits to exploit the uudio n' s potent ial. Th at e nginee r was Edw in Howa rd Armstrong, who inven ted the rege ne rative a mpli fler/de tec tor P in 19 12 at the tender age of 2 1. Th is ci rcuit (a mode rn version of which is shown in Figu re 1.9) employed p ositive feedback (via a " tic kler coil" that coupled some of the output energy hack to the input with the right phase) to boost the gain and Q of the system sim ultaneously. Th us high gain (fo r good se nsitivity) a nd narrow ba nd width (for good se lec ti\'ity) could be obtai n,.-. d rather sim ply from one tu be. Additionally, the nonlin ea ruy of the tube demodulated the signal. Furthermore. ovc rcoupling the ou tput to the input turned the thing into a won derfully compac t RF osc illa tor.

~ His no14rized OU(ebouL: e ntry is actually da ted 3 1 Jan uary 191 J.

"

CHAPTE R 1 A N ONLIN EA R HISTORY O F RADI O

In a 1914 paper e ntitled "Operating Feat ures of the Aud ion,',26 Arms trong published the first co rrect explanation for how the triode worked a nd provided e xperime nial ev ide nce 10 suppo rt his cl aim s. He foll owed thi s pa per with a nother (" So me Recent De velop ments in the Audio n Recei ve r" )27 in w hic h he additionally ex plained the operation of the regenerative am plifie r/detector . a nd showed how 10 make an osci llator ou t of it . Th e paper is a mod el of cla rity a nd quite read ab le even 10 mode m audiences. De Forest. however. was quite upset at Armstrong' s presumptuousness. In a published discussion sec tio n fo llowi ng the pape r. de Forest repeatedly attac ked Arm stro ng. II is clea r from the published exchange that . in sharp contras t with Arrn stron g, de Forest had difficulty with ce rta in bas ic conce pts (e.g.. that the average value o f a sine wave is zero). a nd didn 't even unde rstand how the triod e, his own invent ion (more of a discovery. really) . actually worked. Th e bitter e nmity tha t arose between these two men never wa ned . Armstron g went on to deve lop circ uits that continue to dominate communications sys te ms to this day. Wh ile a mem ber of the U.S. Army Si gna l Corps during World War I. Arm strong became involved with the problem of detecting e nemy planes from a dista nce. a nd pursued the idea of trying to home in on the signals naturally generated by their ignit ion sys te ms (spark tra nsmitte rs ag ain). Un fortunate ly, little useful radiation was found below abo ut 1 MH z, and it was exceedingly difficult with the tu bes ava ilab le a t that tim e to get m uch amplifica tion above that freq uency. In fact. it was o nly with ex traordi nary care that II . J. Round (of bl ue LED fa me) ach ieved use ful gain a t2 Mll z in 19 17, so Arm stron g had his work c ut OUI for him . li e solved the pro ble m by e mp loyin g a principle ori ginally used by Poul sen a nd later eluc idated by Fesse nde n. Wh e n de modulating a co ntinuous wave Elt'l",rim/ World. 12 Decembe r 1914. 27 IRE Pml:t'l' Jin1::J. v, 3. 19 15, PI' . 2 15-47 .

L

1.6 O THER RADIO CIRCU ITS

15

- - - - - --

~

~x

RF Am.·

IF Amp .......... Demod j- . AF Amp ~_" A udio 0",

Tuning contml - - .

FIG URE 1.10 . Superheterodyne receiver block diagram.

sensitivity so that the limiting factor was actually atmospheric noise (wh ich is qui re large in the AM broadcast band) . Furthermore. a single tuning co ntrol was made possible. since the IF amplifier wo rks at a fixed freq uency. He called this system the "superheterody ne" and patented it in 19 17 (see Figure 1.1 0). Although the war ended before Armstrong could use the supe rhet to detec t German planes, he continued to develop it with the aid of se..'eral talent ed engineers, fi nally reducing the num ber of lubes to five from an ori ginal co mplement of ten (good thing. too: the prototype had a total filament current requ irement of ten amps). David Sarnoff of RC A eventually negotiated the purchase of the superhet rights. and RC A thereby came to do minate the radio market by 1930 . The great sensitivity enabled by the inventi on o f the vacuum tube allowed transminer power redu ctions of orders of magni tude while simultaneously increasing useful communica tions distances. Today. 50 kW is co nside red a large amount of power. yetten times this amo unt was the norm r ight after WWI. The 1920s saw greatly accelera ted deve lopment of radio e lectro nics. The war had spurred the refinement of vacuum tubes to an astonis hing degree. with the appearance of improved filament s (l onger life, higher emis sivity, lower powe r requirements), lower ir uerclectrod e capacitance s, higher transcond uctance, and greater power-handling capability, These develop ment s set the stage fo r the invention of many clever circuits, so me designed to challenge the dominan ce of Armstrong's regenerative receiver. 1.6 1.6 .1

OTHER RAD IO CIRCUITS

THE TRF A N D THE NEUTRODYN E

One wildly popular type (If rad io in the early days was the tuned radio -frequency (TRF) rece iver. T he basic TR F circuit typica lly had three RF bandpass stages. eac h tuned separately, and then a stage or two of audio after demod ulation (the latter somelimes accomplishe d with a cry stal diod e). The user thus had to adj ust three or more

16

CHA PTER 1 A N ONLIN EAR HISTO RY O F RA DIO

eN Adjus t eq ual 10

•

I~ /t~ To next stage

B+

O utput bandpass filler

en>

• From pre vious stage

R I. sur r 1ylbia., control

Input bandpass filter

FIG URE 1.11. Basic nevtrodyne amplifier-.

knobs to tun e in each station, Whil e this array of controls may have appealed to the tinkering-di sposed tecbnophile. it was rat her unsuit ed to the average consume r. Oscillation of the T RF stages was also a big problem. ca used by the pa rasitic feedback path provided by the grid- pla te capaci ta nce C"p.2r. Alth ou gh limiting the gain pe r stage wa.. one way to red uce the tend ency to oscillate. the atte ndant degrad a tion in se nsitivity was usuall y unaccept able. T he proble m ca used by ftr was largely eliminated by Harold Wheeler' s inve ntionl'J of the nc utrod yne circuit (see Figu re 1. 11 ) .311 Recognizing the ca use of the prob lem . he inserted a compe nsating ca pacita nce ( C N ), termed the neu tralizing ca pacitor (ac tua lly, "con denser" was the term back then ). Wh en properly adjusted . the condenser fed back a c urre nt e xactly equal in magnitu de but oppo site in phase with that of the plate -to -grid capacitance, so that no inpu t c urrent wa s required to charge the capacita nces. The net resu lt was the suppression of C HI ,' S effec ts, permit ting a la rge increase in gain per stage without oscillat ion." Afte r the war. Westin ghou se acquired the righ ts to A rmstro ng 's rege nera tion pate nt , negotiated licensing agreement s with a limit ed number of radio manufac turer s. a nd the n aggressively pro sec uted those who infringed (wh ic h was just abo ut eve ry body) . To pro tect them selv es,

e

2MIt is len as "an exercise fo r the reader" to show that the real part o f the input imped ance nf an ino.IUl: livc1y loadco.l l:u Ill11l0 1J-l:a lhode a m p li fie r c an I;....,

le ~~

tha ll / l:rt. hc~' au,><; " rt lM;: fc l:o.Ih..t;l. thmug h

C. p • and that this ucgative resistance ca n therefore cause instahility. N Ill' d id this wor k fo r Lo uis Hazeltine, who is frequ ently g iven cre d it for the cucuu. ,\0 Of r;OUNe. it sho uld he noted that Ann .' tru ng·s superheterody ne neatly snh 'e1> the problem by on-

n·:

laining gain al a number of d ifferent freq uencies: Rf . ami AI". T his ap proach also red uces greatly the da nger of UM:iIIat: ion from parasit ic inp ul- ou tpul cou plin g. 31 In some !
17

1.6 O THER RADI O CI RCUITS

A-r

.. AuJ io OUI .

fiGURE 1.12 . Reflex receiver block diagra m.

those "o n the outsid e" organi zed into the Independ ent Radio Manufactu rer s Associ ation and bou ght the rights to Hazelt ine's circuit. Te ns of tho usa nds of ncut rodyne kits and asse mbled conso les were sold in the 1920 s by members of IRMA, all in an anempr to com pete with Armstrong's regenerat ive circuit. Meanwh ile, de Forest was up to his old tricks . He bou ght a compa ny that had a license to make Armstro ng 's rege nenuivc circui ts. A lthoug h he kne w that the licen se was nontransferab le, he nonetheless started to se ll regenera tive radi os until he wa.. caught and th reatened with lawsuits. He eve nt ually skirte d the law by se lling a radio that m ulti be hooked up as a rege nerato r by the cu stom er sim ply by reconne ct ing a few wi res between binding posts that had been co nveniently prov ided for this

purpose.l 2

1.6 .2

THE REFL EX CIRCU IT

The reflex ci rc uit ( Fig ure 1.1 2) e njoye d some prominence in the ea rly 1920 s, but was more popular with hobbyists and ex perime nte rs tha n w ith com me rcia l indu stry . Th e idea behind the reflex is wonderfu l a nd subtle, a nd perhaps e ve n the inve ntor of the circuit himsel f (believed to he Fre nch e nginee r Ma rius Latour .H ) did not full y apprcciatc just how mar velous it was. The bas ic idea was this: pa ss the RF th rou gh some numbe r (say. on e) of amplifier stages, demodu late. and then pa ss the audio back through tho se same a mp lifier s. A give n tube thus simul taneou sly amplified both RF and AF signals.

)2 One anec dotal report ha~ it that de Fo res t su ld receivers with a wire thaI protruded from !:he hac k panel. maned .....ilh a lallclthat ~id so melhing like t'Do 001 cut this wire'; it converts this receiver into a rege neranve one." I have nul fou nd a primary source for this inform ation. bUl it is e ntirel y l·on~i!ll.enl with all we know ahuut de Pore si's charac ter. l) It should be noeed thai AmlStnmg 's SCX'uod pa pe r on the s u~rhelerody ne (pu hlished i ll 1924) contains exam ples of rd leJ; circuit s.

18

CHAPTER 1 A N ONLINE AR HISTORY O F RADIO

The reason that this arrangeme nt made sense beco mes co nvincingly clear only when you co nsider how this co nnec tion allowed the overall system to p O S5eSS a gainbandwidt h produ ct thai exceeded that of the active device itself. Suppose that the vacuum lube in question had a certa in co nstant gain- bandwidth prod uct limit. Further assu me thai the inco ming RF signal was amp lified by a factor G RF over a brickwall passband of bandw idth B. and Ihat the audio signal was also ampli fied by a fac tor G Al' over the same brick wall bandw idth B . The ove rall ga in-bandw idth product was therefore (GRFGAF) B, while the gain- band width produ ct of the combined RF/AF signal processed by the amp lifier was ju st (G RF + G AF) B . For the refl ex circuit to have an adva ntage we need only the product o f the ga ins to exceed the sum of the gains, a criterion that is ea sily satisfied . The reflex circ uit demonstrates that there is nothing fundame ntal about gainbandwidth , and that we are effec tive ly fooled into believing that gain and bandwidth must trade off linearly just because they com monly do . The reflex circuit shows us the error in ou r th inking . For this reason alone, the reflex circuit deserve s more detailed trea tment than it commonly receives.

1.7 ARMSTRONG AND THE SUPERREGENERATOR Armstrong wasn' t co ntent to rest , although after having inven ted both the regenerative and superheterodyne rece ivers he wo uld see m to have had the right. While experime nting with the rege nerator, he noticed tha t under certain co nditions he co uld, for a fl eeting moment , get much greater amplifica tion than normal. fi e investigated further and developed by 1922 a circ uit he called the superregenerator, a circuit that provides so much ga in in a single woe that it ca n amplify therma l and shot noise to audible levels! Perhaps you found the reflex princip le a bit abstru se ; you ain 't seen nothin ' yet. In a superrcgenerator the system is pu rposely made unstable, hut is periodica lly shut down (quenched) to prevent getting stuck in SO ITlC limit cycle . How ca n such a bizarre arrangement provide gain (lots of gain)? Take a loo k at Figure 1.13. which strips the supcrrege nerator 10 its basic etemc nts." Now. during the time that it is active (i .e., the negative resistor is co nnected to the circuit), this second-order bandpass system has a respo nse that grows exponentially with time. Respon se to what'! Why, the initial conditions, of course! A tiny initial voltage will, given sufficie nt time, grow 10 detectable levels in such a system. T he initial voltage co uld conceivably even co me from thermal or shot-noise processes. .14 The classic vacuum tube superregenerator loo ks a lu( like a norma l regencrauve amp lifier. except tha t tbe grid -leak bias nerwork um e cons tant is made very large and the feedhad . (via (he lickler coil) is large enoug h 10 guarantee instahihty . As (he amplitude grows, the grid- leak bias also grows until n curs off Ihe tube. The rube rema ins l;UI off until (he hias decays 10 a value that returns the tulle to Ihe active region. Th us, no separate quenc h oscillator is n«-es.\ ary.

j

1. 7

ARM STRON G AND THE SUPERREGEN ERATO R

19

I.e is Iliited to de~i red inplII signal

Demod .

C,,

.. AuJ io Oul .

•• a

i

Quenched when ope n; Regenerativ e when closed

Qu enc h O",illalor

FIG URE 1.13. Soperregenerative receiver ba ~iC5 .

The problem with all rea l systems is that saturation eventually occurs. and no further amplificat ion is poss ible in such a state. The supe rrcgenerator evade s this problem by period ica lly shutting the syste m down . Th is period ic "q uenching" ca n be made inaud ible if a sufficiently high qu ench frequen cy is chose n. Because of the expo nential gro wth of the signal with time. the superregencrator trades off log of gai n for ba ndw idth . As a bo nus. the unavo ida ble no nlinearity of the vacuum tube can be ex ploited to provide demodulati on of the amplified sig nal! As you might suspect, the superregenemror' s action is so subtle and co mplex that it has never been understood by more than a handful of peop le at a given time. It's a quasipenodicall y time-varyi ng. no nlinea r sys tem that is allowe d 10 go inrermiuently unstable, and Armstrong invented it in 1922. Armst rong sold the paten t rig hts 10 RCA (w ho shared Armstrong's view that the superregenera tor wax the circuit to end all ci rcuits), and bec ame its largest shareholder as a resutt." Alas, the supcrregcnerator never ass umed the dom inant position that he and RCA' s David Sarnoff had envis ioned. Th e reason is simple for us to see now: every superrcge ncmtivc am plifier is fund ament all y also an oscillator . Th erefore, every superregcncrativc recei ver is also a transmi tter thai is capable of ca using interference to nea rby receivers. In addi tio n. the superrege nenu or produ ce s an annoyingly loud hiss (the amplified thermal and shot noise) in the abse nce o f a signal, rather than the rel ative qui et of oth er types of receivers. For these reasons, the superregenerator never took the radio world by storm. The ci rcuit has found wide applica tion in toys. however. When yo u've got to get the most se nsitivity with ab so lutel y the minimum number of ucuve devices, you

)j

In a hil of fort u jtous liming. Armstrong so il! his stock just before the grea t stoc k-market cra sh of 1929.

20

CHA PTE R I

A NONLIN EAR HISTORY O F RA DIO

ca nnot do be tter than the superrege nera tive receiver. Rad io -controlled ca rs, autom anc ga rage - doo r ope ners, and toy walkie-talkies almost invariabl y usc a ci rcuit that co nsists of j usI one transistor ope rating as a supe rregenemtive amplifier/detec tor, and perhaps two ur three more as amplifiers of the demod ulated audio signal (as in a walk ie -tal kie) . The ove rall sensitivity is often of the same orde r as thut provided by a typical superhet. On top of those attributes, it can also demodulate FM through a process known as slope de mod ulation: if o ne tunes the rece iver a bit off frequency so thai the rece iver gain versus frequency is not flat {i.e.• has so me slope. hence the name ). then an incoming FM signal produces a signal in the rece iver whose amp litude varies as the frequency varies: the signal is conv erted into an AM signal that is demod ulated as usual (""it's both a floo r wax (mil a dessert toppin g") . So , if most of the system cos t is associated with the number of active devices, the superregenerative receiv er provides a remarkably economical solution,

1.8 OLEG LOS EYAND THE fiRST SOLlD·STATE AMPlifiER Sure ly one of the most amaz ing (and little- known ) stories from this era is that of self-taught Soviet engineer Oleg Losev and his solid-stale receivers of 1922. Vacuu m tubes were expe nsive then , particu larly in the Soviet Un ion so soo n afte r the revolution. so there was naturall y a great desire to make rad ios on the cheap . Losev' s approach was to investigate the mysteries of cry stals, which by this time we re all but forgotten in the WeM. l ie independ ently rediscovered Round 's carborundum LEDs. and actually published about a half doze n papers on the phenomenon. Be co rrectly ded uced that it was a quantum effect. descr ibing it as the inverse of Einstein's photoe lectric effect. and co rrelated the sho rt waveleng th cuto ff energy with the applied voltage , He eve n noted that the light was emitted from a particular cry stalline boundary (which we would ca ll a ju nction ), and cas t do ubt on a prevailing theory of a thermal origin by showing that the emission could he electronically mod ulated up to
j

T-

-

-

1.9 EPil OG

v,

21

- I hI -2kU t}p.

./ _+----.c"-

•I

Diode ("har,M;leri sl il;~

FIGU Il: E 1.14 . lmev ', crystadyne receiver {$ingle -slagel .

The reaso n almost no o ne in the Un ited Sta les has ever heard of Losev is simple. First , 3 1010:-.1 no one has eve n heard of Arm strong - it seems there isn' t much interest in preserving the nam es o f these pio neers . Plus. most of Lo se... 's pape rs arc in German and Russian . limiting readers hip. Add the ge nerally poor relations betwee n the United Stales and the Soviet Union over most of this ce ntury, and it's actua lly a wonder that unyotlt' knows who Losev was. Lose v himself isn't around beca use he was one of many who slanted 10 death duri ng the terrible siege of Le nin grad. brea thing his last in January of 1942. His colleag ues had advised him 10 leave. hUI he was j ust too interested in finishin g up what he term ed " pro mising expe riments with silicon:' Sadly, all record s of those expe riments have appa rently been lost.

1.9

EPILOG

By the ea rly 1930 s, the supe rhet had been refined to the poin t that :1 single tuning control was all that was req uired. Th e supe rior perform ance and case of use o f the superhet guaranteed its dom inance (as well as that o f RCA ), and virtually eve ry modem receiver, ranging from portable rad io", to radar sets, employs the superheterod yne principle; it seems unlikely tha t this situation will cha nge in the ncar futu re, It is a tribute to Arm strong' s genius tha t a sys tem he conceive d durin g World war I still dominates on the C\lC otthe 2 1st ce ntury. Armstrong, annoyed by the static that plagu es AM rad io, went on to develop (w idehand) frequency mod ula tion . in defiance of theoreticians who declared FM usele ss." Unfortunate ly. Armstrong's life did no t end happi ly. In a sad example o f how our legal system is ofte n ill-equ ipped to deal intelli gent ly with tech nical matters, de Forest

.l6 Bell Laboratories mal~ma l ician Joh n R. Carso n (no known rcunon to the entertainer) had cor-

reedy ~htJw n lhal FM a lway!'> ret.ju ires more bandwid th than A~ 1. disprovin g a prevailing belief In the contrary . Rut he went loti far in declanng FM worthlevs.

I

22

CHA PTER I

A NON LINEAR HISTO RY O F RADIO

cha llenged Arm stro ng's regenera tio n paten t and ultimately prevailed in some of the longest pate nt litigation i ll history (i t lasted twe nty years ). Not long afte r the courts ha nded dow n the final adve rse decision in this case. A rm strong began loc king horn s w ith his fo rme r frie nd Sarnoff a nd RC A in a biller batt le over FM that raged for we ll over another dec ude . His e nergy a nd mon ey all bur go ne. Arm strong comm itted suicide in 1954 at the age of63 on the fort iet h anniversa ry of his demonstration of regen oration 10 Sarnoff. A rm strong's w idow. Maria n. pic ked up the tigh t a nd eve ntually wen! on to wi n e very legal battle; it took fifteen years. De Fore st eve ntually we ntlegit. He moved to Hollywood and worked o n developing so und and co lor for motion pictu res. A few years before he di ed at the ripe old age of 87, he pen ned a c haracte ristically se lf-aggrandizing auto biogra phy tit led The Father of Rad io that sold fewe r than a thou sand copies. l ie also tried to get his wife to write a boo k ca lled J Marri ed a Genifu, bu t she so mehow neve r got around to it.

FURTHER READING Th e stories of de Forest . A rms trong , and Sarnoff are wonderfully recounted by Tom Lewi s in The Empire of the Air, a book thai was turned into a film by Ken Bums for PBS. Although it occasio nally get s into tro uble when it ven tures a technical explana tion . the hum an foc us a nd ric h biographical material tha t Lewi s has unearth ed much more tha n compe nsa tes. (Prof. Lewi s says that ma ny corrections wi ll be incorporated in a la te r paperback edition of his book .) For tho se intere sted in more tec hnical details. there a re two e xcellent books by Hugh A itke n. SyntollY ami Spa rk recounts the earliest days of radiotelegraphy. beginning w ith pre -Hert zian ex peri ments ami e nd ing with Marco ni. Tile Continuous lVm'e takes the story up 10 the 1930 s, covering an; and alternator technology in addition to vacuu m tube s. Cu rio usly, thou gh . Armstro ng is hut a minor figure in A itke n's portrayal s. The story of early crystal de tectors is well told by A . Dou glas in " The Cry stal Detector" (IEEE Spectrum, A pri l 1981 , pp . 64-7) a nd by D. Th ackeray in " When Tube s Beat Cry sta ls: Early Radio Detectors" (IEEE Spectrum, March 1983, pp. 64-9). Mate rial on oth e r carly det ectors is found in a delightful volume by V. Phillips, Early Radio Uin'f' Detectors (Peregrinus. Stevenage. U K. 1980 ). Finally, the story of Losev is recounted by E . Loe bner in "Subhistories of the Light-Emitting Diode" (IEEE Tram . Elec tron Devices, July 1976, pp . 675-99).

1.10

APPENDI X: A VACUUM TUBE PRIMER 1.10 .1

IN TRODUCTION

Sad ly, few e ng inee ring stude nts are ever exposed to the vacuu m tube . Indeed, mos t eng inee ring fac ulty regard the vac uu m tube a qu aint relic . We ll. maybe they're right .

J

1.10 A PP EN DIX: A VACUUM TUBE PRIME R

J\

I

23

=J

I

[}= ~..... e

Cathode

FIGURE

I, 1.5 .

..... ~.....

Plale (t'wing")

Idealized diode ~tructure .

but there are still certain engineering provinces (s uch as high -power RF ) whe re the vacuum lube reigns supreme . Thi s appe ndix is intended 10 provide the necessary background so that an engineer educated in solid-state circuit design can develop at least a superficial familiarity with this historicall y important dev ice . TIle operation of virtua lly all vac uum tubes can be understood rather ea sily once you study the physics o f the vacuum d iode. To simplify the develop ment. we' ll follow a historical path and consider a para llel plate structure rather than the more common coax ial structures. The results are easier to derive bUI still hold generall y,

1.10.2

CATHO DES

Consider the diod e structure s hown in Figu re 1.15. The left-most elec trode is the cathode, whose job is to emit electrons. The plate' s job is to co llect them . Allthe early tubes ( Ed iso n' s and Flem ing' s diod e, and de Forest' s triode a udio n) used directly heated ca thodes, meaning that the light-bulb fi lament did the wor k of emitting electrons. Physically all that happens is that , at high eno ugh tempe ratures. the electrons in the filament material are given enough kinetic energy that they can leave the surface; they literally boi l off. Clearly, materials that emit well at tem per atures be low the meltin g point make the best cathodes. De Forest's first filament s were made of the same carbon variety used in Edison's light bulbs, altho ugh tantalum. which has a high melting point (abou t 3100 K), quickly replaced carbo n. Usefu l emission from tantalum occ urs only if the material is heated to bright incandescence, though , so the early uudion s were pretty power-hungry. Additionally, tantal um lends to cry stallize at high temper ature. and fil ament life is unsatisfactory as a co nsequence of the attendant increasing brittle ness. A typical audion filament had a lifetime as short as 100 - 200 hours. Some uudion s were made with a spare filament that co uld be switched in when the first filament burned out.

24

CHAPTER 1 A N ONLINE A R HISTORY O F RA DIO

Resea rch by W. D. Coolidge (same guy who developed the high-power X-ray tube ) at GE allowed the use of tu ngste n (melt ing point: 3600 K ) as a filame nt materia l. He foun d a way to make filament s out of the unwi el dy stuff (tungste n is not ductile. and hen ce it ord inari ly cannot be drawn into w ires) and opened the path to great improvernenrs in vacuum tube (and light bulb) lon gevity because of the high mel ting point of Ih OlI material. " Unfo rtu nately , luis of heal ing power is required to maintain the operating tem perature of abo ut 24(X) K . and portable (or even luggable ) eq uipme nt ju st co uld not evolve unti l these hea ting req uiremen ts we re reduced. O ne path to impro ve ment (d i5COVcred accide nta lly) is to add a little tho riu m to the tu ngsten . If the temperature is hel d withi n rather narrow limit s (around 19
}7

Tungsten is still used in light kilt><. today.

1.10

APPENDIX: A VA( UUM TUBE PRIM ER

cathode. without worryi ng (much) abo ut the injection of hum that would occur if AC were used in tubes with di rectly heated cathode s. The dra w back to oxide-co ated ca thode s is that they are ex traordinarily sensitive to bombardm ent by positive ion s. A nd to make thin gs worse. the cathodes them selves tend to give off gOis over time. espec ially if overheated. Th us. rat her e luborute procedures must be used to maintain a hard vacuum in tube s using such cat hodes . Aside from pumping ou r the tulle at te mperatures high enough to ca use all the elements to inca nde sce .
1.10 .3

V-I CHA RAC TE RISTICS O F VACUUM TUBES

Now that we' ve taken care of the characteristic s of ca thode s. we tum to
(I)

where "' (x ) is the po tential at poin t x . So lving for the veloc ity as a functio n of .r yields vex)

=

2q "'(x) 'II ..

(21

CHAPTER I

26

A NO N LIN EA R HISTORY O F RADIO

Now, the curre nt density J (i n A/m l ) is j ust the produ ct of the volume charge density p and veloc ity. and must be independe nt of x . Hence we have

J = p(x)v(x) = p (x ) so thai p (x)

= J

t q,,(X) m,

R£' 2

q ,,( x )

.

•

(3)

(4)

Thi.. last equation gives us one relationsh ip betwee n the charge density and the p0tcrniul for a given curre nt density. To so lve for the potential (or charg e den sity) we tum 10 Poisson's equation . which in one-dimensiona l fonn is j ust

d ' ,,(x ) dx'

.o(x)

= ---

(5)

<0

Combi ning Eqn. 4 and Eqn. 5 yields a simple di fferenti al equation fur the potential: (6)

with the boundary co nditions (7)

,,(d ) = V

and £(0)

= _

d"I dx

=

o.

(8)

x =l1

Thi s last bounda ry condition is the result of assuming space cha rge- limited current. The solution is of the foml 1/J (x) = ex" (tru st me). Plugging and chugg ing yields ,,(x) = V (x /d) 4! ' .

(9 )

Now, if this last expression is substituted back into the differential eq uation. we obtain. at long last, the desired V- I (or V-J ) relationship: (10)

where the (geometry- dependent) constant K is known as the perveance and is here given by K

~

<0 d2

j

32q

8 lmr

(1 1) '

1,10 A PPE ND IX: A VA(UUM TUBE PR IMER

27

I •

-+-~=---------- · V

FIGURE 1, 16. V-I charoderis& s of diode (space cha~ imitedl .

The 3j2-power relationship betwee n voltage V and current I (sex Figure 1.16 ) is basic to vacuum tube operation (even for the more common coax ial structure ) and recurs frequently. as we shall soo n see. As stated previously. the V- I characteristic j ust derived assumes thai the current flow is space chargc- Iimited." Tha t is, we assume that the ca thode's ability to supply electrons is not a lim iting factor. In reality, the rate at which a cathode ca n supply electron s is not infinite and depen ds on the cathode temperature. In all real diodes, there exists a certain plate voltage above which the curre nt ceases to follow lhe 3/2-power law becau se of the unavailabi lity of a sufficient supply of electron s. This regime. know n as the emission-lim ited region of operation. is usuall y assoc iated with power dissipa tion sufficient to cause destruction of the device. We will generally ignore operation in the emission-limited regime. although it may be of interest in the analysis o f vacuum tubes ncar the end of their useful life, or in tubes ope rated at lower-than-normal cathode tempe rature. The diode structure we have j ust analyzed is normally incapab le of umpliticarion. However, if we insert a poro us contro l electrode (know n as the grid) between cathode and plate, we can modulate the now of current. If ce rtain eleme ntary co nditions arc met , power gain may be readily obtaine d. Let's sec how this work s. Figure 1. 17 shows a triode tha t is quite similar to the structures in de Forest' s first triode audions. and its operation can he understood as a relatively straightforward ex tension of the diod e. The field that co ntrols the current flow will now depend o n bo th the plate-to-ca thode vo ltage and the grid-to- cathode voltage . lei us assume that we may replace the vo ltage in the diode law with a simple we ighted sum of these two vohages. We then write, using notational co nventions of the era:

}II

And. a\ staled earlier. it also as..\ umes zero initial velocity of elec trons e mitted from the cath ode and ~glec ts contact potential diffe re nces between plare and curhod e. Th is correct ion usually amounts 10 less than a volt and therefore is important on ly Ior low pkue -ro-carho d e voltages.

28

CHAPTER. 1 A N ON LIN EAR HISTORY O F RA DIO

Plate (" wing" )

Grid

Ca thode

FIGURE 1.17. Idealized planor lriod e structure .

"c, _I-_~I-_~I-_~L ..

More nega tive Ec

+~:::.'""':::::... _~=-~:::._------

.. Ell

FIGURE 1.18 . Triod e characteristics.

11'1"~ =

K ( Ec

+

E H) ' / IL

2

,

(1 2)

where K is the triode perveance. E c is the grid-to -cathod e voltage. En is the plateto -cathod e voltage, and In is a ro ughly constant (though geometry-dependent) parumcter known as the amplifica tion factor . Figure 1.18 shows a family of triode characteristics confor ming to this ideal relationship. Physically what goes on is this: elec trons leaving the ca thode feci the infl uence of an electric fi eld that is a function of two vo ltages. Vol! for volt. the more proximate grid exerts a larger influence than the relatively distant plate. If the grid potential is negative then few electrons will be attracted to it, so the vast majority will now onto the plate. Hence, little grid current flows. and there ca n he a very large power gain us a consequence. The negative grid-to -cathode voltage and tiny grid current that characterize normal vacuum tube oper ation is similar to the negati ve gate-to -source voltage and tiny gate current o f depletion-mode n-channcl FETs (fi eld -effect transistors). although this comparison seems a bit heretical to o ld-timers.

j

T

1.1 0 APPEND IX: AVACUU M TUBE PRIMER

29

cathode

FIGURE 1.19. Incremental model

For triode vacuum lube.

The analogy betwee n FETs and vacuum tube s is close enou gh thai even their incremcntal models arc essentially the same (see Figure 1. 19). Approximate eq uations for the transcondu ctance g", (someti mes ca lled the "mutual conductance") and incremental plate resistance rp are readily obtained from the V- I rela tionship already derived: ( 13 )

and ( 14 )

Note thai the prod uct of 8m and T p is simply JI , M ) that If rep resents the opencircuit amplifica tion factor. Add itio nally. note thai the transconductance and plate resistance are only wea k functions (cube roots) of operating poin t. For this reason, vacuum tubes generate less harmonic distortion than othe r devices wor king over a comparable fractio nal range abou t a given ope rating point. Rec ull rhat the ex ponential V- I relationship ofbipol ur transistors leads to a Hncur dcpcndencc o f 1:m on I . and that the square-law depe ndence of dra in current on gate voltage lends to a square -roo t dependence of 1:", on I in PE Ts. The relatively wea k dependence on plate' cur rent in vacuum tubes is apparently at the co re of arguments that vacuum tube amplifi ers are "cleaner" than those made with other types of active devices. It is certainly true that if amplifiers arc driven beyond their linear range that a transistor version is likely to produce more ( perha ps much mo re) distortion than its vacuum tube co unterpart. However, there is co nside rably less mer it to the argument that audible differences still exist when linear ope ration is maintained. The triode ushered in the e lectronic age, making possible transcontinent al telcphone and radiotelephone communications. As the rad io art advanced, it soo n became clea r thai the triode has severe high-frequency limitations. T he main prob lem is the plate-to-grid feedback ca pacitance. since it gets am plified as in the Miller c ffeet. In transistors. we can get aro und the prob lem using cascoding, a tech nique Ihal isolates the output node from the input node so that the input doe sn' t have 10charge a magnifi ed capaci tance. Although this technique could also he used in vacuum tubes.

CHA PTE R I

30

"

A NONLINE A R HISTORY OF RA DIO

AppnJll. imale screen grid voltage

,

Ec Increasing

j !

/

r • FIGU RE l. 2 0 . Tetrode chorocteristics.

(here is a simple r wa y: add anothe r g rid (cal led the scree n grid) be tween the old grid (called the co ntro l grid ) and the plate . If the scree n grid is hel d at a fixed po tential. it acts as a Faraday shield betw een ou tput and input . and shunts the capacitive feed -

hack to an incremental ground. In effect. the ca...coding device is integral with the rest of the vacuum tu be . The sc ree n grid is traditionall y held at a high DC po tential to prevent inhibition of curre nt flow. Besides gelling rid of the Mill er effec t prob lem. the addit ion of the screen grid makes the curre nt flow even less depe ndent on the pla te voltage than before. since the control grid "see s" what' s happe ning
l~

Actually, ot'~ati\'t': resistance beha vior can occur in a triode if the grid is at a higher potential than the plate.

31

1.10 A PPEN DIX: A VACUUM TUBE PR IM ER

• .. Increasing ~:

r:

/

(Scree n and

SU P~'iOf

grid voltages held I;nn, la nl)

• FIGURE 1.21. Pentacle characteristics,

Well, one grid is good, a nd I WO arc bette r, so guess wha t? O ne way to so lve the problem of secondary e mission is 10 add a third grid (called the suppressor grid ), and place it nea rest the plate. Th e suppress or is norm all y hel d at cathod e pote ntial and works as follows. Elec trons leaving the region past the sc ree n grid ha ve a high enough veloc ity that they aren' t go ing to be turn ed around by the suppresso r grid's low potential. So they happily make their wa y to the pla te, and some of them ge nerate secondary electro ns, as before. But now, with the .~ ll p p re s sor grid in place, these secondary electrons are a ttracted bac k to the mo re po sitive plate, and the negat ive resistance regio n of opera tio n is avoided . With the add itional shielding pro vided by the suppresso r grid, the output c urren t depe nds less on the pla te -to -ca thode voltage. Hence, the output resista nce increase s and pe ntodes thu s provide large amplification factors (tho usa nds, compared with a ty pica l triode's value of about te n or twe nty) and low feedback capacita nce ( like 0 .0 1 pF, excl ud ing ex ternal wirin g ca pacita nce) . Large voltage sw ings at the plate are therefore allowed, since there is no longer a concern abo ut negative resista nce (see Figu re 1.2 1). For these rea so ns, pentodes are more effic ient as pow er output devices than tctrod cs. Later, some very clever people at RCA figured OUI a wa y to get the equivale nt of pentode act ion without add ing a n ex plicit suppress or grid. Since the idea is j ust to devise conditions rbat repel seco nda ry electrons hac k to the pla te, you might he able to exploit the natural repulsio n between elec trons to do the same job. Suppose, for example, we conside r a strea m o f electrons flowing be tween two locati on s. At some inte rme diate po int, there ca n be a regi on of ze ro (or eve n nega tive) field if the distance is suffici ently grea t. The effec t of mu tua l repul sion ca n be e nha nced if we bu nch the electrons together. Beam -forming electrodes (see Figure 1.22), working in concert with contro l and screen grids wou nd with equ al pitch and aligned so that the grid wires ove rlap,

CHA PTE R I

32

A N ONLI NE AR HISTORY O F RAD IO

Ca lhude -

Plate ---

I

Electron Beam

\

Contro l Grid

I Screen Grid

f IGURE 1. 22. Beom·ptJ"Wef $trvcture (top view).

force the electrons to flow in shee ts. Th e concc mrutcd electron bea m then genera tes a nega tive field region (a virtual suppress or grid) w ithou t requ iring large electrode spaci ngs . And , as an unexpected bonus. it turns OUI thai the c harac te ristics at low vol tages a re actually superior in some respec ts (the plate CUITCn l a nd output resisranee are higher) 10 those of true pentodes and are thus actually more desirable than " real" pcn todes for powe r a pplica tio ns. Well. this grid mania didn't stop a t the pe ntode , or even the hexode. Vacuum tubes with up 10 seven grids have bee n made. In fact . for decades the basic superhet AM radio (the "All- American Five-Tuber ") had a heptode. whose five grids allowed one tube (usuall y a 12BE6 ) to function as both the local oscillato r and mixer, thus reducing tube cou nt. For trivia's sake. the All-America n Five abo used a 35W4 recrifler for the po wer supply, a 12BA6 IF amplifi er, a 12AV6 triode /duo -diod e as a dem odul ator and audio am plifier, and a 50e5 beam -power audio output tube . Here's some other vacuum tube trivia: for tubes made utter the ea rly 1930s, the fi rst num erals in a U.S. receivin g vacuum tube's type number indicate the nominal filamen t voltage (with o ne exce ption: the " loktals" 411 have numbers beginning with 7, but they are actually 6 -volt tubes 1110st of the time). In the typical superhet men tioned previously, the lube filament voltages sum to about 120 volts, so that no filament transformer wa s required . The last num bers arc supposed to give the total number o f cle me nts. but there was widespread disagree ment on what constituted an clement te.g., whether one should co unt the filamenn . so it is nnly a rough guide at best . T he letters in between simply tell us something about when that lube type was

4" L...klals had a spec ial base that locked the tulles mechanicall y into lhe soc ket 10 preve nt their working IUfl:'oe in mob ile operenon s.

I

1.10

A PPE N DIX: A VACUUM TUBE PRIMER

OJ

registered with RETM A (which later beca me the EIA). No! all registered rube type s were manufactured, so there arc many gaps in the sequence. In CRTs. the first numbers indicate the size of the screen' s diagon al ( in inches in u.s. eRTs and in millimeters e lsewhere). The last seg me nt has the leiter P fo llowed by numbers. The P stands for " phosphor" and the num bers following it tell you what the phosphor characteristics are. For example. P-I is the standard phosphor type for black-and-white T V CKTs, while P22 is tne co mmon type tor color T V rubes. The apex of vacuum lube evo lution was reached with the development (If the tiny nuvisror by RCA. The nuvistor used adva nced metal-end -cera mic construction. and occupied a volume about double that of a TO-5 transistor . A num ber of RC A co lor televisions used them as VHF RF amplifiers in the ea rly 1970 s before transistors finally look over co mpletely, RCA's last vacuum lube rolled o lT the assembly line in Ham so n, New Jersey soon after. markin g the end of about sixty years of vacu um tube manufacturin g and. indeed, the end of an era .

CHAPTER TWO

CHARACTERISTICS OF PASSIVE IC COMPONENTS

2 .1 I N TRODUCTION We 've see n that RF circ uits ge nerally ha ve many passive co mpo nents. Successful design therefore depe nds cri tica lly on a detailed understanding of their characteristics. Since mai nstream integra ted circ uit (Ie ) processes have evolved largely to satisfy the demands of d igital electronics. the RF Ie designer has been left with a limited pa lette of passive devices. For example, inducto rs larger than abo ut lO nH co nsume significant die area and have relatively poor Q (typicall y below 10) and low self-resonant frequency. Ca paci tors with high Q and low temperature coe fficient are ava ilable. but tolerances are relatively loo se (e.g.• orde r of 20% or worse). Addi tion ally. the mo st area -efficie nt ca paci tors also le nd to have high loss and poor voltage coe fficients. Resi stors with low self-capacitance and temperature coefficient are hard to come by. ami on e must also occasional ly co ntend with high volta ge coe fficients, loose to lerances. and a limited range of val ues. In this chapter. we examine IC resistors. ca paci tors. and inductors ( incl uding bond wires. since they are often the best inductor s avai lable), Also, given the ubiqui ty of interconnect, we study its prope rties in de tail since its parasitic» at high frequencies can be quite import ant.

2. 2 RESISTORS There are relatively few good resi stor options in standard CMOS (co mpleme ntary metal-oxide silicon) proce sse s. O ne possibili ty is to use pnlysilicon ("poly" ) interconnect material. since it is more resistive than metal. However, mo st poly these days is sitici ded specifically to red uce resista nce. Resi stivities tend to be in the vicinity of roughly 5- 10 ohms per squ are (within a factor o f abo ut 2-4, usuall y). so po ly is appropriate main ly for moderatel y small-valued resistors. Its tolerance is often poor (e.g., 35%), and the temperature coe fficient. defined as

"

2. 2 RESISTORS

loR TC ,, - -

R aT '

35

(I )

depends on doping and co mpos ition. and is typi ca lly in the neig hbo rhood of J(XXl ppm r c. Unsilici ded po ly ha s a higher resistivity (by appro ximately an ord er of magnitude. dependi ng on doping). and the TC ca n vary wide ly (eve n to zero. in ce rtain cases) as a funct ion of processing det ail s. 11 is usuall y not tightly co ntrolled. so unsilicided poly. if ava ilable as an option at all. frequ en tly possesses very loo se tolerances (e.g.• 50 %). Advanced bipol ar technologies usc self-ali gned poly emitters. so poly resistors are an option there. too . In addi tion to thei r moderate T'C, po ly resistors have a reasonab ly low parasit ic capacitance per un it area and the lowest voltage coe ffic ient of all the resisto r materials available in a standa rd C MOS techn ology. Resistors made from source - drain diffu sio ns are also an option. T he resistiv ities and temperature coe fficie nts are ge nerally similar (within a factor o f 2. typically) to those of silicidcd po lysilicon. with lower TC associated with heavier doping. There is also significant para sitic (junction) capacita nce. and a no ticeable voltage coeffi dent . The former limit s the usefu l frequ ency ran ge of the resistor. while the latter limits the dynam ic range o f voltages that ma y be applied without introd ucin g significant distortion. Addi tio nall y. ca re must be ta ken to avoid forward -biasing ei ther end of the resistor. T hese charac teristic s usually limit the use of di ffused resistors to noncritical circuits. In modern VLS I (very large -scale integrat io n) techno logies, source- drain "dif'fusions" are defined by ion implantation. The source-drain regions formed in this way are qu ite shallow (usua lly no dee per than abo ut 200 -300 nrn. scaling roug hly with channel length ), quite heavily doped . and almost universally silicid ed. leading to moderately low temperat ure coe fficients (order of 500- 1()(X) ppm / "C). Wells may be used for high-value res istors, since res-i s-tivities arc typicall y in the range of 1-10 kO per sq uare . Unfo rtunately. the para sitic capac itance is substantial because of the large-area j unction formed between the well and the substrate; the resuiting resistor has poor initial to lerance (±50 -80%), large tempe rature coe fficie nt (typically about 3000 -j(XXl pp m ;o C. owi ng to the light dopin g), und large voltage coeffici ent. Well resistors mu st therefore be used with ca re. Sometimes, a ~10S tra nsistor is used as a resistor. eve n a variab le one. Wit h a suitable gate- to- source voltage. a compact resistor can be formed . From first-o rde r theory. reca ll that the incremental resistance of a lon g-chann el MOS transistor in the triode region is (2)

Unfortunately, implicit in this equation is that a MO S res istor has loose tolerance (because it de pends on the mob ility and thresho ld) , high temperature coe fficien t

2 .3

CA PACITORS

37

f iG URE 2.1. Example for counting squar~ .

• --- L - - - . Sum-lra le

f iGURE 2.2 . Pa ra llel p1atecopocitor.

In general, the resis tan ce o f the contact ce lls (the "d umbbells" at the ends) must also be taken into acco unt. Although the actual value depends on the details of the contact cell layout. the resistance is typically on the order of half a sq uare.

2 .3

CAPACITORS

All of the interco nnect layers may he used to make trad itio nal para llel plate capacitors (see Figure 2.2). However. ord inary intcrlcvel dielectric tends to he rather thick (OHler of 0.5- 1 /1111 ). precisely to reduce the capacitance bet ween layers. so the capacitance per unit arca is small (a typical value is 5 x IO - ~ pr/ /1m2 ). Additionally, one must he awa re o f the capaci tance formed by the bott om plate and any co nductors (especia lly the substrate) benea th it. Th is parasitic bottom plate ca pacitance is frequently as large as 10- 30 % (or more) o f the main capacitance and ofte n severely limits circuit perfor mance. The standard capacitance for mula . A

w.L

C ::::O: e- = e - /I II

'

(4)

38

CHA PTER 2 CHA RAC TERISTICS O F PASSIV E rc CO M PO N EN TS

FIGUII:E 2.3. Example of Ioterol RuJI. copocitor (k:Jp view).

somewhat undere stimates the capacitance because it does no t take fringing into acco unt, but is accurate as long as the plate dim ension ... are much larger than the plate separation H . In cases where this ineq uality is not well satisfied . a rough first-order correc tion for the fringing may be pro vided by adding 211 to eac h of W and L in com puting the area of the plates:

C :::::: e

(W + 2 H ) · (L + 2 H) 1/

::::::

s

[ WL

II + 2lV + 2L ] .

(5)

One o f the few bits of good news in Ie passive com ponents is that the TC of metalmetal capacitors is quite low, usually in the range o f approximately 30 - 50 ppm j"C. and is dom inated by the TC of the o xide's dielectric constant itself. as dim ensional variations with tempe rature are negligib le. I The total capacitance per unit area can be increased by using more than one pair of interconnect layers. As of this writing. some process technologies offer five metal layers. so that a quadru pling of the capacitance is possible through the usc of a sandwich structure. One can increase the capacitance eve n further by exploiting lateral flux between adja cent metal lines within a given interconnect layer. Allowable adja cent metal spaci ngs have shrunk to values smaller than the spaci ng between layers. so this lateral coupling is substantial. A simple structure that illustrates the general idea is shown in Figure 2.3. where the two term inals of the capacitor are labeled P I and P 2. As can be seen. the " top" and " bottom" plates. co nstructed out of the same metallayer. alterna te to exploit the I J. L. Mc'Creary, " Matc hing Pro perties. and vonage and Tem perature Dependence o f MOS Capaciton; ," IEEE J, S"lid ,SIOIt> Circuits. v. SC- t 6, no, 6, December 19K I , pp, bOK- Ib,

j

2 .3 CA PACITORS

39

lateral flux , Ordinary vertica l Itux ma y al so be ex ploited by arranging the seg me nts of a different metallayer in a co mple mentary pattern. An importa nt a ttribute of a late ral flux capacitor is that the parasit ic bott o m plate capacitance is mu ch sma lle r than fo r a n o rdinary para lle l pl ate structure. since it con..umes less area fo r a given value of tota l ca pac ita nce. Since late ral ca pacitance depen ds o n the to ta l perime te r, the maximum ca paciranee b obtained with layout geo me tr ies thar max im ize the amoun t o f perimeter. A particularly rich so urce of use fu l geometrie s ma y be found in the literature o n Iractals, since they are structures that ma y e nclose a finite area with an infinite perim eter,2 Because of photolithographic limitations , infinite pe rime ter is unattai nable . but largeincrea ses are po ss ible. In certa in cases, capacitance increase s of a factor of 10 or more can be achi eved . A standard alterna tive is to use a MOS ca paci to r. availabl e in C MOS precesses as simply the gate ca pac itance of an ordinary transistor. Even in so me bipo lar processes. a MOS capacitor is still a vailable as a specia l o pt io n. whe re the bonum pla te is typically formed by an e mit ter di ffu sio n, a thinned oxide is the di e lect ric. and met a l o r polysihcon is the top plate. Ca paci tance per unit area de pends o n the die lect ric thickness, but is typicall y in the ra nge of 1- 5 fF{l'1 m 2, o r ro ughl y 20- 100 limes larger than ordinary inte rco nnect ca pacitors. Ge nera lly. a ~tOS capacitor will ha ve a s ma ll, positive tempe rature coeffic ie nt (o n the o rder of 30 pp m /",C ). but o nly if the se miconducto r is d uped to dege neracy. Gale capacito rs the refo re ex hibi t so me what la rge r TC values. When using a gate ca paci to r in a C MOS process. it is important to keep the tra ns istor in strong inversion t i.e.• keep the gate-so urce vo ltage we ll abo ve the threshold ); otherwise, the capac ita nce w ill be sma ll. lossy. a nd hig hly no nlinear. Oc casio na lly, an accumulation-mode MO S FET, co nstruc ted by using n+ so urce- d ra in diffu sion s in an n-well, is made available fo r use as a ca pac ito r to a llev iate this pro blem. Th is typeof capac ito r req uires no thing abo ve w hat is a lrea d y ava ilable in a standa rd C MO S process, but its charac te ristic s may not be co ntrolled (or eve n tra cked ) by the ta b. For both type s of ga te ca paci tor. the Q depen ds o n the cha nnel resistance. :IS detined by Eqn. 2. To de ve lop a c rude. first-or der mode l for such a capac itor , co nsider Figure 2.4. Th is a pproxi matio n g rossly overestima tes the e ffec tive series resistan ce because. in the actual struc ture. po rtio ns of the gate ca paci tance nca r the source and drain connect to tho se termi na ls th ro ugh less resistance than do po rtion s near the ce nter of the channel. ' Th e model identifi es the maxim um resistance - that from the center of the cha nncl to the so urce- drain co nnection - and put s tha t wors t-case va lue

1 A Shahan i er a1 .. "A 12mW. Wide D ynamic Range C MOS Prom-End Ci rcuit for a Po rtable cops Receiver," ISSCC DiRt'·d tI!Tt'chn;cuf Pupt' rs. Slide Su pplement, February 1991. J In fact. it may he shown thaI a be tte r estima te of the eq uivalent resis tance is ac tually on e third the value shown in Figure 2..1.

40

CHA PTER 2

CH A RAC TERI STICS O F PASSIVE

rc COMPO NEN TS

" -+-

-+-

FIGURE 2 . 4 . Evolu tio n of cru de ga te capacitor model .

in se ries with all o f the capacita nce. Neve rtheless. the mode l correc tly predic ts that. to maxim ize () . one should use the minimum allowable device length (10 minimize rd., for a given bias). Furt hermore, one must also exe rcise care in co nnec ting to the device in order to min imi ze addition al resistive losses. Another option is to usc a ju nction capacitance. such as thai formed by a p+ region in an n-well . Since a j unction ca pac ita nce depend s on the applied bias. such a capacitor is oft en ex ploited to mal e elec tronically tuned circuits: d iode s used this way arc known as n lfacl o n (from va riab le reactor). Reca ll tha t the j unction cap aci tance depend s on bia s as follows: 16)

where Cj O is the inc re me ntal capaci ta nce at zero bias. V,.- is the a mount o f forward bias a pplied ac ross the junc tion . l' is the built-i n potential (typically several tenths of a volt). ami " is a parameter thai depe nds on the dop ing pro file. Pur an abru ptly do ped j unctioll. " has u value of 1/ 2. whereas " = 1/ 3 fo r linearl y graded j unetions." As you ca n imagine. j unct ion ca pac itors arc ofte n a JXXlT cho ice if varac tor action is undes ired . Th e capacita nce equation holds well for reverse a nd wea k forward bia ses. but overes timates the true ca paci ta nce for incre asing a mounts of forward bias. A crude a pproximat ion is that the j unct ion ca pacitance in forw ard bias is 2-3 times the zero -bias value. Add itionally. the paramet ers and" should be treated mai nly as curve- fitting paramet ers they may not alway s take on physically reasonable values. J unct ion capacitors also have relative ly large tem peratu re coefficients. varying fro m abo ut 20U ppm /" C atlarge reverse bias to perhaps J(XXl pp m r C at zero bias. It is possible 10 show S that the temper ature coe fficie nt o f a junction ca pacitance may he expressed a pprox ima tely as

e

-c

.. Values of" in e xcess of unity may he roka ined by " hyperabrupt" junctions. but require special pfOcessing not available in standard Ie techoology. S U:-ing infonna tion from ~kCrcary (!\IX footnote I ).

2 ,3

TC

~ (I -

CA PAC ITORS

1l )TC Si

_ ,, [

I ]TC.-. . 1 - VF!¢

41

(7)

where II , tP. and VI' are ~IS in Eqn . 6. TC Si (the TC of €S i ) is aoout 250 ppm r C , and Te. (the TC of the built-i n junction po tential) is doping-dependen t and typically in the range of about - 1000 to - 1500 pp mrC. Unfortunately, the Q of Ie var uctors vari es in an inverse man ner with tbc lun ing range. The asymmetrical doping req uired to provide a large change in capaci tance per unit voltage also guarantees a relatively large se ries resistance . The largest series resistance occurs when the dep leti on region is the narrowest and the cap acitance is therefore the largest . That is, the Q is lowe st when the capaci tance is highest. As a consequence, IC ver uctors mu st he used with care (e.g.• they should not comprise all of a tank's ca pac itance) .

INTERCONNECT CAPACITANC E At radio frequencies. it bec om es especially crucial 10 obtain accu rate values for the parasitic capacitance of interconn ect. The parallel plate formul a from undergradu ale physics often gross ly underestimat es the ca pacita nce beca use one di men sio n is often nor much larger than the distance to the next co nductor layer. The frin ging capacitance is the refore s ignificant. and (he simple formula is often unaccepta bly inaccurate. We will consider three configuratio ns of conductors: Case I is a sing le wire over a conducting plane of infinite ex tent; C ase 2 is a sing le wire betwee n two infinite planes: and Case 3 is a wire between two adjacen t wires all over a single infinite plane. Those who are uninterested in the somewhat tedious derivation s may skip to the summary of the formulas. Finally, kee p in mind that, in all cases co nside red here. a uniform dielectri c is assum ed . The presence of pa ssivation layer s and/or plastic packaging typically increases the capaci tance by amounts ranging from about 10% for the topmostlayers of intercon nect to on ly 1- 2% for the innermost layers. Case 1: Single Cand uctor over Ground Plan e The case of a single, isolated wire over a co nducti ng plane is perhap s the eas iest one to consider first . One formu la for this case. offered by 'ruan." has an intuitive appeal because it is physica lly motivated : it explici tly decomposes the cupacit unce into area and fringe term s by modeli ng progressively as shown in Figure 2.5 (we assume that the wire is infinitely long along the direction perpe ndicular to the plane of this page).

• c. P. Yuan and T. N. Trick, "A Simple formula for the Esnmation

of the Capacuence of TwoDimensionallnlcrco nnccts in VLSI Cucuus," IEEE EIt'clflJn Dt'l 'ict' l A'II ., v, EIl L- 3. 19K2. pp. 39 1-3.

42

CHAPtE R 2

CHARACTER ISTICS O F PASSIVE IC COMPONENTS

4 - .WTfl - .

II

II

H

FIGURE 2.S. Yuan'l decomposi tion of para llel pIote ccpeeiscece into a rea and

fringe comJXlnenb.

As see n in the figu re. the basic idea is 10 obt ain the 101011 ca pacitance as the sum of two parts. O ne compo nent is the familiar are a (fringe less) term proporti on al 10 Wi ll . while the ot her (fringe) contribution invo lves the ca paci tance due to a wire of diameter T. di minished by the ter m proport ional to T/ 2H . Yuan ' s fo rmu la for the capac itance per unit length is thus

C v"'"

IV [ II

~ E - +

2"

Inl l + (211/ T)(I + J I + T! Hl }

- -T] 211 '

(8)

Yuan 's approach work s well as lon g as the ratio WI H is not 100 sma ll. with typical errors in the runge of 5% or 50 . 7 Below lV/H val ues of abour z-j, however. the erro r grows quite ra pidly. Unfo rtunate ly. that often ca n be the regime of interest . particularly when fut ure process tec hno logi es a rc considered. A lso, the subcx pression for the fringe ca paci tanc e is still a hit cumberso me. A nother stra tegy is to aba ndon any physicall y motivated a pproac h a nd di rectly app ly funct ion -fitt ing tech niqu es to the results of two- dimensional (20) field-solv er simula tio ns. O ne formu la resulting from such an c xerci sc was developed by Sakurai:" C Salu rai ~

IV e [ II

O. ISIV

+ - /-/- + 2 .8

(T )o.m] If

'

(9)

where the area contribution (first term ) and frin ge con tributio n (other two te rms ) are show n se parately, as be fore. As w ith Yuan 's form ula , Sa kurai' s equation has inc reas ingly poor ucc uracy at large WI/I ratios, but accura cy superior to YUilIl'S at sma ll WI I f , with the acc uracy of the equations c ross ing ove r in the nei ghborhood of WIH = 2-3, a t least fo r the parti cu lar values conducto r and dielectric thick nesses conside red in Barke 's paper (T = 1. 3 ,1111, If = 0,75 Jlm). Bell er accu racy can be obtained with a formula that is on ly marginally more COIll ple x (a lthough po ssibl y a n incre me nt more co mputa tionally efficie nt) than Sakurai's.

or

7

E , Barke, "Line -to -Ground Capa citance Calc ulation for VLS J: A Comparison,' IE,'EE Trans. Compp. 295-8. T . Sak urai and K . Tam aru , "Si mple Form ulas fur 1\\.'0 - and Three -D imensional Capecna nces," IEEE TrolU. E!1!C1"''' Devices, v. ED-30 , no . 2, February 1983, pp . 183-5.

pUll!r-A idl!J De.f ig" , v. 7, no. 2, February t9811, 5

2 .3 CAPACITORS

43

Tab le 2. 1. Capacitance ofsingle wire over single conducting plane

Methnd

Capac itance for W = 1.36, H = 1.65, T = 0.8

Capacitance for W = 2.311, H = 0. 117, T =0.3

0 . 115 0 .115

o.rco

'IF s.&kurdj Yuan Art'a term

0. 114 0.028

0. 1115 0 . 172 O.lJ'I.l

Such an equation. developed by v.d. Meijs and f okkema ( hereafter referred to as MF) through function fitting, is:" CM F ~ £

T )0 .']] . If ." + ( H [IiIV + 0.77 + 1.06 [(IV)O

(10)

Barke claims that the MF form ula typically yields accuracies better than I % for dimensions appropriate to ICs. Th e simplicity and accuracy of the MF formula are very attractive
• N. v.d. Meijs and 1.T . Fokkema , " Vl SI Circuit Reconseucnon from Mask Topo logy," Integ rllli" t/, v. 2, no. 2. 1984, pp. 85- 119.

.44

CHAPTE R 2 CHARACTERISTIC S O F PASSIVE

......../!.....

t

rc COMPO NENTS

",

mml ii;m

FIGURE 2 .6 . Conductor OfTongemenl for Ccse 2.

Case 2: Wire Sandwkhed between Two Conducting PkJne s

The capacitive load of a single wire be tween two conducting planes ( Figure 2.6) ca n be calculated using the fonn ula for C ase I as a starting poi nt. altho ugh not nccessarily in the way one migh t think . If one doc s the obvious and sums the capacitance of the wire 10 each plane separately, it turns out that the resu lt is a gross overestimate becau se the fringing term is pessimistically co mputed. An impo rtant insig ht is tha t the addition o f a second conducting plane re sults ma inly in a ,.,:diMr;b"rio" o f the fring ing fie ld witho ut greatly a ffec ti ng its mag nitude . This obse-rvation then sugg es ts tha t on e should su m o nly the area terms. and the n ad d a weighted average o f the fringe leon" that we wo uld co mp ute for eac h pl ane sepa rate ly. Ide ally, we 'd li ke an ex p ressio n for th e tota l frin g e term thai gi ves us a prope rly we ighted average of the two frin ge contribution s wh en the two are comparabl e, bu r also co nve rges to j ust o ne o f th ese term s in the lim it as the spac ing to o ne plate approach e s infi nity. O ne re lat ively sim ple type o f ge neral " a ve rag ing" function that has mo re o r le ss the right k ind o f be ha vio r (as " a pproa c hes infinity ) is

( II ) No te that a value o f n = 1 co rres ponds to the co nve ntio na l fo rmu la for co mput ing an average . As an ad hoc fix, suppose we co mp ute the fringe term s se para tely. a pply such an averag ing fu nc tio n. and add th at re sult to the su m ofthe area terms . With a so mewhat ar bitrary c ho ice o f 11 = "" (so that the ex po nen ts are reason able ), the ca pac itance o f a w ire be tween two pl anes is

c :::::

e[w(_1+ _1_) + 0.77 - )']""I]. u, I(-w +-112W)""+ [( -T)' + (T III

+ 0 .89 1

wh ich may be rew ritten as :

III

HI

III

(1 2)

2 .3

45

CA PAC ITORS

Ta ble 2.2. Capacitance of .\ingle wire

between Iwo conducting plane s Capac itance Sakurai sum (If capita nce YII,m sum of capitance Yuan (with ma ximum fringe ) MF Iwith ma ximum fringe ) MF (w ilh weighted fringe ) 20 field -solve r value

IT

II j

0.468 f F/ll m 1)4 34 I' F/ll rn

0.3-I3 IF/ lJrn 0.375 f FIJl m

0.370 f F/ll m O..l ti l f Fh ,m

s

I

f iGURE 2 . 7. Cond uctor o rTOngemeo l for Cc se 3 .

C '"

E[W(_ I + -'liz ) + 0.77 III

l[ (-tt, +-

+ 0. 89 1

\V

I

1 )] " " + T O'S ( -, 1 Hz III

+-, 1 112

)"'''1].

( 13 )

Now let' s com pare the results obtained with thi s fo rmula In tho se c alc ulated by some other methods. S how n in Ta ble 2.2 are va lue s fo r \V = 2.3Mrem. II I = (J .M7 11m, HI 0.48 n m, and T 0 .3 rr m . From the table , we ca n sec that the simple sum-of-c aps me thod ove rest imates the true capacitance by as m uch a s 30% , Yua n' s formula fares co ns ide rably be tter for this particular geo me try: it yie lds 41 value th at is a bo ut 5% lower than the 211 va lue if we arbitrar ily add on ly the larger fringe te rm to the sum of area terms. Th e MF equation (with we ig hted frin ge te rms) g ives us a value that is o nly about 2.5% hig he r than the value co mputed by a fie ld so lver, a nd has the advantage tha t O IlC need n' t keep track of whic h fringe te rm is la rge r.

=

=

Case J : Three Adjacent Wires aver a Single Plane

For the genera l ca se of mu ltiple wire s sa ndw iched be tween two pla nes (Fig ure 2.7 ), we'd like ex pressions fo r the c apacita nce bet ween adjacent pairs of wires as we ll as the total capacita nce M."Cn by e ac h wire. Un fortunate ly, s imple formulas for adjace nt pair capacita nce s don't see m 10 ex l..t ye t . The best we ca n offer is anot he r fo rmul a of

CHAPTER 2 CHARAC TE RISTIC S OF PA SSIVE

rc CO MPONENTS

Tab le 2.3. Three adjacent lines over ground plane Total capacitive load

Me-thud

on center wi re

Sakurai 20 value

O.IS4 f F/ lJrn 0. 155 f F/ l-im

Sa kura i. one that computes the total capacitance of the middle wire of three adjacent o nes over a single conducting plane. Thi s formula expresses this total capacitance as the sum of two components. The

first is simply the ordinary wire-over-ground plane term of Case I. The second is supposedly the ca paci ta nce between the midd le wire and its adjacent ne igh bors. So. it is clai med. ( 14)

where ( 15) and IV C m Ulll.l l ~ e [ O·03fi

+ 0.83 HT-

(T)0222][S] H -'"

0.07 H

(1 6)

Note that the mut ual capaci tance term dies away just a bit faster than the first po wer of the spacing. Add itiona lly. one may presum ab ly use the MF formula for C~i ng1.. if de sired , although Sakurai al most certainly tweaked his coe fficie nts for C mulWl1 so that Crotal would be more or less COI'T« t. If we let \V = 1.36 11m, H = 1.65 nm. T = 0 .8 11 m, and S = 1. 19 11 m, Saku rai' s formula predicts a total ca paci tive load on the center wire that matches clo sely the value given by "Maxwe ll" (a 2D field -solver program); see Tabl e 2.3. Th is level of agre ement is quite good , so it appears that Sakurai' s formula is usefully accura te at least for computing the total loadin g on the center of three wires. Where one can go awry is in the physical interpretation of the individual terms o f his equ ation. Sakurai implies that the term C~in gle is in fact the ca pacitance of the middle line to ground where, in reali ty, the proximity of the two add itional co nductors alters that capac itance in major ways. As a conseq ue nce, the term he calls the coupling capaci tance (C"'UlIISJ> is IIot the capaci tance between adj ace nt pairs of conducrors. as one would naturally assume (and as he see ms to claim). It is only the .m m o f his two term s tha t happens to equal the total ca pacit ive load on the ce nter wire; the part itioning of this total into separa te ground and mutual capacitances is another thing altogethe r.

2.4

IN DUCTORS

Tahle 2.4. Compori.wm of ifl(li l'idlllll capacitance

" ter",.~ for

Case J

Total ca pacita nce

Method

Capacitance tu ground o f cente r wire If F/J./ml

Capac itance betwee n center and outer wires (fF /J./ffil

Sakurai 2D value

0 ,115 IUl56

om9

Ol ~

0. 100

lJ.156

o f center wire

(f F!J.l ml

As a specific exa mple , let 's consider the same co nductor arrangement for whic h Table 2.3 was genera ted and co mpare calculations from Maxwell with tho se of Saku rai. This is shown in Table 2.... As you can see. the sum o f the Sakurai co mponents equals that given by 20 simulations. bur the indiv idua l terms don't agree 'I f all.

2 .4 INDUCTORS

From the point of view of RF circuits. the lac k of a good induc tor is by fa r the mosl conspicuous shortcoming of standard Ie proce sses. Although act ive circuits can sometimes synthesize the equivale nt of an inductor, they always have higher noise, distortion. and powe r consumptio n than "real" inductors made with some number of rums of wire.

2 .4 .1 SPIRA L IN DUCTORS The only widely used on-chip indu ctor is the planar spiral, a square version of which is shown in Figure 2.8. Altho ugh a circ ular spiral is known to provide som ewhat higher Q, ncn -Manhauan geometries are no t supported by many layout too ls and not permitted in many tcchnologies.!'' Thc bulk of the spiral is implement ed in the topm ost ava ilable metal (som etim es twoor more levels stra pped together to redu ce resis tance). and the con nec tion to the center of the spiral is mad e with a crossunder o f some lower level of metal. The induct ance of such a spiral is a complicated functi on of geome try, and acc urate computations require the use of field solvers or G reenhou se' s met hod . I I Howevcr. a (very) crude zeroth-orde r estima te. suitable for quick hand ca lcula tions. is

10 Octagonal spirals are occasionally used as a compromise be twee n square and circular spirals. If necessary. the required d iagonal lines can he made of Manhattan stairstep jog s 10 avoid violation of design rules. II H. M. Greenhouse. " Design of Planar Rectangular Microeleclronic Inductors," IEEE Tmns. Putt s, lIyhrids. and pm·taxing, v. I'H P- IO. no. 2. June 1974 . pp. 101-9. Th is ctas sjc paper lks.I:rihc!ia mldily implemented algorith m for accurately co mputing me inductance.

• CHAPTER 2 CHARACT ERISTICS O F PASSIVE rc CO MPO NENTS

2,

I'"

FIGURE 2. 8 . Sqocre plana r spira l inductor.

(17)

where L is in he nries, " is the number of turn s. and r is the radi us of the spiral in meters. Thi s equatio n typi ca lly yields numbe rs on the high s ide. hut ge nerally within 30% of the co rrec t va lue (and o ften bett er than th at) . For sha pes o the r th an sq uare spira ls. multiply the va lue given by the sq uare spira l fo rmula by the square mo t of the area rutin to obtain a crude es timate of the correc t val ue (sec Eqn. 3ot). Thus. for circular spirals. mult iply the square-spiral value by (;rr/ 4 )O.5 ;:::::: n .89 , and by 0.9 1 for oc tagona l spira ls. Perhaps mo re useful for the approximate design of a sq uare spira l inductor is the following equ ation :

,, ~ [ 1'L ] 1 /3 li o

::: [

PL 1.2 x 10- 6

]113 .

(1 8 )

where P is the windin g pitch in turn s / meter: we have assumed that the permeabili ty is that of free space. To get an idea of how area-inefficien t such a struct ure is. co nside r for example a 120 -nll inductor (t his amount of inductance is conside red small in the context ofd iscre te circuits). made with a spiral pitch P of on e tum eve ry five micrometers. Plugging these numbers into our design formu la . we find that the number o f turns needed is something like 27. cor respondi ng to a required rad ius of around 140 J~ m. The area co nsumed by this ind uctor is equivalent to that of abo ut 8 typica l bo nd pad s; it is 11I(~ t'. C learly. the num ber of such ind uctors mu st he kept 10 a minimu m. and it actually may be more econo mically senviblc 10 co nsider the use of externa l inductors in many instances. In general, pract ical on-chip indu ctances are in the neighbor ho od of 10 nH on down .

2.4 INDUCTORS

/-

49

" a

FIGURE 2 .9. Hollow!>pi rclinductor.

Aslightly more complicated inductance formula thai yields significantly better accuracy. useful in the fine -tuning of an inductor design j ust prior to verifi ca tion via a field solver, is l2 ( 19) where a is the square spiral's mean rad ius. defined as the distance from the center of the inductor to the middle of the windings. C hecks with a fi eld so lver indicate that ibe formula usually has an error of under 5%. Note also that this formula accommodates " hollow" spiral inductors (see Figure 2.9 ) in which one or more of the innermost turns have been remo ved to improve Q. Aside from the large area s po tenti ally consumed. another serious proble m with spiral inductors is their relatively large loss. The DC resistive losses arc exacerbated by the skin effect. which ca uses a nonuniform current distribution in a conductor at RF. The consequence is a reduction in the effec tive cross-section. increasing the series resistance. In addition to the series resistive loss. capacitance to the substrate is another conspicuous problem of on-chip spirals . In silicon technology. the substrate is close hy uypically no more than about 2-5 11 m away) and fairly conductive. crea ting a parallel plate capacito r that resona tes with the inducto r. The resonant freq uency of the LC combination represents the upper useful frequency limit of the inductor. and is ettenso low that the inductor is useless. The pro ximity of the substrate also degrades Q because of the energy coupled into the lossy substrate. Il

H. A. Wheeler. " Simple Ind uctanc e Furm ul.:ls fu r Radio Cui h ." /HE Pml."et.'llinK5. 192K. p. 139K. "The original formula applies to circ ular spimls, and yid Js an \ wers in microhc nncs with dimen~iuns in inches.

50

CHA PTER 2 CHA RAC TERISTICS O F PA SSIVE

L

rc CO M PO N EN TS

RS

Cp

Substrate

FIGURE 2.1 0 .

I'Y\odeI foron-d1 ip ~irol inductor.

An add itional parasitic element is the shunt capacitance across the inductor that arises from the over lap of the cross-under with the rest of the spiral. The lateral capaclrance from tum 10 tum usually has a negligible overall effect because it is the series co nnection of these capacitances thai ultima tely appears acro ss the terminals of the inductor . Figure 2. 10 shows a relatively complete model for on -chip spirals. I ) The model is symmetrical. even though actual spirals are not. Fortunately, the erro r introd uced is negligible in most instance s. An estimate for the series resistance may be ob tained from the followin g eq uation:

«, : : :

I -----:~--= W 'O" . o(l - e I f l ) '

( 20 )

where (J is the conductivity of the materia l. I is the total length of the winding, wand I are the width and thickness of the interconnect. and the skin depth 8 is given by (2 1)

The shunt capacitance CI' is: Cp =

1/ ' W

2

"ox

' -

,

(22)

tux

where II" is the thickness of the oxide betwee n the crossundcr and the main spiral. The capac itance betwee n the spiral and the substrate proper is Cux and is here approximated with a simple parallel plate formula. where the total area is that of the winding: I) P. Vue et aI.• MA Ph)'si-.:al Model for Plana r Spiral Indud ors on Silklln," lt D '" Prot:t't'Jin gs. Dec ember 1996.

2." INDUCTORS

e"

Cu~ =w · I · - . /"

51

(23)

The substrate los s is modeled with R I , which accou nts for two distinct mecha nisms. The fi rst is j ust the loss assoc iated with current fl owing into the substrate through COl' and the seco nd is due to the Ilow of image currents induced in the substrate by currents flowing in the spiral above. T he value of R 1 is given by: (24)

where G Wb is a filling parameter that has the dim ensions of co nductance per area . II is constant for a given substrate materi al and distance of the spiral to the substrate. and has a typical value of abo ut 10 - 7 Sill m2. In addition to co ntributing to loss, the image currents also flow in a direction uppcsite to those o f the ma in inductor , Hence. the e ffect o f the image is to cance l partially the inductance. That undesirable effect is assoc iated with another: as the temperature increases. the substrate resistivity also increa ses, reducin g the effectiveness of the cancellation, As a consequence. the inductance tends to increase so mewhat with temperature, The tem perature coe ffi cient may be as high as - 200 ppm/"C, and improves as the spiral is moved farther awa y from the substrate. The capacitance C I reflects the capac itance of the substrate as we ll as other reactive effects related 10 the image inductance ; it is given by C. :::::

w ., -C

wb --,,--=

2

(2 5)

Aswith G...b , C...b is a filling parameter that is con stant for a given substrate and distance of the spiral to the substrate. A typica l range for C....b is betwee n 10- 3 and 10- 2

fF/" m' . With the foregoing set of equations. it is po ssible to optimize the Q and se lfresonant prope rties of a spiral inductor. One insight gained fro m optimization exercises is that the innermost turns may he rem oved to increase Q , since they contribute negligibly to the total flux while contributing measurably to the total loss, Eve n whe n this and other optimi zation s are perfo rmed. though. one almost invariab ly fi nds that the maximum Q is below 10 (and frequently below 5 ). so these spirals are unsuitab le In many cases. Tbere are some minor refinemen ts that help squeeze a bit more Q out of a given technology. A panemed ground shield I" (see Figure 2.1J ) prevents capacitive coupling 10 the lossy substrate while avoiding a short-circuiting of magnetic flux. Another

•• P. Vue and S , Wong. "On-Chip Spiral Ind ucto rs with Pauem cd Grou nd Shields for Si-Based RA e 's," VLSJ Circu its Sympmium J>ig t'I / o! Tf'chnil' ll l Pupers, June IW7.

52

CHAPTE R 2 CHARACTE RISTICS O F PASSIVE IC CO MPO NENTS

~ -,

1/

DZ

-, 1/ / 1"-

/

;?

I"-

"K:

FI GURE 2.11, Patterned ground ,hield .

adva ntage is thatthe shielding greatly red uces coupling of noise from the substrate 10 the indu ctor . The pena lty paid is a red uction in self-res ona nt freq uency caused by the increase d ca paci tance . If the induc tor is co nstruc ted out of metal layers sufficien tly removed fro m the shiel d layer. thi s penalty can be tolerated in most cases. An effective gro und shield may even be co nstructed out of heavily do ped polysilicon , allow ing one to save the precious metal layers for the inductor itself. O ne may also place alternating wedges of n-well and substrate underneath the inductor 10 push image curre nts dee per into the substrate. Finally. one may employ the hollow spiral layout discu ssed earlier. Wh en used together. these h.e. chniq ucs can sometimes increase Q by 50% or more.

2 .4 .2

BONDW IRE IN DUCTORS

In addition to planar spirals, bondw ires arc also frequently used to muke ind uctors. Becaus e standard bond wires arc I mil (that 's 0 .00 1 inches, or abou t 25 11m) in diamctcr. they have much mo re surface area per length than planar spirals und hence less resistive loss, and therefore higher Q values. Also . they ma y he pla ced well abo vc any conductive planes to reduce parasitic capacit an ces (the reby increasing the sel f-resonant frequency) and decrease the loss du e to induc ed image curre nts. If we may neglect the influen ce of nearby conductors ( i.c.. if we ass ume that the return currents arc infinitely far away). the DC indu ctance of a bondwi re is given by :l ~

For a z-nnn- tong standard bond wire. th is formula yiel ds 2.(X) nH. leading to a handy rule o f thumb that the indu ctance is approximatel y 1 nl l/ fIl lll. Notice thut the

l~ Tho'A RRL n " mlhtJt'*. .

l

American Radio Relay League. Newington. CT, 199 2, pp- 2- I K.

2 .4 IN DUCTORS

53

inductance docs grow faster than linearly with lengt h bec a use there is mutual co upling between parts of the bond wire ( i.e., there is a weak tran sformer action) with a polarity that aids the inductance. From the logarithm ic term. however. we see that ihiseffect is minor. For exa mple. go ing fro m 5 mm to 10 mm changes the DC indu ctance per millimeter from 1.19 nil to 1.33 nil (a t least according to Eq n. 26) . The inductance is s imilarly in..ensltive to the wire diamet er. so even rel ativel y large wire has an inductance on the order of I nH/mm. The inductance value is not necessarily well co ntro lled ( partly because it is wea kly freq uency-dependent in add ition to the obv ious geo metric dc pe ndenci cs j.!" so c ircuin using bondwires must be able to accommodate variat ion in the value o f ind uctance. Despite this limitation. however, bondwire indu ctors have been used for years in a number of co mmerci ally succes ..ful amplifiers. and one recent (althoug h some.... hat impractical ) design ' ? uses several bondwires stitched across a die as part of a high-Q resona tor for an on-chip vol tage-controlled oscillator (VCO). The Q of a bondwirc inductor i.. relatively easy to estimate. The conduct ivity of aluminum is about a x 10 1 S/ m. and the permeability of free space is -t7f x In- 1 11/ m. With these numbers. the skin depth is approximately 2.5 J1m at I G ll z , a hand y num ber to remember . Because the skin depth is s mall compared with the 25-Jl m diameter of a typical bond wire. we ca n readily compute the effec tive re..istance pe r length as R

-J

'" -27frS(J --•

( 27)

For these numbers. one obtai ns a res istance of abou t 125 mQ / mm at I G llz. po ten-

ually allowing the sy nthesis of inductors with Q -values o f 50 at that frequ ency. Because inductive reacta nce grows linearl y with frequ ency while skin loss grows only as the square root . Q.values approaching 100 might he possiblc ut 5 GUl , altho ugh actually achieving such values in pract ice requires extraordinary ca re in mini mizing all loss, especia lly at co ntacts to the induc tor. The temperature coe fficient of a bondwire indu ctor is due to the comhination of two effects. One is simply the linear ex pansion of the wi re with increasing temperalure ; this component ha ... a TC of approximatel y 25 p p rn / ~ C . T he ot her is the change in the contribution of the i nfe rn al flux to the total indu ctance. T he resistance goes upwith tempe rature, ca using the skin depth to increase. increas ing the amount of internal flux (and hence the inducta nce) . The contribution of the inte rnal nux ca n he determined from the fo llowing equation for the internal indu ct ance per un it lengt h of a piece of wire at DC:

16 Wilh aU ltlmaled die auach and bond ing equi pme nt. bowever.jhc rc pca lahilily ( an he excellent, with varicnons hel d 10 withi n 1% or ~t1. 11 J. Cruninckx and M . S teyaert , "A CMOS 1.8GH l Low -Ph ase -Noi se Vollage ·Controlled
CHAPTER 2 CHARACTERISTICS O F PASSIVE rc CO MPO NENTS

54

.~ Mf ~

Ll

B

L 1- M 0000 A

C

A

.

L2

k. : ~

M

{L;L;

L 2- M

nooo

~

8

1:1

c

[[~L

D

FIGURE 2 .1 2 . Cou pled inductors and circuit model.

"0

L ;nl = - . 8rr

(2 8)

Equation 28 yields OJ value of 0.05 nH /mm. so internal indu ctance ev ide ntly acco unts for 5% ofthe total inductance of a typ ical wire at DC. At I GHz. the skin depth is on ly a tenth the diameter o f a bo ndwire . so the intern al inductan ce decreases sub-stantially ( by a factor o f 5 10 10 in this case). The cha nge in intern al inductance with temperature typicall y co ntri butes a TC o f approximat el y 20 -50 ppm l' c. so one may ex pec t the total indu ctance of a bondwire 10 po ssess a TC of roughly 50- 70 ppm / cC. C hange in intern al inductance also influen ces the TC o f a spiral inductor. exacerbatin g the positive TC arising fro m the reduced cance llation by image currents.

Coupled Bondwires The magnetic fields surrounding bo ndwires drop off relatively slowly with distance. As a result. there can be substantial magnetic co upling be tween adjace nt (and even mo re remote) bo ndwires (as well as othe r co nduc tors). A measure of this cou pling is the mutual inductance between them . For two bondwires of equal length. this inductan ce is given ap proxima tely by

1'01[(21)

M ::::: 21t

In D

[)]

- 1+ , '

(2 9)

where I is the length of the bondwires and D is the distance bet ween them , IS For u Hl-m m le ngth a nd a spacing of 1 m m, the mutual indu ctance works out to about 4 ni l . Since the inductance of eac h bundwirc in isolation is about 10 nll . the 4 · nH mutual inductance represen ts a co upling coe fficie nt of 40 %. The logarithm ic dependence o f M on spaci ng means that the co upling decreases rather slowly with di stance, so there call be significant interaction even betwee n alternate pins, for example. One mod el for coupled ind uctors is an inducti ve Tvnerwork in casc ade with an ideal transformer; see Figure 2.12. In lhis m odel . L \ and L 2 are the values each

I' Th is formula ilo ada pted from F. Termon. RaJjn Engi neer 's Jlaml hfl(H;, . McGraw· ll i11. New York ,

194] .

55

2 .4 INDUCTO RS

FIGURE 2 .13. Single· layer solenoid .

inductor has with no c urre nt !lowing in the othe r. Resistive losses, as well as parasiric capacitances. a re not shown in the ideal model of Figure 2. 12, but should he uken into accoun t in criti cal designs. AS:I final note on coupled induc to rs. eve n planar sp irals ma y be used to ma ke uansformers.!" The coupling coefficie nt ca n be as high as O.8...().9 for two spirals directly on top of one unorhcr. and dec reases to ncar ze ro as one ind uctor is s lid laterally untilthey are adjacent.

2.4 . 3 M ISCELL AN EOU S IN DUCTAN CE FO RMUL A S There are a couple of other formulas that are worth knowing. although not all of them have direct relevance to Ie induct ors. T he first is very a ncie nt, a nd it applies to a single-layer ~o le no i d 20 ( Figure 2. 13). For thi s structure. the indu ct ance in micrnnenries is given by

(30) where r and / arc in inches. In S I units , the form ula is (3 1) where a free-space permea bilit y is a..sumcd. T hese formulas pro vide re markably good accuracy (typ ically beli ef than I % ) for single -laye r coils a.. long as the length is greater than the rudiu s.U

It

J. Long et al., "A Low-volta ge Silicon Bipol ar RF Front-End for pe N Recei ver Applications," ISSCC Digest of TechnicalPapers, Febru ary 19')5, pp. 140 - 1.

lD II. A. wh eeler. "Simple Ind uctance Formulas for Radin Coil s," J'mn- rdillgI IIf Ihe IRE. v. 16. IMI. 10, Ocroter 1921'1 . pp. 1391'1-1 400. )1 The

win

best Q if> ohIained when dj a~er itself.

lhe

windin gs are spaced approxima tely by an amoun t equal

10

the

56

CHA PTER 2 CHA RACTERISTICS O F PA SSIV E rc CO M PO N EN TS

Another case of interest is the induct ance of a sing le loop o f wire. Desp ite the simplicity of the structure, there is no exact , clo sed -form expression for its inductance (e lliptic function s arise in the computation of the total flu", ). However. a useful uppm xbuauon is given by (32)

Thi s formula tell s us that a loop of I mm radiu s has an inductance of approx imate ly -t niL A few SJXl! checks with a field solver sugges ts that this formula typi cally underestim ares the inductance for Ie -sized ObjL'CtS. but values are still correc t 10 better than about 25 -30%. In deriving this appro xima tion. the (eas ily calc ulated) flux den sity in the ce nter of the loop is arbitrarily assumed 10 be on e half the average value in the plane of the loop. then the indu ctance is co mputed as simply the ratio of total flux to the current. In view o f the ruther coarse approximation invol ved . it is rem arkable that the formul a work s as well as it typically doe s. Note that. for a single tum and in the limit of zero length . Whc-eler 's fonnula ( F"'1 n. 30 and Eqn. 3 1) co nverges very nearl y to JlO1CT (within 11%). For fussy folks, a better approxi mation is provided by ll (33)

where a is the radiu s of the wire. With this equation. we see that Eqn. 32 strictly hold s o nly fo r an ria ratio of about 20 . To make a crude approxi ma tio n even more so. F"'1n. 32 can be extended 10 no ncircular cases by arguing that all loo ps with equa l area have about the same induelance. regard less of shape. Thu s. we may also wri te

L ~

J1 oJ 1C A.

(34 )

where A is the area of the loop. Acco rding to this formu la. a clo sed contour of l -mm! area has an inductance of about 2.2 nl-l , Wc can check the rea sonableness of these cquations by co nsidering the indu ctance of a loop of ex tremely large radiu s. Since we can treat any suitably short seg ment of such a loop as if it were straight, we can use the equation for the inductance of a loo p to estimate the inductance of a straight piece of wire . We' ve already com puted that a circular loo p of I-111m radius has an inducta nce o f -t nl l, su we have roughl y -t nil per 6.3 mmlcngtb (circumference). whic h is in the same ballpark as the value given by the more accura te formul as.

22 S . Ramo, J. R. Whinnery. and T. Van Duzer . fir1Js m w Wm 'l' J in M,oJ..", Hadin. Wiley. New York. 196.5. p. 3 11.

2.6 APPENDIX : SUMMARY O F CAPAC ITA NCE EQUATIO NS

2.5 SUMMARY This chapter has present ed nume rou s mod els and formulas for on-c hip (a nd o ff-chip) passive elements. A hie rarc hy of fo rm ulas with roug hly co nsta nt simplicity- accuracy products allows one to carry out computati ons at any level of design - rangi ng from initial exploration to final verification - reducing grea tly (e ven obviating) the need for analysis by electromagnetic field so lvers in many instances.

2.6 APPENDIX : SUMMARY Of CAPACITANCE EQUATIONS

II ,

IT

s

t FtG URE 2 .14.

Conductor arrangement for all cases.

Case I: Wire over Sing ~ Conducting P&ane 0 .77 + 1.06 IV C :::::: € [ 11+

1(1VII )"" + (T)"· /I ' I].

OS)

Case 2: Wire between Two Conducting P&anes

c '" e[IV(_1+ _I /11

III

)

+ 0.77

![ (-1+-1)]"."

+ O.H9 1

HI

111

+ Tu. ~

112

(1_, + _,1)""1] . II I

II I

136)

Case 3: Three Adjacent Wires over Single Ground Plan e

137)

C.,n~K: · L

':;::: €

Cmuwal

~

[ e

(T )".'"

1.1 51V 11- + 2.8 -/I IV [ 0,03 II

(T)"."'( S)-U-I]

- 1.3 1 -II

-II

+ 0.83 HT+ 0.585 (TII )"."'][( HS ) - r.,"

]

.

13K)

•

( 39)

where C 'inJ!1c is the ca p..c uuncc fro m the middle wire to grou nd a nd C QlIllual is the capacitance betwee n adjacent pa irs of wi res. The capac ita nce be twee n the oute r pair ti l' wires is generally negli gible ow ing to the Faraday shiel ding provided by the middle conductor.

58

CHAPTE R 2 CHA RAC TERISTICS OF PASSIVE tc CO M PO N EN TS

Also. the formulas for two wires over gro und plane arc similar. The mutual capacitance is the same (and, of co urse. you co unt it only once when computing the total capacitive load on eit her wire); the leon CoinJk' (wh ich then denotes the ca pacitance of either wire to ground ) differs on ly in that the factor 1.31 cha nges to 0 .655.

PROB LE M SET FOR PA SSIVE IC COM PO NEN TS PROBLEM 1 Extend to ci rcular capaci tors the simple fringe correc tion method out lined for s4uare capac itors. Co mpute the ratio of your corrected capacitance to the uncorrected value for the following ratio s of plate spaci ng to diam eter: 0.005 . (UJ!. 0.025, 0.05. 0 .1. Compare your result to the actual correction factors shown in Table 2.5 . Table 2.5. Co rrection f actors for f rillging in ci rcula r capacitors

O,U05

J .023

0.01 0.025 0.05 a.1ll

1.042 1.09-t 1.167 1.286

PRO BLEM 2

(a ) Design a IO-nll square spiral inductor in which the tota l length of the interconnect is 350 l im . the spaci ng between turns is n m, the metal ami ox ide arc both 1-1'1111 thick , and the metal conductivity is 4 x 107 S/ llI. The ox ide has a relative dielectric constant of 3.9 . (b ) Compute the values for the model element" of your design at 1.5 Gil l if one term ina l is grounded. Initially assum e that the substrate is a superconductor. (c) If this ind uctor is used as part of a tank in whic h the external capaci tance and rcsistancc are 500 fF and 10 krl . respecti vely, what is the impedance of the com bin ation at reso nance'! Do no t bother recom puting the mode l parame ter values for the new reso nant freq uen cy. (d) Change your model to use the default values for G ~ub and CMlt>. Usc SPICE 10 determ ine the new imped ance mag nitude at reso nance. Again. do no t bother reco mputing the other model parameter value. Co mpare with your previou s result.

z

PROBLEM SET

59

PROBLEM 3 A parallel resonant ta nk c ircuit is construc ted fro m a a-t urn. 80 /l mx RO-J.{ffi sq uare spira l ind uctor, ami a 5-Ji ffi X5 -ILffi capacitor co nstruc ted fro m two metallayers separated by an aty pica lly thi n O.2-Jl m oxide di electric (E = 3. 9£0).

(a) Initially neglecting fring ing a nd all o the r parusiucs. what is the nom ina l resona nt frequency of this net wo rk?

(bl Now foolish ly assume th at the sub... Irate is a superco nd ucto r. a nd that the bulk of the inducto r is built out of a layer that is J 11 m above the su b... trutc. Igno ring the shunt capacitance of the inductor's c ros sunde r. hut not the ca pacitance to the substrate, what now is the appro ximate parallel reson ant frequency of the network if the substrate is co nnec ted to one ter minal o f the tank? T he ind ucto r w inding s are 8-Jim wide and l -n m thic k . Yo u ma y assume a sym metric a l ind uc to r mod el. Ie) What is the value of the Ind uc tor's effec tive se ries resista nce a t this new resonant frequency if th e in terco nnec t ha s a conducti vity of 5 x 107 S/m? It ma y help to kno w tha t the pe rmeabili ty is about J.26 x 10- 6 II / m. (dl One measure of ho w Iitlle loss a reactive network has is Q. the quality factor. What is the Q of this reso na to r if it is defin ed her e as w L / R a nd is me asu red at the resona nt freq uency'!

PROBLEM 4 A co mmo n proble m, especially in d igita l sys tems . is ho w be st to size interconnect. A w ider line has more capacitance. but lower resistance. so how wide i... wide enough? To put thi s q uestion on a so mew hat q uant itat ive basis. usc the Sa kurai formula for a single cond ucto r o ver a gro und plane and deri ve a n eq ua tio n for the He product of such a line, where R and C arc the to ta l resistance ami ca paci tance. respectively. We will use a simplified ap proach here, and ign ore the effec t of interconnect loading o n whatever ha s to d rive it. Instead. we foc us e ntirely on the interconnect itse lf.

RC . ho w docs the delay increase if the length dou bles? (b) Yourfonnula should show that the RC prod uc t asymptotica lly approaches a min imum va lue as the wid th goes 10 infinity. What wid th produces a de lay J USI 25 % above this minimum value? Express yo ur answe r in term s of the thick ness T and the he ight H abo ve the substrate . (3) If the propagat ion de lay of signal s is proportional to

PROBLEM 5 Junction ca paci tors are normally used in reverse bias as varact ors. To

explore why they are almost never used in (strong} forward bi as, a ssumc tha t the diode behaves as follo ws in the fo rward direct ion: ,.0 "'" - J5 , Vj /vr

•

( P2. 1)

Assume that the thermal vo ltage VT is 25 mY at the o pe rati ng temperat ure . and th at the diode is built in suc h a fashio n that the forw ard c urre nt is I mA a t a junctio n voltage of 0.5 Y.

.

CHAPTER 2 CHARACTERISTICS O F PASSIVE rc COMPON ENTS

60

(a) Calculate the incre mental resistance at ImA .

( b l If the zero -bias capacitance Cj O is 2 pF, what is the capaci tance at the forward bias of U.s V'! Assume an a bru ptly do ped (ste p) j unct ion a nd a r/J o f 0.8 V. (c) Co mpute the reactance of the ca paci ta nce found in part ( b) at J G llz. Doe s the varactor appear main ly resistive or capaci tive at this freq uency? PROBLEM 6 Design a 7.3-oH inductor . You have at you r d isposal a lolal of 6 mm of bondwire and 90() /lm 2 of di e area . Assume tha t the bo ndwire lengt h ca n be contro lled 10 no beli ef tha n 10%. Maxim ize the Q of the resulting indu ctor. subject to the co nstrai ntt hat the final indu cto r value be within 5% of the target value. For simplicity's sa ke. yo u may assume that the followin g p lanar spiral indu ctance formula is ex act :

( P 2.2) PROBL EM 7 Derive an ex pression for the resi stan ce o f int erconnect as a function of temperature for two cases as follow s. (a) The skin depth is very small compared with the cond uctor dime nsion s. ( b) Th e skin depth is very large compared w ith the co nductor dime nsion s. You ma y assum e that the resistivity of the interconnect material is itself PTAT . (c) Us ing you r result to part (a), how muc h variation in

squ are spiral induc tor betwee n - 55 C and Q

Q wo uld yo u expec t for a

+ I 25' C?

PROBLEM B

(a) Deri ve a circu it model for two co upled bondwires. eac h of which is 7 mm in len gth . a nd w hic h a rc sepa rated by 4 mm. You may ignore res ist ive a nd capacitive parasiti cs. (b) Ass ume that on e bondwire is driven by a voltage source thro ugh a res ista nce of 50 n. A lso ass ume that the other bo ndwire is shunted by a 200 -a- load resis rancc. If the voltage source provides a I· Y unit ste p. usc S PICE to plot the vo ltage across the load resistor. (c) Now double the separation to H rnm and re peat . What do yo u co ncl ude abou t the effectivene ss of separation as a means to reduce parasiti c coupling? PROB LEM 9 A IO-kQ po lysilico n resisto r is to he made (lui of ma te ria l w ith a shee t resistivity of 100 a pe r square. (a) Det ermine the minim um dimensions of thi s resistor if the wi dth ca nnot be controlled to a n unce rtainty of bett er than 0 .2 ILm , and if the var iation in resistance due 10 w idth varia tio n must be ke pt be low 5%. For simplicity. assume a straig htforward linea r layout. ( b) De termine the parasitic ca pacita nce to the substra te if the ox ide diel ectric layer (rela tive dielectric constant: 3.9) is l -nm thick . Usin g a sing le He model for

•

PROBLEM SET

61

this structure. what is the app roximate maxim um frequency abo ve which this resistor cease s to appea r predominantly resistive?

PROBLE M 10 Another constraint on conductor dimen sions is impo sed by electro migration effects. At high enoug h current densities. momentum transfer between electrons and metal ato ms can cause physica l motion of parts of the interco nnec t. A nlU1'OWing o f inte rconnec t c auses an increa se in the cu rre nt density, w h ic h acce l-

crates the narrowin g. and so o n until either the resistance increases to unaccept ab le levels or the intercon nect act ually open-ci rcuits. Electromigrution ru les for most com mo nly used interconnect metals usuall y d ictate an upper bound withi n a factor o f 2 of about 10 9 A f m2 DC current (much larger densities are pennined for high -frequency sinuso idal current!'. because little net migration can occ ur in suc h a case). (a) Assuming a ma ximu m allowable current density of 2 x )09 Af m 2 , determine the minimum acceptable interconnect width ca pa ble of suppo rting a current of 100 rnA if the conductor is 0 .5-11 01 thick . lb) Compute the res istance pe r millimeter of yo ur design if the co nductivity is 4 x 101 Sf m. (c) Estimate the parasitic ca paci tance per millimeter if the oxide is I-11m ibic k .

CHAPTER THREE

A REVIEW OF MO S DEVICE PHYSICS

3 . 1 IN TRODU CTION

Thi s chapter focu ses attention on those aspects of transistor behavior that are o f imm edia te relevance to the RF circuit designer. Separation of first-order from higher-order phenomena is emphasized. so there arc man y instances w hen crude approximation.. are presented in the interest of developing insight. As a conseq uence. this review is intended as a supplement 10 , rather than a replacement for, traditional rigorous trea tme nts of the subjec t.

3 .2

A LITTLE HISTORY

Attempts 10 create field -effect transistors (fErs) actually predate the development of bipol ar devices by over twenty years . In Iact. the fi rst parent for a F ET-Iike tran..istor was f il ed in 1925 by Julius Lilienfeld. hut he never co nstruc ted a working device. Before co -inventing the bipolar transistor, William Shock ley also tried to modu late the conductivity of a semiconductor to create a field -effect transistor. As with Lilicnfeld . problem s with his materials system. copper o xide, I prevented suecess. Even utter moving 0 11 to germanium (a much simpler semiconductor to undcrstand than copper ox ide ), Shockley was still unable to make a working FET. In the course of trying to understand the reasons for the spectacular lack of succes s, Shockley's Rell Laboratories colleagues John Bardeen and Walter Brattain stumbled across the point-contact bipolar transistor. the first practical semiconductor amplifier. Unresolved mysteries with thai device (such as negative p, among others) led

Rectifiers mad e of copper oxide had been in use since the [9 2Ch, even though the detailed ope rating princ iples we re nol understood . AmunJ 1976, wnh decades of semiconductor research to SUppt111 hen, Shoc kley wok one last shot
62

3.3 FET $: THE SHORT STORY

63

G"I~

FIG URE 3.1. n-chonnel junction FET.

Shockley to invent the ju ncti on transistor. and the three eve ntually won a Nobel Prize in physics for their work .

3.3 A llhou~h

FET " THE SHORT STO RY

the qu antitative details are a bit co mplicated . the basic idea that underl ies the operation of a FET is simple: Start with a resistor and add a third terminal (the gale) that somehow allows mod ulation of the resistance between the othe r two rcrminals (the source and drain ). If the power expended in dri ving the control terminal is less than that delivered to a load. power gain results. In a junction FET (see Figure 3. I). a reve rse -biased p-n j unction co ntr ols the resistance betwee n the source and drai n terminals . Becau se the width of a dep iction layer depends on bias. a gate voltage variation alters the effective cross-sectional area of the device. thereby modul ating the drain-source resistance . Because the gate is one end of a reverse -biased d iode. the powe r expended in effecting the co ntrol is virtually zer o. and the r ower gain of II j unction FE T is corres pondingly very large . A junction FET is normally conducting. and requi res the applicat ion of a sufbciemly large reverse bias on the gate to shut it off. Because control is effected by altering the extent of the depletion regio n. such FET s are ca lled depletion -mode devices. Although JFETs are not the type of FET s used in mainstream Ie tech nology. the basic idea of conductivity modu lation underlies the operation of the o nes that are : MOSFETs. In the most co mmon type of MOSF ET. the gate is one plate of a ca paci tor sepurated by a thin dielectric from the bulk of a nearly insulating semicond uctor. With no voltage applied to the gate. the transistor is essentially uonconda ctive betwee n the source and drain terminals. When a voltage of sufficient mag nitude is applied to the gate. charge of the opposite polari ty is ind uced in the semiconductor, thereby enhancing the conductivity. T his type of device is thus known as an en hance ment -mode transistor.

CHAPTE R 3 A REV IEW OF MOS DEV ICE PHYSIC S

p O.

//

/

r -;":"-

I

do "

•• p-!lUt..tralt'

_ _ _ _ _J /

FIG URE 3.2 .

n-channel MOSFET.

As with the JFET. the power gain of a MOSFET is qui te large (a t least at DC ); there is virtua lly no power expended in drivin g the gate s ince it is basically a capaci tor.

3.4

MOSfET PHYSICS: THE LONG ·CHANNEL APPROXIMATION

The previous overview leaves out a great man y detail s - we certa inly can' t write any device eq uatio ns based on the material presented so far. for example . We now undertake the task of puuing this subject on
3 ." MO SFET PHYSICS : THE LONG -C HAN NEl APPROXIMATION

65

.... /

I FIGURE 3 .3 . n·chonnel MO SFET I ~ of boundary between triode and saturation).

assumption. the induced inversion charge Is proportlonal ro the gale vo ltage above the ihresbofd. and the induced charge density is co nstant along the channel. However, if we do app ly a positive drain voltage V , rbcn the channel potentia l must increase in some manner from zero at the source end to V :II the drain end. The net voltage available to induce an inversion layer therefore decreases as one ap proaches the drain end of the channel, lienee. we expect the induced channel cha rge density to vary from a maximum at the source (where VA" minu s the channe l potential is largest) to a minimum at the drain end of the channel (where VA" minus the channel potential is smallest), as shown by the shaded region rep resen ting charge density in Figure 3.3 . Specifically, the channel charge density has the follow ing form : Q" (y) = - C. , II V, . - V( y) ) - V.I .

(I )

where QII(Y) is Ihe charge density al position y. C", is £0,/10" and V( y ) is the channel potential at position y . NUll' that we follow the co nvention o f defining the y-direction as along the cha nnel. Note also thai C,,~ is a ca pacitance per 1111;1 area. The minus sign simply refl ects thai the charge is made up of electrons in this NMO S example. Equation I is all we really need to derive the most important eq uations governing ihe terminal characteristics .

3.4 .1 DR AIN CURREN T IN THE LINEAR (TRI ODE) RE GI ON The linear or triode region o f operat ion is defined as one in which V A'S is large enough (or Vd. small enoug h! to gua rantee the formation of an inversion layer the whole distance from source to drain. From our express ion for the cha nnel charge density, we see thai it has a zero value when

66

CHAPTER 3

A RE VI EW O F MO S DEVICE PHYSICS

IV,. -

V ( )' )j -

V, =

( 2)

().

T he c harge den sity thus first becomes ze ro at the d rain e nd at so me part icul ar vo ltage. Th erefore the bo undary (or the triode region is defined by

I V~ ~ - Vd. l - V, = 0

:::=}

Yd.

= V, . - V, =: Vol"",·

(3)

As lo ng as Vol., is smaller th an Volsal ' the device will be in the linear reg io n of o peration. Having deri ved an e xpression for the cha nnel charge a nd defined the linear region of ope ration. we arc now in a position to deri ve an expression for the device current in term s of the te rm inal vari ables. Current is proportion al to charge times veloc ity.

so we've ju st about g OI it: 10

(4 )

= - WQ,, ( y ) t' ( J ).

The velocity allow fields (rem embe r.this is the lon g-eh annc1 app roximation) is simply the prod uct of mobility and electric fie ld . He nce

(5) where \V is the wid th of the device. S ubstituting now fo r the c hanne l c harge de nsity. we o bta in I i) = - \VCo~ (Vg . - V ( y ) - V, I/A " E .

(6)

Next . we not e that the (v- direc ted) e lec tric field £ is simply (m inus) the g radie nt of the vol tage alo ng the cha nne l. Th er efor e, 1/1 = M "Co~ \V { Vg. - V( y ) -

so that

"V.

V,I-

(7)

dy

(8)

Now integrate a lo ng the c hannel and solve for I f):

(91 At la st. we have the following expressio n for the dra in curre nt inthe triode regio n: (1 0)

No te that the rel ation shi p betwee n d rain c urre nt and d rain -to -so urce volta ge is nea rly line ar for smal l Vd•. Thus. a MO SFET in the triode reg io n behaves as a voltage controlled resisto r.

3.4 MO SFE T PHYS ICS: THE t ONG·CHANNE t A PPROX IMATI O N

67

The strong se nsitivity of drain c urre nt 10 drain volta ge is qualit ativel y sim ilar 10 the behavior o f vacuum IUIx triodes, which lend their nam e 10 Ihis regi on o f operation .

3. 4 .2

DR AIN CURRENT IN SATU RATION

when VJ • i.. high e nough so that the inve rsion layer does nor extend all the way from source to dra in. the c han nel is sa id 10 be " pinc hed o ff." In thi... case. the c hannel charge ceases 10 inc rease. c.. using the tot al curre nt 10 rem ai n co nsta nt despite increases in VdJ • Calculating the value of this c urre nt is easy; all we have 10 do is substitute Vd"'l for Vu in our expression for c urre nt: (II )

which simplifies to Ij "Co~ W

I ,) = -

-

2

-

L

( V,fJ -

2

V, ) .

(1 2)

Hence. in saturatio n. the drain c urrent has a sq uare -law de pend ence on the gale "OUTre voltage and is inde pende nt of drain voltage . Becau se vacu um IUIx pc ntodes u hibit a similar Inse nsitivit y of plate c urrent 10 plate vol tage. this reg ime is occa sionally called the pentodc region of operat ion. The transcondu ct ance of suc h a device in sarurauo n is easily found from diffcrentiaung our expression for drain c urrent: ( 13)

which may also be expressed as:

Rm =

IV

2 Jl"C(l ~ L I[) .

( 14 )

Thus. in contrast with bipolar de vices. a long-channel MO SFET' s tran sconductance depends only on the squa re root of the bias c urre nt.

3. 4.3 DYNAMIC ELEMENTS So far. we've co ns idere d onl y DC paramete rs. Le t's now tuke a look. a t the various capacitances a...sociated with MOSF f::.l s. First. since the source and drain regions form reverse -bia sed junctions w ith the substrate, one ex peets the standard j unc tio n capaci ta nce from each o f those regio n...

..

..

CHA PTER 3

A REVIEW O F M OS D EV ICE PHYSIC S

...

.... ... h u ..... ra l ~ 1

FIGURE 3.4 . MO SFET capaci ta nces.

the substrate. These capaci tances are denoted Cjs" and C/J b • as shown in Figure 3.4. where the exte nt of the de pletion region has bee n greatly exa ggerated . The re arc also various parallel plate ca paci tance term s in addition to the j unction capacitances. The capaci tors shown as Co, in Figu re 3.4 represent gate- source ami gale-drain ove rtop capacitances: these arc highly undesirable. but unavoid able. T he source ami drain region!"> diffuse laterally by an amount similar 10 the dep th that they diffu se. li enee. they bloat out a bit du ring precessing and extend underneath the gate elect rode by some amount. As a crude approxima tion. one may lake the amou nt of ove rlap . 1, /1, as 2/3 to 3/4 of the dept h of the source- drain diffu sion s. Hence. 10

(1 5)

where X j is the depth o f the source- drain diffu sion s, e... is the oxide's dielectric constant (abo ut 3.1Jf."l) . and I,, ~ is the ox ide thickness. The par allel plate overlap terms are augmen ted hy frin gin g and thus the "ove rlap" capacitance would be nonzero even in the ab sence of actu al overlap. In this context, one should keep in mind that . in modern devices. the ga te electrode is actua lly thick er than the channel is long, so the relative dimen sions of Figure 3.* are misleading. In fact . since the thickness o f the gate electrode scales little ( if at all). thc "overla p" ca paci tance nuw changes littl e from ge neratio n to ge neration . Another para llel plate capaci tance is the gate- to -channel capac itance CJ:'" Since bo th the source and drain reg ions exte nd into the region undern eath the gate. the effective channel length dec reases by twice the bloat . L n . li enee. the total value of C~e

is

Cxe =

C,,~ lV(L

- 2 L{) .

(1 6)

3," MQ SFET PHYSICS THE LON G · CHANN EL APPROX IMATI ON

69

There is also a ca pacita nce between the cha nnel and the bu lk. C"I>. thut behaves as ajuncfion capacitance as well. Its value is a pproximate ly £5;

C"I> :::::: -

x, W( L -

2L {),

(17)

where XJ is the depth of the depletion layer. w hose value is given by

2£s; qN"",

- - loPs - oPFI·

(I M)

The quantity in the absolu te value symbol is the difference bet ween the surface porential and the Ferm i level in the substrate. In the triod e and satu ratio n regions, thi-, quantity has a magnitude of twic e the Ferm i level. Now the channel is nor a termi nal of the device , so to find how the various capacitive terms contribute to the terminal capacita nces requires knowledge of how the channel charge divides between the so urce and drain . In ge neral. the valu es of the terminal capacita nces depend on the operating regi me bec ause bia.. cond itions affect this partitioning of c harge . For example. w hen there is no inversion cha rge (the device is "ofJ").lhe gate - source a nd ga le - drain ca pac ita nces are just the ove rlap te rm s 10 a reasonable

approximation.

When the device is in the linear regi on the re is an invers ion laye r, and nne may a.....umc thut the so urce and drain share the c hannel c harge equally. Hence. half of Cf€ adds to the ove rlap te rms. Similarly. the Cj.I> a nd Cjal> j unction terms in the linear region. In the saturatio n region. po tential varia tio ns at the drain region don't in fluen ce the channel charge. li enee. there is no contribution 10 C ll d hy CII ,-; the ove rlap term is all there is. The gate-source capacit ance is affected by CII ,.• hUI "d etailed co nside r:Itinns" 2 show thut on ly ubour two thirds of C IIC should he added 10 the ovcrlup te rm. Similarly, e,." contributes nothi ng (0 Cdl> in sat uration. bUI docs contribute two thirds of its value 10 C)/> . The gate- bulk ca pacit ance ma y he ta ke n as zero in bo th triode uud sat uration (the channel charge essentially shields the bu lk fro m what 's hap pen ing al the gate). Wh en the device is off, however. there is a gate vol tage - de pe nde nt capaci tance who se value varies in a rough ly linear manne r belwee n CR,· and the se ries co mbination of CII ,. ami Crb • Below (but nca r) threshold.the value is close r to the se ries combination and approaches a limi ting value of CII ," in deep accumul ation . where the surface majority carrier concen uu tion increases ow ing 10 the positive c harge ind uced by the strong negative gate bias. In dee p acc umula tio n, the surface is strong ly conducting and may

2 Tbe 2/3 factor arises from Ihe rulc ulatjon of channel charge. and inherently ({ln~~ from inlegraling lhr triangular d islrihutinn ass umed in Figure 3.2 in the square-law regime.

70

CHAPTER 3

A REVIEW O F MOS DEVI CE PHYSICS

Table 3. 1. Approximate M OSF F.:T terminal capacitances

C.. C" C, .

a ll

Triode

c. c.

C,el2 + c." C, el2 + C••

C",Crl> / IC,~

+ C~/Io )

Saturation 1C~,_ jJ

+ C'"

C.

0

II

+ C,-b /2 CjJb + C.. / l.

c)." + 2C,,,/3

< c~" < c,~

c.,

c.;

C..

C jJlt

Cj, I>

C1

therefor e be treated as esse ntially a metal. lead ing to a gate- bulk ca pacitance that is the full parallel plat e value. The variation of this capac itance with bius presents one additional op tion fur reo ah zing vaructors. To avoid the need for negat ive supply voltages. the capaci tor may he- built in an n-well using n+ source and drain regions. The temu nal capacitnnces are summarized in Table 3.1.

3.4 .4 HIGH -FRE QUEN CY FIGURES O F MERIT It is perhaps natural 10 attempt 10 characterize mult idim ensional quant ities with a single num ber: laziness is universal . after all. In the speci fic case (If high -frequency performance. two fi gures of merit are particu larly popular. These are Wr and Wma~ . wh ich are the freque ncies at which the curre nt and power gain. respecti vely. art: extrapolutcd to fall to unity. It is wort hwhile 10 review briefly their derivation. since many engineers forget the orig ins and precise meanings of these quan tities and often dra w incorrect inferences as a result. The most comm on expression for CUr assumes that the drain is tcnninutcd in an increment al short circuit while the gate is driven by an ideal current sou rce. As a ' onseque nce of the shorted termination . CUr does nut include informat ion about drainhulk capacit ance. The curre nt- source drive impli es that ser ies gate resistance simhavea stro ng effec t on ilarly has no influen ce on W T. Clearly. both rll and ll rl hig h-freq uency perfo rmance. hut CUr simply ignores this realit y. Furthermore. the gate -to -drain capaci tance is con side red only in the computation o f the inp ut imped ance: its feedforward co ntribution to output current is neglected. With these assumptions. the ratio of drain current to gate current is

e "Ill

id

l

; in

I

g.

;;;::: w ( C RJ

which has a value of unity at a frequ ency

+ C RJ) '

( 19)

3.4 MOSFET PHYS ICS: THE LONG·C HA N N EL A PPROXIMATIO N

71

( 20)

Now the frequency at which the (extrapolated ) current gai n goes to unity has no fundamental importance ; it is simply easy to co mpute. Perhaps more relevant is the frequency at which the maximum power guin is ex trapolated 10 fall to unity. However. computing w ......~ is in general Quite difficult. so we will invoke seve ral simplifying a....umpnons to make an appro ximate deri vation possible. Speci fically. we co mpute the input impedance with an incremen tall y shorted drain and ignore the fccdforward current through C,a. j ust a... in the co mputation of Wr. We do co nside r the fec-'dback fromdrain to gate through C, J in co mputing the outp ut impedance. however, wh ich is important because co mputation of the maximum power ga in requires term ination in a conjugate match. With these assum ption s. we ca n calc ulate the power delivered In the inp ut by the current source drive as simply

;2r P,n = - '"2-s '

(2 1)

where ' , . the series ga le resis tance. is the only dissipative element in the inpu t circ uit. The magnitude of the short-circuit current ga in at high frequ encies is apprcxi mately given by the same express ion used in the computation of Wr:

I'mI'" id

WT ,

(22)

W

II is also straightforward to show that the resistive part of the output impedance is roughly

(23) If the conjugate termination has a conductance of this value then the po wer gain

will be maximized. with ha lf of the gm generator's current going into the co nductance of the termination and the balance info the device itself. The total maximum power gain is therefo re

I ( WT ,

1)2

2" -;;; . ' in ' 2"

I (WT·

Cg J )

·2

(2 4)

' ill ' lI

2 which has a value of unity al the frequency given by

Will"

~

I ~

2" Yr;c;;;.

(25)

72

CHAP TER 3 A REVIEW OF MOS DEVIC E PHYSICS

It I!'> clear that W mn depends on the gate resi stance. so it is more co mprehensive In this regard than ((Jr . Beca use j udicious layout can reduce gale resi stance 10 small values. W mu can be co nsiderably larger than w r for ma ny MOSFETs. The ou tput ca paci tance has no effec t 0 11 W m... beca use it ca n be tuned out w ith a pure indu c ta nce, a nd the refore docs not lim it the amount of pow er tha, may be del ivered 10 a load . Measurements o f both W ol' " and wr are carried our by increasing the freque ncy until a noticeable drop in maximum power ga in Of curre nt ga in occ urs. A simple extra polation 10 unity value the n yields W m.o~ and ((Jr . Because these arc extrapolated values. it is no t necessar ily a give n that o ne may actually co nstruc t practical circ uits operating at , say. W mu ' Th ese figu res of merit should instea d be take n as rough indicatio ns of high-frequency performance capabilit y. 3.4 .5

TE CHN O LOGY SCA LING IN THE LONG ·C HA N N EL LIMI T

Now that we have e xa mined both the sta tic and dy nami c behavior of MO SFETs. we can deri ve a n a pprox ima te express ion fo rw T in term s of operating po int. process param e te rs. a nd dev ice geometry. We ' ve already de rived a n e xpression for g.... so all we need is an ex pression for the requisite capacit a nces . To simplify the deri vat ion. lei us assu me tha t C/P dominates the inp ut ca paci ta nce and is itsel f domina ted by the paralle l plat e capacitance . Th e n. in saturation.

li e nee. (l)r depends on the inver se sq uare or the le ngth . and increases w ith increasing gate- sou rce volt age . Re me mber. tho ugh . tha t this equation holds onl y in the longc hannel regim e.

3 .5

OPERATION IN WEAK INVE RSION (SUBTHRESHO LD)

In sim ple MO SF ET models (s uch as the one we've pre sented so far). the devi ce conducts no curre nt until an inversion layer for ms. However. mobile carriers don 't abru ptly disappear the moment the gate voltage drops be low V" In fac t. exe rcis ing a lillie im agination. one ca n discern a structure re min iscen t of an n-p-n bipolar transistor whe n the device is in the subthreshold regim e. w ith the source und drain regions functioni ng as e mitter and co llec tor. respecti vel y. and the (nor unvcrtcdj bulk behav ing a bit like a base. As V~s drops be low th reshold . the c urre nt decreases in a n ex pone ntial fashion. muc h like a bipolar transistor. Rather than droppi ng ut the 60-mV/ dccade rate of a bipolar. however. the c urre nt in all rea l MO SFETs dro ps more slow ly te.g.•

3.6 MO S DEVICE PHYSICS IN THE SHO RT·C HA N N El RE GIME

73

100 mV/ decade) becau se of the ca paci tive voltage division betwee n gate and source and between source and hulk . Many bipolar analog ci rcu its are ofte n translated into f\.10S form by operating the device" in this reg ime. However. such circuits ex hibit JX"lOr frequ ency respon se because MOSFETs possesx ..ma ll Km ( but good 8m/ I ) in thi.. reg ion o f operation. As a consequence. this regime is not normally of interest in RF design. and will therefore be ignored in the remai nder of this text.

3 .6 MOS DEVICE PHYS ICS IN THE SHORT·CHANNEl REGIME The continuing drive to shrink device geometrie s has res ulted in devices so small that various high -field eff ects become prominent at moderate voltages. The primary high-field effect is that o f velocity saturation. Because of sca tteri ng by high -energy p'opncal") phonons. carrier velocities eve nlUally cease to increase with increasing electric field . In silicon. as the electric field approaches about lOt. Vim. the electron drift veloc ity di..plays a progressively weakening dependence on the field strength and eve ntually saturates at a value of about

10'm/' . In deriving eq uatio ns for lon g-ch annel devices. the saturation drain current is assumed to equal the value of current corres po nding to pinch -off of the channe l. In short-channel devices. the current saturates when the carrier veloc ity docs. To accommodate veloci ty saturation. begin with the long-ch annel equation for drain current in saturatio n: ( 27)

which may be rewritten as I II

JLnC ox W

= - 2- "L ( Vp

-

( 28 )

V/)Vd, al.l .

where the long-chann el Vd, al is denoted Vd, al.1 and is equal to ( Vx.' - V,). As stated earlier. the drai n current saturates when the veloci ty docs. and the vekcit y saturates at smaller vo ltages as the device ge ts shorter. Hence. we expec t V",al todiminish with channelleng th . It may be show n that V".... may be express ed more ge nerally by the following approximation:

v

d......

sothat

"" [

V

( g.

- V

,)

L E] -

II (

,.11 ) -

(V,. -V,)( LE~, )

(V • _ ~'t ) g

+ ( L E....l ·

'9

(- )

• CHA PTER 3

A REVIEW O F M O S DEVI CE PHYSIC S

IV ( V , I f) = -,,"CO, 2- L" g

-

V, ) [ ( V" - V, )

II

( L E....r} J•

(30)

where E 'lal is the field strength at wh ich the carrier veloc ity has dropped to one half the value extrapola ted from low-field mobility. It should be clear from the foregoing equations that the pro minence of shortcha nnel effects depends on the ratio of ( Vg J - V, )/L 10 E...t o If this ratio is small. then the device still behaves as a long device: the actual channel length is irrelevant . All that happens as the device shortens is that less ( Vg • - V, ) (also ca lled "gate overdrive"} is needed for the on set of these e ffects. With the definit ion for Ew., the drain current may be rewritten as

LE~ / 0 = lVCo\ (V, J - V, )voa( [ 1 + ~=-'';-; V, . VI

I

(3 1)

A typical value for E""" is abo ut 4 x 10" Vim . Altho ugh it h so mewhat processdepend ent. we will treat it as constan t in all that follows. For values of ( V". - V, )/ L tha t are large in com pari son to Ewr. . the drain current approac hes the followi ng limit : (32)

Tha t is. the drain current eventually ceases 10 depend on the cltunnel Iength, Furthcrmore. the relationsh ip between drain current and ga te-so urce voltage becomes incre ment ally linear rat her than squ are -law. Let 's du a qui ck calc ulation to obt ain a rough esum ate of the saturation current in the short-channelli mit. In modem processes, t,,~ is about 10 nm or less. so that COl 10 is about 0 .003 5 F/ m. Assum ing a mobility o f 0 .05 111 21VAS. an E""l of 4 x 10 VIm, and a gate overdrive of 2 V, the drain current becomes approximately U.5 rnA per micrometer of gate width . Despite the crude nature of this calculation, actua l devices do behave similarly. although one should keep in mind that channel len gths must be much shorter than 0 .5 11111 in orde r to validat e this estimate at this value of overdrive. Because the gate overdrive is more co mmonly a couple hund red millivolts in analog applications, a reason ab ly usefu l number to keep in mind for rough order -ofmagnitude calculations is that the saturation CUfTC lIt is of the order of 100 mA for every millime ter of gate width for devices operating in the short-channellimit. Keep in mind that this value due s depend on both the gate voltage and Cm . In all modem processes. the minimum allowable channel len gth.. are short enough for these effects to influence device operation in a first-order man ner. However , note that there is no requirement that the ci rcuit designer use minimum-length devices in all cases; one ce rtainly retai ns the option to use devices whose length s are greater than the minimum value. Th is op tion is regularly exercise d when buildin g current sources 10 boo st outp ut resistance.

3.6

MO S DEV ICE PHYSICS IN THE SHO RT-CHA N N EL REGIME

75

3 .6 .1 EffE CT O f V EL OC ITY SATURATIO N O N TR A N SISTOR DYNAM ICS In view of the first-order effec t of veloci ty saturation on the drai n c urre nt , we ought the expression for WT 10 see how device sc aling affec ts high -freque ncy performance in the short-cha nne l regim e . First, let' s compute the limiting tra nsconductance of a short-cha nnel MO S device in saturation:

10 revisit

( 3 3)

Using the same nu mbers as for the limiti ng sa turation c urrent, we find that the transconductance should be rough ly 300 mS per millimeter o f gate width (easy numbers to remember: every thing is of the order o f 100 sornetbings per millimet er ). Note thaI the only pract ical contro l over thi s value ar the disposa l of a dev ice designer is lhmugh the choice of 10 1 10 adj ust C.U (unless a differ ent d ielectric material i ~ u..ed). To simplify calculatio n of W T . assu me (a s befo re) Ihat C~ J do minates the input capacitance. Assu me fun her thai short-c ha nnel effec ts do nOI appreci abl y in flue nce charge sharing, so thai C~, still behaves approxi ma tely as in the long-ch annel lim it: (3 4)

Taking the ratio o f R", 10 C~ J then yields

g",

wT ::::: -

CII S

:::::

! .i\l

I lVL C0 1

3l "Ew =--4 L l

(35 )

We see that the Wr of a short-c ha nnel device thu s depends on I lL , ruther than on Il L 2. Additionally, no te that it does not de pend on bias conditions ( hut kee p in min d that this independ ence holds only in satura tion). nor on ox ide thickn ess or com position, To get a rough feel fo r the numbers, assume a It of 0.05 m1j V-s, an E,~ , of 4 x 10/\

VIm, and an effec tive cha nnel length of 0.5 u rn. With tho se values, Ir wo rks out 10 nearly 50 Gill, (again, this value is very appro ximate). In practice, substantially smaller values lire measured beca use smalle r gale ove rdrives lire used in actu al circuits (so that the device is not operated dee p in the short-channel regime). and als o because the ove rlap ca paci ta nces are nor actually negli gibl e (i n fac t, they arc Irequently of the same ord er as CII , ) , As a consequence , prac tical values of Ir arc about a factor of 3-5 lower. Minimum effec tive c hanne l length s co ntinue to shrink, of course. a nd process technologies j ust mak ing the tra nsiti on from the labo ratory 10 produ cti on pos..ess practical [r values in the neig hbo rhood of 30-5 0 Gil l . T his range of values is similar 10 that offe red by many high -perform ance bipol ar processes. a nd is one reason

76

CHA PTER 3

A REVIEW O F M Q S DEVI CE PHYSICS

that MO S devices a rc incr easingly fou nd in RF applicat ion s previously se rve d o nly by bipo lar or G aA s tech no logies.

3 .6 .2

THRESHOL D REDUCTIO N

We 've a lready seen that highe r drain vol tages ca use c hanne l sho rte ning. resulting in a non ze ro o utput conductance . Wh en the c hannellength is small. the electri c field associ ate d with the d rain voltage may extend e noug h toward the source thai the effec tive thresho ld di minishes. This d rain -induced ba rrier lowerin g ( DIRL . pron ounced "d ibble' tj ce n ca use dramatic inc rease s in subth res hol d c urre nt (kee p in m ind tb e e xpo nentia l sens itivity of the subthres hold current). Ad dit ionall y, it results in a degr adat io n in outpu t cond ucta nce beyond that associated w ith simple cha nnellength modulation. A plot of thre shold voltage as a func tion of c hanne l len gth sho ws a monotonic dec rease in threshold as len gth dec rease s. A l the 0 .5-11 111 le vel. the thre shold red uction can I1c 100- 200 mV over the value in the lon g-ch an nel limit. correspondi ng 10 pote ntia l increases in subthres ho ld current by facto rs of 10 In )(XX). To reduce the pea k c hannel fie ld a nd thereby miti gate high -fie ld e ffec ts. a lightly do ped d rain ( LDD) struc ture is alm ost a lways used in modem devices. In s uch a tran sistor, the do pin g in the drain reg io n is arra nged In ha ve a spa tia l variatio n. prog ress ing fro m re lative ly heavy near the drain contac t to light e r so mew he re in the c ha nne l. In som e cases. the dopi ng profi le result s in o verco mpensation in the se nse tha t highe r dra in vol tages ac tua lly i f/c rease the thres hol d over so me runge of dra in vo ltages before ultim ately dec rea s ing the thresh ol d, No t a ll de vices ex hibit this " reve rse" short-c ha nne l effec t. since its e xiste nce de pend s o n the detailed nature of the doping profile. Additionally, PMO S de vices do not ex hibit hig h-field effec ts as re ad ily as do N MOS tran s istor s because the field stre ngths necessa ry to ca use ho le velocity to saturate arc co nside rably hig her tha n tho se that cause electro n veloc ity saturatio n.

3 .6 .3

SU BSTRATE C U RRE N T

Th e elect ric field near the drain ca n rea c h e xtrao rdinarily large values w ith modcrate voltages in sho rt-c hanne l devic es, As a con sequ e nce. ca rriers ca n acquire e noug h energy be tween sc attering events to cause impac t ionizatio n upon thei r next collisio n. Impact ion izatio n by these "hot" carriers c reate s hole- electron pairs and. in a n N MO S devic e. the holes are co llec ted by the subs trate while the e lec tro ns flow 10 the drain (as usua l). The resulting substra te c urren t is a se nsitive fun ction of the drai n voltage. and this c urre nt re presents a n add itio na l co nd uctance term shunting the d rain to g round . Th is effec t is of g rea test conce rn wh en one is see king the m inimu m o utput co nd ucta nce a t high drain- source volta ges.

3.7

3 .6 .4

OTHER EFFECTS

77

G ATE CURREN T

The same hut electro ns respo nsib le for substrate current can act ua lly cause gate cur(tnl. The charge co mprising this gale current can become trap ped in the ox ide , causingupward threshold shifts in NMOS devices and threshold reductions in PMOS devices. Although this effec t is usefu l if o ne is trying to make non volatile mem ories, it is most objectio nable in ordinary ci rcuits because it degrades lung-term reliability, 3.7 OTHER EFFECTS 3.7.1 CHA N N EL LENG TH M ODU LATIO N In all of the derivation s so far. the saturation curren t is treated as independent of the drai n-source voltage . It would be wonde rful if rea l MOSF ETs actually behaved this way, because we could the n make idea l current sou rces and realize infinite incre mental voltage gain. Alas, precisely because it would be nice if this were the case, it' s ro nbe case. The most significant effect in th is co ntex t is channe l lenK'" matlllltu ;'", . which is analogous to the Early effect in bipolar tran sistors. As the drain vo ltage increases.the extent of the depl etion region near the drain also increases. s hortening the effective channel len gth . Drain current therefore increases with increasing VJ .. and ihis effect yields a non zero OUlpUI conductance . The Ea rly voltage concept still applies reasonably well here, and effective Early vo ltages in MO SFl::.l s lend 10 he- substantially smaller than is common in bipo lar devices. although o ne can increase the Early voltage conside rably by using channels so lon g that any drain space-charge layer variation represents an insignificantl y small fraction of the tota l length .

3 .7.2

BACK-GATE BIA S (··BOD Y EFFECr·,

Another important effe ct is that of back-gate bias (often called the "bod y effect"). Every f\.10 SFET is actually a four-terminal device. and one must reco gni ze that variations in the potentia l of the hulk re lative to the other device terminals will influen ce device characteristics. Althou gh the source and bulk terminals arc usually tied together. there are impo rtant instances when they are not. An example that q uickly comes to mind is a current source- biase d differential pair: the common source con necticn is at a higher po tential than tha t of the bulk in this case, and may move around with input common-mode voltage. As the potential of the bulk beco mes increasingly negative with respect to the source, the depletio n region formed between the channel and the bulk increases in extent. decreasing the amount of negative charge in the channe l. Th is decreased negative charge increases the value of VK~ required 10 form and maintain an inversion layer. Therefore, the thresho ld voltage increases. T his hack -gate bias effect

78

CHAP TER 3

A REVIEW OF M O S DEVIC E PHYSICS

Ic~~

I

gmvJ !J

~ ,

c",

gmbvh, c

C.... bu lk ...... m

FI GURE 3.5 . Incrementol MOSFET model including bock-gote effect lresistive elements not ~hownJ .

(so -ca lled beca use the bulk may be considered anothe r gate term inal} thu s has both large- and small-signal impl icatio ns. Th e influence o f this variation is accounted for in the small-s ignal model by the addi tion of anot he r dependen t c urre nt source, this time cont rolled by the so urce -to -bulk vo ltage ; see Figure 3.5. The bac k-gale transconduct ance K...IJ is ge nerally no larger than about 30% of the main transconductance. and frequentl y abo ut 10% of 8.... Howe ver. these are hardly unive rsa l truths. so on e should alway s c hec k the detailed model s before ma king such assump tions.

Unintended. and much undesired, modu lation o f the source - bulk potential can also occ ur ow ing to static a nd dynam ic sig nal c urre nts flowin g throu gh the substrate. Th is coupli ng ca n cause se rious problems in mixed -signal circuits. Extre mely ca reful a ttention to layout i..-. necessa ry 10 reduce noise prob lem s arising fro m this mechani sm.

3 .7.3

TEMPERATURE VA RIA TIO N

Th ere are two prima ry tcm pcmrure-d ependcnt effe cts in MOS devices. Th e first is a change in threshold. A ltho ugh its preci se be havior de pe nds o n the detailed device de sig n, the th reshold tends to have a TC similar to that of base -emitte r volt age VilE, nam ely, abo ut - 2 mYI"C (within a fac to r of 2). The second effect , that of mobilit y redu cti on w ith increasing tem perature. tends to dominate because of its expo nentia l nature :

( J6)

w here til is some refere nce tem perat ure (e.g .• ] (XJ K ). At a fixed bias. the n. the drai n c urre nt drop s as the te mpe rature increases .

3.8

3.7. 4

TRAN SIT TIM E EFFE CTS

79

N O RM AL-FIELD M O BILITY DEGRADATION

As the gale JX)lcnt ial incre ases. the electron.. in the channel nrc encoumg..." f1lo /Jow closer to the silico n-oxide interface. Rem em ber thai the interface is full of dangling bonds. various ion ic co nta mi nants. and a bandoned cars . A s a conseq ue nce. the re is more scarrering ofcarr iers and rhus a dc-crease in mohility. He m:e, lhe llvllib hle drain current drops below what o ne wo uld expect if mobi lity we re 10 slay fi xed . Since the vertical field is proportio na l to the ga le o verdrive, perhaps it is nor surprising thai the actual drain curre nt is the value give n by the previous equatio ns. mu ltiplied by the following factor:

J +O(Vg s

-

(37)

V, ) '

wbere f:). the no rmal field mobility degradat ion facto r, has a typ ical value in the ra nge of 0.1 -1 V- I. It is device -dependen t. g row ing as to. shrinks. In the absence o f measured da ta. an ex tremely crude es tima te of 0 ca n be o bta ined fro m 0 :::::::

2

X

10- 9 m j V

'0.

.

(38)

Altho ugh there arc ce rtai nly add itiona l effec ts that infl uence the behavio r o f rea l devices (e.g .. variation of lhresho ld voltage a long the c hannel ), thi s ,-h;.pfer has covered the phenomena of g rea les l re leva nce 10 rhe RF circuit designer.

3 .8

TR ANS IT TIME EFFECTS

The lumped mod el s of this c ha pter cl early canno t app ly over an arbi tra rily large Irequency range. As a roug h rul e o f thumb. on e ma y usu ally igno re w ith im punity the true distributed natu re of transistor s up to ro ughly a tenth or fifth of Wr. As freq ucnd es increase. howe ver. lu mped mod els become progressivel y inad equ ate. Th e most conspicuous shortcom ings ma y he traced to a neglect of tran sit tim e effect s. To understand qu a litativel y the most im po rtant im plicat ions of tra nsit time effects. consider app lyin g a ste p in gate -to -so urce voltage. C harge is induced in the c ha nnel. and it dr ifts tow ard the drain . a rriv ing so me time late r ow ing to the finite carr ier velocity. Hence, the transconduct ance has a phas e de lay assoc iated with it. A side effec t o f this de layed tra nsco ndu cta nce is a cha nge in the input imped ance: the delayed feed hack through the gate ca paci tanc e necessaril y prevent s a pure q uad . rature relatio nship betwee n ga le vol tage and ga te curre nt. Va n der Z iel - has sho wn ' that. atleast for long-c ha nnel de vices. the tra nsit de lay ca uses the ad mit tanc e to ha ve a real part thut g rows as the sq uare of freq ue ncy :

) Noisr in So lid StaIr /) f''I';CrS Ilfld Circuits. Wiley. New York . 1986.

80

CHA PTER 3 A REVIEW O F M O S DEVI CE PHYSICS

(39) For a rough calibratio n on the magnitudes implied by Eqn. 39. assume thai K d O is approxima tely equ al to g,.,. Th en . to a c rude approx imatio n.

( )'

g. w g~ ~ "5 w r

.

(40)

Hence. this shunt conducta nce is negligi ble as long as operatio n well below wr is maintained. However. we shall see late r that the the rm al noise associated with this condu ctance mu st be taken into account in any proper noise figure calculation.

3.9 SUMMARY We' ve see n how long - and short-c hanne l device s ex hibit differe nt behavior, and that these difference s are ca used by the variation in mobility w ith electric field . What distingu ishes " long" from "short" is actually a function of electric field stre ngths . Since e lectric field is de pende nt on len gth. lo nge r devices do not exhibi t these high-field effec ts a... readily as do shorte r ones.

3 .10 APPENDIX : O.S -I'm IEVEL·3 SPICE MODELS Th e following set of mode ls ilio fairly typ ical for a O.5-ILm (draw n) process tec hnology . Beca use leve l-S models are quasie mpi r ical. not all of the param e te rs may take on physically reasonable values. They ha ve been adj usted here to provide a reasonab le til to measur ed de vice V- I characteristics . as we ll as 10 lim ited dynami c data inferred frum ring oscillator frequ e ncy measurem e nts. It should be mentioned that there arc many other SPIC E MOSF ET model sets in e xis tence . the most rece nt o ne being B5 1M3. Th e newe r models pro vide bett er acc uracy at the ex pense of an expone ntial grow th in the nu mbe r of parameters. not all of whic h arc physically based . In the interest of providin g rea sonable accu racy with the simplest model s. we will ma ke e xtensive use in this te xtboo k of the relatively primitive level- S models prese nted here.

' SPICE LEVEL3 PARA METERS .MODEL NMOS NMOS LEVEL= 3 PII' = O.7 TOX= 9.5E-Il9 XJ=O.2lJ T1'G = 1 + V1D=0.7 DELTA= 8.8E-0 1 LD ~ 5 E - 08 KP=1.56E-Il4 + UO= 420 TIl ETA= 2.3E- 01 RS II ~ 2 .0 E +()() GAMMA= 0.62 + NSUB= 1.40E+ 17 NFS=7.20E+ J J VMAX= 1.8E+ 05 ETA=2. J25E-02 + KAPPA =J E-OJ CGDO=3.0E- 10 CGSO= 3.0E- JO

PROBLE M SET

81

+ CG BO~4 . 5E - JO CJ= 5.50E-04 MJ= O.6 CJSW=3 E- 1O + MJSW=O.35 PB= 1.1 • wen ewdrewn-Dcha.w, T he suggested Delta.W is 3.80E- 07 .MODEL PMOS PMOS LEVEL= 3 PIII ~O . 7 TOX=9. 5E- 09 XJ= O.2 U TI'G= - I + VTO= - O.950 DELTA= 2.5E-O I LD= 7E-OS KI'= 4.SE-05

+ UO= 130 TH ETA= 2.0E - OI

RSH = 2.5E+ On G A ~1 ~l A =().52

+ NSUB= I.OE+ 17 NFS ~6.50E + I I VMAX= 3.0E+05 ETA=2 .5E-02 + KAPPA= g.OE+nl1 CG DO ~ 3.5 E - JO CGSO= 3.5E- JO + CGBO=4.5E-1O CJ= 9.5E- 04 MJ=O.5 CJSW=2 E- 1O + MJSW=O.25 PB= I • Weff=Wdrawn-Delta _W. The suggested Dclt a.W is 3.66E- 07 A very brief explanatio n of the mod el param eters is provid ed in Table 3.2. The parameters NRS. NR D. AS . AD. PS. and PD are all speci fied in the device description line. not in the model set itself. bec ause they depe nd on the d imen ...ion s of the device.

PROB LEM SET f OR M a S DEVI CE PHYSICS PROBLEM 1 In long-channel devices. C It J typ ically dominates over CIIJ • Th is problem investigates whether this remai ns the case for short-channel devices. (a) Using the approximatio n L f) ::::: ~ xJ' deri ve an ex pression for Cltd/C ItJ in terms of channelle ngth L and xJ' (b) Plot C,J /C" versus L for Xj = 50 nm . 150 nm and 250 nrn. and for L ranging fr om 0.5 J1 m to 5 J1 m. What de yo u ded uce about the scaling o f these capacitances with chan nel len gth ?

PROBLEM 2 The curre nt mirror is a ubiq uitous subcircuit. However. merel y spec ifying the current ratio alone doe s not constrain the design suffici ently to determ ine individual device dimension s. so other criteria mu st he conside red. In circuits that require a high degree of isolation. one criterion is to minimi ze the sensitivity to substrare voltage fluctuation s. To illustra te the nat ure of the problem. compare two s imple 1:1 NMOS current mirrors. The reference curre nt for bo th is 100 J1 A . wh ile the device widths in the two mirrors are 10 J1 m and 100 11m . (a) Assume that noise co up led from a near by ci rcu it may be mode led as a 1 MHz. lOO-mV amplitude sinusoidal m ilage generator co nnected between ground and the body (bulk) terminal of the NMOS transistors. Assume that we measure the output current as it flows int o a DC source of 2 V. Which mirror will exhibit worse sensitivity to the substrate fluctuatio n? Explain your answer in term s of parameters of the four -ter minal incremental mod el. 4

82

CH A PTER 3

A REV l f W Q F M O S DEV ICE PHYSICS

Table 3.2. SPICE tevel-B mode l parameters Parameter name

"III TOX XJ

C ' ," VCllli,,"a l sy mbo l

S urface po tential in strong inven;ion

'., x,

Gale oxide unck ness

Source - drain j unction de pth Gate material pobrity: 0 for AI, - I if same as subtrate, oppos ite suh!.lra le

TPG VTO

Desc ription

v"

+ I if

Thres hold at V•• = 0

DELTA

Mode ls threshold de pendence on width

CO

So urce -drain late ral d iffusion. for comput ing Lof! ; n(1t used to ca lculate ove rlap capecirarces

01'

Process uanscoeducta nce coefficient

UO

u»

low-field ca nier mobili ty at surtece

TIf ETA

Vertical-field mo bilit y degradation factllT

RSU

Source ---drain d iffusio n ..hect res istance; muh iplieti by NRS and NRD 10 obt ain rotal s oerce and drain ohm ic resistance. respectively

GA M MA

NSUB

y

N", orNv

Bod y-effect coefficien t Eq uivalent substrate dupi ng Past surface state censuy

NFS

VMAX

"~

M axi mum carri er drift vejocuy

ETA

Mudels cha nges in thres hold due tn Vd , varianons

K A PPA

" K

ca lX)

C. d O

Gate - drain ove rlap capac itance pe r width

CGSO CGBO

C•• n

Gate-source over lap cap aci tance pe r width

c_ ~o

Gate- bulk overlap capacitance per lengt h

Models effects uf channel-leng th modulation

CJ

C PI

Ze ro- bias bulk bottom ca paci tance pe r unit so urce - drain area ; m ultiplied by AS and A D to obtain lo tal bottom capac itance of source an d drain III V.n = Vdb = 0

MJ

nl j

Bott om source - dr ain j un ctio n g ruding coe fficient

CJ S W

Cjs""

Ze ro-bia s sidewall j unct ion r npacitance pe r un it peri me ter of source- drain adjace nt 10 field ; mu ltiplied by PS and PO 10 obtain total side wall capaci tance

MJSW

M ) s.....

Sioewan j uncncn grading coemc teut

PB

"j

Bulk j unctio n porential barri er used to com pete j unctio n capac itance for other than lew bia s

PROBLEM SET

83

(bl Verify yo ur a nswe r us ing the Le vel -3 S PICE models of this c ha pte r. a nd find from simulations the ac tua l incrementa l co mpone nt of the o utput cu rre nt fo r the two mirrors.

PROBLEM 3 Deri ve a more ge ne ra l ex pressio n for wr that is valid in both the sho rtand long-channel regi me s. ra the r than only in e ithe r li mit , by first de riving a ge neral expression for the transcondu ctance. You may neg lect back -gate bias crt ect . PROBlEM 4 Thi s problem inve stiga tes the importa nce of vario us correc tio n facto rs on the V- I characteristics o f MOSFETs . (a) First assume operation in the lon g-channel regime an d plot the drain c urrent as a function of dra in- source volt age as the gate vo lta ge is varied in 200 -mV steps from 0 to 3 V. Assume a mobi lity of 0 .05 m 2/ V-s a nd a ' 0 \ o f 10 nm. an d let Vd • range from 0 to 5 V. Assume lV/L = 10 . (b) Now take ve loc ity sa turatio n into acco unt. Assu me that L£ 'W,l is 1.75 V. a nd replot as in pa rt (a). By what fac tor has the ma xim um d ra in current diminished? fc) Now take into acco unt ve rtical-field mobil ity degradation . Assume that 0 is 0.2 V- I and replot. By what factor has the ma ximum dra in c urre nt now diminished. relative 10 the lon g-chan nel case "?

PROBLEM S If the ga te -to -so urce overd rive volt..ge of a n NMOS trun sisror is he ld constant at J V, then w hut is the percentage c hange in device tran scond uctance as the temperature increase s fru m )(XJ K to .ton K"? PROBLEM 6 The vo ltage de pen den cy of gate ca paci tance ca n lead to distortion a nd other errors in a nalog circ uits. To e xplore thi s idea further, co nside r the amplifier circuit shown in Figu re 3.6 . Here , ca pac itor C 3 repre se nts the gate c apacitance of a transistor inside the up-amp. whose o pe n-loop ga in is G. S uppose we w ish to use this circuit as an inverting buffer. so we in itially se lec t C 1 = C2 = C .

c, C, " IN

-1h---"-I-'

FtGURE 3.6 . Ampti ~er with voltoge-$6l'lsitive ca pacitance. (I) Derive a n ex pressio n fo r the ga in e rror. de fined as the d ifference in the magn i-

tudes of the input a nd o utput . Le t the input vo ltage be I V.

84

CHAPTER 3 A RE VI EW O F M OS DEVICE PHYSICS

( b) Clearly, the nominal value of gain error ca n in princip le be reduced to zero - for example. by a suitable adjust ment o f the feedback capacitor. However, such an adj ustment cannot compensate for voltage-depende nt capaci tance variations in C3. Specifically. suppose C J has a voltage sensitivity as follow s:

where C M is the capac itance value at zero bias and a is a first-order voltage sensitivity coe fficie nt. Derive an express ion for the minimum acceptable ga in G if the variation in ga in error is to be kept below I part in JO~ as the input voltage swings frum 0 to 5 V.

PROBLEM 7 Cha nnel length mod ulation (CLM ) is one effect that ca n ca use the output current to depend on drain voltage in saturation. To investiga te CLM in a very approximate way, assume that the region ncar the drain is in depl etion . Hence, the effec tive len gth of the channel (the portion in inversion ) is smaller tha n the physical len gth by an amo unt eq ual to the ex tent of this depletion region. Furthermore, the amount of dep letion depends on the drain voltage, thus leading to a nonzero output conductance in saturation. In order to simplify the developm ent , assume the square -law characteristics IJ. Co~ W 2 if) = -2-I - ( V~ , - V, ) , -d f

where

L ~JT

= L-

/j .

In tum , usc a simple formula for the depletion layer 's extent:

1= Derive an express ion for the output cond uctance from these equations.

PROBLEM 8 C urrent mirrors are often designed to provide a current ratio other than unity. In principle. this ratio call be set by adj usting either width or length but, in practice, the length is almost never varied to provide the desired current ratio. Explain. PROBLEM 9 As d iscussed in the text. the magnitude of the threshold voltage increases if the source- bulk junction is increasingly reverse -biased . Thi s effect ca n be modeled form ally a.. follows:

where the parameter y is the bod y-effect coe fficient.

as

PROBL EM SET

(a) In the inverting am plifier ci rcui t shown in Figu re 3.7. what is the output voltage if the input is grounded'! Assume thaI the substrate is grounded and la ke bod y effect into acco unt. ~C"l

,

=HIl) ",A / ~' -

v,nom = O.7 V

W!L=2

2'5 = O.6 V

Y: 0 .5V " 2

W!L : 4

FIGURE 3. 7. Inverting amplifier.

Ill) Calculate the output voltage w he n the input is co nnected to the 5 -V supply.

PR08 LEM 10 One way to model the cumula tive effects ofD IBI. and ch an nel-length modulation is 10 multipl y the sta ndard eq uations by the following co rrection factor :

( PJ .5) The parameter ), (and its rec iproc al. the Early voltage ) thus accou nts for the ob served increase indrain current in saturation as drain voltage increases. Since f\10SFETs are rhus imperfect current sources in Saluta tion. one mu st be careful in choos ing device dimensions and biasing cond itions. For a simple square -law device . determi ne the appropriate length and width to providea drain current within I % of I mA as the drain voltage ranges from 3 V to 5 V. Assume a mobility of 0.05 m2j v-s. a COl of 3.5 mF j m 2• a gate overdri ve of 1.5 V. and aA of 0.1 V- I.

CHAPTER FOUR

PASSIVE RLC NETWORKS

4 .1 INTRODUCTION On e c harac te ristic o f RF circuits is the re lative ly large ratio of passive to ac tive compon cnt s. In stark contras t with di gital VLS I ci rcuits (or eve n with o ther anal og circuits, such as cp-um ps), many of those passive co mpo nents may be indu cto rs or even tra nsformer s. Th is cha pter ho pes 10 co nvey so me underlying int uition that is useful in the design of RLC network s. As we build up thai intuition, we "tl begin 10 understand the many good reaso ns fur the prepondera nce of R LC networks in RF circuits. Amo ng the most compe lling of these are thai they can be u...ed 10 mat ch or otherwise modify impeda nces (i mportant for efficie nt power transfer, for example), cancel tran sistor parasitic s to provide hig h gain at high freq uenc ies, and filter o ut unw anted sig na ls. To unde rsta nd how R LC net works may confer these and ot he r benefits. le t' s revisit so me s imple second-order exam ples fro m undergradu ate introductory ne twork theory, By loo kin g at how the se networks be have fro m a couple o f di fferen t viewpoi nts. we'H build up intuition th at will pro ve useful in unde rstanding network s of m uc h higher o rde r.

4 .2

PARAllEL RLC TANK

Let 's j ust ju mp rig ht into the study of a paralle l R I.C circuit. As you probably know, thi s circuit ex hibits resonant behavio r: we ' ll see what th is implies moment aril y, This circui t is al so ofte n ca lled a tank cirrflif l (o r sim ply tank) . We begin by studyi ng its comp lex impedance. or more direc tly, it ... admittance (mo re co nvenient for a para llel net wo rk ); see Figu re 4 . 1. Fo r thi s network , we know tha i the adm ittan ce is sim ply I So -called either hy analogy with acousnc resonantors or becau se it l>lnre s energy. as does a lank of water.

86

4 .2

87

PAR ALLEL RiC TANK

'.

L V....I

FIGURE 4 .1. Pa rallel Rl.C tonk circuit.

Y=

G + j w C + _._1_ = G + j (WC ]wL

__1_) .

wL

(I )

From inspection of the netw ork (or of the eq uation), It 's easy to see thai the admittance goes to infinity both at DC
_I ) =0 ( wnc - wu L

1

~ !lJo = .jL C·

( 2)

A frequently handy rule of thumb is thai a l-nH inductor and a l -pF capacitor resonate at 5 GHz to an excellent (be tter than I %) approximation. Knowing thi s o ne datum allows rapid computation of the resonant frequency of any other L C combination.' In any event, at resonance. the admittance is purely real and equal to G by vinue of the cancellation of the reactive term s. To say that the reactive terms cancel at reso nance is certainly correct, but a little glib. As we' ll sec shortly, the indi vidual currents in the induc tive and capac itive branches ca n be surprisingly large, although they cancel each other as far as the external world is concerne d . We' ll also see that this augmented current is a sign that a downward impedance transformation has taken place, a phenomenon we will ofte n exploit. To explore these behaviors more fully and describe them in the most generally useful way. we need to introduce another param eter. Q.

2 Some aueoes use the leon "antireson ant" for paral lel reson ances. and reserve the term "resonant" rorseries circuits. We will use the leon " reson am" 10 indicate a ll)' cence ucno n o f ind uctive and capecmvereectances , wberber in sene s or parallel circuits. On thuse occas ion... whe n sor nedi...tinction is neres..~, we' ll jU\1say "se ries reson ant" or " parallel reson ant," J AI IO'Ner frequencies, it may be more coovemem to remember thai a 1' jJ. H inductance and a l -nF capacitance resonat e at .5 MHz.

CHAPTER 4 PASSIVE RlC NETWO RKS

8B

4.2. 1 Q Aside from the reso nant frequency itself anot her important descript ive parameter is the qualityfucsor, or simply Q. Different [ bu t equiv alent) defin ition s of Q abound. hul the most fundamental one is as follows: energy Moret!

Q =:

W == ::::=='7''":::= average power dis sipated

(3)

Note that Q is dimensionless, and that it i.. proportional to the ratio of energy stored 10 the energy lost. per unit time. Th is definition is fundament al beca use it says nothing speci fic about what stores or dissipates the energy. So. as we' ll see later on, it applies perfectl y we ll even to di stributed systems. such as microw ave resona nt cavities, where it is not possible to identify individual inductances. capa citances , and resistances . It should also he clear that the notion of Q app lies both to reso nant and nonresonant sys tems, so one may talk o f the Q of an RC circuit. Let' s now usc this definition to derive express ions for the Q of our parallel Rl. C circuit at resonance, At the resonant frequency. which we ' ll denote by w" , the voltage aero...s the network. is simply linR . Rec all that energy in such a network sloshes back and forth between the inductor and capacitor. with a co nstant sum at resonance . As a co nsequence, the peak energy stored in either the ca pacitor or inductor is equal 10 the total energy store d in the netwo rk at any given lime . Since we happen to know the peak capacitor voltage at resona nce (it's just r.... R). it's most convenient to usc it to co mpute the network energy: (41

Now we need 10 com pute the average power d issipated . Again , this computation is easy 10 carry OUI at reso na nce, since the network degenerates to a simple resistance there . The average po wer dissipated in the resistor at resonance is therefore simply

Pavg =

!/;kR .

(5)

The Q of the network at resonan ce is then

Q=

£1"1 w U- ,-

l avg

1

= -J LC

R

4C (/pk R) 2 I

2

'2 /pkR

=

J L/C'

(6)

The quantit y J '-Ie has the d imensions of resistance, and is sometimes culled the characteristic impedance of the network." It is significant beca use it is equal 10 the magni tude o f the capacitive and inductive reactances at reso nance. as is easily shown: ~

This term is usually applied to transmission lines, hUI has a certain importance even in lumped I'IClwort S.

89

4.2 PARAllEL RLC TA N K

L

12el

[[

= 12,.1 = woL = ./LC = YC'

(7)

We will find that this q uantity recu rs with some frequency.' so keep it in mind . Before we continue. let's see if our equation for Q makes sense. As the parellel resistance goes to infinity, Q does, too . Thi s behavior see ms reasonable since . in the limit of infinite resistance. the network degener ates to a pure L C system. With only purely reactive elements in the net work , there is no way for energy to dissipate. and Q should go to infi nity. just as the eq uatio n says it should. Plus, Q also increases as the impedance of the reactive e lements decreases (by decreasing LIC). since the pure resis tance becomes less significant compared with the reactive impedances. For completeness, we may derive a cou ple of additional expressions for the Q of ourparallel RL C network at resonance :

R

R

Q = - - = --=wo RC. IZL. cl woL

4.2 .2

(8 )

BRAN CH CURRENTS AT RE SONAN CE

As mentioned earlier. the inductive and ca pacitive branch currents at resonance ca n differ signifi cantly from the ove rall network current (which is simply due to the parallel resistance). Let' s now com pute the magnitude of these curre nts. Again, at resonance. the voltage ac ross the netw ork is JmR. Since the inductive and capacitive reactances are eq ual at reso nance, the inductive and capacitive branch currents will be equal in mag nit ude :

Ihl

I/" IR I/" IR,/LC R = Il el = -IVI =-= = 1/,,1v r.-= = QI /,"I· Z woL L LIC

(9)

That is, the current flowing in the inductive and capacitive branches is Q times as large as the net current. Hence, if Q = 1000 and we drive the network at resonance with a one-ampere current source, that one ampere will flow through the resistor, but one thousand amp eres will How through the inductor and ca pacitor (until they vaporize ). From this simple example, yo u can well appreciate the incompl eteness of simply Slating that the inductor and capacitor cance l at resonance!

I You may recall that the characteristic im peda nce of a uansmission line is given by the same ellpression, where L and C are Interp reted as the indu ctance and capacitance per Mi, l('rIg/h. We will Clplore the connect ion terwee n finite . lum ped LC networks and inli nilc, di slribul ed syslelTU more fully in Chapter 5.

CHAPTE R 4 PA SSIVE RL C NETWORKS

90

4 .2 .3

BANDWIDTH AND Q

We 've already ded uced the beh avior of the net work at freque nci es fur re move d from resonance. and we 've also taken a detai led look at the net work ' s be havior at resonance. Let's now exa mine the beh avior of the tank circuit at frequ e ncies s lightly displ aced from resonance to round out our analysis. FiN . let w = Wo + f::1w . Then we ma y rewrite our expression for the admitta nce as

Y = G

+ -j

wL

2

(w L C - I ) = G

j + -(2t:J.wwo + (t:J.w) 2 IL C.

wL

( 10)

For sui tab ly sma ll displ acement s about Wo (that is. values of t:J.w that a re small relat ive 10

wo) , this expression s implifies to

y

~

G

+ j2Ct:J.w.

(II)

Th is adm ittance behavior is exactly the same as that of a resistor of value R in paralle l w ith a ca pac ito r of value 2C. exce pt w ith t:J.w re plac ing us, Hence, the shape of the ad mitta nce c urve for (s mall) pos itive displacement s about the resonant freq uency is the sa me as that o f a paralle l RC networ k . a nd we may defin e a half-ba ndwidth as sim ply 1/ 2RC by an alogy with the Re case. Wh y "Jullf-ba ndw idth? " Well , from the sy m metry of the appro ximate admittance funct ion derived previou sly. the shape for displacements be low resonance will be the mirror ima ge of that above resonance, so thai the total -3- d B bandwidth is just II RC . Somethi ng in terest ing happe ns w hen we normalize this bandwidth to the reso nant frequ ency: ( 12)

On ce again , Q has popped into the picture : the fractional bandwidth is simply IIQ· For a given resonant freque ncy, then , higher Q implies narrower bandwidt h , From the foregoi ng deve lop ment s you ca n see that Q is an im portant parameter, a nd we shall find in ma ny subsequent exam ples that signifi ca nt analytical (and synthetic ) advanta ges often accrue from focu sing Oil it.

4 . 2.4

RING ING AND Q

Although we have focu sed on the frequency- domain behavior of R LC networks. perhaps it is worthwhile her e to mention at lea st one important tim e -dom ain property. TIle res onant exchange of e nergy betwee n the ind uctor and the capaci tor leads to an impulse response w ith the fa mi liar damped oscillator y c harac teristic . Since Q is a measure of the rare of e nergy loss. o ne would ex pec t a highe r Q to be associated with more persistcm ringin g than a lowe r Q , To place this intui tion on a more quantitative

.4•.4 OTHER RESONANT RLC NETWORK S

91

basis, recall that the impulse response is a damped sinusoid with an expo nential envelope. It is straightforward to show that the time constan t of this envelope is 2RC,

sothat (13)

The time constant may be rew r it ten (using Eqn. 12) 10 yield V (t )

ex

\t(1t' - (I/ T)l ll'/ Q I .

(14 )

Thus, the amplitude decays to 1f t' of ill'. initial value in Q / ll cycles. Bt..c ause an exponential decays to abou t 4% of its initial value in II cycles. Q is roughly equal 10 the number of cycle s of ringing. This handy rule of thumb is extre mely useful for rapidly estimating Q from experimental impulse (or step) response da ta .

4 .3

SERIES RIC NETWORKS

We may follow an exactl y analogous dual approach to dedu ce the properties of seriel'. RLC circuits. The details of the deri vations are relatively uninteresti ng. so here we simply present the relevant observations and equ ations. The resonant condition corres po nds aga in to the frequency where the ca pacitance and inductance cancel. Rather than resu lting in an admiuance minimum. though . resonance here results in an impedance minimum . with a value of R . The eq uation for Q involves the same terms as for the parallel case . but in reciprocal form :

Q = J L/C R

( 15)

At resonance . (he vo ltage across either the inductor or capaci tor is Q time!'. a!'. great as thai across the resistor. Thu s. if a series R LC network with a Q o f IfXXl is driven at resonance with a one -volt source, the resistor will have that one volt across it . but a thrilling one thousand volts will appea r across the induc tor and capacitor." You are encouraged to explore indepe nden tly and in mo re detail the du ality of series and parallel resonan t circ uits.

4 .4

OT HER RESONANT RIC NETWORKS

Purely parallel or series R LC networks rarely exist in practice. so it's import an t 10 take a look at configurations that might he more realistically representative . Considcr. for example. (he case sketched in Figure 4 ,2. Becau se inductors tend 10 be

• By the way. such resonem voltage mag nification is the fundamental ha~i s for Testa co il operation. With such techniques . Tesla was ahle In generate an es timated 5--tO MV in 1899 !

92

CHAP TER 4

PASSIVE RL C NETWORKS

FIGURE 4 . 2 . Not-quite -parallel

Rl.C tonk circuit.

significa ntly lossie r than ca paci tors. the model shown in the figu re is often a more realistic approx imation to typical para llel R LC circ uits. Since we've already analyzed the purely parallel RLC network in de tail, it would be nice if we co uld re -use as much of this wo rk as possib le. So. let' s co nvert the circuit of Figure 4.2 10 a purely paralle l R L C network by replaci ng the series L R section with a pa rallel one. Clearly . such a substitution cannot be valid in gene ral. but over a suitab ly restricted frequ ency ran ge (e.g.• near resonance) the eq uivalence is prelly reason ab le. To show this formall y, let 's equate the impedances of the series and parallel I. R sections:

If we equate rea l parts. and note thai Q = R pl woL p = woLsl Rs. 7 we obtain R p = Rs ( Q l

+ 1).

(1 7)

Similarly. equating imaginary par ts yields Ll' = L s (

Q 2 + 1) Ql '

( 18)

We may also deri ve a similar set of eq uations for computing series and parallel Re equivalents: R p = Rs(Q 2 + I ),

c,

=

CS (Q~~ I).

( 19) ( 20)

Lee s pause for a mom en t and look at these tran sformation formulas. Upo n closer examination. it' s cle ar that we may express them in a universal Conn that applies to bot h RC and LR networks:

1 If lhe series and paralle l sect ions are 10 be equivalent. then their Qs certainly musl be equivalent.

4 . .5

Rl C NETWORK S AS IMPE DANCE TRANSFORMERS

93

Z S= Rs + j X s

FIGURE 4 .3 . Nef\IIfork lor maximum power

transfertheorem. (2 1)

and

XI' = Ks (

Q2+ Q2

1)

'

(22)

where X is the imaginary part o f the impedance. Th is way, one need only remember a single pair of " universal" formulas in order to co nvert any " impure" R LC network into a purely parallel (or series ) o ne that is straightforward to ana lyze. However. one must bear in mind that the equivalences hold only over unarm»' range off requencies centered about woo

4 .5

RIC NETWOR KS AS IMPEDANCE TRANSFORMERS

The relative abundance of power gain at low freq uencies allows designe rs 10 treat it essenually as an infinite resou rce . Design specifications are thus often expressed simply in terms of a voltage gain, for exa mple. without any explicit reference to or co ncern forpower gain. Hence. circuit design at low freq uencies usuall y proceeds in blissful ignorance of the maximum powe r transfer theorem derived in every undergraduate network theory cour se. In striking contrast with that insouciance, RF circ uit design is frequently preoccupied with power ga in beca use of its relative scarcity. Impedance transfomilng networks thus play a prominent role in the radio frequency domain.

4 . S.1 THE MAXIMUM PO WER TR AN SfER THE OREM Tounderstand more explici tly the value of impedance transformers. we now review the maximum power transfer theorem (see Figure 4.3) . The prob lem is this: Given afiud sourer impedance Zs. what load impedance ZL maxim ize s the power deliv ered to the load? The power delivered to the load impedan ce is entirely due to RL , since reactive element s do not di ssipate power . Hence. the power deli vered is simply (23)

CHAPTER 4 PASSIVE RL C N ETWORKS

0000

•

",---<~

C ~~

FIGURE 4 .4 . Upward impeda nce tron ~lormer .

0000

c

"I'

FIGURE 4 . 5 . Downward impeda nce tra nsformer.

where V R and Vs arc the rms voltages across the load resi.. . tance and source. rcspeclively. To maximize the power delivered 10 Rt.. it's clear thut XL and X s . . hould be inverses so that they sum to zero. In add ition. maximizin g Eqn. 23 under that condilion leads to the result that R I . should equal Rs .llence. the maximum power transfer from a fi xed source impedance to a load occu rs when the load and source impedances are complex conjugates. Having established mathematically the condition for maximum power transfer. we now con.. .ider practical method.. . for ach ieving it. 4 .5.2 THE l -M ATCH The multip lication by Q of voltages or currents in resonant R L C networks hints at their impeda nce -modifying potential. Indeed. the series- parallel RC/LR network conversion formulas developed in the previous section actually show this property explicitly, To make this clearer. consider Ol K C again the circuit of Figure 4.2. redrawn slightly as Figure 4.4 . Here we treat Rs as a load resistance for the network. When this resistance is viewed acro ss the capaci tor. it is transformed to an equivalent R p accord ing to the for mulas deve loped in the previous section. From inspection of those " universal" equations. it is d ear that Rp will always be larger than Rs . so the network of Figure 4.4 transforms resistanccs upward. To gel a downward impedance co nversion. j ust interchange port s (5)Ce Figure 4.5). There is a nice. intuitive way to keep truck of which way the imped ance transformation goes. For example. if the circui t in Figure 4.5 is driven by a test voltage

4 .5 RiC N ETWO RKS AS IMPEDANCE TRA NSFORMERS

9'

source. the result is a parallel RLC net work since the Thevcnin res istance of the lest source is zero. Nowthe ind uctive CUITCnl in a paralle l RL C net wo rk a t resonance is Q times as large as the c urre nt throu g h R», T his incr ea...e in c urre nt is see n by the source and may be inte rpre ted as a red uc tio n in resistan ce. II should abo be cle ar tha t interchanging the indu ctor a nd ca paci to r doe sn' t a lter tbe transforma tio n ratio. altho ugh othe r co nside nuions ma y dictate whethe r o ne chooses a high-pass or low -pass co nf igunu io n. This circuit is known a.. a n I.-match because of its shape ( perha ps yo u have to be lying on your side a nd d ysle xic 10 see this ). and ha s the attri bute of si mplicity, Ho wever. there are only two degre es o f freedom (o ne can c hoose o nly L a nd C ). Hence. once the impedance transfo rma tion ratio a nd re son ant freq ue ncy have been speci fied. network Q is au tomatic ally determined. If yo u wa nt a di ffe rent value o f Q the n yo u must use a netwo rk that offers add itional degrees o f free do m: we ' ll study so me of Ihese shortly. Asa final note o n the Lmsrch . the " unive rsal" eq uations ca n be s implified if Q ~ » 1. If this inequ ality is satisfied. then th e following approx ima te eq ua tio ns hold :

, (woR!';I C )' = RI

R p ~ Rs Q ~ = Rs

s

Ls . -C

(24)

which may be rewritte n as

(25) where Zo is the c harac teristic impedance of the network. as dis c ussed in Sectio n 4 .2.1. One may also deduce (hat Q is approximate ly (he square roo t of the tran sformation ratio: (2 6)

Finally. the reactances don 't vary much in und ergoing the transfo rmat ion :

x,

~

x;

(27)

As long as Q is greate r than about :3 or 4. the error incu rred will he und er abo ut 10%. If Q is greater than 10. the maximu m e rror w ill he in the ne ighborhood of I % o r so. Hence. for qu ick . back -of-the -en velope calculation s, these s implified eq uations are adequate. Fina l design values ca n be co mputed using the full " universa l" eq uatio ns.

4.5 .3

THE ". ·M ATC H

As already di scu ssed . o ne limita tio n of the L ma rc h is thai o ne ca n ...pccify o nly two of center frequency. im peda nce tran sfo rm ation rati o. a nd Q. To acquire a thin! deg ree of freedom. o ne can employ the net work shown in Figure 4. 6.

96

CHAPTE R" PASSIVE RlC NETWO R. KS

L

fi GURE 4 .6 . The }I" ·ma tch .

L,

0000

FIGURE 4 .7. x- mc tch us cc scod e of l-mctch es.

This circuit is known as a x -rnetch. again because o f its shape. The most exped ient way 10 understand thi... matching network is 10 view it as two Lmatches connectcd in cascade, one thai transforms down and one that transforms up; sec Figure 4 .7. He re. the load resista nce HI' is tra nsform ed 10 a lowe r resistance (known as the image or intermediate resistance, here denoted HI ) at the junctio n of the two ind uctance s. The ima ge resista nce is then transform ed up to a value H in by a second L-rnalch section. Now. it may see m a bit silly to usc one Lcsection 10 go down. then anot her to go back up. However. we have gained the add itio nal degree of freedo m we were seeking. Recall that , for an Lmatch , Q is fixed at a value roughly equal to the square root of the transformati on ratio. Typically, the Q o f an L-match isn' t particularly high because huge transformation ratios are infrequently required. The a -match decouples Q from the transformation ratio by introducing an intermediate resistance value 10 tra nsform to, allowing us 10 achieve much higher Q than is generally available from an Lmatch . eve n if the overall transformation ratio isn't particularly large. Because we now have three degrees of freedo m (the two ca pacitances and the sum of the two inductances), we can independently specify center frequency, Q (or bandwidth), and overa ll impedance tran sfor mation rat io. However, as with the L· match (or any other kind of match), impractical or inconvenie nt component values can result , and MIme creativity or compro mise may be required to generate a sensible design. In many instance s. cascad ing severa l matching networks may be helpful. In order 10 derive the design equations, first transform the parallel He subnerwork o f the right-hand Lcsection into its series equivulent, as shown in Figure 4.8.

...

---- --- ~---- ----------- -

4 . 5 RL C NETWORKS AS IMPEDANCE TRANSFORMER S

R,n-'"

fi GURE 4 .8 .

L,

L2

0000

0000

R ,_

=f

97

i .,

JR '

Jl' -match with tron ~formed right-hand heetian.

When we replace the OUIPUI para llel He network with its se ries eq uivalent , the ser ies resistance is. of CUUI"l\C. R,. Hence. th e Q of th e rig ht-hand L section may be

written as

woL,

~ =

~ Vii; - 1 = Qri~tM .

(28)

me

Al same time. recognize thai the left-hand Lcsecrion a lso sees a resi stan ce of HI at the cent er freque ncy. Th e refor e. its Q is given by

r;;:- =

woL\ - - = f --,;: R, ~ R,

11

Q ld l '

(29)

The overall network Q is simply (30)

Equation 30 allow s us 10 find the image res ista nce . g ive n Q a nd the transformation resistances. On ce H, is co mputed. the tot al ind uc tance is qui ck ly fo und :

QR, L l + 1,2 = - -.

(3 1)

Wo

The values of the capaci tors arc also fou nd readil y:

C1 = C2 =

Q lefl ,

wn R m Q ri ghl .

w oRp

(32)

(3 )

As a practical meucr. no te tha t find ing HI from Bqn. 30 ge nera lly req uires itera lion. A good starti ng value can be o bta ined by assuming that Q is large (eve n if it isn't). In that case, RI is approxima tely given by

(.fR;, + .,fR;)2 Q2

(34)

~

98

CHAPTE R 4 PA SSIVE RLC NETWORKS

o~oo [ ] ~~o C

1 C2

1 r

RS

-FIGURE 4.9. T-molch.

If Q is very large. or if you're j ust doi ng so me preli minary "coc ktai l nap kin" ca lculations. then ite ration ma y not e ven be necessary. And that's all the re is 10 it. As a parting note, one final bit of trivia dese rves mentio n. An add itional reason that the x -ma tch is po pular is that the parasitic capacitances of whatev er connec ts 10 it can he ab sorbed into the network design. This property is particu larly valuable beca use capaci ta nce is the dom inant parasitic el e me nt in ma ny pract ical cases.

4 .5 .4 THE T·MATCH The a -match results fro m casc adi ng two Leecrions in one particu lar way. Connect ing up the L-scctions a nother wa y leads 10 the dual of the a-match, as show n in Fig. ure 4 .9. Here. what would be a s ingle capac itor in a practical im plementa tion has been decomposed explicitly into two separate ones. The ( parallel ) image resistance is seen across these capacitors. either loo king to the right or loo king to the left as in the a -match . The design eq uatio ns are readil y derived by following an approach analogo us to that used for the a -m atch . The overall net work Q is simply

(3 5)

from w hich the ima ge resistance may he found. Then: (36)

(37) (38)

The 'l-mat ch is part icu larl y use ful whe n the source and term ination para sitic» arc primarily indu ctive in nature. allow ing them to be absorbed into the network .

4 .5 Rl C NETWORKS AS IMPE DANCE TRANSFORMERS

99

c,

F1GU RE 4 .10.

TapPOd capaeitol'" ( MOnOU

as a matching nefWorlc. .

4. 5 .5 TAPPED CAPACITOR RESO NATOR AS AN IMPEDAN CE MATCHING NETWOR K Tapped resonator s share w ith the n - and "l-mut ch the ability to se t cente r frequency, Q. and transformation ratio . As a co nseq ue nce, they are a lso co m mon idioms in RF design. An example of such a ci rcuit is the tapped ca paci to r tank , frequently used in oscillator- since it combines a reso nator with impe dance nun..form ation a nd so a llow s one to couple energy o ut of the ta nk wi tho ut degradi ng Q excessively . See Figu re 4 . 10 . Theoperation of thi s impedance tra nsformer is be st understood as simply the consequence of the capacit ive vol tage divider. A vo ltage reduction in a perfec tly lossless network must be accompanied by a n impedance red uc tion proportional to the sq uare of the voltage attenuation if po we r is 10 he co nse rved. Th is ne twork is not perfec tly lossless. but we do ex pect th e impedan ce tran sformation ratio to be rou g hly

( 39)

so

that the network transfo rm s a re sista nce R in do wn ward to a va lue R2, o r a resis-

urce R2 upward to a value R m . To confirm this e xpecta tio n. let us a nalyze the resistive ly loaded ca paci tive divider ill isolation. The ad mitt ance of the co mbinatio n is re ad ily found after a litt le la bor: (40)

The real part is (4 1) At sufficiently high freque ncies, the eq uiva le nt shunt co nd ucta nce simplifies to

(4 2 )

100

CHA PTER 4

PA SSIV E RLC N ETW O RKS

as an ticipat ed . Equation 42 also defines a fac tor , /I , which is the turn s ra tio of an ideal transfo rmer that wo uld yield the sa me resistance transformatio n as the ca pacitive d ivider . The conce pt of an eq uivale nt turn s ratio w ill prove partic ularly useful in unifying the trea tme nt of vari ous osc illa tors . For the sa ke of co mpleteness. we also com pute the imagi nary pa rt of the adm ittance: B _ we I + w 3Ric.c;!(Cl + C2) (43) m w ;!Ri(C. + C2)2 + I which . at sufficie ntly high frequ en cies. approac hes a lim iting value of

C 1C2

B,n ::::: ur - C,

+ C2

=

tu

»

Ceq .

(44)

Not surpri sing ly, the resi stively load ed tap ped ca pacitor net wor k presents a suscepta nce equal to that o f a series combination of the two capac ita nces. The foregoin g series of eq uatio ns serves we ll for analys is and pa rticularly to develop design intuition. Equation s 42 and 4-1 are also ex tre mely useful for first-cut. bac k-of-the-enve lope designs. Howe ver. to carry out a detai led design requires a bit more labor. To derive a series of equations more suitable for design, we now apply parallel - se ries tran sformations un til we massage the ne twor k into so me th ing we already know. In broad outline. what we ' ll do is equivalent to the fo llow ing procedu re. Transform the paralle l R 2 C 2 combi natio n into its series coun terpart . Th en combine C . w ith the tra nsformed ca paci ta nce to yield a s ingle se ries Re in parallel w ith the indu c tor. Next. trans form that se ries Re in to its paralle l equivalent. T he parallel R thus fou nd is se t equa l to R m . Carrying out tha t strategy. we first de termi ne the required ne twork Q: (45)

where we interpret the bandw idth as the frequ ency span ove r which the impedance tran sfo rm ation ratio is to re mai n roughly consta nt. We the n note that one expression for the network Q is R

Q = - '-" .

(46)

t. = Rin

(47)

wol..

so tha t •

wnQ

Ne xt. transform the pa rallel RC sec tion into its series equivalent. The series reo sis ror has a value give n by

R,

Qi+ and

I

(48)

4. 5 RLC NETWORKS AS IMPEDA NCE TRANSFO RMERS

101

(4 9)

where Q2 is the Q of the parallel RC sec tion. We then recognize that the series resister may also be viewed as the result of transforming R jq : Ri ..

Rs = Q 2 + I '

(5U)

Equaling the IWO expressions for Rs and so lving for Q2 yields

.'!2(Q 2 + I) - I.

(5 1)

R"

The original parallel RC has a Q of

Q2 = WOR2C2.

(52)

sowe may write

(53)

The only remaining undetermined e lemenl is C I • To derive an eq uation for its value, fi rst express the series co mbination of C 1 with ClS as a single capacitance:

:c2~S_ Cc>.j = :::"C,-"C C 1 + C 2S

(54 )

The network Q then can be expressed as C 1 + C2S WO R 2S C 1C 2S

(5 5)

Solving for C 1 yields one possible formula for the last unknown capac itor :

C _ C, (Q l + I) 1- QQ2 - Qi '

(56)

The derivations are a lillie cumberso me. but the ideas behind them are very simple. Onceyou are well versed in this parallel- series transformat ion business. fi guring out how tank circuits transform impedances is conceptually straightforward ; it's j ust the execution that may be a hit involved .

102

CHAPTER 4

Rin

PASSIV E RL C NETWO RKS

c=

----.

fi GURE 4 .11. Tapped inductor resonator.

4 .5 .6 TAPPED INDUCTO R M ATCH For the sa ke of co mplete ness. we now co nside r briefly the ta pped inductor resonator as a marching net work (see Figure 4 .11). A s you m ight e xpec t , its behavio r is quite similar 10 that o f its tappe d capacitor cou nterpart . We won ' t go through a detailed derivation of the design eq uations since they're es se ntia lly the sa me as for th e tapped ca paci to r case, hut we can immed iatel y make the following observation: R2 must be less than R in beca use. once agai n. we have a voltage divider. As in Sec tion 4 .5.5. we may proc eed w ith th e followi ng de rivat ion s. First . determine the net work Q. Th e n. since

(571 C= -

Q - .

(58)

WOH in

Next. transform the parallel R L section into its series equivalen t. The series resistor has a value given by

Rs =

R,

=-;--;Q ~ + r:

(59)

while the inductor transforms to a value (60)

We may also consider R s to be the resu lt of transforming R in :

R _ S -

R in

Ql

+1

(6 1)

Equate the two expressions for R s and solve for Qz:

R,

-

«; ( Q' +

I ) -I .

which is the same ex pressio n as for the ta pped capacitor network.

l

(62)

4 ,5 RL C NETWORKS AS IMPEDANCE TRANSFORMERS

FIG U RE 4 .12.

Having found

Doubl,H opped

nKOnon l

103

match.

0 2. we may write (63)

(6-1)

Tow ive for the last unknown. note that we may express

0 as

Q = wolL I + L 25 I.

(65)

R>s

Solving for L . ultimately yields L

L IQ Q 2 1=

2

Qi J

oi + I

•

(66)

thus completing the design.

4 . 5 .7

DOUBL E-TAPPED RE SO N ATOR

In certain instances, impractica l or inco nvenient component values may he needed in the various transfor mation networ ks considered so far. Quit e often, this problem appears in the form of excessive required capacitance . Obtai ning add itional degrees uf freedom 10 mitigate this proble m requi res the use o f networks with additional elements. One such network is the double -tapped reso nator , disp layed in Figure 4 . 12. Th is circuit boosts R2 10 a larger effective paralle l resistance across the who le tank than in a standard tapped capacitor network. and then reduces this parallel resistance by the lapped inductors to the desired value R in • Th is techniq ue therefore increases the required inductance and simultaneously reduces the requ ired capac itance. potentially bringing both closer to comfortab ly realizable values.

104

CHAPTER.4 PASSIVE Ri C NETWO RKS

L

R, -

•

R,

---<..

f iG URE 4 .13 . L-motch ottempt.

Th e double-tapped matc hing net work has fou r compo nents and so provides four degrees of free do m. We may now spec ify ce nter frequ ency. imped ance tra nsformalion ratio . Q (bandwidth) , and. say. total inducta nce (or ca pacitance). Na tura lly. dcrivario n of a de sig n proc edure is left as a n exercise fur the reade r. As you might have surmised. focu sing on Q invari ance a nd se rie s-parallel tran sformalio ns is the key to the derivations. As a final parti ng note on this particular matching ne two rk . it should be me ntioned tha i - although it solves a thorny practical prob lem in princi ple - the finite Q of physicully rea lizable inductors (and. 10 a lesser exte nt. o f ca paci tors ) impo ses a bo und on the a mount of impro vement that one can obtai n in pract ice.

4 .6 EXAMPLES Let's work out a few examples 10 ma ke sure tha i the design procedu res are clear. We ' ll conside r Lmatch . x -march . tap ped capac itor. a nd tapped ind uc tor networks in trying 10 mee t the followi ng speci f ications: ce nte r freq ue ncy o f I Gi ll . RI = 50 R2 = 5 n, and a bandwid th of 25 MHz .

n.

4 .6 .1 l · MATCH ATT EM PT More paramete rs than ca n generally be sc i by the Lrnatch arc specified in this design. Neverth eless. we ' ll proceed with an atte mpt as fo llows. The net work is sketched in Pigure 4.1J . ( I) The tra nsfor mat ion ratio fixes the value of Q as

Q=

'Itt:: II; - 1 = 3.

(67)

He re. our estim ate o f ba ndwidth based on Q wo uld be ra ther crude. but ce rta inly sufficie nt to reveal our inability to meet the 25 -M Uz bandwidth spec ificatio n. We need a Q of a round 4(). so we 're a n order of magnitude awa y from mee ting the requ ireme nt. (2 ) We 'v e computed Q fro m the impedance ra tio : le t's equate it to the Q co mputed wit h the inductor impeda nce to ena ble a computatio n of the required ind ucta nce:

Q = wo - -L R2

105

4 .6

EXA M PL ES

~

QR~ ~~9 H L = - - ;:::;: _._' n .

(68)

Wo

(3) Nnw let' s use ye t another, hut still eq uivalent, ex press ion fo r Q to find the required capacita nce value:

Q = WOR IC

,. C

=

~

wOR l

= 9.55 pF.

These values are quite pract icall y reali zable in Ie form (much more on this subject later), bn the desig n has excessive bandwidt h. A nd that ' s about all there is for the Lmatch.

4 .6 . 2

~ · M A TC H

We follow the procedu re outli ned in Section 4.5.3 . (I ) The required Q is Mill 40. (2) A Q of 40 is large e no ugh to cal c ula te the ima ge resistance wi th the ap proxi mate formula

R , ;:::;:

(.JR::. +2./R..)' P In

;:::;: 0 .0 54 Q .

Q

(70,

(3) The capacitors arc ca lcula ted as foll ows: Qldl

C. = - -

wO R in

;:::;: 305 pr ,

Q righl

C, = - - " 96.8 pF. wo R p

(7 1) ( 72)

(4) Finally, (he requ ired inducta nce is

QR,

L = - - ::::: 0.344 nl f.

w"

(73)

In any Ie technology, these values are highl y incon veni ent. in par ticular. the large lotal capacitance consumes excessive die are a . Add itionally, the inductance is low enough thai it may be d iffic ult to realize accurately a nd with low loss. Althou gh we've been able 10 meet the design requirement s in pri nci ple, we still don' t have a practical design if the network is to be co mp letel y inte gra ted .

4 .6 . 3 TAPPE D CA PAC ITOR MATC H Here we proceed as in Section 4 .5.5. (I) The required Q is still 40 . (2) First, we compute L :

• CHAPTE R 4 PA SSIVE RLC N ETWORKS

106

L

~

R io

-woQ

~

(7~ )

0. 199 nlt .

(3 ) Next. we find the bottom capacitor. C2:

c, =

~ ( Q' +

Q,

1) - 1

R io

- - = -'-----=---;:-- - '" woR z

~Ol

pF.

(7 5)

(4) We fin ish by findin g the other capaci tor' s value:

c I

= C,( Q! + I) '" 186 F. QQ, _ Q! p

(76)

As you c a ll SCC, the lapped ca paci tor reson ator has unfo rtunately yielded a no less impractical sci o f values.

4.6 .4

TAPPED IN DUCTOR M ATCH

If we carry out a procedu re an alogou s to our design exe rcise of Section 4 .6 .3, we ohlain C :::::: 127 pF. L . :::::: 136 pll . and L z ::::;: 63 pH . Althoug h the total ca pacitance has decreased substantially to a value that would be co nsidere d only somewhat o neroua. the indi vidual inductances arc probably impract icall y small. Agai n. even if they could he accura tely reali zed (say. by a suitably short piece of metallization), the lypicallossiness of on-chip interco nnect would mostlikely prevent attaining a Q of 40. There are a few remedies that ca n be a pplied to improve the situation. Recogn ize that the prob lem fundamentall y ste rns from the high Q value sought; it is prob ably easiest to conside r reflecting N2 across the capacitor so that it becomes N in • lIi gh Q fur this parallel tank impli es a small characteristic net work imped ance (small LIC ratio) relative to H 2 • and a small Z o/ Hz ratio in tum implies small L and large C . If. for a given tran sformation ratio. we could boo st the effective parallel res ista nce. then we could employ a larger Li e ratio to achieve the same Q and therefore increase the requi red induct ance to practica l values. Hence we need an upward impedance transformer within the tran sformer. Use of a do uble -tapped resonator is therefore one possibl e so lutio n to this difficulty. Another poss ibili ty is to cascade on e or more add itiona l impedance tran sformers. The reade r is once again invited to pursue these opt ions independen tly.

PRO BLEM SET FO R PA SSIVE RLC NE TWOR KS

PROBL EM 1 An important theorem states that the load impedance should equal the co mplex co nj ugate of the so urce impedance in order 10 maximi ze pow er transfer.

PR0 8LE M SET

10 7

This SOI.I nd ~ simple eno ugh . but enginee rs so meti mes dra w incorrec t inferences about how to satisfy this co nd itio n. Specifically. co nsid er two a pproaches to match ing a purely resistive 75-0 load to a source whose im pedance at some freq uency happens III be 50 + j 10. Downwardly mob ile eng inee r A , afte r readi ng abo ut the maximu m power transfer theorem , du tifull y add s a 25-0 resistor and a capaci tor o f - j 10 Q in series with the source. whereas enginee r B offers a similar so lution but rep laces the 25-Q resistor with an a ppro priately des ign ed Lmurch . Quantitatively compare the two approaches by computing explici tly the ratio o f powers delivered to the 75-0 load by these two solutio ns. assuming equal Thevenin source voltages. Qualitat ively ex plain the reaso n for the difference. and why engineer A can look forward III a long career at Su bOptima l Produ ct s. Inc. PROB LEM 2 In add ition to maximizing power tran sfer, impedance -tran sforming networks are widely used simply to enable a speci fic amo unt of power to be deliv ered to a load. An import ant ex ample is. found in the output stage o f a tra nsmitter where. owing to supply vo ltage limitat ion s. a downward impedance tra nsform ation of the antenna resistance is necessary. A common load im pedance is 50 Q . Suppose we wish to deliv er I W of pow er into such a load at I G Hz. but the power amplifier has a maximu m peak-to-pea k s inusoidal voltage swing of only 6.33 V bec a use of vario us losses and tran..Istor brea kdown problem s. Design the foll owing match ing networks to allow that I W 10 he delivered. Usc low-pass versions in all cases. and assume that all reac tive clements are ideal (if only that were true . . . ).

(a) Lmarch. (b) a-match ( Q = 10). (c) Tenatch ( Q = 10).

(d) Tapped capacitor ( Q = 10 ). (e) If the maximum allowable on -chip ca pacitance is 2
PRO BLEM 3 As we'll see , reso nant circuits are also ind ispen sab le for allowi ng am plifiers to function at high frequ encies. To investigate this property in a crude way. consider a standard common -source amplifi er whose transistor is mode led with a hybrid-". model that neglects C gd and r".oo t incl udes rK • C 8 J • and Cd" t rhe junction capacitance from the drain to the substrate. here at source pote ntial). (a) Assume a source resista nce equa l to rll • and ca ll the load res ista nce RI-. Deri ve an expression for the vo ltage ga in of the amplifier. At w hat frequ ency WI doc s the magnitude of the voltage gain go to unity 'l

108

CHAPTER.4

PASSIVE RLC NETWO RKS

( b) Now ass ume thai we add ind uctan ce in se ries with the source to ma ximi ze the voltage de veloped ac ross C &J at freq uency W I. What is a n ex pression for this indu cta nce. L s '! Wh at is the new voltag e ga in at W I? (c ) In addition to th e input series inductance. su ppose we pl ace an inductance from the drain to gro und to resonate o ut the dr ain capac ita nce a t WI . What is an expression fo r thi s ind uctance. L OUl ? Now wh at is the voltage ga in at WI'!

PROBLEM 4 An idea l dipo le receiving ante nna that is muc h sho rte r than a wavele ngth may be modeled as a vo ltage-d rive n se ries Re ne two rk. whe re the porti o n of R tha t is du e to rad iation is given approximat e ly by

R,., '" 395(1/ 1. )' .

( P4.1)

where I is the length of the an te nna and )" is the wa vele ngth. This eq uatio n wo rks reaso nab ly we ll up to I I )" ratios of abo ut 1/ 4. (a) First ass ume that the o nly resistance in the ant enna model is this radi atio n resistance. Further assume that the va lue of the eq uivalen t voltage ge nerator is simply the rece ived E -field strength (which yo u may assume is fixed) multip lied by the a ntenna length I . G ive n a n "optim um" passive impedance ma tc hi ng netwo rk interpo sed between the a ntenna and so me fixed resistive load Rt , w hat is the ma ximum powe r Iransfer efficie ncy. defin ed as the rat io of power delivered to RL to the total dissipated in the syste m. as a functio n of (nonnalized) a ntenna len gth ? A nsweri ng th is q uestion does not req uire a numerical val ue for R L • (b) Now suppose that the an te nna has some addit ionalloss. represented by a resistance RJ in seri es w ith the radiatio n resista nce term. Does yo ur a nswe r cha nge? Ho w'!

PROBLEM S Design an I.-m atch to matc h a IO·Q so urce to a 75-0 load. Assume that the ce nter freq ue ncy is 150 M flL. PROBLE M 6 It was me ntio ned e arlier tha t an Lmatch has o nly two deg rees of freedo rn. Hence, o nce ce nte r freq uency and impe da nce trunsformation ratio a rc chos en, th e Q (and the refore the hand width) are fixed . The a -matc h adds o ne degree of freedom. allowing independent c ho ice of all three param e ter s. Re -do Pro ble m 5 with the additional con strain t th at the tota l bandwid th (If the matc h be I S M Hz. PROBLEM 7 Disa pp ointed by lac klu ster sa les of the ir lo w-ban d wid th-hig h-offset o p-amps- Fromage'Icc h (now a proud sub sidiary of SuhOplimal Product s. Inc.) has decided to ho p o nto the wireless bandwagon. Th ei r savvy market re search departmen t (a gu y narnr..-d Earl) concl udes that they sho uld en te r the A M radi o mark e t. On e of their ci rcui t design problems involves a common-so urce a mplifier with a tuned load that is resonant at the tradi tio nal (tho ug h subo ptima l. o f co urse ) imerrnedi ate frequency (IF) o f 0.1 55 kHz. For reaso ns that are never made cl ear to yo u, the

PROBLEM SET

109

drain load resistance at reson ance mU ~1 be five kilohms.1l AI the sa me time. the output olmis stage must ultimat ely d rive a 5-Q load. (al Assume infinite OUlPUI resista nce fo r the tra nsisto r a nd initially as sume infinite Q for all reactive el em e nts. Furt her assume that the tran sistor ha~ ze ro drain gale capacita nce . Devi se an Lmatch 10 sa tisfy the design req uire me nts. Make reasonable approxi ma tio ns. (b) What is the [0Ia l - 3-d8 ba ndwidth of this circuit? (c) Now suppose that the indu ctor found in pari (a) actu all y possessed a Q of 100 at the desired ce nter freque ncy. Find the eq uivale nt se ries res ista nce of the inducror and the n redesign the ma tching net wor k . gi ven the assu mption thai the inductor 's series res ista nce remains co nsta nt . (dl What is the new band width? (e) What if the transistor were made in a terrible pr ocess technology and actually had a 5-pF C xd ? Would th is affec t the perfo rma nce of this stage very much? Assume that the transistor 's ga le is dri ven fro m essentially zero impedance. and ihat is zero.

r,

PROBLEM 8 The Lmutch pro vides onl y two degrees of freedom. Hence. o nce the impedance transformat ion ratio a nd reso nant freq ue ncy have bee n c hose n. Q is de termined automatically; you get w hereve r Q yo u ge t . Usua lly. the Q is fa irly low. fcading 10 a reasonably broad band match . A lthou g h this property is freq uen tly desirable. il i... often bette r 10 have ind epen dent control over a llthree parameters . lOll Design a a-march 10 satisfy the design req uire me nt s of Pro blem 7. Initia lly assume the same ideal co nditio ns and a desired lota l - 3- d B bandwidth of 10 kHz . (b) Inevitably. the inducto r fou nd in part (a) again has a Q of 100. Pirst find the equivalen t series resista nce of the ind uctor. then redesign the m arc hin g ne two rk to accommodate this additiona l loss mechani sm. Are a ll the values po siti ve rea l?

PROBLEM 9 Yet another ma tc hin g networ k is the lapped rea cta nce netwo rk . It shares with the x -netwo rk the ability In set ind e pend ently the tran sformation ratio , Q, and center freq uency. Repeat Prob lem 8(a), now usin g a tapped capaci to r ma rc hing network. To keep the des ig n procedure as unco mpl icated as possib le. yo u may assume large Q fro m the ou tse t a nd use the a ppro xi ma te formu las thai a pply in thai regime. PROBLEM 10 As stated in Problem 4, the impedan ce of a dipo le antenna thai is much shorter than a wave length may be modeled as a voltage -d rive n series RC net\lori:. where the po rtio n of R th ai is due to rad iatio n is give n appro xima tely by

I

NUl kilo-ohms. Similarly. megohms. not mega..ohms.

110

CHA pt ER 4 PASSIVE Rl C N ETWO RKS

( P4.2l where I is the length of the antenna and Ais the wavelength . Suppose that the antenna ca pacitance is 15 pF, and assume a 1/ ).. ratio o f 0. 1. (a) What value of induc tance is req uired 10 reso nate out the antenna capacitance at a frequency of 30 MHl '! What inductor Q is required if its effec tive series resistance is not to exceed I0% of the antenna resistance'! (b) Now suppose you want 10 connect th is antenna to a receiver whose input impedance is 50 n . Design the appropriate Lmatch . You may unrealistically assume infinite inductor Q. What is the bandwi dth given these assumption s? PROBLEM 11 Design a a-network to match a source impedance of 5 - j30 n 10 a 50-0 resistive load . If the Q of the net work is 100. what is the current in each element of the matching. network when I \V is delivered to the load ? PROBLEM 12 Consider the network sho wn in Figure 4 .14 . Using the method of series-parallel transformations. simplify the network to find the impedance al 11 M) MHz when C = I pF. L = 10 nil . = 15 Q. and HI' = I kQ .

«,

c

Rp

FIGURE 4 .14 . Lon y RLC

networl.: .

PROBLEM 13 In the circuit shown in Figure 4 . 15. calculate the Q and sketch the magni tude and phase respo nse for R = I, 10. and J(XI n. AIM) sketch the step response for each of these three ca ses.

R

=c

"mIT

FIGURE 4 .15. RLC I'\etwork .

PROBLEM 14 The electrical mod el for a typical qu artz cry sta l is a series R L C circuit in parallel with a shunt capaci tor. Co. Although Co docs model a physical capac Hance. the R L C co mpo nents model electro mec" alliCfl/ properties and therefore mOlYtake on values not practica lly reali zable with ord inary inductors and capaci tors.

PROBLEM SE T

A typical IO-MHz crys tal has a se ries resistance of 30 Coof 2pF.

111

n, a Q of 100.000, and a

(a) Neglecting Co, what values for the se ries capacitance and inductance
PRO ILEM 15 (a) Using the energy defi nition for Q. show tha t an alternate form ula for Q for one-port is: Q = ImIZ )

Re[Z]'

OJ

( N.)

\\0here

Z is the imped ance o f the net work . (b) Derive the analogous form ula in terms of adm ittances. PR08LEM 16 Th is problem considers the issue of match ing to a nonlinear load . As seen in numerou s examples ( particularly in the histori cal chapters) but mit disru..sed in much detail. a very co mmon AM demodulator is the envelope de /ector. This circuit is evidently nonlinear beca use o f the presen ce o f the d iod e; M'C Figure 4.16.

FIGURE 4 .16, Envelope detector.

The problem is to determi ne the effective resistance presented by the envelope detector fi.e.cthe circuitry to the rig ht o ft he indicated bou nda ry ), This prob lem is so mewhat diffi cult 10 solve exactly, but useful a pproximations ca n be obtained by invoking several simplifying assumption s. Becau se different assumptions lead to slightly different answers. please usc only the assumptions given. eve n tho ugh some of them may seem dubious. You ' ll j ust have to trust us that, by some rnirucle of num erolo gy, the errors introduced ca ncel sO that the final answe r is rou ghly correc t.

11 2

CHAP TE R 4

PASSI V E RiC N ETW O RKS

(a) Let the input voltage tli n be A CO S wI. and initiall y assu me that Rs is zero. Assume further that the voltage dropped across the d iode is negligibly small whenever the diode is forward-biased. Note that this assumption is contrary 10 the: "0 .6- V·· rule that we commonly apply in large-signal analyses. In effect. we are assum ing thai R is sufficiently large that the current through the d iode is quite small. In that CU!'iC, little voltage is dropped across the diode. Finally. as sume that the He prod uct is much larger than the period o f the sinusoidal drive. With these assump tions. sketch approximately the o utput voltage lIow and the diode current waveforms in steady stale. At this po int , it is nol necessary 10 identify features quantitatively; approxi mate shapes suffice. Figure 4.17 shows the input waveform in hashed form for convenience .

A

\"

!

,/

\ '"

fi GURE 4 .17. Approximate ovlput voltage a nd d iode current For ~..Iape

de<.do<.

(b) Derive an expressio n for when the diode turns on (with our refere nce convention, the peak input occ urs at t = 0 ). To simplify the derivation. feel free to assume that e- ·I :::::l 1 - .r for small .r , and also that cos r ::::: I - x 2/ 2 for small x . It's sufficient to tell us when. in the j ir.\" !,l'ri nd after ' = 0 , the diode turns on. It may also help to remember that ClI S .r = cos t - x ) . (c) Assume, somewhat erroneously, that the diode turn s off the moment the input reaches its pea k. As measured entirely by the capacitor vo ltage, what is the total energy (in joules) supplied by the input so urce per cycle ? (d) Invoking energy equivalence is one way to define an effective resistance for a nonlinear load. What we mean is this: A sinusoidal voltage source d irectly loaded by a resistance RI. delivers a certain amount of energy 10 that resistive load per cycle . What eq uivalent value of RI. con nected directly to the source would consume the same amount o f energy per cycle as ca lculated in part (c)1 Express your answe r in terms of R .

PROBLEM SET

11 3

Thus. this is the value one should use in designing match ing network s III dri ve an envelope detector. if maximi zing power gain is impo rtant. As a la..t trivia lI11tC. this consideration is of greatest significance when designing "ze ro- power" receivers (e.g.•crystal radios), since the only source of energy is the inco ming wave itself. and maximizing power transfer is therefore critically important.

CHAPTER FIVE

DISTRIBUTED SYSTEMS

5.1 INTROD UCTIO N There are two impo rtant regimes of operating frequency. distin gu ished by whether o ne may treat circuit elements as " lumped" or d istribu ted. The fuzzy boundary be(ween these two reg imes co ncerns the rat io of the physical dimensions of the circuit rel ative 10 the shortest wavelength of interest. AI high enoug h freq uencie s. the size (If the circuit clements becomes comparable 10 the wavelen gths. and one cannot employ wit h impunity intuiti on derived fromlumped -circuit theory. Wires must then be treated as the tran smi ssion lines that they truly arc , Kirch hoff' s " laws" no lon ger hold ge nerally. and identifi cat ion o f R , L . and C ceases to he ob vious (or even possible). Thanks 10 the s mall dim ensions involved in (Cs. tho ugh. it turns out that we can largely ignore tra nsm issio n-line effects well into the g igahertz range, at least on. chip. So . in this text, we will foc us primarily on lum ped-param et er descriptions of our circuits. For the sake of completeness, however , we should be a little less cava. lier (we'l l still be cavalie r. j ust less so) and spend some time talk ing a bout how and where OIlC draws a bo undary betw een lum ped and distributed dom ains . In order to do this proper ly, we need to revisit (briefly) Maxwell's equations.

MA XW El l A N D KI RCHHO ff Many students (and many practi cin g engineers. unfortunately) forget that Kirch hoff's vo ltage and curre nt " laws" are approximations that hol d only in the lumped regime (w hic h we have yet to define) , They are deri vable from Maxwell's eq uatio ns if we assume quasistatic behavior, there by eliminating thc coupling terms that give rise to the wave equation. To understand wha t all this means, let's rev iew Maxwell's eq uations (for free space) in differen tial fonn :

'"

5 . 1 INTRO OUCTIO N

11 5

= O.

I I)

'V . EoE = p .

(2)

Q ' ILo ll

8E

V x II = 1+£0-

al

.

an

v x E = - J.l. o- . 01

(3) (4)

The fi rst equat ion says that there is no net magnetic charge {i.e.• there are no magnetic monopoles), If there were net magnetic charge. it wou ld ca use divergence in lhe magnetic fie ld. We wo n't be using that eq ua tio n at a ll. The second eq uation states that both "ordinary" curre nt and the lime rate of change of the e lectric field produce the same effect on the magnetic field . The term that invol ves the deri vative of the electric field is the famou s displacemen! (capacitive) current term that Maxwell pulled out of thin air 10 produ ce the wave eq uation. Finally. the fourth equation ( Faraday 's law) says that a changing magneti c field causes curl in the elec tric field . Wave behavior arises fundamentally beca use of the coupling term s in the last two equations: A change in 10: causes a change in II . which causes a cha nge in E. and so on. lf we were to set either 11 (J or £ 0 10 zerothe coupling term s would disappear and no wave equation would result; ci rcuit analysis could then proceed tin a quasistatic (oreven static) basis. As a specific example, selling li n to zero makes the electric field curl-free. allowing E to he expressed as the gradient of a pot ent ial (within a minus sign here or there) . II then follows identicall y thai the line integral of the E-field (wh ich is the voltage) around any closed path is zero:

v=

J

E · '" = J I- V¢) . '" = o.

(5 )

This is merely the ficld -thcureticu l express ion of Kirch hoff' s voltage law ( KYL). To derive KCL (K irchhoff's current law), we proceed in the same manner hut now set Eo equal to ze ro. T hen . the curl of II depends on ly on the current density J, attowing us to write:

V · J = V · ( V x II ) = O.

(6)

That is, the divergence of J is ident ica lly zero. No divergence means no net current buildup (or loss) al a nod e.

116

CHAPTER 5 DISTR IBUTED SYSTEMS

Ofco urse. neithe r Ito no r Eo is actu all y zero. To show that the forego ing is not hopele ssly irre levant as a co nseq ue nce , recall that the speed o f ligh t ca n be ex pressed as'

c = I/ J /i OEO.

(11

Setti ng It o o r EO to zero is there fo re equiva len t to sc ni ng the speed o f lig ht to infinity. Hen ce , KC L a nd KVL are the re sult of assuming in finitel y fast propagati on ; we eXJX'Ct the m to hol d reaso na bly we ll as lon g as the physical d ime nsions of the circui t c le me nts are sma ll compared with a wavelength so tha t the finiten ess of the spee d of ligh t is not noticeabl e:

1 « A.

(8)

whe re I is the le ngth of a circuit element and A is the sho rte st wavelength of interest . To develop a gu t fee l for what thi s constra int me ans numerically. consider an Ie subc ircuit wh ose lon gest d imen sio n is I mm . If we arbitrarily say that " much less thanv mcans " a factor of at lea st 10 sma lle r than:' the n such a su bcircuit can be treated as lum ped if the hig hest -frequency signa l o n the chip ha s a wavelen gth g reater than roug hly I em. In free space. thi s wavelength corresponds to a freque nc y of about 30 GHz. O n chip. the freq ue ncy limit decrea ses a bit because the re lati ve d ielectric constants for s ilico n an d s ilicon dio xid e are we ll a bove uni ty ( 11.7 an d 3.9. resp.). bu t it s ho uld he clear that a full tran smi ssion -lin e treatme nt of the on-chip desig n and an a lysis problem is ge ne ra lly unnece ssa ry unt il fa irly hig h freque ncies are reached. In summary. the boundary be tween lumped c irc uit theo ry (w here KVL and KCL hol d, a nd w he re o ne ca n identify R. L , a nd C) a nd d istributed sys te ms (w here KVLlKC L don 't hold , a nd whe re R, L . a nd C ca n' t al way s be loc a lized) de pends o n the size oj the circuit element relative to the shortest wavelength of inte rest, If the ci rcuit cl em e nt (and interco n nec t is ce rtai nly a circuit e leme nt in thi s context) is very short co mpared wi th a wave le ngth. we ca n use trad itio na l lumped co nce pts and inc ur lillie e rro r. If not , then use of lumped ide as is inap propriate. Because the dime nsio ns o f CM OS ci rcuit blocks arc typicall y much sho rte r than a wavel en gth, we sha ll glibly ig no re transmission -lin e effec ts for most of thi s textbook .

5 .2

LINK BETWEEN LUMPED AND DISTRIBUTED REGIMES

We now turn to th e problem o f ex te nd ing into the di stributed regime the design intu itio n develo ped in thc lumped regime. The motivati o n is more tha n merely

I App lying the "d uck le ~l" version of Occam 's raror (" i f il walks like a duck and quacks like a duel , it Iflusl he a d uck" ), Maxwell showed thai lighl and elecrromag ncnc waver. arc the same thing. After a ll, if it travels ai lhe' speed of Iighl and reflects like light , il mu ~t he light. MO!>t wou ld agree tbattbc deri vatnm o f Mnwe ll"r.eq uatio ns represe nts the cnl\lm ing inrcllec tuat achie vement of~ 19th century.

5 .3

DRIVING -PO INT IMPEDA NCE O F ITER ATE D STRUCTURES

117

pedagogical for . as we shall see. ex tremely valuable de-ign in.. . ighrs emerge fro m lhis exerci se. An important e xa mple is thai delay, instead o f gain. ma y be traded for

bandwidth . Interconnec t is an exa mple (If a system that may he trea ted success fully at lower frequencies as. say. a simple He line. With thai type of mind sct . red uction of Re " parasitics" in order (0 inc rease b andwidth becomes a major preoccupa tio n oft he ci rcuit and syste m desig ner ( pa rticularly of the Ie designe r). Unfo rtunately. red uctio n of parasitics be low so me min imu m a mo unt is pract ically im po ssible. Intuiti on fro m lumped -circuit design would therefore (mis)lead us into think ing thai the bandwidth is limited by these irreducible pa rasitics. Fortunately, a proper treatment of int crcon ncct as a transmission line. rather than as a finite lumped He net work , reve als othe rwise. We find that we ma y still convey sig nals with exceed ing ly large bandwi dth a.~ lung
5 .3

DRIVING ·POINT IMPEDANCE OF ITERATED STRUCTURES

We begin by study ing the dri ving-po int imped ance of uniform. iterated struc tures. It's important to note thai certain nonu nifo rm structures Ie .g.• expo ne ntially tapered transmission lines 2 ) have exceedi ngly usefu l propert ies. hut we'Illimi t tbe pre se nt discussion to a co nsidera tio n of uniform structures on ly, Specifically. co nside r the infinite ladd er net work show n in Figur e 5, 1, Even tho ugh resistor symbols arc used here, they represent arbit rar y impedan ces. To find the dri ving-point impedance of this network without summi ng an infinite series. note that the impedance to the ri!!:ht of node C is the sallie as that to the right of B, and the same as that to the right of A .3 Th erefore. we may write

2 For tho!.c of you who are cunoo s, the e,;po nentially tapered line allows uoe to achieve a broadband impa:l.:mce match instead of the narnl\\ toand impeda nce match that it quart er-wave transformer pRl\'ides. The transformati on ratio can he controlled by choic e of taper constants. I This is an earrerrely useful technique for analy zing such structures, hut a surpris ingly large per~ge of engineers nave never heard or it , Of perhaps don 'l remem ber it In any event , it certain ly yvn a IIcmendnu s amo unt of labor over a ITNlfe straightfoward appl'Oll('h , which wou ld require SlImming \'arious infinue series.

CHAPTER. 5 DISTRIBUTED SYS TEMS

l IB

7" _

.

: _ Z-l E A_Y_ZF _"L:_ E ' _CLY _ _Z_:::::::::::::::::::: FIGURE 5 .1.

Lodder network.

Z," = Z + [(I / y) [I Z,"1. which expands to

Zin = Z

+

Z inl Y IIY + Z;n

Solving for Zin yields:

Z = Z ± J Z' ,n

+ 4(ZI Y)

2

= Z 2

[I±

/ 1+ 4 ] . ZY

In the speci al case where Z = IIY = R ,

. ('+J5)

Z m=

2

R :::;:: 1.61 8R.

This ratio of Z;n to R is known as the golden ratio (or golden .\'('CI ;OIl ). and ,,00·....5 up in contexts as diverse as the aesthet ics of Greek geometers. Renaissance an and architecture. and solutions to several interesting (but largely useless) network theory problems.

IDEA L TR AN SMI SSION LIN E A S IN FINI TE LADD ER NETW O RK

Let's now consider the more general case of the input impedance in the limit where IZ YI « 1, and where we continue to disallow negative values of Zin. In that case. we can simplify the result to

We see that if Z I Y happe ns to be frequency-independent , then the input impedance will also be freq uency-i ndependent." One impo rtant example of a network of this type is the mode l for an ideal transmission line. In the case of a lossless line. Z = sL 4 ladder networks with lh i_~ pmpe rty are called "conslanl-.I; " lines . ...fnce Z / Y

L

=

.1; 2 for a con slanlt.

5 .4 TRANSM ISSION li NES IN MORE DETAIL

119

Ld:

0000

FIGURE 5. 2 . Lumped RIC model of tran!>mi, !>ion-line wgmenl .

in~ nile!>ima t

and Y = $C. where Land C represent differenti al ( in the mathematical sense ) ci rcuir elements. The input impedance (ca lled the characteristic imped ance Zo) fur an ideal. lossless infinite trans mission line i ~ therefore ( 14 )

Because Y. the admitta nce of an infinitesimal capacitance. approaches zero as the length of the d ifferentia l element approaches zero. while the reactance o f the differential inductance elem ent approac hes zero at the same time. the ratio l /YZ approac hes infinity and so sat isfies the inequ ality necessary to validate our deri vation. The result - jhat we arc left with a purel y rea l input im pedance for an infinitely lon g tran smission line- should be a familiar one. but perhaps this part icular path to it might not be. An often-asked qu estion concerns the fare of the energy we launch dow n a transmission line. If the impedance is pu rely rea l. then the line sho uld beha ve as a resistor and should dissipate energy like a resistor . But the line is co mposed of purely reactive (and hence dis sipationless) elements. so there would appear to be a parado x. The resolutiun is that the energy doesn' t end up as heal if the line is tru ly infinite. The energy ju st keeps traveling down the line forever. and so is lost to the ex ternal world just as if it had heated up a resi stor and its environs; the line acts like a black hole for energy.

S .4 TRANSMI SSION LINE S IN MORE DETAIL The previous section ex amined the impedance behavior of a lossless infinite line. We now extend our de rivation of (he characteristic impedance Zu to accommodat e toss. We also introd uce an addit ion al descriptive parameter. the propaga tion cOlIJ tw lI y.

S .4 .1 LU MPED MO Del FO R l O SSY TRANSMISSION LI NE To derive the relevant parameters o f a lossy line. conside r an infinites imally short pieceof line. of length d z, as shown in Figu re 5.2. Here. the elements L . R. C . and G are all quantities per unit leng,h and simply represent a specific exa mple of the more general case co nside red earlier.

CHAPTER 5 DISTR IBUTED SYSTE MS

120

Th e inductance accoun ts for the energy stored in the magn etic field around the line. while the series resistance accou nts for the inevitable energy loss (such as due 10 xkin effec t) thut all ordinary conductors ex hibit. The shunt capacitance models the encrgy stored in the electric field surrounding the line. and the shunt co nductance acc ounts for the loss due 10 mechanisms such as ord inar y ohmic leakage as well as loss in the line's dielectri c material.

5. 4 .2 CHA RACTERISTIC IM PEDANCE O F A LO S5Y TRAN SMI SSION LINE To compute the impedance of a lossy line, we follow a method exac tly analogo us 10 that in Sec tion 5.3 :

Z o =Zd, +((I / Yd z) IIZ ol =Zdz +

Z"

I

+ ( Y d ;:)ZIl

(15)

We w ill co nside r the limiting behavio r o f Ihis ex pres sion as d :. ap proac hes zero. so we may U ~ the first-ord er binomial ex pans ion o f I / ( I + .t ):

l o=

Zo z d :. + ,-,,...,,:;'-;-:-:;;1 + ( Y d z )Z o

~ Z dz

+ Zol l -

( Y d :,)Z ol =

Zu + tI:.( Z - YZ 5 ).

( 16)

Can ce lling Z n from both sides , we see that the final term in paren theses must equal zero. The characte ristic impedance is thus

· = Vrrz = z,

R + jl.JJ /. G + j wC

( 17)

If the resistive ter ms arc negligib le (or if Re ju st happen s to equal G L ), the equation for Zo collapses to the resu lt we de rived ea rlier:

z, =

J I.;G.

( 18)

Becaus e the impedance approaches ../LIC at sufficie ntly high freque ncy, independent of R ur G . it is sometimes known as the transient or 1'1I1.w- impedance.

5 .4 .3 THE PROPAGATIO N CO N STAN T In add ition to the characteristic impedance. one other important descriptive parameter is the propagation constant, usuall y denoted by y . Whereas the cha racteris tic impcdancc tell s us the ratio of voltage to curre nt at anyone point nn an infinitely long line, the pro paga tion constant enab les us to say so mething about the ratio of vo ltages

121

5 .4 TRA N SM ISSIO N LINES IN MORE DETAi l

(orcurrents) between any two points on such a line. That i.., y quantities the line's attenuation properties. Consider the voltages at the two ports of a given subsection. The ratio of these voltages is readily computed from the ordinary voltage divider relationship:

v -

V

/H I -

"

I

I

z"

II (1 / I' d , ) Z d :. + IZo II (I /Y d .;:>! .

e19 )

Thus,

V... 1 V.

Zd:

ZO II (I / Yd z) + [Zo [I (1 / 1'd ,)1

Zu Z oZY(J ;:)2 + Z u + Z eI: '

(20)

Because we will use this expression in the limit of very small d z. we may discard the term proportional to (,/:) 2 and again use the binomial expansion of 1/ ( I + xl 10 preserve only the first-order dependence on d : (remembe r. we' re engineers - the wbole universe is first-order to us lj. This yield.. Vlttl V.

-

::I:::

Zo = 1 d "" 1 1 + (Z/Z o) z Zo + Zd:

.!... z, d : =

1-

.jzr d z.

(2 1)

Despueour glibness. the net em u in these approximations actually docs co nverge to zero in the limit of zero dt , Let us rewrite the previous equation as a difference equation:

=,

V... 1 = V,,(I - v Z Y, z )

==:::}

V.. , - V.

d,

=

= - v Z Y V" .

(22)

Inthe limit of zero dz . the difference equation becomes a differential eq uation:
=

-~- " z r v.

t1z

( 23)

The solution to this first-order differential equation should he familiar: (2 4 )

= 0) times an exponential factor. The exponent is conventionally written as - yz so thut. atlast . (25) y = .JZr = ! eR + j", L )( G + j",C ). That is, the voltage at any position : is simply the voltage \;(1 Ohio: voltage at z

To develop a better feel for the significance of the propagation constant, tirst note lhat y will becomplex in general. Hence. we may express y explicitly as the sum of real and imaginary parts: y

=!

eR

+ j", L)(G + j ",C) = " + jfi .

e261

CHAPTE Il. 5 DISTRIBU TED SYSTEMS

122

Th en (27)

The first exponen tial term becomes smaller as distance increases: it represe nts the pure une nuauon of the line. T he seco nd ex po nential factor has a unit mag nitude and co ntributes on ly phase.

5 .4. 4

RELA TIO NS HIP O f y TO LINE PARA M ETER S

To relate the constants a and f3 explicitly to tran smi ssion -line parameters. we make usc of a coup le of identities. First. recall thai we may ex pre ss a co mplex nu mber in both ex pone ntial ( po lar) and rectangular form as follow s: M e }· = M cos ¢

+ j M sin ¢ .

Here. M is the magn itude of the complex number and ¢ is its ph ase. Th e po lar fonn allows us to co mpute the squ are roo t of a co mplex number with case (th ank s 10 Euler):

Th e IU!'i.1 facroid we need 10 recall fro m undergrad ua te math is a pair of half-angle iden titie s:

cos(4'/2) =

J~ (1 + cos ¢ )

(30)

sin(q,/ 2 ) =

!
(31)

and

Now, y is the square rout of a complex nu mber: y =

hi' =

= / ( RG -

+ j w L)( G + jwC ) w' LC) + j w ( LG + RC ).

/ (R

Making me of o ur identit ies and turnin g the crank a few revo lutions, we obta in:

• = Jj[ /w 4( LC) l

+ w' [(LG) ' + ( RC )l l + ( RG)l + ( RG

_ "" LC )]

and

p=

Jj[ / w' ( LC )' + w' f(LG) ' + ( RC)' I + ( RG )l - ( RG - w' LC )J.

These last two expression s may a ppear cumbe rsome, but that's only because they arc. We may s implify them considera bly if the prod uct RG is small co mpared with the othe r terms. In such a case, the attenuation con sta nt may be written as

5 ,4 TR A N SMISSION LINES IN MORE DETA il

u '"

i ll j ",'I L C )' + ", ' IILC) ' + IRC )')

"" L C].

123

(35)

which , after a certain amount of blood letting. further simplifi es In (36 )

This. in turn. may be further appro ximated by

1371 Thus, the attenuation per length will he small as long as the resi stan ce per length is small compared with Zoo and if the cond uctance per length is small compared with Yo. Turning our atten tion now In the eq uation for fi. we have (3 8)

In the limit of zero loss (both G and R = 0 ), these expressions simplify In tr

= Re lY) = 0

(39)

and

fI = lOl ly ) = ", .JLG.

(4 0)

Hence. a losslcss line doesn't attenuate (no big surprise). Since the uttcnuution is the same (zero) at all frequen c ies. a lossless line has no bandw idth limit. In addilion. the propagation conslant has an imaginary part that is exactly propo rtio nal to frequency. Since the delay of a system is simply (minus) the derivative of phase with frequency. the delay of a losslc ss line is a constant. independent of frequency: Tlk 1ay

iI = -~ ¢ ( w) = ow

il - (- fJz) = J LC:.. ilw

(4 1)

We loan now appreciate the remarkable property of distributed systems alluded 10 in the introduction: The capacita nce and inductance do not dirt'ctly cuus c (/ band~' idf" redllctioll. They result only in a propagation de lay. If we were to increase the inductance or capacitance per unit length, the delay would increase bUI bandwidth iilkally) would not change, This behavior is quite different from what one observes with low-order lumped networks. Also in stark contrast with low-order lumped networks. a transmission line may erhibit aj rt'qlletlcY' ;lIdependt'1U delay. as seen here. This property is extremely de\iralJle. for il implies that all Fourier components of a signal will be delayed by precisely the same amount of time; pulse shapes will be preserved, We have j ust seen

12'

CHAPTE R S DISTRIB UTED SYSTEMS

that a losslcss line has this pro perty of zero d ispersion. Since all real lines exhibit non zero loss. thou gh . must we accept dispe rsion (nonuniform del ays) in practice? Fortunate ly. as Heavisidc! first po inted out. the a nswe r is No . If we exerci se some contro l over the line constants. we can still obtain a uniform !!TOUP delay ('l 'n l with a (OJ,f y line ( OI l leas in princi ple). In parti c ular. Hea vis ide d iscovered that choosi ng Re equ al 10 G L (or. equivalently, c hoosing lite L JR time consta nt of the se ries impeda nce Z equal to the C/ O tim e con star n of the shunt ad mitta nce Y ) leads to a consta nt group del ay. Ther e is non zero attenu ation . of course (can'! gel rid of that. unfortunately ). but lite consta nt group delay mean s tha t pu lses onl y gel smalle r as they travel down the line: they don't smear out (disperse). Showin g thut Hea viside was correc t isn' t too hard. Selling RC a nd G L equal in our exact e xpre.. .sio ns for a and fJ yields

a = Rel y l = j RG

(42)

a nd

fJ =

Im ly ] = wjLe .

(43)

Note that the e xpression for fJ i.. . the sa me as that for a lo........less line, a nd therefore also leads to the slime freq uency-independent del ay. A lthough the atte nua tion is no longe r zero, it co ntinues to be frcqu en cy-ind e pendent: the handwidth is still infini te as long as we c hoo se 1.1R = CI G , Furtherm ore, the cha rac ter istic impedance becomes ex actly equal to .JLI C at all frequencies. rath er than approaching thi s value asymptotically at high frequ enci es, Sett ing LG = RC is best accomplished by increasing eithe r L or C. rather than by increasin g R or G , becau se thc latte r strate gy incr eases the a ttenuat io n ( presu mably lin und esirable effec t). M ichael Pupin of Colu mb ia Uni versity, fol lowin g throu gh on the imp lica tio ns of Hea viside's wo rk. suggested the add ition o f lumped inductances periodically along tele phone tran sm i.. .sion lines tu red uce signal dispersion . Such "Pupin coils" permitted significa ntly improved tele phon y in the 19 20 s and 1930 s."

5 .5 BEHAVIOR OF FINITE -LENGTH TRANSMISSION LINES Now that we 've ded uced a nu mbe r of im po rta nt pro perties of tra nsm issio n lines or infinite length . it's lime to co nsider what happen s when we term ina te finite-length lines in arbitrary impedances. S By the way, he was the first In usc vector calc ulus to cas t Maxwell's equations in mod ern form, and ('I

the o ne who introduced the U M' o f Laplac e trans forms ro M>1\l C ctrcun prtlh lem ~ , Ala... lhe use of lumped ind uctances meod uces a hamlwidlh limitation lhal true, d i..tributed lines do not have, Sinc e haoollo' idth 300 channel capacity are c1o..cly related, all of the Pupin C'o ih (which had been installed at great e,, ~n'>e ) even tually had to he remo ved (at great ('"pense ) to permit an increase in the number of ca ll.. carried by each line .

S. S BE HAV IO RO F FIN ITE-l ENG TH TRA N SMI SSIO N LIN ES

125

5 .5 . 1 TRA N SMISSIO N LIN E W ITH M ATCHED TE RMIN ATION The driving-po int im peda nce of a n in finitely long line is s imply ZooS uppose we cut the line somewhere. d isc ard the infinitely lo ng rema inder. a nd re place it with a single lumped impeda nce of value ZooTh e driving-poi nt im pedance rnu...I re main Z o; there's no way for the measureme nt apparatus to distin gui sh the lum ped impeda nce from the line it re places. Hence. a signa l applied to the line simply n a ve ls do wn the fi nite segment of line. event ually ge ts 10 the res isto r. he ats it up. a nd contributes 10 global warming.

5 .5 .2 TRAN SMISSION LINE WIT H A RBITRARY TERMINATION In general, a transmission line will nOI be terminated in preci se ly u s c harac teristic impedance. A signal traveli ng do wn the line mainta ins a rat io o f voltage to c urre nt that is equal (o f co urse) to Z o until it e nco unters the loa d imped a nce . The tcrminanon impedance imposes its own particula r ra tio of m ilage to c urre nt. ho we ver. and the only way to reconcile the con flict is for so me of the sig na l tu reflect bad toward thesource. To distinguish forward ( incide nt) qu antities fro m the reflected ones, we will usc the subscripts i a nd r , respecti ve ly, If £j a nd I; are the inc ide nt vo ltage and c urre nt. then if !'> clear that

z., =

E I,

---.: .

(44 )

At the load e nd of things. the m ismatch in impedances give s rise to a reflected voltage and curre nt, We still have a line ar sys tem. so the to tal vo ltage at a ny po int on the system is the superpositio n (If the inc ide nt and reflected voltages. Sim ilarly. the net current is a lso the superpos itio n of th e incide nt a nd reflected c urren ts. Beca use the current com ponents tra vel in o pposite dir ec tio ns. the superpos itio n her e res ults in a subtraction. T hus. we have

Z _ :: £ '--'+'--£;:.c' f. - / ., I,

(4 5 )

We may rewrite Ihis last eq uat ion to show an ex plicit proport ion alit y ttl Zu as follows:

_ e. w e, _

ZL -

I ; - I,

£,[1+£,1£,] _ [I +£,1£,]

- -

I,

I

I,/ Ii

- Zo

I

, . 1, / 1,

The ratio of reflec ted to inc ide nt qu an tities at the load e nd of the line is called and will generally be com plex . Using Fz. . the expression for ZL becomes:

( 46 )

rL .

CHAPTER S DISTR IBU TED SYSTE MS

126

Z =Z [1 +tO , /E,] = Z [1 +fL] L

(I

I _ 1, /1;

0

rL

I -

•

(47)

So lvi ng for r r yields

fL

ZL - Zo

-

(481

--;'--,---,;"

- Z,

+ Zo

If the load impedance eq uals the charcc terisuc impedance of the line. then the reflection coefficient will be zero. If a line is terminated in ei ther a short or an open. then the reflection coe fficient will have a magn itude of unity: this value is the m axim um magni tude it ca n have (for a pure ly passive system such as this on e. anyway). We ma y ge nera lize the conc e pt of the re flec tion coe ffic ie nt so that it is the ratio ot the reflec ted and inci de nt qu an tities a t a ny arbitra ry po int a long the line:

(491 Here we follow the convention of defining z = 0 at the load end of the line and loca ting the dri vin g so urce at z = - I. With thi s co nvention. the voltage a nd c urrent a an y po int z alo ng the line may be ex pressed as: Vt z) = V;e- YZ + V, e Yz • I ( z)

=

I, e - Y ~ -

I r(' Y<' .

As a lways. the imped ance at an y po int z is sim ply the rat io of voltage to c urre nt:

Substi tut ing for

rt. and doin g a whole heck of a lot of c ru nching yields ZI,)

Z"

+ e Yl ) + (t, - YZ _

Zt. (e - Y;:

Z,

= - : ; " - - - - -Z

---.!:. (e - Y ~ - (, YZ,

+k

-Y;:

e Y<' )

-

+ (' Y;:)

Z" Writi ng this ex pression in a more com pact for m. we have

Z (z)

ZL - - tanh y ' Zo ..

z;-= 1 -

ZL - - ta nh y z

Z" In the special case wh er e the attenuatio n is negligibl e (as is co mmo nly assumed to pe rmit tractable analysis ), a co nside rable s implificatio n results:

5.7

Z (z)

-

Z"

ARTIFICIAL LIN ES

ZL - j l:.mfJ:. Z L cos f3: Z" = -=---;;--= ZL Z o cos fi :. 1 - j - ta n fJz

127

-

jZ n sin fJz j Z L s in fi z

(55)

Z"

Here, z is the ac tua l c oordinate va lue a nd will a lways be zero o r negat ive. As a final co mme nt , note that thi... ex press ion is pe riod ic. Such be havior is strictly

observed only in losslcss line s, o f course, but prac tical line s w ill be ha ve sim ila rly as long a.. the loss is neg ligible. Pe riod icit y im pl ie s that o ne need cons ide r the impedance behavio r on ly ove r so me finite sec tio n (specifica lly. a htllf-wave le ngt h) of line. This observation is exploited in (he construc tion of the Smi th chan , a brie f stud y o f which is taken up in Chapter 6.

5 .6 SUMMAR YOF TRANSMIS SION ·lIN E EQUATI O NS we've seen that both Ihe character istic impedance a nd the propagat ion constant are simple function s of the pe r-len gth serie s impedance a nd sh unt admittan ce :

z,, = y =

If:

../ZY =

=

j (R

R + j w l. G + j wC '

+ j w L)(G + j wC ).

(56)

(5 7)

Using these parameters - in conjunctio n with the de finition of retlcc tion coefficic nt -, allows us to deve lop an eq uation for the dri vin g-point im pedan ce of a lossy line terminated in an arbitrary impe dan ce . In the c a se of a lo ssle ss (o r negli gibly It)ssy ) line. theexpressio n for im pedan ce la ke s on a reaso nably sim ple a nd pe riod ic for m, set ting the stage for d iscu ssio n o f the Sm ith c hart .

5 . 7 ARTIFI CIAL LIN ES We've just see n thai an infinite ladder net work of infinitesimally sma ll ind uctors a nd capacitors has a pure ly real input impeda nce o ver an infinite ba ndw idth . Altho ug h structures thut are infi nite ly lon g arc so mew ha t incon veni en t to reulizc. we can a lways terminate a finite len gth of line in its c harac te ristic im pedan ce. Ene rgy, being relatively easy to fool. cannot dist ing ui sh be twee n rea l tra nsmissio n line a nd a rcsistor equal to the cburactcris tlc im peda nce , so the dr iving-poi nt impedan ce of the properly terminated finite line re main s the sa me as that of the infinite line , a nd still over an intinile ba ndwidth . Theft are instances w he n we migh t wish to a pproximate :I continuous rmn sm ission line by a finite lumped networ k . M ot ivati on s fo r doing so may inclu de con venience of reahzan on o r g re ate r contro l over line constants . Howe ver. use o f a finite lumped approx imation g ua rantee s that the c harac te ristics of such an artific ial

CHAPTER S DISTRI BUTE D SYSTE MS

128

FIGURE 5.3 . lumped delay line. line ca nnot match those of an idea l line ove r an infinite h:md width .7 The design of c irc uits thai e mploy lum ped line s mu st ta ke this ba nd width Iiminuion into accou nt. On e im portant UM: of a rtifici allines i.. in the synthe sis of de lay line s; see Figure 5.3. Here . we use L C Locctions 10 synthes ize o ur line. As in the continuo us ca se, the driving-point impe dance is j ust (58)

wh ile the del ay pe r sec tio n is

r.; =

J LC.

(59)

The value (If a lumped delay line is that o ne may o btain large del ays without having to usc, say. a kilomet er of coax ia l c able.

5 .7.1 CUTOf f fREQU ENCY O f LUMPED LINES Unlike the distributed line. the lumped line presents a real. con stant impedance only over a finite ba ndwid th . Even tua lly, the input impedance becomes pure ly reacti ve! indi cating that rea l po wer c an he de livered nei ther to the line nor to an y load connected til the o the r e nd of the line. Th e freq uenc y at which (hi!'; occurs is known as the line 's cutoff freq uen cy. an d is rea d ily found by using the form ula for the input im ped ance of an infinite (but lumped ) L C line" repri se d he re from Section 5.3 for co nve nie nce :

Here . let Y

Z,, =!'2: [I±jl+4]. ZY

160)

= jwC and Z = j w L. Then the input impedance is (611

7 One ca~y ..... ay to see thi-, i~ III recognize that a true tran ~mi s.~ton line. being a delay element. pro". i~ unbou nded phase shift as the freq uency appruac hcs infin ity. A IUIl1 pt."lJ line can pro vide on ly . finite phase shift because of the tini te num ber of energy stora ge elements and hence a finite number

of poles. 8 From inspec tjr m(If the networ k. it should becle ar th"t dnvmg- pomt im pedance eve ntually colla pses tn that IIf the inp ut induc tor, since the capacitors ac t ultimately like short s"

5.7

ARTIFICIAL LI NES

129

FIGURE 5 .4 . One cho ice for termina ting lumped lines.

FIGUIU 5.S. Alternative choice for terminating lumped line s.

At sufficiently low freq uencies. the term under the- radi cal has a ncr nega tive value. The resulting imaginary te rm. when mu ltiplied by the jw L j2 fact or. provides the real component of the input impedance. As the frequency increases. howeve r. the magnitu de of the term und er the radi cal sign eventually becomes zero. At and abo ve this frequency, the input impe da nce is purely imaginary. and no pow er can be delivered to the line. Th e c utoff frequ ency is therefore given by Wh

2 = - -. J I.C

(62)

Since the lumped line's c harac te ristics begin to degrade- well be low the c utoff frequency. one must usually se lec t a c utoff we ll above the highest frequency of in ter est . Satisfying this requirement is part icularly import ant if good pul se fideli ty is necessary. In designing artifici allines. the L j C ratio is c hose n to pro vide the desired line impedance. while the LC prod uct is cho se n s mall e no ugh to prov ide a high e nough c utoff frequency to allow the lim' to approximate ideal behavior over the desired hand~ idth. If a specified overal l time delay is requi red . the first IWO requi rem en ts defin e the minimum number of sec tions that mus t be used .

5.7.2

TE RMIN ATIN G LUMPED LIN ES

There's always a question as to how one terminates the circ uit of Figu re 5.4. For example. one choice is 10 e nd in a capac ita nce and sim ply term ina te across it. Another cboice (see Figure 5.5) is to e nd in an inductance . A ltho ugh both of these c hoices VIill work after a fashion. a bett er ultemau ve is to compromise by usin g a half-section at buh ends of the line. as shown in Fig ure 5.6.

130

CHA PTER 5 DISTRIBUT ED SYS TEMS

FIGURE 5 .6 . Holh.ection$ For line lermina tion.

L,

FIGURE 5 .7. m-derived hoJf ·~ for line termination .

Such a co mpromise extends the bandw idth over the choices of Figu res 5A and 5.5 . Each half-sec tion co ntributes half the delay of a full section. so puttin g one on eac h end adds the del ay of a full section. Furthermore. and more important. a half-section has twice the cutoff frequ ency of a full sec tion. which is precisely why OCUcr bandwidth is obtained .

S. 7.3 m· DERIV ED HALF· SECTIO NS The port ir npcdau cc of the L C half-secti on begin s to increa se significantly at about 30 - 40% o f the cutoff frequ ency owin g to the parallel resonance form ed by the output capac itance and the rest of the reactance it sees. Thi s behavio r can be moderated by the usc of half-sections that are only margin ally more elabo rate than the single L C pai r. Speci fically. if the capaci tor is rep laced by a series L C branch , then the frequ ency range over which the impedance stays rough ly co nstant can be increased eve n further because the decreasing impedance of the series reson ant branch helps offset the increasing impedance. A simple net work that achieves the desired result is shown in Figur e 5.7. The elemen t values are given by the following equations: L1 =

mL

2 '

(63)

I _ ml

L2=

L.

(6-1)

(65)

L

PROBL EM SET

131

A networ k mod ified in this ma nner ls called a n »r-d eriv ed half-sect ion becau se, for any value of the paramete r m , the no minal c haracte ristic impedance rem ains the same as that o f the s imple L C hal f-sec tion . This ca n he ve rified by direct substitutio n into Eqn. on. Th e im ped ance stays roughly consta nt up to about 85 % of (he c utoff frequency for a value of m equal 10 roughly 0.6. Th is c hoice is ther efore a common one.

5.8 SUMMARY We have identified a fuzzy boundary between the lum ped and distributed reg imes.

and found that lumped concepts may he exte nded into the di stribut ed regime. In carrying out that exte nsion. we ha ve discovered that the re are several. pe rhaps many. ways (0 trade ga in for delay rather than band width . A<; a final observation on this subject. perhaps it is wo rthwhile to reiter ate (hat avoiding a straight gain-bandw idth tradeoff requ ires a gross departure from single- pole dynamics. He nce. all the struc lures we've seen that trade ga in for del ay involve many energy storage elements. Another way to loo k al thi s issue is to recogni ze (hal if we arc to trade de lay fo r a nyjhin g then we must have the ability to provide large delays. But large del ays imply a large amount of phase cha nge per unit freque ncy, and if we arc (0 operure ove r a large bandwid th the n the to ta l phase cha nge req uired is ve ry large. Again, this need for large a mo unts of phase shift necessari ly implies (hal ma ny pole s (and hence many inductors a nd ca pacitors) wi ll he required, resulting in the rel atively complicated networ ks we' ve seen." If o ne is pursuing operation ove r the largest po ssible bandwidth, howeve r, usc of these distribut ed conce pts is all hut ma nda tory , As we shall see in the chapter on high -freq uency a mp lifier design, dis tribu ted conce pts ma y be applied to active circuits to allow the realization o f amp lifiers w ith exce ptionally large bandwidth by tradi ng delay in exchange for the improved band width .

PRO BLE M SET FOR DISTR IBUTE D SYSTE MS PROBL EM 1 (a) What is the impedance loo king into a lossless qua rte r-wa ve tra nsmission line tcr-

minuted with a resistive load? Let 2 0 be the line's cha racteristic impedance a nd RL the load resistance. (b) Given your answer to (a), propose a me thod fo r matc hing an 80 -0 load to a 20 · n source at 500 MliL. 9 Anu ct'JIl ion i\ the superregeneranve am plifier. There. (he sampled nature o f lhe syste m effectively CilU>e'l illia'iot'll of lhe single stage 10 acr in a way lhat is similar ttl a ca..caJc ur such stages.

CHAPTE R S DISTRI BUTE D SYSTE MS

132

PROBLEM 2

(a ) Ca lculate the input im ped ance of a lossless tran sm issio n line of 11.6 wuve length in ex te nt. and who se c haracteris tic impedance is 50 Q . if the lo...d impedance is

60.3 + j 4 1.51l . ( b) Repeal (a ) fur a lossy line whose attenuation is 3. 1 dB pe r waveleng th . PROBLEM 3 Th is problem considers an ex tre mely use ful property of tra nsmission lines th at arc )./8 in ex te nt . Deri ve a n ex press io n fo r the input impcdance of a lossless ).. /8 line of c ha rac teristic impeda nce Zn . te rmi nated in an a rbitrary impedance R + j X . Com me nt o n potential a pplicatio ns for wh ut yo u have di sco vered . PROB LEM 4 In much di:..c rete RF work.fhc standing wave ratio (SW R) is used beca use it is easier to measure tha n th e re flec tio n coe fficie nt . The two q uantities are rel ated. howe ver : SW R - I

WLI=SWR+ I ·

( P5.11

(a ) Show that SWR is the rmio of' the peak voltage a mpluudc 10 the mini mum volta ge amplitude alo ng a lossless line . (b) Wh at is the SW R o f a pro per ly terminated line? Of a shorted line? An opencircuited line? (c ) Wh at arc the reflectio n coeffic ie nt and S W R for a 50-Q line termi nated in.t5 fl?

PROBLEM 5 De sign a n a rtifici alli ne wi th a cbamc reris tic imped a nce of 75 n and a c utoff freq uency of J G Hz . Use S PICE In plot the ga in a nd phase of the line when tcrmimucd in 75 Q, as we ll as the input impedance ( bot h mag nitu de and phase) from DC to 2 Gill . Would yo ur de sig n he suitable for an on -chip im plem en tation? Expl ain.

PROBLEM 6

Consider an

He diffus iontine, a transmi ssion

line in whi ch there is no

indu ctance.

(a) Deri ve expressions for a , {j. and the c harac teristic imp edance . ( 0) Deriv e an ex pres sion for the input imp edance of such a line when it is termi nated in a n o pe n ci rcuit. (c ) Using yo ur answe r to part (b ), show th at the real part of the input impe dance at low frequ en cies is eq ua l to o ne third the total resistance . Th is eq uality allows om: to deri ve a single- pole lu mped He mod e l to ap proximate (c rudely) the fully distribut ed cnsc by simply substituting the to tal ca pncit auce and tine third the tot al resistan ce il110 the model.

PROBLEM 7 An ideal losslc ss tra nsm iss ion line with indu ctance of 3 nil /mill and of le ngth f is te rmina ted by a 6O-Q re sisto r. A voltage ste p is a pplied a t the input

PROB LEM SET

133

at time' = O. What le ngth o f line and cha racteris tic impedance do we need to provide a 5 ~ n s del ay and a maximum amplitude across the load resistor? What is the capacitance pe r unit leng th o f your line?

PROBLEM 8 (a) Suppose we construct a simple 3.2-ns lumped delay line out of ten identical tow-pass L · ~ l i oil'. , where each inductance IS :m nH and eac h ca pacnence Is 20 jJr pF. To two significant dig its. what is the characteris tic im pedance of this line? ( b) What is the cutoff freq uency in hertz? (c ) Using SPICE . plot the frequency respon se (magnitude and phase) of this line from DC to twice the frequency found in ( h). Drive the line with a source whose resistance is the value found in part (a). ami terminat e it with another resistance of this value, (d) Plot the respo nse to a unit step. (e) Now use simple half-sections on the ends. Replot as in (e ) (up to the same frequency as before ) and (d) . (f) Replace the simple half-sections with m· deriVl'll half-sections. with III equal 10 0.6. Replot as in (e) and Id ). Any improvement in bandwidth'! Comment on the appearance o f the step res pon se relative to thai in part (c) .

PROBLEM 9 Construct a discrete art ificia l line with tive simple L C sections. The IOO-pF capacitors yo u are allowed to use ha ppen to have a series parasitic indu ctance of 2 nH. while the l oo -JIII ind uctor s at yo ur disposal have a parasitic shunt capacitance of 5 pF. Plot the freq uency respo nse of this line when it is term inated in its characteristic impedance. Doe s the line behave as you expect? Explain. Resi mulare without thc para sitic clements and compare. PROBLEM 10 Dielectri c loss can ge nerally he neglected in the low-gigahert z range for on -chip lines. However. substrate and conductor losses can he quite significant. In order to reduce the former . microstrip lines are som etimes used . Such transmis sion lines use a co nductor over a ground plane. Occasion ally. even striplines arc em ployed, whic h are shielded structures in which a conductor is sandwiched between two ground planes. Using the form ula from C hapter 2 for the capacitance o f such lines. estimate the inductance pe r unit length for the two ca..-es if the relative dielectric constant is 3.9. Fro m those values. derive an ex pression for the characteris tic impcd ance for the two types of line. Using your formulas. ap pro ximately what range of characteristic impedances could one expec t for on-chip lines?

CHAPTER SIX

THE SMITH CHART AND S- PAR AME TE RS

6 .1

INTROD UCTION

Th e subject o f C MOS RF integrated circ uit design reside s at the convergence of two ve ry differe nt e ngi neering tradit ions. Th e design of microwave circuits and sys te ms has its origins in an era where devices and interconnect were usually ton large to allow a lumped descriptio n. Furt hermore, the lac k of suita bly detailed mode ls a nd compatible com puu nional rools forced e ng ineers 10 tre at syste ms as two-po rt " blac k bo xes" with frequency- do mai n gra phical method s. Th e Ie design com mun ity. on the other ha nd. has relied on the de velopme nt of detailed device model s for use with simulalion too ls thai allow bo th freque ncy- a nd time -domain analysis. As a consequence, e nginee rs who work with traditional RF design tec hniq ues and those sc hool ed in conventional Ie design ofte n lind it difficult to conve rse . Clearly, a sy nthes is of these two trad itions is required . Analog Ie designer s acc ustomed to worki ng w ith lower-frequ ency circuits tend to have, at nest . only a passin g familiari ty with two staples of tradit ional RF design : Sm ith c harts and Scparamet crs (vscatteri ng" parameters). A lthough Smith charts tod ay a rc less relevant as a computational aid than they once were . RF instrum entation co ntinues to present dat a in Smith-cha rt form . Furt hermore. these dat a arc oft en Svpurumctcr c haracte rizations of two -ports. so it is importa nt. e ven in the " modern" era , to know something about Smit h cha rts and Scpa rumctc rs. Thi s c hapter thu s pro vides a brief de riva tion of the Smith cha rt, alon g with a n e xpla nation of w hy S-para mc te rs WOIl out ove r other paramet er se ts (e.g ., imped ance or admittance) to desc ribe microwave two -ports.

6 .2

THE SM ITH CHA RT

Recall from C ha pte r 5 the express ion for the reflec tio n coefficie nt in terms o f the no rmalized load impedance: 134

6 .2 THE SM ITH CHART

r

ZL -- I = ~ Z,,-O_

Z -!: + I

(I)

Zo

Th e relatio nshi p between the no rmalized load impeda nce and I" is hi-uniq ue: knowing o ne is eq uivale nt to knowin g the o ther. This observatio n is importa nt bel;au~ thc falllilidJ cu r vc:-o uf the S m ith c han a rc aimpl y a ploui ug, in the F vplane, of co ntours of constant re sistance and reactan ce . A natura l q uestion to ask is wh y on e sho uld go ro the trouble of esse ntially plo tting impedance in a no nrec tili ncar coordinate syste m. since it's ce rtai nly more straighrfo rward to plot the real and imagi nary pans of im pedance directly in sta ndard C artesian coordinate s. There are a t lea st two good rea so ns fo r the non-ob viou s c hoice . On e is tha t trying to plot an in finite imped ance directly poses se lf-ev iden t prac tica l problem s. Plot ling r instead neatly hand les impedances of arbitrary magnitude because 11'1 canno t exceed unity for passive loads. Th e o ther reason is that . as shown in Ch apter 5. r repeats e very ha lf-wa ve le ngth whe n a lossless transmission line is ter m inated in a fixed imped ance . lie nee. plotting I' is a natural and co mpact wa y to e ncode this periodic behavio r. Much of the co mputatio na l power of the S m ith c hart de rives fro m this encoding. a llowing eng ineers to determi ne ra pid ly the le ngth of line needed to transform a n impedance to a parti cular va lue. for exam ple. The re lat io nship betwee n impeda nce a nd r given in Eq n. I may be co nside red a mapping of o ne co mplex numhcr into ano ther . In this case . it is a spec ial type of mapping know n as a bilinear transformation beca use it is a ratio of two linea r functions. A mo ng the va rio us pro perti es of bilinear transfo rma tion s. a pa rticularly relc vanr o ne is that c irc les re ma in ci rcles whe n mapped. In this co nte xt. a line is considcred a ci rcle of infinite rad ius. lie nee. ci rcles and lines map into e ithe r circl es o r lines. With the aid o f Eqn. I. it is stra ightforw ard to sho w that the imagi nary axis of the Z-plane ma ps into the unit ci rcle in the r -pla nc. while o ther lines of co nstant rcsisranee in the z-ptanc ma p into ci rcle s of va rying di am eter that a re a ll tangent at the point r = I: sec Fig ure 6.1. Lines of constant re ac tance are orthogona l to lines of co nstant resistance in the z -pbme. and this o rtho go nality is preserved in the map ping. Since lines map to lines or circles. we ex pect co nstant-reac tance lines to transform to the c ircular arcs shown in Figu re 6 .2. Th e Smith c hart is just the plotting o f bo th co ns tant-res ista nce and co nstant-reactance co nto urs in the I' -pl ane witho ut the ex plici t prese nce o f the r -plane axes. Because. as mentio ned earlier. the primary ro le of the Smith c hart these days is as a standard way to prese nt impedance (o r re flectance ) data , it is wo rthw hile tak ing a lillie time to develop a fa miliari ty with it. Th e ce nte r of the Smith c hart correspo nds

136

CHAPTER 6 TH E SMITH CHARTAND S-PARAMETE RS

Iml..~I" lll'e plaor Im[Z )

" -I,I.m e

Im[1"/

RqZJ

/

URlt cin: k

FIGURE 6 .1. Mapping of COf!s!onl-resiskJnce lines in Z-pkJne to cirde s in f -plane.

lmll )

Im[rJ

:::::: R..IZJ

FIGURE 6 .2. Mapping of constant-reacta nce lines in Z-plane to contou rs in F vplc ne.

to zero reflection coeffi cien t ami, therefo re, a resistance equal to the normalizing impedan ce. The bottom half of the Z -plane map s into the bottom half of the unit circle in the Fcplane. und thu s capacit ive impedances arc alway s fou nd there. Simil arly. the top hal f of the Z- planc correspo nds to the top half of the unit circle anti inductive impedances. Progressivel y smaller circl es Df constant resis tance co rte...polld In progressively larger resistance values. The poi nt r = - I corresponds 10 zero resistance (or reactance), and the poin t r = I co rresponds 10 inlinite resistance (or reactan ce). As a simple. but speci fi c. exa mple. let us plot the impedance o f a series He network in whic h the resistance is 100 Q and the capacitance is 25 pf-: all normalized

6 . 2 THE SM ITH CHA RT

'" .' '" .~ . ~Y=: "

137

'.

..., ."., .."~B .. ., " '\ " -00 _ 00 00• . . . .. ..... "- ......... ".. .. ... , ,, " '" . .... ...,. . ..... ...." . .. ..... '., " . " .. " .': ' .. ...... . , .. .. .. '.. ,., ', . .. " .. .. .. .. , ... ... .. .. .. . .. .. .. """" '.

_-

~

"

'-,~"""

,,~

'

"

~

'

.....

"

,

..

.

FIGURE 6 .3 . The Smith chart w ith series RC example.

to a 50-Q sys tem. Since the impedance is the sum of a real part (eq ual to the resistance) and an imag inary part (eq ual to the ca pacitive reactance ). the co rrespond ing loc us in the Fcplane mustlie along the circl e of constant resistan ce fo r which R = 2. The reactive part varies from minu s infinity at DC to zero at infinite freq uency. Since it is always negat ive in s ign. the locu s must be ju st the bottom half of Ihat constant resistance circle . traversed clock wise from r = I as frequency increa..es, as seen in Figure 6.3. There are numerou s otbe r propertie s of Smith charts. and the types of com putations that may he per for med graphically and rapid ly with them are truly remarkable.

138

CHAPTER 6 THE SMITH CHARTAND S-PA RAMETfRS

I,

- ~

r- - - - - - - - - - l--.';. " +

+ V,

T~o-I·ort

V,

FIGU RE 6 . 4 . Port variable definitions.

However , since machine computation has largely displaced that role of Smith charts. we direct the interested reader to Smith's papers for further applications.'

6 .3 S· PARAMET ERS System s can be characterized in numerous ways. To simplify ana lysis and perhaps elucidate important design criteria . it is often valuable to use macrosco pic dcscripno ns. which preserve input-output behavior but discard detai ls of the internal structure of the system. AI lower frequencies. the most common representations use impedance or admittance parameters, or perhaps a mixture of the two (called. sensibly enough, hybr id parameters). Impedance parameter s ulfo w one to express port voltages in tenus of port currents. For the two -po rt shown in Figure 6.4, the relevant equations are:

v, = V2 =

+ Z \2/2, Z2, / I + Z21.12.

Z II / \

(ll (31

It is most convenient to open-circuit the po rts in success ion to determine the various Z parameters experimentally, beca use various term s then become zero. For instance, determinatio n of Z I I is easies t when the ou tput port is open-circuited because the second term in Eqn. 2 is zero under that condition. Driving the input port with a current source and measuring the resulting voltage at the input allows direct cornputation of Z ll . Sim ilarly. open -circuiting the input port. dri ving the output with a current source. and measuring VI allows determination of Z 12 . Short-circu it cond itions are used to determine admittance parameters. and a cornbination of open- and short-circuit conditions allows determ ination of hybrid parameters. The popularity of these representatio ns for characterizing systems at low frequencies traces directly to the ease with which one may determine the parameters expe rimentally.

I For exa mple. see P. II. Smith. M An Improved Trunsmissjon Line Calculator ," Electronics. v. 17. January 1944. p. D O.

139

6 .3 S· PARAMETE RS

E i2

~ - z.

. /W-

Two-I'on

E' 2

FIGURE 6 .5 . S-pa ra melet' part varia ble definitions.

At high frequencies. however. it is quite difficult to provide adequate shorts or opens. particularly over a broad freq uenc y range. Furthermore. active high-frequency circuits are frequentl y rather fussy about the impedances into which they opcrute. and may oscillate or even expire when terminated in open or short circuits. A different set of two -port parameters is therefore required to evade these experimental problems. Called scattering parameters (or simply S-parame ters ), they exploi t the fact that a line terminat ed in its characteri stic impeda nce gives rise to no reflection s, lnierconrecnons between the instrumentation and the system und er test can therefore be of a comfortable length . since no short or ope n circuit needs to be provided ; this

greatly simplifies fixturing. As implied earlier. terminat ing po rts in open or short circuits is a co nven ience for lower-frequency two -port descriptions because variou s terms then become zero. simplifying the math . Sca tteri ng parameters retain this desirable property by defin. ) l'Olwf.:t' ing input and output variables in terms of incide nt and reflected (sc ane n.-d WaI'I'J , rather than pt)rt volt ages or curre nts (w hich are difficu lt to define uniquel y at high frequencies. anyway). As can be seen in Figure 6.5. the source and load termi nation s are 7.0 • With the input and output variable s defined as shown, the two -po rt relat ion s may be writte n as

hi = b2

=

+ S12U2, S2111 1 + Sn Q 2.

SIIQ I

(4)

(5)

where

= Ei l / ~'

(6)

a 2 =Ej d ~.

(7)

b, = E,,/IZo. h, = E,,/IZo.

(H)

al

(9)

The nonnalization by the square roo t of 2 0 is a convenience that makes the square of the magnitude of the vario us 0 " and b" equal to the power of the co rresponding incident or reflected wave.

140

CHAPTE R 6 THE SMI TH CHART AND S·PARAMETERS

Driving the inpu t port with the output po rt termin ated in Zu se ts tl 2 equal to zero, and allow s us to determine the fo llowing paramete rs: 5 11

~ = ti-hil = -Ed = I I. Ed

(10)

S~ l

E r2 /1 2 = -=-. II I Ei l

( II)

•

Thus, .~ll is, simply the input reflectio n coe fficien t. while .~2 1 is a so rt of ga in since it relates an output wave to a n input wave . Speci fically. its magnitu de sq uared is called the forward transducer po wer gai n wit h Zn us so urce a nd load impedance . Si mi larly. terminating the input po rt and driving the outpu t JXI" yield..

02

rzz = -

11 2

hi

S 12

= -

(/ 2

= -EE -22 = I~2. r

(12)

i

e.,

=-. E j2

(Ill

Here. we see that $ 22 is the output reflection coefficient: S 12 is the reve rse tra n..rni..sio n. whose magnitude squared is the reverse transdu cer powe r gain with 2 0 as source and load imped ance. On ce a two -JXII1 hOI!. bee n c ha racte rized w ith Svparume tcrs . direct des ign of syste ms may proceed in princi ple w ithout knowing a nything about the internal wor kings Ilf the two -port . For example . gain equations a nd stability c rite ria can be rec u..t in te rms of Scpuram e ter s.2 f lowcver. it is im portant to keep in mind that a macroscopic approach ncccssurily disc ard" pote ntially impo rtant information. such as se nsitivity ttl pa ram e ter or process varia tion. For thi s reason. Svparume ter measurements are ofte n used ttl derive e leme nt values for model s whose topologies ha w bee n det erm ined from tirst prin cipl es or physic al rea soning. To summarize. the reasons thai S-para meter s have become near ly uni versal in high -frequ en cy wo rk ar c that "zcrov-le ngth fixturing cables a rc unnecessar y. there is no need to syn thes ize a short or ope n circuit. a nd termi natin g the two -port in Zn greatl y redu ces the pot ential for oscil lation .

6 .4

APPENDI X: A SHO RT NOTE ON UNIT S

Th e inability to identify unique vo ltages and c urre nts in dis tributed sys tems. co upled with rhc Rr en glncc r' s preoccu pation with power gain. hus nuu le power the natural quantity O il which to foc us in RF ci rc uits and syste ms , Power level s arc expressed

2 A represe ntative reference that covers this topic is G , Gu n/alcl . Mk mlHn'(' Tmn_~;,\I.. r Arnf'fiJif'n. Znd ed.. Pre nuce -Hall, Englewood Ch ITs. NJ. 1'197.

6 .5

A PPEND IX: W HY 50 (O R 75)

m

141

in waus. of cou rse. hUI wh..1 ca n co nfuse and fru stra te the uninit iated arc the va rio us dec ibel versions. For e xam ple. "d llm" is qui te co m mo nly used . The " til" sig nifies that the O-dB referen ce is o ne milliwatt , while a " d BW" is refer en ced III o ne wall . If the reference impedance le ve l is 50 n. then 0 dRm corresponds In a vol tage of about

223 mV nns. As clear as the se defin itio ns a re. there are so me w ho actively insist o n confusing

volts with warts. redefi ning () d Hm In me an 223 mV rms regardle ss of lil t' impedance {,,\.,t. Nor on ly is this redefinition unnecessary, fn r o ne m ay a lways define a dil V. il is aIM> dangerou s. A.; we wi ll Sl."C. critical performance mea ... urcs, suc h as linearity and noise figure. intimately invo lve true po.....er ratios. particularly in study ing cascaJcJ systems. Confusin g po wcr wit h vo ltage ratios leads In g ross e rro rs . He nce. throughoo t this text . 0 d Um tru ly rncan -, (Inc milliwatt . and 0 d.UV means on e volt . Always, With thai out of the way. we return to so me definitions. Common when discu....ing n oise or distortion product s in oscillators o r power amplifier s i... "due," wb cre the "c" bere signifies thur the O·d U reference i!oo the power of the ca rrie r, Finally. a good way 10 start a fislflg ht among microwave e ng inee rs is III arguc over the pronunciation of the pre fix "giga ," G ive n the Greek o rig in. both gs sho uld he pronounced as in " g igg le:' the choice now advocated by both ANS I (America n Na tional Standard s Institute ] and the IEEE ( Instit ute of Electrical a nd Elec tron ic s Engineers). However, there is still a siza ble co ntinge nt who pro no unce the first X as in

"giant." 6.5

APPENDI X: WH Y 5 0 lOR 75) fl?

MI)!oot RF instrum e nts and coax ial cables have standardized im pedances {I f eithe r 50 or 75 ohms, It is easy 10 infer from the ubiquity of these impedances thut there is something sac red about the se values. a nd that th ey sho uld the re fore he used e ven in Ie designs. ln thi .. appe ndix . we e xpla in where the se number s ca rne fro m in the first place to sec whe n it doc s a nd doc s not make se nse to use those impedances.

6 . 5 .1

POWER -HANDLIN G CA PA BILITY

Consider a coaxial cable with a ll uir diclectnc. Ther e wi ll he. of co urse . so me vo ltage at which the d ielectric break s down . For a fixed rati o of inner a nd o uter co nd uctor d iamcrcrs. a sma ller inne r co nduc tor w ill lo wer thi s breakdown vo ltage tbccuu se o f the lighter radius of curva ture) . which wo uld tend to decrea se the cable's power-h an dl ing capability. To increu..c the breukdow n vo ltage. o ne could increa se the inn er diame te r. How ever.the c ha rac teristic impe dance wo uld the n increa se. w hich would le nd to re duce the power de live rable to a load , Beca use of the se two competing effe cts. there is a well-defi ned nuio of conductor d iame ters thai maximize s the po wer, ha nd ling capability (If a coa xia l cuble .

14'

CHAPTE R 6 THE SMI TH CHART AND S' PARA METE RS

Havi ng es tablished thai a maximum exists. we need to dredge up a couple of equation s 10 lind the actual dimen sion s that lead to this maximum . Speci ficall y, we need an equation for the pea k electric field between the co nductors . and anot her for the characte ristic impedance of a coaxial cable:

v

(14)

Emu = a In(b/ a ) ' C7:

2 0 = V Jil t: ·

In(b/a) 60 2Jt

::::::-

rr:- · In(b/a),

v E,

(15)

where a a nd b a re the inner a nd oute r radius, respective ly; e, is the rel at ive di electric constant. which. for our air line case. is esse ntially unity, The next step is 10 recognize that the maxim um power deliverabl e 10 a load is proport ion al to V 2/ Z o. Using our eq uaticns. fhat translates 10

Now lake the derivative. set it equal to zero, and pray for a maximu m instead of a minimum: dP =

da

.'!..- [a 2 In (~)]

da

a

= () _

~ = ,;e.

a

(17)

Plugging this ratio back into our equation for the characteristic impedance gives us a value of 30 Q . That is. to maximize the power-handling capability of an air dielectric transmission line of a given outer diameter, we want to select the di mensions 10 give us a 2 0 of 30 Q . Hut wait 30 doc s not equal 50. even for relatively large values o f 30. So it appears we have not yet answered our origi nal question. We r.eed In consider one more factor: cable attenuation.

6 .5 .2

ATTEN UATIO N

In the main body of the chapter, we derived a general express ion for the attenuation co nstant of a transmission line that account s fur both d ielectric and conductor losses. It may be shown (but we won't show it) that the atte nuation per length due to dielectric loss is practically independent of conductor dim ensions. Simplifying our equation to account only for the attenuation due to resistive loss gives us R a '" - - . 2Z o

(18)

where R is the series resistance per unit length. At sufficiently high freq uencies (the regime we're concerned with at the moment), R is due mainly 10 the skin effect. To

6 .5

APPEND IX: W HY 50 tO R 75)

m

' 43

reduce R. we would wa nt to inc rease the d iamete r of the inne r conduc to r ( 10 ge tmore "skin"). bUI that wo uld lend Itl reduce Z o at the sa me lime. a nd it' s not clea r how to .....in. Again. we see a compe tition betwe e n two opposin g effect s. a nd we expec t the optimum 10 occur once more at a specific value of bill and . he nce. at a spec ific Z oo Just as before. we invoke some eq uations to ge t 10 a n actual numerical result. T he only new one we need here is a n e xpression for the resistance R . If we make the usual assumption that c urre nt flows uni form l y in a Ihin cyl inde r of a thi ckness equal to the skin depth &. we can wri te

R 0"

_I_ [~ + ~ ] . 2l"f&a a

( 19 )

b

...bere 0 is the conductivity of the wire and & is the same as always :

(20)

wnh these equ ations. the atte nuation consta nt may be expressed as :

R

(21)

u = - - o"

2Z"

Taking the deriva tive. setting it eq ual 10 zero, and now pra ying fo r a minim um instead

of a maximum. we have

du

- = 0 =:?

ria

~ [~ + ~J = 0 da In(b) -

=:? In

(b) -;;

= I

"

+ h'

(22)

a

Atter iteration . this yields a value of abo ut 3.6 fo r bta , corres ponding to a Zo o f about 77 n. Now we have all the info rmat ion we need . First off cab le T V equi pmen t is based on a 75- 0 worl d because it corresponds (nearly) to minimum loss. Power levels the re are low. so power -ha ndling ca pab ility i_~ not an issue. So why is the sta ndard there 75 a nd not 77 ohms'! Because e ng inee rs like round num bers. Thi s affin ity is also the reason fo r 50 0 (a t last) . Si nce 77 n gives us min imum loss a nd 30 0 gives us ma xim um power-han dling ca pability. a reasonabl e compromise is the averag e - which . after roundi ng. gives us 50 O . Andthat's it.

CHAPTE R 6 TH E SMIT H CHARTAN D S· PARAMETE RS

6.5 .3 SUMMARY Now thai we understand how the macroscopic universe came In choose 50 n. it should be clear thai ,one should feel free 10 c hoo se very differe nt impedance levels in ' In integra ted ci rcuit. RF circuit performance in ICs is rurely dominated either by the power -hand ling or a tte nuation cha rac teri stics of the interconnec t. As a resuh . K engineers have the luxury of selecting va..tly diff erent impe da nce level s than would be the norm in disc rete design. Indeed. gi ven the rela tively large power consumed in trying 10 drive 50 -0 levels . it should be evident thai 50 Q is generally 10 be avoided in most IC design s - except. pe rhaps. at interfaces with the out side wor ld.

PR OBLEM SET FOR SMIT H CHART AN D S· PARAMETERS PROBLEM 1 As discussed in the text. the Smith chart is nothing more than a mapping of contours o f constant resistance and reactance fro m the im ped ance plane 10 the rellccta ncc plane. (a) Explicitly derive express ions for these map pings. ( bl Show that circ les map into circles. PROBLEM 2 Can one plot a Smith chartlocus fur a los!>y transmission line? Explain. If it is po ssible. plot an example. PROBLEM 3 TIle bilinear map ping thutleuds to the Smith chart maps an infinitely large domain into a finite. periodic runge. Show specifically that it is periodic in a It(/({-wavelcngth . PROBLEM 4 Since the Smith chart is period ic in a half-wavelength . it is easy to see that quarter-wavelength lines posses!' an impedance reciproca tion property. Demonstmtc how one may exploit this property to convert imped ances to admitt ance. and vice versa. with case on the Smith chart. PRO BLE M 5 The various two-port representations arc. of course. equivalent to each other since they ultimately descri be the same system. Show this explicitly by converting impedance parameters to Scparameter s. and vice versa . PROBLEM 6 Convert hybrid parameter s to Scpa rume te rs and vice versa . PROBLEM 7 Convert admittance parameters to Scparnmerers and vice versa . PROBLEM 8 Plot . on a Sm ith chart. the impeda nce o f a ser ies R l. C network. Let the normalized resistance be unity. Choose the inductance and. ca paci tance to yield a co ntour that is neith er too large nor too small fur comfortable plotting.

PRO BLEM SET

PROBL EM 9 It is u rule of thumb thai plots on a S mith chan always go clockwise as frequency increases. Defend this rule. if possible. Are there exceptions'! PROBLEM 10 II is observed that networks whose Smith trajectories are far from the origin (of the r -planc) arc narrowband. Explain.

CHAPTER SEVEN

BANDWIDTH ESTIMATION TECHNIQUES

7.1 INTRODUCTION Finding the - 3· <.I B bandwidth of an ar bit rary linearnetwork can be a difficult problem in genera l. Conside r. for example. the standard recipe for computing bandwidth:

( I) deri ve the input- o utput transfer functio n (using node equ ations. for exa mple): = jw; (3) find the magnitude of the resulting expres sion; (4) set the magnit ude = 1/ ../2 o f the "midband" value; and (5) solve for w .

(2) se t s

It doesn ' t take a great dea l of insigh t to recog nize that ex plic it computation (by hand) of the - 3- d8 bandwidth using this method is generally impractical for all but the simplest systems. In particular. the order o f the denomi nator polynomial obtained in step (I ) is eq ual to the number of poles (natural freq uencies), which in turn eq uals the number of degrees of freedom (m easu red, say. by the number of initial conditions one may independe ntly spec ify ), whic h in turn eq uals the number of indepen de nt energy storage elements (c.g., L or C) , which in turn ca n he as large as the number of e nergy sto rage element s (phcwl ). Thu s, a ne twork with 11 ca paci tors might require the equivalent of findi ng the roo ts of an ut h-ordc r po lynomial. If 11 excee ds j ust four, no algebraic closed-form so lution ex ists. Even if II = 2, it might be labor-intensive to obta in the flnal numericul res ult. Machine com puta tio n is cheap a nd gelling c heaper all the time, so perhaps the analysis of net works doe sn't present muc h of a proble m. However. we are inte rested in developi ng des;g" insight . so that if a simula tor tell s us that there is a problem the n we have so me idea o f what to do about it. We the refore see k methods that are reason ab ly sim ple to a pply yet convey the de sired insight . eve n if they yield answe~ that might be a pprox imate. Simulators ca n then be used 10 provide final quantitative veri fica tio n. 146

7.2 THE M ET HOD O F O PEN ·C IRCU IT TIM E CO NSTA NTS

147

Two such a pprox imate met hod s a re ope n- a nd sho rt-c irc uit time co ns tan ts. T he former method provides a n estimate of the hig h-frequ e ncy m lloff po int. the latter of the low-frequency po int . T hese me thod s are val uable beca use they ide ntify w hich elements are re sponsible fo r the ban d width limi tation . This informa tio n a lone is o fte n sufficient to sugges t what mod ifica tions sho uld be tried ne xt.

7. 2 THE METHOD OF OPEN - CIRCUIT TIM E CO NS TA N TS

The method of ope n-c ircui t time c o nsta nts ( OCTS). also kno wn as "zero value" rime constants. was de veloped in the mid - l960s a t M IT. As we shall sec. thi s powerful technique allows us to e stima te the ba ndwidth of a system almost by inspection. an d sometimes with surprisingly good accuracy. Mo re important , an d unlike typica l ci rcuit simulation programs. OCr s identify which elements art' responsible fo r bandIf idth li",itarioflS. Th e great valu e of thi s pro pe rty in the design of amplifiers ha rd ly needs expression. To begin deve lopment of th is method , let us consider all -pole tra nsfer functions only. Such a sys tem funct ion may be writte n a s foll ows : V.,( S)

\',(s )

ao

(I )

=

where the various tim e c o nsta nts may o r may no t be rea l. Multiplyin g out the terms in the de no minator lead s 10 a pol ynomiul rha r we shall express as

(2) where the coefficient h n is simply the produc t of a ll the time consta nts a nd hI is the sum of all the time co nsta nts . ( In ge ne ra l. the coeffic ient o f the ,\. j term is co mp uted h)' forming all uniq ue product s of the II time co nsta nts ta ken j at a time and sum min g aJl Il !fj !in - j)! suc h produ cts.) We now assert that , ncar the - 3- d B frequ e ncy. the first-o rder term ty pica lly dom-

inates over the hig he r-or de r te rm s, so that (pe rhaps ) to a rea so nable app ro ximation we have

_V_ ,,_ (.,_ ,) ;:::: ,-_a~",.--, V; (.I') h IS J

+

a"

0::::." ,)< + I ·

(.3)

The bandwidth of ou r o rig ina l sys tem in radi an freq uency as estimated by this firstorder approximatio n is then simply the rec iproc al of the e ffec tive time constant: (4)

Before proceed ing furt he r, we sho uld consider the co nd itions under which o ur ncglecr of the higher -order term s is ju stified . Let us exa mine the de nomi na tor of the

148

CHAPTE R 7 BAND W IDTH ES TIMATIO N TECH NI Q UES

transfer function near our est imate of W h , For the sake of sim plicity. we start with a second-o rde r po lynomi al with pure ly re a l r oot s. Now. al o ur csn mated - 3-d H frequ e ncy, the o rigina l denominat or po lynom ial is

15) No te that the magnitu de of the second term is unity (why '!). A s a co nseq ue nce. both

and

16) (7)

have magnitudes no grea te r than unit y. Thus. the product of these terms (which is ..- quelto the ma gnitude of the le adin g term of th e polynomial) must he sma ll com. pared to th e ma gnitude of the sec o nd (first-o rde r) term . Th e worst case (K'C U rs when the two time co nsta nts are equal. and eve n then the sec o nd-orde r term is o nly one fourth as large as the f irst-ord er term. Extending the se argu ment s In pol ynomi als of hig her order reveals that the es timate o f the bandwidth based sim ply o n the coefficie nt b j is ge ne rally reaso nable s ince the firsr-onjcr term generally docs dominate the denominator. Furtherm o re. the ba ndwidth estimate is usually co nse rva tive in the se nse thai the actual bandwidth will almost always be tJt least (1.\ hi1:h as estimated InU...,

by this method.

So far. a ll we' ve done is sho w that a first -order esti mate of the bandwidth is possible if nne is given the sum of the pole time constan ts ( = b l ) . A las. such in formation is al most ne ve r ava ilable. a ppare ntly ca sting se rio us dou bt on the value of o ur e ntire e nte rp r ise. s ince the wh ole poin t was 10 avo id things suc h as di rect comp utatio n (If the pol e loc ations in the first place,

Fortunatel y, il is possible 10 re late the desired timc -conxtam sum. hi. to (mo re or less ) ea sily computed netwo rk quanti ties. Th e new reci pe is thu s as foll o ws. Conside r an arbit rary lincur net work co mprising: o nly re sisto rs. sources (dependent or independen t). and III ca pac ito rs. Th e n: ( I ) co mpute the effective resistan ce R jo faci ng eac h j th ca paci tor with a ll of the

oth e r ca pacit o rs re mo ved (ope n-circuited . hence the nam e ): (2) fo rm the product Tj o = RjoC j (the subscript () refe rs to the open -circuit co ndition) fo r e ac h ca paci to r: (3) SUlll all III such "open-circuit" time constan ts. Remarkabl y.the sum o f open-ci rcuit time co nstants fo n ned in step (3) is in fact precise ly eq ual to the sum hi of the pu le time consla nts. a resu lt pro ved by R. B. Ad ler (sec a lso Sectio n 7.4 ). Th us. at last . we ha ve

18)

7.2 THE M ET HOD O F O PEN·C IRC UIT TIM E CO NSTA NTS

149

7.2 .1 O BSERVATIO NS A N D IN TE RP RETATION S

The method of OCr !'> is relat ively simple to apply because each time -co nsta nt ca lculation involves the computation of j ust a single resista nce. a lthou g h o ne must be wary of the impedance -modifyin g po ten tia l of de pen de nt so urces (such a.s the transcon ductance of a transisto r mode l). In an y eve nt. the a mo unt of co mputa tio n req uired is typically substantially (i ndeed. o ften fanta stica lly) less than that needed fur an exact solution. The greatest value of the tec hnique lies in its iden tificat ion of those cle ments implicated in band width limitation s - that is. those el em ent s whose associated opencircuit time constants dominate the SUIll. Thi s kno wled ge ca n guide the designer 10 effect appropri ate modifi cations to circuit va lues o r even suggest whol esal e topologjcal changes. In contras t. SPIC E and other typi cal simulators o nly provide a numcrical value for the handwidth while conveyi ng lillie or nothing a bout wh at the designer can do to alter the perfor mance in a desired di rection. The origin of this property of OCr !'> ma y he regarded intuitivel y as follows. The Tl'dpmcol of etlt·h j th open-circuit lim e n m MOIII is the bandwidth IIwI the cirr:" ;1 lrllufJ exhib it if that j th capacitor wert' the onlv capaci tor i n the system. Th us, ea c h limeconstan t represen ts a local btmd",idlh degrada tio n tenll. Th e me thod of OC r s then stales that the linea r co mbina tio n of these indi vidu al. loc a l lim ita tio ns yields an estimate of the to tal band widt h . Th e value of OCr s de rives d irec tly fro m the identifi cation and approx ima te qu a ntificati o n of the loca l band width bonle nccks. 7.2 .2

ACC URACY O f OC"

One must be careful no t to place too much faith in the ability of OCr s to pro vide accurate estimates of hand width in all cases. Th is situa tion sho uld hardl y he surprising in view of the rath er brutal truncation to first order of the de nom inator po lynomial. Howeve r, there are num e rou s co ndi tions under whic h OC r es timates a re fair ly reasonable, as see n in Sec tion 7.2. It should he clear that an OC r bandwid th es timate is in fac t e xact for a first-order network since 110 truncat ion of terms is invol ved there. No t surprisingly. then . the OCr estimate will be q uite acc urate if a net work of hig her or der hap pens to he dominated by o ne pole (that is. if one pole is m uc h lowe r in freq uency tha n a ll of the o ther poles). There are ma ny sys te ms of practica l interest . such as operat ional am plifie rs, that are designed to have a do minant sing le pol e, a nd thus for whic h OCr estima tes are quite acc urate. Unfortunately, there arc so ma ny other cond itio ns unde r which OCr s give poor estimates that so me cavea t is necessary, For e xamp le. com ple x poles qui te co mmonly arise ( intentio nally or ot he rwi se ) in the de sign of wid eba nd multistage a mp lifiers. Often the physica l o rigi n of these co mplex po le!'> can he traced to the inte rac tion of

I SO

CHA PTER 7 BANDW IDTH ESTIMATI ON TECHN IQ UES

the prim ari ly capacitive input impedance of on e stage (as in a common-source configura tion} with the inducti ve co mponent of the o utput impedance of so urce followers. The reason that the presence of complex. pules Up~IS OCt estimates is as follows: The coe fficient hi is the sum of the pole time co nstants. and thus ignores the imagina ry part s of complex poles since they m ust a ppear in co njugate pai rs . Ho wever. the true bandwid th of. say. a two -pole system does depend o n both the real and Imaginary parts, As a result , gross error s in OCr estimates are not uncommon if complex po les are present in abundance. The nature and magnitu de o f the proble m are best illustrated with an example. Co nside r the simplest possible case. a two -pole tran sfer function: If (s )

=

_,2 2~, [ 2w.. + -w" + I

I'

The OC r bandwidth estimate is found from the coefficie nt of the

(9)

J

term : (10)

it may be shown that the actu al bandwidth is ( II)

In this part icul ar case. we see thai the OCr es timate predi cts monotonically increa sing bandwidth as the damping ratio S approac hes zero. while the actual bandwidth asymptotically approaches about 1.5Sw" . Thu s, it is possible for OC r estimates to he 0pl imi.u ic - in this case, wildly so . At a { of about 0.35, OCr es timates are correc t; for any higher damping ratio , OC r s are pessimistic . Fortunately, the po les of amplifiers are usually designed to have relatively high da mping ratios (to co ntro l ove rshoo t and ringing in the step respon se. and to minimize peaking in the frequen cy response) , so for most practical situations, OC r estima tes are pessimistic. Since it is ge nerally not possible to tell by inspection of a network if complex poles will he an issue. one must always keep in mind that thc primary value of a CTS is in identifying those portions o f a circuit that control the bandw idth, rather than in provid ing accura te bandwi dth estimates. Circuit simulators will take ca re of the latter task .

7.2 .3 O THER IMPORTA NT CO N SIDERATIO NS Although application of open-circuit time co nstants is rea sonabl y straightforward, there are one or two final issues that deserve con siderat ion. An extremely important idea is that not all ca pacitors in a network belong in the OCr calculations. For insta nce. fairly large coupling capac itors are freq uently used in discret e designs to connect the output of one stage to the input of the next without the bias poi nt of one stage

7.2 THE METHOD O F O PEN -CIRCUIT TIM E CO N STA N TS

151

upsetting thai of the other. Blind application of the OC r method would lead one to conclude erro neo usly that the larger this capacitor. the lower the bandw idth (time constants here often co rrespond to the aud io range, sugge sting thai large band widt hs are not possible). Fortun ately. real circuit behavior defies these implications. The prohlem sterns from the presence of zeros associated with the coupling capacitors. Reca ll that the assumed form for the system function consists of poles on ly. Since all zeros are thus assumed at infin itely high frequency. it is hard ly surprising that the presence of low-frequency zeros confounds our estimates of band width . The solution is 10 prepmce.H the networ k prior to application of the OC r method. Thatis. recognize that the coupling capacitors are effectively short circuits relative to the impedances around them at freq uencies near the upper bandwidth limit. Thus. one 11I11)' apl,/)' De n onty to models that are appropriate to high-frequency regime. It is usually obvious which capacit or s are to he ignored (con sidere d short circuits). hut there are occa sions when one is not so sure . In these cases . a simple thought experiment usually suffices to decide the issue. Now OC r s urc concerned only with those capacitors that limit high-frequ ency gain. As a COl1Sl'LJ ucIKe, the remova l (i.e., the open-circuiting ) o f a capacitor thar belongs in the OC r calculation should result in an increase in high-frequen cy gain. The test, therefore, is to co nsider exciting the network at some high frequency and imagining what would happen to the gain if the capacitor in question were ope n-circuited . If the ga in would go up then the ca pacitor belongs in the OC r ca lculation. since we infer from the thought experiment thn the capacitor doe s indeed limit the high-freq uency gain. If the gain would not change (or even decrease. as in the coupling capacitor case) upo n removal. that ca pacitor should probably be short-c ircuited. The nec essary co nclus ions can usuall y be reached without takin g pencil to paper. One last issue that deserves some attent ion co ncerns the relatio nship bet ween the individual open-circuit time constants and the lime co nstants of the poles. We have asserted (without formal proof) onl y that the sli ms of these time co nstants are equ al 10 each other. One mw.-t theref ore resist the temptation to rquute an open-circuit lime fllII,I'talll with a co rrespo nding pole location . Indeed. the number of poles may not even equal the number of capacitors (con sider the trivial case of two capacitors in parallel). Since the number of open -circuit time constants und the number of poles may he unequal, one clea rly can not expec t each OC r 10 eq ualthe time constant of a pole in general.

me

7.2.4

SO M E USEFU L FORMULA S

wh encomputing open-circuit lime constants for tran sistor amp lifiers. care is req uired because the feedback. action of the 1:,., generator modi fies resistances. As a co nscqeence. one should ex plicitly apply a test source (choose the type that will most directly allow computation of V6 f) to derive ex pression s for the effective resistances.

152

CHAPTER 7 BANDWI DTH ESTIM ATION TE CHN IQ UES

FIGUR E 7. 1. Increm enk:JI model for open-circuit resistc oce calculatio ns.

To illustrate a ge neral met hod . we will deri ve fo rm ulas for the resistances facin g Cr' and Cr d . To simplify the de rivations. we wi ll ignore body eff ec t and output restslance . However. co mplete formulas including bo th of those effects are provided at the e nd of this c ha pter for reference . De riva tio ns are (surprise!) left as a n e xercise fo r the reader. Co nsider OJ comprehe nsive model fo r a MO S FET with ex terna l resis tan ces added in se ries with eac h terminal (exce pt for the substrate . w hic h is our ground refere nce ). as show n in Figure 7. 1. A ltho ugh this model ex plicitly incl udes the back-gale tran sco ndu cta nce K... b and output re sista nce r o o we will not use them in this first set of derivations. In all of the SPICE runs that follow. however. the com plete model will be used. First, let' s co mpute the resistance facin g C1P' A pplying a test voltage source t', (since that cho ice direct ly fixes the value of JJss ) , we exploit supe rpos ition (once t'~J is known . we may trea t the tran scon ductance ge nera tor as an indepe ndent c urrent source of valu e g ", l', ) to o bta in. whe n all is said and done. V, I,

= -;;R'-s-'+'--;; R-£ +

so that the equivalen t resistance facing

e

g .,

(12)

is given by

(131 Hence. r io is the sum of the resistors divided by I + R", R E • Now, to com pute the resistance facin g C sd ' use a test current source (you may try a test voltage source. but you' ll regret it). T he inter ven ing algebra is a little involved. but the resistance Illay be ex pressed in the foll owing mnemonic form : (I ~ I

7.2

THE M ET HO D O F O PEN ·C IRC UIT TIM E CO N STA NTS

153

where ' left is the resistance bet ween the left terminal a nd g ro und , ' nghl is defined between the right term inal and g ro und. and .l:", .ell is the effective transconductance [defi ned as the ratio o f curre nt fro m the de pe nde nt curre nt so urce to the vol tage between the lefl termina l a nd g ro und). For o ur model , we have

I J5) (16)

8... .elf = 8", . I

+ g... R E

(1 1 )

Af(Cr a lillie prac tice, these eq ua tions w ill help yo u to zip throu gh band widt h calc ulations.

7.2 .5

A DESIG N EXA M PL E

We've seen that the me thod of open-circuit time co nstants promi se s

(0

simplify de-

sign while conveying important insight. Let's no w carry out a n ac tua l desig n to sec if itlive s up to this promi se . Suppose we want a n am plifier w ith the foll owin g specific ations: m ilage gain magnitude: > 18 d B (o r a bo ut a facto r o f 8 ); -3-d8 band width : > 450 MH z (i mp lies a maximum OC r sum of ......350 ps ). Furthermore. assume that we m ust meet these speci fica tio ns w ith a 2·kQ so urce resistance driving the input and a t -p f ca pacitive load o n the o utput. In a truly practi cal design. there wo uld usua lly be addition al specifications (such as ma ximum a llo wed power consumption , dyn am ic ra nge , etc.). hut we' ll keep the design space restricted for now,

Further suppo se that we arc to meet th ese specifications with tnm sistors fro m the 0.5-11 III (drawn) techn olog y described in Chapter 3. To simplify the process. Il'I us uscjust one size of device, and ju st one bias current for a ll tran sistors. In u bette r design. of course. nne wo uld generally usc different biases and differ e nt device sizes. but we need to impo se some a rbitrar y co nstraints if we are to co mplete our task in fini te space! Arbitrurily selec ting a per-t ra nsistor bias c urre nt of 3 mA , a 150 -ll m- wide N I\10S transistor in this process tec hno logy has the followi ng approx ima te e leme nt values whenopenuing in saturatio n:

c,. = 220 fF,

C.b =

'0= 2 kO ,

no f F,

C gd = 45 f F,

g",= 12 mS.

g ", />

C d />

= 1.8 mS.

= 90 fF:

154

CHAPTER 7 BA ND W IDTH ESTIM ATIO N TECHNI Q UES

C t "" lpF

· We' ll see later why this i..n't labeled MI

FIGURE 7.2 . First-cut de1ign (bia sing not shown).

Rs '" 2kU

C t . Cdh2"" l.U9pF

Rt "" '-_--'-':;=="----'----'-'''-''---' I k
FIGURE 7. 3 , Incremental mode l a f first·pass de1ign.

Even though some of the capaci tances are bias vohage -depende m . we will assume that they are co nstant at the values shown. The on ly way 10 start a design is, we ll. to start . Put something (al most anything) down . It 's easier to edit than to create, so virtu ally any rea sonable initial condition is acceptable. A few simple calculations will let you know fairly q uickly if yo u're on the right track , and you ca n alway s obsess later abou t the part iculars. So. let's start with the co mmo n-source (CS ) co nfiguration (after all. it provides voltage ga in, and has a mod erately high inp ut impedance) . In all that follows, we "Il neglect the details of how biasing is laken care o f (since we ' re focu sing on dynamic perfo rmance issues), but be aware that any practical design must include careful attention to the bias problem. Recalling that (neg lecting bod y effect) the voltage ga in from gate to drain of a basic CS amplifier is - Rm Rc , and bei ng mindful that we do have to worry about gain loss from the additionalloadi ng by the transistor 's own output resistance, let 's shoot for a gmR" produ ct that is 50% larger than the gain specification. With the resulting choice of 12 fur 8m R" , we find that we must selec t Rs. = I kQ , Ou r ci rcuit then appears as shown in Figu re 7.2. The correspo nding incremen tal model is then as sketched in Figu re 7.3. Note that the source - bulk potent ial is zero. so the back-gate tran sco nductance conm bures zero current and the source - bulk ca paci tance is shorted out. From the model. it' s ea sy to see that the low-freq uency voltage ga in j ust barely meets specifications: (18) Av = - g,.d Rt. II r,,) = - 8.

7.2 THE METHOD O F O PEN ·C IRC UIT TIM E CO N STA NTS

155

5V

fiGURE 7.4 . Secood pan :

cascade amplifier.

~ow,let' s

estimate the bandwidth

Td h2

sec j ust how bad the news there is:

= Caa r-na = TL

RW "

10

C db2 ( R L

II ro) = 0.06 ns:

= Cd RL II ro) = 0.67 ns: I

CO.... -t ns + 0.84 ns + 0.06 ns + 0.67 ns)

(21) (22)

" SOO Mrps.

(23)

We see that our bandw idth is abo ut 79 MHz (S PICE says 86 MHz), so we 're quite a hitshy of our goal o f 450 Mll z. Now, who's the big culprit? From our four calc ulated time constants, we see that there are two similar-sized ones. The larger of these is associated with the drain- gate capacitance. CNd 2 , even though that capacitance is numerically the smalles t, beca use its effec t is Miller-multiplied hy the gain. So, if we are to improve bandwidth , we must figure out how to mitigate the Miller effect. Recall that the Miller effect arises from connecu ng a capacitance across two nodes that have an inverti ng voltage gain between them. So, one po ssible solution would be 10 distribute the gain amon g N stages. rather than try to get all of our gain out of onestage. You are encouraged to explore this prom ising option independ ently. Another possibility is 10 iso late (somehow) the offending capaci tance so thut it no longer appears across a gain stage. We will pursue this approach as we attempt to gel all of our gain in one stage . The cascode amplifier eliminates the Miller effect precisely by performing this iso lation . Co nsider the circ uit shown in Figure 7.4. A.~ usual, the value of VB1AS is not terribl y critical. It j ust has to be high enough to guarantee that M2 stays in saturation. and low enough to guarantee that M3 stays in

156

CHAPTER 7

BAN DW IDTH ESTIM ATION TECHN IQ UES

saturatio n. A va lue of 2.3 V satisfies these co nditio ns com fortably in this particu lar case . and is the value used in the S PICE sim ulatio ns. Befo re plunging mindlessly ahe ad into a pile of computations. le t' s think about how th is ci rcuit works. T he input vol tage is co nverted into an o utput c urre nt by transisto r Mz ( i.e.• M 2 is a trunscond uct or ). Transistor ,\'1) merely tmn...fe rs thi s current to the outputload resisto r. Now. the output is at the d rain (If M.l. while the input is at the gate of M 2 • and the re is no ca paci ta nce directl y acrm.s the two nodes. Hence. there is very little Mill er mu ltipli cation. a nd we e xpect a s ig nifica nt improve me nt in band width .' Th e iso lat ion provided by cascod ing al so ha s a be nefici al e ff ect on the ga in. Voh· age c hanges a t the d ra in of M .t have hard ly any effec t on the dra in c urre nt of M2. Hence. the o utput c urrent c hanges little. A n equivalent state me nt is that the output resistance has inc reased . In th is particul ar case. the inc rea se is enough to e liminate the e ffec t of ro for a ll practical purpose s. We would therefo re e xpect a ga in very ncar - 12. a nd S PICE simulatio ns show that it is abou t - I I. If th is exce ss gain holds up as the design evolve s. it lIlay be trad ed ofT for improved ba ndwid th . if needed or desired . Returning to open-c irc uit time -con stant estim ates of ba ndwi d th . draw the model co rrespo nd ing to this ca scode co nnectio n. and calc ulate the resist ance facin g each capacita nce. Our o f laziness. the re will he no mo re inc rem ent a l model s from he re on out. so yo u're o n yo ur o wn no w ("so me asse mbly requ ired" ): r",. 2

= C#JZrllfZ = (220 fF )(R s ) = 0 .44 ns

T/(dZ = Clld2rlld2 ::::: (4 5

(unc hanged );

(24)

fF)(R.~ + _ 1- + g",2R.\. _ I_ )

::::: 0.14 ns

g... J

g... .l

(be ttert j.

(25)

Equation 25 is a hit approximate becau se we arc negle c ting the eff ec t of Nm h .l and (
r" in thi s ca lculatio n

t"xd = C x,d r x<)

~

(220 fF )( I /~"' J ) ~ 0.018

0 ...

(26)

mew ).

( 27)

I To com ple te the arg umen t. note that .he gai n betwee n the gale and drai n tlf M: is - I. i... no! multiplied by very much at all.

MI

that C. n

7.2 THE METHOD O F OPEN ·CI RC UIT TIME CO N STANTS

157

5V RI. = IkU V B1AS

S I( " IN

1

2U 1

'']--

..

"1 2

r-!

-

3mA

- ,-

-

. . \lO CT

TC,-=I pf 7

FIGURE 7.5 . Third pass: coscode a mplifier with oefput sccrce follower.

T. /))

=

TJ /)J

Tdo 2

=

(130 fF)( I / g",} ) ~ 0.011 ns

C . /)) T./)} :::::

=

C d /)) Tdb ) ::::::

c.Jb2 T,Jb2 :0::: TL

=

(90 f F HRd :o::: 0.090 ns

(new) . (new ).

(29)

{better).

(30)

(90 f F)O / gm} ) :O::: 0.008 ns

CLTL

RW '"

=

CL R L

I ( 1.75 ns)

= I ns (worsel). '" 570 Mrps.

(28)

(31)

(32)

Wilh a new {est ima te...d) ban d width of a bout 90 Mill (S PICE says 109 MH z). we ca n see thar rhe cascode co nncc tio n has given us a substantia l improve me nt in band width . BUI we still have a lo ng way to go . Looking at the new big offender . we sec that it involves the load ca paci tance Ct.· Driving it w ith such a hig h ( I -H 2: ) resistan ce is ob vio usly the prob le m. so we sho uld be able to red uce that time co nstant to a sma ll value with a so urce foll ower; this is shown in Figure 7.5. Again. we' ll ignore bia sing det ails. J ust ass ume th ai we put a c urre nt so urce (o r a plain old resisto r) in the so urce leg of M-4 to bias it to 3 rnA . For purposes of timeconstant calculatio ns, we' ll see that the resistance of the hias network is easily made negligible, so it doesn't rea lly matt er w ha t we assume. The source follower doc s not qu ite have unit ga in beca use there is a c apaci tive voltage divisio n between Cg ,' 4 a nd c.b4 . A careful ca lculation. verified by S PICE. reveals that the gai n has d rop ped from - I I to - 9.5. Fort unate ly. this value is still in excess of the desired value. Calculating the tim e co nstan ts for this iter ation yields the following li.st: r RJ 2 = OA-I ns

(unchanged) ,

(33)

= 0 . 14 ns

(uncha nged ),

(34)

t gJ2

158

CHAPTE R 7 BA.NDW IDTH ESTIM ATIO N TECHN IQ UES

,v R S '" 2l U \ IN

' ) --

- ,-

-

"

\ 'OIJT

+

FIGURE 7.6 . Fourth pa~~ : cascode a mp/ifier with twa

source follovven .

t. ua = (UX)S ns (unchange d).

t f/J 4

f , . ..

=

(35)

t . 1>3

= D.O I I ns

(unchanged).

(36)

t A" )

= 0.0 18 ns

(unchanged) .

(37)

t RJ J

= O.()45 ns

(u nc han ged) .

( 38)

t dbJ

=

(u nc hanged) ,

(39)

{U)l)O

C f/ d J r RJ 4 ~

ns

(45 fF )( Rd

~

0.045 ns (new) ,

= C, .4r"... :::::: (220 fF H I/ g..... ) :::::: U.018 ns (new ).

(40)

(41)

Th is Jast eq uation is a hit mo re approximate than usu al because of the neglect of 8mb 4 w ith a I-k G! drivin g re sista nce (the co rrec t value is about 0 .045 us). A more care ful derivation shows tha t the resistance in Eq n. 4 1 sho uld be multiplied by about ( I + K",I>4 R L ) . so one may ne glec t thi s fac tor o nly if 8 mh RL is muc h smaller- than unity. Fortun ately. th is part icu lar lime co nstan t is no t dominant in thi s case, so the large percentage error in f A"" has an insignificant effect on the overall tim e-constant sum. Continuing: T. b 4

=

C .' b 4rJb4 :::::

(I 30 f FH I/Xm.d : >: : O.OII ns

rt. = CLr L :::::: (1 pF )(l / g",-t ) :::::: 0.080 ns BW :::::

I :::::: 1.1 Grp s . (0.906 ns)

(new).

( bette r!),

(42) (43) (4-1)

So now we 're up 10 a bout 175 M i ll (S PICE says 222 f\.11Il); we "only" need to pick up anothe r fa cto r u f abo ut 2 in ban d width . Lo ok ing over our tarest li ~l of rime' (...o nsta nts. we see that r~ , 2 dominates by far because o f th e 2-kQ so urce resistance. On e o bvio us remed y is to add a n input buffer to red uce the re sistan ce driving the ga le- so urce capacita nce of M2: see Fig ure 7.6.

7.2 THE METHOD O F O PEN -CIRCUIT TIME CO NSTA N TS

' 59

A recompu tation of the gain reveals that the slight attenuation of the added source follower takes us down to a gain of - 8, leaving us with no more marg in. With these cha nges , we expect In get pretty close 10 the desired bandwid th because r, .2 is about half the total, and we ca n prob ab ly drop it to ncar zero. Reco mputing the lime co nstants, we obtain:

f ll'J1

=

f d1>2

= 0.008 ns (unchanged "

(45)

f , bl

= 0.0 1I ns

(unchanged) .

(46)

f ,.sl

= 0.018 ns

(unchanged ),

(47)

T, d .1

= O.().t5 ns (unchanged).

(48)

f dbl

= 0.090 ns (unchanged) ,

(49)

f ,.d "

= 0.045 ns

(uncha nged) ,

(50)

f ,.. ..

= 0.018 ns

(unc hanged),

(5 1)

f , b"

= 0.0 11 ns

(unchanged) ,

(52)

tt.

= 0.080 ns

(unchanged),

(53)

C.,d , • •1 ::::::

(220 fF )( I/ g",d :::::: 0.018 ns

(new) .

(54)

Again, Ihis last calculation is in error beca use 8",1>1 Rs is 3.6, so the time constant really ought to he multiplied by ( I + 3.6) = 4.6 to yield about 0.083 ns. A more careful ca lcul atio n that also takes ' ot into account reveals that the lime co nstant here is in fact about 0 ,086 ns. We now have f, rll

=

f , 1>1

= Cs ,M,b1 :::::: (13 0 fF )( I/ g",d :::::: O.DI I ns

f l/., 2

=

f i/d 2

=

Cllr/l ' 1/d1 :::::: (45 fF )(Rs ) :::::: 0.09 ns

C I/J2 r l/_,2 ::::::

CI/,l2 r ,lJ2 ::::::

(new), (new),

(56)

( belled ),

(57 )

(220 fF )(I I.J:md :::::: oms ns

(45 fF)[ _I-

+ _ 1_ +

8 ", 1

:::::: 0.01/ ns BW ",

8m 3

8"'3(_I_)(_I_)J

(betterl). 1

(0.4 74 ns)

(55)

'" 2. 1 G'P"

8ml

8 m3

(58) (59)

The estimated bandwidth has now increased 10 about 340 MHz . Ow ing to the conservative natu re of the estimate. it is reaso nable to expect the actu al bandwidth to be quite close to our goal. In fact . SPICE simulations show that the bandw idth is about 540 MHz. well in excess of the desired value. If desired. so me of this excess bandwidth could he exchanged for increased gain.

160

CHAPTER 1 RANDWIDTH ESTIM ATION TE CHN IQ UES

Suppo se, though , Ihat SPICE were to confirm yo ur wo rst fears. and you lind that the a mplifier just doesn 't quite ma ke it. Arc the re any other modifica tions that you could try ? Th e an swer , of course. is Yes. O ne optio n was passed over earlier: distribut e the gain alllong se veral stages. Us ing two or mo re stages, it would he a trivial matter to hea l the bandwidt h speci tlcrulon by a handy marg in. Th ese trick s a rc by no mean s the on ly one x, and we w ill spend a co nside rable a mount of lime in Chapter 8 ex plorin g some importa nt altern ative met hod s. However. to w he t your appetit e a nd stimulate MIme thinking, he re are some vag ue allusions to othe r possibilities . T he method of open-circ uit time con stants assumes an all-po le tra nsfer function. and gives more acc urate answe rs if all the po les are real. Co nside r the effect of purpo sefully violating the...e assum ptions by allow ing zeros and /o r co mplex poles, Careful place ment of zeros (antipol es) or complex poles will e xte nd the ba ndwidth. although the frequ ency respo nse may 11 0 longe r be monotonic. O ne way to form com plex pole.. if> throu gh feedb ac k {j ust think of a two -po le roo t locu s, for exa mple]. or th rou gh the resonance of inducto rs (real or synthetic) with capacitors. Th e surprisin gly large band widt h of the last ci rc uit in the c hain of design ite ration!'> is largely du e to the to nn at ion of complex pole s arising from the inter action of the gate-source capacita nce of M1 w ith the ;IIJI/('lil'(' output im pedance of so urce followe r M ,. Zeros ca n be formed . for exam ple, by ca pacitors in par allel with ...o urce bypass resi...tors. As we' ll sec, a j udicious chok e of ca paci tor value ca n ca u..e thi.. ze ro to cance l a ban dwidth-li miti ng po le. Anothe r possibilit y, made most pract ical in diff eren tia l sy...tcm s, is to use positive feedback to ge ne rate negative ca paci ta nces. These nega tive capaci tance s ca n cancel po sitive o nes to yield band width increases. O f course . there is a c hance of unsrable behavior that m ust be ca refully watc hed . but thi s method , called neutralization, can yiel d useful bandwidth im prove me nts. We will turn to a det ailed e xami nation of these themes very soo n.

7.2 .6 SUMMARYO F O PEN·C IRC UIT TI ME CO NSTANTS We ha ve see n tha t the me thod of open -ci rc uit time constants is an extremely valuable too l fur design ing am plifiers for good dyna mic perfo rmance . mainl y because of its ab ility 10 iden tify the proble m areas of the circ uit. Because of the trem end ous insights gained with very mod est effort. we are gene rally w illing 10 over look its qu antitat ive limitations, such as the ofte n highl y conse rvative nature of the es tima ted handwidth . As lo ng as we take ca re to usc the method on ly with models tha t apply to the high-frequency regi me. we are assured of reasonab le answe rs. As a parting re mar k. it should be noted that the influence of inductances can be inco rpo rated as well into the meth od . altho ugh with ge nerally unsatisfac tor y results for rea sons that w ill he explained ...hortl y,

7. 3 THE M ETHO D O F SHO RT·C IRC UI T TIM E CO NSTA N TS

16 1

The must intuit ive way to unde rstand how one incorpo rate s the effect of ind uctances on bandw idth is 10 recall that each timc-co nstar uterm represents an individ ual. local co ntributio n to the band widt h limitation : we Ireat the sys tem
7.3

THE METHOD Of SHORT·CIRCUIT TIME CONSTANTS 7.3 .1

INT RODU CTION

We've already seen how the meth od of ope n-ci rcuit rime constants allows us to estimate the high-frequ ency - 3-d 8 point of an arbitrarily co mplex sys tem by decemposing thc bandwidth computation into a successio n of first-orde r culcukuions. Each (If the time co nstants rep resents a local handwidth degrada tion term, and the sum of these individual degrada tion terms equals the recipr ocal of the overall bandwidth . As we saw, open-circuit time constan ts are valuable beca use they identify which e tcmenrs limit the bandwi dth . Suppose that. instead of estimating: the high -frequ ency - 3-d B po int . we wanted to find the /ow· frequency - 3-d B po int of an AC-coupled system. How would we calculate how large the co upling capaci tors must be in order to achieve a s pecified

,.2

CHAPTE R 7 BANDWIDTH ESTIM ATIO N TECHNIQUES

low-freq uency breakpoint '! Fortunately. we may invoke a procedure that is analogo us to the method of open-circ uit timc co nsta nts. Th is dua l technique is known as the method of short-circuit time constants (SCr s).

7.3 .2

BACKGROUND

In the method of open-circuit time constants, we assumed that the zeros of the netwo rk were all at infinitely high freq uency. so that the tran sfer functi on co nsisted only o f poles. In the ca se of short-circuit time constants, we instead assum e tha t all of the zeros are at the origin. and that there are as many poles as zero s. Thu s. me co rrespo nding system function may he wri tten as follows: (60)

(S +SI)(S + S2 ) ' " (s +S,,) '

where the various pole frequ encies may or may not be real, and k is simply a constant to fix tip the scale factor . Multip lyin g out the tenus in the de nomin ator leads to a polynom ial thai we shall express as (61) where the coe fficient b l is the sum of all of the pole frequencies, and b" is the product of all of the pol e freq uencie s. ( In general. the coe fficie nt o f the sj term is co mputed by form ing all unique products of the " frequ encies taken j at a time and summing all " l/j !('l - j)! such products. j We now assert that. near the low-frequency -3- d B breakpoi nt . the higher-order terms do minate the denomin ator. so that we obtain

b "

Vo (S )

Vj(s ) :::;:: s"

b

+ b.s..- l

=

S

+ L::=l .f; •

(621

The low frequency - 3-d H poi nt of o ur origina l system in radian freq uency as esumated by this first-ord er approx imation is then simply the sum of the pole frequencies: , W I ::::::

hi =

L

Sj

=

WI .rot·

(63)

; =1

Befo re proceeding furt her. we should conside r the co nd ition s under which our neglect of the lower-order terms is ju stified . Let us examine the denom inator of the tran sfer function nca r our estimate of W I . For the sake o f simplicity, we co nsider a second-order polyno mial with purel y rea l roo ts. SI and S2. Now, at our estimated -3-d B freq uency, the ori ginal denomi nator pol ynomial is

(64)

7.3

THE METHO D OF SHO RT·CIRC UIT TIME CO NS TANTS

163

Sub..tin ning our expre..sion for the esum ated - 3-d B point ......e obtain

(65) Clearly. the 13..t tcrm is small co mpared w ith the magnitudes of the other terms. Thus. the neglect o f all but the two highest-order terms invol ves little error. The worst case occurs when the two pole freque ncies arc equal. and even then the error is not terrihly large, Extendin g these arguments to polynomials o f higher order reveals that the estimate o f the low-frequency cutoff based simply on the coefficie nt hI is generally reasonable. since the higher-order terms do in fact dominate the denominator. Furthermore. the low-frequency cutoff estimate is conservative in the sense that the actual cutofffrequellcy will almost always be as low a.f or lower than estimated by this method. So far. all we've done is show that a first-order estimate of'thc band w idth is po ssible ifone is given the sum of the pole frequencies ( = b l ) . Of course. if we kne w the pole frequencies then wc could co mpute this sum d irectly. Fortunately, as was the case with open-eircuit time constants. it is possible to relate the desired pole -frequency sum. hI. to (more or less) easily computed network qu antities. The recipe is thus as follows. Consider an arbitrary linear network comprising only resistors. sources (dependent or independent). and III capacitors, Then: (I) compute the effec tive resistance Rj~ facing each j th capacitor w ith allof the other capacitors sllOrt-circuited (the subscript s refers to the short-circuit condition for each capacitor); (2) compute the "short-circuit frequency" I / ( RjsC j ) ; (3) sum all m such short-c ircuit frequencies.

The sum of the reciprocal short-circuit time constants formed in step (3) turns out to be precisely equal 10 the sum b l of the pole frequencies. Thus. at last . .....e have (66)

7.3 . 3 O BSERVATIO N S AN D INTERPRETATION S

The method of sers is relatively simple to apply for precisely the same reaso ns that OCr s are easy to apply - namely. eac h time -constant calculation invol ves the computation of just a single resistance. although one must agai n be wary of the impedance-modifying potential of dependent sou rces. In any event. the amount of computation required still is typically substantially less than that needed for an exact solution. Again. the greatest value of the technique lies in its identification of those elements implicated in bandwidth limitation s, The reciprocal of each j th short-circuit time

164

CHAPTER 7 BAN DW IDTH ESTIMATION TECHN IQ UES

co nstant t o,' the tow-frequency - 3-dH Im'ok"" ;,,, that the cinuit wau kl exhibit if that j lh capacitor 11'1'1'(' the olfly capac ito r ill the system, The method o f S e T S the n states that the linear co m binatio n of the se indi vidu al , loc ulh minuions yields a n es timate of the overa ll - -'-dB point . The value of SeT S derives d irec tly from the identification a nd approxi ma te q uant ificat ion of the loc al degradat ion te rm s. Althou gh the developm ent so far has co nside red only c apacita nces . ind uctances also can be incorponued into the method . Howeve r. thei r presen ce oft en co mplicates significa ntly the deci sion of whic h reac tive cle me nts really be lon g in the computation. The mo st in tuit ive way to under stand ho w o ne incorporates the e ffect o f indueranees Oil bandwidth is 10 rec all that each reci procal tim e -c on stant term represent s a n ind ivid ual. local contri bu tio n to the lo w-frequen cy cutoff '; we tre at the system at each step o f th e ca lculatio n as if th at jth reactive e lement were th e sole lim iting one. So, ev ide ntly, o ne treats a ll o f the inductors as open ci rcui ts whe n comput ing the appropriate e ffec tive resista nces . Th e R IL fre q uencies arc then ad ded to the various II He freq uen ci es to yield the tota l es tima ted low-freq uen cy cu to ff po int.

7.3 .4 ACCURACY O F SCn As with OC r s. truncati on o f the denominator po lynomial me ans tha t on e mu st be ca re ful not to place too much faith in the ab ility of SC TS to provide acc ur ate estimates o f W I in a ll cases. This c ave at notwithsrundi ng. it sho uld be cl ea r that .111 SCT es timate is in fact e xact for a first-order network since I/O truncation of term s is involved there. No l surp ris ing ly. the n, the SCT es timate w ill be qu ite accurate if a network of high er orde r happe ns to be domina ted by o ne po le ( her e . tha t mcuns that one pol e is much hiKher in freq ue ncy than a ll the o ther pol e s ).

7.3 . 5 O THER IMPO RTANT CO NS IDERATIO NS A ltho ug h ap plication of sho rt-c ircuit time cons rams i.s prcu y stra ightfo rwa rd, there arc o ne or two tine points tha t me rit discu ssion. As wit h OCr s, no t all capacito rs in a net work belong ill the SCt c alculatio ns. For instance. the capacit ors in a transistor mod el almost never be lon g. Hlind applica tio n o f the SCt method wo uld lead 10 c urious (erroneo us) re sults (an d a who le heap ofextra calculutions ). Th e problem is easily understood if yo u remem ber th at we assumed that uII the zeros arc at the ori g in, and that the num be r o f po les eq ua ls the num ber of ze ros. so that the ga in in the limit of in finite ly high freq uen cy is tlar. not ze ro. We violate these assumptions rather seve re ly if we incl ude all the stuffth at cause s hi gh -freq uency ro llolf {i.c.• a ll ihe stuff tha i OCr s worry abou t). The so lutio n is 10 preprocess the ne twor k prior 10 a pplication o f the SeT method . That is. recogni ze thut a ll the capaci tors thut limit hig h freq ue ncy gain a rc effectively open circuits re lative to the impedances aro und them at freq uencies ncar W I . Th us. onr mav tlpply SC u tit/I)'to motlefs thtlt art' tll'l'mpritltt' 10 the iow-frrqu encv n'Kimt',

7.3

THE METHOD O F SHO RT-CIRC UIT TIM E CO NSTA NTS

16'

" 2

'" c,

FIGURE 7.7. Ccsccde amp/iller.

It is usually o bvio us whi c h ca pac ito rs are to be ignored (considered open ci rcuit s) . but there are occa sion!' when o ne is not so sure. In these cases . a simple thou gh t experiment usually suffices to dec ide the issue. Now SC t s a re co nce rned o nly with those capaci tors that lim it low-frequen cy gain. As a co nseq ue nce. the rem o val {i.e ., the open-circuit ing) of a ca paci to r th at be longs in the SC t ca lculation sho uld resul t in a decrease in low -freq uency ga in. T he test. the re fo re , is to c o nside r exciting the network 0.1 1so me low freq uenc y a nd ima gini ng wha t wo uld happe n to the ga in if the capacito r in questio n we re ta ke n o ut of the circ uit (ope n-c ircuited ). If the ga in wo uld dec rease then the ca paci tor belongs in the SC Tcalc ulatio n. since we infer from the re o sult of the thought e xpe riment that the capac ito r does ind eed lim it the lo w-freque ncy gain. If the gain wo uld no t c hange (o r e ven incr ease ) upon remo va l. tha t capaci to r should he open-c ircuited and le ft o ut of the co mputation. Again . as with DCrs . the necessary co ncl usions ca ll usua lly he rea ched witho ut ta king penci l to paper. To under scor e these issues. le t's co nsider a specific example, the c ascod c um plifi er. As see n in the acco mpanyi ng sc he matic ( Fig ure 7.7 ). the re lire th ree capacito rs. The input co upling ca paci to r, C n, rem o ves allY DC from the input s ig na l to preven t upsetting the bias o f the amplifie r. So urce bypass ca pac itor Ct· is chose n to sho rt the source of M I to g ro und at a ll sig na l freq uencies to restore the ga in lost by the so urce degeneration resistor. Bias byp ass ca pac ito r C b g uara ntees that the gate of M l is a n incrementa l g ro und a t high freq ue ncies to kee p the o pen-ci rc uit time -con sta nt sum small. Let's usc o ur tho ug ht -expe rime nt tec hniq ue to ded uce whic h of these three capacitors belo ngs in the SC r calc ula tio n. If we begin with C in • we note tha t the low frequency gain doc s dec rease ( 10 zero. in fact) if we ta ke it o ut of the c ircuit . Hence. it belongs in the ca lculation. Simi la rly, C E be longs in the ca lcula tion bec ause its removal also red uces the low-freque nc y ga in.

166

CHA PTE R 7 BA N DW IDTH ESTIM ATION TECHN IQ UES

What about C,,? What happens to the low-freq uency gain if we lake it out of the circuit? Th e an swer can he ei ther trivial or Ion J eep to fathom. depend ing on how you approac h the question. Th e easiest way 10 gel 10 the answer is In recogn ize that M l is a dev ice that converts an incoming vo ltage to a n inc rement al drain current. All M 2 does is lake this cu rre nt a nd pass it 0 11 10 the output load resisto r. Therefore. whether or no t the gate of M2 is an incre ment al ground is irre le vant. a nd the removal of C" will therefore have essentially no effect on the low-freq ue ncy ga in. Thus. C. doe s I W f belong in the calc ulation. Th e importa nce of not blindly a pplyi ng the meth od ca nnot be overe mphasized . In fact , on e of the earliest expositio ns of the me thod erroneously includes Cb •2 On e last issue that deserves some atte ntio n concern" the rela tion ship between the reciprocals of the indivi dual short-circuit lime constant s and the pole frequ encies. We have asserted (again without formal proof) only that the sums of these frequencies are equal to each other. Th erefo re. just as with ope n-circ uit tim e constan ts. one mu st resist the temptation to equ at e each reciprocal short-ci rc uit time consta nt with a corresponding pole frequ enc y. Since the number of short-ci rc uit tim e constants and the nu mber o f poles may not e ve n be eq ua l. one ca nnot e xpect eac h S'Cr to equal the time co nsta nt of a pole in general.

7.3 .6 SUM MA RY A ND CO NC LUDING REMARKS We ha ve see n that the me thod of short-circ uit time consta nts sha res w ith its dual. the method of open-circuit time constants. a num ber of adva ntages and disadvan tages. It is an invaluable too l for designin g a mplifiers because of its ability to ide ntify the pr oblem areas of the circuit. Because of the tre me ndous insights gained with extrem el y mod est effort , we a re ge nera lly willi ng to over look its qu a ntitative limitetiona , suc h as the often high ly co nservative nature of the est imated low-freq uency c utoff poi nt. As long as we take care to app ly the met hod only to model s that apply tothe low-freq ue ncy regime. we are assured of reaso na ble an swe rs. In concl usion, the method of short-ci rcuit time consta nts helps on e design circuits to satisfy a given low-freque ncy cutoff spec ification. Despi te the quantitati ve shortco mings of the method . the valuable intuit ion provided a nd the labor sa ved arc more than sufficie nt com pensa tion.

7.4 FURTH ER READI NG Fo r a proo f of the equality of the sum of open -circ uit a nd pole time constants. see P. E. Gra y a nd C. L. Sea rle. Electronic Principles ( Wiley. New York. 1969 . PI'. 531-5).

2 P. E. Gray and C. L. Sear le. Electronic Principles, Wiley, New York. 1%<1. PJI. 542- 6 .

7. 5 IU5ET IME . DELAY. AND BAN DWIDT H

16 7

By the way, this wo rk has been extended 10 allow the exact co mputation o f all the poles of a net work . It involves the computation of various cross -prod ucts of ope nand .\l lOrf.C ircuit lime constants to obtain the coefficients of all the powers of s in the denominator of the transfer functio n. O riginally developed by B. L. Cochrun and A.Grabel, it was simplified by S. Rosen stark , but the method is still sufficiently cumbersome (fro m a band-calculation viewpoint) thai the insight-to -wor k ratio is usually unfavorably small. However, it occa sionally proves usefu l (especially if you choose ttl automate the procedu re by writing your ow n code). For more infonnation, see Cochrun and G rabe l's paper, "A M ethod for the Dete rmination o fthe Transfer Function of Electronic C ircuits" (IEEE Tram . Circuit Theory, v. C T· 20, no. I , January 1973, pp. 16 - 20 ), and Rosen srark's book , Feedback Amplifier Principles (Macmillan, New Yurko 1986, pp. 67- 77 ).

7.5

RISETlME , DElAY, AND BANDWIDTH 7.5 .1 INTRO DUCT IO N

The method of open-circuit rime co nstants allows one to estimate the overa ll ba ndwidth from local RC prod ucts. In this sec tion. we develop a number of ways to esrimale bandwidth from time -dom ain par ameters . In thi.. connection, on e occe..ional ly encounters various rules of thu mb, such as "bandwidth limes risetime eq uals 2.2: ' "risetimes add quad raticall y" or " buy low, se ll high ." As useful as they are, however, they aren't entirely reliable. To identify when these rules o f thumb hold, we now turn to their formal derivation. We start by deri ving a rule that appears trivial, obvio us. and irrelevant: T he total delay of a casc ade of sys tems is the sum of the individual delays. The reason for starting here is to introd uce some ana lytical techniques and insights that have far broader applicability. As always. don' t worry too much about all the mathematical minutiae; the deri velions are provided simply for co mpleteness. Those interested primarily in the application of these relationships may skip the interven ing math and simply take noll' of the results.

7.5 .2

DELAY O F SYSTEMS IN CASCADE

We shall see that many analytical advantages acc rue from definin g delay (and later, risetime) in term s of moments o/' /'l' impulse response. As seen in Figure 7.8, one delay measure is the time it takes for the impulse respon se to reach its "center of mass," that is, the normali zed value of its first mom ent : o

T

J::" ,h (t) d,

=f~h(n d"

(67)

168

CHAPTE R 7 BANDW IDTH ESTIMA TION TECHN IQUES

hf I)

TD

FI GURE 7.8. Illustrative impul!le response.

Thi s quantity is al so known as the Elmore delay in some of the lite rature. after the fellow who first used this mo ment-ba sed approach ..l Th is part icu lar measure of time delay derives much of its uti lity fro m the fact that it is readi ly rel ated to a number of Fou rier tra n..form identities. allowing us to exp loit the full pow er (If linea r syste m theo ry. Speci fically, the first mome nt is

1

I

~ thlt )dl = - ~I (I H (j ) J ~lr

_.,..,

elf

(681

.

f __ n

i:

Th e normali zation factor is s imply the DC ga in. (69)

h (t)dt = 11(0 ).

so that

,

70

:::::;:

J':x. th (r ) d l 00

L"" 11(1) dt

= - .

1

d

j2rr ll(O) df

_I

I/ ( j)

.

(701

1=0

Using this definiti on . line linus that the Elmore delay for a s ingle- pole low-pass sys-

tem is ju st the po le time co nstant, T . Now thai Mon sieur Fouri er has graciously helped us out by pro vid ing a definition o f time delay pur ely in te rm s of his tran sforms. the derivation becomes straightfor ward. Spec ifically, conside r two sy ste ms with impu lse re sponse s " l(l) and h 2(t ) with correspondin g Fourier transform s 111( / ) a nd 112(/ ). From ba sic linear system

.1 W, C. Elmore, " The Transient Respon se of Damped Linear Net.....llrb w ith I' ••rnc ular Regard 10 \\'idehand Amptlhel1o: - J. AI,pl. Phy.f ., v. 19. January 19·UI. pp . 55- nl.

7. 5 RISETlM E, DElAY, A N D BANDWI DTH

169

theory. we know thut the Fourier tran sform of the se two sys te ms in casc ade is j ustthe product of the indi vidu al tran sfo rms, so thor II tlll = H I Ill · The o vc rull tirnc delay is therefore T /) . t<>1

=

(7 1)

which we may ex pa nd to obt ain ( 72)

From this we immed iatel y (well o kay. may be no r qu ite immedim t'ly ) not e that T/).I"e

= 1'01 + TD!.

(7 3 )

which was to he show n. We see that usc of thi s part icu lar definition of lime delay has led us til the intuitively satisfying resu lt Iha t the o vera ll delay of a cascade of sys te ms is simply the sum of the ind ividu al del ays .

7.5 .3

RISETIME O F SYSTEMS IN CASCADE

Deriving a riscrimc additio n ru le prese nt s a so me wha t more sig nifica nt c ha lle nge. In particular. it turn s ou t that de vel opment s based on the conventio na l I O- (}O% dcl inition of riseume a rc a lmost ce rtai nly doo med to fa il bec ause o f the a na lyt ica l d ifficulties invol ved wit h co mbi ning expo ne nt ials of d iffering time co nsta nts. S ince Ihi s measure of risc tim c is arb itrary a nyw ay. we migh t as we ll sec k an nltcm ati vc ( hut still arbitrary) de finition o f risetimc that pe rm its tractabl e a nalysis . Just as we e mployed the first mom ent of the impul se respon se in defining the tim e delay, we lind the sec o nd mom ent use ful in defin ing the risctimc . Referring ugnin to Figure 7,H. note that the qu a ntity ~ T is a mea sure of th e d uration o f the impulse response and hen ce also a mea sure of the rise tim c of the ste p resp o nse (si nce the step response is the integ ra l oft he im pulse re spo nse). S pec ifica lly. ~ T is tw ice the "radius o r gyra tio n" about the "ce nter o r mass" (Tf) o f 11(1) , Reca lling .~ Ol IlC dusty relationships from first-year calculus, we lind that

(-/1"T )' --_[ r/00:,/ h 1t) dt - (T,, )' ) -

-00 I1(1) d I

(74 )

.

Again this de finition a llo ws the usc o r Fourier tm nsforrn ide ntities . In pa rticular,

00 t' h (t)dt =

1

- "<)

so thai

-" 1 ''dI'' H( f) I ( _ JT) (

f =U

.

(75 )

CHAPTE R 7 BANDW IDTH ESTIM ATION TECH N IQU ES

170

(76)

Here we have made use of the equation for de lay developed in Secti on 7.5.2. Simplifying ( !). we ob tain

The Elmore risctimc for a sing le- po le low-pa ss syste m is 2r . Proceeding as in Secti on 7.5.2. we now co nsider two systems. each with its own risetime. Then I

'n>e.....

(78)

After a ...mall algebraic miracle. this leads to the desired resu lt at last (79)

Thus we see that the sq ua res of the ind ividua l rise times add linearly to yield the squ are of the ove rall ri setime. Stated alternat ively. the individu al n setimes add in root- sum -squared (rss) fashion to yie ld the ove rall risct imc: (80)

Now that we've derived these result s. we should spe nd SOIllC tim e discu ssing condition s under which the foregoi ng form ulas may yield un sati sfactory estimates of delay or risetim e. In particular. co nside r w hat ha ppens 10the ca lculated delay and risetime if the integral of h (l ) is nearl y zero . This situation might arise. for example. if h(1 ) oscillates more or less eve nly a bout zero. Since the integral of h (1) appears as a normali zing factor in the denominator of ou r expressions for delay and risetime, inappropriate values for these qua ntities may result . To handle this d ifficulty one might propose a modi ficution of ou r definitions so that the delay and r isetime de pend on the moments of the .\'q llu re [nr perhaps the abso lute value ) of h (l) . " It is left as an exercise for the reade r" to show that such moditications resu lt in exceedingly unpleasant express ions that arc cumbersome 10 U
7.5

IlISETlME, DelAY, AND 6AN DW IDTH

Vjtl

FIGURE 7. 9.

RC low-pa n filler and

17 \

=I - exp(-tl RC )

~tep response,

best when the impulse respon se is unipolar [or , equivalen tly, when the step respon se is monoton ic) . If the individu al syste ms satisfy this requirement then the rela tion ships derived here will ho ld we ll. The greater the departure from the step-respo nse monotonicir y co nditi o n. the less a ppropriate the use of these formulas.

7.5.4

A IVERY SHO RT) APPLICA TIO N O F THE RISETIM E A DDITION RULE

Aside from perm itting one to predic t the riscd me of a cuscadc of systems. the risetime addition ru le may be used to ex tend the limits of instrumentation. Consider. for example. tryi ng to measure the risetime of a syste m whose bandwidth is about the same as that of the instrumen tation. Specifically, ass ume that an oscillo sco pe with a known nsetimc of 5 ns displays a value of 6 ns for the risctim e of a sys tem unde r test. Using the r isetime addition ru le. we can infer thai the true system r isetime is about J .J ns. saving us the trouble and expense of try ing 10 make this measu rem e nt with equipment thut is faster still.

7.5 .5

BANDW IDTH-R ISETIME REL ATIO N S

We now lake up the pro ble m of examining the ru le o f thu mb tha t led us In this endeavor in the first place : (8 1)

where W_ ld R is the - 3-d R bandwidt h in radians per seco nd and ' ri.., is the 10- 90% ri setimc in response to a step. Where does this ru le come from? Conside r our o ld friend. the simple RC low-pass filter displayed in Figure 7.9. Given the equation for the respo nse 10 a unit voltage step. it is straightforward 10 compute the 10- 90% riseume: 'ri ..c

(0.9)

= RC In -

0.\

::::: 2.2 RC .

( 82)

Note that this value is about 10 % higher than the Elm ore rise time co mputed e a r1 ier . In addition 10 the riserime. we alread y know th ut th e - 3-dB ba ndwidth ( in radian s per second) is simply 1/ RC . Hence . the band width-ri setim e product is in fact abo ut 2.2. as the rule Mates.

172

CHAPTER 7 BANDWID TH ESTIMATIO N TE CHN IQ UES

Since it wa:... de rived for a first-order case . sho uld we expect the ru le to hold ge nerally for systems o f arbitrary order? Well . let's loo k at a co uple of ot her ca.. cs. Consider. for example. the step response of a two -pole system: (83)

where

4' = tan

-1[/I={2] ,

I~I

.

The - 3-d B ba nd w id th of Ihis syste m is give n by : (85)

Let 's usc these formulas to ex plore wha t happens as we change (. For the extreme case of a dampin g ratio of zero. the risctimc und band wid th are I

I,

. -,

. _,

_ 1.02

I( -_ U =w"- l SIO 0.9 - 510 0.11 ...., -tv"- . w"I(=o~ 1.55w" .

1861 (87)

so that the co rres po ndi ng bundwidth-risetime product is

or about 72 % of the value obtained for the first-o rde r case . For a re ason ably we ll-da mped syste m. we migh t expect closer agreement with lhe first-order re sult . As a spec ific examp le, if we se t ( = 1/ .J2 thc n the riscti me and bandwidt h are 2. 14 I ,, - (89) r I;= l/.ti w" and

I

whl , = I/ J2 = w".

(90)

so th at (t)hf r l ' =l / .ti

~ 2. 14 ,

(91)

or a value within a small pcrccrnage of the first-order resul t. No te that the product of band width a nd Elmo re rise timc is 2.0 for a s ingle -pu le syste m. In ge neral. the bandwid rh- riseti me product will ra nge betwee n 2 a nd 2.2 if thc systc m is we ll da mped to r more precisely, if the impulse respo nse is unipol ar so that the step re sponse is monotonic. fo r the sa me reasons thai prevailed in the mome nt-based ex pres sio ns fo r riseume and time de lay ); the product wi ll decrease if the system is not very well da mped. However. even in the case nf no damping at all . we have seen thai the ban dwi dt h-risetime product still doc s nu t dev iate th at much .

7.5 RISET IME, DEl AY. AND BANDWIDTH

173

Becau ..e most syste ms of prac tica l inte rest are ge nerally wel l dam ped. we can expectt hcir ba ndwi drh- risetime produ c t to be about 2.2. T herefore. measu rem en t of 1hc step re sponse risetime is often an expedient wa y to obtai n a re aso nably acc urate estima te of the handwidth: o nly one experiment ha s to he performed. and step excitation s are often easie r to ge nerate th an sine wave s.'

7.5 .6 OPEN ·CIRCUIT TI M E CO NSTA N TS. RISE TIME ADD ITIO N. AND BANDWIDTH SHR INKAGE As we have see n. ba ndwidth and rise timc have a ro ug hly co nsta r u product (at least for systems th at a re " we ll be have d") , In additio n. the risctimes o f casc aded sys tems increase in root -sum -squ ared fushion . From these two re lat io nships. we can ded uce a bandwidt h shrinkage law. It is instru ctive 10 co mpare the results of this exe rcise with the ban d width shrinkage law de rived in Chapt er g, Consider a cascade of N ide ntical a mp lifier s, each of which is sing le -po le with a time co nstant r , Combi ning the r isetim e add ition rule w ith the bandwidth-ri set imc relationship yie lds I I BW ~ - - (92)

lr:. ~ r 2

r./N·

-

Compare that ap proximate re sul t with the more exact (hut still approxi mate ) rela tionship

""In 2 0 .833 RW ::::: - - "' - -

, ..IN

(93)

, ..IN

(see Chapter 8). A s ca n he see n. the fun ction al dependence o n N is the sa me; the

equatio ns differ o nly by a relatively sma ll mult iplica tive fac t ur. ~ Note that the me lhod o f open-circuit tim e constants wo uld pred ict q uite a diffe re nt result. Since the e ffec tive lime con stant is found there by sum ming a ll the indi vid ual time constants. the OC r · es tima ted hand width wou ld be

BW ::::: -

I

,N

.

(94)

The difference is sig nifica nt, and underscores ye t aga in how the use of open-circuit lime constants ca n lead to ex tre mely pessimistic estimates of bandwidth if a sing le pole docs not dominate the transfer fun ction.

• AI knlo freqeeucjes, anyway. , It ..hould he mentioned thai one con sequence of the difference ~I .... een the Elmore and 10- 90'}ri'>('ti ~

i§ that Elmore somc....,hat undcreMimalel; the IO- 91>'l for identical sl
ri ~lime

growth . A bencr esnmare

• 17A

CHAPTE R 7 8A N DW IDTH ESTIMATION TECHNIQUES

7.6 SUMMARY The method s of open- and short-circuit time constants allow us 10 estimate rapidly the upper and lower - J · d B frequencies. almost by inspection of the network. As long as the ci rcuit satisfies the assumpt ions well. the met hod s yield reason ably accurate answers. More important. however, is the valuable design insight provided. Ano ther way 10 estimate bandw idth is from a measuremen t of risetime. we've seen that moment s of the impulse response allow us to exploit the power of linear system theory 10 show thai delays add linearly and that riseti mes add in roo t-sumsquared fashion. Furt hermore. we've seen that the prod uct o f band width and risetime is roughly constant and approximately equal to 2.2. For all of these relations. accuracy is greatest when the step respo nse is monotonic. Thai is. as long as the step respo nse has negligible overshoot and /or ringing. the results derived here will hold well. If these conditions are not well satisfied then all bets are off. Therefore. do not fait into the trap of believing that these rules of thumb are exact and universally applicable. As long as we keep this cavea t in mind. we can use these relationships to extend significantly the boundaries of our instrumentation. or make quantitative inferences abo ut frequency-domain performa nce from time -do main measurement" (or vice versa) when the necessary conditions arc well satis fi ed (as they often. but not always. are ).

PR OBLEM SET FOR BANDW IDTH ESTIMATIO N PROBLEM I Consider a truly differential amplifier whose pules are all known to be purely real. The differential gain has a measured band width of w ~ . and the response appears very much like a single- pole system. hut direct applicatio n o f the method of open -circuit rime constants yields an estimate that is (I n ' by about a factor of 2 in the low direction. Identify the likely source of the problem and descri be how to fi x it. PROBLEM 2 It was asserted in the cha pter that the method of open-circuit time constants is always conservative for an all-pole system in which all the poles are real. Prove this assertion. PRO BLEM 3 One might wonder how a single-pole He model could ever adeq uately describe anything that is truly a higher-order system. In order to explore o ne aspec t of this question. use open-circuit time constants to derive. for arbitrary n, a bandwidth estimate for the He network shown in Figure 7.10.

PROBLEM SET

IUn

'.

Rln

175

IUn

--·-··-------~~:;~re ,. secli()n~ total, so CJn

0 ,.

FIG URE 7.10 .

CJn that R is the lutiil restsraoce and C is the lotal ca pacitance .

RC Iodder network.

(a) What is the estimated bandwidt h of the network in the limit uf infinite n? Him: II may helpful to know that the sum of the fi rst II integers is 11(" + 1)/ 2. (b) For the speci fi c case o f" = 4 , co mpare the ope n-circu it time-co nstant estimate of bandwidt h with the result o f SPICE simulations.

PROBLEM 4 Use open-circuit time constants to estimate the bandwidth of a highpass, single-po le Re network, and compare with an analytical value obtained from inspection ofthe actual transfer function. The estimate, of course, is gross ly in error. Explai n why. PROBLEM S Derive the rule that states: The drain - gate capacita nce sees an effecrive open-circuit resistance given mnemonically by " r lrfl + r o s hl + g "" etT r Jeft roght'" where g ....etT is the ratio of short-c ircuit drain current 10 the gate voltage (not gale to-source voltage) . In th is case, neil her a lest voltage nor a te1
2

176

CHAPTER 7 BAN DWIDTH ESTIMATIO N TECHNIQUES

constants were unequ al? Specifically, what if Dil l' tim e constant were dominant [ i.e.• much larger than the fest)?

PROBLEM 8 It was mentioned in the chapter thai it was nor correct 10 include in the short-circuit time -constant calculation the gale bypass capacitor for a cascoding device. Show explicitly why by deriving formally the current transfer function of the co mmon-gale amplifier dep icted in Figure 7. 11. You may neglect all reactive parasitic s in the derivat ion .

FIGU RE 7. 11. Common·go te omplifier for short-circuit time-con!Jont calculation .

(a) What is the actual transfer function? Sketch a Bode plot of magnitude and phase. (11 ) From the transfer function. cxpluln why the value of gate bypass capacitance is esse ntially irre levant fur determining the low-frequency breakpo int. PROBLEM 9 we've seen that feedback ca n mod ify resistances considerably (as with. e.g.. the Miller effec t). especially within a device. O f course. this impedancemodifying property is hardly a local phenome non. To underscore this po int , calculate the resistance fucing the- capacitor in the circuit of Figure 7. 12. Here. the input signal t 'i n consists of a DC bias and an incremental signal term .

FIGURE 7. 12 . Feedbac k-bia sed a mpl i ~ 6f

for nnisklOce calculation. (a ) Derive an expression for the resistance facing the capacitor. (h ) Wh at circuit consequence ensues from a capacitor that is too small?

PROBLEM SET

177

(e) If all the resistors are 10 -1.; 0 and G is io-. what value o f capacitor is req uired to prod uce a low-freq uency breakpoi nt of 20 Hz? Is th is capacitor value co nsistent with an integrated realization? PROBLEM 10 The "s uperbuffer" has been suggested as a mea ns to boot strap nut the gate- drain capacitance of a follower and thereby increase bandwidth . To demon ~trate this action ex plicitly. derive ex press ions for the open -circuit time constants of the circuit show n in Figure 7. 13. You may neglec t bod y effec t as well :1.0; all resistive parasiric s.

fiG URE 7. 13 . Superbu ffer For open-circuit time -co ru lanl colculatioo.

Compare your derivation with the time -constant sum obtained with All and / 2 removed . Discuss the con ditions unde r which usc of the supcrbuffcr is bencfi ciul.

CHAPTER EIGHT

HIGH -FREQUENCY AMPLIFIER DESIGN

8 .1

INTRODUCTION

The design of amplifiers at high freq uenci es involves more detailcd co nsiderauons than at lower frequencies. One simply ha... to work harder 10 obtain the requisite performa nce when a pproaching the inherent limitat ion s of the devices themselves. Add itiona lly. the elTcet of eve r-prese nt parasitic ca paci tances and inductances can impose se rious constrai nts on achievable pe rforma nce. At lower frequencies. the met hod of open-circ uit lime constants is a powerful intuit ive aid in the design of high -bandwidth amp lifiers. Unfortunately. by focu sing on minimizi ng various RC prod ucts. it leads one 10 a relativel y n
8 .2 ZEROS A S BAN DWI DTH EN HA NCER S

179

L

R ) - - - - ,- -

vour

FIGURE 8 ,1. Shunt~peok ed a mplifier.

amplifiers. As we' II see, it is co ns iderably easier 10 obtain gain if it is to he provid ed only over a narrow band ce ntered about so me nominal operating freq uency. We'll conclude with a more ge neral co nsideration of the problem of designing am plifiers with large gai n-bandwidth prod ucts. We 'll find that the co mmo nly held belief that gain and bandw idth must tra de off linearl y is false. Instead. it is po ssi ble In consuuct networ ks in which ga in trades off more with delay, allowi ng much greater fleribility in con structing amplifiers with large gai n- ba ndwidth products. We begin with the study of a simple exa mple. the shunt-peaked amplifier. whose behavior is nOI well predicted by open-circ uit time co nstants. 8.2

ZEROS A S BANDWIDTH ENHANCERS

8 .2 .1

THE SHU N T· PEAK ED A M PLIFIER

Back in the 1930 s. when television was be ing developed . one probl em of critical importance was that of designing am plifiers with a reasonably flat respo nse over the 4-MHz video band width . Although obtaining this bandwidth seems trivial tod ay, it was challenging with the devices ava ilable at the time. Ju st 10 make thin gs mor e difficult, the amplifier also had to he cheap enoug h for usc in a mass-market consumer item, so the number of vacuum tubes had to be kept 10 an absolute minimum . A technique that satisfied this requirement oflarge band width at low cost is known as shunt peaking, ami il was used in co untless television sets at lea st up to the 197{ls . Stripped to its essentials, a shunt-peaked amplifier is sketched in Figure 8.1. This amplifi eris a standard co mmo n-so urce configuration. with the add ition of the inductance. If we assume thaI the transistor is idea l, then the only e lements that control the bandwidth are R, L , and C . T he capacitance C may be taken to rep resent all the loading on the output node, inclu ding that of a subseq uent stage (pe rhaps arising from the input capacitance o f anot her transistor. for exa mple) . The resistance R is the effective load resistance at that node and the ind uctor pro vides the band width enhancement, as we now proceed to demon strate .

180

CHAPTER 8 HIG H-FREQ UEN CY AMPLI FIER DESIGN

L

r FIGURE 8.2 . Model

of shunt-peaked ampli'ief.

Given ou r assu mptions. we ma y mod el the amplifie r for small signa ls as shown in Figure 8.2. II".. clear from the mod el that the transfer function I ,...... /i,,, is j ust the im pedance of the RL C network . so it should be straig htforw ard to ana lyze. Befort' laun ch ing into a detailed derivation. thou gh . let' s think about why adding an inductor this way should give us a ban dwidth e xte nsion. Firs t. we know thai the gain of a purely resistive ly loaded co mmon-so urce amplitier is pm portioual to g... R L • We also know that whe n a capac itive load is added, the gain eve ntually falls off ax freque ncy inc reases because the capac itor 's impedance dimin ishes. Th e addi tion o f an induct a nce in series with the load resistor provides an impedance com ponent that increases with frequen cy {i.e .• it introdu ces a zero). which hel ps offse t the dec reasi ng impeda nce of the ca paci ta nce. lea ving a net impeda nce that re mains roughly co n...ta nt over a broader frequency range than that of lhe orig inal HC net work . A n equivalent lime -domain inte rpretati on may be pro vided by co nside ring the uep response. Th e indu ctor delays c urre nt flow throu gh the branch contai ning the resistor, ma king more cu rren t a vailable for cha rging the ca paci tor. reducing the riseume. To the exten tt hat a fast er rise time implies a grea te r bandw idth , a n a ppro pria te choice of induc to r therefore inc reases the ba ndwid th . For mally. the impe da nce of the R LC ne twork may be writte n .IS

Z(.,) = c.,L

+ R ) II _I

=

.I"C

: [JU. / RJ + II . s LC + .IR C + I

(I)

In addition to a ze ro, the re ar e two poles ( possibly complex) , definitely vio lating the conditions for ope n-circuit tim e constants. Since the ga in of the amplifier is the prod uct of f.[", and the magn itude of Z( s), lei ' s now compute the latte r as a func tio n of frequency :

(wL/ RJ2 + I (J)2 L C

)2 + {w RC) 2'

(2)

Notice that , in co ntrast with the sim ple RC ca se, the re is a term in the numerator (from the zero) that increases wit h increasing freque ncy. Purt hen norc.fhe I - w 2 LC

8.2

181

ZERO S A S BA N DW IDTH EN HA NC ER S

term in the de no minato r contributes to a n incr ease in 121 for frequ encies below the LC reso nance as wel l. Both of these terms ex te nd bandwidth. Unfortunate ly, this last equation is not nearly as useful for des ign as it is fur a nalysis; we don' t have a n e xplicit guide tha t tell s us how to se lect I. given R and C . for example. As will he shown. the re is no s ing le "optimum" value of ind uct a nce, but we can narrow the runge of possibi lities by imposing one or two a rbitra ry ( but nonetheless se ns ible ) require me nts. To facilitate subseque nt de rivatio ns. we introd uce a fac to r m . defin ed as the rutio of the Re and 1.1R lime consta nts: NC

(31

m = - - .

LIN

Then. our transfer function becomes Z(,)

= (., L

I

+ N) II -sC

=

R( rs + l )

,

S 2 r • 11I

+ 51 111 + I

.

(4 )

where r = L I R . The magnitude of the impedance, norm ali zed to the OC value ( = R ) as a function

of frequency, is then (wr)2

IZ ( jw )1

N \0

+I

(5 1

(I

that

w,

-=

(- 2",' +

111

)

+1 +

n,' )' 2 ( - 2 + ", + 1 + 111 ,

(6)

where WI is the old - 3-d B frequ ency ( = l i Re ), and W I is the new. The prob le m. then. is to choo se a value of In that leads to so me desired be havior. Maximizing the band wid th is one obvious possibility. Aft er a certain amount of effort, one finds that this maxim um occu rs at a value of m

=.[i ::::::

1.4 1,

(7)

which exte nds the ba ndwidth to a value about 1.85 times as large as the uncom pe nsated bandwidth. An yone who bas labored to mee t a tough bandwidth speci fication can well appreciate the value of nearly dou bling ba ndwid th th rough the add ition of a single inducta nce at no increase in powe r. Unfortunately, howeve r. this choice o f III leads to near ly " 20% peak in the frequency response, a value ofte n considered undesirab ly high . To rnod crute the peaking, one might see k a ba ndw idth other than the cbscl ure ma ximum hy inc reasing 111 .

18'

CHAPTER 8 HIGH · FRE QU ENCY AMP LIFIER DESIGN

One specific choice is to set the magnitude of the impedance equal to R at a frequency eq ual to the uncompensated bandwidt h. So lving for this condition yields a value o f 2 for nI , with a correspo nding bandwidth of

w =wl J I +../5 ~ 1.8wl.

(8)

Hence. the bandwidth in this case is still quite close to the maximu m. Further culeuJation shows that the peaking is substantially reduced. to abo ut 3%. The arb itrary choice that lead s tothis result is frequently used becau se it yields such a significant bandwidth enhancement without excessive frequency respo nse peaking. However. there are many cases where one desires the frequency response to he COIll· pletely free of pea king. Thu s. perhaps one might seek the value of m that maximizes the bandwidth , subject to the con straint o f no peakin g. The conditions for such maximal flatness may be found throu gh the following general techniqu e: Form an expression for the frequency response magnitude (or, as is frequ ently more co nvenient, the square of the mag nitude) , and maximize the number of derivat ives whose value is zero at DC, Carrying out this method manu ally is frequent ly labor -intensive. but in this particular example, a straightforward calc ulation reveals that the magic value of III is

m =I + ../2 ~ 2Al.

(91

which leads to a band width that is about 1.72 times as large as the uupeaked case. Hence. at least for the shunt-peaked amplifier. both a max imally Hat response and a substantial band width extension ca n be obtained simultaneo usly. In othe r situatio ns. there may be a speci fication on the time response of the amplifier, rather than on its frequency response, One example of practical interest is an oscilloscope deflec tion amplifi er, whose time response (characteri zed. say. by the: step or pulse response) must be " well behaved." That is. not only must we amplify uniforml y the various spectral compo nents of the signal over as large a bandwidth ax practical. hut the phase relationships among its Fourier components must be preserved as well. If the spectral components do not expe rience equ al delay (measured in absolute rime. not degrees), potentially severe distortion of the waveshepe can occur. Such " phase d istort ion" is objec tionable for the bit errors it can cause in digital systems . or for its obvious negative implicatio ns for the fidelity of analog instrumentatio n such as 0~c i IJOSC(lpe5 . To quantify thi s type o f d istortion. first consider the phase behavior of a pure time delay. If all frequencies arc delaye d hy an equal amount of time, then this fix ed amount of time delay must represent a linearly increasing amount of phase shift as frequency increases. Pharse distortion will be minimized if the deviation from this ideal linear phase shift is minimized .

8.2

ZEROS A S BAN DWI DTH EN HA NCERS

183

Evidently. then. we wish 10 exuminc the delay as a function of frequency. If this delay is the same for all frequencies. we will have no phase distortion (other than the change in shape thai results from the ord inary filtering any band limited amplifier provides). Formally, the delay is defined as follows: T/) (w )

d¢ E

(10)

--.

dw

where ¢ is the phase shift o f the amplifier at freq uency w . Unfortunately. it is impossible fur a networ k of finite order to provide a co nstant time delay ove r an infinite bandwidth. since infinite phase shift would ultimately be required whereas poles and zeros cont ribute only bounded amounts o f phase shift. All we can do in practice. then, is to provide an app roximation to a constant delay over some fin ite bandwidth. By analogy with the frequency response case, we sec that a maximally flat time delay will result if we maximize the number of derivatives of 1i J(w) whose value is zero at DC. Again, this method is general. Because of the invol vement of urcrangents in expressing the phase shift due to poles and zeros, computing the relevant derivatives is generally quite a bit more unpleasant than in the magnitude case. Even for our shunt-peaked amplifier. which is only second-order. the amount of labor is tremendous. Ultimate ly. however. one mayderive the following cubic equation for nr (co mputatio nal aids are of tremendou s benefit here ): (I I )

whose relevant root is

3+ J5]'/l + [3_2,15 ]" :::::: 3. 10, I+[ 2 3

nr =

(1 2)

corresponding to a bandwidth improvement factor of a hit under 1.6. Since the conditions for maximally fl al frequency response and maximally flat time delay do not coincide, one must compromise (this situation is hardly limited to the example of the shunt-peaked amplifier, of course). We therefore sec that , depending on requirements. there is a range of useful inductance values; see Table 8. 1. A larger L (smaller m ) gives a larger band width extensio n but poorer pulse fidelity, whereas a smaller L yields less bandwidth improvemen t hut beuer pulse response.

8 .2 .2 SHUNT PE AKING , A DESIGN EXAMPL E Even though shunt peaking traces its ori gins (O 4 · ~tII l video ampli fi ers from the 1930s, it is a useful trick even in the modem era for the same reasons it was originail)' valued: it allows one to squee ze the maximum performance from a given technology. This observation is particularly relevant given the acknow ledged inferiorityof CMOS. Furthermore. it is especiall y important to note that the techn ique does

CHA PTE R 8 HIGH - FREQ UENCY A M Pli FIER DESIGN

lB 4

Tabl e 8. 1. Shunt-peaking mmmilry Normali zed Con di tion

- 104 1 2 - 2.4 1 - 3.1

Ma ..imum ba nd width

IZI = R @ w "" lIN e Ma xima jly flat frequency respon se fk",' grou p dela y No sh unt peul ing

oc

Normaliz ed ba nd wid th

peak fre quency respon se

- 1.K5 - 1.8

Ll 9 I.OJ I 1

...... 1.12 - 1.tI

1

nor require a high-Q peaking inductor. and is therefore qu ite compatible with Ie realiza tions. To underscore this point . co nside r the problem of designin g a 1.5-GHz co mmo n-source broadb and ampli fier intended to prov ide gai n 10 a bloc k of phasemod ulated channels. In this application. then. phase linearity is important . so we will c hoose III = 3. 1 for best gro up de lay uniformity, Let the 10la1ca paci tive loading o n the drai n be J.5 pF (fro m bo th the tra nsistor and loa di ng by interconnec t a nd subseq ue nt stages ), and ass ume that the loa d resistance canno t be made sma lle r tha n 100 n witho ut inc reasin g by a n unacceptab le a mount the po we r co ns ume d to keep ga in constant . If the ba nd wid th is e ntirely co ntroJlcd by the o utput nod e the n the ba ndw id th of the a mp lilie r is just a bit o ver I G i ll.• so mewhat shy of the 1.5-GHz goal. If we assume that the minimum acce ptable resista nce is used. then the required shunt-peaking ind ucto r is read ily calculat ed as

L

~

R' C

- - = 4 .8 nIL 3. 1

( 13)

A 4 .S-nH pl anar spira l ind uctor is read ily implemented in sta ndard C MOS technologics: the amplifier appea rs in Figure S.3. With this ind uctor . thc estimated bandwid th incre ases to approximatel y 1.7 GHz. co m fo rta bly in e xcess of the requiremcn t . Aga in. this improve me nt is o bta ined w ithout increasing the power co nsumed by the stage . Fin ally. note that the Q of the dr ain netwo rk is appro xima tely 0.5 at 1.7 Gil l . so ind ucto rs with mod est Q (such as IC spira l indu cto rs) suffice.

8 .2 .3

MOR E O N Z ERO S A S BA N DW IDTH ENH AN CERS

we've se e n fro m

the shunt-peaked amplifier ex a mple that ze ros arc q uite useful. despit e the ir neglec t by the meth od of open -c ircui t time co nstan ts . To illustrate the ir utility with ano the r sim ple ( bu t re levant) exa mple. conside r an oscilloscope pro be. Contrary 10 w hat o ne m ig ht think . it is most e mpha tica lly " 01 just a glorified

l

8. 2 ZEROS A S BAN DWI DTH EN HA N CERS

185

IOClU

R

~

YIN

cb

J-.- - - r--- "

I

~

TC 7

vo ur

df :

I .5 PF

fiGURE 8. 3. Shunt·peokedamplifierexample.

Probe Tip .. _ _, _ , 9 me g

;.It' c

pt'OI>e

fIGURE 8.4 . Simplified oscilloscope/probe model.

piece of wire with a lip on one end and a connec tor on the other. Th ink about this fact: Most " Iu : I" probe s (so -ca lled beca use they provide a Iactor-o f- l0 ane nuauonj present a IO-MQ impedance to the circuit under lest , yet may provide a band width of 200 MHz . BUI intuition wo uld suggest that the maximum allowable capacitance con sistent with this band width is abo ut 80 aF (O.OR f'F}! So, how can probes provide such a large bandwidth while presenting a IO-MQ impedance? The answer is that the combination of a probe and oscilloscope isn' t an "ordinary" He network. A simplified model of the scope /probe combination is shown in Figure 8.4. T he 1MQ resistor represents the oscilloscope input resistance; C ,mpc represents the scope's input capacitance. Inside the probe, there is a 9-MQ resistor to provide the necessary In:1 uuenuatinn at low frequencie s. However, to avoid the tremendous band width degradation that would result from usc of only a simple c -rneg resistor. the probe also has a capacitor in parallel with thai v -meg resistor. At high frequencies. the 10: I allenuation is actually provided by the ca pac itive voltage divider. II may be show n (and you should show it] that when the top He eq uals the bottom RC. the attenuation is exacrly a factor of In, independent of f requency. There is a zero thai cancels precise ly theslow pole. lead ing to a tr.ansfer function that has no band width limitation. Th is case thus represents the ultimate in band width "e nhancement,"

186

CHAPYfII 8 HIGH -FR EQUEN CY AMP LIF IER DESIGN

" 2

a,

1'-----_ ~

'r

FIGURE 8 . 5 . Possible step responses of pole-zero dou blet.

Because it is impossible 10 guarantee exact cancellation with fixed elements. all 10 : I probes have an adj ustable ca pac itor. To appreciate more co mpletely the necessity for such an adj ustmen t. let' s evaluate the effect of imperfect pole -ze ro cancellation with the following transfer function: 1/( ,)

au + I = '---"-"'''rs

+J

(1 41

For simplicity's sake. note that no at tenuation factor is incl uded in this expression. Th us. the ideal value of the co nstant U' is unity, so that H ( s ) is ideally unity at all frequencies. lei's now cons ider the step response of this system (o ften known as a pote-m» doub let) . The initial- and final-value theorems tell us that the initia l value is n . while the final value is unit y. Since state evolves expo nentially with the time co nstant of the po le,' we can rap idly s ketch a couple possible step responses, one with al < I and one with a 2 > I; see Figur e 8.5 . We see that the respo nse jumps im mediately to a , hut se ttles down (or up, as the case may be) 10 the final value. with a tim e co nstant of the po le. Ir a happen s to equal un ity. the respo nse reac hes final value in zero time. In all practical circuits, of course. add ition al pole s limit the risetime to a non zero value. hut the ge neral idea should be clear from this exa mple. From Figure 8.5, it is easy to see the impo rta nce of adju sting the ca pacitor 10 avoid gross measurem ent erro rs. T his calib ratio n is most easily performed by exa mining 1 For :l.o me inellplicahle reasonthere sel;'ms to he it Iair amou nt of confus ion about this poinl. 11le pKSC'lI("e of the leru merely allen the initi al e rror . hut th i~ effi lf alwa ys <;("111("; to the final Villut w ith a time consta nt u f just t he pole.

8.2

ZERO S AS BAN DW IDTH EN HA NCERS

187

FIGUII:E 8 .6 . Zero-pecked common-source amplifier.

the respon se to a square wave slow enough to see the stow settling due to the doub let's pole. and adj usting the capacitor for the flattest time response. This notion of cancelling a po le with a zero ca n be used to extend the bandwid th of active circu its as well . of co urse . Thi s type of ca ncellation may be implemented as in the degenerated com mo n-source am plifier o f Figure 8.6. Here. C is 110 1 cho sen large enough to behave as a short at all frequencies o f interest. Instead. it is chosen just large enough to begin shorting out R when C L begins to short out RI.. It is therefore relatively stmighrforward to understand that ideal compensation should resuit when RC ~ Ri C I .. As with the oscilloscope probe exa mple. proper adjustment is necessary to ob tain the best response . 8 .2 .4 TW O -PO RT BA N DW IDTH EN HA N CEMENT Shunt peaking is a form of bandwidth enhancement in which a one-port network is connected between the amplifier proper and the capacitive load. Although the near doubling of bandwidth provided is impressive. it is po ssible to do bcuer still by employing a two -port network between amp lifier and load . One may augment shunt peaking by additionally separating the load capaci tance from the output capacita nce of the device. If ,I series inductor is used to perform this separation. the overall result is a comhination of shunt and series peaking; sec Figure 8.7. ft may be shown that . in the absence of L r, the maximum bandwidth is J2 times that of the uncompensated cuse. Series peaking actually predates shunt peaking. but its bandwidth boost comes en tirely from the peaking provided by complex po les; there is no zero to help things along. The smaller bandwidth improvement guaranteed the relative obscurity of series peaking once shunt peaki ng was invented. For the sake of completeness. the desired inductance if only series peaking is used is given by

R' C

L2= -

- . m

( 15 )

188

CHAPTER 8 HIGH· FREQ UENCY A Ml'liFIE R DESIGN

FIGURE 8 . 7. Amplifier with ~hunt and ser~ peaking_

FIGURE 8 .8 . Shunt and double ·series peaking

where til = 2 corresponds to the maximum bandwidth and maximally flat amplitude case . A ch oic e of 111 = :I lead s to ma xi ma lly nat g ro up de lay and a bandwidth boost factor of about 1.36. We won't spen d a ny more tim e a nalyz ing th is co mbina tio n because it is an intermedi ate step o n the way 10 a much bett er hand wid th ex te ns io n meth od . T he next evolut ion ary step is to ad d an ind ucta nce be twe e n the device an d the rest o f tile net work, a s sho wn in Figu re 8.8 . Th is com bin ation ( I f shunt a nd series pea king work s as follow s. J ust as in the step re spon se of an or d inary shu nt-peaked um pli ficr . the now of curre nt int u the ]O:IUresistor co ntin ues 10 he de ferred by the acti o n of L r. This actio n a lone speeds up the charging o tthe load ca paci ta nce . In add itio n to Ih'.1 mech ani sm , th e tran sistor initially has to d rive o nly its own o utput c apacita nce for so me time. becau se L "~ del ays the divers io n of c urre nt into the rest of the net wor k . lien ee , r isctime at the d rain imp roves. whic h we again inte rpre t as implyin g an im proved ban d wid th . So me time afte r the d rain volta ge has rise n significantly. the volt age across thc load ca paci ta nce hegins

8 .2

ZERO S A S BAN DWID TH ENHANCERS

189

to rise as current finally starts to tlow through L 2. lienee. such a net wor k charges the capacitances serially in time, rath er than in parallel. The tradeo ff is an increased delay in exc ha nge fo r the improve d bandwidth . We will see that this bandwidth - delay tradeoff is a recu rre nt theme. To save die area . the co mbination of three inductors can be real ized conveniently as a pair of magnet ically coupled inductors ( i.e.• a transform er). s ince the equivalent circuit model of such a con nectio n is precisel y the arrangement we seek . These may be implem en ted as a pair of spiral indu ctors that have bee n placed on top o f each other and offset appropriately to obtain the de sired amo unt of coupling. A furthe r improvement is poss ible if a small bridging capaci tance is added across the indu ctors to create a parallel resona nce. The increa sed circulating currents assoc iated with the resonance help to push the bandw idth out even furt her. After even more agony than suffered in deri vin g the eq uations for the shu nt-pea king case. one may show that the coupled induct ance s should eac h have a value given hy (16)

where l. is interpreted as the primary or secon dary inducta nce with the other windi ng open-ci rcuited . Hence , this is the value o f ind uctance used in design ing and laying out each spiral. for example. The bridgi ng cupucit ancc should have a value of

_CL [ ~ + ]

Cc -

4

I

Ii. •

II 7)

It may also he shown that a coupling coe fficie nt of 1/ 3 yields a Bune rwort h-typc (maximally flat amplitude) response. while a k of 1/ 2 leads to maximally flat group delay. These coupling coefficients are not part icularl y large and are therefore readi ly obtained in pract ice . Two adj acent bo ndw ire s typi call y have coupling coe ffic ients in this range. for example. Applying these conditions then lead s to the amplifier of Figure 8.9. The result ing network is ca lled a " Tvcoil" ( because of the way the schematic is drawn] and has been used for ove r fort y years in osc illosco pe circ uitry. The T-coil is ca pable o f almost triplin g the bandwidth (the theoretical maximum is a 2 ./2 improvement, or about 2.83 x , ob tained with the Butt erworth condition ) if the o utput capacitance of the device is neg ligib ly s mall compared with the load capacitance. It may he shown that the bandwidth is maximi zed if the j unction of the two inductors dri ves the hig her-capaci tance nod e. In Figure 8.9. we have assu med that the load capacita nce is larger than the outpu t capaci tance ofthe tran sistor. The dra in and load capaci tance connect ions may he reversed if the ou tput ca paci ta nce happens to exceed the load capaci tance.

190

CHA PTE R 8 HIGH -FRE QUE N CY A M PLIFIER D ESIG N

FIGURE 8 .9. Amplifier with h oil bandwidth enhancement.

C c = O.l ZSpF

t,

~

"

I'---.-~.

'om

FIGURE 8 . 10 . T-coi! ba ndw idth enha ncement example.

As a final rcnnem cm , slime additio nal compe nsatio n for the output capacitance of the tra nsistor may he provided by adding more inducta nce in se ries with it, effective ly providin g more ser ies pea king . A nea rly eq uivale nt result may be ob tained merel y by tappi ng the indu c tor s at othe r tha n their midpo int ( in this case, closer to

the load resistor endj. Mod ifying our ea rlier shunt-pe aked design ex ample yield s the amplifier o f Figure 8. 10. where , 10 keep the co mpar ison fair, we have continued to assume that we desire a ma xi mally flat group del ay. As can be see n. the to ta l inductance has doubled . Howeve r, since the two indu ctors are on top o f eac h o ther w ith a small offse t , the additiona l area is modest (on the orde r of 50 %). In add ition , the J 25- f F bridg ing ca pacita nce mi gh t he pro vided as a n inhe rent byp rodu c t of the ove rlapping ind uc tor layout.

8 . 3 THE SHUNT-SERIES AMPLifiER

'9'

The theo retica l bandwi dth impro vem en t factor provided by this circuit is about 2.7.2 Hence. roughly a 2.7-G l ll bandwidth can be expec ted. substa ntiall y better than the

1.7-GII7. bandwid th of the shunt-peaked case. It is important to underscore that this improvement is obtained without an increase in power. We will later apprecia te this structure as itse lf an intermedia te evolutionary step on the way to a co mpletely "d istributed amplifier " (to he d iscussed shortly). in whic h parasitic ca pacitances are absorbed into structures that trade gain for delay rather than for band width . (For example. consider a transmission line - it consists o f inductance and ca pacitance. but these eleme nts impose no limit on ba ndwidth becau se the capacitances are charged seria lly in time.) In the mean time. it may be con s idered simply as a more sophisticated way to di vert current away fro m the load resistor and into the load ca pac itance.

8.3 THE SHUNT-SERIES AMPLIFIER In contrast with the open-loop architectures we've studied so far. an alternative approach to the design of' hroadband amplifiers is to usc negative feedbac k . One particularly useful broad band circuit that e m ploy s negative feedback is the shunt-series amplifier. Its name derives fro m the usc of a combi nation of shunt and series feed back. and its utility derives from the relative constancy ofi nput and output impedances over a broad frequency range (which makes cascadi ng much less complicated ). as well as from its case of design. In ad dition . the dual feedbac k loops confer the usual benefit s normally as..oc iated with negative feedback - namel y. a reduced dependency on device parameters. improved distortion. broader band width and a nicer complexion. Stripped of biasing details. the shunt- series amplifier is depic ted in Figure 8.11. where Rs now denotes the resistance of the input ...ou rce and HI. i ~ the load re..isranee. Thu s. the amplifier core just consists of Rf", HI. and the transi.... or. To understand how this amplifier works. initially assume thut HI is large enough (re lative to the reciproc al of the transistor 's gm ) so that it degenerates the overa ll transcondu ctance 10 appro ximately l/N I . Since HI is in series with the input and outp ut loops. the degeneration by HJ i.\ the "se ries" contribution to the name o f thi.. amplifier. To continue the analysis. assume also that R F i~ large enough so that its loading on the output node may he neglected . With these assumptions. the m ilage gain of the amplifier from the gate 10 the drain is approximately - Rt. / RI. Although we have assumed that RF has but a minor effect on gain. it has a controlling influence on the input and output resistance. Spec ifically, if reduces both 2 Again. this value assumes that the output ca pacitance of the transistor i ~ 11l'gligih ly smal l comp ared with the load capacitor. If t hi ~ inequality is lint well satis fied. additional series compe nsation will be required to achieve bandwidth hornits of lhis order .

192

CHA PTE R 8 HIGH · FREQ UENCY A M PLIFIER DESIG N

R(~ll

r

Rm

FIGURE 8 .11. Shvn~ie~ amplifier Ibia~i"9 001$hownl.

quantities through the {shunt ] feedback it provides. Add itionally. the redu ction of input and output resistances helps 10 increase the bandwidth still further by redu cing the open -circ uit tim e -con stant sum. Tn com pute the input res istan ce Rmo we usc the Iac t that the gain from gate to drain i.. app roximatel y - RL! R,. If. as seems reasonab le. we ma y neglec t gale curre nt . the n the inp ut resistance is du e e ntirely 10 c urre nt flow ing throu gh Hr . Applying a test volt age source at the gale terminal allows us to compute the effec tive resisranee in the usual way. Just as in the classic Miller effect. connecting an impedance across two node s that have a n inverting gain between the m resul ts in a red uction of impedance. Formally. R ill is given by

Rin

=

-,.-:R.:c'e'.-_ '" ~--"R:..,~_ 1 - A i'

1 + R1j R 1'

(18)

where A i' is the vol tage gain from ga te to drain. Nnw, to compute the output resistance, apply a test voltage source to the drain nod e und again take the rat io of R

l ' I~'t

ttl i t~'I :

' -:.+.,:R .c'e.-' ;:::::: - -:-,R.:c'e.fl ut -

1 + Rs/R 1

_---'R .c,e.-,_ 1 + Rs /R )'

( 19)

If the so urce and load resist ances are equa l (a pa rticu lar ly common situatio n in discrete realization s). the n the denomi nators of Eq n. 18 and Eqn . IY arc approximat ely eq ual. Since the num erators are al so approx imately equal. it foll ow!' that H in a nd Rnw arc themse lves nea rly equal. If R .~ = RL = R the n we Illay write : Rout :::::: Rill ::::::

R, se 1+ R/ R,

(2 0)

T he ea se with which this am plifier pro vides a s imulta neous impedance match OIl bo th input a nd output ports accounts in part for its popu larit y. O nce the impedance

8. 3

193

TH E SHUN T-S ERIES AM PLIF 1ER

level and gain arc chose n. the required value of the feedback resistor is easily deterrnined . Coupling knowled ge o f the load resistance with the req uired gain leads quickly tn the necessary value of R I . To complete the design. a suitable device width and bias point must he chosen. Ge nerally. these choices arc made to ensure sufficient gill to valid••rc the assu mption s used in develo pin g the foregoing sci of eq uations.

DETAILED DESIGN O F SHUNT-SERIES AMPLI FI ER The forego ing presen tation outlines the first-ord er behavior of the shunt-series amplifier in order to hel p the develop ment o f design intuitio n. To carry out a more detailed design. however, we now conside r some of the sec o nd-order factors neglec t..x .l in the previous section. Low-Frequency Ga in and Input-Output R e~stances We start by computing the gain from gate to drain since it all ows us to find the input and output resistances easily. Once the gate-to- drain ga in and input resistance are known, (he overa ll gain is trivially found from the vo ltage divider relationship. First, recall that the effective transco nductan ce of a comroon-sou rce amplifier with source degeneration is g... .df

R. = 1 + R. R I ·

(2 1)

Note from Eqn . 2 1 that the effective transconductance is appro ximate ly II RI as lon g as g", R1 is much larger than unity. Applying a test voltage from gate to ground causes a drain current 10 no w through both the load and feedback resistors. So me frac tion of the lest voltage also feed .. fur ward directl y 10 the output . Supe rposition allows us to treat each o f these co ntributions to the outpu t volt ugc separately: 1J""t

=

V - l: m.c11 lest

Rf,R,. RF + RI. +

«, VIC'l

(22)

RF + R,.

Solving for the gain yields

A V = ~ = _ Hl . [ IJl t , 1

RI

1

1+ l/I:",R )

] .[

1

1 + RdR F

] . [1 -

1 ]. 1:"' ,t'il RF

(231

AI/hough not the most co mpact expression. Eqn. 23 shows the gain derived earlier from first-order theory mu ltiplied by three factors (in bracket s). each of which is ideally unity. The first "nonideal" factor reflects the influence of finite Mm o n the effective transconductance. Wh ile ,l(m.cff approaches I/ RI in the limit o f large Mm RI ' this fi rst factor shows qua ntitatively the effect of nonin finite RmRl . T he seco nd term is the result of the load ing by RF on the output node. As long as RF is substantially larger tha n the load re..istance R/. . the gain reduction is small.

CHAPTE R 8 HIGH ·fREQUEN CY AMPLIFIER DESIGN

19'

Th e final gain reduction factor is du e to feedforwa rd of the input sig nal to the output. Thi s feedforward reduces the ga in because the ordinary ga in path invert s but the feedforward path docs no t . Hence. the fcedforward term part iall y ca ncel s the desired OUlpUI. Th e tra nsconducta nce of the fcedforward leon is IIR ,:; he nce. as long as this par asit ic transco nduc tance is s ma ll compared w ith the des ired tra nsconductance Km .df. the ga in loss is negligibl e . Having exam ined the com ple te ga in eq uat io n term by term. we no w presen t a much more co mpact ( but still e xact) e xpress io n. useful fo r ca lculatio ns to fo llow: (24)

whe re Ht: is simply the reciprocal of the effective tra nscondu cta nce. Th e up shot is simply that. in orde r 10 obtain the desired gain, one mu st choose a value of HI (or HE) that is somewhat smaller than anticipated on the basis of the first-order equations. Now that we ha ve a com plete ex press ion (t wo. eve n) for the low -frequency gain. we ca n ob tain a more accura te value for the resista nce between gate and ground :

H,

(2l )

1 - Av ' whic h . after using Eqn. 24, become s

(26)

In ge neral. one design s speci f ically for a part icul ar valu e of gai n. Assuming suecess a t achieving that goal. the value o f feed bac k resistance necessa ry to produce a desired input res ista nce is readily found sim ply from Eqn . 25 . The outp ut resistance {i.e., as see n by R i. ) is also simple to find. Aga in. we a pply a test voltage source 10 the d rain node a nd co mpute the rat io of lest voltage to test cu rre nt. Performing this exercise yie lds Vic,, '

R Oll i

= --r' leM

=

I

=

Rf: (R F

+ Rs}

Rf." + Rs

(27)

Compa ring the ex press ions for inpu t and output resistance, we sec that if Rs and RL a re equal (as is co mmonly the case) the n R m a nd Roul wi ll also he precise ly eq ual. Thi s happy coincide nce is on e reason for the tremen dou s popularit y of this topology! } It should he 110100thaI this tupolog y is also wjdely used in Ihe bipolar form in which il wa~ fin.! realized. There is a mmor d ilTerence in lhat finite f:J causes the input and output resistance to be somewh at uneq ual. alt hough the error is small fur typical "a llies of fl. The input resistance is smaller by a factor of approximately I - 1/2p . while Ihe outpu t resistance is higher by a factor (If al'ltJUI I + 1/ 2Ji . The gain is also slightly lower. by a factor o f ahe m! I - 21P.

8.3

195

THE SH UN T- SERIES AMPLIFI ER

It should he emphasi zed that . when carryi ng OUI a design (as opposed

analysis). the desired gain is known. Hence. if Ihe input and output resistances are to be equal, selection of the fee db ack resistor is trivial fro m Eqn. 25. TIle value o f HI is then chosen 10 provide the COffeCI gain, completing the design . 10

Bandwidth and Input-Output Impedances Having presented exact express ions for various low-frequency quantit ies (gain and input-output resistances). we now derive approximate expressions for the bandwidth as well as the input and ou tput impe dances of this amplifier. Before plowing through a slew o f eq uations, let's see if we can anticipate the qualna tive behavior o f these quantities. Because this amplifier is a low-order system. we expect gain and bandwidth to trade off more or less linearly. Furthermore. precisely because it is a low-o rder system. an open -circuit time -constant estimate of bandwidth should be reasonab ly acc urate. We also expectthe input impedance to possess a capacitive compo nent. partly because of the presence of C 6 J • bur also beca use of the augmentation of Cfld by the Miller effect . The output impedance. on the other hand. could behave differently because the shunt fee dback that reduces the output resistance becomes less effec tive as frequency Increases. As a result. the output impeda nce could actually rise with frequency. leadi ng 10 an ind uctive co mponent in the output impedance . Having made those predic tions, let us proceed with a calculation of the opencircuit time-co nstant sum. To simplify the development. a..sume that the only device capacitances arc C Il ., and C/td' Furthermore. neglect the series gale resistance. Finally, assume that the so urce and load resistance bo th equal a value R . The effective resistance facing Cild is clearly Rp in parallel with a resistance given by (2K)

so thatthe resistance is: HI-"

II

( Ns

+ NI. + Rm.df HSNtJ

= HF II R(2 + 1<",, <'I1R ),

(29 )

After substitution for Hr . this becomes R( I - A v l

II

R(2 + gm" ff R) ,

(301

Note that, in the limit of large gain. the resistance facing C/(d approaches (3 1)

as might be anticipated from considering the Miller effect. Computing the resistance facing CflJ is somewhat more invo lved. hut ultimately one may derive the following express ion:

196

CHAPTER 8 HIGH -FREQ UENCY A M PLI FIER DESIG N

R(R F

+ R + 2 Nd + R I H,.·

(2 R + R F)( J + Km R J> + g", R2'

(32)

In the limit of large gain. Eqn. 32 sim plifies to

R I (33)

HI 8m

No ll.' thai the ratio HIH, is approximate ly the magnitud e o f the ga in (fro m ga le 10 drain). Bec ause both ope n-circu it resistances are then roughly proportional to gain. the gam- ba ndwid th product of the sh unt-serie s amplifier is approxi mately co nsta nt. The estimated band width of the amp lifier in this limit is therefore

[ (C" + RC,,)]-'

BW ", IA .,J ..-

- 2-

(34)

Having derived an approximate exp ression for the bandwidth . we now consider the input im ped a nce. A s stated ea rlier. the input im pedance should po sse ss a c apacitive component because o f eft, and the M iller-multiplied C ftJ . A crude upproxima lion 10 the total ca pacitance may be obta ined simply by a ssumi ng that the impeda nce at the ga te co ntrols the band width of the amplifier. T hat is. ass ume that the time co nstant o f the amplifier 's pole is the prod uct of the so urce resistance Rs ( = R ) and the ca paci tance at that node. With that ass umption. the effective inp ut ca p••cha nce is jus t the bracketed term o f Eqn. 3-1 d ivided by R : "" G~ y

G in ....... -

-

gmR I

+ C ll d -jA2v- l .

(35 )

In almost all practical case s. the Miller-augm en ted G il d do minates. Note that the presence of this capaci tanc e. which effectively appears between gute and ground. makes it impossible to achieve a perfect inpu t im pedance match at all freq uencies . Furthermore. as the frequency increases. CII' progressively shorts out and so connec ts the source degeneration resistance HI to the gutc nod e. IIcnce. even the input resistance tends to degrade as well, di mini shing as the frequ ency increases. These effects cun be mitigated to a certain extent by using some simple techni ques. First . an L match cun he used to tran sform the resist ive par i up 10 the desired level. such as 50 Q . at some nominal frequency (gene rall y a litt le beyond where the qu ality of the match has begun 10 degrad e notic eab ly). Of the possible type s of Luuuches. the hest choice is usually one that pla ces an inductance in series with the gate and a shunt ca pacita nce acro ss the amplifier input ; such a net wor k becom es transparent at low frequencies. where no correc tion is required . The series inductance (If the L-match ge nerally leaves a residu al inductive com ponent. T his inductance is easily compensa ted by simply augmenting the ...hunt ca pacitance of the Lnet work . With this compensation. the frequency range over wh ich a reasonably good input match is obtained ca n often be doubled .

8 .4 8A N DW IDTH ENHANCEMEN T W ITH

t, DOU8l ERS

197

To co mpute the OUlpUI impedan ce, apply a test voltage source to the drain and calculare the ratio of the resr m ilage 10 the curre nt Ihal it suppli es . In the limit of high gain, one finds that the output imped ance includes an inductive component whose value is approxi mately (3" 1 where C Kd has been neglected. In order to develop a deeper understand ing uf the or ig ins of this inductan ce, note that the gate voltage is some fraction of the lest voltage applied to the drain. Speci fically. the gale voltage is an attenuated and low-pass- filtered version of the applied drain voltage due to the ca pacitance at the gate. Hence. the gate m ilage lags behind the voltage at the druin. The transistor then co nverts the lagg ing gate voltage into a lagging drain current. From the viewpoin t of the test so urce, it must supply a current with a component that lags the applied vo ltage. Thi s phase relationship betwee n voltage and current is characteristic of an inductance. From this insight , we can asscss the effect of neglectin g C Kd . Since C Nd supplies a leading co mpone nt of voltage at the gate, it tend s to offset the inductive effect. As a result, the output inductance actually observed can he cons iderably smaller than rho upper bound estimated by Bqn. 311 if C l
2 rrfr = ~=-= CIi .• + CXd

Loosely speaking, then, IT is the ratio of transconductance (0 input capacitance. If a way could be fou nd IU, say. decrease input ca pacitance without decreasing transconductance. IT would increase. The ordinary differe ntia l pair may be considered an Ir doubler by this de finition, for the device capacita nces are in series as far as a di ffe rential input is concerne d . Hence. the differential input ca pacitance is one half that uf each transistor. The d ifIerenrial transconductance. on the other hand. is unchanged because, altho ugh the input voltage divides eq ually between the two transisto rs. the differential output current is twice the current in eac h device . Hence. the overal l Mage transconductance is equal to thut of each transistor, and a doub ling o f IT results; sec Figure 8. 12.

198

CHA PTE R 8 HIGH -f REQ UENCY AM PLIf iER. DES IG N

i n = i. - i 2 il

i2

FIGURE 8. 12. Di~entior pa ir as doubler.

'1

dra in

I. "0

source

FIGU RE 8 . 13. Darlington pa ir 05 f T dou bler.

Because it is not always co nve nie nt to arr ange for diffe rential signal pat hs. it is som etimes de sirable to synthes ize a sing le -e nde d !r doubler. T he differe ntial pair can be co nverted into a single- e nded do ub ler by inter changi ng the gate a nd source connection s on one device. Th e atte nda nt polarity reve rsal allow s us to take the output as the sum (rather than diff ere nce) of the two drain c urre nts, so merely lyi ng the two drains toget her com pletes the tran sformation , as show n in Figu re ILl3. One may recognize the result as to po logi ca lly identical to a Da rlington pair. An importan t d istinc tion . howe ver. is that both transistors should he biased to roughly the ,W ill i' curre nt to ma ke the device irs approx ima tely equal. Th ere are numerous llICthlllb fur sutb fy ing thi s biu s req uire ment . bu t o ne part ic ularl y simple and co nven ie nt urm nge me nt is the Cl\IOS version of a bipo lar circu it developed by Car l Battjes of Tektroni x:" M.'C Figure 8. 14 . By using mi rror M 2-M 3 • transistors Atl and ~

" Mono lithic Widd>and Amplifier." U.S. Patent 114.236. 119. grar ued 2S November 191$0.

8 .5 TUN ED A MPLIFIERS

199

source

FIGURE 8 . 14 . 8attie~ ' r doubler.

MJ are guaranteed to operate with substantial ly equal drain curre nts. Becau se the capacitances of M2 and ,\(3 are in parallel. thou gh . this circuit does not quite provide a doubling of Ir -T he ac tua l increase is abo ut a factor of 1.5. With Ir doubling circuits it is o ften possib le to obtai n 50% increases in bandwidth. althoug h the exact improvement de pend s on nu merou s and varia ble facto rs. Chief among these is how dependen t on f r the cir cu it's band width happens to he. and how much source - bu lk paras itic ca pacit ance there is. Clearly. if the bandwid th is limited by something else (e.g.• en externalload ca pacitor interacting with a load resistance). increa sed It will prov idc little improvement . Nevertheless. f r doublers are valuable for pushing bandwi dth beyond wha t onc would normal ly believe are the limits of a given tech no logy.

8 .5 TUNED AMPLI FIE RS 8 .5 .1 IN TRO DUCTIO N We' ve already seen that the design of broa dband amplifi ers can be guided by the method of ope n-circuit tim e constants. with a possible assist from band widt h cxte nsion tricks such as shunt peak ing. However. it is nol alway s necessary (or eve n desirable) 10 provide gain over a large frequen cy range . Often. all that is needed is ga in over a narrow freq uency range ce ntered about some high freq uency. Such tuned amplifiers are used exte nsive ly in co mmunications circuits to provi de selective amplification o f wa nted signals and a degree of filtering of unwanted signals. As we' lI see shortly. eliminating the requi rement for broadband operation allow s one to obtain substantia l ga in at relatively high frequ encies. That is. to zeroth order. the effort required to ge t a ga in of ICX) ove r a bandwidth of I MHz is roughly inde pendent of the center freq uency about whic h that band widt h is ob tained ; the difficulty

200

CHAPTER 8 HIGH -FREQU ENCY AM PLIFIER DESIGN

FIGURE 8.15. Amplifier with single tuned

load.

in obtaining a speci fied gai n-bandwidth product is approximatel y constant and independent of cent er frequ ency (within ce rtain limit s). Furthermore. the power requ ired to obtain this gain can be considerably less for a narrowband imp lementation . This la...t co nsideration is particularly important when designing portable equipment. where battery life is a major concern.

8.5 .2 COM MON -SO URC E A M PLI FIER W ITH SINGLE TU NED LO A D To understand why the gain-bandwidth prod uct should be roughly indepe ndent of ce nter freq uency. co nsider the am plifier shown in Figure 8. 15 ( biasing details have been om itted) . If we dri ve fro m a zero -impedance so urce las shown). and if we can neglect series gate resistance. then the drain-ga te capa citance CII•1 may be absorbed into the capacitance C. In that ca se, we ca n mod e l the circuit as an ideal transconduc tor dri ving a parallel HL C la nk . At low frequencies. the inductor i.s a short and the increment al gain is zero. whereas at high frequen cies, the gain goes to zero beca use the ca paci tor acts as a short. At the reso nant freque ncy of the tank , the gain becomes simply J.:mH since the inductor und capaci tor cancel. For this ci rcuit the total - 3- dB band widt h is. us usual . simply I/ RC . li enee. the produ ct of gain (mea sured ut resonance ) and ban dwidth is ju st I J.:m G · BW =g", R · = - . RC C

(38)

For this example. with all o fit s simplifying as sumptions. we obta in a gain- bandwidth product thai is independent of ce nter freq uency, as advert ised . To underscore the pro found im plications ti l' this last stateme nt. co nsider two altern ative methods for obtaining a gain of JOOO at 10.7 Mil l. (e.g.. for thc IF section of an FM rad io ). We co uld attempt a broadband amplifier design. which would requi re us to ac hieve a gai n- ba ndwidth prod uct of over In Gill (not a trivially accomplished goal). O r we could recognize that. fur the FM radio example. we need unly

8,5 TU N ED A MPLIFI ERS

FIGURE 8. 16 . Ampli fier with single tuned

201

lcod.

obtain this ga in over a 200 -kHz bandwidth. s in whic h case we on ly have to achie ve something Iike a 2(X)-Mllz gain-bandwidth prod uct. a considerably easier task. The fundamental differe nce between these two approaches is. o f course, due to the cancellation of the load capacitan ce by the indu ctor in the tuned am plifier. As long as we have direct acce ss to the term inals of any parasitic ca paci tance (and can make them appea r across the tank), we can resonate out this ca pacitance wit h an appropriate choice of inductance and obtain a constant gain- bandwid th prod uct at any arbitrary center frequ ency, Naturally, real ci rcuits don't work qu ite as neatl y; we s uspect that we prob ab ly won't he able to get gain at 100 T t tz from Jell -Ow transistors. for example, no mat ter how good ou r inductor is. But it rem ain s true that, as long as we sec k ce nter frequ encies that arc reaso na ble," tuned loads allow us to obtain roughly constant gain-bandwid th product.

8.5 .3

DETAilED ANALYSIS O F THE TUNED AMP LIFIER

The analysis j ust perfo rmed invokes many simplifying assumptions. In partic ula r. the choice of a zero source resistance and zero gate resistance allo wed us to absorb the drain- gate capacit ance into the tank network . permitting the inductance 10 offset its eff ects. Since C~
, This value applie s 10 commercial brua<.k a"l FM radio ; your mileag e may vary" ~ We' ll quantify jhis better a hnle later, hul fur nuw pretend lhal " reasonable" means " rea~lInahl y ....'ell belO\lo"uJr .M

~--- --- -

202

CHAPTER 8 HIGH -FREQ U ENCY AM PLIFIER DESIG N

C, d \ ';n

•

C,.

C

R

L

FIGUR E 8.17. Incrementol model for circuit.

Using this model. we can co mpute two important impedance s (actu all y ad mittances, to be preci se). First. we'll find the equ ivalent admitta nce seen 10 the left of the R LC lank; then we'll find the admittance see n to the right of the source resistan ce Ns. In carry ing out this analysis. it is bette r 10 apply a test ,'olragi' so urce across the lank to find the equivalent adm ittance seen 10 its left . Rem ember. yo u' ll get the same answer whether you use a lest "u llage or a test curre nt (ass uming yo u make no errors. or at least the same errors). but a lest volt age is more convenient here bec ause it most d irec tly fixes the value of v"J' the voltage that det ermines the value of the co ntrolled source . The preci se details are somewhat messy and essentia lly unreward ing. but the end re...uh is that the ad mittance seen by the tank consists of an eq uivalent resisranee (w hich we ' ll ignore for now) in parallel with an equivalent capaci tance. Th is capacitance is given by (39)

Notice that Cet! can be fairly large. Thi s is actua lly an alterna tive ma nifestation of the Miller effect. nnw viewed from the output port . So me fract ion o fthe voltage app lied to the drain a ppea rs across V g5 , where it exc ites the gm generator . The resultin g c urrent adds to that through the capacitors and mu st be supplied by the test source. so the so urce see s a lower impedance. One component of that curre nt is due to a simple ca paci tive voltage di vider ami is thu s in phase with the applied voltage . It therefore re pre sents a resistive load on the tank. ca using a gain redu ction . A nother component of the curre nt leads the applied vo ltag e and therefore represent s an additio nal ca pac itive load on the tan k . The addi tional ca paci tive load ing by Cl"(j shifts do wnward the resonant frequency o f the output tank . Altho ugh this shift ca n be compensa ted by a suitable adj ustme nt o f the indu ctance. it is ge nera lly inad visabl e to operate in a regime where the resonant freq uency de pends critically on poorly co ntrolled . poorly chara cterized. and potentiall y unstable transi stor parasitics. It is therefore desirable 10 select C relat ivel y large compared wit h the expec ted vari ation in param eters. 1'00 that the total tank ca paci tance

,... I

I

8.6 NEUTRALIZATIO N A ND UN ILATER A LIZA TIO N

203

remains fairly independent of process and operating point . The unfortu nate tradeoff is a red uc tio n in the gai n-bandwi d th prod uct for a given transcond ucta nce . A more se rio us effec t o f C, J beco me s ap parent whe n we conside r the inp ut impedance (cr. more directl y. the input admittance ). Since the inte rme di ate deta ils are again of lillie U50e o utside o f deriving the o ne bit of trivia we're a bo ut to state. we' ll simply present the re sult: Yin

=

)'L)'F

YL+YF

+

g ",YF

)'l.+YF

.

(40 )

where Yin is the ad m itta nce see n 10 the right of the so urce resistance Rs. JI.' is the adrmnance of el/d. and Yl- is the ad mitt ance of th e R l. C la nk (we have sc i rg 10 ze ro )." If, as is often the case, the ma gnitude of the feed back ad mitt a nce YF is small compared (0 that of )'1., then we may wri te:

(41 ) The significa nce o f this result becomes appa rent w he n yo u o bse rve that )' L has a net negative ima ginary part at freque nci es where the tank loo ks indu ct ive {i.e.• below resonance ). so that the seco nd te rm o n the right-ha nd side of the eq uation (and the re fore )' in) can ha ve a negative real part ; that is. the input of the ci rcuit can act as if a negative resistor were connected to it. Having nega tive resistance s around ca n e nco urage oscillation (whic h is j ust fine ift his is yo ur intent, but mo re typically is no t). We ce rtainly have all of the necessary ingredients: ind ucta nce. ca paci ta nce . a nd negative resistance . If there were no e ltd . there wo uld be no such pro blem . The difficulty w ith e ltd. the n. is th at it co uples the input and o utput circuits in potentially de leteri ou s ways. It loa d.. the o utput tank and decrea ses ga in. dctu nes the output tank . a nd ca n cause instability Thi s lan er problem is particularly se vere if line anernprs to add a tu ned c ircuit to the input as we ll. Furt hermo re. even be fore true instability sets in. the interaction of tun ed circui ts at both port s m ay make it e xtre mely challenging to ac hie ve proper tuning. Unfortunately. C /(d w ill a lways he no nzer o ( in fact . it is typically about 30- 50% of the main gate ca pac ita nce. so it is hardly negligibl e ). To mitigate its vario us undesirable effec ts the refore req uires the usc of som e to pologi cal trick s.

8.6 NEUTRALI ZATION AND UNll ATER ALIZATION One strategy de rives naturally from recog nizing that the problem stems from co upling the input and o utp ut po rts. Removing the coupling sho uld therefore be of ben efit . , To avoid obscuring the argument any further . we have re gjected the Iran~hl or ' s output admi nancc in Ihis oevetcome ra: it may he absorhed into YL if a more U K I analysis is de sired.

20'

CHAPTER 8

HIGH · FR EQUE N CY AMP LIFIER DESIGN

e----- ~ ' our

FIGURE 8 .18 . Coscode c mptifiel" with single tuned

Iood.

T

~C ~~ L t '''--,------r

~

V B1AS

•

fiGURE 8 .19. Source -coupled ampli fier with single runed load.

Th is decoupling of output from input should feci fumi liar - it is precisely what climinates the Mill er effect from common -so urce amplifier s. ami what wor ks there wo rks here as we ll. Sec Figure H.tS . By pro vidin g isolat ion bet ween input and output ports w ith the co mmo n-gate stage. we e lim inate (or at least g rea tly suppres s) dctuning and the pot ent ial for instab ility. thus allow ing the attai nme nt of larger gain- bandwidth

products. Anothe r !l llx l]Og y th at ac hieve s these obj ectives is the so urce- co uple d a mplifier {whic h may he viewed as a source foll ower driving u conun on-g urc stage ). as shown in Figu re H.l 9. On ce again . thi s structure isolates the ou tput fromt he inpu t and therefore docs not sutter as se riously from the instabilit y a nd lk tuning problems of the simple l,:0Il111Um-Sl,IUrce stage. Hoth the cuscodc and source-coupled umphfi cr bc have similarly wi th regard to isolation . The ca scode prov ide s rough ly tw ice the gain fur a given total c urre nt
8 .6

N EUTIl: A LlZATlO N AND UN llATERALIZATlO N

205

T f----- .

'our

If It

V FIGURE 8 .20 . Neutralized common-source amplifier,

requires less total supply voltage (since the two transistors aren't stacked as in the cascod c). The choice of which topo logy to use is usually based on such considc rations of headroom and gain. The circuits of Figures 8. 18 and 8. 19 arc exa mples o f nearly " unilateral" amplifiers. that is. one s in which signals can now on ly one way ove r large bandwidths. You can well apprec iate the value of unilarcralization; aside fro m co nferring the circuit benefits we' ve already discussed. it makes analysi.. and design much easier by reducing or eliminating unintended and unde sired feedback . If we cannot (or choose not to) eliminate undesired feed back, anothe r approach is to ca ncel it to the maximum possible extent. Since thi s ca nce llation is rare ly perfeci over large band widths, (his approach is generally ca lled " neurralization'" to distingui..h it fro m more broadband unilateralizano n techniques that do not depend on cancellations. The cla ssic neutralized amplifier is shown in Figure 8.20. Notice thai the inductor has been replaced by something slightly mo re complex: a ta pped inductor. or l lU W · trausfonner. By symmetry. the vo ltages at the top and bouom o f the inductor arc exactly 180" out of phase in the connection shown." Therefore. the drain voltage and the voltage at the top of neutra lizing capacitor C N are 180<1 nut of phase. Now. if the undesired coupling from drain to gate is due only to Cx" then. by symmetry, selection of eNequal to C Nd guarantees that there i.. no net feedb ack from drain to gate ! The current through the neutra lizing ca pacitor is equal in magnitude and opposite in sign

I ,,;"culralil.alion was de\"elllped for A M broadcast radios in Ihe 1920 s hy Harold Wheeler while working for Louis Hazeltine. His ill\'clltiol'l allow edthe attainment of large. staMe gains from tuned RF amplifiers. and Ihus red uced jhe number uf ga in stages laOll hence the num ber of vacu um tubes) ~uiret.l in a typical radi o. permimng significant cos t reduc tions over many rival app roac hes. • Noll' lhat autotransformers are not strictly necessary here. They are merely an hisloricallycommon and C
CHAPTER 8 HIGH · FREQ UENCY A MPLIFIER DESIG N

I'BIAS

-1

FIG URE 8 .2 1. NaItrolized ccenroco-sccrce ampli fier

(more practical lOr ICsl.

to that through Cgd ; we have removed the coupling fro m output to input by adding mo re co uplin g from output to inp ut (it ' s j ust out of phase so that the m 'l co upling is ze ro).

Neutralization was originally Implemented with tapped transformers, but the poor quality of (and large area consumed by) on -ch ip transformers makes this particular method unattractive for Ie imple mentation. Ob serve. however. tha t the tapped transformer is used simply to obtain a signal inversion. Since inversion s are easily obtained other ways, pract ical neu tralized Ie amplifiers are still rea lizable. One topo logy uses a differential pair 10 obviate the need for a uun sformcr. as seen in Figure 8.2 1. Because perfec t neut ralizat ion with these techniques depends 0 11 feeding back a current thai ispreciufy the same as that through Cgd , the neutralizing ca pacitor e N must match Cf/d precisely. Unfort unately, Cgd is somewhat voltage -dependent . Perhaps because o f the difficulty of providing precise ca ncellation in the face of this variability, neutralization has foun d limited app lication in semiconductor amplifiers. Vacuum tubes. with the ir highl y linea r and re latively constan t co upling capacitances, were muc h better can didates for usc of this technique. Nevertheless, with sufficient dili gence, it is possible to obtain usefully large gain- bandwidth improvement s in semiconductor-based amplifiers using neutraliza tion.

8.7 CASCADED AMPLIFIERS So far in our study o f high-frequency amplifiers. we 've loo ked at open-circuit time constants, shunt peaking, peak ing with zeros. tuned amplifiers. unilareralization. and neu tralization - but all mainly in the context of single-stage circuits. However, it is

8. 7

207

CA SCA DED A M PLIFIERS

frequently the cas e that we can' t ge t e noug h ga in o ut of o ne stage. Th e q uestio n o f how many stages o ne sho uld use then natu rall y arises. Furthermo re. if eac h stage has a certain ba nd width , what ba nd width will the o verall a mplifie r ha ve '? Finally, is there some optimum number o f stages o ne sho uld use to maxim ize the o vera ll band wid th at a given gain in a give n technol ogy? To a nswer these qu estion s. we now co nsider

the properties of cascaded a mplifie rs.

8 .7.1 BAN DWIDTH SHR INKAG E Let's suppose that e ac h a mplifier stage has a unit DC ga in ( 10 s implify the math marginally) and a sing le po le . Th e a mplifier 's tra nsfe r fu nc tio n is the n Il ( s )

1

= --. u + 1

(42)

Acascade of 11 such a mplifiers w ill therefor e have an o verall tra nsfer functio n of A( s )

= ( _'_ )" . rs + I

(43)

We find the ba ndw id th in the standa rd way by co mputing the magnitu de of the transfer function and solving for the - 3· dB m lloff freq ue ncy:

IAUw)1=

1Cr'+ ,) I"~ ~.

so that

I ( J (w r)2

+I

)"

~

I

Ji .

(45 1

Clearing radica ls yields

[(wr)'

+ I]" =

2.

(46)

- I.

(47 )

and solving fo r the bandwidth atlast gives us (I)

=

~ j2 1//1 r

That is, the bandwidth of the o ve ra ll a mplifier is the band width of e ac h srugc . multiplied by some funny factor. As 11 ap proaches infinit y, (he o ve ra ll bandwidth tends

toward zero. The preci se form of the band wid th shrinkage is perhaps a lilli e hard 10 see fro m II , tho ug h . we can simplify the term unde r the rad ica l sig n 10 make the re lation ship substa ntially clearer. Ma rhe mauc ians wo uld suggest using

ihis formula . For large

208

CHAPTE R 8 HIG H · FREQUE N CY AM PLIFIER DESIG N

Table 8.2. Bandwidth vrrsns II

n

Actual BW ' norma lized '

Apprcxmuue BW (norm alized )

1 0.6.13 0.5 10 0.435 0.386 0.350 0.323 0 .30 1

0.833 0.589

l b.?

OA8 1

5.7 ,.,

I 2 3

,, 6 7

,

u.ue

» Error

,'>,

,.

0.372 O.J.lO 0.315

.l ll 2.9

O.2~

2.3

2.5

a series expa nsion of 2" " a nd then using only the tirst coup le o f term s. An equivalent {albe it roundabout) meth od is to ex plo it the somewhat better-known expansion for e' {sa cx plxJ) . We begin by recogni zing that (./8)

Then . for large

II,

we can write I

1 + -1 0 2.

(49)

"

We thus deri ve the inte restin g result tha t the ban dw idth bcbu ves approxima tely as foll ows:

1 z (t) = _J2 1/n _ I ~ _1h - 1 0 2 : : =: -D.K3) -,

r

r

1/

r/ii

'50)

T hat is. the ba ndw idth shrinks as the inverse square root of the numbe r of stages, at least in the lim it of large n. Befo re going furth er, we mig ht wa nt to get a bet ter feel for how muc h of an approxi ma tion is invo lved here, especially since we 're going to usc th is resu lt later on. C learl y, the for mu la is olfhy about 17% for n = I, We hope that the error decreases rapid ly, a nd it doc s, as Table 8.2 shows. Observe that the e rro r- dro ps be low 5% pretty quic kly, so the npp ro xir nate bandwidth shrinkage formu la is reason ably accurate. Note also tha t the a ppro ximation underestimat es the true band width by a bit. We can also deduce from the eq uation and the tab le that the meth od of open -circuit time co nsta nts docs a prett y rotte n job of estimating ba ndwidth for this cascade of identi ca l a mplifie rs. In this case, we have 1/ iden tical po les, so tha t we wo uld predict

8. 7 CASC A DED AM PLIFIER S

209

from the OC r method that bandwidt h goes d irect ly as 1/ 11 when . in fact. it goes a... the reci procal square root . Now thai we' ve deri ved this result . we can usc it that maximizes overa ll sys te m ba ndw idth .

8.7.2

10

determine the gain pe r stage

OP TIM UM GA IN PER STAG E

With the bandwidth shrinkage formula . we 're in a position to identi fy the op timum strategy to maximize bandwidth in a casc ade d ampl ifier, given a stated gain requirement and technolo gy constraints. Again. we' Il a..sumc Ihat all the stages are iden tical (because if one were slower than any other . it would represent the handwidth bottl eneck for the whole amplifier), each with a single pol e whose /rt'CJ'It'IICy depends i nversely on /he .f'a~w gai n . That is, each stage has a con stant ga in- ba ndw idth produ ct. so thai stage gai n and hand width trade off linearly. Our goa l is to find the number of stage s that . for a given overall gain requirement . maximizes the bandwid th (a nd hence the overall gai n-bandwidth product ). Assume that the overall gain is to be G, so that eac h ampli fier stage must have a gain of G Il ,.. If eac h stage ha.. the same gai n-bandwidth product (dr . then Ihe single-stage bandw idth will be wr BW",, = - , - . G In

(51)

From the approximate ba ndwidth shrinkage formula . we may wr ite the foll owi ng expression fur the bandwidth o f the total ampli fier: (dr .Ji02 BWt< '1 ~ - - . - - .

c v-

.;n

(52 )

The reciprocal of the bandwidth (somew hat handier for what we'I l do shortly) is therefore (53)

We'll now maximize the total bandwidth by minimizi ng its reciproc al: (54)

Taking the derivative. cancelling terms. and solving yiel d...

(55)

21 0

CHAPTER B HIGH -FREQ UEN CY AMPliF IER DESIG N

Table 8.3. Max imu m HW and G . BW versus G Maximum HW G 10

20

'0

n

".•

6.0 7.8 9 .2 10.6

100 200 '00

12.4

IlXIO

13.8

morrnahzedj

G . HW

0 .24 0.21 0. 18 0.17 0.16 0.14 0.14

2..1 4.2 9.0 17 32

70 140

According to this analysis. the ga in per Mage sho uld the refo re be chosen as the square root of e if we want to maxi mize the overa ll ba ndwid th . 10 The numbe r o f stages co rres pond ing to this optim um is n = 2 InG.

(56)

and the o veral l ba ndw id th co rre spond ing to this co nd itio n is

RWI ,.. =

«) 1 •

n2 2e . In G

~

0 .357w 1 ::::::

Jiil"G

•

(51)

From th is last express io n. we c an see tha t the o ve ral l band width is relat ive ly insensitive to the va lue of overa ll gain whe n Ihis o ptimum is c hose n. In fuct , the produ ct of ban d width and the square mot ofthe log of ga in is con stant . Perh aps this inse nsitivity is aga in best illu st nued in tabu lar form ; see Table 8.3. Ig no ring the minor pra c tica l de ta il of nonintege r values of I I . we ca n set: thai the bandwidth c ha nges by le ss than a fac tor of 2 eve n thou gh the gain cha nges by a fartor of 100 . C lea rly, eve n tho ugh each ampl ifier stag e has a co nsta nt ga in- ba ndwidth product . the overa ll a mp lifier doe s no t. In fac t. the gain- band width prod uc t actually gro ws witho ut bound as II incr ea se s. beca use a ca scade of th is type trad es o fT bandwidth for the sq uare root of the log of ga in.

10

A ~i m i l ar derivation, but usin g open-circun lime constant s rather than the band widt h shrinkage formula . leads IIftC 10 predict an op tim um gain per stage (If (' in' lead of its sq uare root. 'The d if. terence is not l>igni lican t beca use the optimu m con ditions are relanve ly "ttar," that is, the ovcnll oondwidth is nut ove rly se nsitive to the preci se value of gain per l>tagc . Spccitically. it is easy 10 show that. if one uses a gai n o f (' per stage, the bandwidth fu r large /I dcgradc~ toy a [actor of only ~. or ahou l 0 .116.

8 ,7

211

CASC A DED AM PLIFIERS

--lLJL

v IS

I ---c0o-,---.-i C

IDEAL

LPF

_ VOln

FIGURE 8,22 , Superregenerative amplifier.

It shouldn' t take much reflection to co nclude that a constant gain-bandwidth prod-

is really only a propert y o f single-pole syste ms. That many commonly encountered systems are dom inated by a small number o f poles h ay. one) has led to the widely held misconcept ion that gain and band width mu st trade off linearly simply because they often do. However . we've seen that thi s relationship breaks down dramatically when the order of the system grow s to large values. Thi s observation is quite useful. for it suggests that one may purposefully construct system.. of high order specifically to decouple gain fro m bandwidt h . We will ex ploit this observation to construct amplifiers that trad e bandwidth for only weak functions of gain.

uct

8 . 7. 3 THE SUPERREGEN ERATIV E AMP LIFIER

Back in the 192()-.. Edwin II. Armstrong developed the superregene rative amplifier for radio. It em ployed enough po sitive feedback to drive an amplifier into a specia l imerminent oscilla tory condition and. in so doing. enabled the attainment of spcctncular amounts of gain fro m a single stage . It was the first circuit 10 violate. in a major way. the linear gain- bandw idth tradeoff " law." Although his implementation was actually a bandpa ss amplifier. we will anal yze the correspo nding low-pass version; its gain- bandwidth characteristics are the same as its bandpa ss progenitor. A greatly simplified supcrrege nerative am plifi er is shown in Figure 8.22. Notice that there is a negative resistor in thi s system. We can always synthesize the equivalent of a negative resistance usi ng active devices, so invoking its existence here is perfectly realistic. As a consequence of the negative resistance. the Re time constant has a negative value. and the pule is in the right-half s-planc. There is therefore an ex po nential growth in the capacitor voltage whenever the sampling switch is open. 111e longer we wail before closing the switch again. the greater the gain. by an expo nentially growing factor. Aquantitative analysis of this amplifier is straightforward. Assume thut the switch closes for an infinitesim ally short time. and that (he input source is capable of instantly charging up the capacitor. It is not necessary even to approach these conditio ns in

212

CHA PTE R 8 HIGH · f REQ UENCY AMPLI FIER DESIGN

rea lity, hut accepting the se assumptions simplifies the an alysis withou t int rod ucing any funda mental errors.

When the switch opens, the capacitor voltage ramps up exponentially from the initial voltage (which is L'in) : (58)

Thi.. ex ponential grow th is a llo wed 10 co ntinue fo r a pe riod T , a nd the ex ponentially g rowi ng sig na l is a....eragcd by an ide-al low-pass fille r. lie nee.

-"
= -I

T

i

T

u

t~", t'

IINC

dt =

RC TINe {to - IWin' T

-

(59)

If the lim e co ns ta nt RC is sho rt com pare d wi th th e sa m pling peri od T. we ge t a gain factor that is expon ent ially related to that ratio. To invo lve bandwi dth ex plicitly. not e that - since we have a sa mpled syste m - we mu st satisfy the Nyq uist sa mpling c rite rio n. He nce. we m ust c hoose 1/ T hig he r than twice the highe st frequency co mpone nt of the input signal. Thai i.., we mu st have

I BW < - . 2T

(1i01

The refo re . the product of ban dwidth a nd the log of the gain fur thi s type of a mplifier IS

BW - Jn G = -I-

2RC

1 + -2T

(He) T '

In -

(61)

Thus. we sec thut the su pc rrege ne rative a mplifier trades off ba ndwidth for the log of ga in, to a reason able ap proximation . As in the case of a ca sc ade of a mplifiers. Ihe implication is tha t the ban d width c hange s little as one varies the ga in o ver a large ra nge. In fuct . if o ne is willing to e nd ure a suffic ie ntly long rege ne ratio n inte rva l. the ga in can he mad e enormously large und so a llow the overa ll ga in- bandw id th produ ct to e xce ed tha t of the ac tive dcvice ts) invol ved . Ano the r differen ce bet ween the superreg en erativ e amplifier a nd a c ascade of conve ntiona l . .ing lc -p olc amplifiers is that it acc omplishc s thi .. gain- band wid th tradeoff wit h just one RC . The period ica lly time -vary ing nat ure of thi s ump lilic r e ndows it with so me o f the propertie s of a high -order sys tem with o nly OIl C e ne rgy -sto rage clemen t. with a bandpas s ver sion o f this amplifier , Armstrong was able to o bta in so much ga in from a sing le vac uum tube tha t he co uld amplify fundam ental noi se sources (w hich we' ll study very soon) to a ud ible levels . Il l' an d RCA q uite rea so nahly assumcd that this rem arkable property wo uld be ex tre mely usefu l in rad ios . Th e supc rrcgc ncrativc a mplifi er is no t used very m uc h these da ys . how eve r. for a va riety of pract ical rea so ns. C hief among the m is thut it is an oscill ator. and RF versio ns of the supc rregenerulor arc ac tua lly parasitic u unsmiuers thut ca n ca use interference. Additio nally. e ve n in the abse nce of a sig na l, the supe rreg c ne raro r am plifies

8 ,1 CA SCAD ED AMPLif i ER S

213

nnio;,e to audible and annoying leve ls. These cha rac teris tics ha ve limit ed supe rregcnerative circuit s to rela tivel y low -tech applications such a s c hild re n's wa lk ie -ta lkies nhe most inexpensive o nes. they're easy to spot because they have a ,..h aructcri stic. annoying hiss eve n w he n no sig nal is be ing recei ved). Neve rthele ss, it is 'I fuscinating amplific a tio n prin ciple s ince it allows o ne to obtai n. in a si ng le stag e , an o verall gain- bandw id th product th at ca n g rea tly exceed the Wr of the device invol ved , -c

8 .7.4 A CO NU NDRUM we've now stud ied severa l types of a mplifiers (an d there are ot he rs) that trade off bandwidth for on ly weak func tio ns of ga in. It sho uld be clear from these exa mples thaI the notio n of 'I fixed gai n-band widt h prod uct is badly flawed. But most e ngineers recog nize that the re's no free lunch , and sta rt look ing fo r other things thaI might be trad ing off with ga in, IItum s o ut tha t the most important of these param e ters is delay. It is so me time s ( but not a lway s) the case that the re is a stro nge r tradeoff between ga in a nd del ay tha n the re is bet wee n gai n a nd bandwidth . In man y a pplications (suc h as in TV or o ptica l tibe r sys te ms ) co mm unication is a one -way affai r, so delay is frequently more tolerable tha n limited bandw idth . It's nice to know that there are prac tical cases w he re we c an im prove a paramete r that we do c are abo ut by degrading a param eter that we do n't. Having acc epted that the co up ling be tween ga in ami ba ndwid th is ev ide ntly we ak at best (as we saw in Cha pter 5), we migh t rea so nab ly nsk what gain or ba ndwid th could be obtai ned if we were willi ng to to lerate a n arb itrary del ay in the respo nse. The surprising cnswer t in prin ci ple , a nyway) , is th at we c.o uld obtain arbitrarily large gains at a fixed bandwidth if we didn't care about de lay. We 've a lready see n ex am ples of gain-de lay rradeoffs. The T-coil compensa tor , as well 3.\ ca scaded a nd superrege ne rative a mplifiers. ex hibit larger gains or impro ved bandwidth OIl the ex pe nse of longer del ays . In the cas e o f the c asc aded a mplifier. adding stages inc rease s gain but a lso incr eas es de lay (a lthough at a slowe r ra te tha n the gain increase) . Simi larl y, len g then ing the rege nera tio n interval in the supe rregenerative am piiller inc rease s the gain (ex po ne ntially ) but a lso increase s (ob vio usly ) the delay. To under sta nd ho w o ne might synt hes ize circui ts tha t ex hibit a mo re di rect bandwidth-delay tradeoff tha n we've ide nt ified thu s fa r. reca ll that the reciprocal of rise lime is a mea su re of ba ndwid th . Now imagine 'In am plifie r that save!'> a ll of the energy in an input !'otep for so me lon g period of time wit hout produci ng any ou tput . then dumps it all at o nce to the output , yi eld ing a very fast riseti me (= high handIlo'i dlh). S uc h an am pl ifie r wo uld ind eed trade de lay for ba ndwid th d irec tly, Fro m the forego ing de scri ption, we see that suc h a n am plifier mu st ha ve the ability to pro... vide large de lay s (10 a llow this tradeoff in the first place ) ove r a la rge ba nd width . This require me nt in tum suggests tha t netwo rks of rel ativel y hig h order a re need ed,

»

CHA PIER 8 H IGH - FREQ U EN CY AM PLIFIER DESIG N

2"

(Pieces o r e ansmlssion line used to create a tapped de lay line)

R = Z"

'

...

FIGURE 8 .23 . DiWiWted amplifier .

since u pole (or zero) appro ximates a lime delay only over a limited frequency interval. To proceed thus requires a more detailed understandi ng of networks of high o rder. A s we' ll St.' C in the ne xt sec tion. transm ission lines a nd the ir IUIllJX.' d approximations are particu larly valuable in th is co ntext, allowing one 10 construct amplifiers with bandwidths approaching ta r .

8.7 .5 THE DISTR IBUTED AMPLI FIER Without questinn, the most elegant exploitation of distributed concepts is the distributed ampli fier inven ted by W. S. Percival of the United Kingdom in 1936. Ill' appa rently did n't talk about it "cry much. though. and wides pread aware ness of this scheme had to awa it the publication in 1 9~ 8 of a landmark paper by Glnz ton. Hewlett, Jasbcrg, and Nne .11 In the abstract to their paper , the authors note that " the ordinary co ncept of ' maximu m band width- gain product' docs not ap pl y to this dis tributed amplifier." Let's sec how thi s structure achieves u gain-for-delay trudcoff'without affecting bandwidth. As ca n be see n in Figure H.23, inputs to the transistors arc supplied by a tapped delay line. and the outputs of the transistors arc fcd into another lapped delay line. Although simple sections are shown, the best performance is obtai ned whe n »r-derived or 'It-coi l sec tions are employed, as discussed earlier. A vo ltage step applied to the input propagates do wn the inp ut line, causing a step to appea r at eac h transistor in succession. Each transistor ge nerates a current equal

II E . L . Gm zron . W. R . Hewle tt. J . H . Ja...berg, and J. D. Nne. "Distributed Am plification," Proc. IHE. A ugU:'
8 .7 CASCA DED AMPLIFI ER S

215

to its Km multiplied by the value of the input step. and the currents of all the transistors ultimately sum in time coherence if the delays of the input and output lines are matched , Since each tap on rhe OUTpUI line presents an impedance of Zo/2, the overa ll gain is Av

'I I:.. ZO

= ---. 2

(62)

In contrast with ordi nary am plifier cascades. this amplifier has an overall gain that depends linearly Oil the number of stages and therefo re can ope rate at frequencies where eac h stage actually has a gain smaller than unity, Co nsequently. the distributed amplifi er can operate at substantially higher frequ encies than can co nventiona l amplifiers. Furthermore. since the delay is also proport ional to the number of Mages. this amplifier does trade gain fur delay; bandwidth does not factor into the trade -off in any direct. obvious way. Another way to look at this amplifi er is to recognize that o ne source of band width limi tation in conventional amplifiers is the drop in input impedance with increasing frequency that accom panies inpu t ca pacitance. Here . however. we absorb the device's input capacitance into the constants of the tapped dela y line. 12 Hence. until the cutoff frequ ency o f the line itself is approached. the input impedan ce remains constant and equal 10 Zoo Similarly, the output ca pac itance o f the devices can be absorbed into the output line. Since input ca pacitances are usually larger than output capacitances. so me signifi cant adj ustment of line constants is necessary to guarantee matched delays. Lest you think that 10 achieve a practically usefullevel of balance is possible only in theo ry. you should know that vacuum tube distributed amplifiers were success fully usedin many Tektron ix ~ o scilloscope s fur many years (their mudel 5 J 3 was the first (0 use this type of amplifier). The amplifiers were used in the final vertical de fl ection stage. and typicall y involved six or seven " matched" pairs of vacuum tube s. Band widths of roughly wd2 were routinely achieved, so that lOO-Mlb general-purpose oscilloscope s were available hy around 1960,u Given these attributes. one may reasonably ask why this type of ampli fier is not ubiquitous toda y. Part of the reaso n is that it is rather power hungry, since many stages are required to provide a given gain. Another is that the active devices thai supplanted

12 We are assum ing that the input impe dance of the device look s capacitive at high freque ncies, However. this assump tion is not always satisfied, andthe departure Inunthis assumption must be taken into acco uot io prac tical de signs if good res ults are to he achie ved . 13 Tbe distributed amplifiers in the Tek tro nix 585A toO-M Hz nsc illoscope used 6DJ8 duo -triodes, which have hs of roughly JOn Mtb , The delay lines w ere Ct>l11~ of T-coils. .... nic h pro vide ~UC1' hand .... iLlth than ordi nary m -deri\'oo lumped appro~irnaliol1 s ,

• CHAPTE R 8 HIGH -FRE QU ENCY AM PLIFIER DESIGN

2 16

vacuum tubes, bi po la r trans isto rs, have severa l c harac te ristics that ma ke the m unsuit-

able for usc in distributed amplifiers. The biggest offender is the parasitic base reo

r".

...iste nce. whic h spo ils line Q a nd the re fo re deg rades the line. Bipolar d istributed am plifie rs co nseq uently acq uired an unsa vory repu tation. Fina lly. the lumped lines co uld not he integ rated until very rece ntly. whe n devices im proved e no ugh M) that

frequencies of ope ration increa sed to a range where fu lly integrated lines become pract ica l. Distributed amplifier s a ll but disappea red as a co nseq ue nce . They lin ally made th eir rea ppearance in abo ut 1980 , whe n wo rke rs in Ga As techno logy rediscoven..-d the princi ple. Since thut time, distributed amplifiers have been con stru cted in a variety of co mpo und semiconductor tec hnologi es. with InP versions ac hie vi ng IOO -G li z band widths. The o ne CM O S impleme nta tion to date ex hibits ncarly a 5-G I I7 bandwidth in a O.S-Il m technology using bondwirc inductors. Substantially higher band widt hs may he ex pected as C r-.fO S process technology continues to improve,

8. 8 SUM M A RY

We have see n thai purpo sefu l violations of the conditions assumed in the development of'ope n-c ircui t time con stan ts can lead In sig ni ficant bandwidth improvements. O nly simple net work s are used . and the tec hniq ues do not req uire an inc rease in power, Both shunt peaking and peaking throu gh zeros provide these im provements. albeit at some cost in pulse response fideli ty. A tradeo ff exi sts between the amount of handwidt h increase and pulse dis tortion. butt he large improvements obtained for trivia l effort make these met hod s wo rth conside ring. We 've see n thut use of tuned load s allows the attainment of es sentially the same gai n band width at hig h freq uencies as at low frequencies by ex ploiting the resonant cance llation of parasitic and explicit ca pacitances by inductances. We 've also see n that co upling from output to input can severely limit the practically attainable gai n- bandwid th produ cts by loadi ng and detuning the output tank. destabilizing the ampli fier. and modifying port impedances that co mplicate the casc ading of stages. Dctunin g and destab ilization can he suppres sed greatly through the use of unilate ral topologies (such as the casc ode or source -coupled amplifier) that provide isola tion between output and input port s ove r a wide frequ ency range. or thro ugh the use of neut ralization to cancel the undesired feed back over some freq uency and operating po int range. The notion of a fixed gainbandwidt h prod uct was shown to be false, as demo nstrated in the examples o f casc ades o f ordinary amplifiers. as well as of the superregenerative amplifiers. which all exhibi t more of a ga in- delay than a gain-bandwidth trade off. In this point of view, then. we see that series peak ing. shunt pea king. and Ttc oil compensatio ns. which e ffectively distribute the load, may be thou ght of as a

PROBLEM SET

217

sequence of ever-bcne r approx imations on the way to the distributed amp lifier, wh ich distributes the active device as we ll a ~ the outp ut load.

PROB LEM SET FO R HI GH -FRE QUEN CY AMPLIFI ER DESIG N PROBLEM 1 Plot gain and phase for the shunt- peaked netwo rk ...how n in Figure 8.24. Le t m ta ke on values from I 10 5 in steps of 0 ,2.

H' C L = m

f iG URE 8.24. Shunt-pea ked RtC netwofk.

PROB LEM 2 Deri ve the equ alio n... for serie... peaking ( Fig ure 8.25). Sh ow formally what conditi ons maximize bandwid th and yield maximally flat response and time delay, For these three con di tion.... pro vide expressions for the bandwidth achieved. Verify your answers with ... imulntions.

,

L = R C m

FIGURE 8 . 25. Series-pea ked RtC network .

PROBLEM 3 Unfortunat ely, the rel ativel y large parasitic capacit ances of typical MOSFETs can prevent fr do ubl ers from working as well us on e would like. Explore this idea further by simulating both the Darlington and Bunjes doublers. using any model set (e.g ., the level-S SPIC E mod els provi ded in C hapte r 3) . Terminate each in an incrementa) short circuit ( prov ided by a 2·V DC source). and measure the incremental current gai n as frequ en cy increases. Determine fr with a first-order extrapolation to the unit ga in frequency. Com pare with the Ir value for a sing le device at the same bia s current . Which par asitic capacita nces ca use the disappointment ( if there is one)?

21'

CHA PT ER 8 HIGH -FREQ UENCY A M PLIFIER DESIGN

PROBLEM 4 Following a development parallel to the o ne shown in the chapter. derive an express ion for the gain- bandw idth be havior of an RLC bandpass superregcnerauve amplifier.

PROBLEM 5 Provide an ex plicit expression that shows under what conditions )'i. may have a negat ive rea l part in simple commo n-source amplifiers with tuned load. If you were to build such an amplifier and d iscover that Y in was negat ive. what remedies co uld you apply? OITer two spec ific solutions. PROBLEM 6 Design a single -stage tuned amplifier to meet the followi ng smallsignal specifications: [voltage gain]: > 50. measured at the ce nter freq uency ; tota l bandwidt h ( - 3-uR): > 1 MHz; center frequ ency: 75 M Hz; sou rce resistance : 500 ; load : 10 pF. pure ly capaciti ....e. Use the process characteristics from C hapter 3. Assume thut thc (external) inductor ' s self-resonant frequ ency is high enough to be neglected . You may also assume that the inductor Q is 200 at the ce nter frequency of75 MH z, PROBLEM 7 Th is probl em explores a number o f impedance transform ation issues that are of interest in high-frequ ency des ign. (a) High-speed follow ers have a tendency 10 ring Of c....en oscillate when driving capacitive loads. Your task is to identify the conditions that may cause this pr oblem and to propo se a solution. First co nside r a source follower as shown in Figure 8.26. Assum e thai the dra in- gate ca pacitance eM is zero , as is the parasitic gate resi stance rr - Derive an ex pression for the incremental inp ut imped ance as seen by the source, 111. What is the real part of yo ur answer?

'. FIGU RE 8 ,2 6 . Capocitively

Iooded source fol lower.

PROBLEM SET

21 9

(b) Over whal range of load capaci tance can the real pa n of the inpu t impedance he negative?

(c) For a load capacitance know n to fall within the range of part ( b). mod ify the ci rcuit ~) that the source VI always see s an imped ance whose real part is positive. keeping in mind that the fo llower is intended for high -speed operation. Pro vide formulas to allow co mputation of the values of any added co mponents. (d) Now co nsider a CE or CS ci rcuit with indu ctive source dege neration. Thi s inductance may arise uninten tion all y from unavoidable wiring parasitics. or may be inserted purpo sefu lly. In any case. deri ve an expression for the input imped ance of the circuit depi cted in Figure 8.27. What is the rea l part of the inpu t impedance?

L

FIGURE 8.27. Inductively

loaded source fol lower. PROBLEM 8 Consider the zero -peaked amp lifier shown in Figure 8.28.

"

'O\lT

f IGURE 8 . 28. Zero-peaked amplifier (b io ~i n9 not ~ hown ).

(a) Derive
220

CHAPTE R 8 H IG H- FR EQ UEN CY A MPLIFIER DESIG N

(b ) Choose C 1 so that the zero cancels the output po le. What is the bandwidth of the overall amplifier if g", R t = 9? Express you r answer in terms o f the output pole freq uency. II R2C 2 • Recognize that the output pole frequency is the bandwidth of the unpeuked amplifier. PRO BLE M 9 Thi s pro blem concerns some practical d ifficulties in high-frequency amp lifier design. Conside r a co mmon-gate amplifie r (Figure 8.29) in whic h parasitic inductance is explicitly mod eled (assume that the tran sistor is magically biased into the forwa rd-active region ). Since the drain is tied direc tly to V /)fh this is a low-gain circuit. so use an appropriately simplified incre mental model (e.g.• neglect r..).

FIG URE 8. 29. Common-gate amplifier with porosiric inductance,

(a) Assuming that one may neglec t alljll nctio ll capacitances as well as r ll • derive an ex press ion for the input impedance (as a functi on of ,5). Make no ot her approximations. (h) Sin ce the increm en tal model is valid only for frequencies we ll be low to r , your answer 10 part (a) applies only to tha t restric ted freq uency range. Simplify your answer to part (a) accordingly. and deri ve ex pressions for the reul and imaginary parts of the impedance when .5 = jw. For f!l t' rest of this I'mhfel1l , assume that the simplified equations apply. (C) It should he clea r that you can alway s model any impedance as the series conneclio n of a resistor and a rea ctive clement. For this equivalent inp ut im pedance of this part icular circuit. what is that reactive eleme nt, and wha t is the expression fur its value? (d ) A typical value for the inductance of a straight piece of wire is about I nll /mm . As a consequence . it is diffi cult to construc t any rea l circuits with parasitic indu ctances much smaller than several nnnohenr jes. Su ppo se' we happen In have 10 nit of total parasitic inductance be twee n the base term inal and ground. and suppose further that the value of C~ ,. is 10 pF ( ycs, this impli es a ga rgantuan device) . Abo ve what freq uency J eril wo uld there I'olelltilllly be a stability pro blem at the inp ut? You need to preserve only two significant d igits in your answer. Explain why there co uld be a stability proble m above J eril'

PROBLEM SET

221

(c) Suppo...e now thai we drive the circuit w ith a small -sig na l sinu...oidal volt age source w hose Th e veni n resista nce is 50 Q (purely real) . at a freq ue ncy of 31m" Suppose furt he r Ih:'11 I/x", ha ppe ns to be 5 Q :.II ou r particular bias point. Wh al is the ratio of the vo ltage at the so urce 10 tha t of the in put vo ltage'!

PROBLEM 10 De sign a fully integrated amplifier with the followi ng small-signal :r.pccs: [volt age gainl: > 10. measured at "modera tely low" freque ncy ; bandwidth (-3 dB ): > 500 MH z; source resi stance: 50 Q : load : I pE purel y capacitive ; maximum freq uency respo nse peak: < 10%; to tal supply powe r: < 50 mW . Use device model s fro m Chapter 3. Assume thai the supply m ilage is 3.3 V. You may usc up In 20 nil of on-c hip indu ctance and 5 nH of bood wire inducta nce . Assume on -c hip spirals have a Q of 5 at I G Hz . a nd prete nd Ihat the correspondi ng effective se ries resista nce remains consta nt at all freq uencies. Ignore self-resonance of all inducto rs. You are also permitted up 10 20 pF of on -ch ip ca paci ta nce. You may assume that the ca p acitor is ideal in all respects.

PROBLEM 11 Thi s proble m explo res in greater detai l the notion of non uniform group delay as a so urce of distortion . (a) Plot the delay versus freque ncy fur a single HC high -pass filter. T he freq uency axis should he normal ized 10 II He. (b) Co nsider a second-orde r low-pass sec tio n whose tran sfer function is lI( s ) =

sz , 2" + [ Wn

QWn

+I

]-1

( PH. I )

Repeat part (a) for Q = 0.5 . I. and 2. now normali zing freq uency to (1)n ' (c) For the shunt-pea ked am plifie r. we noted that there is a trad eoff between ma ximum ba ndw idth improvem ent s a nd pul se fidelity. Recall that its ideal tra nsfer function is A(.» = K.(,L

I

+ R) II -sC

~

K. RI.,I L / Rj+ I I

"T,;-;;--';-'-;";;-c--;-

s LC + .IRC + I ., ( L/R) + I = A II -,-;';;:';-"-';",-':--;, ' LC + .,RC + I

( P8.2)

Repeal pa rt (a) for the followin g ra tios of RC 10 1. / R: 2. 2.5 . 3. 3.5. Plot yo ur results as a fu nction of wlwl . where W I is II HC . the uncompensa ted amplifi er 's band w idth . You may find it helpful 10 ma ke usc of the result s of parts (a) a nd (b). Phases of cascade d linea r sys te ms add. so thei r dela ys also udd.

222

CHAP TER 8

HIGH -FRE QUEN CY AMP LIFIER DESIGN

PRO BLEM 12 Out of laz iness, most eng inee rs chouse the common-gate device in a cascode 10 be of the same size as the common-source device . Howeve r. this choice is almost neve r the optimum. In the questions that follow, you may neglec t bod y effect but do fWI neglect parasitic device ca paci tances.

(a ) Disc uss what happen s as the casc cd ing device is made progressivel y narrow er than the com mon-source device. Assu me that the same bias curre nt flows through both (again. this cho ice is not necessaril y optima l). (11) Discu ss wha l happens as the cascod ing device is made progressively wider. (C) Given yo ur insights from (a) and (b), desc ribe a form al procedure for finding the optimal s ize of the cascodi ng de vice to ma ximi ze bandwidt h .

CHAPTER NINE

VOLTAGE REFERENCES AND BIASING

9.1 IN TRO DUCTIO N The pre....iou s chapter o n a mp lifier de sign ge nerally ignored the issu e of ge nerating suitable bia... voltages or c urre nts. Th is neg lect wa s by conscious desig n in orde r to minimize cl utte r in the c ircuit di agrams. In thi s c hapte r we fina lly take up the study of this important topic , foc using o n a variety of way s 10 ge nerate voltages a nd cu rre nts that are rel ati vel y inde pendent of supply ....ol tage a nd temperature. Because C MOS offers relatively limited o ptions fo r rea lizing bias circ uits. we'H see that so me of the most useful biasing idio m s a re actually those based o n bipo lar c ircuits . A parasitic bipolar device exi sts in every C MOS technology and may be used . fo r exam ple. in a bandgap volt agc refe rence. Even tho ug h the c harac te ristics o f parasitic transistors arc far from idea l. the performance of bias c ircui ts made with suc h de....ices is fre quently vastly superior to th at of " pure" C MO S bias circui ts. In what fo llow s. it is wort hw hile to kee p in mind that any voltagc we produce must depend o n some collectio n of param eters that ultim atel y have the di men sions of a voltage (such as kT/q. for example) . Similarly. any curre nt we prod uce mu st depend 011 parameters that ultim atel y have the dimen sion s of c urre nt (such as VIR) . Although see m ingly ob ....iou s a nd trivial sta teme nts. we ' ll see thatthey life ex tre mely useful guide s for the design of sta ble referen ces.

9.2

REVIEW OF DIODE BEHAVIOR

Although the volt agc across a fo rward -b iased diode is re lati vel y inse nsitive to c urre nt because of the logarithmi c de pendence of diode cu rre nt on diode vo ltage . its variation with temperatu re is sig nifica nt. To unde rstand the preci se nature of the temperature dependence. recall that the diode voltage may be expressed as

Vp = nVT In ( 223

~;) .

(I)

CHAPTER 9 VOlTAGE RE f ERE NCES AND BIASING

where Vr is the thermal voltage k Tl q and II . the ideality factor. is typically between I and 1.5 in diodes. Transistor VRES (BE denotes " base emitter") conform more closely 10 the " ideal diode law" than do ordinary diodes. :1.0 we will assign " a value of unity in all thai follow s. It is frequ ent ly ( bu t incorrectly) inferred from Eqn . I that Vo has a po sitive Te beca use of its pro po rtio nality 10 Vr . The fly in the o intme nt is that Is itsel f has an exponential temperature depe nden ce. and this alters the situation considerably, To clari fy maners. co nsider the following q uasicmpirica l express ion for Is:

( V"'' )

Is = In cxp - V r

(2)

.

where 10 is som e proc ess- and geometry- dependent current I (1 0 is typically around 20 orders of magnitude larger than Is at roo m temperature. so In is much larger than typica l values o f I ,l ) , and where VGn is the bandgap voltage (abo ut 1.2 v j cx trupolated to abso lute zero. Using this detailed ex pression for Is , we can ex pand the equat ion for V/J as follows.?

(I,,) '"

Vo = VGn - Vr In -

.

(31

Thus. we see that the j unctio n vo ltage dec reases linear ly fro m a value of VGlI. as seen in the plot of VI) ven..us temperature at co nstant diode curre nt ( Figure 9. 1). Note that this eq uatio n tell s us that Vo always eq uals V(;o at ab solute zero. ' Furth ermore. it's easy to sec that the tem per ature coeffi cient at any te mperature is simply

dV" = dT

(41

With the assu mption of constant 10 • the temperature coefficien t is inde pendent of temperature and equ al to a bout - 2 mVj K. Th is linea rly decreasing behavior is known as C TAT . for "c om pleme ntary to absol ute temperature." Note that the voltage does depend ( logarithmically) on diod e current, so the temperature coe fficien t also de pends somewhat on the diode current, with lower currents associat ed with higher tempe rature coe fficie nts.

I It also depends weakly o n temperature. t>tll wc'Il defe r a detailed disc ussion ahOtll lhe behavior of 10 until Section t,l5 . l The m inu s silln is nnl an e ITOr. Just re m e rnbcr thnt lilt> :ul"umt'nl of Ihe log Ilt> re is lyr ica lly mUt:h larger than unity. J Again , this value is an e~l l1l rolal ed one . II musl he stressed that the behavior of real juocnonv al roth exnemes IIf tempe rature will differ from thai shown ; tbe equalions prescnlW lo't' validity at extreme ly (;(lld temperatures (say. < IlX' K ' because of earner free ze -oct (i.e.. the dopants fail to ionize) and band gap variation with temperature, and at hig h tem peratu res ( > -l5H.... j(.. ' K I because the silico n !!I!CS intrinsic.

225

9. .4 SUPPLY· INDEPENDENT BIAS CIRCUIT S

Vn VGO - 1.2 _.-

lower l l.l - - - " bOO . typ.

FIGURE 9. 1.

T ( kd vi n ~ )

Approximate behav ior of VD versus tempere ture.

Although a Vfr based reference can provide an output that depends very little on supply milage. the CTAT behavior mayor may not he acceptable. depending on the applicatio n. However. we shall see that the CTAT behavio r of a Vo is particularly valuable for usc in a class of references based on the bandgap voltage VaG. we'tl rake up the detailed study of bandgap reference s in Section 9 .5.

9.3

DIODES AND BIPOLAR TRANSISTORS IN CMOS TECHNOLOGY

The most fl exible opt ion for realizi ng diodes and bipola r transistors in s t a n da rd C ~ I O S technology derives fro m the parasitic substrate p- n- p transistor availab le in n-wcll processes. The p + source - drain diffusions serve as the emitter. the n-well as the base. and the substrate as the collector. To red uce series base resistance. it is advisable to surround co mpletely the emitter with n+ diffu sions placed as clo se to the emitter as the desig n rules allow. as sugges ted by Figure 9.2. As with its counterpart in inexpensive bipol ar processes. the substrate p-n- p in CMOS technology can only he used in circuits that allow the co llector to he at substrate potentia l. Fortunately. there are numerous circuits thai satisfy this cond ition. For example. a simple voltage "reference" ca n he co nstructed with this device connected as a grounded diod e. where the emitte r is the anode and the ca thode is the base and co llector (substrure) tied together.

9. 4 SUPPLY·INDEPENDENT BIAS CIRCUITS In order to minimize sensitivity 10 power supply variations. it is desirab le to derive the bias currents for reference voltages from the refere nce voltages them selves. rather than directly from the power supply. Alt hough it may see m a violation of some

226

CHAPlfR 9 VOLTAGE REfE RENC ES A N D BIA SING

..mill....

ro lln1or

.,

" . ·_rli

FIGURE 9.2. Paro sitic substrote p-o-p in n-well CMO S lnol drawn to sca le).

R

• v"

I" R

In

Vl)(: l) , = R

R,

fi GURE 9. 3. Self-biased reference.

fundarrernall aw (the " no free lunch" pri nciple). it is possible to arrange for this condition. To illu strate how one may acco mplish this feat . con sider the circuit of Figure 9.3. As you ca n see. the current through the diod e de pe nds on the diod e volta ge itself. rather than on the supply voltage . This technique thus pro vides exce lle nt supply voltage inde pe ndence. An important practical note is that a start-up network is always necessary in selfbiased circuits beca use there arc two slates, one which is stable in the conventional sense and another in which all curre nts arc zero.' The start-up network guarantees thai the circuit gels out of the undesired metastable state. Most practical implem entations of the sel f-biased circuit dispense with the opamp, as seen in Figure 9.4 . The PMOS mirror ? enforces equality o f the NMOS drain

4 Even though the ail-zero stille is metastable, practical circuits are found in this slate a madtJeningly large port ion of the time . A start-up network is therefo re mandator y for reliable operation. S Better mirrors would genera lly he used in practice to avoid supply-dependent mirror ratios: simple one s are shllwn III reduce schcmanc clu tter.

9.5 BAND GAP VO LTAGE RE FE RENCE

227

V DU

Sian -up

network

• R

FIGURE 9.4 . Alternative self..bios.ed reference.

currents. and hence that of the NMOS V, I ' Thus. the diod e voltage appears across R; the corresponding current is the same in both hal ves of the mirror and is therefore the bias current of the diod e itself. Thu s. as in the op- amp versio n o f this circuit. the diode provides its own bias current . The self-biase d circuit of Figure 9.4 is qu ite versatile. It should be clear that the diode may be replaced by a variety of elements. For examp le. a diod e -co nnected MOSF ET would prod uce a bias current of V,./ R. or a zener d iode (if available ) could he used instead. As we' Il see in the next sect ion. the self-biase d circuit is particularly useful in realizing bandgap voltage references in CMOS technology.

9.5

BAND GAP VOlTAG E REF EREN CE

Because IC tech nolo gy directl y offers no reference voltages that are inherently co nstant. the only practical option is to combine two voltages with precisel y com plementary tem peratu re behav ior. Thus. the gene ral recipe for ma king tempera ture independent refere nces is to add a voltage that goes up wit h rcrnpcruturc to one that goes down with tem perature. If the two slopes ca ncel. the sum will he independent of temperature. Without question. the most elegant realization of this idea is the bandgap voltage reference. It produ ces an output voltage that is traceable to fundamen tal constants and therefore relatively insensitive to variations in process. temperatu re. and supply, The first wide ly used bandgap voltage reference was designed by Bob Widlar in the hugely popular and revo lutionary LM309 5-Y regulator IC from National Semiconductor. It was the first refere nce whose initial acc uracy was good enough to eliminate the requirement for adj ustmen t by the end user. T hus. o nly three terminals were needed (allowing use of inexpensive transistor packages). making this part as easy to use as one co uld bope . To understand qua ntitatively how bandga p references wor k. we need 10 re -examine the detailed be havio r o f ju nction voltage with tem peratu re. Since transistor junctio ns

228

CHAPTER 9 VOLTAG E REFE RENCES AND 81ASING

\'O E

..

- - - - - - : almo1>1 perfectly crAT (hut ~ 11lJlC' ;~ current dependcr uj

hllOK, lyp .

fi GURE 9.5 . V8E ver sus tempera ture .

ex hibit mo re nearly ide al cha rac teristics tha n o rdi nary di odes, we will assume handgap implem ent at ion s that usc transistor s. A plo t of VBI' versus te mperatu re is sketched in Figure 9 .5." Recall tha i Va E is near ly perfectly C TAT Ii.e.. it goes do wn linearl y with te mperature ). Now suppose we add 10 thi s C TAT VISEa vo ltage tha i is pe rfectly proportional to a bsolute te mperature ( JYTAT). If we c hoose the s lo pe of the PTAT term eq ual in magnitude to that o f thc CTAT ter m, the sum will be inc..lcpe ndcm o f'te mpera rure (see Fig. ure 9.6) . We sec that so mething funny happens above ahoul6(X) K . but the fact thai the principle fa ils at tem peratures high e nough to me lt lead is rare ly a prac tica l concern. No te thatt he additio n of a ?TAT a nd C TAT voltage in th e pro per ratio yie lds an o utput eq ualto the ban dgap voltage (ex trapolated to 0 K ). indepe nde nt o f temperatu re. S tated ano ther way. if we adjust the JYfAT cornponcr u to ma ke the o utput voltage eq ua l to V(; (l a t a ny te mperature , then the o utput voltage wi ll eq ua l VG O at all temperatures - at least in thi s slightly sim plified pictu re. At this poi nt , it's nat ura l to co nsi de r ho w o ne obtain s a JTfAT vo ltage. since this wh ole concept re lics on having on e aroun d . Let 's sta rt with the fam iliar eq uatio n for

VilE: VRE =Vl' ln( l

c)

Is

.

(5)

Using this ex pres sio n, we can re adi ly co mpute the di fference in two VRES for identica l trans isto rs o perating ut two differen t values o f collecto r c urre nt (ur , mo re ge ne ra lly. for tran sistors ma de in the sa me proc ess, o perating a t two diffe rent values of collecto r c urrent de nsity):

Ii

Again . keep in mind that this plot of VII" is a slight fielitln because we have neglected the small curvature caused tly the weal temperature dependence nf I n. The coerecnon is seco od-onjer. and we will take care of thi s lillie deta il shor1 ly.

9.5

v VGO · l .2 -

229

BAN DGA P VO LTAGE REF EREN CE

CTAT + PTAT lenn =con~ l an l "VGO

---

/'

~---_//

<, !'TAT

T ( kcl vin ~ 1

FIGUR E 9.6 . Illustration

of bandgap reference principle.

(6)

The misleadi ng Is Icon drops OUI. so we ca n concl ude co nfidently that 6 VIIE tru ly is PTAT if the co llector curre nt densities are in a fixed ratio . Th us. while eac h VRE is nearly C TAT. the difference between two VilES is perfectly PTAT.

9.5 .1

CLASSIC BA NDG AP REFEREN CE

Now that we've gOI all the ingred ien ts. all that rem ains is to sum the C TAT VilE term with the righ t amount of PTAT 6VRE • Although on e could imagine a num ber of methods for doing so . the Brokaw ce ll is a particularl y elega nt (and accurate) imple me ntation of the bandgap reference. The classic bipol ar implementation is shown in Figure 9.7 (again , basic mirrors arc shown for simplicity's sake); we' ll mod ify this circuit shortly for imple men tation in C MOS techn ology. As we shall see , the output voltage is the sum of a ITfAT voltage and a Vlili . Here, QI and Q2 ope rate at u fixed current den sity ratio of m ( > 1) set. for exa mple. by rauoi ng the emitt er areas. Now, by KVL, the vo ltage acro ss R2 is the difference in VRESof QI and Q 2, and is therefore PTAT and equalto Vr In m . Assuming that the TC of R2 is negligib ly small. the current passing through it will also be PTAT. Furthcrmore. the current throu gh R J is simply twice that through R 2• since the two collector currents arc equal." Therefore. the voltage drop ac ross the entire resisto r string is purely PTAT. Finall y, the output vol tage is ju st this ITAT voltage plus the VlIE of Q2. as ad vertised. With proper choice of R I and R 2• the output voltage will have

1 We are oc:glccling em ' N due 10 mismatch. nonzero base currents. and finite Early voltage .

230

CHAPTER 9 VO LTAGE RE FERENC ES AN D BIA SIN G

Start -up network

" 1 = AVllE/R2 }l-f-c=-~VOllT

RI

FIGURE 9.7. Classic Brokaw

bandgap

reFereoce circuit.

zero TC, A1'> a free bo nus, a F'TAT voltage is ava ilable at the e mitters of Q J and Qz. providi ng the rmometer outpu ts.

Design Example To carry out an actu al design, we need some characterization dat a for our process. As a speci fic exam ple. suppose we go to the lab a nd find tha i VOE = 0 .65 V al 300 K a nd ICX) ILA for a transistor of Q 2'S s ize. Furt he rmore, le t III = 8. Th is choic e" of m sell' AVRE = 53.8 mV (a number comfortably larger than any offsets that we expect) at JOO K. Since we definitel y know the value of VBE at 100 JLA . a prudent choice for the co llector currents would be thi s value of 100 Il A . and this choice then fixes the value of H 2 = AVOE/ IOO ILA = 538 Q . Now. since we want the outpu t volta ge to be 1.2 V, the drop acro ss HI mu st be 1.2 - VRE - .o.VBE = OA96 V. Finall y. notin g that the c urre nt thro ugh HI is twice that through H2 , we concl ude that we should choose HI = 0.496 V/ 200 J.l A = 2.48 H 2, co mpleting the design. You may have not iced that the collec tor currents in the Brokaw cell are not constant (i n fact , they are IYfAT if we assume that the resistors have zero TC). To see why this docs not invalid ate all we 've done so far (in fact, it is be neficial), it is now time to take care of a few details - namel y, those invol ving the tem perature depe ndency of 10 • A qu asiempi rical expression for 10 is

lo = At RT ' ,

II II is importa nt thai Q z be laid nUl as eight insta nces of Q I 10 guarantee Ihal Q1 behaves as eight parallel devices of Q I'S sile o H Q I is placed al the ce nter " f a com mon -remrojd arrangement. errors due 10 process variation will be minimized.

231

9.5 8AN DGAP VOLTAGE REF ERE NCE

" In: GREATLY exaggerated

'to - 1.2 -

curva ture

• Ideal (shown for reference )

T (kelv ins )

FIGURE 9,8 .

V~ ver sus tempera ture .

where A t· is the emitter area . B is a process-depe ndent co nstant . T is the absolute temperature, and r is a process-dependen t quantity we ' ll call the ClIn 'CIturt' coeffide nt. For the relatively dee p. diffused emitters of o lder bipo lar processes. r typically has a value between 2 and 3, while for the shallow. implanted (and very heavily doped") diffusions that are mo re common in mod ern CMOS and high-speed bipol ar processes. r typically ranges from 4 to 6 . With this eq uation for 10. we can express VRE as follows:

(8) Plotting in Figure 9.!'! as before. we can see why it is rea souuble to call the parameter , thc curvature coe fficient (aside from the euphonious alliteration). Because the argument of the Jog is not quite independent of T , the temperature coefficie nt of V8 E is not quite co nstan t. lead ing to a small dep artu re from CTAT behavior for VUE. Add itionally, we 've see n at least one implement ation of a bandgap reference in which the collector current is also not constant. So, let' s compute the actual TC that results if, in addition to the tem perature dependence of 10, we also consider a co llector current that varie s as the n th power of T :

so that

BE

C )-

- - = -k [In( dT q A EBP "

]

( r - II ) ,

(10 )

, Bandgap narro wing and nonlinearit y in !he heavily doped emme rs are probabl y n:lioponsibl~ for lhe high values o f r.

CHAPTER 9 VO LTAGE RE feR EN CES A ND BIA SIN G

232

whic h we may rewrite in a somew hat lowe r-e ntro py form. as in Sec tion 9.2:

I I I) Note that the curvature term disappears if r = n . and we're left with the same expression fur the temperature coe fficient as derived earlier. In the Brokaw cell." = I. which reduces the effec t o f. but docs not ca ncel. r (remem ber. r is typica lly a minimum of 2••md CUll ran ge up to abou t e). G raphic ally, think ofth e increa sing. co llector curre nt with tem perature as struightenlng out the V B!; curve. The next qu est ion is: How dues the curvature term affect the bandgap reference itself! Th e most ex ped ient a nswe r co mes fro m deriving the condition fo r net zero TC. Suppo se we ca ll G Vr the PTAT com po nent that we add 10 V H ~;' Th eil, the TC of the PTAT co mpo ne nt may be wri tte n as G Vr / T a nd the co ndition for ze ro TC i ~ therefore

d V IlI ;

dT

G VT

T

-- +-- = O ~

G=

[Vau -

V UE

+ (r

- n)VT I

VT

( 12)

whic h corre spond.. 10 an output voltage o f ( 13)

Th is la..t eq uatio n de pend .. on Vr and therefore imp lies that the output m ilage ca nnot have zero TC at all tem peratu res; the best we can do is ac hieve zero TC at one temperature. Furthermore, in orde r to achi e ve thi s ze ro TC condition at that one tempe rature, we IK~ to adju..t the output to a vo ltage higher than VGO by a n am o unt eq ual to (r - " )Vr . Fortunatel y. this co rrection term is rela tive ly small, typ ically a mounting to j ust te ns of millivolts out of a tota l that is greater than a volt . He nce, the output need only he trimmed to a value a fe w pe rce nt greater than VGQ at the temperature whe re zero TC i.. desired (gen erally, the ce nte r o f the opera ting te mpe ra ture range). At this po int, we 'd like to quan tify the errors that are caused by the curvature. Unfortunately, altho ugh the equat ions we 'v e developed ...o fa r are valu able for design, they a ren' t quite suitable for anal ysis. To deri ve one that is, let us c hoo se the factor m :-on that the outpu t vol tage has zero TC a t so me tem perature we'H call TN (for the reference te mpe rature) . Th rou ghou t. we'f usc the subsc ript R to de note a variable' s value a t this reference temperatu re. With thi... notationa l convention. we may express V''''I as follo ws: Vom(T)

T

= VGo + -

TN

T

(r - Il )Vr N - -

TN

(T) • TN

(r - ,, )Vr Nln -

or , as some prefer . (15)

233

9.5 BA N DG A P VO LTAG E REFER ENCE

T:lhle 9 . 1. OUlpUI voltage as j unct ion

r -II

, ,, 3

of T

v,.... ~

V"", @

V~ @

T "" - 5S' C

T "" SO"C

T = 150 C

1.22b 1.252 L27o,l

1.22S 1.256 1.2S4 1.] 12 1.339

1.227 1.2.53 1.2XO 1.307

1.30 5 1.331

am} r - n Maximum error

1.333

2 mV

, mv j mV

7 mV K mV

Not e that this eq uatio n ha s the right limiting be havior : when T = Tit. it yields the OOlpUI voltage co rrespo nding 10 the zero- TC co nd ition. A lso note that , if we were able 10 arra nge fo r Ihe coll ector curr ents to vary as Tit with II = r, then the output voltage would have u rn TC Oil all temperatures if v,..., were adj usted to a value Ve o at any te mperature. Th i.. last observation is Oil the core of ma ny efforts to sy nthesize curvatu re -corrected bandgap refere nces. Eve n wit hou t e labora te c urva ture - correc tion met bod s. though , the Brokaw ce ll (where II = I in the classic implement ation ) provides outsta nd ing performance. because the c urvature inhere nt in bipol ars is simply nOI all that bad . and the Bro kaw cell contributes little e rror of its ow n. To illustra te this po int , let's co mpute the uctual erro r one could eXJX"Ct. over a tem peratu re runge of - 5 5~ C to + 15CrC. if Tit is chosen as + 50 "C :1011 if the qu antity r - II ranges from I 10 5; sec Table 9. I. As is evident. the 10la l c hange in OUlpUI vo ltage is less than a percent ove r the e nlire tem peratu re range, eve n with relatively large values o f r - " . Furthermore. the output is a maxi mum at the refere nce tem perature, a nd drops o ff above a nd below this temperatu re in a quasiparabolic manner. As a conseq uence, setting TN eq ual to the center oft he desired operating tem peratu re range near ly minimize s the ma xim um deviation fromth e value at TN. As a final note, the le vel s of performance in Table 9. 1 assume a n ideal sce nario in which second-orde r e rrors (d ue to device mism atch , non zero resis tor TC , fi drift , etc.) arc ign ored . Ac tua l perfo rmance will he so mewha t worse in prac tice owing to the com bined effec t of these sources. C areful layout of all devices is ma nda tory in order to minimiz e e rro rs.

9.5 .2

BANDG AP REFEREN CE S IN CMO S TE CHN OLOG Y

The classic Brokaw cell uses bipol ar tran sistors in whic h alt device te rmi nals float . so it ca nnot be im pleme nted direct ly in this fonn in C MOS techn ology. Rearranging

234

CHAPTE R 9 VOlTAGE REFERE NCES AN D BIA SIN G

Voo

Start-up network

FIGURE 9.9. CMOS bandgap reference .

to accommodate the restrictions placed on the parasitic substrate p-n-p yields the circuit show n in Figure 9 .9. Transistor Q 2 is designed to have m times the em itter area of Q J. The quad of CMOS tran sistors enforc es equ al emitter curre nts. so thai the collector current density ratio is approximate ly m . Imp licit in the last statement is that this circuit has a greater sensitivity to f3 than the original Brokaw ce ll. This unfortunate co nseq uence of bei ng forced to usc the substrate p-u-p's leads to larger errors than the classic bandgap ce ll. particul arly because fi is rare ly large enough to he ignored (values of 5- 10 are typical) . Neverthel ess. eve n a poorly performing ba ndgap reference is considerably superior to anything that ca n he built out of pure CMOS co mpo nents. Chous ing co mpone nt values for this circuit proceed s in a manner quit e similar to that for the cla ssic ce ll. Begin by specifying a reference temperature 7H at which the TC is to be zero. For illustrative purposes. assume tha t this temperature is to be 350 K . The next step is to ca lculate the target output vo ltage at this reference temper ature. As menti oned previously. the shallow. heavily-d oped p + diffu sion s used to make the emitters lead to relatively large curvature coefficients. with r typi call y 4 or 5. If no device model s are availab le. a reasonab le starti ng po int is to assume a value of 4 for the quant ity r - 11, Hence. the target output vo ltage sho uld he

v,..... =

VG U + ( r - n)Vr :::::- 1.32 V.

(16)

Now assume that we have selected 100 I1.A for the ind ividual emitter currents. and that the larger transistor. Q 2 . has a VBE of 0 .65 V at this current lit TR _ Then R1 is simply Vout - VBE2 = 6.7 H 2. (11)

"

9.6

CO NSTA N T- g", BIA S

235

V I)Il

Slilrl-Ur

I REf' N

I

TI REFP

nctwor

M2 R2

fiGURE 9.10. Ba ~c comtant-g .. reference .

If we assume an III equal to 8 then VS EI is abo ut 63 mV larger than V S F.2 ' lienee. VOUI

-

VBEI

It

= 6.07kQ .

(1 8)

thus completing the design. As a fi nal comment o n this circuit. one usually finds that the bias current is relatively constant with temperature bec a use resistors typically have a positive TC. which offsets the PTAT tendency of the core design. Thu s. currents from a mirror slaved to the PMOS mirro r will be ro ughly constant. The precise TC obtained may be adjusted through a suitable cho ice of resistor values if the ultim ate goal is to generate a bias current rather than a reference voltage.

9.6 CO NSTANT-g m BIA S A constant current or constant voltage is often desirable, but this is not always the case. Important examples include situations in which it is the transconductance that must he held constant•.such as in the low-noise amplifiers described in Chapter I I. A circuit whose bias current correspo nds to a 1:"1 that is inversely proportiona l to a refe rence resistance is a mod ificat ion of the self-biased CMOS quad of transistors we've already seen; this is shown in Figure 9. 10. The transconductance of M J must eq ual that of the combination of M 2 and R 2 • To show that this equality holds. consider cutting the loop at the gate of M 2 . Applying a test incremental voltage at that gate produces so me current io • equal to gm .riH tIl V~~le, which is mirrored back to MI. That current is converted hack into a voltage i"/ g,,,1 by the d iode-connected MI . A self-consistent solution is possible only if that voltage equals the original gate drive of M 2. Thus.

i

.:

g"' l

1:""nghl

- o = P, ale = - --

==>

g", .ri ghl

= g", l ·

(\ 9)

236

CHA PTER 9

Start-IX

VO LTAGE REFEREN CES A ND BIA SIN G

,

I Rl1'N

IlC."IW\l

I J REI-1'

R2

FIG URE 9. 11. Improved cooskml-g", reference.

111c Ieft- ami r ight-half transco ndu cta nces are the re fore eq ua l wh e n th e circu it is ope rat ing. Now . if ,\(2 is ex tre me ly w ide . the n the tran sconduct ance of the right-hand circ uit will be ro ug hly I/ H2 and thus M I w ill have a tran sconduc tan ce of th is value as well. as Loan be seen from the foll owin g eq uat io n: Km.ri ght

=

gm 2

R

2

+I•

(20,

w hich is a pproximatel y equal 10 I/H z for gm2R2 muc h greate r than unit y. Noll.' that this equa tio n holds for bo th long- and short-c ha nne l de vices. Bl.'CilUSC this ci rcuit depend.... n n a refere nce resistan ce. the qua lity o f the reference is dependentmai nly o n the q ual ity of the res istor . In nonc rit ical applications, ordinary on-c hip resistances (c.g.. un silici dcd pnl y) ma y suffice. w hereas more dem anding a pplic ations may require the usc of a n e xternal resistance. Aside from resistor (a nd PMOS mi rro r) errors. the ci rc uit (If Figure 9 ,10 suffers from depen de nce on M2' S tran scondu ctanc e . A s lightly more e laborate a rra nge ment can com pletel y eliminate thi s prob lem ; see Figure 9 .1 1. He re. the tran scond uctan ce of the op -a mp-M2 combinat ion is preci sely equal to I/R2. independentl y of M2. Th e re maining e rrors a rc the n due to resis tor and PMOS mirror e rrors. Because the nutput imped ance of shor t-cha nnel MOS FET s is nut purticulurly high . relatively long devices should be used in the PMOS mirror. Further impruvcmcnts ma y he obtai ned by forcin g the PM OS devices to havc equal Vns. as shown in Figure 9. 12. Th e add ed o p-amn forces the two PMOS de v ice s to pos· sess equal dram-to -source voltages, there by eli minat ing e rrors due to c han nel -length modulat ion and DIBL. With this type of circuit. one may red uce greatly the varia tion in .':",-de pc ndc nt param e te rs - such as gain. input im ped ance. and noise figure of LNAs - as te mpe rature. processin g, and supply m ilage vary.

PROBLEM SET

237

Slart-ur

nctwor

I REB > I REf:'Il

.2

fiGU RE 9.12. Minimum-error coosronl-g", reference.

9.7

SUM M A RY

We have see n thai the self- biased cell is quite versatile. permit ting the generation of currents proportion al 10 the ratio of a volt age in on e branch to the res istance in the other. The vol tage may be provide d by a variety of elements. such ;IS a forwa rd-biased junction. Al though a VBf. by itself has limited utility as a volt age reference because of its negative TC . its CTAT be hav ior is valua ble in compensating the M'AT !iVBf. in a bandga p reference c ircuit to yield an outpu t roughly equal to V" o wit h extremel y srnall tempc rurure variation. Even when parasitic bipola rs are used in an otherwise Cl\10S ci rcuit. the bandgap principle allows the synthes is of more accu rate and stable voltages or curre nts than is possible with ordi nary C MOS circ uits, Finally. a comtant-Xm bias circuit was presented . allowing the stable biasing of such transco nductance -sen sitive ci rcuits as filters and LNAs.

PRO BLEM SET FOR VOLTAG E REFERE N CE S AND BIA SIN G PROBLEM 1 111i s prob lem explores in more detai l the characteristics of the constant-Km reference. Refer to Figur e 9.13. Rather than ma king M2 extremely wide to make the tran scondu ctance mainly proportlonul ro the co nductance of H2. another optio n is availa ble if the transistors are al l long-channel devices. Show this formally by deri ving an ex press ion fur the transco nductance of M l if the M2 is S times as wide as M I. To simplify the deri vation. neglect body effec t and assume that the Pl\IOS mirro r is an ideal I: I mirror.

23.

CHAPTE R 9 VOLTAGE RE FER ENCES AND BIASING

Start-up netw ork

MI

M2

R2

FIGURE 9. 13. Basic conslont-g", rele-eoce.

PROBLEM 2 In the im proved co nstum-g; reference o f Figure 9. 14. investigate the effec t of PMOS mirror errors ca used by no nzero output conductance. Model the PMOS devices as square -law. and assum e thai they are ideal except for a channellength mod ulation coe fficie nt of 0 .1 V- I, Derive an expression for the transcondu ctance of M I.

Start,ue nerwo r

J""",

I

R2

fiGURE 9.14 . Improved constont-g", reference.

PROBLEM 3 Usi ng the simple circuit o f the first prob lem. select device sizes and resistor value to produce an output current sink of 250 J1.A and a tran scon ductance of 1 illS at 3(X) K. Usc the level-S dev ice parameters fro m Cha pter 3 and simulate with SPICE to verify that the design works as desired . PROBLEM 4

T he equations for the C MOS-compa lible bandgap reference do nUl take finite p-n-p p into acco unt. Unfortunately. typical values for P of such transistors are oftcn below 10. Re- dcnve the ex pression for the output voltage inclu ding fJ. PROBLEM S In this prob lem. we co nsider the settling behavior o f a noni dcal voltage reference in respo nse to a tran sient d isturbance. Co nsider the popular ci rcuit

..

- - - - - - - - - - - - - - - - - - - ----239

PROBLEM SET

shown in Figure 9, 15. Assume that each transistor is IO-Jl m wide and that Vo n is 3 V. Use the level-J dev ice models given in Chapter 3. VDO

FIGURE 9.15. vg<·based YOltoge reference.

(a) Choose I ,d to make the output voltage I V. (b) What is the low -freq uency incremen tal output resistance? (e) Now consider what hap pen s if the reference voltage must drive a total load of 3 pF, and if a di..turbance happen s 10 bump me output voltage 10 1.5 V in I ns (e.g.• with a fast-actin g current source). Calculate the settling time 10 I% of the original value of I V (yo u may neglec t body effect in your hand calculations) ami verify yuur answer with S PICE . Explain discrepa ncies quantitatively. PROBLEM 6 In low-voltage circu its. it becomes difficult or impractical to use ordinary cascod e structures 10 increase the output resistance of current sources . Alternate cascodi ng techniques can be used to redu ce the voltage req uired. however. An example is sketched in Figure 9. 16.

f-------""1I---f------~Irl M5

:~

Il

f l GURf 9.16 .

MJ

t..ow-heodroom cascade.

242

CHAPTER 9 VOLTAG E REFERENCE S A N D BIA SIN G

fa) Explain the operation of the startup network . (b) When the bandgap is operating, M9 is supposed 10 be otT. In order to ensure that M9 does not disturb the operation of the bandgap ce ll. its V"s should be at least 400 mV below the threshold. What is the smalles t ratio of (WILh to (WIL)s that would satisfy this requirement? (c) Simulate this circuit by ramping up the supply voltage slowly from zero. At what supply voltage does the circuit "snap" into ope ration?

CHAPTER TEN

NOISE

10.1 IN TRODUCTION The sensitivity of co mmunications systems is limited by noise. The broadest definition of noise as "everyth ing except the desired signal" is most emphatically '10 1 what we will use here. however. because it does n OI separate. say. artificial noise sources {e.g.• 6()· Hz power-line hum ) from more fundamental (and therefore irreducible) so urces of noise thai we discuss in this chapter. Thai these fundamental noise sources exist was widely appreciated only after the invention of the vacuum tube amplifier. when engineers fi nally had access to enough gain 10 make these noise so urces noticea ble. II became ob vious that simply cascading more amplifiers eventually produces no furthe r improvemen t in sens itivity because a mysterious noise exists that is amplified along with the signal. In audio systems. this noise is recognizable as a cont inuous hiss while, in video, the noise manifests itself as the characteristic "s now" o f analog TV systems. The noise sources remained mysterious until H . Nyquist, J. R, Johnson and W, Schottky I published a series of papers that explained where the noise comes from and how much of it to expect. We now tum to an examination of the noise sources they identified,

10.2 THER MAL NO ISE Johnson - was the first to report careful measurements o f noise in resistors. and his colleague Nyquist- explained them as a consequence of Brownian motion: thermally l Th is name is freq uen tly misspell ed in English-language publieatiuns. II is nut all thai uncommon to see " Sbot key," "Sho llkey," or "Schonkey,' Wh ile .. e're at it, "Schnnn,' a" in the trigger , is also rni!>.s pelled quite utte n. with co mmon inco rrect rende rings being "Shmiu" and " Schmidt," 2 "Thermal As itation of Elec trici ty in Conductors ," Ph.vs. Rev., Y, 32, July 192H, pp . 97- 10lJ. ) "Therma l Agitation of Elec tric Charge in Con ductors ," Ph)'s. Rev., v, 32, Ju ly 1928, pp. 110· 13.

243

CHAPTER 10 N O ISE

agitated charge carriers in a conductor constitute a randoml y vary ing curre nt that g ives rise 10 a tand e m voltage (via O hm's law). In honor of these fellows. thermal noise is often culled Johnson noise or. less frequ ently. Nyq uist no ise . Because the no ise process is random. one ca nnot iden tify a specific value of voltage at a particular time ( in fact . the ampli tude has a Ga ussia n distribution), and the on ly reCO UN: is to characte rize the noise with statistical measures, suc h as the meansquare or root -mean-square values. Beca use o f the thermal origin. we wou ld eXJX'C I a depe ndence on the absolute tem per ature. It turns out that thermal no ise power is exactly proportio nal to T (the astute migh t even g uess thai it is propo rtiona l to kT) . Speci fica lly. a q ua ntity called the avaitabte noise pOI\'er is give n by (I )

w here k is Bol tz mann' s co nsta nt (about 1.38 x 10 - 2.1 J/ K ), T is the absol ute temperat ure in kelv ins. and 6! is the noise ba nd width in hertz (eq uivale nt brick wall bandw id th) ove r whi c h the me asure me nt is made. We will cla rify sho rtly what is meam by the terms "ava ilable noise po wer" and " noise bandwidth:' but for no w simply note tha t the no ise so urce is ve ry broadba nd (i nfi nite ly ~(), in fact , in the simplified picture prese nted here"). so that the to ta l noi se po..... er depe nds o n the me asu rement band w idth . Noll' that thi s e q ua tio n tell s us tha t the ava ilable noise powe r has a spec tra l density that is independent offrequency. and that the to tal po we r thus g row s w ith ba ndwidth w itho ut limi t . T his is a bit of a lie. bUI it is true e noug h for all ba ndwidths of interest 10 the electrica l engineer . With Eqn. 1. we can co mpute tha t the ava ilable noi se power ove r a I-li z bandwid th is abo ut 4 x 10 - 21 W (or - 174 dHm ) 5 at m om te mperatur e. Further note that the co nstancy of the noise den sity impli es that the thermal noi se power is the same over any given absolu te band wid th . Th erefor e the noi se po wer in the interval betw een I MH z and 2 Mllz is the sa me as bet ween I G Hz and U Xll G ll z. Because of th is co nsta ncy, thermal noi se is ofte n described as " w hile: ' by a nalogy with white ligh t. However . the analog y is not e xac t. since white light co nsists of co nstant energy pe r wavelength whereas w hite noise has co nstant e nergy per hertz . Th e te rm "ava ilable noi se power" is sim ply the ma xi mu m po we r that ca n he delive red to a load. Recall that the co nditio n for ma ximum po we r tran sfer (for a resistive net wor k) is eq ua lity of the load a nd so urce resistances . This suggests the uxe of the net wo rk shown in Figure 10.1 to co mpute the available noise power.

Th e imp lication that the available power grow s without bou nd as the bandwidth approac hes infinity ...hould him at the reed III mod ify this fonn ula . We will tal e care o f thi... de tail shortly. S Recall tha t the reference level for 0 dBm i... one milliw att , 4

10. 2 THERM AL N OISE

245

R

i H f OJ;l ! C;; _ ! R '... .. . ..

• •• • • • • _ . _J

FIGURE 10.1. Network for compuli ng the thermal noise of a resistor.

The model of the no isy resistor is enclosed within the dashed bo x. and is here shown as a noise voltage generator in series with the resistor itself T he power delivered by this noisy resistor to another resistor of eq ual value is by defi nition the available noi se power:

(2) where e" is the ope n-circuit rms noise vo ltage generated by the resistor R over the bandwidth t:J.j at a given temperature. The mean-square ope n-c ircuit noise voltage is therefore (3) = 4 kTR t:J.I .

t';

A couple of use ful rules of thumb emerge from plugging some numbe rs into this At room tempera ture. a I· H 2 resistor ge ncnuc s abo ur a nY o f rms noise over a bandwidth o f I l b . wh ile a 50 -Q resistor generates a bit over 0.9 nY (Mime people just ca ll it I nY, since it's easier to rem ember) of nn s noise over that same bandwidth . These values are perhaps amo ng the very few worth co mmuting to me mory. Just reme mber that the rms volta ge is proportional to the square root ofthe bandwidth (and resi..ranee) when scaling these numbers to any bandwidth or resistance. In many cases. the noise is specified in term s of the s pectral density ruther than a total value. and is fou nd simply by dividin g the mean -square noise value by ti j (or the rms noise value by the square root of tif) . Th us, a l-kQ resistor has an rms noise spectral den sity of about 4 nY/ JIfl: (funny units, yes) or a mean-square noise density of appro ximately 1.6 x 10 - 17 y 2/ IJz . For therm al noise, the spectral density is a constant thai depends only on temperature (and Boltzman n's constant), and is independe nt of freq uency," In the more ge neral case. the density may vary wit h frequency, and o ne then uses the term "spot noise density" to underscore thut the stated density applies o nly at some spec ified spot in the spectru m. We've seen that every physical resistor has a noise sou rce associa ted with it. In the Theveni n rep resentation of Eqn. I, there is a voltage source in series with the resistor. Altern ati vely, we may construct a Norton equivalent model in which a noise Ia~t equation.

• Set foo tnote 4 . however.

246

CHA PTER 10 N OI SE

-;2 = _~ = ._ U.Tl!J. f = 4 tTGli f _ • R2 R

~ = 4lcTRtif

FIGURE 10. 2 . Reshtor thermal noise models.

c urrent so urce shunts the resistor. Bot h mode ls are valid. o f course . a nd the choice of which to usc in a give n situatio n is driven by pract ica l consideratio ns . suc h as computa tional co nvenience or intuitive value . The two noise models for a resistor are d isplayed in Figure 10.2. Note thai the

polarity indicatio ns on the noise voltage source and the arrow lin the noise current source are simply references. They do not imply that the noise has a particular consta nr po larity (i n fact . the noi se has a zero mean ). Abc note that . since the noi se arises from the random the rma l agitation of charge in the conductor, the only ways to red uce the noise o f a gi ven resistance are 10 keep the te mpe ra ture as low as possible a nd to limi t the bandwidt h to the mi nimum useful value as well. Beyo nd these re medies. the re is nothing that ca n be do ne about therm al noise. Now. what abo ut this " brickwall" ba ndwi dth business? Th e d ist incti on is made to unde rscore that the noise ba ndw idth 6f ge ne rally is not the sa me as the - 3-dB band width . Rat he r. the noise bandwid th is that of a perfec t. brick wall (rec ta ngular) filter that po ssesses the sa me area and peak value as the actua l power gain-versusfrequency c haracteristic of the sys te m. including that of the measuremen t apparatus. Th e noise ba ndwidt h is the refo re I

I1f

= IHpkl'

[N

10

(4)

11/(f)I' df.

wher e /l pk is the peak value of the magnitude of the filt er voltage tran sfe r function fI (f ). This normali zation concept allow s comparisons to he made on a sta ndard basis. As a speci fic exam ple, consider a sing le -po le R C low-pass filter . We know that the - J -d B bandwidth (i n hert z) is simply l j 2:;rRC , but the equivalent noi se bandwidth is COIllPUh.'d as

I

6f

[~ [

sa j'i'j'i Jo

I

]

I

(21rfRC)2 + I tlf = 2:;rRC arct;m 2J1IRC I;' n

I

= '2 f " H= 4 RC '

Il)

10 . 2 THERMAL NO ISE

We see thai a single- pole low-pas s filler (LPF) has a no ise band width thai is abo ut 1.57 limes the -3- d B band width . Thai the noise bandwidth exceeds the -3- d8 bandwidth makes sense. since the lazy rolloff of a single-pole filter allows spectral components of noise beyond the filler 's - 3-d B frequency to contribute significantly 10 the output energy of the filte r. A similar calculation shows thai a cr itically damped , second-order low-pass filler has a no ise bandwidth that is ap proxi mately 1.22 limes the - 3-uR bandwidth . In general. as the rolloff bcc omes steeper. the - 3-d 8 and noi se bandwidths tend to converge. Th is behavio r should seem reasonable. since sleeper rolloffs imply that the fi ller characteristics more closely approximate those of an ideal bric kwal l filter.' Now it' s time to tic up a few loo se end!'> . It turn s OUI that Ihe spectral density of thermal noise actually increases with frequency, ruther than remaining constant. This result follows from a more detailed derivation that takes into account the actual distributio n of ca rrier energies, modified by considerations related to the Heisenberg uncertainly principle." A more general ex pressio n for the thermal noise voltage is as follows:

l('" =

low""! n

( lo W)

(6)

corh - - . 4 :l lkT

where h is Planck's constant. aoou16.62 x IO- J.l J-s. Although it is not immed iately obvious, Ihis new equation docs not completely invalidate everything we did earlier. In fact , there is hard ly any difference between this equation and 4 kT R!::J. j for frequencies below abou t 80 Tllz at room temperalure. The correction is thus negligible for all reasonable (non-optical") frequ encies, so electrical engineers ordinarily do n' t need to know abo ut it. However, the birth o f quantum theory actually traces hac k to Planck's reso lution of a parad ox (known as the " ultraviolet catastrophe") assoc iated with the assumption of a co nstant spectra l density in the co ntext of the spectru m of blackbody radiation. Th is connection is so strong that one so urce of thermal noise in a wireless co mmunications system is the noise associated with the blackbod y radiation of the object on which the antenn a is

focu scd.!''

J

I

An adduionalcalculauon shows that a second-orde r lPF with { = I/ ..{i (the minimum value that avoids peaking in lhe freq uency response) ha.s a noise bandwidth that is onl y ahout 11% larger lhan the -3,d B bandwidth. TIle more lightty damped pole pair gives a frequency resporl'>e shape lhat has a rlaner gain cnaracterisnc in the pa...' hund and faster inilial rollofT heyond the -3-d " frequency th;m its more heavily damped cou sin, mus mOl\" c1o'>ely a ppm~ i mali ng a brickwall filter. H. Heffner, "TIle Fundamental Noise Limit of li near Amplifiers : ' IRE Ju ly 1962, pr .

rroc..

l ~ -II .

II For reference, visible light spans about an octave. from a b out -lOO TH z to !lOll 1'111.. Amo Penxias and Rebe rt Wilson of BellLabs understood very well exac tly how much noise a microwave receiver should exhihil. Whi le meticulously tracking down a sruhbom exce ss noise

10

246

CHA PTER 10 N O ISE

Some other issues demand attcmion as well. First. some good news: purely reactive cleme nts generate no thermal noise. Now. all real ca pac itors and indu ctors are lossy 10 so me e xte nt. a nd this loss implies that the impedance has a real part . Th is real resi stance doc s genera te thennal noise. hu t the pur ely capacit ive or indu ctive part s of the impe da nce do not. How ever. note tha t noi sy c urrents flowing through any impedance. reactive or real. will give rise to a noisy voltage . It is also important 10 recognize thai thermal noise is not associated with every e leme nt represe nted sc he matically with a res istor symbo l. A n importan t e xa mple is fou nd in bipo lar transistor models. where r" is a fict itiou s resistance in the se nse that it is the result of a linearization of the buse-eminer j unction's exponential V- I characteristic; it does I /II( generate thermal no ise . However, the parasitic resistance terms. such as rh and rc- do .

THERMAL N OI SE IN MOSFET, Dra in Curre nl Noise

Since FETs ( both jun ction and MOS ) arc essentially voltage -contro lled resistors. they exhibit thermal noise. In the triode region of ope ration particula rly, one would expect noise commensurate with the resistance value. Indeed. detailed theo retical considerations II lead to the following ex pression for the drain curre nt noise of FETs: (7)

where KJ O is the drain- source co nductance at zero \ '0 $ ' The parameter y ha.. a value of unity at zero V/lS and. in long devices. decreases toward a value of 2/3 in saturation. Note that the drain current noise at zero V" ,\" is precisely that of an ordinary conductance of value KJf) . Unfo rtu nately, measu rements show that short-channel'? NMOS devices in suturetion exhibit noise far in excess of values predicted by long-chan nel theory. sometimes by large factors ( y is typically 2-3. but can he co nsiderably larger ). Since the origin of this exces s is carrier heatin g by the large electric fields present in short devices, it is important to usc the minimum practica l drain voltages, PMOS devices do not

so urce the y initially annbu ted 10 their eq uipmem , the)' di sco ve red the hack grou nd nncrow ave radialinn that uniformly suff uses the universe. In an encou nter with some cu~mu lllg i1o" 1ohortly a nerwaru . lhey lea rned tha t the energy Il f tim. raJ iOll illll dglCL"U wel l ..... h h a prcJ i..:ti" n ( by R. II , Dicke ) of the cncrfY of Ihe ec hoes of the Rig R,m g. Fh r their work , Penzias and wil son recei ved the 19 78 Nt>he11'ril,( in physics. Not a bad reward for undc ", la nd ing noi se . Let ma r he it I~-.on to us all.

II A . van der Zie!. " Thermal Noise in Field Effttl Trolfl 1oi ~t ()P.;." Proc. IEEE. A u gu ~1 1% 2. pp, 1801 - t 2. U A s stared in Chdplc r J. " shtlrt-channd" 10huuld he interpreted as "high elecmc field."

249

10 . 2 THERMA L N O ISE

fiGUR E 10. 3 . Gate noise circuit model

(aftervon de- liell. exhibit as much excess broadband noise un til considerably highe r field s arc reached . so they ma y be used when their o ther c harac te ristics allow it . l l

GoteNoise In addition to drain c urre nt noise. the th ermal agita tio n o f c hannel charge ha s another important co nseq uence : ga te noi se . Th e fluctu atin g cha nne l potential couples ca pacitively into the gate terminal. leading to a noi sy gate current. Ahhou gh th is noi se h negligibl e at low frequenci es. it ca n dominat e at radi o frequencies. Van dcr Zi ell~ has show n that the gate no ise ma y be e xpressed as

(Hl where the para me ter gR is (9)

Van der Zicl gives a va lue of 4/3 (twice y ) for the ga le noise coe tflcier n.ji . in lo ngchannel de vices. The circu it mod el fo r gale noi se tha t fo llows directly fro m Eq n. Sand Eqn. 9 is a conductance co nnec ted between gale and so urce. shunted by a noi se cur rent source (see Figure IOJ ). Th e gat e noise c urrent clearly has a spec tral de nsity that is not constant. In fac t. it increases wit h freq uency. so pe rhap s it o ught to he ca lled " blue noise" to co ntinue the optica l analogy. For those who prefer not to ana lyze syste ms th ai have blue no ise so urces, it is possible to recast the mode l in :1 form with a noise voltage so urce th at possesses a constant spectra l de ns i t y. l ~ To derive th is a lternative mode l. first tran sform the purallel Re netwo rk into a n equivalen t series Re net wor k. If o ne ass umes a reason abl y high Q. then the capac ita nce stays ro ug hly consta nt du rin g the tra nsform atio n. The parallel resistance becomes a se ries re sistance w hose va lue is

I) G. Ktimtl\"ic..:h et al.. "Ph ys jcu l M,)(Jcling 0 1" Enhanc ed Hig h-Frequ enc y D ra in and Gale C urre n! Noi'\e in Shon-Ch aune l MOSFETs." Pmc('('l!i ngJ oj (hI' Fi,.~( IIllt'nlt'.I;gll uj Mi.ud.MtJ
,'I'

cuus, May IQQ7.

2SO

(HAPTER 10

V ng

N O ISE

,

r

gate ~ .T!

source o

e

"

FI GURE 10.... Ahemotive gale noise circuit model. I

1 """,;-;I r~ = K/i . Q 2 + I ::::: Kx . Q 2 = 5gr/1) ,

( 10)

which is inde penden t of freq uency. Finally, eq uate the short-c ircuit curre nts of the original net work and the transformed versio n. aga in w ith the assu mptio n o f high Q. Th e eq uivalent series noise voltage so urce is then fo und to be ( II)

which possesses a consta nt spectral density. Hence. this alternative ga le noise model co nsists of elements whose values are indepe ndent of frequen cy; SoL'C Figure l OA. Althoug h the noise behavior of lo ng-cha nne l devices is fairly we ll understood. the precise behavior o f J in the short-channel reg ime is unknown at present. Given that both the gale no ise and dra in noise share a common origi n, however. it is probably re asonable as a c rude app ro ximation 10 assum e that (j co ntinues to be a bout twice as large as y . Hence. just as y is typically 2- 3 for short-channel NMOS devices, (j may he taken as 4-6 , Because the two noi se so urces do share a co mmo n o rigin, Ihey are also co rrela ted. Th at is. there is a component o f the gale noi se cu rre nt th at is pro porti ona l to the drain c urre nt o n an instanta neou s basi s. We w ill ex plore the implicat ion s of this co rrelatio n. bot h q uantitativel y a nd qu a litative ly. in C hap ter II .

10.3

SHOT NOISE

Ano ther noise mechani sm . kno wn as sho t noise . was first descri bed a nd ex plained by Scho ttky in 19 18. 1f> It is therefore occasiona lly known as Sc hottky noise in recognition of his ac hie veme nt. Th e fundamen ta l basis for sho t noi se is the g ra nular nature of the e lectro nic charge . but ho w this g ra nularity tran sla tes into noise i... perhaps not as stra ig htfo rwa rd as one might thi nk .

If> "Otoer sp onnme Strom sch w ankungen in versc mede ee n Blectrizuatsleuem" ["O n SrontancOll ~ Current Huouauon s in Various Elcc mcal Conductors" ], AtlIUlIt'1i tit'( Physik . v. 57. 19 18, pp. 54 1---67.

10.3

SHO T NO ISE

251

Two conditions must be sati sfied for sbot noise to occu r. There must be a direct current flow and there must also be a potential barrier ove r which the charge carriers hop. The second condition tells us that ordinary. linear resistors do nor generate shot noise. despite the quantized nature of elect ron ic charge. The fact that charge co mes in discrete bundles means that there are d iscontinuous pulses of current every lime an e lectron hops an energy harrier. It is the randomness of the arrival times that gives rise 10 the whiteness of shot noise. If all the carriers hopped simultaneously, shot noi se would have a much more benign character. as we' ll see in a later example. We would expec t the shot noise current 10 depe nd on the charge of the elec tron (since smaller charge would resul t in less lumpiness and therefore less noise ), the lola) OC curren t now (le ss curren t also means fewer lumps), and the bandwidth {j ust as with thermal noise ). In fact , shot noi se does depend on all of those quan tities, as seen in the following equation:

j; = 2ql octo. j ,

( 12)

where T,; is the rms noise curre nt, q is the elec tronic charge (abo ut 1.6 x 10- 19 e ), IDc is the OC current in amperes, and !::J.j is again the noise bandwidt h in hertz . Note thai, like thermal noise, shot noise ( ideally) is white, and has an amplitude that possesses a Gaussian distribution. As a reference po int. the rms current noise density is approximately 18 pA / ..IiiZ for a l· mA value of trcThe requirement for a potential barrier impl ies that shot noise will on ly be a..sociared with nonlinear device s, although not all nonlinear devices necessarily exhibit shot noise. For exa mple. wherea.. both the base and collector currents tire sources of shot noise in a hipolar tran sistor because potential barriers are definitely involved there (two j unctio ns) , only the DC gale leakage current of FETs (bo th MOS and juneliontypes of FETs) contributes shot noise. Because thi..gale current is normally very small, it is rarely a significant noise source (sad ly, though, the same cannot be said of base c urrent). 17 There arc some additional obs ervation s worth noting at this point. As with the case of thermal noise, the spectral density does not remain constant to infinite Irequency. However , significant departures from simple theory typically do nOI occ ur within the useful band width of the devices, and we will consequently assume a constant spectral density in all thai follows. II was also mentio ned that the rando mness of the arrival times imparts to shot noise its characteri stic whitene ss. To underscore the importance of the rundonmess. consider what the shot noise spectrum would look like if all the carriers were 10 hop

17 However, as mentioned previously, the thermally agilaled channl'l charge induces noisy gate currents that can te important at radio frequencie s.

252

CHAPTER 10 NO ISE

the po tential har rier s imultaneously, and if we could neglect the averaging effects of nonze ro transit tim e. More speci fically, suppose we were 10 ge nerate a 1.6 -mA curren t by having len well -trai ned elec trons hop the harrier together every femt osecond. The spec trum of the curre nt would be periodic. with impulses spaced at multi ples of an inverse femtosecond, or I PHz (thai'S I O I ~ Hz). Therefore . " 0 shot noi se would he evide nt until the first noise com po nent was reached, the frequ ency of wh ich exceeds tha t of visib le light! Alas. co nvinc ing el ectrons to ex hibit this level of cooperation is somewhat difficult . and shot no ise ap pears 10 he here to stay. Finally. the term "shot" is not a co rruptio n of "Schottky:' as some occasionally asse rt. It is s imply that if yo u hook up an audio system to a source of shot noi se biased at a very low curren t, the resul ting sound is much like thai of buc kshot (pe llets) dropping onto a hard surface .

10 .4

FL I CKER NOI SE

Withou t question. the mo st mysteriou s type o f noise is flicker noise (also known as II[ no ise or pink III noise). No universal mechanism for flicker noise has been identified. yet it is ubiquitous . Phenom en a thai have no ob vious co nnection. suc h as cell mem brane potentials. the earth's rotation rate. galactic radia lion noi se. and transistor noise all have fluctu ations with a II [ characte r. As the lem l " I/ F ' suggests. the noi se is charac terized by a s pec tral de nsity that increases, apparently without limit , as frequen cy decreases. Measurements have verified this behavior in electronic syste ms down tua small fraction of a rrucro hert z . One unfortu nate implication of the increasing no ise wi th decreasi ng freq uency is the failure of avera ging ( bandlimiting) 10 improve meusureme ut accuracy. since the noise power increases j ust as fast as the ave rag ing in terval. Beca use of the lack of a unifying theo ry, mathematical ex pression s fo r II[ noise invariably cont uin variou s emp irica l paramet ers ( in con trast with the theoreti ca l c1eanliness of the equatio ns fo r therma l and shot noise ). as can he see n in the following eq uation :

-

K

N ' = -rz c f.

I"

(13)

Here N is the rills noise (either voltage or current), K is an empirica l parameter thut is device -speci fic (a nd ge nerally also bias-dependent), and II is an expo nent that is usually ( hut not always) close to unit y. A question that oft en arises in connection with Il f noise conce rns the infini ty at UC implied by a 1/ f functional depe ndency. II's instruct ive \0 carry out a calculation with typica l nu mbe rs 10 Sl'C why there is no problem. practi cally spea king . 1M An uptkal l>)' l>lcm that eccer nua tes energy al the lower vil>i tole freq ue ncies reddens white Iit:ht , \0 1// noiM' is " pi nk" toy an alugy,

10 .4 FLICKER N OI SE

253

First. let the parameter " have its commonly occurring value of unity. Then, integrate the densit y to find the total noise in a frequency band bound ed by a lower frequency I I and an upper frequency I,,:

-N ' = fI,l>1Kdf

= K In

(I,) 11 .

(14 )

This equation leiIs us that the total mea n-square noi se depends o n the log of the frequency ratio. rather than simply on the frequency difference (as in thermal and shol noise). Hcnce. fhc mean -squa re value o f 1/1 noi se is the same for equal frequency ratios ; there is thus a certain con stant amo unt of mea n-square noise per decade o f frequency, say, or some specific amo unt of nTI S noise per m il t dec ade of frequency (yes, funny units again for rms quantities). As a specific numerical example, suppose measurements on an amplifier reveal thai its Iff noise has a density of 10 J1V nn s per root decade. Thu s, Ior the lti-decudc frequency interval below I Hz, the total 1/1 noise would hi.' ju st four times larger. or 40 IN rms. Recognize that 16 decades below I hertz is eq ual to one cycle about e\'ery 320 million years. I" and you have to concede thai " OC" infinities are simply not a practical problem . The resolution of the apparen t paradox thus lies in recognizing that true DC imp lies an infinitely long observation interval . and that humans and the electronic age have been around for on ly a finite lime. For any fi nite observation interval. the inf inities simply don't materialize.

10.4 .1 fLI CKER N O ISE IN RESISTORS Ric ker noise also shows up in ord inary res istor s. where it is often called "excess noise," since this noise is in addition to what is expected from thermal noise considerations, It is found that a resistor ex hibits 1/1 noise only when there is DC current fl owing thro ugh it, with the noise increasing with the current. In the discrete world, garden-variety carbon composition resistors arc the most conspicuous offenders, while metal-fil m and wirewc und resistors exhibit the smullest amounts (If exces s

nmse. The current-depende nt excess noise of ca rbon composition resistors has been explained by so me as the result of the random formation and extinction Il l' "mic ro -arcs" among neighbo ring carbon granules." Curhonfi/m resistors, which arc made differently, exhibit much less exces s noise than do carbo n co mposition types. Whatever IQ Antl t~r uscful4u.mtily ttl keep in mind is Ihe number of seconds in a year. ....hich uo an excette m epproxtrmnom is the square mut of lOIS or abour 32 million. 20 C. D. Motchenbacber and F. C. Fitchen, UJw-Nfli_~(' E/('ctn mic De sign , Wiley. New Yurk. 1973.

p. I72.

CHA PTE R 10

25A

N O ISE

the explanat ion . it is ce rtainly true that excess no ise increases with the DC bias. so one should minim ize the DC drop across a resistor . Th e foll owing a pproximate express ion shows expl icitl y the dependency of this noise on va rious parameter s: 2" K R~ 2 e = -·-· V 6J II

fA

'

115)

whe re A is the area o f the resistor. R o is the shee t res istivity. V is the voltage across the resistor. a nd K is a material-specific parameter . For diffused and ion -implanted resistors. K has a value of roughly 5 x 10 - 211 5 2 · m 2 , w hereas for thick -film resislOP; (no l normally ava ilable in CMOS proce sse s). K is about an orde r of magnitude larger .U

10.4 . 2 FLI CKER N OI SE IN MO SFET, In electronic devices. Ilf noi se arise s from a number of differe nt mecha nisms. a nd is most pro min ent in devices thai are sensitive 10 surface phenomena . Hence. MO SFETs ex hibit significantly more 1// noise than do bipolar devices. On e means of com pariso n is 10 speci fy a "come r frequency:' where the 1// and thermal or shot noise compo nents are equal. A ll other thin gs hel d equal. a lower 1// corne r implies less 101011 noise. It is relatively trivial to build bipolar devi ces whose 1// comers are below lens or hu ndreds of hertz . a nd many MOS de vices routinely ex hibit 1// corne rs of te ns o f kilohert z to a megahert z or more. C harge trapping pheno me na a re usually invoked to ex plain 1// noise in transistors. Some ty pes of defec ts and certain impurities (mos t plent iful nt a surface or interface of some kind) ca n randomly trap and release charge . T he trapping time!'> arc distributed in a way that can lead to a 1/ f noise spec trum in bo th MOS and bipolar transistors. Since MO SF ETs a re surface devices (a t least in the way that they arc conve ntionally fabr ica ted). they exhibit thi s type of noise to a muc h greater degree than bipo lar transistors (whic h are bulk dev ices). Large r MOSF ET s ex hibit less Il f noise because their la rge r gate capac itance smooths the fluct uation s in channel charge . Hence. if good II / noise performance is to be obtai ned from MO SFETs. the largest practical device sizes mu st be used (for a given Nm ) ' The mean-squ are 1// drain noise c urrent is give n by , K -;1 = _. K g;;. . 61 : : : : - . w 1, . A . 6/ f . " f IVL C'0' 21 K . Laker ill1J W. San'iCn . IJe.~iKn

Yor k. 19%.

L

IIIAnalog /nl egnJlf' d Ci'l"Il;h
(16)

M d Jraw·l lill. N",

10.5

POPCO RN N O ISE

255

where A is the a rea of the gate ( = WL) a nd K is a dcvic c-spcci tic consta nt. Th us, for a fixed transconductance. a larger gate area a nd a thinner dielectric red uce this noise term. For PMO S dev ices. K is typically abo ut 10 - 28 C 21m2 • w hereas for NMOS devices it is about 50 times la rge r.22 One should keep in mind that these consta nts vary conside ra bly from process to process. a nd e ve n from run to run . so the values of K given he re should be treated as c rude estimates. In part icu lar, the superior II f performance of PM OS devices may be a temporary situa tion. as it is due to the usc of buried c hanne ls that ma y cease to be widel y used in the futu re.

10.4 .3 FLI CKE R N OI SE IN JUNCTION S Forward -hi ased junctions also exhibit 1/1 noise. The noise is proportio nal to the bias current a nd inve rsel y proportional to the j unction area: (1 7 )

where the consta nt K typically has a value of around JO- lS A· m 2 • O nce again. however. conside rable varia tion fro m process to process is nor uncommon.P Flic ker noise in bipolar tra nsistors is attributed e ntirely 10 the base- e mitter junetion (since it is the only one in forward bias ). It has bee n es tablis hed ex perime ntally that on ly the base c urre nt e xhibits 1/1 noise.

10.5

POPCORN NO ISE

Another type of noise thai ca n pla gue se miconductors is known as popcorn noise (also called burst noise. bista ble noise. and random telegraph sig nals. RTS ). It is understood eve n more poorly tha n 1/ f noise. and shares with 1/ f no ise a se nsitivity 10 conta mination. Go ld-dopc d 24 bipo lar tra nsistors exhibit the high est levels of hur st noise. sugges ting a particular sensitivity to contamination by metal ion s speci fically, although not all popcorn nois e may be the resu lt of metal ion contamination. Th is noise was first observed in point -contact d iodes. but has also bee n see n in ordinary junc tion a nd tunnel diod es. some types of resisto rs. a nd both dis cre te a nd integrated ci rc uit j unction trans istors. Burst noise is cha racterized by its rnultimodal (most often bimoda l) and hence no n-Gaussian a mplitude di stribution . That is. the noise sw itc hes betwee n two or more discre te values. but al ra ndo m times. T he

n

Laker and Sanse n, op. en.

n Laker and Sansen , op. cit. 24 Tbe purp ose ful red uct ion of carrier lifetime by gold do ping is occas ionally U'iC."tJ in bipo lardc..-ices [0 speed recovery from saturation ,

' 54

CHAPTE R 10 NOISE

the explanat ion. it is ce rtainly true tha t exce ss noise increases wit h the DC bias. su on e should minimize the DC drop across a resistor . The fo llowing approximate ex pression shows ex plici tly the dependency of this noise on variou s parameters:

-2 K R2 2 e = - . --....9 • V 6 [ " fA '

(15)

where A is the area of the res isto r. No is the sheet resistivity, V is the vo ltage across the resistor. and K is a material -specific parameter . For diffused and ion -implanted resistors. K ha s a value o f roughl y 5 x 10- 211 5 2 · m 2 • whereas.. for thick -film resistors (not norma lly available in C MOS processes ). K is about an order of magn itude larger. 21

10.4 .2

FLICKER N O ISE IN MOS FEh

In electr onic devices. 1/ f noi se arises frum a number o f differe nt mech anisms. and is most pro minent in devices that are se nsitive tu surface phenome na . Hence. MOSF ET s exhibit significantly mo re II f noise than do bipol ar devices. One means of comparison is to speci fy a "come r frequency," where the II f und ther mal or shot noise compo nents are equal. All other things held eq ual. a lower 1/ 1 corne r implies less total noise. It is relati vely trivial to build bipol ar devices whose 1/ 1 comers are be low ten s or hundreds o f hertz . and many MOS devices routinel y exhibi t 1/1 corners of tens of kilohert z to a megahertz or more. C harge trap ping phenomena are usually invoked to ex plain II! noi se in transistors. So me type s of defects and ce rtain impurities (most plentiful at a surface or interface of some kind) ca n ran domly trap and release charge. The trapping limes are distribut ed in a way that can lead to a 1/ 1 noise spec tru m in both MOS and bipolar tra nsistors. Since MOSFETs are surface devices (a t least in the way that they are co nventiona lly fabrica ted) . they exhibit Ihis type of noise to a much greater degree than bipo lar transistors (w hich are bulk device s). Larger MOSF ETs exhibit less III no ise beca use the ir larger gate ca pacita nce smoo ths the fluctuation s in channel charge . li enee. if good 1/1 noise performance is to he obtained from MOSF ETs. the large st practica l device sizes mu st be used (for a gi ven g",). The mean-square 1/ 1 drain noise curre nt is given by

-2

K

"

f

i = - .

g~

IVL C '

,"

. !J.f ::::::: K . W 2 • A . !J.I T f .

(16)

21 K. Laker and W. Sansen. fk ,f ign of And/f/g lntrgrnted Circuiu and S.nt~",.f. Mc-Graw-Hill, N",'

York, I996.

10.5

POPCORN NO ISE

255

where A is the area of the gate ( = WL ) and K is a device -specific constant . Th us, for a fixed transconducta nce. a larger ga le area and a thin ner dielectric red uce this noise term, For PMOS devices, K is typically a bout 10- 211 C 2/ m2, whereas for NMOS de vices it is abo ut 50 lim es larger.2.2 On e should kee p in mind that these constants vary considerably from process to process, und even from run to run , so the value!'> of K given here should he treated as crud e estimates. In particular, the superior III performance of PM OS devices may be :1 temporar y situatio n, a... it is due tn the use of buried channels that may cease to he widel y used in the futu re.

10.4 .3

fli CKER N OI SE IN JUN CTION S

Forward- biasedjunc tion s also ex hibit Il f noi se. The noi se is proportional to the bias current and inversel y proponiona lto the j unctio n area : (1 7 )

where the constant K typi cally has a value of aro und 10 - 25 A_m 2 . Once agai n. bowever, co nsidera ble variation fro m process to process is not uncommon? ' Ri cker noise in bipo lar tran sistors is attributed entirel y to the basc -emin er junction (since it is the on ly one in forward bia s). It has bee n established expe rimentally thai only the base curre nt exhibits II f noise.

10. 5 POPCORN NOIS E Another type of noise that can plague semiconductors is known as popcorn noise (also called burst no ise. bistable noise. and rando m telegraph signals. RTS ). It is understood even mor e poorly than II f noi se. and shares with II I noise a se nsitivity to contamination. Gol d-doped -" bipolar tran sistors exhibit the highest levels of hurst noise. sugg esting u pa rtic ular se nsitivity to contamination by metal ion s specifically, although nol all popcorn noi se may be the result of metal ion contamination. This noise was first observed in poin t-contact diodes, but has also been see n in ordinary j unct ion and tunnel diodes, some type s of resistors. and bot h discret e and integrated circuit junction transistors. Burst no ise is chaructenzcd by its mu ltimodal (most ofte n bimod al) and hence non-G aussian amplit ude d istribution . Th at is. the noise switche s between two or more discrete values, but at rando m times. The

I2 Laker and Sansen , ofl. c it. II Laker and Sansen. np. cit. 2.01 1be purposeful reduction of carrier lifetime by gold dopi ng is ecc asio najly used in bipola r devic es 10 speed recovery from saturation.

CHAPT ER 10

256

N O ISE

switc hing inte rval s lend to be in the aud io ra nge Ie .g.• 10 JI S o n up ), and the po pp ing so und that is he ard when a hurst noi se so urce is connected to a n audio sys te m is why this is known as "popcorn" noise. A s a prac tical mail er . describing popcorn noise mathem aticall y is nor a terri bly useful exerci!M:.2.:'i since it is so variable. So me de vices ex hibit little o r no popc orn noise. wh ile o thers - nominally fabrica ted the same way - ma y show large a mo unts of it. In all cases, meticulous cleanliness in processing is the key to co ntrolling po pcorn no ise, and des cribin g it with qu asicmpirical eq uation s therefor e ha s limited practical va lue. But. for the sa ke of co mplete ness, here's a n eq uatio n for it anyway:

-

N' -

K

I

+
(18)

Here K is an e mpirica l, device - a nd fabri cation-d ependent (and, again. ge nera lly bia s-dependent ) co nsta nt, and f ,. is a corner frequ ency bel ow whi c h the burs t noise de nsity flauen s out , For frequenci es we ll above f .... the total mean-squ are no ise between I I a nd J. is (19)

You will probably never need to use Eq n. 19.

10 .6 CLASSICAL TWO -PORT N O ISE THEO RY Having deve lope d detailed no ise model s for ind ividua l dev ices, we now turn 10 a mac roscopi c description of noise in two -ports. Foc using on such syste m noise models can grea tly simplify a na lysis a nd lead to the acq uis ition of useful desig n insig ht.

10.6.1 N O I SE FACTOR A usefu l measure of the noise perfor mance of a syste m is the noi se factor. usu ally denoted F. To define it and un derstand why it is usefu l. con side r a noi sy (but linear) two -port d riven by a sou rce that ha..... an admi tta nce Y. and a n eq uiva lent shunt noise curre nt T; (see Figure 10.5 ). If we are concerned onl y with o ve ra ll input-outpu t be havior. it is a n unnecessary com plicatio n 10 keep track of a ll of the intern a l noi se so urces. Fort unatel y, the ne t effeet of a ll of those so urces can he rep re sen ted by just o ne pa ir of ex ternal so urces: a noise voltage and a noi se c urren t. Th is huge simplificatio n a llo ws rapi d evaluation of

.~ My sincere apul ogi el> tu lhe aUIh<.lf!> o f the ma ny excellent dtssenauon s and papers on the phe -

no menon. RUI I Sland lly my uatcnem.

10 .6 CLA SSICA LTWO 'PORT NO ISE THEORY

I 2 ~n 1_: No'.,

.'. ( 0

~

" .."1;(

Noi<.eles..\ 2-Port

FIG U RE 10 .6 . Equivalent noise model.

how the source admittance affec ts the overall noise perfo rma nce . As a co nsequen ce. we can identify the criteria one must satisfy for opli ~ um noise performance . The noise factor is defined as • f es

total output no ise power

out put noise du e 10 inp ut source

•

(20j

where. by conven tion. the source is at a tempe rature of 290 K.2h Th e noise factor is a measure of the degradation in signal-to- noise ratio that a system introdu ces. The larger the degradation. the larger the noise factor. If a system adds no noise of its own then the total output noise is due entirely to the source , and the noise factor is therefore uni ty. In the model of Figure 10.6, all of the noise appear s as inputs to the noiseless network, so we may co mpute the noise figure there. A calculation based direct ly 0 11 Bqn. 20 requires the computation of the total power due to all of the sources , and dividing thai result by the powe r due to the input source . An equivalent. and sim pler . method is tu compute the total short- circuit mea n-sq uare noise current. and divide that total by the short- circuit mean-square noise curren t due to the input source. This alternative method is equivalent because the individual power comn bution s arc proportional In the s hort-ci rcuit mean -sq uare current, with a proportio nality constant

:!&

You mighl wo nder why a relanv ely t"txI12l)() K is the reference ie mpererure. The reason i.. simply Ihal l;T i...then ..H ill x 10 - 21 J and engineers like round numbers.

258

CHA PTE R 10

N O ISE

(w hich involves the curre nt division ratio betwee n the so urce and two -port) that is the same for all of the term s. In carry ing (lui thi s comp utatio n. o ne generally encounters the proble m of co mbining noi se sources that have vary ing degree s of correlation with o ne ano ther. In the spec ial case of zero correlation. the individual " o wers superpose. For example. if we assume . as see ms rea sonable. thai the noise powers of the source and of the two -port are uncorrela ted . then the expression for noise figure becomes

(21) No te th ai . althoug h we have assumed that the noise of the so urce is uncorrelated with the two equivalent noi se generators of the two -port . Eqn. 2 1 does not assume that the IWO-!XlTt's generators ate also uncorrelat ed with each o ther. In order 10 accommoda te the po ssibility o f corre latio ns between e" and i" . express i" a.'i the sum of two compo ne nts. On e, ie, is correlated with e lt • and the other, i• . i..n't :

., = ie

+ ill'

(22)

Since it' is correl ated with e". it may be treated a... proportional to it, through a consta nt whuse dimen sio ns a re those of a n admitta nce: (23)

the consta nt Ye is kno wn as the correlation udmittance. Combining Eqn. 2 1, Eq n. 22 , a nd Eq n. 23. the noise factor becomes (24)

'111e expression in Eq n. 24 con tains three indepe nde nt noise so urces, eac h of which may be treat ed a.'i thermal noise produ ced by an equivalent resistance or co nductance (whet her or not such a resista nce or con duc tance act ually is the sou rce of the noise):

"

R = n " - 4 k Tt> f '

(251

i' G = .. " - 4kTt> j'

(26)

,'1,

(27)

Using these eq uivalences. the e xpression for noise fac tor ca n be writte n purely in te rms of Impedances a nd admittances:

259

10 .6 CLA SSICA LTW O -PO RT NOI SE THEORY

2R

F = I + G" + lYe + Y. 1 " G,

",,-+.:.:G,,,,,),,'..:+...:.:(8" ,,-+ .:.:8"'.:)..: ' 1.:8,,,. = 1+ .:G","_+'-'.I("G G, .

( 28)

where we have ex plicitly deco mpo sed each admitta nce into a sum of a conductance G and a susceptance H.

10.6 .2 O PTIMUM SOURC E ADM ITTA NC E Once a given two-port 's noise has bee n characterized with its four noi se parameters (Ge. Heo R" . and G..I, Eqn . 28 allows us to identify the general cond itions for min i-

mizing the noi se factor. Takin g the first deri vative with respect to the source admittance and setting it equal to zero yields (29J (3 0 )

Hence. to minimize the noise factor. the so urce susceptance should be made eq ual to the inverse of the corre latio n susceptance. while the source conductance should be set equal to the value in Eqn . 30 . The noise factor correspondi ng to this choice is found by d irect substitution of Eqn. 29 and Eqn. 30 into Eqn. 28: F nlin

= I + 2R" IG'1fll

+ Gel = I + 2 R" [

G" R"

+ G,7, + G,.] .

(3 1)

We may also express the noise factor in terms of F ll li ll and the source admittance :

_

R" I(G. - G 2 + ( B,. - B ) 2 I. opt) u pl G,

F - Fmin + -

(32)

Thus. co ntours of constant noise factor arc circles ce ntered abo ut (GoP! ' ll"pd in the admittance planc.U It is important to recogn ize that. althoug h mini mizing the noi se factor has something of the flavor of maximizing power transfer. the source adm ittances leadi ng to these co nditions are ge nerally not the same. as is appare nt by inspecti on of Eqn . 29 and Eqn. 30. For exa mple. there is no reason to expect the correlation susceptance to equal the input susce ptance (except by co incidence) . As a consequence . on e must generally acce pt less than maximum power gain if noise performance is to be oprimired . and vice versa . 27 They are also circles wbe n plot ted o n a Smith chan because the mepping between the IWtl plan es is a hilinear rransfon nauo n, which prese rves circles.

260

CHAPTE R 10

N O ISE

10.6 . 3 LI M ITATIO N S O F CLA SSICA L N O ISE O PTIMIZ ATIO N T he cl assic al theo ry just prese nted implicitl y assumes thai one is give n a device with partic ular. fixed c harac teri stics. and defin es the . . ource admittance thut will yield the minimum noise figu re given suc h a device. Although one sta rts wit h fixed devices in discre te RF design. the freedom to c hoos e device di me nsion s in Ie rea lizat io ns points ou t OJ se rious shortcom ing of the classical a pproach: Th e re a rc no spec ific guidelines abou t what dev ice size w ill minimi ze noise. Purt hcrrno re . power consumption is frequ ently OJ parameter of grea t interes t (even an ob sessive one in many po rtable applica tions ). but JX}\~..er is simply not conside red OIl all in cla ssica l noise optimization. We w ill rerum to these themes in great detai l in the c hapter on LNA design . hut for now sim ply be aware of the incomple teness of the classical approach .

10.6 .4 N O ISE FIGURE A ND N OI SE TEMPER ATURE In add itio n III noi se fac tor. other f igures of merit that o ften c ro p up in the literature are noise figure a nd noi..e te mperature . The not ..e figure {NF l is s imply the noise fac tor expressed in decibcl s.P Noise tem perat ure. TN. is an alterna tive wa y of express ing the effec t of an amplifie r's noise conmburion. a nd is de fined as the inc rease in te mpera ture req uired of the so urce res ista nce for it 10 accoun t for all of the ou tput noise at the refe rence tem perat ure Tref (w hich is 290 K ). 11 is related 10 the noise factor as follows:

TN 1+ - => TN = TR:f · (f' - I ). Tre,

(33)

A n a mpl ifier that add s no noise of its own has a noise tem pe ratu re of zero kel vins. No ise tem perature is part icula rly useful for descr ibin g the perfo rma nce of cascaded am plitie rs (a s discu ssed further in Chapter 18) and those w hose nois e factor is quit e close to unit y (or whose noise figure is very clo se to 0 dB ). s ince the no ise tempera ture o ffe rs a higher-resolut ion descr ipt io n of no ise performance in such cases. Thi s ca n he see n in Tab le 10.1. Noise figures in the ra nge of 2- 3 dB a rc ge nerally considered very good. with values around or bel ow I d l! considered outsta nding.

10.7 EXAMPLES OF NOISE CALCULATIONS Here arc some e xamples of noi se calculations to tie up a few loo se e nds and ge nerally reinforce this materi al. 2ll Just to complicate matters, the definitions for noise factor and noise ti!!urc arc ~ .....ilcht..d in sotne texts.

10. 7 EXAMPLES OF NOISE CALCU LAT IO NS

261

Tabl e 10 .1. Noise fig ure. noise [actor. W IlJ noise temperatu re NF tJ BI

F

0.5

1.122 1.1-18 1. 175 1.202 1.230 1.259 1.21018 1.3 18

0 .6

0 .7 0 .8 H.t)

1.11 1.1

J.2 1.5 2.0 2.5 3.0

3.5

I A I3 1.585 1.778 1.995 2.1]9

T.\

(K )

35.-1 -13.0 5tl.1 58 .7 66 8 75. 1 lB .b

92 .3 120 170

226

m

359

~ v,.. R I ~ R2

fi GURE 10 . 7. Noisy resistive network.

Examp le 1

Quite often. a network will cons ist of a number of indi vidua l noise sources . Thi s ex ample looks at a way to simplify calculations by com bining individual noise sources before diving into messy math. Co nsider the noisy resistive network shown in Figure 10.7. Let's co mpute the total noise as mea sured across the output terminal s. We'll perform the co mputation two ways. First, we compute the nu s noise voltage that each resistor contributes to the output. Then. ass uming that eac h resistor generates noise that is unco rrehucd with tluu of the other (a reasonable assum ption here, one would hope). we com bine the noise sources in root-sum -squared t rss) fashion to find the nn s value of the overall noise. Proceedi ng in this man ner yields the fo llowing sequence of com putations:

(35)

• 262

CHAPTE R 10 NOI SE

FIGURE 10.8 . Capocitively Iooded noi sy resistor.

Combining the individual noise so urces yields. after some crunching: (36)

Examinatio n of the final result tells us that one may save a little labor by simply combining the resistances into o ne equ ivalent resistance at the outset , and then

co mputing the rms noise o f that single equivalent resistance. Having illustrated the longer way. you can well app reciate the utility of fi rst co mbining resistances before plunging into a noise calculatio n, part icularly for more complex networks.

Example 2 Suppose thai the only bandwidth limitation in making resistor noise measurements were the ever-present stray capacitance of any physical setup. Derive an expression for the mean-square noise for a network consisting o f a resistor R shunted by a capacitance C. The circuit under co nsideration is thus as sketched in Figure 10.8. where e" represcnts the thermal noise of the resistor. Reca ll that a single- po le RC filter has a noise band width that is rr/2 times the - 3-d B bandw idth. and that the - 3-dB bandwidth in turn is 1/ 2 :rRC Hz (telling us that the noise band width is j ust 1/ 4RC Hz). Hence. we may compute the mean -squ are output voltage noi se without too much trouble: (37)

Th us. we see that the mea n-squa re noise voltage is independent o f the resistance in this case. The reason that the resistance drops out is that a larger resistance has a proportionally larger noise source. but also a proportiona lly smaller bandwidth. so that the total mean-square noise voltage remai ns constant for a given capacitance. A deepe r insight is that kT represents the maximum available therm al noi se energy. while the mea n-square energy in a capacitor is simply C V 2. Equating the two and solving for the mean-square voltage leads to the derived expression - namely.

i ric.

10 . 8 A HAN DY RUL E O F THUMB

2.3

Example 3 For a number o f years at an advanced technological university in the north eastern United Slates, an unint en tionall y cruel hoax was perpe trated o n a success ion o f unsuspecting undergraduates by an equally unsuspecting lecturer. The stude nts were assigned, for their bachelor 's thesis, the task of designing and building an amplifier with a - ~- d R ha nd width of I Ml iz and a midband vo ltage ga in of IOf) . The so urce resistance was given as ranging between 100 H2 and I M Q , and the supply voltage was the old standard ± 15 V. Assuming that these hapless stude nts could so lve the t'fIomWl1S stability problem s associated with tryin g to build an amplifier with a gain-bandwidth product of I T ll z {an extremely d ifficult prob lem in its own right ), let' s see if the thennal noise in the so urce resistance impo ses any signifi cant fundament al limits. Generou sly choos ing the lower limit of source resistance , we noll' that the therm al noise of a IOO-kQ resistor has an rms den sity of 40 nV/ JiTZ at room tem peratu re. We expec t the noise band width to be greater than the - 3-d8 bandwidth of 1 MHz , but we' ll co ntinue to he genero us and ass ume that the noise bandwid th is also abo ut I MHz. The total rm s noise co ntributed by the resistor at the input of the amplifi er is therefore about 40 IN, mean ing that the noise at the output of this gain-o f- Ju" amplifier is calcul ated to be 40 V nu s if we co ntinue to pretend that all of the output noise is due solely to the source resistor. Even with this unrealistically generous assumption, we note tha r rbe ca lculated rms output noise exceeds the total s upply voltage by about 33%, so that the ampli fier would spend most of its time slamming from mil to rail j UM from the noise o f the inp ut source resistance alone! And naturally, the real situation would be even wo rse, since the ampli fier would nec essarily contribute some additional noise of its own. The task , as assigned. is impossible unless cryogenic means are made available! This example underscor es the impo rtance o f first carry ing out back -of-the -envelope sanity checks before investing ulot of design effort .

10 . B A HANDY RULE OF THUM B

Making measurements of noise can he rather invol ved if it is 10 be done acc urately. Typically, a spectrum analyzer is required to determ ine the noise density as a function of frequency, or the output noise is compared with a ca librated no ise source. Occasiona lly. though, there are situations in which the noise is known 10 he Gau ssian to a reasonab le approximation (or simply assumed so ), and all tha t is desired is a rough es timate of the noise. In such ca ses, it can be suffi cient to look at the time waveform of the noise on an oscilloscope and estimate the peak-to -peak value. If the amplitude distribution is trul y Ga ussian. the waveform will exceed six times the nns value on ly about 0.3% of the time. so simply dividing an eye ball estimate of the peak-to-peak value by abo ut six will usua lly yie ld a reasonable estimate of the rms value of the noise .

CHAPTER 10 NO ISE

264

10.9 TYPI CAL NOISE PERFORMANCE To de vclo p an appreciation for wha t noise perfo rmance on e ma y e xpec t in practice. here arc a number of exa mple s. General- purpose circ uits tend 10 be somewhat noisy. For examp le, the very popular 74 1 or -a mp has poor noise performance , with an equivalent input no ise vu llage density of aro und 20 nV/ J'Hl (abou l lhat o f a 10-H2 resistor) and an eq uivalent inpur current noise den sity of approxi matel y 100 fA/.,.Ii"il (co nsistent with the 7~ 1\ input bias curre nt o f 100 nA J. Both the voltage and current noise typically ex hibit III co mers somewhe re betwee n 100 Hz and I kHz . TIle relativel y large voltage noise of the 74 1 can he attributed 10 the U!'C of act ive devices as load s. since active devices essentially a mp lify thei r OW II mremal nois e. More modern am plifie rs have much bet ter noise performance . For example, the D P-27 from A nalog Devices e xhibits a noise vo ltage dens ity of around 3 nV/ Jfil a nd inpu t current noi se that is similar tot hat of a 74 1, wi th mo st of the improvements a ttributable to the U!'.C of ordi nary resistive loads in the first stage and a reduction in parusuic base resi stance. Process mod ifications also result in a spec tac ular ly low 1/ f comer of about 3 Hz for the voltage noise. Becau se a major so urce of input c urre nt noise is simply the sho t noi se associat ed w ith the in put c urre nt itself. one would ex pec t FET-input a mplifiers to fare bett er in this regard . So it is no t too surpris ing that the OP-2 IS JF ET-input op-amp exhibits 10 fA / ../iTI. of input c urrent noise densi ty (a t room tem peratu re), about a factor of 20 be tter than the 74 1. lr is rather challe ng ing to design low-noise a mplifier s for 50-Q RF sys tems. since the thermal noi se associated with the source impedance is so small. In particular, it is c ritically important to min imile the paru suic base res istance (fo r bipolar impl ementations) or gate resistance (fur FEr circ uits). For exam ple . a 50 -Q ba se res istance already makes it im possible to achi eve a no ise figure bett er than 3 dB . To obtain sutlicicmly s mall ga te or ba se resistan ces, it is ge nerally necessary to use a parall el combination of ma ny sma ller unit devices. Des pite these difficu lties. silicon bipo lar RF LNAs (l ow-no ise am plifier s) with noise ligures of around 2 dB at I G Ill arc available from a nu mbe r of so urce s. Th is level of noi se pe rfo rmance is imp ressive beca use a no ise figu re o f 2 dB implies an ab so lute maximum par usiuc base resista nce of about 30 Q . Sta ted a nothe r wa y. the equivalent input voltage noise of the amplifier must he be low 0 .7 nV/ sIHZ - well below whut is com monplace for general -purpose circuits suc h as up-amps. Recently re pon ed CMUS LNA s III the gigahe rtz ran ge also show great promise. w ith noise figures in the nei ghborhood of 3 dB a nd about 10 mW of power consumption at 1.5 G lIz .;N As we ' ll sec in the next c hapter. the level of performance 2~

Shaeffe r and Lee. op. cu.

10.10 A PPENDIX · N OI SE M ODELS

2.'

demonstrated so fur by no mea ns represen ts (he best that may be ach ieved . Fu rther scaling and more ca reful des ign wi ll resu lt in even better performance. Note that noise f igure is (he common way (0 report no ise performance in RF systems since the reference impedance level is known. wh ile equivalent noise vo ltage and current generators are more commonly used to desc ribe the noise of general purpose build ing blocks that might be used with a wide range of source impedances.

10.10 APPENDIX : NOISE MODElS This appe ndix sum marizes (he noi se models prese nted earlier.

t';

-;>

4UI1!

,-~ = -R 2 = -R- = 4£: TGI1!

FIGURE 10 . 9. Re~i~tor thermal noise modeI~.

The following excess mean-square noise voltage is add ed (tithe mean-squ are thermal noise voltage in Figu re 10 .9 if I{f noi se is of interest: (3 R)

; 2 = 2qh x'l1/

"

K,

+ - 11/

t:

FI GUR E 10.10 . Diode noise model.

For the diode noise model (Figure 10. 10 ). the

111 term may he ex panded as follows:

K f ; 2 = _ . _ . ilJ. } f Aj

(39)

266

CHAPTER 10 NOISE

, - -,--

h -r-r--r-r-

'

• KI i;" = 2q 1iflJ./ + j !:J./ j;~

= 2q 1cl:!.j

FIGURE 10 .11. Bipolar transistor nohe

model.

The more detailed 1/ f bebavicr o f Eqn . 39 applies also to the bipol ar mod el of Figure 10 .11. which neglects the thermal noise of r b. In many practical ampl ifiers. however. this noise is qu ite important, and may even domin ate .

.' , "

.~ c"

s

--

i;d = .u r ygJo/i./

K,

+ j l!J./

v';J = 4 k TlJr~ !:J.f I

r, =-5}(dll FIGU RE 10 .12 . MO S noise model.

For the MOS model of Figure 10 . 12, the

-i 2 = _K .

,

g,~

1 WLC".2

II f term may he ex panded as follows:

'I '-" -K f

• U

•

2 ' WT

A

'I,

. U

(40)

PROB LEM SET FOR N O ISE PROBLEM 1 Show formal ly that the noise figure o f a resistive attcn uator at 290 K is equal 10 its attenuation . In your answer. define carefully what "a ttenuation" specifi ca lly refers 10 .

PR08LEM SET

267

PROBLEM 2 Derive a formula for the overall noise fi gure of a cascade of systems. as shuw n in Figure 10.13. Here. eac h no ise factor P is com puted with respect to the output impedance o f the previous stage. Furthermore. eac h power gain G is the available power gain - that is. the power availab le under matched co nditions.

FIGURE 10.13 . Cc scc ded systems for computation of no ise figure .

From your formula . what do you deduce about the relative contributions to noise fi gure of curlier versus later stages? PROBLEM 3 Derive an ex pression for the low-frequency noise fi gure o f a bipolar transistor. Use the model of Figure 10 .11 , but with r b and its thermal noise included. Ignore flicker noise, and assume that all noise sources in the model are uncorrelated. Assume that the source resistance is R•. Hint: Since there are only three noise sources in the device. it is probably not worth deriving a two -port equivalent noise rnodel in this case. The equation you are abo ut to derive is frequentl y ca lled Nielsen's equation. after the fellow who first publls.hed it. PROBLEM 4

Using your answer 10 Prohlem 3. derive ex press ions for F min and

R opl'

PROBL EM 5 Bec ause low input current is a highly desirable attribute in many amplifiers. designers have evolved numerous techn iques for achieving this goal. The problem is much more challenging fur bipol ar amplifiers because of the fundamentalneed for base current. One obvious method is simply to use transistors with large ~ , and operate the input stage at a low current. Un fort unately, the former trades off with base resistance (among others), and the latter choice degrad es Km and Ir, so gain and speed ca n suffer. An alternative that is frequ entl y used is to ca nce l the base current with an internal current mirror. so that the external world docs not have to supply it. That way.the input stage can be biased at a relatively large current without causing a large current 011 the input termin als.

(a) What is the ( low-freq uency) input shot noise current density if no input curre nt cancellation is used and the base current is 100 nA? We are looking on ly at the pure shot noise due to the base current; do not worry abo ut re fl ectin g any other noise sources to the input. (b) Now ~ uppose that we succeed in cancelling nearly all of the input current by using a 99-nA internal curre nt so urce so that the input current (as determi ned

CHAPIER 10 NO ISE

2.8

by external measure men ts ) decrea ses 10 I nA . Wh at is the input shot noise cur-

re nt density in this case? Commen t on the advantages and di sadvantages of the c urre nt-c ance llation method . PROBLEM 6 A common pro blem is how to c hoose a n a mplifier for be st system noise perfo rma nce . Manufacturers migh t supply data abou t the equi valent input noise voltage and c urre nt, hut e nginee rs so met imes dra w incorrec t infere nce s from this information, par ticularly jf they read too much meanin g into the term "optimu m impedance." Suppose we ha ve a c ho ice betwee n two a mp lifiers . They hoth have 10 flV/Jill inp ut noise volt age de nsity, hut a mp lifier A has 50 fA j ./ilZ input noise c urre nt density while a mp lifier B has t w ice as m uc h . (a) Wh at is the o ptim um so urce resistance for eac h ampli fier? You ma y ignore correlation betwee n thc noi se sources. ( b) If the source resistance happe ns In be 100 kO , which op -ump should you use? Assume that (eve r-elusive) ideal. br oadba nd. arbitra ry-ra tio. fosslcss transformers are ava ilable. (c l For your c ho ice in ( b). what is the best possible sys te m noise figure? PRO BLEM 7

We' ve see n that on e se rious pro blem w ith RF IC s is the lack of high-Q inductors . Since low Q is ca used by dissipation a nd act ive feedbac k can compcnsate for energy loss, the re have bee n many propo sals over the yea rs for variou s active induc to r sc he mes . One (bu t by no means the on ly ) problem common to all of the se active sc hemes is that of limited dynamic mn ge . Th at is. it is not suffici ent merely to sy nthesize an cleme nt with inductive small-signal impeda nce. It is also importa nt to ha ve a large dyna mic runge . b ou nded fo r s mall signals by the noise floo r and for large signals by the lineari ty ce iling . Th is problem e xplores some noise prope rties of acti ve induc tor s. Altho ugh we wiII exami ne one particular circuit configuration, the outlin es of the result apply quite ge nerally to all activ e indu cto r ci rcuits, leading to the depressing concl usion that active ind uctors have limi ted utilit y, (a) First conside r, fo r compa riso n purpo ses. .1 pa ssive paralle l R L C ta nk . what is the eq uivale nt shunt mean-squ are noise c urrent de nsity'! Th e L and C are ideal losslcss clements. ( b) Now consid e r a magica l two-port that rec iproca tes im ped ances" Such an element is called a gyrator, and its prope rties were first e xplored by the Dutch theorist R. D. H _ Tetlegen of Ph ilips . Im peda nce rec iproc ation is po te ntial ly relevant because Ie processes give us good capacitors. Combining gy ra tors and good capacit ors might then give us good ind uctors (so goe s the cl assic argument).

269

PROBLEM SET

On e simple way to make an active inductor is. with the circuit shown in Figure 10. 14. (A quick history note : Th is circ uit deri ves from the n?lIc lallCt' IIIbe connectio n devised in the 19JOs. for e lectro nic tuning and FM generutio n.) Note that biasing details. arc not show n: simply assume thai the transistor is somehow hiao;ed 10 produce a speci fied value o f transcooductaoce 8....

R

c ,f-=---1:[1

gm

l

=•

'7 FIGURE 10 .14 . A ctive inductor.

(e)

(d)

(el

if)

(g)

Assume thai the Re freq uency is much lower than the ope rating freq uency of the inductor. Further assume that the resistance R is itsel f much larger than the reactance of the ind uctor at nil frequenc ies of interest. and that the transistor model has no reac tances and no parasitic resistances. With these assumptions. derive an express ion for the ind uctance. Derive an expression for the Q of this inductor as a function o f frequency if the capacitance is entirely due 10 the C•• of the transistor . Otherwise, make U "C of the same assumptions as in part ( b). Express your answer in term s of WT . Assume that all of the noise of this circuit co mes from just two sou rces: the resistor R and the drain current noise of the MOSF ET. That is, neglect gate current noise. As a consequence , the calculation will underestimate the actual noise somewhat. Even so, we ' ll see that the news is depressing enough. What is the most general expres sion for the short-c ircuit dra in cu rrent noise density d ue solely to the MOS F ET itself ? Whal is the component of short -ci rcuit drain cu rrent noise densit y due 10 the ther mal noise of the resistor? What is the short-circuit curren t noise density d ue dire ctly to the resistor itself? Con tinue to ass ume that the Rell • frequency is much lower than the opera ting frequency of the inductor. Assume that these noise sources are uncorrelated. and incorporate your expression for Q derived in part (c) to derive a ge neral ex pression for the total shortcircuit noise curren t de nsity.

Comparing your answers 10 (a ) and (g) , you should he able 10 draw some co nelusions abou t the noise properties of active inducto rs. Further acknowledg ing large-signal limita tions. as well, nne must conclude that active inductors have serious problems that are difficu lt - perhap s even impossible - to evade.

• 270

CHAPTER 10 N O ISE

PROBLEM 8 Deri ve a ge neral expre ss ion for the noi se band width of a second-order low-pass filter. Using your formula . veri fy that a critically dam ped second-order fil -

ler has a noise bandwidth thai is about 1.22 times the - 3-d8 bandwidth . PROBLEM 9 A s ing le -pole a mplifier has a voltage gain o f 1000 and a - 3-d8 bandw idth of J kll z. (a) Wh at is the tot al mean -squ are output noise voltage if the input no ise spec tral density is IO- IS V 2j Hz andtlat ? Ass ume that this noise source complete ly models all the noise in the sys te m. (b ) Repeal part (a) if the input noise spec tral density is not flat and in feet has the following behavior:

;;z = 10-"(10 t;"j I+ - f-Hl) .

( r lO.l )

PROBL EM 10 Explain unde r what co nditions one may add noise sources in rootsum-sq ua red fashion. Derive a more ge ne ral "addition" law. PRO BLEM 11 Consider the mod el shown in Figure 10. 15 for a sample -and-hold circ uit. Th e res istor model s the finite on-resista nce of the sa mpling switc h. Since that resistance is thermally noisy. the sam pled-and-held voltag e w ill also be noisy. Compute the mean -squ are value of the output no ise.

f iGURE 10 . IS. Sample-and·hoId cirCUIt.

PROBLEM 12 Suppose that you are g iven a single -e nded am plifie r that possesses a certain equivale nt input no ise voltage a nd c urre nt. You may assum e that these are uncorrc latcd with c uch other. De mon stra te that a d ifferent ial amplifie r constructed with two of these single -e nded a mp lifie r!'> achieve!'> the same min imum noise figure bu t at twice the powe r consumption. In addi tion. speci fy the optimum differential source impedance for the d ifferen tial am plifie r in terms of the optim um single- ended source impedance for the sin gle -e nded LNA . PROBLEM 13 Th e relat ionship between reflec tion coe ffic ient and admittance is given by the following bilinear transform ati on.

I -Y r -- -1 + -Y '

( r lO.2)

PROBLEM SET

271

where Y is the udmi nance. normalized to the characteristic impeda nce of the system (e.g.• 50 0 ). Using this relationship. recast the equation for noise factor. ( PIO.3) in term s of the real anJ ima gin ary pans of the reflec tion coefficient ing contours still circles? Are they still co ncen tric'!

r . Arc the result-

PROB LEM 14 For a simple CMOS differential pair biased with a single-transistor MOS current source. find the equivalent mean-square drain current noi se o f o ne uu nsistorin the pair in term s of the drain curre nt noise sources o f the three tn•ansistors. the iransccnductances o f the differential devices. and the output resistance of the current source. You may ignore the output resistance of the differen tial device s and ass ume thai the gates arc driven by a low-resistance source. Do 11t H ass ume that the pair is in the balanced state.

CHAPTER ELEVEN

LNA DESIGN

11.1

INTRODUCTION

T he first stage of a receiver is typicall y a low-noise amplifi er ( LNA). whose main funct ion is to provide enough gain to overcome the no ise of subseq uent stages (such as a mixer ). Aside from providing this ga in while add ing as lillie noise as po ssible, an LNA should acco mmod ate large signals without distort ion . and freq uentl y must also present a spec ific imped ance, such as 50 Q . to the inpu t so urce . Thi s last con sideration is part icu lar ly importan t if a passive filter preced es the LNA . since the transfer charac terist ics of many filter!' are quite se nsiti ve 10 the quality of the termination. In principl e. one can obtain the minimum noi se figure from a given device by using the op tim um source impedance defi ned by the fou r noise param eters: C e , Be' Rn, and G" . Th is cla ssical approach ha s im porta nt shortcomings . ho wever, as described in the previou s c hapter. For exa mple. the so urce impedance that min imizes the noise figu re ge nerally differs. perha ps considerably. from that w hich maximizes the power gain. Hence. it is po ssible for poor ga in and a had input match to accompany a good noise figure. Add itionally. power consumptio n is a n important consideration in many a pplications. hut classicalnoise opti miza tion simply igno res pow er consumption altoget he r. Finally, such a n approach presumes that one is give n a de vice with fixed cha racteristics . and thu s off ers no ex plicit guida nce e n how best 10 exe rcise the Ie des igner 's freedom to tailor device geometries, To de velop a design stra tegy that ba lances ga in. input ir npcdunce . no i...e figure, and power consum ptio n. we will de rive analytical expressions for the fo ur noise parame ters d irec tly fro m the de vice noise mod el. a nd the n exa mine seve ral LNA arc hitectures. As we 'll sec, insigh ts gained from that exercise allow us to design na rrowband LNA s wi th ncar-mi nimum noi se figure. along w ith an excelle nt impeda nce match and good po wcr gain, all within a power budge tthat is speci fied a priori. An imporra nt co llateral result is a s imple form ula that yiel ds the optimu m device w idth for a given tec hnology. source impedance. and operating frequ en cy,

272

11.2 DER IVATIO N OF M O SFET TWO - PORT N OI SE PARA METERS

11.2

273

DERIVATION OF MOSFET TWO -PORT NOISE PARAMETERS

Recall that the MOSF ET noise mod el consists of two sources. The mean-square drain curren t noise is (I ) i;d = 4 kT ygdOtJ. j ;

me gale current no ise is where W

2

C2

'>

g~= --.

5gdo

(3)

Further recall that the gate no ise is correlated with the drain noi se. with a correlation coefficient defined formal ly as c ==

(4)

The long-chan nel value of c is theoret ically j 0 .395. Becau se its value in the shortchannel regime is not know n at prese nt . we will assume that c remains at its longchannel value in all of the nu merical exa mples that follow. We will also neg lect C, d to simplify the derivation . Wh ile the achievable no ise figure is little affected by CKd • me input impeda nce can be a strong functio n of Cgd. and (his effec t must be taken into acco unt when design ing the inpu t matchi ng network . To derive the four equ ivalent two -port noise parameters. repeated here for COllVC-

mence. e' " " - 4kTt;!'

R =

G =

)' U

" - a r s] :

Y, == -i" = G,, + }'B",

(51 (fi)

(7)

e"

we fi rst reflectthe two fundament al MOSF ET noise source s bad 10 the input port as a different pair of equivalent input generators (one voltage and one curre nt source). The equivalent input noise voltage ge nerator accounts for the o utput noise ohserved when the input port is short-c ircuited ( incrementally speaking ). To determi ne its value. reflec t (he drain curre nt no ise back to the input as a noise voltage and recognize that the ratio of these qu an tities is s imply 8m. T hus.

~

e; =

i; J

g;,

4 kT ygdOtJ. j

=

g;,

(8)

27<

CHAPTER 11 l NA DESIGN

from which it is apparent that the equivalent input noise voltage is co mpletely correlated . and ill phase. with the drain curre nt noi se. Thus. we can immediately det ermine thai e2 R = II = ygJ O (9) " - ~ kT6 f -g-;-' T he equivalent input noise vo ltage ge nerator by itsel f doe s not fully acco unt for the drain curre nt noi se. however. beca use a noi sy drain curre nt also flows eve n when the inpu t is open-circ uited and induced ga le current noise is ignored. Under this openci rcuit co nd ition. di viding the drain current noi se by the transconductance yields an equ ivalent input voltage which , when mult iplied in turn by the inpu t admittance. gives us the value of an equivalent input curre nt noise that completes the modeling of ;,,<1 : T ;;d (jW CIf~ ) 2 4k Tygd06f(jwC6J)~ - ,. , i"l

=

2

g",

~

=

K;;,

= e,,(jwC/f J) .

(10)

In this step of the deri vation . we have assumed that the input ad mittance of a MOSF ET is pu rely ca paci tive. This ass umption is a good ap proximatio n for freq uencies well below to r , if a ppropria te high-freq uency layout practice is obse rved 10 minim ize gale resista nce . Give n this assumption . Eqn . 10 show s that the input no ise current i ll l is in qu adratu re, and therefore co mplete ly co rrel ated. with the eq uivalen t input noise vo ltage e ll ' Th e total equivale nt inp ut current noise is the sum of the reflected drain noise co ntributio n of Eqn . 10 and the induced ga le curre nt noise. The induced gale noise current itself con sists of two terms. One. which we 'Jl denote i ll /fC ' i!'. fully correlated with the dra in curre nt noise, while the ot her. ;"/f'" is co mpletely uncorrcl ated with the drain curre nt noise. li enee. we may express the correlatio n admitta nce as follows:

· C6 ' + = jW

0.

o . K" -. - ' /" l "J

= l 'W Cg.

+ K", '

'-, _

- . -.-.

(I I)

/,,,1

To express Y.. in a more useful form. we need to incorporate the gate no ise correlation factor ex plicitly. To do MI . we must mani pul ate the last term of Eq n. II in ways that will initia lly appear mys terious . First , we express it in terms of cross-correlations by multiplying bo th nu mera tor and denominat or by the co njuga te o f the drain noise current and then averagi ng eac h: (12)

The last eq uali ty. in which ;"If replaces i-, ... is valid because the uncorrelated portion of the gate noise current necessa rily co ntributes no thing to the cross -c orrela tion.

11.2

DER IVATIO N OF MO SFET TWO - PO RT NOI SE PA RA M ETE RS

275

Using Eq n. I I. we may write the correlation adm ittance as

(1 3)

which. in turn. may he expressed as

( 14)

which explains all of the maneu vering. since the correlation coe ffic ient has finally made an explicit appearance . Substituting for the term under the radical yields (1 5)

If we assume that c co ntinues 10 he pure ly imaginary, even in the short-c hannel regime. we finally obtain a useful express ion for the correlation admitta nce:

Y, = jw C, .

+ jwC, . gz. . Ie l! 8 Kd O

5y

= j wc, . ( 1 + a iel! 8 ) . 5y

(16)

where we have used the substitutio n

g. a= - .

(17)

gdO

Since a is unity for lon g-chann el devi ce s and progressively decrea ses as channel lengths shrink . it is o ne measure of the dep arture fro m the lon g-chann el reg ime . We see from Eqn. 16 that the correlation admittance is purel y imag inary. so that G( = 0. 1 More significant. however, is the fact tha t Y,. does not equal the admittance of C,•. allhough it is some multiple of it. Hence. nile cannot maximize po wer tra nsfer and minimi ze noise figure simultaneo usly. To investigate furt her the important implications of this impossi bility, though . we need 10 derive the last remaining noise parameter, G" . Using the definitio n of the correla tion coe fficient. we may ex press the indu ced gate noise as follows:

I Again. this conclusion i.. based on .. neglect uf ..ny resistive term at the input.

276

CHAPTER 11 lN A DESIG N

Table 11 . 1. Summary of MOSFET

two -port 1I0 ; St' pa rameters Parameter

Expresemn

R. &w2c i . 1I - Icf ) jgJ O

Th e ve ry last te rm in Eqn . 18 is the uncorretated port ion o f the ga le noise current. so that . finally.

From Eqn. 20, we sec that the optimum source susce ptance is esse ntially inductive in c haructcr. e xce pt that it has the wro ng freq uen cy be havior. Helice. achieving a broadband noise match is fund am e nta lly difficult.

Continuingthe real part of the optimum source ad mittance is 8 5y ( I - lel'),

(21)

and the m inimu m no ise figure is given by

f;",, =1 + 2R.' G,,_ + G,) "' J +

2r .w- J ~=----,= y8( 1 lei' ), v 5wr

(221

In Eq n. 22. the ap pro xi ma tio n is exac t if o ne trea ts W r as sim ply the rat io of g. 10 CR' . Not e thai if there were no ga te c urre nt noise (i.e.• if d were ze ro ). the minimu m noise fig ure wo uld be 0 d B. T hat un realistic predi ct io n a lo ne sho uld be e nough

11.3 l NA TOPOLOGIES: POWER MATCH VER SUS NO ISE MATCH

277

Tahle 11.2. Estimated 1'~nlll ( y

= 2, D = 4)

"" /'"

Fm,n (dB)

20

0.5 00 0 .9 1.0

" '0 5

10 suspect that ga te noise rnu..1 indeed cx i..t . Also no ll' thai . in princi ple , incre asin g the co rre latio n be t wecn drain a nd gale c urre nt noi se wo uld improve noi se figu re , although correlation coefficients unrealistically ncar unit y wo uld he req uired to e ffec t large redu ct io ns in noi se figu re. Anoth er importan t o h..crvatio n is that im prove ment s in WT th ai acc ompany technology scaling abo im prove the noi se f igure at any give n frequency. To under ..c ore this poi m . fe t us as ..ign nu meri ca l values to the par ameter s in Eqn. 22 . However . because the detailed behavior of so me o f these para meters in the short-channel regime is unknown, we will ha ve 10 ma ke so me ed uc ated g uesses 10 a rrive at estimates of Fmin• As mentioned in Cha pter 10. measure men ts of y reveal that it ca n be 2- 3 tim es larger in sho rt de vice s than predi ct ed by lon g-ch annel theory. No published values of {; in the sho rt-c hannel regime ex ist. unfortunately, so we will a ssu me thut it , ro o. is augmented hy a factor o f 2- 3. Becau se a single mec ha nism (th ermally dri ven c han nel charge fluctuations) fundamentall y gives rise to both ga te an d drain noise. thi s last assumpt ion migh t not he grossly in error. Finally, let lIS as sume thai lei remains equal to O.:W5. eve n in the short-chan ne l reg ime . Table 11.2 shows 1"l11 in as a fun ction of normalized frequen cy if shor t-c ha nnel e ffects cause a tripling in y and D. To the ex te nt that the a ssu mp tio ns made arc rea son able. it is encourag ing that exce llent no ise figures are possible . eve n with increa sed y and J, for frequencie s wel l below (lJr . Having deri ved e xpre ssion s for the noi se parameter s and 1'~tl i n , we now co nsider in detail the c ircuit im plic at io ns of those eq uatio ns.

11.3

LNA TOPOLOGIES : POWER MATCH VERSUS NOISE MATCH

The deriva tio ns of the previ ous sec tio n sho w that. for a MO SFET. the so urce impedance (hat yields mini mum no ise fac tor is ind uctive in cha rac te r and ge ne rally unrelated to the co nd itio ns thai ma xim ize power tra nsfer. Furthe rmore. the input impedance of a MO SF ET is inh ere ntly ca paci tive . M) pro viding a good match 10 a 50- Q

278

CHAPTER 11

lN A DESIGN

'OtJT

f iGURE 11.1. Common-source a mplifier with ~unt input resistor lbi-mi ng not shown).

so urce w ithout tic-grad ing noise performance would appea r 10 be difficul t. Since prese nting a known resistive impedance to the e xtern al world is almost always a c ritical requireme nt of LNAs. we will first ex amine a nu mber o f ci rc uit topol ogies that accomplish this fea t and the n narrow the field o f contenders by evaluating their noise properties. O ne straig htforward approa c h to pro vid ing a reaso nably broad band 50·n lenni -

nation is simply 10 put a 50 -0 resistor across the input terminals of a common-source amplifi er; this is shown in Figure II.I. Unfortun ately. the resistor R. adds thermal noise o f its own. and attenuates the signal ( by a factor of 2) ahead of the tran sistor. The co mbination of these two effects ge nerally produces unacceptably high noise fi gures.? More formally, it is straightforwa rd to establish the followi ng lower bou nd on the noise fi gure of this circ uit: 4y

I

F > 2+ - ·- a 8mR '

(23)

where Rs = HI = R. T his bo und app lies only in the low-frequency limit and ignores gate current noise alto get her. Naturally, the noise fi gure is worse at higher frequencies and when gate noise is taken into account, Th e shunt- series amp lifier. described in Cha pter 8, is another circuit that provides a broa dba nd reul inpu t impedance . Since it doc s not reduce the signal with a noisy attenua tor before ampl ifying, we expect its noise figure 10 be substantially better than that of the circuit of Figure 11. 1. The am plifier sketched in Figure 11 .2 suffers from fewer problem s than the previous circuit, yel the resist ive feedba ck network continues to generate thermal noise of its own , and also fails tu present to the uuu ststor a ll Impe dance tluu eq uals Zopt al all frequen cies (perhaps at any frequency). As a conseq ue nce . the overall amp lifier's

2 A ~ a specihc exa mple. one rece ntly puhlir.hed SOD-M ill CMOS amplifier has a noi ~ figure in ercess or 11dB (repor ted as 6 dB by ignoring the anenuauon and noi..e or R 1 ) ,

11.3 l N A TOPOLOG IES: POWEll: MATCH VER SUS N O ISE M ATCH

279

RoUl

..J vix

R,

R,

r

R,

,7

"If IL

Roo

,

v OUT

R, 7

FIGUIl:f 11.2 . ShunHeries ampli~er (bia sing not shownl.

FIGU RE 11 . 3 . Common-gote amplifier (bia sing nat shown ).

noise fi gure. while usually much better than thai of Figure J I. J. sn ll ge nerally exceeds the device Fmin by a considerable amount (typically a few decibels). Nonctbeless. the broadband capability of this circuit is frequen tly enough o f a compensating advantage that the shunt-series amplifier is found in many LNA appli cations. even though its noise figure is nor the minimum possible.' Another method for rea lizing a resistive inp ut impedan ce is 10 use a com mon-gate confi guration. Since the resistance look ing into the source terminal is I/ Mm. a proper selection of device size and hias current can provide the desired 50-n resistance : sec Figure 11.3. As with the circ uit of Figure 11.1. it is straightfo rward to establish the following lower bound (Ill the noise figure oft he common-gate amplifi er (again. at low trcquend es and neglecting gate current noise): F 2: I

+ yla.

(241

This hound assumes that the resistance loo king into the source term inal is adjusted to equal the source resistance. and is about 2.2 dB in the long-ch annel limit and .t .8 dB

) Ckarly. the nohe ngure o f this circuit ca n be low e noug h to be quite practical. f(Jr it is used in an in' ltUment lhat measure!io noise figure ( ~ 897I1A . made by Il e.....len -Packard ), as one c umpl~.

280

CHAPTER 11 lNA DESIGN

c

- A(s)

FIGURE 11.4 . Impedance tron~formalion model.

I

for sho rt de vices (yj u = 2) . The noise figure wi ll be sig nificantly worse a t high frequen cies and when ga te c urren t nois e is take n into accou nt. A ll three ofthe preceding topo logies suffe r noise figure degradation from the prese nce of noisy resista nces in the sig nal path ( inclu ding c hanne l resistances. as in the case o f the common-gate amplifi er ). Fortunately, one ca n pro vide a resistive input impeda nce wit hout resistor s. contrary to intuition. The first hint of this possibility act ually traces hack to the V:U': UUIll tube era , su a brief hist orical digression is in order. Both vacuum lubes and MOSF ETs are devices with nomi nally capaciti ve input impeda nces. Th e key word. however. is " nominally:' for if the inpu t impedances actually were purd y capac itive then the input co uld neve r consume a ny power , and the power ga in wo uld necessaril y he infinite even at infinitely high frequency. Such a result defies com mon se nse (und ex perim ent J. so one might surmis e thai the input impeda nce IIlUSt po ..sess a res ist ive com ponent. A MOSF ET 's !!
-i

.-; ,C"',"I +-;---;A7(,"')I

Here. we are implicitly assu ming that tilt' drain is l ~ nninil lcd in a short .

(251

11.3 l N A TOPOlOG IES: POWER MATCH V ERSUS NO ISE MATCH

Now let

A(.{)

28 1

huve gain and phase shift: 5 (26)

then

1

1

Zoo = ~~-'----~~ - ~~-----'-----,...,--,-. jw Cl I

+ Aoe

i4l 1

j wC[ 1 + Ao(cos 4> - jsin 4» 1

(27)

Collecting terms and foc u..ing on the denominator. we obtain Yin = jwC( I

+ A o co~ 4> 1 + A ow e

sin 4> .

(28J

from.... hich it is appurenr rhat the inpu t admittance indeed possesses a real part whose value depen ds on the phase lag ¢ . With zero phase lag. the admittance is pure ly capacitive. as anticipated from quasistatic analyses. If the more realistic scenario o f a nonzero phase lag is co nsidered . the equivalent shunt conductance is see n to increase with freq uency. Perh aps it is no surprise that measurements show that the phase lag itself grows with freq uency. and the eq uivalent shunt conductance typically increases a!\. the square of frequency. to a good approximation. Transit time effects abo cause a resistive component of input impedance in vacuum rubes. where the phenomenon was first observed. Because of the finite veloci ty of charge. then. a real term is .111 unavoidable reality in charge -co ntro lled devices such as vacuum tubes and FET!\.. In the context of low-noise amplifier s. we ac tually seek to enhance this effec t. for it can he used to create a re..isti ve input impedance without the noise of real re..isrors. Fromthe foregoi ng. it is clear that one po ssible method is to modify the device (e.g.• elongate it) in order to enhance transit time effects directly. However . thi!\. approach has the undesir able side e ffect of degrading high-frequency gain. A better method is to employ inductive source degenera tion . With such an inductance, current !low lags behind an applied gate voltage , behavior which is qualitatively similar to the mechanism described . An impo rtant adva ntage of this method is that one then has control over the value of the real part of the impedance through choke of inductance, as is clea r from computing the input resistance of the circ uit shown in Figure 11.5. To simplify the analysis. co nsider a device model that incl udes only a trnnsconducumce and a gate- source capacitance. In that case. it is not hard to show that the input impedance has the following form : (2 9 )

s Tbe form ~hnw n is chosen effect a, wen.

~ i mpt y

for convenience. Any function with a phase I
282

CHAPTER 11

LNA DESIGN

I.

, 'our

L

FlG U" 11 .5 . l..doctNely degenerored common-soorce amplifier.

Hence. the input impedance is thai of a series R L C network . with a resistive term that is directly proporti onal to the inductance value. More generally, an arbitrary source degeneration impedance Z is modified by a factor equal to ItJ(jw) + I) when reflected to the gate circuit. where tJ(jw) is the current gain: P(j w ) = -. . (30)

.

WT

JW

The current gain magnitude goes to unity at wr as it should. and has a capacitive phase angle because of CK~ ' Hence. for the general case.

Z,,(jW ) =

-._1- + IP(jw) j W C s'

+ uz =

_._1_

j WCK~

+ Z + [WT] Z.

(31)

jW

Note thai capacitive degeneratio n con tributes a negative resistance to the input impedance." Hence. any source- to-s ubstrate ca paci tance off sets the positive resistance Irom inductive degeneration. It is important 10 take this effec t into acco unt in any actual design. Whatever the value of this resistive term. it is important to em phasize that it does not bring with it the thermal noise of an ord inary resistor beca use a pure reactance is noiseless. We may therefore ex ploit this property to provide a specified input impedance without degradi ng the noise perfo rmance of the amplifie r. The form of Bqn. 29 clearly shows thai the input impedance i .~ purely resistive at only one frequency (at resonance), however . so this method can only provide a narrowband impedance match . Fortun ately. there are numerous instances when narrowband operation is not only acce ptable hut actually desirab le. so inductive degenerution is ce rtainly a valuable technique. Th e LNA topol ogy we will examine for the rest of this chapter is therefore as shown in f igure 11.6. to Ca pacinvely loaded source followers are infamous for their }Xll.lr sta t'o ility. Th is negative inpul re~istanlX i:;. fundame ntally responsible. and explains why add ing so me povinve Il.'sislance in series wuh the gale circuit helps solve the prob lem.

11.3 LNA TOPO LO G IES: POWE R MATCH VER SUS N OI SE M ATC H

R. " IN -Y'N

283

19

0000

MI

1,

FIG URE 11.6 . Narrowband lNA with ind uctive source degeneration lbia ~jng not !>hewnl.

The inductance L J is chosen to provide the desired input resistance (eq ual 10 RJ • the so urce resistance ). Since the input impedance is purely resistive only at resois needed to nance. an additional degree o f freedo m. provid ed by indu ctance guarantee this cond ition. Now. ar resona nce. the gate -to -source voltage is Q times as large as the inpu t voltage. The overall Mage tra nscondu cta nce G... under this condition is therefore

L,.

( 32)

where we have used the appro ximation that WT is the ratio of g...l 10 C/w Noll' thai the overull rransconductance is independent o f the device tn.ansconduclance. Th is result is the co nseq uence of two co mpeting effects thut cancel precisely. Consider narrowing M J • for exampl e. without changing any bias vo ltages. The device transconductance would then decrease by the same factor as the width. However. the gate capacitance would also shrink by the same factor. and the inductances would have to increase (aga in, by the same factor) to maintain resonance. Since the ratio of inductance 10 capac itance increases. the Q of the input network must increase. Th e increase in Q cancels precisely the reduction ill device transcond uctance. so that the overall transconductance remains unchanged. The question remains as to the size of M j . One might argue that the width of M l should he selected with the aid of Eqn. 2 1, which ex presses GU l'l as a functio n of gate capacitance. Selling G,>pl equa l to the source conductance then yield.... the "optimum" value of C,p . which. in turn. allows us to compute the necessary device width. The best way 10 illustrate thc practical shortco mings of this approa ch is with a numerical example. Suppose we wish (0 design an LNA for U.\C in a 50-a system ut 1.57542 G Uz .7 Using Eqn. 2 1. we fi nd that the required value o f Cg • is

7 This frequency corres ponds 10one used by the G lobal Posuioning System (GPS ).

• 284

CHA PTER II LN A DESIG N

.

G"",

u
-

5y

(33)

,

( I - lei' )

where we have continued to usc y = 2. /} = 4 , and lei = 0 .395. a nd have additionally assumed that a is only a lillie less than unity. Ty pic al gate ove rdriv es" in analog circuits are usually low e nough that U' is no t very much smalle r tha n unity; values in the ran ge of 0.8 to 0 .9 a re nul uncom mon . Eq uation 33 ass umes a n a of abo ut 0.85. pa rtly in keeping with this observation hut ma inly 10 kee p the nu mbers simple. Beca use. In a first approxima tion. device capaci tance is about I pF per millimeter, a device large e nough 10 produ ce the requ ired value of CR~ would he roughly .f mm wide. Furtherm ore. the bias current for such a large device typically would

he large ulso (well over 100 mA , typically). Hence. even though the noise fi gure would correspond very closely to F lll in • the power consumed wou ld he unacceptably high for virtually any upplic••tion . Since power consumption is an important practical constrai nt. the most generally useful noise optimization technique must consider power a prio ri. Although amplifiers designed with an explicit power constraint will necessarily exhibit higher noise figures than could be obtain ed if infinite JXlwer consumption were permitted. we should put such tradeo tfs on a rational basis 10 balance gain. noise. power. and input match in a controlled manner. 11.4 POW ER · CONSTRAINED NOI SE OPTIMIZATION To develop the desired noise op timization tech nique. we must express noise fi gure in a way that lakes power consumption explicitly into acco unt. Given a specifi ed bound o n power consumption. the method should then yield the optimum device that minimizes noise. Although the detailed derivations are so mewhat complex. the end results are remarkably simple. Readers interested primarily in applying the method arc invited to skip to the end of this sectio n. We start with the general express ion for noise figure as given by classical noise theory: R (34) F = f~nin + G: 1(G ., - G"l't) 2 + ( H.< - H"I'I)2j. The goal here is ultimately to refo rm ulat e the expression for noise figure in terms of power co nsumption. Once we derive such an equation . we' II minimi ze it subject to the constraint of fixed power and then solve for the width o f the transistor that COITtspends to this o ptimum conduron. To simplify the development . let us assume that the source susceptance 8 , is chosen sufficiently close to 8 '111 that we Illay neg lect the d ifference between the two. We MRecallthat gale overdrive is defined as ( Vl •

-

~', ) . the gall:' VOIt,lgC in excess

nfthe thres hold.

11.4 POWE R-CO NSTRAINED N O ISE O PTIMI ZATIO N

285

will justify this step formall y at a later tirne. Given this ass um ption. the expression for noise figure red uces to (35 )

Next. ream•rnge the express ion for Gopt {Eqn. 2 1) to define a parameter with the dimensions o f" a qu ality factor. Th is maneuver will help reduce clu tter in the eq uations to come:

G,opl

--

w C~ .

~ a

s (I _ lei')

~

_

5y

Q"",.

(36)

To accommoda te the possibili ty o f operation with source co nductances other than Gape, we 011",,) define a similar Q in whic h GO{'( is repl aced by G•• the actua l source conducta nce : (37)

Now re-ex press Eqn. 35 using Eqn . 2 1. Eqn. 22. and the noise parameters of Table

11.1 :

(3K)

The para meters a , 1.:"" Q"pI , and Q. in Eqn. 38 are linked to power d issipation. We need to make the lin kage explici t, however, and rewrite those term s direct ly in terms of power. To do so , first recall that a simple ex pression for the dra in current is ( 39)

which may be rewritten as

p'

/0 = WL C,, ~ V "'lI E "'!I -- '

I+ p

(4 0)

where (4 1)

and ~

V'"

(4 2)

LE<>a1

Given Eqn. 40, the power d issipatio n can be written as follows: (4 3 )

'86

CHAPTER. 11 l NA DESIGN

Furthermore, the transconductance 8", can be found by d ifferentiati ng Eqn . 40. After a lillie rcnrrungcm cnt . this muy be expressed as

X'" =

[ I+ P/2 ][ ( I +p)2

IV] [

I.l. " C<>~ LVoJ

=

lX

IV]

Jl nCoxt:V,I<1

= a gJU·

(.wI

Ano ther of the param ete rs of Eqn. 38 linked 10 po we r is Q•. Recall th ai Q, is a function of e fl S' which in tum is a functio n of device width . Equation 40 may be sol ved fur lV, and the resulting express ion sub..utut ed into the equation for QJ' whh the fo llowi ng res ult:

(451 w here

3 V.~D~D~'~'~::..:£~~ :: Po =-2 wR.

(46)

With the a id of these e xpres sio ns. the noise figu re (Eqn. 35) ca n be written in term.. o f p and P D ' '! Minimi zing the resulting equation is co mplex enough that it is

best solved graphically in the general case if an exact answer is desired. More insight. h OWC\'C f . is provided by an approxima tio n that hold s if p « I. Fort unately. that inequali ty fails to hold o nly in high-power circuits . a reg ime in wh ich we art uninte rested . Assuming o the r than high-po wer o pera tio n. the n. the minimum noise figu re occurs whe n

(47)

S ubstitut ing Eqn. 47 into Eqn. 45 yie lds the value til' Q. that leads to the powerco nstrai ned min imum noi se figure: (4R)

Ca refu l insp ect ion of Bqn . 48 reveal s that it is q uite inse nsitive to the particular value of the co rrela tion coe fficient, and se nsitive on ly to the rati o of 8 to y. As sugges ted ea rlier . a ltho ug h the indi vidual va lues of /) and y m ay c hange due to hoi carrier effects. the ir ratio m ay vary muc h less. Hence, the numerical value of 4 for Q"" given in Bqn. 48 ma y he reasonably invariant. A mo re exac t anal ysis reveals that the optimum value is typical ly cl oser to 4 .5. hUI th e achic vnhle no ise figure is relative ly inse nsitive to values of Q ,

~

D. K. ShaclTcr and T. II . Lee. "A I.5V. I.5GHz C MOS l ow Noise Amplifier." IEEE J. SolidS/art Cin·uitJ. May 1997. Unfortunately. the result is a ratio o f !>illh-urdcr polynom ials.

II ."

POWE R· CO N STRAIN ED N OI SE OPTI MIZATIO N

287

Table 11 .3. Estimated Fm,n {II/(I Fm,n I' ( y = 2. 8 = 4. a = 0 .85) wr/ w

F",," (dB )

20

0.5

1.1

15 10

lUi

1.4 1.9

F""nf' (dB )

0.9 1.6

5

3.3

0.1 dB or less over this range.!" Lo wer values lead 10 circ uits thai are less sensitive to parameter variations. ..... bile less die area is consumed by transistors correspo nding to higher values of Q JP ' Once Q , p has been determined. it is a simple matter to provide. at last. an expression for the widt h of the optimu m device : (491

Equation 49 assumes a Q.
Fmin/,

~ I + 2.4 !::. [~]. "

(50)

WT

It is instructive 10 compare Eqn. 50 with the absolute minimum possible. give n hy Eqn. 22. to see precisely how much noise figure degradat ion must be tolerated in exchange for the reduced power consumption: 15 11

Comparisons o f F m,n and F min p are shown in Table 11.3. We see that. ti t a fixed normalizedfre quency, the d ifference is typically between 0.5 d B and I d B. It would appear. then. that excellent noise fi gures are still possible . 10 Shae ffe r and Le e, np. cit.

288

CHAPTE R 11

l NA DESIGN

Thi s part icu lar noise optimizatio n method is attr act ive beca use it balances all paramete rs of intere st. An exce llent match is guaranteed by the ind uctive source degeneration techniq ue. while providing nearly the bes t noi se figure po ssible in a given tech nolo gy with a specified power di ssipation.'! Th e resona nt cond ition a t the input also assures good gain OIl the sa me lime. since the effective stage transco nductance is proportional to wT/w and the noise figu re involves a ratio of the same quantities. li enee. bot h improve s imul taneously as W T increases. The design procedure im plied by the develo pments in this section is fortunately con siderably simple r to app ly than to deriv e. First . usc Eqn . 49 10 determ ine the neeess a ry de vice w idth , Th en . bias the device w ith the a mount of cu rre nt allowed by the power co nstrai nt. Next . select the value of source degene rati ng indu c ta nce to pro vide the desired inpu t matc h. usin g the value of Wr that corres ponds to the bias conditions. Th e expect ed noise figure can then he computed from Eq n. 50. Finally. add sufficie nt induct ance in series with the gate 10 brin g the inpu t loop into resonance at the desi red.opera ting frequency, comple ting the des ign. Having outli ned the basic procedure. we now exa mine some illustrat ive exa mples to high light addi tio nal design co ns ide ratio ns.

11 . 5 DESIGN EXAMPL ES 11.5 .1 SIN GLE · EN DED lN A T he basic input circuit has already been desc ribed . so to comple te the design largely requires only the add itio n of bias a nd output ci rcui try. For narrowband a pplications. it is advantageo us to tune out the output capac ita nce to incre ase ga in. He nce. a typical sing le- ended LNA might appear as show n in Figure 11.7. 12 C ascoding tra nsistor M 2 is used to red uce the interaction o f the tune d output with the tuned inp ut . and to reduce the e ffect of M I'S NJ • Th e total node ca pacitance a t the druin of M2 reso nates w ith induct ance L J bo th 10 inc rease gain a t the center freque ncy and simulta neously 10 pro vide a n add itional level o f highly desirable band pass filte ring. The input and outpu t resonances a rc commonl y (a nd arbi trarily) set eq ual to eac h othe r but . if desired. ca n he offset from eac h othe r to yield a flutter and broader respon se. Transistor M .l essen tially forms a c urrent mirror with M I . a nd its width is so me small fractio n of Ml 's w idth to minimi ze the powe r overhead (If the bias circuit. The

e

I

I

II Tht- q ualrher " nearly" rella :h lhe factlhal lhe c eretanon susceptance is a bn larger jhan wC•• t-ee Table t I , l l. hut the difference is le....s tha n 25'l- , One might compromise betwee n opIimum gain and optimum noi se and cb oo-e a total ind octarw;.-e that is ahnut 10000 Iarger than for opt imum gain. so that the sysle m's nn mm al reson ant freq uency ts about 5'l below the operating freq umcy. These num bers are ge ncrahy smalle r man com pooe m tole rance s, and thus may be more of an 3I;ade rnic than a practic al iss ue. 12 Tluv example is ada pted Immthe pape r by Shadfcr and Lee (o p. cu.j.

289

11 .5 DESIGN EXAMPLES

1-,_.

vour

R It IAS

Rs

ell

Lit

YIN -Y#---j ~L.,~-~1

FIGURE 11 .7. Single·ended lNA .

current through .\1.\ is. se t by the supply voltage and R n:f in co njunction with the ~ J of J'.1j . The resistor R IlIAS is chosen large enough that ils equivalent noise current is small enough 10 he ignored . In a 50-Q sys tem, val ues of several hundred ohms 10 a kilohm or so arc adequate. Here. Jet us select a value of 2 kQ . To co mplete the biasing, DC bloc king capacitor C 11 mus t he present 10 prevent upsetting the gate-to-source bias of M r, The value of C 11 is chosen 10 have a negligible reacta nce at the signal frequency, and is so metimes impleme nted as an off-chip component. depe nding on the value requ ired and die area co nstraints. let us determine co mpo nent values and device sizes assu ming operatio n at JU Grp s. a 50-0 source resistance . and a 5-mA bias curre nt for "" r- Assume also tha t. for the O.5-lim technology co ns ide red here. L d f is 0 .35 li m and C 0 1 is 3.8 mF/ m 2. 1j Given this information. we may ca lculate the optimum widt h o f the main input transistor All as ap proximately 50U JHll . At a bias current of 5 mA , (J)T i... abou t 35 Grps for a transistor of this s ize in this techno logy, and the factor a is fou nd to he 0.85. Assum ing that y is 2, we may read ily compute the minimum noise tlgu rc for this amplifier from Eqn . 5 1 as approx imately 2.2 dB . Let us assume for 11I1W that this value is accept abl e. If it were not . we would have no reco urse other than II I incrca...e power in ord er to increase W T . Havi ng computed ia r , we next find the value of the source degenera tio n induct ance. I ogenerate a real pan o f 50 Q , L s Ill U:.1 be a pproximately I A nU (neglecting ClIJ 's effeci on impedance ), a value that may be realize d as either a bondwire o f about l .a-mm length or as an on -chip planar spiral. G iven the difficulty of red ucing parusiric induelances below value s o f thai order anyway, the cho ice o f a bondwire seems reasonable .

u These numbers are typical for O.5-IUII h lruwnJ technologie s.

...

CHA PTE R II

LNA DESIG N

1.5V

I kU

FIGURE 11 . 8 . Complete 1.5-GHz, 8·mW

single·ended LNA.

e

In ord er to compute L g • we need 10 know g •• which . for our chosen device. is approximatel y 0 .67 pF. Resonating this capacitan ce at 10 Grp s requires a total inductance o f j UM under 15 ni l. so that L g mu st he appro ximately 13.6 nj l , which is somew hat large. Not only would a planar spiral of this value cons ume more die area than would probably be consiste nt with an economical de sign. its lossincss would also degrade the noi se figure significantly. li enee a co mpromise is pruden t: realize o nly part of the total with a planar spiral. and the rest wit h a bondwire. Alternat ively, use a bo ndw irc and an external inductor , the latter realiz ed as PC board interconnect, perhaps. If the packaging is ame nable . excellent repeatability is possible because of the relative d imensional stability and reprod ucibil ity of a PC hoa rd trace. Computation of the outp ut tuning inductance value requ ires knowledge of the total ca pacitive loading. Suppose that. for this example. a 7-nH indu ctor resonat es with that capacitanc e at our IO-G rps operating freq uency. Th e required inductance is small eno ugh that a planar spiral inductor is a rea sona ble choice here . Co mpleti ng the design req uires specification of the DC blockin g capacitor. which we will select as In pE T his value changes the effective series capaci tance from 0.67 pF to O.b3 pF, and thu s shifts the resonant freq uency by about 3%. Th is small deviation is usually to lerab le. give n that the uncertainti es in co mpo nent values and model parameters arc frequ ently much larger . The finished LNA then appears a!\ show n ill Figu re 11.8. The bias transistor 's width and curre nt are arb itrari ly chose n as a tenth of that of the ma in tran sistor . Even though a simple reference current so urce is shown. it may be advantageou s to use a co nstant -g; bias source to stabilize gain and input imped ance over tempe rurure and supply (sec Chapter 9) .

11 . 5

291

DESIG N EXAM PLE S

The cascodi ng transistor is chose n here to have the same width as the main device. This choice is co mmon. but it is so mewhat arbitrary and thus not necessari ly ideal, Two competing co nsiderations co nstrain the size of the cascodi ng transistor. Th e gate- drain overlap capacitance ca n reduce the impedance loo king into the ga te and drain of M l considerably. degrad ing bo th the noise perfo rma nce and input match. It is a straightforward matter to show that . for equal-sized co mmon-source and cuscod ing devices. the resistive co mponent at the inp ut is given by (52)

The degenerating inductance must therefore be increase d

compensate. To suppress these co nseq uences of the M iller effect , one would nonnally desire a relatively large cascodi ng device in order to redu ce the ga in of the co mmon-source transistor. However. the parasitic source ca pacitance assoc iated with a large device effectively incre ases the amplification of the casc cdi ng device' s own internal noise at high freq uencies. Merging the source region of the cascoding transistor with the drain region of the common-source device is effective in reducing ma ny of these problems. and is most readi ly imp lem ented by making the two devices eq ual. One should also con sider the possibility of pulling ex tra current out of the source of M1 to increase selec tively the bias curre nt of just the ca scodi ng device 10 increase its g",. Measurem ents made on ac tual amplifiers essentially identical to this example typically revea l noise figures somewhat in excess of the calculated minimu m o f 2.2 dB , owing to various parasitic eff ects. To minimize the noise figure degradation. it is particularly important to ensure that the resistive part of the input impeda nce is big enough by guaranteeing a suffici ently large degeneration inducta nce and by mitigating the Mill er effect. Failure 10 do so elevates the noise figure - not only becau se of the deviation from Gup! ' but abo because it exace rbates the effect of any noise due to loss in the gate inducta nce. As a consequence. it is critical to keep the ga le inductor ' s Q as high as possible. and it is also advi sable to use a somewhat larger source inductance than the theoretical value. since the noise consequences of excessive inductance are much less serious than for insuffi cient ind uctance. Following this advice can easily keep NF degradation below 0.5 d lt , whereas ignoring it ca n j ust as easily increase noise f igure by over 1.5 d B. Another po ten tial source of NF degradation is "e pi noise: ' the thermal no ise of the effective resistance betwee n the bod y of the transistor and ground. T he equivalent noise voltage of that resistance mod ulates the back gale. producing a mean-square drain noise current component whose value is given by 10

(53)

292

CHAPTER 11

lNA DESIGN

Compa ring this e xtrinsic noise contribution with the drain noise caused by the thermal no ise of N. mod ulating the top gate. it is d ea r thai we must satis fy the following inequ alit y:

,

'

g..." R",t> «g;" R•.

(54 )

Bec ause uncorrelar ed noise adds in root -sum-squared fashion . a 10% increase occ urs whe n the e pi noise power is about 20% as large as thaI du e 10 N•. Interpreting this 1: .5 ratio as the mean ing of "much less than: ' the inequ ality becomes (55)

To develop a fC'C'1 for how much of a practical problem there is in satisfying Eqn. 55. ass ume pessimis tically thai the ratio of to p-ga te to bad -gale transcond uct ance is only ..liD. The ineq uality then red uces 10 the foll o....-i ng req uiremen t: (56)

It is ge nerally nol difficult 10 satisfy this inequ ality comfortably by surrou ndi ng the tran sistor with man y substrate co ntac ts. In extre me cases. howe ver. it may be necessary to break up the main device into se veral subsections. eac h of whic h is surro unded with a large quantity of substrate tap s to guara ntee a negli gible contribution by epi

norse. 11.5 .2

DIFFER EN TIA L LN A

T he single- e nded LNA a rchitec ture has a t least on e important shortcoming. a nd thai is its se nsitivity to parasitic ground induc ta nce . It is d ear from the sc he matic of Figure 11.8 thut the ground re turn of the signal source is supposed to be at the same pot e ntia l a.s the botto m of the source dege nerat ing inductor, Howe ver. there is inevita bly a differe nce in these po tent ials beca use the re is always som e non zero impeda nce betwee n the two po ints. Since 1.4 nil is not a lot of induc tance. small a mounts of additio nal parasit ic reactan ce bet ween the two gro unds ca n have a large effect on a mplifier perfo rmance. It can e ven allow signal-depende nt curre nts from subsequent stages tomodulate "g round," formi ng a parasitic feed bac k loop that destabilizes the amplifier. Dedicating addi tional ground pin s or using sophisticated pack ages ca n mitigate this pr oblem . hut such mea sur es are eithe r o f limited effec tive ness or so mew hat costly. An alternative is to e xplo it the increm ental ground loc ated a t the symmc rncat point of a differe ntial structure. Sec Figure 11.9. where the source dege nerating indu ctances ret urn to a virtual ground (for diffe rential sig nals). Any parasitic reacta nce in series w ith the bias c urre nt source is largely irre leva nt. since a c urre nt source in se ries with an impedance is still a c urre nt source. li enee. the rea l part of the inp ut imped ance is

4

T I

11.5 DESIGN EXAMPLES

293

I I

I

FIGURE 11.9. Differential lNA l ~implifi edl .

controlled only by L . and is unaffect ed by para sincs in the current source's, ground return path . Anot her importum attribute of the differential co nnec tion is. of C()U~. ifs ab ilit y 10 reject com mon- mode disturbances.':' Th is consideration is part icularly importan t in milled-signa l upplicutions. where bot h the supply and substrate voltages m:lY he noisy. To maximize common-mode rejection at high frequenci es, if is critically importanr for the luyou t to be absolutely as symmetrica l as possible. II is importanr to kee p in mind Ihal - for equ al total powerco nsumption - the noise fi gure oft his amp lifier is higher tha n its single -ended co unterpart. Spcciticully. rbe power consumed is twice thai o f a sing le-ended amplifie r to achiev e the sume noise figure. Offsetting this disad vantage is the improved linearity thai attend s divid ing the input voltag e between two devices. Hence, the dynami c ran ge increa ses even though the noise figu re remains constant (aga in, assuming twi ce the power consumption). The circuit of Figure 11.9 ignores biasing details - in pa rticular, it doesn' t show how the DC ga te po tentia ls are established . Furthermore, the gate bias for the cascoding transistors is the supply voltage, and such a cho ice may impress enough volta ge across the bonom transi stors to increa se the values of y and ~ at higher supply vo ltages, To mitigate such hot -el ectron no ise degrada tion , it is prudent to usc only eno ugh drain bias 10 keep tra nsistors co mfortably ou t of the triode region, and no more. One of many possible met hods for achiev ing the desi red resu lts is shown in Figu re 11.10. In this ci rcuit . tra nsistors M 1 through M4 are the I.NA core of the previou s, 14 To ensu re the truth til"this ...tate men r, the spiral inductors.o f btllh hah-circ uns shtluld be laid out with lhe same onenta tjon , not murore d: oth erwi se , ind uctivel y cou pled ntliSl.' wi ll not appear a.' a common -mode signal.

CHAPTER 11

'"

LNA DESIGN

I.SV

LIC~ FIGURE 11 .10. Complete 12-mW, 1.5-GHz differential LNA .

sim pl ified sc hema tic . Th e o utput of thi s first stage is AC-co uplcd throug h Cc 10 M1 and M M. whic h provide additio nal gain. To save power . the CUITe n t th roug h the ~'C. o nd ga in stage is reused 10 supply the four co re tran sisto rs. He nce. the o utput tuning indu ctors L d re turn to the co m mo n so urce co nnectio n o f M 1 and M H ra ther than to the posi tive supply. To keep a ll of the transisto rs in the ...tack in saturatio n with a low supply voltage. a common-mod e bias feedbac k loop se ts the drain voltage of M l and M l eq ual to a fixed traction of VR"I. To the ext en t that Vd
11.6

LINEARITY AN D LARG E-SIGN A L PE RFO RM AN CE

295

(5 7 )

where the voltages arc referen ced to the common source co nnection o f the input

pair. Transistor M'l guarantees sta rt-up of the bias loop by provid ing some default ga te voltage for the cuscod ing tran sistor s until the op-amp has a chance to act . Finally. because (If the mode st performan ce req uired of the up-a mp. negligihly low bias currents may be used there. 11.6

LINEARITY AND LARGE· SIGNAL PERFO RM A N CE

In addition to noi se tigure. gain, and input match , lineari ty is an important consideration because an LNA must do more than simply ampl ify signal s witho ut adding much noi se. It must also remain linear eve n when stro ng signals arc being received . In particular. the LNA mu st main tain linear operation when receivi ng a weak signal in the presen ce of a stro ng interfering one; otherwise. a variety of pathologie s may resu lt. These conseq uences ofintermodulat ion distortion inclu de desensit izati on (abo known as block ing) and cross-mod ulation. Block ing occurs whe n the ir ucrmodulation products cau sed by the strong interferer swamp out the desired wea k signal. wherea s cross- mod ulatio n results whe n nonlinear interar.-tion transfers the modulation of one signal to the carrier of another . Both effec ts arc undesirable . o f course. so another respon sibi lity of the LNA designer is to miti gate these problems to the maximum practical ex tent. The LNA design proc edure described in this chapter doe s not address linearity directly. so we nnw de velop som e methods for evaluating the large-signal performance of amplifi ers. with a focus on the acquisition of design insight. As we ' ll sec. althoug h the narrowband LNA topol ogy achieves its good noise perfo r mance somewhat at the expe nse of linearity, the tradeoff is not serious enough to prevent the realizat ion of LNAs with more than enough dynam ic ran ge til satisfy demanding applications. Although there arc many mea sures of linearity, the most co mmonly used arc third order intercept (11'3) and I-d B co mpress ion point ( PI dB). III To relat e these measures to readily calc ulated ci rcuit and device param eters. suppose that the amplifier's out put signal may he represented by a po wer series, 17 Furtherm ore. assume that we will evaluate these measures with signals sma ll enoug h that trunca ting the series after the cubic term Introduces negligibl e error :

If> In direct.c onccrsjon (hnmod yne) receivers. the second- orde r intercept is more important. L1 We are ulso assu ming lhal in put and out put are related through an anh ystcrenc ( mc mlJry lcs~) process . A more accurate method would em ploy Volterra w ries , for ex ample. but jhe res ulting comptcl:ily obsl;ures m uch of lhe design insigh t we are see king.

I

,.,

CHAPTER 11

/' ( V rx -

l NA DESIGN

+ V ) :::::: Co + (' IV + C2V 1 + C3V ·, ,

(58)

where Eqn . 58 describes the spec if ic case of a tran sconductance . Now consider two sinusoid al inp ut signals of eq ual a mplitude but slightly di ffere nt freq ue ncies: (59)

Subst ituting Eq n. 59 into Eq n. 58 allows us. after sim plifica tion a nd collec tion of terms, to iden tify the compone nts of the output spec trum." Th e DC and fund amental co mpo nents are as fo llow...: (60)

Noll' thai the quad ratic factor in the ex pansion contributes a DC term thai adds to the ou tput bias. T he cubic fac tor uugme nts the fu nda me ntal term , bUI by a factor proportion al 10 the c ube of the am plitude . and thu s contributes more tha n a s imple increase in gain. In general. DC shifts com e (mill eve n power s in the series expa nsion, while fund am en talte rm s come from odd factors. There are also sec ond and third harm onic terms. ca used by the qu adratic and cubic fact ors in the se ries ex pans ion. respectively : (61)

In ge neral. nth harmonics come from nt h-o rder fac tors. Har monic di stortion products. bein g of much highe r frequ e ncies than the funda mental. are usuall y attenuated enough in tuned a mp lifier s so that othe r nonlinear produ cts dom inat e. Th e quad ratic term also contributes a second-orde r intcrmodulation ( 1M) produ ct. as in a mixer ( M.'C C ha pter 12): (62)

As with the ha rmon ic d istor tion produc ts. these SUIll and difference frequ ency ter ms are effect ively atte nuated in narrowba nd a mplifiers if W I a nd W 2 a re near ly equal. as assumed here. Finally. the cubic term gives rise 10 third-o rder inrcrmodulatio n product s: ( ~ C ~ A 3 )[ COS (W I

+ 2(2)( + COS(W I -

2( 2) (

+ cos(2co l + CO 2)! + cos(lco l -

( 2)1 ).

(63 )

18 This de rivatio n mal e" considerable usc o f the fnllnw ing lrigllnomclric iJ cntily: rcos .e )(Cl)'; y ) = [cu s f .r

+ y ) + cn' (.l

- )')J/ 2,

11.6 LIN EA RITY AND LA RG E· SIGN AL PE RFORM A NC E

297

Note that these prod ucts grow as the cube of the drive am plit ude . In general, the amplitude of an nt h-order J~l produ ct is proporti on al to the 11 th power of the drive amplitude. The sum frequen cy third-order 1M term s are of diminished importance in tuned amplifiers because they typicall y lie far enough out of hand to be significantly unen uated. The diffe rence frequ ency components. however. can be qui te troubl esome. since their freq uenci es may lie in hand if WI and Wz differ by on ly a small amoun t (as would be the case of a signal and an adjace nt channel interferer. for example) . It is for this reason thai the third-ord er intercept is all important measure Ill' linearity. II is struighrforwurd from the foregoin g seq uence of equatio ns to co mpute the input-referred third-o rder intercept po int ( II P3) by setting the amplitude o f the 1M3 products equa l to the amplitude o f the fundame ntal: ( 6-1)

v. here we have ass umed only a weak departure from linearit y in ex pressing the Iundamental output amplitude . It is important to emphasize thatthe intercept is an extrapolated value becau se the correspo nd ing amp litudes co mputed from Eqn . 64 arc almost always so large that truncating the series after the third -order term introduces significant error. In both simulations and experi ment. the interc ep t is evaluated by extrapolating trends observed with relatively small amplitude inputs. Since Eqn . 64 yields the square of the voltage amplitude. dividin g by twice the input resistance R, gives us the power at which the ex trapo lated equalit y of IMJ and fundamental terms occ urs:

II P J =

~3 I~ I -I C3 N,

.

(65 )

Figure 11.11 summarizes the linearity definitions. It is cu stomary to plo t the ou tput powers as a function of the power of eac h of the two (eq ual-am plitude) input tone s. rather than their sum. Since third-order products grow as the cube of the drive amplitude . they have a slope that is three times that of the first-ord er output when plotted on logarithmic scales. as in Fig ure l l.ll. Note that . in the figure. the I-d B compress ion point occurs at a lower input power than II P3. Thi s genera l relationship is nearly alway s the case (by a healthy margin) in practical amplifiers. Having de fined the lineari ty measure s. we now conside r ways to es timate IIP J with and withoutt he aid of Eqn. 65 .

11 .6 .1 METHOD S FOR ESTIM ATIN G IP3 One way to find IP3 is through a tran.. . lent simulation in wh ich two sinuso idal input signals of equa l amplitude and near ly eq ua l frequency drive the amp lifier. The

CHAPTER 11

298

Ou tput Power (dB)

f (-

' Id " ....

Output (dB

Comr-:essionr nml

lNA DESI GN

Fir st-ord er Out put

'"

ThiN ·Onl<' bnercept

, ..." ! -: , t " ,// ~Thjrd-OnJcr 1M term .. ....

I IdB

Input

li P]

PO'M/er (d B )

FIGURE 11.11. Illustrat ion of LNA performance pcrcmeters.

third -o rder intermodulation products of the outpu t spectru m are compared with the fundamental term as the input amplitude varies and the intercept is computed. Altho ugh simple in prin cip le. there are several significant practical difficulties with the metho d . First, since the distortion products may he seve ral ord ers of magnitude smaller than the fundamen tal term s. numerica l noise of the simulator ca n eas ily duminare the output unless exceptio nally tight tole rances are imposed.! " A closely related con ...ideration is thai the time steps mu st be eouallv spaced and small enough not to introd uce artefacts in the output spect ru m." When these co nditio ns arc satisfied. the simulations typically exec ute quite slowly and gene rate large output files. Pure frequ ency-dom ain simulators (e.g.. harmon ic balance too ls) can compute IP3 in much less time. hut arc currently less widely ava ilable tha n time -domain simulators such as S PICE . Equati on 65 offers a simple ex pressio n for the third -ord er intercept in term s ur the ratio of two of the power series coe fficients, and thu s sugge sts a ll altcrn ative method tha t might he suitable for hand calc ulations. O ne is rarel y given these coefficie nts directly. but it is a straig btforward maner 10 determine them if an analytical expression for the transfer characteri stic is availabl e. Even without such an expression. there is an ex tremel y simple procedu re, ea si ly imp lemen ted in "ord inary" simulators suc h as SPICE . that allows rapid estimation of 11'3. Th is technique, which we'll ca ll the three-point 1I1l'1/wd, exploits the fact thai knowi ng the incre mental gain at

1'1 Tole rances must be much tighter, in fact , than the "accu rate" default op tion... commonly ottered 20 Thi s requirement s h;~ m~ from the a~..umpnon , made b)' all FFT algorithms used to)' practical Slm· uhuors . thai the time sample... arc uniformly spaced.

11 .6

LINEARITY AN D LARGE -SIGNA L PER FORMA N CE

299

three different input amplitudes is sufficient to determi ne the three coe fficients ( ' 1. ca. and Cl . 21 To derive the three -point method. Mart with the series ex pansion that relates input and output: (66)

Tbc increme ntal gai n [transconduct anc e) is the derivative of Eqn. 58:

(67) Although any three different values of l' would suffice in princip le. part icularly convenient ones are O. V , and - V. where these voltages are interpreted as deviations from the OC bias value. With those choices. one obtains the following expressions for the corresponding incremen tal gain s: (68)

g(O) :::::: CI.

g( V) :::::: CI

+ 2C2V + 3q V 2•

(69)

(70)

g ( - V ) ::::l CI - 2c2V + 3c l V 2.

Solving for the coe fficie nts yield.. . c\

C2 CJ

=

=

= g (O),

(7 1)

g( V) - g( - V )

(72 )

4V

g ( V ) +g ( - V) -2g (0 )

(13 )

6V'

Substituting into Bqn. 65 these last three equatio ns for the coe fficients then gives u.. . the desired expression for II P3 in term s of the three incremental g a i n.~ : 22

l i P) = 4V ' R.,

_I

g (O) g ( V)+ g( - V ) - 2g (0 )

I_

(74 )

Finding II P3 with Bqn. 74 is much faster than through a transient simulation hccause determining the incremental gains involves such lillie computation for either a simulator or a human . The three -point met hod is thus particularly valuable for rapidly estimating II P3 in the carly stages of a design.

11 Thi ~ mc.'l hud i ~ lin adaplmion of a classic tech nique from the vacuum luhe era that allow s esnm alion of hdrnlllllic distortion . 12 fla\'ing determined all of the coe fficiems in term s of readil y mea sured ga ins, it is easy ttl derive similar cJl [lre~ion s fur harmunic and seco nd-oroer 1M di...IOI1ion. The I..ue r quantity is c!>Jllocially relevam for duecr-convers lo n rece ivers .

.. CHAPTE R 11

300

LNA DESIGN

Table 11.4. Approximate

ttr J (d llm ) versus

v,,,, lIud Q. for single -ended LNA ex{/ ml'/~'

v.. U.1 U.2 0 .) (U

0.5 I.U 2.1 3.U

II P.\ @ Q. = I

IIPJ @ Q. = 2

II P3 @

II P) fliJ

li P)

Q. = 3

Q. = ..

Q. = 5

In.n 17.2 17.8 18A 18.9 21

10.6 11.2 11.8 12..1 12.9 15 19 21

7.1 7.7 8.3 8.9 9A 115 15.5 17.5

"

27

(d>

". 6 ' .2 ' .8

2.6 .1.2 3.8

6A

"A

6 .9 9 13

" .9 7 11 13

"

Having derived the three -point met hod . we now appl y it to a n a pproximate analyticalrnodcl for the tra nscondu cta nce of a sho rt- cha nnel MO SFET . We the n use that result to estimate the I IP3 of the na rro wband LNA .

11.6.2

LNA LINEA RIT Y W ITH SHO RT-CHA N N EL MO SFET

An appro ximate a nalytical expression for the tran sco ndu c tance of a sho rt-channel MO SF h'" i!> give n by Eq n. 39 (ne g lect ing norma l fiel d mobility deg radatio n):

Km =

][ IV ] p) 2 11 " C"~ L V.od ' [(11++ p/2

whe re p =

VIIJ

-

V,

L E'~l

= -

V....-t -

(751 (761

i e:

Beca use ve loci ty saturation ca uses the tran sconductan ce to approach a co nstant value the overdrive incr eas e s, the designer ca n im prov e I IPJ hy increa sing the overdrive vo ltage if the inp ut sig na l is mea sur ed acro ss the gate and so urce termin als.

:I S

In the narrowhand LNA architecture, the input voltage is multiplied hy the Q of the input ci rcuit be for e appea ring be twe e n gate a nd so urce. li enee , the third-order intercep t (refe rred to the amp lifier input) in suc h a case is reduced by a factor of Q2, Thus, to r the narro whand LNA . the fol lo wing expressio n fill" I IPJ hold s :

4V' liP }

=

I

Q ; R, . g ( V )

g (O)

+ g(

V)

(77 )

To see a ppro xim ately what interc epts mi gh t be practically ac hie vabl e. assume that R , is 50 Q . thut E..... is 4 X 106 Vi m. and thut Q. ranges from I 10 5 in a process tech nolog y fo r w hic h L eff i!'> 0 .35 u rn. Using those numbe rs in Eq n. 39 a nd Eqn. 77 3110 ws us 10 ge ne rate a table o f (ve ry) a ppro xlmatc J I P3 va lue s (see Table 1104 ).

11 .6

LIN EA RITY AN D LA RG E-SIGN AL PE RFORMA NCE

301

Because o f the cascade (If assumptions used, the values in Table I 1.4 must he considered highly app roxirnate. In particular. neglect of'ncar-threshold effects in Eqn. 39 causes a significant overestimate of II P3 at low overdrives, and neglect o f vertical field mobility reductio n causes a moderate underestimate at high overdrives. It is importanr ro emphasize that these errors are a result of using an approx imate analytica l device model. and do nor reflect a fundamental limitation of the three -point method nselr. Even so. practical single -stage LNAs in which the transistor 's Lrn is 0 .35 Jim exhibit II P3 values that are frequently within 1- 3 dB of the numbers shown. It is also worth noting that d ifferential amplifiers will have 3-dB- beller IIP3 than shown in the table bec ause such amplifiers divide the total input signa l eq ually betwee n two input transistors, all other factors held constant. It is clear fm m Table I I A that ope ration with a Q. greater than unity degrad es IIP J , so that power-optlmizc d. single- ended LNAs typica lly exhibit IIP3 values within a few decibels o f abo ut 5 dam. Those values app ly to single -stage LNAs only, howevet, because the linearity of a multistage am plifier is generally limited by other than the first stage. Hence . the numbers tabulated represe nt limiting values that multistage designs ca n only approach . Seco nd-stage linearity limitatio ns can easily reduce li P) by 5- 10 dB.

Because linearity is so tightly co upled to the overdrive voltage. improvements in IIP3 generally co me at the expense of e ither increased powe r (i f Q. is fi xed ) or reduced gain (i f Q. is reduced or if feedback is used ). If nne is willing to tolera te increased po wer dissi pation, exce ptio nally high 11 1'3 values may be ac hieved. As a specific example. note that a IO-1 5·dBm II P3 is possible for ove rdrives between 500 mV and I V for Q. in the 2- 4 range. The foregoi ng derivatio ns focu s on II P3, which may be considered a low-level measure of linearity because it is meas ured or computed with signal am plitudes small enough to cause insignificant departures from linear operation; the intercept is an extrapotated value. The other commonly used linearity measure, the I-d B comp ression point. chaructcrizes a higb-lcvellimitauon because it descr ibes when the /I( '1/1ll1 out. put power is 26/,*, below a linear extrapolation of small-signal beha vior. The output I-dB compression point is primaril y a function of the total bias current and availablc supply voltage. und is thus roughly inde pendent of channel length if those quantities are held constant . As channel lengths decrease, however , the small-signal linear ity of devices improves owing to the increasing prominence of velocity saturation. so IIP3 increases ;IS devices scale. Because these two measures thus characterize linearity in two distinct regimes, there is no fixed re lationship between them. Nevertheless, one ca n use the cubic power series approximatio n and analytic device models to estimate that IIP3 is typicallyaboul 10- 15 dB beyond the I-d B compression point for current h..-chnolog ies. One can expec t the differences between the two measu res to increase beyond that range as channel lengths shrink.

302

CHAPTER II

l NA DESIGN

11 .7 SPURIOUS·FREE DYNAMIC RANGE So far. we have identified two ge neral limits on allowable input signal am plitu des. The no ise figu re defin es a lower bo und, while distorti on sets an upper bou nd. Loo sely speaking. the n. amplifier s can accommodate signals rangin g fro m the noise floor to some linea rity limit. Using a dynam ic range measure helps designers a void the pitfall of improving one parameter (e.g.• noi se figure ) while inad vert entl y destroyi ng another. Thi s idea has been put on a quan titati ve basis throu gh a par ame ter known a.. the spurious- free dynamic ra nge (SF DR). Th e tenn "spurious" mean s " und esired: ' and is o ften shortened 10"spur."2.\ In the contex t of LNAs. it usually refers 10 the third -order prod ucts. hut may oc casionally apply to other undesired output spectral compo nents. To understand the rationale behind using SF DR as a specific measure of dynamic range. define as a mor e general measure the lesser of signal-to -noise or signal-todis tortion ratio. and evalua te this measure as on e varies the amplitude of the two tones applied to the amplifier. As the inpu t amplitude increases from zero. the first-order out put initially has a negative signal-to- noise ratio but eventually emerges fmm the no ise floor . Bec au se third-ord er distortion depends on the cube o f the inp ut amplitude. 1M3 prod ucts will he well below the noi se floor at this point for any practical amplifier. li enee. the dynamic ran ge improves for a while as the input signal co ntinues to increase, since the desired output increases while the undesired output ( here, the noi se) stays fixed . Eventually, however , the third -ord er 1M terms also emerge from the noi se 11 00r. Beyond that inp ut level. the dynamic range decreases, since the 1M3 terms gmw three times as fast ( O il a d B basis) as the first-o rder output. The SF DR is defined as the signal-to- noise ratio correspo ndi ng to the input amplitude at which an undesired prod uct ( here. the third-or der 1M power ) j ust eq uals the noise power, and is there fore the maximu m dyna mic range that an ampli fier exhibits in the foregoin g ex periment. Thi s is de picted in Figure 11.12. To incorporate explici tly the noise figu re and IlP 3 in an expression for S FDR. first define N o; as the inp ut-referred noise po wer in deci bels. Then. since the third-order 1M produ cts have a slope of 3 Oil a dB scale, the input power below IIP 3 at which the input-referred 1M3 power equals N" is given by

A PI =

IIP3 - N m 3

(78)

(aga in. all po wers are expressed in dcci bels ). The SF DR is just the d ifference betwee n the input power implied by Eqn. 78 and No;:

2.1 Occasionally (and erroneou sly). "spurii'' is used for the plural. even though "spurious" is nul a Latin word ; "spurs" is the preferred plural.

TI

11. 8

303

SUM M A RY

Om put

Power (d B )

IIP3 •• _

}

•••••• _•••••••• _••••• _._••••••••••••••••• O U1Jl1.l1

nl.li~

fiG URE 11.12 . Spurioudree dynamic range (third·Ofdefl.

SFDR = (IIP3 - !J. P,) - N"i IIP 3 - N,,; 2 = IIP 3 - No, = - (II P3 - N,,,) ;

3

3

(79)

all power quantities a rc again in dec ibe ls. Note that the input -referred noi se pow er (i n warts this lime ) is "imply the noise (actor F times the noise power kTli.f. Note also thai output-referred qu antiti es may be used in Eq n. 79 beca use the same ga in fac tor scales bo th terms. It is satisfy ing that SFD R is ind eed bounded on one e nd by lIP) and ont he ot her by the noise 110m. as argued qualitative ly at the begin ning of this sec tio n. The factor 2/ 3 comes into play because of the part icular way in which the limits are deli ned .

11 .8 SUMMARY We've seen that power- con strain ed noise optimization leads to a particular topo logy and a well-defined device size that is a function only of source resista nce. process technology, and operating frequ ency, Within that spec ified hound on power cons um plion, the procedure yields
• CHAPTE R 11 lNA DESIGN

revea l rea sonably good ag ree me nt with predictions. Reason able agr eem ent ma y gcncrally be expected as long a o; the de vice is opera ted we ll be low W T. If be lief linearity is req uire d. either po wer consum ptio n o r gain mu st deg rade in exc hange for Ihe im proved linea rity. Fo r exa mple . the overd rive voltage ma y be increase d. the inpu t Q decreased . or negat ive feed bac k may be e mployed. Th e theory also a llow s us to concl ude that continued sc a ling of dev ice Sill' S w ill imp ro ve the linearity obtained on a given pow er budget. Finall y. co mbining the sig na l a mplitude limita tio ns imp lied by the noi se and dis. tortion figures of meri t yields a mea su re of the ma ximum dynamic ran ge of an umpl ifier . the spurio us-free d ynamic ra nge.

PROBLEM SET FO R LN A DESIGN PR08 LEM I Let us rev isit the single -e nded LNA with ind uctive so urce dege nerat ion ( Fig ure 11 .13). We wish to e xam ine the effect o f parasitic capacita nce at the drain o f M 1 (and the source of M 2). For this problem. suppose tha t the two devices ha ve the sa me wid th . e ven though this choice may not he optimal.

R BlAS

I{s

" IN

ell

L~

~r--'-'''''''-l l

FIGURE 11.13. Single-eoded LNA.

(a) Su ppo se initi ally that the two transistor s arc la id o ut as physically separate devices. For thi s case, the drain-to -bulk ca pacnance of MI may be similar In magnitude 10 elf' -Calc ulate th e o utput noi se contri bution o f M 2 under these circumsta nces. (b) Now suppose that the two transisto rs ar c laid (Jut suc h that their so urce and dra in reg io ns a re shared . This layou t arrangeme nt reduces the parasitic ca pac itances

305

PRO BLEM SET

by a fac tor o f about 2. Recalculat e the o utput noise co ntribution of Al2 for this case . Compare wit h your answer 10 pan (a). PROB LEM 2 A nother source of noise figure degradation in the circuit of Problem I is the CRJ of M j • To sec how this c apac ita nce ca n adversely a ffec t no ise figure. calculare the impedance presented to the so urce of .'1,12 as a function of R •• How docs this impedance etfect the noise contribution of ,\1 2 as the resistance uecreascs? PROB LEM 3 Most o r the low-no ise amplifiers presented in this chapter are narrowband amplifiers. In some practical situations . a narrowban d ampt ilier stage may be followed by a widehand stage (e .g.• a source follower) . Assume that a narrowband LNA with a noise bandwidth o f 100 MH z precedes a wid eband source foll ower with a noise bandwi dth of 2 GHL. Assuming that the LNA po ssesses a 15-dH voltage gain and a 2-d H noi se figure. calculate the necessary transconductance o f the source follower such that it co ntributes no more than 0.5 dH to the overall no ise figure. PROBLEM 4 Using the approxima te II PJ values in Table 11.4. calc ulate a numerical value for the peak SF DR of the single-e nde d LNA design exa mple. Vou may assume that the linearity is limited by d istorti o n in the input device M I. Hint: You may infer the proper overdrive vol tage from the specification of a given. PROBLE M 5 Cons ider the resistivel y shunted co mmon-so urce "L" NA pic tured in

Figure 11.14 .

",

VO ti T

FtGU RE 11.14 . Common-source o mpl j~er w ith shunt inp ut resistor.

(a) Derive an expression for the no ise figu re (factor ) forthis ampliticr in the ab sence of gale noise. Neglec t drai n- ga te and drainhulk capac itances and cpi noise. (b) Re-derive the no ise factor. now takin g gate noise into accou nt. PROB LEM 6 Consider the shunt- series broadband LNA shown in Figure) 1.1 5.

(a) Derive an express ion for the noise figure (factor) for this ampl ifier in the ab sence of gale noi se. Neglect druin-gare and drain- bulk capacitances and epi noise. (b) Re-dcrivc the noise factor. now taking gure noise into acco unt.

... 306

CHAPTE R 11

LNA DESIG N

R UllI

FIGURE 11.15 . Shunt-series ompl;~er lbiasing not !>hown).

PROBLEM 7

Consider the common-gate broadband LNA (Fi gu re 11.16).

' 01:'

fi GURE 11.16 . Common-gate amp lifier lbiosing not shown ).

(a ) De rive a n ex pre ssio n for the no ise fig ure (factor) for thi s a mplifier in the abse nce of gate no ise. Neglect drain-gale and drain- bulk ca paci tances and c pi noise. (0) Re -deri ve the noise factor. now takin g gute noi se into account.

PROBLEM 8 r ut the ci rcui t of Problem I. derive an expression for the noise factor co ntribution of the bias resistor R BIAS ' Assume that the ga te termina l of the diodeco nnecte d referen ce transistor M3 is terminated in a negligibl y low impedance. PROBLEM 9 Pollowing an ap proac h simila r to th at used 10 estima te liP) , deri ve an expre ssion for the .w'contl-o rder inte rcept II P2, in terms o f coeffic ients of the series ex pa nsio n. for the induct ivel y degenera ted LNA.

PROBL EM 10 Conside r a simple bipolar common-emitter amplifier (Fi g ure 11.17).

,

I 'ocr

"N-(

~

FIGURE 11.17. Bipo lar common -em itter a mpliher.

PRO BLEM SET

307

(a) Deri ve ex press ions for the low-frequ ency equivalent input no ise voltage and current for this amplifie r. Ignore flicker noise. but co nsider base resistance. (b) If this amp lifier is driven by a source whose resistance is R. , deri ve an cx pression for the noise factor . (c) What bias current minim izes the noise facto r for this circuit if no other co mponent s may be used '?

• C H A PTE R TW ELV E

MIX ER S

12. 1 IN TRODUCTION Must circuit analysis proceeds with the assump tion s of linearity and time invariance. Violations Ill' those assumptions, if consi dered at all. an' usually treated as undesirable. However. the high performance of modem co mmunications equipme nt actually depe nds critically on the presence o f at lea:..t one cle ment thai fails to satisfy linear time inva riance. As noted in C hapter I. the superheterodyne I receiver uses a mi ller 10 perform nn impo rtan t freque ncy tran slation o f signals. Armstro ng's invention has been the domina nt arc hitec tu re for 70 years because this frequ ency tran slation selves man y prob le ms in one fell swoo p (see Figure 12.1).2 In this architecture, the mixer tran slates an incoming RF signal 10 a lower Irequen cy;' known as the intennedia le frequ ency (IF ). Although A rmstrong originally soug ht this, frequency lowe ring simply to ma ke it eas ier to obtain the req uisite gain, ot her xignitic an t ad vantages acc rue as we ll. As one example. tun ing is nnw acccmphshed by varying the freq uency of a local oscillator, rat her than by vary ing the ce nter frequ ency of a mult ipol e ba ndpass filter. Thus. instead of adj usting several L C networks in tandem to tune to a desired sig nal. one simply varies a s ingle LC com bin ation to change the fnequ ency of a loc al oscill ator ( 1,0 ). The intermediate frequency stages can then usc fixed bandpass filters. Se lec tivity is therefore determined

I Why "supcrvhetcrodyne? The reason is that I'e sscndc n had alre ady invented something culled the "heterodyne," and Arm stron g had 10 dislingui!>h hi!> invention In un Fessen de n's . 1 Proving ucccss ha.. man y l:llht-n lll,h ilc fai lure is de\'Clnpmcn l vigorously, J Actually , nne may also tranvlare to a hiXhu frequency. hut we will defer a d i scu s.~ i()n of lhat C~ 10 Ch apter 18.

308

12.2 MI XER FUNDA MENTALS

~

~~

Rf Amp'

l-J

309

-.. Out.

Tumn g co ntrol - - .

· npliunal

FIGURE 12.1. Superheterodyne receiver block diag ram.

by these fi xed-frequency IF fillers. which are much easier to reali ze than variable-

frequency fi lters. Additionally, the overall gain o f the system is distributed over a number of different frequency hands ( RF, IF. and baseband ). so thut the required receiver gain (typically 120 - 140 dB on a power basis) ca n be obtained witho ut much worry about potential oscillations arising from parasitic feedback loops. These important attributes explain why the superheterodyne architecture still dom inates. over 70 years after its invention.

12. 2 MIXER FUNDAMENTALS Since linear, time-invariant systems cannot produce outpu ts with spectral compcnents not present at the input, mixers must he either nonlinear or time -varying elernents in order 10 provide frequency tran slation . Historically, many devices (e.g.• electrolytic cells. magnetic ribbons. brain tissue. and rusty scissors - in addition 10 mort traditional devices such as vacuum tubes and transistors) ope rating on a host of diverse princip les have been used. de monstrating that virtuall y anv nonlinear element can he used as a mixer:' At the core of all mixers presently in usc is a multiplication of two signals in the time domain. T he fundam enta l usefulness of multiplication may he understood from examination of the following trigono metric identity: ( A COSWt/ )( H c o SW 2/ )=

AB

2 (coS(W, - W2)t +COS(WI+W2)/1 ·

(I)

~1 u lti pl icati o n

thus results in output signals att he sum and difference frequencies of the input , signals whose am plitudes arc propo rtiona l tot he produ ct Ill' the RF and LO amplitudes. Hence. if the 1.0 ampl itude is constant (:IS it usually is), any amp litude

~

OfC\lUl'C, some nontmeaones wun bener than <.ll.hers. so we wilt focus (lll the m<.lfe pracnc al types.

310

CHAPTE R 12 MIXE RS

mod ulation in the RF signa l is tran sfe rred to the IF signal. By a similar mechanism. an undesired transfer o f mod ulation from o ne signa l to anothe r ca n a lso occ ur through nonlinea r inte rac tion in both mi xers a nd amplifiers . In that co ntex t the resu lt is called c ro.u- mod ulution. as men tion ,...d in C hapter II . and its suppressio n throu gh improved

linearity is an important design considera tion. Havin g reco gn ized the fund a mental ro le of multiplication. we now e numerate and define the most significant cha racteristics of mixers.

12. 2 .1

CO NVER SIO N GA IN

On e im port ant mi xer c haracte ristic is co nvers io n gain (o r loss), whic h is defined as the ra tio of the desired IF o ut put 10 the va lue (If the RF input . For the multiplier described by Eqn . I. the convers ion gain is the re fore the IF o utput, A B/ 2. divided by A (if tha t is the a mpli tude of the RF input). He nce. the co nve rsio n gai n in this example is H/ 2, or ha lf the LO am plit ude. Co nvers io n gain. if e xpresse d as a power ratio. can be greater than unity in active mixers; passive m ixe rs are ge nera lly capable o nly of voltage o r curre nt gain at best.s Conversion ga in in excess of unity is often conve nie nt since the m ixe r then provides a mplifi cati on a long with the freq ue ncy tra nslat io n. Howe ver. it does nOI necessarily follow that se nsitivity im proves. since noise figure mu st a lso be co nside red . For this rea so n. passive m ixer s may offer superior pe rform ance in so me cases despite their co nve rsion luss.

12. 2. 2 N OI SE FIGURE: SSB VERS US DSB Noi se fig ure is defined as o ne might ex pect: it's the sig nal-to-nois e ra tio (SNR) at the input ( RF ) port divid ed by the SNR at the o utput (IF) po rt . There's an important subtlety, how ever . th at ofte n trip s up both th e uni nitiated and a substantial frecuon of practici ng e ngineers. To apprecia te th is difficulty, we first need 10 make an impor tent o bse rvatio n: In a typica l mi xer . the re are ac tua lly two input freq ue nci es that ",'ill ge ne rate a give n inte rmedi ate freq uency. O ne is the desired RF signa l, a nd the other is ca lled thc image signa l. In the co ntex t of m ixers. the se two signals are frequently re ferred to collectively as side bands . The reaso n thut two suc h freq uen cies exist is that the IF is simply the magnitlkh of the diffe rence bet wee n the RF a nd LO freque nc ies. Hence. sig na ls both above am i below W t .! ) hy an amount eq ual to the IF will produ ce I F o utputs of the same freq ue ncy. The two input freq uenci es are the re fore se parated by 2WIF. As a specific

~ An exception

is a c!.:ts.\ uf system... know n as parametri c co nverters or parametr ic amplifiers. in which pow er fro m the LO i... trun...ferre d to the IF through reactive nonli nea r interaction (typically with vuract ors j.fhu... ma king pow er gain 1X1...... ible .

12. 2 MIXER FUNDAMENTALS

311

numerical example . !'> uppose that ou r syste m's IF is I(X) Mll z a nd we wi sh to tune to a signal at 900 1\.l l l z by se lec ting a n LO frequ ency o f I Gl lz . Aside from the desired 9OO-~ I H z RF input . a 1.I -GH z image signal w ill also produce a difference freq ue ncy component at the IF of 100 Mll z. The exis tence of an image freque ncy com plica tes no ise figure computa tio ns because noise origimuing in bo th the desire d a nd im age frequ e ncies therefore becomes IF noise. yet ther e is ge nerally no desired sig nal at the image freq ue ncy. In the usual case whe re the desired sig nal exists at only one freq ue ncy, the noise figure that one measu res is called the single-sideband noi se figu re (SS B NF); the rare r case• .... here both the "main" RF and image signals contain useful informatio n. leads to a double-sideband (DSB ) noise figu re. Clearly. the SSB noise figure will he greater than for the DSB case. since both have the same IF noi se bur the former has signal pow er in only a single sideband . He nce. ihe SSB NF will normal ly be 3 dB highe r than the OSH NE t> Unfortunatel y. OSH NF is repon ed muc h more oft e n because it is num erica lly s maller a nd rhus falsel y conveys the impression of bet ter performa nce. eve n though there are few communications syste ms for whic h OSB N F is an appropriate figure of merit ." Freque ntly, a noise figure is stated w itho ut any indicatio n as to whe the r it is a OSR or SS B valu e. In such cases. one may usually as sume that a OSB figure is being quoted . Noise figure s for mixe rs te nd to he co nsiderably higher than tho se for a mp lifiers because noise fro m freq uencies other than a t the des ired RF ca n mix down 10 the IF. Represe ntative values for SSS noise figu res range from 10 dB 10 15 d B or more. 11 is main ly because of thi s larger mi xer noise thai one uses LNAs in a receiver . If the LNA has sufficie nt ga in then the signal will be a mplified to levels well above the noise of the mi xer a nd subseq uent stages. so the overall recei ver NF will be dominated by the LNA instead of the mixer. If mi xer s were no t as noisy as the y are. the n ihe need for LNAs would diminish conside rably. We w ill return 10 thi s the me in the chapter on recei ver a rchitectures (Cha pter 18).

12.2 .3

LIN EA RI TY A N D ISOL ATION

Dynamic range req uire ments in modern. high -performa nce tel ecom m unica tions sysrems are qu ite se vere. frequentl y e xceedi ng SOd S a nd approachi ng IOOdS in many instances. As discu ssed in the previous c hapt er. the floo r is es tablished by the noi se figure. which conveys some thi ng abo ut how small a signal may be proc essed . where as

• TIns 3-dB difference a'>.\ umes that the conversion gain 10 IWO eq ual .side bands is the same . AIlhough this assumptio n is usuall y well sa tisfied. it need 00( he . 1 Two importan t exce ptions in which bo th side bands contain useful infonn al ion are rad io asrronomy (as in the measureme nts of the ec hoes uf lhe Big Bang ) an.t dir ect-con version rece ivers (see

Chapter 18).

p 312

CHAPTE R 12 MIXER S

the ceilin g is set by the onset of se vere nonl inearities that accompany large input signals . As with amp lifiers. the compress ion po int is one mea...urc of this dynamic range ceiling. and is defi ned the same way. Ideally, we would like the IF output 10 be propo rtional to the RF inpu t sig nal amp litude : this is the se nse in which we inter pret the term " linearity" in the con te xt of mixers. Howeve r, as with amp lifiers

(and virtually any other physical system), real mixers have some limit beyond which the output has a sublincar dependence Oil the input. The co mp ressio n po int is the val ue of RF s ig n a l~ at whic h a calibrated departure from the ideal linear curve oc· curs. Usua lly, a I-dB (or, more rare ly. a 3· d B) compression value is speci fied. One may specify eit he r the inp ut or outpu t signa l strength at wh ich this co mpression occurs, together with the co nversion gain, to allow fair co mparisons among different mixers. The two -tone third -ord er interc ept is also used 10 charac terize mixer linearity. A two -tone intermodulation test is a relevant way 10 evaluate mixer performance beca use it mim ics the real-world sce na rio in whic h both a desired signal and a potential inte rferer (perhaps at a frequency j ust one cha nnel away) feed a mixer input. Ideally, each of two superposed RF inp uts would be translated in freq uency without interacting with each other. Of cours e, practical mixers will always exhibit some irnermodulation eff ects. and the output of the mixer will thu s contain frequency-translated vers ions of third-order 1M co mpo nents whose frequencies arc 2WRFI ± WRF2 and 2WRF2± WRFI. The diffe rence freq uency terms may heterod yne in to compo nents that lie within the IF passba nd and arc therefore ge nerally the troublesome on es, while the sum frequency signa ls ca n usually be filtered OUL As a measure o f the degree of departure from linear mixing behavior. one can plot the desired OUI PUt and the third -order 1M outp ut as a function of input RF level. The third-order interce pt is the extrapo lated intersec tion o f these two curves. In ge neral, the higher the intercep t, the more linear the mixe r. Aga in, one ought to specify whe ther the intercep t is input- or output-referred. :I 'i well as the conversion ga in, to permit fair co mparisons amo ng mixers. Additionally, it is customary to abbreviate the intercept as I P 3, or perhaps II P3 or a l P ) (for input and output third-order intercept po int, respectively). These definit ions arc summarized in Figure 12.2. Cubic no nlineari ty can also cause trouble with a 5;"8 Ie RF inp ut. As a specific ex ample. co nsider building a low -cost AM rad io. The standard IF for Al\.t radios happens . unfortunatel y, to be 4 5 5 k H z ( mai n ly for hi st oric al reasons ], Tuning in a

8 Slime man ufacturers la nd au tho~) repo rt an fill/p ili co mpres siun point . If the co nveo ton gain at that porm is known then the figure ca n he re flected back 10 the input p oint . Sadly. many insist on burying that bit (If informatjon, making il extremely difficult ro perform fair compa r isons of mixer performance. We will always state explicitly whether the figure is an input or output parameter.

12.2 MIX ER FUNDAMENTALS

313

IF OUlpul

I'o\\" cr I dB

O IP)

I ...-/:"' .: ---- Third -Order Imerccpe

" ", !

Desired

' ..

OUlpul

/CI. __

-

J

Down -Con vert ed Th ird-Or-der J\I Term

, I d B liP] Compression

Potnt (lnpul )

FIGURE 12.2. Definitloo of mixer linea rity porometen.

station at 9 10 U il ia legitim ate A M radio freq uency ) req uire s that the l.0 he set to 1365 k ll l .9 Th e cubic nonlinearit y co uld ge nerate a compo nent at 2Ct1RI' - W I.O. which in thi.. case happe ns to coinci de with our IF of 455 kl lz . One might he te mpte d to assert that such a co mpo nent is not a problem because it adds to the desired output. O ne therefore might even be tem pted to co nsider this an asset. Howevcr. fhc third-order 1M prod ucts have amplitudes tha t are no longer proportional to the inp ut signal amplitude . Hence. they re presen t amplit ude d isto rtion that can corrupt the "co rrect" output (we're talki ng about an amplit ude -mod ulated signal. after all). Even if the exact numerological co incidence of the foregoing exa mple docs not occur, various third-o rder 1M term s can possess frequ encies within the pa ssban d of the IF amplifie r. ultimately degrading signal-to- noise or signal-to- distortion rollins. Another pa rameter of great pract ical import ance is isolation. It is ge nerally desirable to min im ize interacti on amo ng the RF, IF, and LO ports. For instance. since the LO sig nal powe r is generally quit e large compared with that of the RF s ignal, any LO feedthrou gh to the IF o utput might ca use problems at subse quent stages in the signal processing chain. T his pro blem is exace rbated if the IF and LO freq uencies are similar, so that filtering is ineffecti ve. Even reverse isolat ion i.~ importa nt in many instance s. since poo r reverse iso lation might permi tt he strong LO s ignal (or its harmonics ) to wor k its way back to the antenna . where it ca n rudiutc und cau se interference 10 ot her rece ivers. ~

A local oscillator frequency of 455 I.lIz also works, bul is. a les..\ practical ch oice because such ~ Iow-~ ide injection" req uires the joc al osc illator 10rune over a large r range than if lhe LO frequenciev were above the desired RE

3 14

CHAPTER 12 MIX ERS

FIGURE 12. 3 . General two-port IlOfllinearity.

12 . 3 NONLINEAR SYSTEMS AS LINEAR MIXERS We now conside r how 10 impl em ent the mult iplication that is the heart of mixing actio n. Some mixe rs direc tly im plement a m ultiplic ation. w hile othe rs provide it inci de ntally th roug h a nonl inearity. We follow an histo rica l path and first examine a ge ne ral two-port nonlin curity.!" since mix ers of thai type predate those designed spec ifically 10 behave as multiplie rs. Sec Figure 12.3. If the nonlinearit y is "well-be haved " (i n the math em atical se nse) , we can describe the input-output relationship wit h a se ries e xpans ion: N

»oor =

L

Cn ( VI N ) tl .

(2)

11 .. 0

Using an N th -orde r non linearity as a mixer req uires the signal L' IN to be the sum o f the RF input and the local osci llator signals. In ge ne ral. the ou tput will consist of three types of produ cts: DC terms, harm onics of the inp uts, and inrermodulation prod uct s of those har monics. I I Not al l of thcse spectral compo ne nts a re desirable, so part of the c hallenge in mixe r design is to devise topo logies that inherently generate few undesired te rm s. Even-orde r non linear factors in Eqn . 2 contribute DC term s: these are read ily fil tered out by AC coupling. if desired . Harmo nic te rm s. at mw w a nd mWRF. extend from the fundame nta l (m = I) all the wa y up to the N th harmonic. As with the OC Ico ns. they are also oft e n relatively easy to tiller out because their frequ e ncies are usually well aw ay from the desired IF. Th e Inrermcdulation ( 1M) products are the vari ous sum a nd di ffe rence frequency terms. Th ese have frequ encies e xpressible as PWRF± qw LO. where integers p and q arc grea ter than zero and sum 10 values up 10 N . Onl y the second -orde r inrermodulation term ( p = q = I) is normally de~ ired . 12 Unfortunatel y, other 1M prod ucts might ha ve frequ encie s d ose tu the de... ired IF, making them difficult to re move, as we shall

10 We will shortly see the ad vantages of three -port mixers. II Keep in mind that fundamental are harmonics. 12 The order of a give n 1M term i the sum of p and q . M) a second -orde r 1M produc t arises from the l(uad ralic term in tbe sen e... expan...ion.

- -- - ------ - - - - 12.3

NO NLINE AR SYSTE MS AS LI NEAR MIX ERS

315

sec. Since it is ge nerally true that high -order no nlincun ries ( i.e.• large val ues o f N in

the power series expansion) tend to generate more of these undesirable terms.13 mixers should approximate square-law behavior (the lowest-order no nlinea rity) if they only have one input port (as shown in Figure 12.3). We now con sider the specific properties of a square -law mixer 10 identi fy its advantages over higher-order non linear mixers. TWO ·POR T EXAMP LE : SQUARE ·LAW MI XER To see explicitly where the desired multiplication arises in a square- law mixer. note that the only non zero coe fficients in the series expansion arc the CI and ( '2 terms." If we then assume thnt the input signal " IN is the sum of two sinuscid s. (3)

men the output of this mixer may be ex pressed as the sum of three distinct compo-

nents:

(4) where Vfund

1' '''/WlC

=

+ VLOCO"'(WLOI »). 2 e2 l! VRF CO!'.( WR Ft) ) 2 + lr'ro COS(W to t) 1 1.

= c l l vRf COS(W ltf l )

v"''''''s = 2 qv RFv LOlcos(w RFI) lI cus(WLOI »).

(5) (6) ( 7)

The fundam en tal terms arc simply scaled versions of the original inpu ts. and therefore represent no useful mixer output ; they must be removed by filtering. The v....uancomponents similarly represent no useful mixer output. as is evide nt from the following special case of Eqn. J: [cn:-wl j2 = ~ I I + cos 2wl ).

(8)

Thus. we see that the V"lUM e co mponen ts co ntribute a DC offset as well as second harmonics of the input signals. These also mu st generally be rem oved by filtering. The useful output com es from the Vc....... co mpone nts because of the mult iplication evident in Eqn. 7. Using Eqn. I. we may rewrite I'"""", in a form thai shows the mixing action more clearly:

(9) II A" with most sweeping generalitie s. there arc exceptio ns 10 this one. In building frequency mulupliers, for example. high-order harmonic nonline urities arc extremely useful. In mixer design, however. il is usually uue lIlal high-order nn niinearil ies are undesirah le. l~ There may also be a nonzero DC term (i.e.• Co may be nonze m ], hut this com ponen t is easi ly TCJIllWN by tlllering. so we will i" ntlfe it at the ou tset to rtduct' cqualion d ulln.

p 316

CHAPTER 12 MIX ER S

c

L

FIGURE 12 .4 . Square-low MOSFET

mixer himplifiedl. For a fixed LO amplitude. the IF output amplitude is linearly proportional to the RF input amplitude. That is. this nonli nea rity impleme nts a linear mi xing. since the output is proportional to the input. The conversion gain for this non linearity is read ily foun d fro m Eqn. 9 : G.. =

C2 t 'RF t'W

=

C2L' LO .

( 10,

L' RF

J ust as any other ga in pa ramete r. co nversion gain may be a dimen sion less quantity (or a transcondu ctance. trnnsrcsistancc. CIC.). It is customary in disc rete designs to express co nversion gain as a pow er ratio (or its decibel equivalent), hut the unequal in put and output impe dance levels in typical Ie mixers mak es a voltage or current conversion gain appropri ate also. To avoid confusion. of course. it is essential to state explicitly the type of !!ain. t 5 As asserted earlier, the square-law mixer 's advantages are that the undesired spectral component s are usually at a frequency qui te di fferent frum the intermediate frequency. and thus readily removed. For this reason , two -port mixers are often designed to conform to square-law behavior to the maximum practical extent. Excellent square-law mixers may he realized with long-channel MOSFET s, or a pproximat ed by virtually any other type of nonlinearity in which the quadratic term dom inates; see Figure 12.4. In this simplified schematic. the bias. RF. and LO terms are shown as driving the gate in series. The summation of RF and LO signals can be accompli shed in practical circuits with resistive or react ive summers. Because the RF and LO signals arc in series. there is poor isolation betwee n them. An alternative (htil funct ionally eq uivale nn arrangement that reduces the effect the relatively large LO signal o n the RF port is shown in Figure 12.5. The RF signal

or

l~ Allton frequently. published " powe r" gain ligures arc essentially vo ltage gain measuremen ts and

are ther efore gross ly in error if the input and output impe danc e levels ditTer signific antly, as they oft en do . It see ms necessary 10 em phasiz e Ihal walls and volts are not the Mi me.

12.3 N ON LINEAR SYSTEMS A S LIN EAR M IXER S

C

317

L

CD vRf - "4 h~~ 1 RR IAS V HIAS ~

fiGU RE 12. 5 . Squore -IawMOSF ET mixer {a lternative con~ gurotion l .

drives the gate directly (through a DC-blocking capacitor). while the LO drives the source terminal . Thi s way. the gate -to -so urce voltage is the sum o f ground -referenced LO and RF signals. The bias current is set directly with a current source. and the DC gate m ilage is determ ined by the value of VBIAS _ Resistor H OIAS is chosen large enough to avoid excess ive loading. and also 10 minimize its noise comr ibuuon. In deriving an express ion for conversion gain. we use our assumption thai the device is long enough (and biased appropriate ly) 10 allow us 10 ex press the drain current as follows: i ,) = ( I I) Short-channel ( high-field) devices are more linear as a res ult o f velocity saturation. and thus arc generally infe rior 10 long devices as mixers. III If the gate-source voltage VIl'" is the sum o f RF, LO . and bias terms, then we may write . Il ~ W 2 1/) = 2 1. {Y HI AS + [ VRf COS(WRft) - vWCOS(WLOt) ! - Vrl ". (1 2 ) from which one may readil y find thai the co nversion gain (h ere. a tnm sconductan ce) is simply

(1 31 This square-law device thus has a conversion transconductance that is independ ent of bias. 17 It is still dependen t on tem perature (through mobility variatio n) and LO drive amplitude, however. The reader is reminded once again that "short-charme r' actually meuns " high- held." Hence. even ~short " devices may still behave quadratically for suitably small d rain- source voltages. I' This independence of bias hold" o nly in the square -law regi me. Enough hias must jhcrefore he supplied It) guarantee this condition. Hence. V8 1A S is not permined to equa l zero. In fact, it must be chosen lurge enoug h 10 guarantee that the gore-source vo ltage always exceeds the threshold Voltage, since a MOSF ET behaves exponentially in weak inversio n. 16

-

31 8

CHAPTER 12 MIXER S

Because perfect square- law behavior is not necessary 10 ob tain mixin g act ion. /.II can be a bipo lar tran sistor , for exa mple, becau se the quad ratic fac tor in the series expension for the expo ne ntial ic - L' BE relation ship dominates over a limited range of input ampli tudes. Prec isely because many nonlineanries are well approx ima ted by a sq uare -law sha pe over so me suitably restricted interval. one ca n es timate the conversion ga in for ot he r no nlinear de vices use d as mixe rs o nce the value of the qu adratic coe fficie nt (c 2) is found . To under score this point , lei 's est imate the conversion gain for one more nonlinea r element. a bipol ar tran sistor. (onven tan Gain of 0 Single Bipolar Tran,i , for Mixer To simplify the calculation. lei us co ntinue to ignore dynami c effect s, Th en we can usc the expo nentia l L' BE law:

Expa nsion of this fami liar relati onship up to the second-order term yields !"

-' , [ "IN+ - ("IS)']

ic :::::: tc I + -

Vr

By inspecti on (well, almost). ( '2

I 2

Vr

8m = --. 2Vr

(15)

(16)

so that a n estimate of the conve rsion ga in is: (17)

Th e conve rs ion gain here is a tra nsconduc tance proport ion al bo th to the sta ndard increme ntal tran sconductan ce. and to the ratio o f the local osci llato r drive am plitude to the therm al voltage . TIle conversion gain for a bipo lar tran sistor is the refore dependent on hi as curre nt. LO am pli tude. and tem peratu re. As in the co rres po nd ing derivation for a MOSFET, the forego ing com putation ignores parasiti c se ries base and e mille r resistances, Th ese resista nces ca n linearize the trunsistor a nd therefore weak e n mixer action. T ho ughtful device layout is thus mandatory to min imize thi s effect. 12 .4

M ULTIPLIER-BASED M IX ERS

We have see n that nonhneariti es produ ce mixing inci de ntal ly throu gh the multiphcations they provide, Preci scl y be cause the mul tipli cat io n is on ly inciden tal. these nonlincaritie s usually gene rate a host of und esired spec tral com pone nts. Addit iona lly, 18 We have implicitly assumed that lhe:' base-emitter drivecontains a DC component as we u ;b!be RF and LO components. M) thdt t c is nunzero.

12 .4 MUlTiPlIER·BASED MIXER S

319

FIGURE 12.6 . Single·balanced mixer .

since two -port mixers have on ly o ne input po rt. the RF and LO signals are ge nerally not well isolated from each other. Th is lack of isola tion ca n cause the problems mentioned earlier. such as overloadi ng of IF ampl ifiers. as well as radi ation of the LO signal (or ih harmonic s) hack out through the ant enna . Mixers based directl y on multiplication generally ex hibit supe rior perfo rma nce because they ideally generate only the desired intermodulation produc t. Furthermore. because the inputs to a multiplier enter at separate port s. there ca n be a high degree of iso lation amo ng all three signals ( RF. LO. and If). Finally. C f\.10 S tech nology provides exc ellent switches. and one can imp lement outstanding mult ipliers with switches .

12.4 .1 SINGLE· BA LANC ED M IXER One extremely co mmon family of multipliers first converts the incoming RF vo ltage into a current and then performs a mult iplication in the current domai n. The simplest multiplier ce ll of this type is sketched in Figure 12.6. 1'1 In this mixer. ut o is chose n large enou gh so thai the transistor s alternatel y sw itch (commutate) all of the tail current from one side to the other at the LO freq uency?" The tail current is therefore effectively multiplied by a square wave whose frequ ency is that of the loc al osci llator :

(\ HI Because a square wave con sists of od d harmon ics of the fundam enta l. mu ltip lication of the tai l current by the square wave results in an output spectrum that appears

19 Mixen; uf this general kind are one n lumped together and ca lled Gilbe rt mi.. . ers, bu t nnly so me actually are. True Gilbert multipliers function en tire ly in the curre nt do ma in. Jeferri n~ the prtlh lem of V- I conversion hy ass uming Ihal all variables arc already available in the form of currents . See Barri e Gilbe rt's lan dmark paper, "A Prec ise Four-Qu ad ran t M ultiplie r with Su hna nl ~....'Ctllld Response," I EEE J. Sf/fid·Stlllt' Cin:lfiu. December 1968 . pp. 365-73 . 20 One may etso interc hange the role s o f LO and RF m pi n. bul the resulung mi....er ha~ lower com ersion gain and W(Jr.< nui"C pe rformance. among ot her deficiencies. A more tklailctJ d i!\Cu!\,..io n of Ibis iss ue is def erre d In a larer sec tio n.

p

320

CHAPTE R 12 MIXER S

as shown in Figure 12.7 ( WII:F is he re c hose n atyp ica lly low com pared with Ww 10 red uce clu tier in the gra ph). The output thus co nsists of sum and differen ce co mponents , each the result of an odd ha rmonic of the LO mixing wit h the RF signal. In add ition, odd harmoni cs of the LO appear dir ect ly in the ou tput as a co nseque nce of the DC bias c urre nt mu ltip lying with the LO signal. Because of the prese nce of the LQ in the output spec trum. this type o f mi xer is known as a singte-bakmced mixe r. Double -ba lanced mixer s. whic h wc'I l study short ly. ex ploit sy m me try to re move the undesired out put LO component through cancell atio n. Altho ugh the current source of Figure 12.6 inclu des a component that is perfectly proponional tu the RF input sig nal. V- I converte rs of all real mixer s ore imperfect. He nce. an impor tant design c halle nge is to maxim ize the linearity of the RF transconductance . Linearity is most com monly enhanced thr ough so me type of so urce dege neration. in bot h com mon-ga te and com mon-source tran sconduct ors: see Figure 12.8. T he com mon -ga te circ uit uses the source resistance R, to linearize the transfer characte ristic. Thi s linea rization is mo st effective if the admittance loo king into the source terminal uf the transis tor i... much la rger than the conduc tance of R p In that case. the tra nscondu cta nce (If the stage approac hes 1/ R., . Inductive degeneration is usually preferre d ove r resistive dege neration for several rea...ons.2l A n induc tance hOI... ne ithe r the rmal noise to degrade noi se figure nor DC voltage drop to diminish supply headroo m. T his last conside ration is part ic ularly relc..'ant for low-vol tage-J ew -po wer app lica tions. Finally. the inc reasi ng reactance o f a n ind uctor with inc reas ing frequ ency helps to a ttenuate high freq uency harm onic and intcrmod ulation com pone nts. A more complete sing le- bala nced mixe r thaI incorpo ra tes a lineari zed tmnsconductancc is shown in Figure 12.9. Th e value of VOlA.'. es tablis hes the bias current of the ce ll. wh ile Rfj is c hose n large e noug h not to load down the gate ci rcuit (and also to red uce it" noise contribution). Th e RF signal is applied to the ga te through a DC blocking capaci tor ell . In practice. a filter wo uld he used to re move the LO a nd et her undesired spect ral compo ne nts fro m the output. The conversion transcon duc tance of this mix e r ca n he estima ted by assu min g that the Lo- drive n transistors behave as perfect switc hes . Th e n the differe ntial output curre nt may be rega rded as the result of mu ltiplying the drain c urre nt of M . by a unit-am plitude square wave. Since the am plit ude of the fun dament al compo nent of a sq uare wave is -tIlT times the a mplitude of the square wa ve. we mOlY wr ite : (19)

21 Capacitive degeneration has also been tried, hut is marked ly inferior 10 inductive degeneration beca use il increases noi'ot' and d h tnn i"n at high frequencies.

12.4 MULTIPLIER -BA SED MIXER S

(1)1-0

I !, I

I

3 lll U 1

SUll.U

7 UJlt l

321

..........

'"

FIGURE 12.7. Repre!>eOlotive outpvl spedrvm ol single'balonced mixer.

( 'IIUIlIlon·-'.;o u r ce

FIGURE 12.8 . Rf tronsconducton for mixers.

IF Ouiput

,jG12 ~~

fiGURE 12,9 . Single -ba lanced mixer with linearized transconductance,

322

CHAPffR 12 MIXER S

IF Oul

FIGURE 12. 10. Active double·balanced mixer.

where g... is the tran sconductance of the V- I converter and G r is itself a transco nductance. The coe fficie nt is 2/ rr rather than 4/ rr because the IF signal is d ivided evenly betwee n sum and difference co mpo nents.

12.4 . 2 ACTIVE DO UBL E·BA LAN CED M IXER To prevent the LO prod ucts from gett ing to the out put in the first place. two singleba lanced circu its may be combined 10 produ ce a dou ble-ba lanced mixer; see Figure 12.10. We assum e o nce aga in that the LO drive is large enough 10 make the differential pa irs act like current-steering switches . Note that the two single-balanced mixers are connected in anti para lle l as far as the LO is co ncerned but in parallel for the RF signal. Therefore. the LO term s sum to zero in the ou tput, whe reas the co nverted RF signal is doubled in the output. Th is mixer thus provides a high degree o f LO-IF isola tion . cas ing filtering requirement s ut the output . If care is taken in layou t. IC reuliza tions of this circuit routinely provide 40 dB of LO-IF isolation. with values in excess of 60 dB possible. As in the single- balanced active mixer, the dynam ic range is limited in part by the linearity of the V- I converter in the RF port of the mixer. So. most of the design effo rt is spe nt atte mpting to find better ways of provid ing this V- I co nversion. The ba..ic linearizing tech niques used in the single- balanced mixer may be ada pted to the double-balanced ca se, as shown in Figure 12. 11 In low-voltage application s, the DC current source can be replaced by a parallel L C tan k to create a zero- headroom AC cu rrent so urce . The reso na nt frequ ency of the t.ank should be chosen to prov ide rejection of whatever co mmon-mode co mponent is most objectionable. If several such co mpo nents exi st, one may usc series co mbinations of par alle l t.C tanks. With such a cho ice, a co mplete dou ble -balanced mixer appears :1.\ she w n in Figure 12. 12. The ex press ion for the co nversion transcond ucranee is the same as for the single -balanced case.

12.4 MU LT IP LIER -BASED MIXER S

323

FIGURE 12.11 . linearized differential RF tronscanductor for double-balanced mixer.

IF Out

' L<"

~=!=~_

L

c

FIGURE 12.12. Minimum supply-heodraam double-balanced mixer.

Noise Figure af Gilbert-Type Mixers Computing the noise figu re of mixers is difficult because of the cyc losrationary nature of the noise sources. O ne tech nique involves characterization of the time-varyin g impulse response. arguing that a mixer is at least linear, if not tirnc-invuriaut .P Although the method is acc urate and quite suitable for analysis, it co nveys limite d design insight. Nonet heless, we ca n identify severa l important noise sources and make general recommendations about how to minimi ze noi se figure. One noise source is certainly the transcond uctor itself, so that its noi se figure establishes a lowe r bound on the mixer noi se figure. Th e same approach used in COI11puting LNA noise figu re may be used to co mpu te the tran sconductor noise figu re.

n

C. D. Hull and R . G . Meyer. "A Systemanc Ap proa ch to Ihe Analy!>is of Noise in Mi xers," IEEE Trons. Cirrujl.~ m id S.\'SIr""'.~ I. V. 40. no. 12. December 1993, pp. lJ()9-19.

CHAPlER 12 MIXER S

The diffe rential pair also degrades noise performance in a numb er of way s. One nois e figure contribution arises fro m impe rfect switc hing. w hich causes atten uation of the sig nal current. Hence. one challenge in such millers is to design the switches (a nd a..sociated La dri ve ) to pro vide as little atten uatio n as possible . A nother NF contribution of the switching transistors arise s from the interval of time in which both transistors conduct curre nt and hence generate noise. Additionally. any nois e in the La is also magnified during this active gain interval. Minimizing the simultaneous conduction interval reduces this degradatio n. so suffi cient LO drive mu...t he supplied to make the differential pair ap proximate idea l. infi nitely fast switches to the maximum practical extent. Fina lly. the 3-d B attenuation inhere nt in ignoring either the sum or d ifferenc e signal automatica lly degrades noise figu re (by 3 dB ) s ince the noise cannot he discarded so readily, As a resu lt. practical currentmod e mi xers typically ellhihit SS B noise figures of at least 10 dB . with value s more frequ en tly in the neighborhood of 15 dB .

Uneority 01 GilberHype Mixen The I P3 of th is type of mi xer is bounded by that of the tmnsconductor. Ml the threepo int method used to estimat e the 11'3 of ord inary amplifiers may also be used here to estimate the IP3 of the trunsconduc ror. If the LO- driven tnmsistors act a." good switches then the overall mixer IP3 ge nerally differs little from that of the transconductor. To guarantee good switching. it is important to note thar -. although sufficient LO drive is necessary - exc essi ve LO driv e is 10 be avoided . To underst and the reason thai excessive LO drive is a liability rath er tha n an asset. co nsider the effect of ever-prese nt ca paci tive parasitic loading on the common-source connec tion of a differentia l pair. As each gate is dri ven far beyond what ' s necessary for good switching, the com mon-source voltage is similarly overdri ven. A spike in curre nt results. In ellu eme cases. this spike can cause transistors 10 leave the saturation region. Even if that doc s nol occu r. the output spectrum can bec ome dominated by the co mponents ari sing fro m the spikes , rather than the downco nverted RF. Hence. one should use only enough LO drive to guarantee reliable switching. and no more.

A Short Note on Simulotion of Mixer IP3 with Time-Domoin Simulotou Co mmon circuit simulators. such as SPIC E . prov ide accurate mixer simu latio ns only reluctantl y, if at all. The problem stems from two fundamental so urces : Th e wide dynamic range of s ignals in a mi xer forces the use of far tighter num eri cal to lerances than arc adequate for " normal" circuit simulat ions: and the large span of frequencies of import ant spectral components force s long simulation times. li enee. obtaining an acc urate value for IP 3 from a transient simulatio n. for exam ple. is usual ly qu ite challenging. Furthermore. a correct noise figu re s imulation for C ~ 1 0S mixers is not possible at all with comme rci ally avail able too ls. beca use the device noise mod els

12.4 MU lTI PlIER·BA SED M IXER S

325

presently in use are incorrect. The reader is therefore ca utioned to treat mixer simulation resul ts with great ske pticis m. Because eve n the "accurate" options availa ble in so me simulation tools are orders of magnit ude too loo se to he useful fo r IP3 simulations, one specific action that mitigates some of these proble ms is to tighten to lerances progressively unt il the simulation resul ts stop changing significantly. In parti cular. the behav ior of the 1M3 compo nent ill an IP 3 simulation is an extreme ly sensitive indic ator of whe ther the tolerances are sufficiently tigh t. If the 1M3 terms do nut exhibit a +3 slope (o n a dB scale I. chances are high that the tolerances are too loo se. One mu st also make sure that the amplitudes of the two input tones are cho sen small enough ( i.e .• we ll be low either the compress ion or intercept poi nts ) to guarantee quasili near operatio n of the mixer; otherw ise. higher-orde r terms in the no nlinearity will contribute significantly to the outp ut and co nfound the results. In the early phases of design . the three -poin t method may be applied to the transconductor to estimate its IP 3 witho ut having to suffer the ago ny of a trans ient s imulation. Another subtle consideration is to guarantee equal time spaci ng in the transient simulation. s ince FFT algorit hm s gener ally assume uniform sampli ng. Beca use some simulators use ada ptive time ste pping to spee d up co nvergence . significant spec tral artefact s can arise when computing the FFT. O ne may se t the time step to a tin y frac tion of the fastest time interval of in terest to assure co nvergence without resort to adaptive time stepping. As an example. one might have to use a tim e step (parameter "delma x' in HSP ICE 2J ) that is three orders of magnitude smaller than the period of the RF signal. Hence. for a l -G l tz RF input . one might nee d to use a I-ps tim e ste p. It is this co mbinatio n of iteration . tight lime ste p. and numericul rolc runce problem... that causes I P3 simulations to execute so slowly.24

Additional Linearization Techniques Because the linearity of these current-mod e mixers is contro lled primarily by the quality of the transcond uctance . it is wurthwh ile to consider addit ional way s to ex tend linearity, Philosophically. there arc four methods fo r doin g so: predis tortion . feedback. feedforward. and piecewise a pprox imatio n. These techni ques can he used alone or in com bination. What fo llows is a representative ( but hardl y exha ustive) set of exa mples of these method s. Pred istort ion casc ades two no nli nearitie s tha t are inverses o f each ot her. and it shares wit h feedforward the need for care ful match ing. Predistortion is actua lly nearly

13 HSPICE is a trademark of the Meta-Software Corporation. Z4 MeasLl nn }! these qu antiti es in the labora tory alStl req uires some ca re. As with the simulation. the amp litudes of the IWu input ume s must he low enough to avoid excit ation of higher-o rder nunlin eartncs (whi ch wo uld cause a slope of other rhun +3) hUI still sufficie ntly larger than the noise floor.

326

CHAPTER 12 MIXER S

FIGURE 12 .13 . MOSFET CTOss-quod.

ubiquitous. as it is the princi ple underlying the operation of c urrent mirrors. In a mirror . a n input CUTTenl is convened 10 a gate -to -so urce voltage through so me non linear function that is the n undone to prod uce an output current exactly proportional to the input . Prcdi storticn is also fu ndamental to the operatio n of true Gi lbert m ixers. where a pa ir o f j unc tions computes the inverse hyperbolic tange nt o f a n input differential c urre nt, and a d iffe rentia l pai r subseq ue ntly undoes that no nlinearity. Negat ive feedback co mputes a n est imate of e rro r, inve rts it .und add s it bac k to the input. thereby helping to cancel the errors thai distortion represe nts. The reduction in di sto rtion is large as long as the loop tran smi ssion magnitude is large. Because a negati ve feed back syste m co mputes the e rro r a po ste rio ri, the o verall closed-loop ba ndw id th must be kept a s ma ll fracti on of the inh erent band wid th ca pabilities of the ele me nts co mprising the sys tem: o the rwise . the a po steri ori es timate will be irrelevant a t best and destabilizing at wo rst. Th e series feedback exam ples o f thi s c hapter arc p opular me thods for linea rizi ng high-frequ e ncy trunsccnductor s. Co ntrary to comm on prej ud ice. po sitive feed bac k ca nnot be precl uded as a line arizing techn ique. Furtherm or e, s ince loo p transmission magnitudes mu st be less than unit y 10 g uar antee stability, the ban dwidth pen al ty is much less se vere than for negat ive fee d back . A s an illu strative exa mple, th e cross-quad, adapted fro m its bipo lar progenito r, uses positi ve feedback to sy nthesize a virtua l short-c ircuit: see Figu re 12. 13. To show that this co nnec tion prese nts a short circuit to an applied current, i;n. consi der how the voltages at the sources o f M I and M2 cha nge as iin cha nges , As i; n incr eases. the gate- so urce voltages o f M2 and M J incr ease hy an eq ua l amount, while those of M I and M .I similarly decrease. Th e voltage at nod e A is:

(20) Si m ilarly, the volt age at node B is:

12.4

MUlTlPlIER·8A SED MIXERS

~

M~

t-

327

V illAS

FIGURE 12 . 14 . Cro n- quod lra nKoodudor.

(2 1)

Thai is. the voltage at eac h source term inal is below Vil lAS by an amount eq ual to the sum of a high Y,~ and a low Y,p Hence , the two so urce voltages arc always eq ual; the circuit synthes izes a virtual short circuit.2j Such a short c ircuit can be used to shift the burden of linearit y away from acti ve elements to a passive clemen t. such as a resistance ; this is shown in Figure 12.1-1 . Because nodes A and B are at the same pote ntial. the current injected into A is equal to lJ in / R ~ . T his injected current is thus perfect ly propor tional to the inpu t voltage. and is recovered as a ditTerential output current at the dra ins of M .l and M 4 • A variation of the cross-quad applies the input voltage across the gales of the top pair. as see n in Figure 12. 15. The value of the transconductance is still eq ual 10 the conductance o f R• . Feedforward is another linearization tec hnique: it co mputes an estimate of the error at the same lime the system processes the signal. thereb y evading the bandw idth and stability problem s of negative feedback . However. the error co mpu tation and ca ncellation then depend on match ing, so the maximu m practi cal distorti on reduction tends to be substantially less than generally attain able with negative feedbac k . Feedforward is most attractive ur high freq uencies. where negat ive feed hack beco mes less effective owing to the in....ufficien cy of loop tran smission. An exa mple of feedforward co rrec tion applied 10 a rran sconductor is an ada ptation of Pat Qu inn ' s bipol ar "cascomp" circuit 26 ( Fig ure 12.16 ). As ca n be see n. this uansconductor consis ts of a cascoded different ial pair to which an add itional differential pair has been added . So me linea rization is pro vided by the source degeneration

15 This analysis neglects body effec t. Practical implemcnunion s do not work quite ideally as 3 result. :!6 "Feedforw ard Amplifier: ' U.S. Patent fl 4 ,1 ~fdl44. issued 27 March 1919. reissued 1984 .

328

CHAPTER 12 MIXER S

, I

VI~

-1

M4

FIGURE 12 .15 . Alterna te connection of cross-quad tronsconductor.

, I

M

II

V OIAS

FIGURE 12 .16 . MOSF ET oncomp.

resistor R. hut significa nt nonlinearity remains in the transcondu ctance of inner differ ent ial pair M l-Ml . To see this ex plic itly. co nside r th at th e voltage across the resistor is the inpu t voltage minus the di fference in gate- to -source voltages of M ] and M 1:

(221 The goa l is to have a differential output cu rrent precisel y proportional to t 'ill ' so any 6l'.oP repr esent .. an em ir. The cascod ing pair po ssesses the same .6.l'!l'J as the inpUI pai r. which is mea..ured by the inne r d ifferential pair. A current proportional to this error is subtracted from the main current to linearize the transconductance. The name "c uscomp" derives from this combination of a casc ode and error compensatio n. Although the inn er pair is shown as an ord inary different ial pair for simplicity, it is frequently adva ntageous to Jincurize it to increa se the error correction range.

12.4 MU LT IPLIER -BASED MIXER S

329

FIG URE 12.17. CMOS 9...cell.

Anothe r nonfcedback approach is piecewise approximation . wh ich e xploit s the observation thai virtually any syste m is linear ove r so me suff iciently small runge . II divides responsibility fo r linearity a mong several sys tems . eac h of which is active only ove r a small enough range so Ihal the compos ite exh ibits linearity ove r a n extended range. Gilbert 's bipolar "multi-ta nh.. 27 a rra nge me nt is a n exa mple of piecewise a pproximarion. In MO S form . it appears as shown in Figure 12. 17. Each of the three d ifferential pairs behaves as a reaso na bly linea r rransconducn mcc ove r a n inp ut voltage range ce nte red about VB. (l, a nd - VB. respecti vely, For input voltage s nca r zero. the transconductance is pro vided by the middle pa ir. a nd is roughly constant for small enough VI:-l. As the input vol tage de viates significantly from zero. the tail c urren t eventually stee rs ulmost com pletely 10 one side of the middl e pair. Howeve r. with an appr opriate sel ectio n of bins voltage VB. one of the c uter pairs take s ove r and COIItjnues to contribute an increase in outpu t c urrent; see Figure 12.1K. The ove rall tran scon ductance is the sum of the individual offse ttranscon duc tances. and can he made roughly constant ove r an alm ost arbitrarily large range by using a sufficient number- of add itiona l different ial pairs. eac h offset appropriate ly. Th e tra deoff is a n incre ase in power dissipat ion and inpu t capac ita nce .

12. 4 . 3

PO TEN TIO M ETRIC M IXER S

Gilbert-type mixe rs first convert a n inco ming RF vol tage into a curre nt throu gh a transconductor. whose linearity and noise figure se t a firm bound on the ove ral l mixer lineari ty uno no ise fig ure . An al ternative to usi ng voltage -cont rolled curre nt

%7

The name de rives fro m the fact that the Iran ...fer charac iert suc o f a bipolar diIT<': t<': ntial pai r i_s a hypertoo lic tange m .

CHAPTE R 12 MIXE RS

330

VB

IS

FIGU RE 12 . 18 . Illustration of linearization by piecewise opproximat1on.

~'lI: f

'I I

+

• •

I ~ Out pu l

'w '14

FIG URE 12 .19. Poten tiometric mixer.

sources in V- I converters is to use voltage -c ontrolle d resistances. For example. consider varying the resistance of a triode -region MOSF ET in a manner inversely proport ional to the inco ming RF signal. If the voltage between drain and source is maintained at a fixed value, the current !lowing through the device will be a faithful replica of the RF voltage, and if Vd., varies with the LO then the current will be propo rtional to the prod uct of the LO and RF signals. One possible implementation of this idea is sketched in Figure 12.19.28 The fou r MOSF ETs perform the mixing. wh ile the ca pac ito rs re move the sum freq uency compo ne nt us we ll

a~

hig he r-order

prod uct s.

2K J. Crols and M. Steyacrt, "A 1.50 lfz Highly Linear C MOS Downconverslon Mh er," 1EE£ J. S" liJ-SM f' CirruilS. v. 30. no. 7. July 1995. pp. 736 ~ 2 .

12."

33 1

MULTIPliER· BA SED MIX ER S

The RF input drives the gates uf the transistors, while the La drives the sources. A simplified analysis assumes thai the resistances of the transistors are inversely proportional to the RF signal. In thai case , the current through the devices is

vro

i m = - - ::::: ur o ' r d.1

\V

J.l C,,~ - I ( V RF -

L

Vr) - VLO ) ::::: K · l'LO '

l ' rf .

( 23)

Because the current is then the result of a multipli cati on of the RF .1I1d La signals, there are co mpo nents at the sum and d ifference freq uencies. as desired. T his current fl ows through the feed back resistors so that the IF signal is available as an output voltage. The op-amp need only have eno ugh bandwidth to handle the difference frequency component, since the sum compo nent is filtered out by the four capac itances. Note that, for good linearit y. the gate overdrive must grea tly exceed ur.o- Hence, VRF must po ssess a sufficiently large DC compo nent to satisfy this inequality for as large a value of V r f thOlI must he ucccmmoda rcd. Practical mixers of this type may exhibit good lineari ty (e.g., 40 dum IIP3) bUI high noise figures (e.g.• 30 dB ). The high no ise figures are the result o f the resistive thermal no ise of the input F ETs (which is worst when the signal levels are small) and the difficulty o f pro viding a good noise match with the broad band op-amp. As a co nsequence. the overall dynamic range of this type of mixer is typically about the same as conventional Gilbe rt-type current-mode mixers.

12.4 .4

PASSIVE DO UBLE·BALAN CED MI XER

Sofar, we've exa mined active mixers only. with their attendant need for linear transconduction. However , passive mixers have some attractive prope rties, such as the potential for extremely low-power operation. Co nsidering thai C MOS techno logy offers exce llent switches, high-performance multip liers based on switching are naturally realized in C MOS form. In the active mixers considere d so far, representations of the RF signal in the form of currents, rather than the RF voltages themselves, are effectively multiplied by a square-wave version of the local oscillator. An alternative that avoid" the V- I conversion problem is to switch the RF signal directly in the voltage domain. Th is option is considerably eas ier 10 exerci se in CMOS than bipolar form. which is why bipo lar mixers are almost exclusively of the active. current-mode type. The simplest passive co mmutating C MOS mixer consists of four switches in a bridge contigurarion (see Figure 12.20 ). The switches arc driven by local oscilla lor signals in antiphase. so that only one diagonal pair of transistors is conducting at any given time. When M I and "'4 are on. VIF eq uals l'R F. and when "'2 and Mj are conducting, l' IF eq uals - VRf. A fully equivalent de scription is that this mixer multiplies the incoming RF signal by a unit-am plitude square wave whose frequency is

332

CHAPTER 12 MIXER S

I L0 -1 MI ~-

rn

1

M4

MJ

~ U)

I FIGURE 12. 2 0 . Simple double -balan ced pa ssive CMOS mixer.

tha i o f the local oscillator. Hence . the output co ntain s ma ny mixi ng prod ucts that result from the odd -harmonic Fouri er components of the square wa\"c.;!9 Luckily. these arc oft en readily filtered out . as di scussed previou sly. The voltage conversion gain o f this ba sic ce ll is easy to compute from the foregoing descri ption. Assuming mult iplication by a unit-amplitu de square wave. we may immediately write G c = 2/rr .

(24)

Here, the 21lt factor again result s from splitting the IF energy evenly between the sum and differen ce com poncms.:" In pract ice, the actual voltage co nvers io n ga in may differ somewhat from 2/1f beC
/1/(1)] .

2" This suu nuon is the same as with the curre nt-mod e mixers. however. A I ~, eve n harmonics Oh M 1.0 ten us may be non ze ro if the UU lYcy cle uf the ~4 uare wave i ~ not exactly 50%. lIJ If we a~Sll me equ a l source and loa
.'1

~-----------------"""! 12.4 MUlTIPLI ER -BASEDMIX ERS

333

The fun cti on RT(l' is the tim e -varyin g Tb cvenm -cquivalcm cond uctance as viewed from the IF po rt . while XT """ and g, a re the ma xim um and ave rage values. re spcclively. of NT(l) . Th e mixing function, m(I) . is defined by flI er )

=

g U) - g (l - ho /2 ) gU I

+ g (1

where g(t) is conductance of ea ch switch and

TIO / 21 Til )

.

(26)

is the period of the LO drive. The

mixing function has no DC co mponent. is periodic in Tux. and has on ly odd harmonic conte nt becaus e of its half-wa ve sy m metry. TIll' Fouri er tran sform of the lin>! bracketed term in Bqu. 25 hns a value of l in at the LO frequency for a sq uare-wave dri ve (as asse rted e arl ier). and a value of 1/ 2 for a sinusoidal drive. so the effective mixing functio n indeed contributes a higher conversion ga in for a square-wave dri ve . Ho wever. the se co nd bracketed te rm is un ity for a squa re -wave dri ve rbccuu se the peak a nd average c o nd uctances a rc eq ua l) hut tr/2 for a sinusoi da l dr ive. Th e ove ral l co nversio n ga in is g reate r with a sinusoida l drive because the second term more th an compen sa tes for the s maller con tribution by the (eff ec tive ] mixing fun ction . Th e difference is not particu larly large, however. with a sinuso ida l d rive, the co nve rsio n ga in is H/ -I (-2. 1 dB ). compared w ith the 2/7t ga in (-3. 92 dB ) obtained w ith the sq uare-wave dri ve. Because of the spectru m of the (e ffect ive) mi xing fun ction. unde sirable produ ct s can appear at the IF port of this type of mixer. Th e subject of filterin g the refore deserve s careful co nsideration, es pec ially in con nec tio n wit h the issue of input and output termination s. In disc rete designs, the so urce a nd load impedance!'> arc usu all y real and well -de fined (50 n, for exa mple ), but the so urces a nd loads for IC mix ers are usuall y on -chip and not al a ll standardized . Far from a liabilit y, thi s lack of standardiza tion is a degrc..e of free do m that the IC engi neer ca n ex ploit 10 improve performance. As a specific example, reactive so urce and load te rminat io ns might he preferable because they do no t ge nerate noise. Becau se it is diffi cult to obtai n broadband operation with reactances, narrowb and o peratio n is im plied for most pract ical mixers with reactive term ination s. Fo rtunate ly, ther e are man y appl ications for whic h this restric tio n is no t a se rio us limitat ion . A somewhat more elaborate pa ssive mixer that ex plo its this freedom of so urce and load termina tion s appe ars as Figure 12.21. 32 No ll' th at this mi xe r as sumes a capacitive load. represent ed as C L in the sc he matic . Thi s assumpt ion reflect s the ty pical situation in full y integrat ed C MOS circuit s. and stands in co ntras t with the resisti ve terminations commo n in di sc rete design s. A capac itive load ge nerate , no therma l noise of its own. a nd also hel ps filter out high -frequ e ncy noise and di stort io n.

32 This example is udaptcd from A. Shah an ] et al.. "A 12mW wloc Dynumic Range C MOS FmutEnd fur a Portabl e G PS Receiver ." ISSC(' J>iKf'.Hof TechnicalPapers. February 11}l;l7. pp. ] 6 K- 9.

334

CHA PTER 12 MIXERS

FIGURE 12. 21. low ·noise, na rrowband pc ssive mixer.

T he input net wor k consists of an L marcb in casc ade with a parallel lank . The L matc h . comprising 1.1 and part of the tank capacitance, provides an impedance transform ation thai moderatel y boo sts the RF signal voltage 10 help reduce the voltage conversion lo ss. The parallel ra nk. fo r med by L ) a nd C3 + Ct . fille rs ou r-o f-band noi..e and distortio n components present at the input and ge nerated by the mixer it-

self. Resistor RI sets the common-mod e potentia l for the input circuit. Because any nonline ar ity in the ta nk ca paci tance redu ce s I P3. C 3 is be st im plemen ted as a meta l-metal ca paci to r. To redu ce the area co ns ume d. a good c ho ice is 10 use a lateral

flux or fractal ca paci tor. Because of the sm all vo ltage boo st provid ed by the Lmatc h . the vo lta ge con vers io n loss c a n be so mewha t be tte r than the 3.92 d B that a simple switch brid ge would exhibit idea lly. As an ex am ple, o ne im ple mentat io n in a (>.35· /110 techn o logy exbibit s a 3. 6· d B vo lta ge co nve rsio n lo ss with a 1.6 · G Hz RF and 1.4 -G li z LOY Both noi se figure a nd I P3 are stnmg fun cti o ns of the LO d rive , since the resislance o f the switches in the "o n" state mu st be ke pt low and co nsta nt to opt imize bo th pa ramete rs . T he IP 3 is also a fun ction o f the am o unt of vol tage boost provided by the Lvmatch . T his boost muy he adj usted dow nward to trade co nversion ga in fo r improved I P3 an d. in so me case s, it ma y he appro priate to remo ve the L· ma tc h a lto gethe r. Typical SS B noi se figu res o f 10 d B a nd inpu t IPJ of 10 d8 m are readily achievab le w ith a n LO drive a mp litude o f 300 mV.34 As a c rude estimale . the DSB noi se figu re nf this type of mix e r is a pprox ima te ly eq ua l to the power atten ua tion. As a final not e on the noi se performance o f th is ty pe of mi xe r, it is pe rhaps useful to o bse rve that the a bse nce o f DC bias c urre nt im plie s the abs e nce o f 1/ f no ise . Th is

n A. Shahani et ul, [ Feb ruary IW7 ). op. c it. 34 This value applies to a sine-wave LO.

335

12 .5 SUBSAMPllNG MIXER S

-.",

'"--H:--f-+++-cH-+++-H,--+-,++ -H- , 0. .

f iGU RE 12 . 2 2 . lI1 u$lra tion

\.

of w6ampling.

property is particularly valuable in direct conversio n receiver arc hitectures. where such noise is often dominant. To reduce the power co nsumed by the LO drivers. the gate capacitance o f the switches may be resona ted with an inducto r (fo r narrowha nd upplicauonsj . resulting in a power reduction by a factor of Q 2. It is tri·vial to redu ce the power to the order of a milliwatt or less. even at gigahertz frequenci es.

12 . S SUBS AMPLING MI XER S The high quality of C MOS switches has also been explo ited to realize what are somelimes called subsamp ling mixers. Thi s type of mixer exploits the observ ation that the information bandw idth of mod ulation is necessarily lower than the carri er freq uency. Hence, one may satisfy the Nyquist criterio n with a sampling rate that is also lower than the carr ier freq uen cy, effecting downccn version in the process. As ca n be seen in Figure J2.22. the higher-frequency signal is sampled at the instants indica ted by the dots. while the downconverted signal is shown as the lowerfrequency rec onstruction. The theoretical advantage of this approach is that it may be easier to realize samplers that operate at a frequency well below that o f the incoming RF signal. From the figure. it should be clear that a properly designed track-and-hold circuit ( Figure 12.23) serves ,IS a subsampling mixer." In the sample (trac k) mode. transistors AI , through Ms are turned on while trunsistors Alb and Al7 are placed in the "off" state. Devices M ), M 4 , and Ms put a voltage equal to the common-mode voltage 11.'\'1.'1 VC M on the right-hand terminals of the sampiing capacitors, while input switches M l and M2 connec t the capacitors to the RF input signal. Because Mr. and M7 are open. the op-amp is irrelevant in this trucking mode. and the tracking bandwidth is simply set by the RC time constant for med by the total switch resistance and sampling (and para sitic) ca pacitance. Beca use the M This exam ple is ada pted from P. Chan er al.• MA High t)' Linear t -Guz C MOS Dow ncon version Mi:\er," 1£ £ £ J. Solid-Sluu Circuits. December 1993.

-

CHAPTER. 12 MIXER S

336

~ 0,

---L 'I I

+

KF Input

'" FI GURE 12. 23. TrlXk-ond-hoId subsampling mixer lsimpli~edl .

system opcr..tes open-loop in this mod e. it is easy to obtain trackin g ba ndwidths far in excess of what can he achi eved with a feed back struc ture. For examp le. it is trivial

In obtain trucking bandwidths greater than I G Hz in a l-n m technology, In the hold mode. all switch states are reversed. so that the only conducting transistors arc the two feedb ack devices M 6 and M 7 • In this mode. the circuit degenerates to a pair of charged capaci tors feed ing hack aro und the op- amp. Th e se ttling lime of this sys te m need only be fast relative to the (slow) sampling pe riod . rather than to the RF sig nal pe riod . Thus. the bandwidth penalty associated with feedback is no r se rious. Al though a subsarnpler is clocke d at a rel ati vel y low frequency, the sampler must still possess good tim e resolution or else sam pling errors resu lt. Therefore. beyond an adequate tracking bandwidth , one mu st also have low a pertu re ji tte r ( i.e., low uncertainty in the sampling instant s), and this requirem ent plac es ex traordinary demands on the phase noise of the sampli ng cl ock . Hence. eve n though the freq uency of the sampling cl oc k need only satisfy the Nyq uist criterion upplfcd to the modu le1;01/ bandwid th . its absolute time j itter must be a tiny fract ion oft he carrier period. A nother prob lem is that the sampling operation converts mor e thun j ust the signal. Noise
12.6 APPENDIX ; DIO DE· RING MIXER S

337

FIGURE 12 .24 . Simple diode mixer.

is difficult in practi ce to reali ze LNAs that pro vide simultaneou sly high gain and high linearity. so again overall (sys te m) dynami c ran ge ma y act uall y suffe r. As a result of ihese proble ms. one mu st take great care in applying subsa mph ng .

12.6 APPENDIX : DIODE ·RING MI XERS This appendix conside rs a num ber of passive mixers tha t are common in discret e implementatio ns. Th e four-di ode dou ble -bal anced mi xer has particularly good characteristics. a nd i.. nearl y ub iquit ou s in high -performance di screte eq uipme nt.

12 .6 .1 SIN GLE·DIOD E M IXER The simplest and olde st pa ssive mixe r uses a single: diode. as see n in Figure 12.24. In this circuit. the outp ut R LC lank is tuned 10 the desired IF. a nd " IN is the sum of RF. LO. and DC bias components. Th e nonlinear V- I characteristic of the diode provides diode c urre nts at a num ber of ha rmonic a nd intc rmodulati on frequen cies. a nd ihe lank selects o nly those a t the IF. II is tempting 10 reject this circuit as hopelessly unsoph isticated . It doe s nOI provide any isolation. a nd it doe sn't pro vide a ny co nversion gain. for exam ple . However. at the highest frequ en cie s. it may be difficult to exploit othe r types of non lineariries. and such simple mi xe rs may be suitable. In fact , all of the detccrors" for radarsets developed in WWIl were sin gle -d iode circ uits.17 Addi tionally. many early l,;HF television tune rs also used mixe rs o f this type . Much of the mod ern wo rk in the millime ter -wa ve bands simply would no t be po ssibl e witho ut suc h mi xer s. As another not e o n this cir cuit . it can be used as a c rude demodulator for AM signals if the input signal is the AM signal (a t either RF or IF ). When uMX1 in thi s manner. the output induct or is re moved e ntirely, no LO is used. and a simple RC ne twork prov ides the out put filte ring. Millions of "crystal" radio sets used this type of detector ( know n in this context as an envelope detector ). a nd even most AM superheterodyne radios buill today use a s ingle-d iode de mod ulator. • We will use the term s "d etector " and "delllodulillor " imerchangeably. '" The birth uf modern se miconductor technolog y ( an he traced d irectly to the development of microwave diode s for radar. By the end u f WWII. point- contact microwave diode s capahle uf operation we ll into the gij!aheRl range became widely available.

.

33B

CHAPTER 12 MIXERS

•

- -' IF

FIGURE 12. 25 . Single-balanced diode mixer.

12.6 . 2 TW O -DIO DE MI XER S The re are severa l othe r ways 10 use diod es a s m ixe rs . A'!io we ' ll see. it will appe ar thai a diod e bridge can he used as j ust about anything , dependi ng o n w hich terminals are defined as input a nd o utput a nd which way the diodes po inl. JII With t w o diod es, it 's po ssibl e 10 construc t a s ingle -balanced mixer. In this case. o ne ma y o bta in isolation betwee n LO and IF. but there is pour RF-I F Iso lation;

see Figure 12.25. Assume that the LO dri ve is sufficient 10 ma ke the diodes act a.~ sw itches. rega rdle ss o f the magnitu de of the RF input. With a positive va lue for t't.o . bo th d iodes wi ll be o n (no te the reference dots on the tra nsformer windings), effectively connecting l' RF to the If output. When I' LD goes negative . the diodes ope n-circuit and discormcct ugs. Hence. this mixer ac ts the same as the acti ve commutating mixer studied previously. The poor RF-IF iso lation should be self-evident from the co mment that the diodes connect the RF and IF ports together whenever the d iodes are on. Similarly, it should be evidentt hat symmetry guarantee s excellent RF-La iso latio n. Whenever the diodes arc on, the RF voltage can on ly develop a common-mode voltage across the transformer wind ings, so no voltage ca n he ind uced at the La po rt .

12.6 .3

DOUB lE ·BA lAN CED DIODE M IXER

By adding two more diodes and one more transformer. we can construct a do ublebalanced mixer to provide isolation amo ng all ports (sec Figure 12.26 ), Once again, assume that the La drive is sufficient 10 ca use the diodes to act as switches. In the circuit shown, the left pair of diodes is on whenever the L a dr ive is negative, whereas the right pai r o f diodes is on whenever the L a drive is positive.

.~H

Diude s can even he used to provide gain by ellplt1iting the nonlinear juncuon capacitance to make a thing known as a para metric amplifier . The nonlinearity can he used tu tran sfer energy from a loca l oscillator ( known as the pump in par-amp parlance ) 10 the signal. instead of the more COItven uonal uu nsfcr of power from a DC SOUin' to the signal frequenc y, Parametric amplifiers can he extremely low-none devices. since on ly pure reactances are needed 10 make them wl.-k ,

12.6 APPENDIX : DIO DE·RING MIX ER S

339

'-------'-" Otlll m fi GURE 12. 2 6 . Double ·balanced diode mixe r.

Wi th the LO drive po sitive. the voltage at " Right Mid" must be zero by symmelry, since the center tap of the input tran sform er is tied to ground. Thu s. VIF equals tJRF (again. note the po lari ty dot s). With the LO drive negative. it is "Le ft Mid " that has a zero po tent ial. and L' IF equals - l' MF. Hence. this mixer effectively multiplies t'Rf by a unit-amp litude square wave whose frequ ency is that of the LO. Isolation is gua ranteed by the symmetry o f the circuit. The LO drive forces a zero potential at e ither the top or bottom terminal of the output transformer. as noted previously. If the RF input is zero . there wi ll he no IF o utput. Hence. thi s co nfiguration provides LO- IF isolation . Similarly. we can sho w LO--RF isolati on by considering a zero IF input . Since. again. there is a zero potentia l at either the top or bottom terminal of the outpu t transformer, there will he no primary voltage and therefore no secondary voltage . These passive mixers arc availab le in d iscrete form, and perform exceptionally well, The up per limit on the dynam ic range is typica lly constrained by diode break down. and isolation is a funct ion o f the matching levels achieved . With a single quad of diodes. typi cal dou ble -balanced mixers routinely achieve conversion losses in the neighborh ood of 6 dB and isolation of at least 30 d R. and can acco mmod ate RF inputs o f up to I d Bm at the l ·dD co mpression point while requiring an LO drive of 7 dR m. Higher RF level... can be acco mmodated if series connections of diodes are used in place of each diod e of Figu re 12.26. the dra wback being an increased LO dri ve requirement to guarantee switching oper ation of the diodes. Using a total of 16 diodes. for ex ample. extends the RF input range to around 9 dBm but also requ ires a who pping 13 d Bm o f LO drive.

12 . 6 .4

FIN A L N OTE O N DIODE M IXER S

When actually using such mixers. one should be aware that it is critica lly impon ant to terminate all pon s in the proper cha racteristic impeda nce - not only at the RF, IF. and desired LO frequencies. but at (he image freq uenc ies as we ll. If only narrowband terminations are used, it is possible for reflections o f various inrerm cdulution prod ucts

d

340

CHAPTE R 12 MIXE RS

to degrade performance seri ously. Hence, it is genera lly insufficient merely 10 use a standard R LC ta nk as an output bandpass filter without all intermediate buffering stage to guarantee a broadband resistive termination . Failure to satisfy this condition can be the so urce of many perpl exing phenomen a .

PROBLEM SET FO R M IXERS PROBLEM 1 Us ing the device mod els from C hapter 3. design a single- balanced mixer with induct ive so urce degeneration to achieve an II P3 (If + 6 dHm . What i\ the co nversion transco nductance '! PROBLEM 2 We' ve seen the utility of synthe siz ing a virtu al ..hurt -circui t for Iinearizing transconduct ances. Su ppo se that so meone were to pro pose the alternative circuit of Figure 12.27.

FIG URE 12 . 2 7. Common·gote tronscond uctor w ith gate inductance (bia sing not shown).

(a) First deriv e an expression for the incremental impedanc e look ing into the source terminal. ( b) Select a value of indu ctance that makes the rea l part o f the input impedance zero. Are additio nal cle ment s need ed at the in put po rt to make the imaginary part also zero? If so, then w hat are they, and what arc the express ions for their values? (c) Having synthesized the virtual short, is this curre nt recoverable as a drain current sigmll? If so. sketch how this cir cu it would he used to mak e a mixer. PRO BLEM 3 Simulate the version of the cros s-quad transcon duct or in which the input signal drives the gales. Use minimum -length devices with a width of I00 1~ 1Il, a total bias current of'" mA (supplied fro m ideal curre nt sources), and a resista nce of 2(K) Q . Usc the model pa rameters for the process desc ribed in Chapter 3. (a) Initially ignore bod y etTL'CI. and measure the low-freq uency transconductance wit h the output terminated in-t -v DC so urces. Is it what yo u expect" Explain.

3A'

PRO BLEM SET

(b) Now incl ude body effect and re peat . Comment on any d iffe rences, and also determi ne the inpu t com mon-mode vol tage range over which the transcond uctance is roughly consta nt. (c) Find the input-referred third-ord er in tercept from simulations using two equalamplitude tones. at 95 MH z and 105 MHz, that po ssess a common-mode value in the cente r of the runge found in ( bl . Noll' that this value mu st he repo rted as a volt age, since the input impedance is nOI speci fied. Be sure to ob serve the simulation caveats discussed in the cha pter to avoid incorrect values. (dl Compare your answe r to (e) with an estima te ob tained from the three -po int method .

PROBLEM 4 In this problem. we assess the rel ative merits of square-wave and s inusoidallo cul osci llator waveforms. Co nside r the speci fic case of a n ideal multiplier used as a mixer in which the RF signal is A sin(w luit) .

(al Which type of LO yields a higher co nversion gai n if they bo th have the same amplitude ? (b) Compa re the relat ive req uirements on post mixi ng filters for the two LO sig nals.

PROBLEM S Co nsider the double-balanced passive mi xer shown in Figure 12.2R.

Rs /2

\lRF(? Rsl2

1.0

1

M,l

'I I

f- ~

' IF

~ 1J'l

~-

'''r~ CO

w-j[

'" I

FIGU RE 12 , 2 8 , Simple double -ba lanced pc ssive

CMOS mixer . (a) If the IF port is terminated in a resistance eq ual 10 R•• what is the co nversion gain? Assume infinitely fast switching. and neg lect switch resistance . (b) Because o f the effect ive mult iplication by a square wave . the I F OUIPUI co ntai ns components at freq uenci es in add ition to the desired one. Sket ch the approximate output spectrum if the RF input is a sing le frequency sinuso id, and discuss filtering requi remen ts. (e) How would your answer to ( b) change if the La drive did not possess a perfect50% d uty cycle'! Pur your answer on a more qu antit ative ba sis by ex plici tly

• CHAPTER 12 MIXER S

342

de rivin g an ex pression fo r the spectru m of it 11011-50% du ty cycle sq uare wave. Express yo ur answ er in terms of D, the du ty ra tio ( here , idea lly 0 .5) . What docs yo ur answe r tell yo u about the need for sy m metry in device switching'! PRO BLEM 6 In the prev ious problem , sw itc h resis tance was neglected . a de ficiency we now reparr. (a) Derive a n ex pression fur the ma ximum acceptable sw itch res ista nce if the degradation in conversion ga in (relative to the ideu! ca se) is no t to exceed J dR . ( h) Provide an expression fo r de vice width co rres po nding 10 yo ur a nswer in (a). For simplici ty. you may assume sq uare -law beha vior . Expre ss yo ur formu la in terms of the LO gale overdrive. (e) Provide an expression for the total power co n..umed by the La in driving the

capacitance of all the switches as computed from your answer to part (b). PROBLEM 7 The conversion gain equations and three -point method for estimat-

ing mixer II PJ from transconductor IIP3 assume that the current-mode . LO-driven switches are perfect. However. as mentioned in the text. it is possible (perhaps even easy) for improper LO drive to cause degradat ion of conversion gain and significan t distortion. To explore the issue of imperfect switching in more detail. consider the simple differenti al 11.10 5 current switch used in a single -balanced mixer (see Figure 12.29). S imulate this circuit with an ideal current sou rce drive. as shown in the fi gure. Let the DC bias current be I mA oand select an RF current amplitude of 100 IIA . With devices (use the level-S models from Chapter 3) that are each I DO -JI m wide. plot the amplitude of the I (I-MHz IF compo nent (measured as a diffe rential output current into DC voltage sources of a value larger than the com mon mode of the LO drive) as the LO drive amplitude increa ses from 0 V to I V in 1£Xl-mV steps. How does the conversion gain vary? What docs this experiment tell you about how one should select the LO amplitude?

WRF '" 2x( IOOMll z)

»to = 2x(IIO.\1I 1z ) .. foe + IRf cos(WJlFl)

FIGURE 12.29. Single-balanced mixer. PROBLEM 8

Repeat the previous problem . but instead of focusing on co nversion gain. look at the output spec trum al other than the IF. What co nclusions can you

PROBLEM SET

343

draw? In particular. explore the effect of the source- hulk capacitance of the rransistors. Set them to zeru and resimulate. Co mpare and co mment. Design a thR--e"pair "mu lti-tanh" tranM:onducto r using the device models from Chapter 3. The speci fi cations are a total allowed bia.. current of 5 rnA and a low-frequency transco nductance of 20 mS. with a maximum ripple of ± 10%. Veri fy your design with simulations.

PR08LEM 9

CHAPTER THIRTEEN

RF POWER AMPLIFIERS

13 .1 INT RODUCTION In this chapter. we study the problem of delivering RF power efficiently 10a load. As we'll disco ver \'cry quickly, scaled- up version s of the small-signal amplifiers we've studied so far are fundam entally incapab le of high efficiency, and oth er app roaches must be considered. As usual, tradeoff'sare involved. this time among linearity, power gain. output po wer. and efficiency, Power amp lifiers ( PAs) may he divided into seve ral ca tegories, dependin g on whether they ' re broadb and or narrowb and , and whether they 're inte nded for linear or constant-envelope operation . A linear amp lifier is simply one that prod uces an ou tput thai is Intended to be a faithful re plica of the input . A co nsta nt-e nve lope amplifier . o n the other hand. is one tha t ge nera lly produces a n o utput whose a mplitude is ide all y inde pendent of the input. A s we 'Il see . consta nt-e nvelope a mplifiers provide the highest e fficie nc y. so for consta nt-a mplitude s ignals (such as FM ). they are perfect ly suitable. Th e origina l analog (" A M PS" ' ) ce llular te le phones usc FM . and thei r trc nsm ute r o utpu t stages arc ge nerally of the co nsta nt-e nve lo pe type .! Other communic utio ns syste ms e mploy ampl itude mod ulatio n. however , a nd therefo re demand much g reate r linearity. Th c variety of power a mp lifier topo logi es re flects the inabi lity of an y single ci rc uit to satisfy al l req uire me nts. 13 . 2 GEN ERAL CONS IDE RATION S Contrary tn what one's intuition m ight suggest . th e ma ximum power transfer theorem is largel y u seless in the design o f pow er a mp lifiers. On e m ino r reaso n is th ai il 1 Advanced Mobile Phone Service. 2 11 should he mentione d that con stant-e nvelope amplifiers can provide linear operation. hut require tec hniqu es that are difficult with tC components. We will have more to say on this subject in lhe sectiun on Class C ampli ticrs .

13 . 3 CLA SS A , A B, B, A N D C PO WE R AMP LIFIERS

345

BFL

BFC

C

R,.

FIG URE 13 . 1. General power anplifie,- model.

isn't entirely clear how to define impedances in a large -signal. no nlinear system, A more important rea son is that even if we were able to solve that little problem and subsequently arra nge for a conju gate match. the eff iciency would he only 50 c;;. hecause eq ual amounts of powe r are then dissipat ed in the source and load . In man y cases. this value i.. unacceptably low. As an extreme ( but reali stic) example. consider the pro blem of delivering 50 kW into an antenna if the am plifier i.. only 50% efficient. The circuit dissipa tion wo uld be 50 kw as we ll. presenting a rather challenging thermal management problem. Even in the low-power do main of portable communications devices s uch as ce llular phones. high efficiency is extremely desirable to extend battery life or reduce battery weight. lienee. instead of limiting efficiency to 50% by maximi zing JX)wer tran sfer, one generally designs a PA to deli ver a specified amount of power into a load with the highest possible efficiency co nsistent with acce ptable pow er gain and linearity. To see how one may achieve these goal s by blithely ignoring the maxim um power transfer theorem. we now consider a class ic power amp lifier topology.

13.3 CLASS A, AB, B, AND C POWER AMPLIFIERS There arc four types of power amp lifiers. distingu ished primarily by bias cond itions. thai may he termed "class ic" because of their historical precedence. T hese arc labeled Class A, AB. B. and C, and atl four may be understood by studying the single mod el sketched in Figu re 13.1. 3 In this general model. the resistor R i. represents the load into which we arc to deliver the output powe r. A " big. fat" inductance, BFL , feeds DC power to the drain. and is assumed large eno ugh so that the current through it is substami ally conslant. The drain is co nnected to a tank ci rcuit through ca p acitor liFe ttl prevent any

) Many var iation s on this 11k-me exist, but the operating features o f all of the m may still he unde rstood with thi s model.

• CHAPTER 13 IF POWER AMPLIF IERS

DC dissipation in the loud . O ne adva ntage of this particular co nfigura tion is that the transistor 's outpu t capaci tance ca ll be absorbed into the lank , as in a conventional sma ll-s ignal am plifier. Anothe r is that the filtering provided by the lan k cuts down on out-of-ha nd emissio ns caused by ever-prese nt non lineunti es. Thi s considerarion is part icu larly important here because we are no lon ger restricting ourselves to small-signal operation and mu st therefore expect some distortion. To simplify analysis. we assum e that the lank has a high enough Q that the voltage across the tank is we ll approx imated by a sinuso id. ev e n if it is fed by non sinusoidal c urre nts . Thi s assumptio n nece ssarily impl ies narrowband ope ratio n. A lthou g h bro ad band power amplifiers are certainly also of inter e st. we w ill limi t the pre sent di scussion to the narro wband cas e. 13 .3.1

CLA SS A AMPLIFIER S

The Cl ass A powe r am plifie r is j ust a sta ndard. text book s mall-s igna l a mplifier on ste ro ids . Th e a ssumption in C lass A de sign ( indeed , its de li ning c ha racteristic) is tha t bias leve ls are chose n so th at th e tran sistor operate s (quusi-) linearl y. For a bipolar rea liza tio n. thi s co nd ition is sa tisfied by av o id ing cutoff an d saturatio n; for MOS implementatio ns. the tra nsisto r is kept in the pen tode (saturation") region of operation. The primary d isti nct ion be twee n C lass A po wer ampl ifiers a nd sma ll-s ignal amplifie rs is that the signa l currents in a IJA are a subs ta ntia l fractio n of the bias level. a nd on e wo uld therefo re ex pec t pot en tia lly se rio us d istorti on . In narrowband opera tio n, as implied by the ge ne ra l c irc uit model . a tank ci rcui t so lves the d istortion problem pot e nt ially assoc luted with suc h large sw ings so that . overall, linear ope ration prevail s. A ltho ug h linearity is ce rta inly desirable. the Class A a mplifier pro vide s it at the ex pe nse o f e ffic iency beca use there is alway s dissipation d ue to the bias c urrent , even whe n there is no signa l. To unde rsta nd q uant itatively why the e fficie ncy is poor. assume that the drai n c urre nt is re aso nahly well approximated by :

ln = / lJc + irf sin wut .

(I )

wh ere IDe is the bias c urre nt . irf is the a mplitude of the sig na l co m ponent of the drain cu rre nt . a nd Wo is th e signa l freque ncy (an d aIM) the re sona nt freque ncy of the ta nk). Altho ugh we have g libly ign or ed di stort ion . th e errors introd uced are not seriou s e no ug h to inva lida te what foll ows. Th e outpu t voltage is s imply the product of a sig nal c urre nt a nd the load resisran ee. Sin ce the big, fat ind uctor UPI. force s a su bstantially co nsta nt c urr en t through 4 It i~ unfonunale indeed thaI the word "sa luralion.. ha... op Poli.ing mea nings for MU S and bipoillf devices .

t3 . 3 ( LASS A, AB, B, AND ( POW ER AMPLI f iERS

347

\ 'I)S

Joe

FIGURE 13 . 2. Droin voltage a nd current far ideal Clan A ampl i6er.

it. KCL tells us that the signal current is no ne ot her than the signal compo nent of the drain current. T herefore ,

v" = - irfR sin wof.

(2)

Finally, the dra in vo ltage is the sum of the DC drain voltage and the signal voltage . The big, fat ind uctor BFL presen ts a DC short, so the drain voltage swings symmetrically about VDI) . ~ The drain voltage and curre nt are therefore o ffset sinusoids that are 180" out of phase with eac h other, as shown in Figure 13.2. If it isn't clear from the eq uations , it should be clear from the figure that the transistor always di ssipates power because the product o f drain current and dra in voltage is always positive. To evaluate this d issipa tion qu antitativel y, compute the efficiency by first ca lculating the signal power deli vered to the resistor R:

I'rf

=

., R ',(

T'

(3)

Next. compute the DC power supplied to the amplifier. lei us assume that the quiescent drain curren t. toe. is made j ust large enough to guarantee tha t the tran sistor does not ever cut otT. T hat is. (4)

, This is not a typo graphica l error . 11k: drain actually swings /lb. " " tIN: JlOI'itive supply, One way 10 argue that thi'l must he the case i'l to recog nize that an ilkal inductor can not have any OC von age across it (otberw ise, infinite cu rrents wou l,J eventually flow). The refore. if the drain voltage s.... ings beluw the supply. it must also swing above it. Th is kind of thinking is particularly helpful in deducing the characteristics of various types of switched-mode p ower conven ers,

p

348

CHAPT ER 13 RF PO W ER AM PLIFIERS

so thai the inpu t DC power is: (5)

The rat io o f RF output power to DC input pow e r is a measure of effic ie ncy (usua lly called the d rain efficie ncy) , a nd is give n by: Prr

'1= - - = PDe

i~ ( RI2 ) i rf R = - - . irrV[)f) 2VlJn

(6)

Now. the ab solu te ma ximu m that the produ ct i rf R can have is Von . Th e refo re, the maxim um theor etical d rain efficie ncy is j ust 50%. If one makes d ue a llo wance for non zero min imu m Vm . variatio n in bias co nditio ns. no nidca l dri ve a mplitude. and inevitable losse s in the filler a nd imercon nect , values substa ntially sma ller tha n 50% ofte n resu lt - particul arl y at lower supply voltages. where V/).~. on represent s a larger frac tio n of Vo/). Conseque ntly, drain e fficie ncies of 30 -35'1- a rc not a t all unusual for prac tical Class A a mplifie rs. Aside from e fficie ncy, anot he r importa nt co nside ration is the stress on the output tran sistor. In a Class A amplifier. the maxim um drain-to -source voltage is 2V[)f), wh ile the peak drain curre nt has a value of 2V/)/)! H. He nce. the device mu st be able to with...ta nd peak voltages and c urre nts of these mag nitudes. e ven though both maxima do not occur simul taneously, Si nce scaling trends in IC process techno logy force reductions in breakdown voltage. the des ign of PAs becomes more difficult with each pa..s ing generation. O ne co mmon way to quantify the rela tive stress on the de vices is to define another type of efficie ncy, called the "nor mali zed power o utput ca pa bility,' which is simply the ratio of the actu al output power to the product of the maximu m device voltage and current. For this type of am pli fier. the ma xim um value of thi s di me nsion less fi gure of me rit is

P"

Vh,, / (ZR)

I

(ZV/w )(ZV/w ! R )

8

(7)

T he C lass A amplifier thus provides linearity at the cost of low efficie ncy a nd relutively large de vice stresses . For this re.ISOII , Cl ass A a mplifiers are ra re in RF power a pplica tio ns" and rel at ively ra re in a ud io po we r applicat ion s (pa rtic ularly so a t the higher pow er levels) for the reasons cit ed," It is impo rtant to unde rscore once again that the 50% e fficie ncy value represents an uppe r lim it. If the d ra in sw ing is less than Ihc ma ximum assumed in th... forego ing, and if the re a re additional losses a nyw here el se, the efficiency drops. As the

h

1

Exce pt, perhaps. in low-lcvelnpplicanons . or in the ea rly Mages (If a cesceoe. An exce puo n is the high-end audio crowd. of COI.If"C, for whom power consumption is o ften ml( a con strain t .

13.3

Cl ASS A , A B, B. AND C PO WER AMPLIf i ERS

3<.

"os

FIGURE 13. 3. Drain voltage and current for ideal Clan 6 amplifier .

swing approaches zero, the dra in efficie ncy also app roache s zero because the signal power deli vered 10 the load goes 10 zero while the transistor continues to bum DC power.

13.3 .2 CLASS B A MPLIf iERS A cl ue to how one migh t achieve higher efficie ncy than a Cla ss A amplifier is actu-

ally implicit in the waveforms of Fig ure 13.2. II sho uld he clear that if the bias were arranged to reduce the fraction o f a cycle over which drai n current and drain voltage are simultaneously non zero. transistor dissipation would diminish . In the CIa.... B amp lifie r. the bias is arra nged 10 shut olT the output device half of every cycle. Obviously, a gro ss departure from linear operation results. and a high Q resonator i.. absolut el y mandatory in order to obtain an acceptable approxi mation to a sinusoidal output voltage . Although the single-transistor version of a C lass H amp lifier is what we' ll anaIyzehere. it should be men tion ed tluu most practical C lass 13 amp lifiers arc push-pul l config urations of two tran sistors (more on this topic later ). For this amplifier. then. wc assume that the drain current is sinusoi da l for one half-cycle and zero for the oth er half-cycle:

in = itf !'lo i n wo t

for

; 1)

>

o.

(8)

The output tank filters out the harmonics of (his curren t. leaving a sinuso idal druin voltage as in the C lass A ampli fier . The drain current and drain voltage therefore appear approximately as shown in Figur e 13.3.

CHAPT ER 13 RF PO WER A M PLIFIERS

350

To co mpute the outpu t voltage . we first find the fundament al compo nent of the drain current and then multiply this curre nt by the IOOld resistance:

21,TI' irr< sin (Vo' )(sin Wo l ) dt =

hund = T

(9)

0 Un ::::::

;' f

.

(10)

T Rsmwo t .

Since the maximu m possib le value of u" is maximum value of t« is

V Of) ,

it is clear fro m Eqn . 10 thai the (1 1)

The peak drain curre nt and maximum output voltage are therefore the same as for the Class A amplitlc r.1I Computing the drain efficie ncy as before. we fi rst calculate the output power as:

v'

P,,= 2R " '

11 21

where l'n is the a mplitude of the s ig nal ac ross the load res isto r. The ma xim um value of the am plitude rema ins VOl) . so the ma ximu m o utput power is

VAn

p o . mu = 2R '

I1J)

Co mputing th e DC input po wer requi res co mputation of the ave rage d ra in current:

~ tv

2Voo . 2Von = -TI l.oT/2 - smwot d t = --, R JrR

(14)

so that the OC pow er supplied is

2VJv rrR

Puc = - - .

115)

Finally, the maxim um drain efficiency for a C lass B amplifier is '1

Po n = -'= -:::::: 0 .785. P oe 4 mn

(16)

The drain efficiency is thu s co nside rably higher than for the Cl a..s A PA . Continuing with o u r hypot het ica l example of a :"iO-kW tra nsmi tter, the device di." ipal inn would

8 1111'ass umption of ha lf-sinusoida l current pulses is. necessarily. an appro ximation. TIle drain current in practica l cuc uits differs mainly in Ihal the transition to and from zero c urrent is nlll abrupt. Hence jhe true devi ce dissipanon is somewhat gre ater, and Ihe e fficienc y somewhat jcwer. uhaa predicted by ideallhen ry.

13. 3 CLASS A , A B, B, A ND C POWER A MPLIFIERS

351

diminish to less than one third of its previous value, from 50 kW to under 14 kw , Howeve r. a~ with the Cl ass A am plifie r, the ac tua l efficiency of a ny practical imp lementat ion will be somew hat lowe r tha n given by the a nalysis show n here ow ing to effe cts that we have negl ected . None theless. it re mains true that , al l othe r things held equal. the C las s B a mplifier offer s substantially higher efficie ncy than its C lass A COUSin .

The normalized power ca pability of this amph ner i.'o 1/8, the xamc as for the Clas s A , since the outp ut powe r, max im um drain vchage. and ma ximum drain c urrent are the sa me ." With the C lass B am plifier, we have acce pted disto rtion in exchange for a significant improvement in efficiency. Since this tradeoff is effec ted by redu cin g the fraction of a period tha t the transi sto r conduc ts curre nt, it is natu ral to ask whe ther furthe r im provem ent s might be po ssi ble hy redu cin g the conductio n a ngle eve n mo re. Exploration of this idea leads to the C las s C am plifie r.

13 . 3 .3

THE CLASS C AMPLI FIER

In a Cia.... C PA, the gate hias i.. arranged to cause the transisto r to conduct less than half the time. Consequently, the drain c urre nt consists of a pe riodi c train of pulses. It is traditional to app roximat e these pul ses by the top pieces of si nusoid s to faci lita te a direct annlysi s.!'' Specifically, o ne assumes that the drain c urre nt is of the fo llow ing form: (i 7)

where thc offset trc . which is a nalogo us to the bia s c urre nt in a linear a mplifie r, is actually negative for a Class C a mplifie r. Of course, the overall drain c urre nt ;D is always pos itive or zero. That is. the dnain current is a piece o f a sine wave when the tra nsistor is activ e, and zero when the tran sistor is in cutoff . Wc co ntinue to ass ume that the tra nsistor behaves at all times as a curre nt so urce (high ou tput

impedance)."! Beca use we still have a high- Q output ta nk, the voltage acTOSS the load re mains substantial ly sinusoidal. T he drain volt age a nd drain c urre nt the refore a ppea r as de picted in Figure 13A . In the deri vat ions that follow, don ' t worry about being ah le to replicate all of the de tai ls. As we ' ll see , how one designs such amplifiers differs

9 A rwo -transisror push-pull C lass B amplifier has a flo rmalil eu power c.lp;ltJilily that is twice as large. 10 See e.g. Krauss. Hos rian. and Raab. Solid ·Slale Had io Hnginrrring. Wiley. New York , 19X I. II violation of thi s ass umption leads 10 an exceedingly com plex situation. Unfortunate ly, many tJip'lllar Class C amplifie rs force the transistors into saiurutiull fur so me fr..cuon of a period .

• CH APTER 13

352

RF POWER A M PLIFIERS

V DU

lOr ideal Ckm

FIG UII:E 13 .4 . Drain voltage and correot

C ompli~ef .

.--. 2 '~

fiGU RE 13 .5. Detail of d ra in current waveform.

considerably from what the equatio n... imply. so focus instead Oil the general conclusions reached rather than the minutiae. We begin by so lving for the tota l angle over which the drain current is nonzero. In order to red uce the number of steps needed 10 arri ve at the an swer, we first rewrite the express ion for the drain current in terms o f a cos ine rather than a sine: in = I De

+ i , f CtlS W lIl .

in > O.

(18)

C learly, s uc h a m aneu ve r changes nothi ng since the tim e o rig in is arb itrary anyw ay.

With this mod ification. the current pulses appear as shown in Figure 13.5. Se tting the curre nt equa l 10 zero and solving for the total co nduction angle 241 yields

2<1> = 2 .

cos-1(_'.DC). ',f

( 19)

13.3

353

CLASS A , AB, B, AN D C POWER AMPLI FIER S

which may he solved for the " bias c urre nt" 3!'> foll ows: ( 20)

l oc = - i rf cos 4>. We arc no w in a position to compute the ave rage dra in current:

-ill = - 1 2Jr

J-

(I De

+ i' f cns ¢ )d¢

I-

= - I 2f/J / l>c + - I fi rf sinf/JI 2Jf

_ ¢>

2Jf

.

(2 1)

_ ¢>

After substitutio n with the ex pres sio n for Inc . thi s yie ld !'> :

t« .

o cos o] .

I nc = - Ism ¢ -

(22)

IT

We will usc thi s ex press io n shortly in deriving an eq uatio n for the effi ciency as a functio n of cond uct ion a ng le. The other quantity we need is a ge nera l expression for th e pow er deliv ered tu the load. As with the C lass B casco th is deri vat ion is simplified because of the hig h· Q tank circuit. so we need to co mpute o nly the fundamen ta l term in the Fou rier se ries :

i1und

2[,'

=-

T

II

I

in sin wot J t = - (4 / DC sin ¢ 2Jf

+ 2i' f¢ + ;' f s in ¢) .

(23)

Substituting for I lle . we obtain h UM =

~ (2¢ -

(24)

s in 2¢ ).

ZIT

With o ur e xpre ssio n for the fundamental currenrt hrougb th e load . we ca n ea..i1y derive an eq uation fo r the m aximum o utput voltage swing: Vou = i,r -

R

(25)

(2¢ - sin 2¢) .

2IT

allowing us 10 solve for the curre nt . " f

i rf

in te rms of Von : 2lTVo /l sin 2¢)

(26)

= R( 2¢

The peak drain curren t is the sum of i« and the bias ter m:

if), p~ = which simplifies

i' f [sin ¢ - ¢ cos ¢ IT

2JrV f) R (2¢ - s in 2¢ ) .

O I + ."..~"'-""::..,..,.

( 27 )

10

2JfVOD [ i" .pk = 7."",","'-""~:;c I R (2¢ - sin 2¢ )

+

(s in ¢ - ¢ cos

n

¢)].

(28)

35<

CHAPtE R 13 Rf PO WER AMPlif iERS

For a fixed out put voltage. the peak dr ain current approaches infi nity as the pulsewidth decreases toward zero . The dra in efficiency is readily calc ulated with the equ ations we 've j ust derived: '/ma> =

2tP - sin 2$ . -ttsin tP - tP cos tP)

.

(29)

As the conduction angle shrinks toward zero, the effi ciency approac hes 100 %. While thi s sounds prom ising. the output power lliso tends toward zero at the same time, since the fundamental component in the ever-narro wing slivers o f drain current shrinks as we ll. Furthermore, it is clear fru m the equation for peak drain current that the normalized power-handling ca pability of the Cla s... C amplifier approaches zero a!'> the co nduction angle approac hes zero . All of these tradeoffs force the attai nme nt of less than )()(Yk effic iency in practice. since we generally want a reaso nab le amount of outpu t powe r as well a!'> high efficie ncy. Having endured the forego ing derivations . one might be disap po inted that they are typicall y not used very much in the actual process of designing a Class C PA. One reason is that there arc few convenien t choices for the gale bias, with zero volts a particularly con venien t one. The signal co mpo nent of the gutc drive is then chosen sufficiently large to produce the desired output power. T he co nduction angle and efficiency therefore arc usually not ex plicit design parurneters. but simply the COIl."f' quence... o f the cho ice o f zero bias and output powe r. Another rea son is that the ass umptions made (e.g.. sinuso idal current spikes. current source behavior of the transistor ) are not always satisfied well enough 10 trust the eq uations quantitati vely. Again. the prim ary virtue in ca rrying out the exercise is to develop some ge neral intuiti on use ful for design - mainl y, thai the efficiency can be large. but at the cost of reduced powe r-handling capa bility, gain. and linearity.

13 . 3 .4

THE C LASS A B AM PLIFI ER

We have see n that Class A amplifiers conduct 100% of the time, Class B amplifi ers 50% of the time. and Class C PAs somewhere between 0 and 50% of the time. The Class AB amplifier, as its name sugges ts. conducts somewhere betwee n 50% and ItN)% (If a cycle. depend ing on the bias levels chose n. As a result . its efficiency and lineari ty are interm ediat e betwee n those of a Class A and Class B amplifier. Thi .. co mpromise is frequ entl y satisfactory. as one may infer from the populari ty of this PA . We do not need ttl undertake a separate derivation of eq uations for this amplifier because the equations fur the C lass C case also apply here (they also include rhe Class A and Cla ss B case ). The only differen ce is that the bias current is positive, rather than nega tive.

13.4 ClASS 0 AM PLIFIERS

355

Vou

~.

"

FIGURE 13. 6 . da~~ Oampli~ef.

13 .4

CLASS D AMP LIFIER S

The PAs prese nted so far use the active device as a co ntroll ed curre nt source. Anot her approach is 10 use the device as a swi tch. the reaso ning being thai a switch ide ally dissipates no pO\~..cr. for there is ei ther zero voltage across it or zero curre nt through it. Since the swi tch's V- I product is therefore always zero, the tran ..ictor dissi pates no power and the efficie ncy mu st be I OOlk . One type of amplif ier that exploits this observation is the C lass 0 amplifier. At fi rst glance (sec Figure 13.6 ), il loo ks the same as a push-pull . tran sformer-cou pled version of a C lass B amplifier. In contrast with the parallel tan ks we' ve typica lly seen. a series R LC net work is used in the output of this amplifier. since switch-mod e amplifiers are the du als of the current-mode amplifiers stud ied earlier. As a consequence. thc output filters are also du als of each other. TIle input connection guarantee s that only one transistor is dri ven o n at a given time. with one tran sistor hand ling the positive hal f-cycles and the ot her the negative half-cycles, j ust as in a push-pull C lass B. The difference here is that the trunsistors are driven hard enough to make them act like switches. rat her than as linear (or

quasilincur) umpliflers. Because of the switching action , each primary terminal of the output tran sformer

T2 is alternately dri ven to ground. yie lding a square-wave voltage acros s the primary (and therefore across the secondary) winding. When one drain goes to zero vults. transformer uctio n force s the other drain to a vo ltage of 2V"" . T he output fill er allows only the fundamen tal compo nent of this square wave to now into the load. Since only fundamental curren ts now in the secondary ci rcuit . the primary current issinusoidal a.. well. As a co nsequence, each switch see s a sinusoid fur the ha lf-c ycle that it is on. and the transformer curre nt and voltage therefore appea r as in Figures 13.7 and 13.8. Beca use the tran sistors act like switches, the theoret ical efficiency of theClass 0 amplifier is 100 %.

CHAPTE R 13 RF POWER AM PLIFIER S

356

,I

~

-

r-

FIGURE 13 . 7. M , drai n voltage and correot for- idea l d(n~ 0 amplj~ef .

2

-

-P.L-_----'.L-_----'.L-_----'.L-

t

FIGURE 13 .8 . T2 secondary voltage a nd curre nt For ideal Closs 0 amp lifier .

The normali zed power ha ndlin g of this a mplifier happ e ns 10 he ll

P" VOS. on • i/J _p~

(30)

whic h is be tter tha n a C lass B push- pull lind much bet ter than a C lass A amplifier. Of course. the Class D nmpli tie r cannot normally provide linear mod ufunon. but it docs provide po te ntially high effici ency a nd docs not stress the devices very much .

11 It may hell" to kl:'l:r in mi nd that tile am plitu de o f tho: fundamen tal L1.lllll'unl' nl III a square WiI\'\' is

4/ " limes the arnphtude ot the square wave.

13. 5

CL ASS E AMPLIF IER S

357

One practical probl em with this (or any oth er switching ) PA is that there is no such thing as a perfect switch , Nonzero saturation vol tage guaran tees ...tatic dissipulion in the switches , while finite sw itching speeds imply that the switch V- I prod uct i!'o nonzero during the tmn sirions. Hence, switch-mode PAs functio n well only at frequenci es substantially below fr oFurthermore. a particularly se rious redu cti on in efficiency can result in bipol ar imp lementatio ns if , due to charge storage in sa turation. one transi stor fail .. to tum complete ly otfbefore the other turn s Oil. Transfor mer action then atte mpts to apply the full supply voltage acres.. the device that i.. not yet off, and the V- I prod uct ca n be qui te large.

13 .5 CL ASS E AMPLI FI ERS As we' ve seen, u...ing transi stors as swi tches ha s the pote ntial for pro viding greatly improved efficie ncy, but it 's not always trivial to realize that poten tial in practice due 10 imperfec tion s in real switches. The associated dissipat ion degrades efficie ncy, To prevent gross losses. the switches mu st be qui re fast relative to the freq uency uf operation. At high carrier frequencies, it becomes increasingly difficult to sa tisfy thi.. requiremen t. If there were a way to modi fy the circ uit to forc e a zero swi tch voltage for a no nze ro interval of time abo ut the in..rant of s witching, the dissipat ion would decrease. The Class E amplifier uses a high -ord er reactive network that provides enoug h degree s of freedom to sha pe the switch volt age to have bo th zero value mill zero slope at switch turn-o n, thu s reducin g ...witch losses. Unfortunately, it doc... nothing for the turn-off transition, which is often the mo re trou blesom e edge, at least in bipolar design s. A nother issue . as we ' ll see later, is that the Class E amplifier has rather poo r normalized power-han dling capability (worse, in fact, than a C lass A amplifier), requ iring the usc of rather oversized devices to deli ver a given amo unt of power to a load , despite the high potential efficiency (theoret ically 100 % with ideal switc bcs ) of this topology. The primary virt ue o f the Cla..s E amplifier is that it is straightforward to design . Unlike typica l Class C amplifiers, pra ctical implementation s require little postd csign tweaking to obtain sat isfactory op era tion. With that preamble ou t of the way, lei 's take a loo k at the C lass E topology sketched in Figure 13.9 . As in ea rlier examples. the R FL simply provides a DC pat h to the supply and a pproxi ma tes an open ci rcuital RF. Note add itio nally that the capaci tor C j is conveniently positione d, for any device output ca paci ta nce can he ab sorbed into it. Deri vation o f the design eq uations is more involved than it's wo rth to do here, bur eager readers arc di rected to the c1assic paper by the So kals for details. I J The design equations arc as foll ows: U N. O. Sokal and A. D. Sokal . "C lass E. a New C tass of High-Efhcicncy Tuned Si n~le -Esllkd Power Amphtie rs," JEt E 1. Soli,J·S1I.lte Circuits, v. SC- 10, June 1975, pp. 16K--76.

CHAPTER. 13 RF POWER AMPLIFI ER S

358

FIGURE 13.9. doss E ompli~er .

QR L = .

(l l)

w

C1 =

I w R(;r2/ 4

+ 1)( :r/ 2)

::::::. w( R · 5.447)

(S.Q\.I7)( 1 + Q -1.42) .

C' :::::: C, - -

-

2.08

(ll) (33)

For max imum effic iency, one desires the ma ximum Q consistent wit h the desired bandwidth . In pract ice. the ac hievable Q will oft en be subs ta ntially lower than the value thut would limit band w idth significantly. On ce the Q is chosen. des ign of the C las... E PA proceeds in a straightforward man ner. using the equations given. Unfortu nately. computatio n o f drain current and voltage waveforms is difficult. However . they loo k so mething like the graphs in Figure 13. 10 when everyth ing is tuned up. Noll.' tha t the drai n voltage has zero slope at turn-on. although the current is nea rly a ma ximum when the switch turn s off li en ee. switc h dissipation ca n be sig nifica nt durin g tha t tra nsition if the sw itc h isn' t infinitel y fa st (a s is the case with most switc hes you' re likel y 10e ncou nte r). This dissipation can offsetmuch of the impro vement obt ••ined by red uci ng the dissip ation during the tra nsition to the "on" state. Addi tio na lly. no te that each of the wuvefonn s has a rather dramatic peak-toaverage ratio . In fact . a detailed ana lysis shows thai the pea k drai n voltage is approximatel y 3.6VlJ /) , w hile the peak drain c urre nt is rou ghl y 1.7V/)fJ JR. Th e maxim um output power deli ver ed 10 the load is "

_

2

() - 1 + JT 2J4

.

V2 V2 1m ~ 0.577 . ..J2Q R R

(341

Th e normalized power out put capabili ty is ther efore

P" lOllS.,,,, ' i 1J . p~

(lS)

As you ca n sec. the Class E is more de manding of its switch speci fications than even a C lass A a mplifier.

13.6

359

CLA SS F AM PLIFIERS

:

.. ........

! ....

. \

\

. .......

FIG URE 13.10 . Waveforms for Clem E ampli~er.

Becau se of the poor power capability and the red uced efficiency due to switch turn off lru>seS. 14 practical impleme ntations of the Cla ss E amplifier do not exhibit significantly supe rior efficiency to well -executed designs of other types (e.g.• the Cla...s F amplifier to he described next) . Add itio nally. beca use (If the relatively large switch stresses. C lass E amplifiers do no t scale gracefully with the trend toward lower-power (and hence lower-breakdow n voltage ) technologies. For these reasons. Class E amplifiers have not fou nd wide application despite a great flurry ofinterest that followed publicatio n of the first several pape rs on the am plifier. 13 .6 CLA SS F AMPLIFIERS

Implicit in the design of C lass E ampli fiers is the co ncept o f exploiting the prope rties of reacti ve terminati ons to shape the switch voltage and current waveforms 10 advantage. Perhap s the most elegant ex pression of this co ncept is fou nd in the C lass F amplifier; SL'e Figure 13.11. Here. the output tank is tuned to resonance at the carrier frequency and is assumed to have a high enough Q to act as a sho rt circuit at all frequencies outside o f the desired bandwidth . 14 The rather large peak drain current also degrades efficiency in practice , strce all real switches have nonzero -on" vo ltages.

CHAPTE R 13 RF POWER AMPlif iER S

360

~

11I~i

~

z; :;: h LdIO"''' C

t--:::

FIGURE 13 .11. Clou F ompli ~er.

The len gth (If the tran smi ssion line is chose n to be precisely a quarter -wavelen gth at the carrier frequ ency. Recall that a q uar ter-wa velen g th piece of line has a n " impeda nce reciprocation " property. That is, the input imped ance of such a line is proportio nal to the- reci proca l of the terminat ion impedance:

Z2 2 m = ZO.

(36)

I.

We may deduce from this eq uat io n th at a 'mil-wave length pi ec e of line pre sen ts an input impeda nce eq ual to the load impedance. since two quarter-wave sec tions give us two reciprocat ion s that undo each other. With that q uick review out of the wa y. we can figure o ut the nature o f the impedance see n by the drain. At the carrier freq ue ncy. the d rain sees a pure resistance of R L = Zo. s ince the ta nk is an open c ircuit the re. and the tran smi ssio n line is therefore terminated in its c ha rac teristic imped ance . At the sec o nd harmo nic of the carrier. the d rain sees a short . because the tan k is a short at a ll frequen cies away fro m the ca rrier (and its modu latio n side ba nds ). so the tra nsm issio n line now appears as a half-wavelength pie ce of line. C learly. the dra in sees a sho rt at till eve n harm on ics of the carr ier, since the tra nsmi ssion line appcan as so me integer mu ltiple of a hal f-wave le ngth at a ll eve n harm onics. Co nversely, the d rain sees an ope n c ircuit at a ll odd harmo nics of the carrier , because the tank still appears as a sho rt c ircuit: the transm ission line appears as a n od d multip le of a q uarter-wa vel ength and therefore prov ide s a net reci proc ation of the load impedance . Now , if the tra nsisto r is assumed 10 act as a switch , the rea ctive ter min ation s guar tha t all o f the odd harmo nics of the dra in volta ge will see no load (other than th at associ ated wit h the transistor 's own OUIPUI im ped ance ). a nd hence a sq uare-wave voltage ideall y resu lts at the drain (recall th ai a sq uare wave with 50% duty ratio has onl y odd harmonics ). Bec ause of the open-ci rcuit cond ition imposed by the transmi ssion line at all odd harm o nics above the fundamenta l. the only CUITCnt thai flow s into the line is at the

13 .6 CLAS S F AM PLIFIERS

361

IS ~

'" t -

FIGURE 13. 12. Drain vo/toge and current

far ideal doss F am plifier .

fundamental frequency. Hence. the dra in curre nt is a sinuso id whe n the transistor is on. And. of co urse. the tan k guam r uces thai the output "ullage is a s inusoid eve n thou gh the tran sistor is o n for on ly a ha lf-cycl e (as in a Class B amplifier). By cle verly arrangi ng for the square-w ave m ilage IU SL'C no load at all freq uencies abo ve the fundamental. the switc h current is ideally 7£ fU bot h at switch turn -on and turn-off tim es. The high effici encie s possible are sugges ted hy the wavefo rms depicted in Figu re 13. J 2. T he total peak-to -peak drain voltage is seen to be twice the supply voltage . T herefore. the peak -to -peak voltage of the fund amcnrul co mpo nent

of l'OS is (4/1l' )2Vll o .

(3 7)

Note that the fundame ntal has a pea k-to -peak value that actually exceeds the total swing, than ks 10 the ma gic of Fouri er transforms. Nnw, since only the fundam ental compo nent survives to dri ve the load, the ou tput power delivered is p _ [(4/ Jr) VOO ] 2 (3Xl ,, 2R lllJS

Since the switch dissipates no power, we can concl ude that the C lass F amplifier is capable of 100 % e fficiency in principle. In pract ice. one can obtain efficiency superior to that o f C lass E amplifiers. Additionally. the C lass F P/\ has substant ially better normalized powe r-handli ng capability. since the maximum vo ltage is j ust twice the supply while the peak drain curre nt is

.

' I> pi< .

2Von -l V"" =- .-1l' = 1l'R _8 . --. R

The normalized po wer hand ling ca pability is therefore

(30 )

362

CHA PTER 13

RF POw Ell. AMPLIF IERS

I'

"

(-10)

l'O S".., · iIJ.pl

or exactly hal f that of the C lass D amplifie r. In some respect s. the Class F amplifier may he co nsidered equivale nt to a sing le- ended C lass D amplifier. II should be emphasized again that switched- mode am plilicrs - suc h as the Class D. E. o r F amplifier - are in herently cons tant -envelope amplifiers . Th ai is. they do not pro vide an ou tput that is proportionalto the drive. Hence. linear mod ulat ion ca nnot normally be ob tained from these amplifiers. Some co mmunicatio ns systems usc modulation meth od s that involve amp litude modulation Ie.g ., QA M) 10 im prove spec trum utilization , and linear operation is therefore necessary. At present, this requi rement has forced the use of Class AB amplifiers . with a correspo nd ing red uctio n in efficiency relati ve to co nsta nt-envelope PA topologies. A ge neral met hod for provid ing linear operation at co nsta nt-e nvelope effic iencies remain s elusive. In Sectio n 13.7 we will conside r in more detail the problem (If modulatin g power amplifiers .

ALTERNATIV E CLA SS F TOPO LOGY The topol ogy show n in Figu re 13. 11 is elegant, hut the tran smission line nmy be inco nveniently long in man y a pplica tions . Furthermore. the benefits of an infinite (or nearly infinite) imped ance at odd harmonics ot her than the fundament al are somewhat undermin ed in pract ice by the transistor 's own ou tput capaci tance . Hence, a lumped approxi mation freq uentl y performs nearl y as we ll as the tran smission -line version .

To create such a lumped approximat ion . re place the tran sm issio n line with a numocr of parallel resonant filters connec ted in se ries. Each of these resonators is tuned to a different odd harmon ic of the carrie r frequency. Qu ite often , s imply one tank tuned to 3cvo is sufficie nt . Significa nt im pro vement in efficie ncy is rarel y noted beyon d the u..e of the two tan ks show n in Figure 13.13.

13.7

MODULATION OF POWE R AMPLIFI ER S 13. 7.1 CLA SS A , AB, B, C. E, F

Modulat ing a Class A or B amplifier is straightforward bec au se the output vo ltage is direct ly proport ional to the amp litude of the signal co mpone nt of the drain current. i rf . li enee. if i rl is itself propo rtional to the inpu t dr ive. linear mod ulation results. A good approx imatio n to this proport ionalit y is readi ly achieved with sho rt-c hannel

13 .7 MODULATI ON O F POWER AMPLIFIERS

8 1'1-

363

tu ned to 3to u tu ned 10 501u tu ned to OJu

FIGURE 13.13 . Alternati ve Clem F ompli ~er.

~1 0 S

devices. which possess co nsta nt transconductance with sufficient gate volta ge. Bipolar devices ca n provide reasonab le linearit y as a result of series base resistance. either exte rna lly provided or simply that of the device itself The Class C amplifier po ses a more significant challenge. as may be apprecia ted by studying the eq uation for the OUlpUI current derived ea rlie r:

.

irf .(-4 , 1 2H

lfund = -

.

-

., ....

sm _ ) .

(4 1)

Despite appearances. the fundamental compo nent of the current through the resislive load is ge nerally 1101 linearl y proportional 10 i-t, beca use the trigonomet ric tenn in parentheses is also a function of i rf . 15 Hence. Class C amplifiers do nOI normally provide linear modulation capability and are therefore generally unsuitable for urnphtudc mod ulat ion. at least when a modu lated carrier drives the gate ci rcuit . To mod ulate a Cla ss C (or E or F) amplifier. one may superimpose the mod ularing signal on the supply voltage. a technique known as drai n modulation. As lon g as the amp litude of the mod ulati ng signal is no t made too large co mpared with the supply voltage. reaso nably linear mod ulation can result. Finally. with all of these methods, negative feedbac k may he employed In reduce distortion. To relax the requirements on gainbandwidt h prod uct for the feedback loop. one may sample the output signal. dem odu late it. and usc the demodu lated signaltu close the loop; see Figure 13.14 . Such architectures are often distingu ished by the demodul atio n method used . For exa mple. if the demodulator consists of a pair of mixers driven with carriers in quad rature. the PA is sometimes said to be linearized through Cartesian f eedback . We will have more In say on the issue of quadrature mixers in the cha pter on archit ectu res.

I} 11Iat is. wilh the exce ption of Cla ss A or R operation. For me olher casel>. proport io nali ty betw ee n dri ~e

and response does no' occ ur M J thatlinear modulation is noI an mherenr property,

CHA PTER 13 RF POWER AM PLI FIERS

364

m"dllh,ti"n

n...n .........llOl ....

FIGURE 13.14. Negative feedboc~ for impr ovi rog

modulation linearity.

13 . 7. 2 PULS EWIDTH MODULATION Another technique for obtaining nearl y linear modulation is throu gh the use of pulscwidth modulat ion (PWM). Amp lifiers using thi .. technique arc occ asionally known as Class S amplifiers. although this terminology is by no means universal. Such amplifiers do not perform mod ulation throu gh variation of drive amp litude. Rather . it is acco mplished by co ntro lling the duty cycle of co nstant-amplitude drive pulses. The pulses are filtered so that the output power is proportional to the input duty cycle, and the goal of linear operation at high e ffic iency is achieved in principle. Although PWM works well at relatively low freq uencies (c.g.• for switching po.....er conve rters lip 10 the low-megahertz range). it is fairly useless at the gigahertz carrier frequ encies of ce llular telephones. The reason is not terribly profound. Cons ider, for exa mple. the problem of ac hieving mod ulation ove r a 10: I runge at a ea rne r of IGHz. With a halt-period o f 500 ps. modulation to 10% o f the maximum value requires the gcncnuion of 50 -ps pulses. Even if we were able to gene rate such narrow pulses (\'eJ)' difficult ). it is unlikely that the switch would actually tum o n completely. leading to large dissipation . Therefore, ope ration ofPWM amplifiers uvcr a large dynamic range of output power is esse ntially hopeless at high frequencies. Stated another way, the switch (and its drive circuitry ) has tu he n times lu ster than in a non-PWM umpliticr. where 11 is the desired dynami c range. As a result , it become s increasin gly diffic ult to usc pulscwidth modul ation once ca rrier frequencies exceed roughly 10 MHz. 13 . 8 SUM M A RY OF PA CHA RACTE RISTICS We have SI..'Ctl that Class A amplifiers offer good lineari ty bur JXJlJr powcr-handli ng ca.. pabilit y (0 . 125 on a normalize d basisj and low efficiency (50% at absolut e maximuml. Class B umpliticrs improve on the effic iency (to .....78.5% at best) by redu cing the fmction o f a period during which the transistor is active. while maint aining the potential for linear modul ation .

13. 9 RF PA DESIG N EXA M PLE S

365

Class C amplifiers offer efficiencies approaching 100%, bu t both normali zed power-handlin g capability and powe r gain app roach zero at the sa me time . Also. they sacrifice linear operation to obtain the improve ments in efficiency. Additionally. bipolar Cla...s C amplifi ers actually do not satisfy many of the assumptions used in the derivations. and are difficult to design and construct as a consequence; MOS and vacuum tube implementations lend to be less troublesome in (his regard. Amplifiers based on switching concepts do not readil y provide linear modulation capability either. bUI theoretically offer 100 % efficiency at nonzero power-handling capability. Although such perfection is unrealizable. at least the limitation is 0 0 1 an inherent property o f the topology. Class D amplifiers offe r a normalized power-handlin g capability of approxi mately 0.16. but suffer from potentially large vcrowbur" dissipation due to noninfinite switching speeds. Class E PAs solve the dissipation problem for the tum-on transition. but actually have worse dissipation in the tum -off transition and also have terrible normalized power-handling ca pability (0 .098). Clar..s F amplifiers have only half the power-handlin g cap ability of Class D amplifiers yet howe no potential for crowbar problems. since only one device is involved. However, o ne potential dra wback is the need for a relatively complex reactive network (transmission line or its lumped equivalent) . Finally, the problem of obtaining linear mod ulation from switching amplifiers was noted to be an unsol ved one for the time being. Pulsewidth mod ulation requires unreasonably fast dri ve circuitry and switches, and is therefore completely impractical for gigahertz-speed carriers.

13. 9

RF PA DESIG N EXA M PLES

Suppose we want to design a linear amplifier for usc in a l -G j lz communications system. The requirements are to supply I W into 50 Q _ Assume thai a 3.3-V DC power supply is available. We must specify importa nt device parameters. compute all component values, and estimate drain efficiency. Important I/ O / I' : In the examples thai follow, we will make arbitrary assumptions about the minimum values o f V , ),\" in the "on" stare to simplify design procedures. These assumptions must always be checked and modified in real designs.

13. 9.1 CLA SS A AMP LIFIER DES IG N EXAMPLE

First. see if Ihe supply voltage is large enough to allow I W to he delivered to the load without an imped ance tmnsfonn ation:

v2

(3 1) 2

2R

2 .50

1' . = -.!2!!. = - '' - :::::: () I W m.u

.

.

(4 2)

CHAPTER 13 RF POWE R AMPLIFI ERS

Clearly. the supply voltage is not nearl y suff icient. so un impedance tra nsformation i ~ req uire d . Th e maximum value o f the tra nsfo rmed resista nce is

R

max

v' (33)' =----!1Q...=-"-:::::: 5 4 0 2 r1Tl>l~ 2.1 . .

(43)

In practice. the 50 -0 load resi sta nce wo uld have to he transform ed to an eve n lower value ow ing to such una voi dable di ssip ative mech ani sm s as vo ltage d rops in the tran...istor a nd power loss in the filter and interconnect. Let us arbitrari ly ass ume that designing for an effec tive load of 4 0 compe nsates sufficiently for these losses. Th is assum ption must he chec ked in an y actual design, of co urse. a nd the tran sfo rmed resista nce red uced furth er if the o utp ut po wer is lower than desired . Alternatively, if the o ut put power is more than suffi cient. the tran sfo rmed resistance may be increased to improve efficiency . With a 4-n loa d. the peak RF curre nt will not exceed VIJIJ I R = 825 mA . a nd the DC d rain c urre nt bia s must be sci to ap pro ximat ely thi s value. Si nce the pea k d rain cu rre nt is the sum of the hias a nd peak RF current . the transisto r mu st he designed to ...upply ubou t l .bS A with m ini mum voltage drop. In this case. we ca n tolerate a red uction in effec tive Von of on ly a co uple hundred mi llivolts o r thereabouts. M) the "o n" res ista nce of the tra nsi stor mu st be kept below ro ughly 200 ron . In a typi cal O.5-tlm C MO S technology, device widt h... of seve ra l millimeters wo uld therefore be req uired. If we assume that I W is in fact ult imatel y deli vered to the load after alllosses have been taken into account . the drain efficie ncy wo uld thcn he

'/ = -

Po

r oc

=

I 0 .8 25 A . 3.3

v

:::::: 37%.

(44)

Th erefo re, when the amplifier is supplying thai I W to the load , the tran sistor will he dissipa ting a bout 1.7 W, so the packagin g and hcut sinking must be designed to keep die te mpe ratu res acceptably low with thi s dissipa tio n. Howe ver. the news is ac tua lly ,",'O rM: than suggested by the fore going co m putation . The C lass A amplifier ex hibits wor se effic iency as the o utput swing decreases. since th ere is always d issipatio n du e to the DC bias curre nt eve n when the amp lifier is de livering zero RF ou tput powe r. Hence . if a po wer level be low the 1-W target is de livered. the di ss ipa tion of the transisto r ca n be substan tially greater than 1.7 W.ln the worst case , wit h no RF input , the transistor dissipates the power as sociated with its DC bias - fo r th is particul ar exa mple. a value of abo ut 2.7 W. lienee. if the input dri ve is ever permitted to dis appea r. the packagin g must he made ca pable of di ssipating this g reater figu re, rather than the 1.7-W va lue co mputed ear lier. For a Class A amplifie r. then. the worst therm al problem s are associated with ze ro input signal. As a sid e no te. it s ho uld be mentioned that significant imp rovem ent s could be obrain e d by d ynam ically varying the bias as a fu nc tion of the req uired power level. In that case, the tran sistor di ssip at ion ca n he reduced sig nifica ntly. leading to large

13 .9

RF PA DESIGN EXAM PLE S

367

improvemen ts in etticicncy at lower power levels. Thi s type of ada ptive C lass A amplifier has been co nstructed. and efficiency similar to thut obtai ned with Cla ss B amplifiers was obta i n~d . lh The s mall co mplication associ ated wi th performing the adaptation has apparently been enoug h of a barrier to preven t its wide spread adoption, but this topology becomes more attrac tive at higher frequencies beca use of the higher ga in offered by Cla..s A over C lass B amplifiers .' ? To rou nd o ut the design , we need to speci fy co mponen t values for the ou tput fil ter and matchin g network . Assume that the output filter is a s imple paralle l L C. and that the requ ired Q is a bout 10 . The correspo ndi ng IOO -M Hz bandwidt h at a center frequency o f I Gllz mea ns that the reac tance of the L and C must he 5 {} in order to give us the required Q of 10. Hence we choose

X, = 5 .

~

L=

5 = 0.80 nH 2;rr . I GHz

and

Xc = 5

=

I C = -;--0;----;-=;- ~ 3 1.8 pF.

5 · 2;rr . I GHz

Recall that, at reson ance, eac h reactive clement in the parallel tank circulates an RF current that is Q times as large as that flowing in the load . Hence, the L and C (alld associated interconnecn must be a ble to withstand 2 A of pea k current flow in this particular design. III Next, we need to choose the s ize of the inductor, HFL ( known as an RF c"oke because it "c hokes off " the flow of RF curre nt thro ugh it), so that its reactance is large enough . If we arbitra rily choose a factor of 10 as. " large enoug h," we can read ily compute the req uired value . I" For this RF choke , we wi..h its reactance to he at least about len tim es as large as the 4 -0 resistance of the lank at reson ance :

16 A. Sateh and D. Cox. "Imp roving the Power-Added Efficiency of PET Amp lifiers Opcmtmg with varying-Envelope Signals," IEEE Trans. lfil' nJWIlI 't' Theory and Tn /mil /I/O, -e. J I. no. I. January 19K3, pp. 5 1- 0, 11 Since the transistor is in c utoff half the time in a Class B ur nplifier, it provide s roughly half the gain of a C lass A umplificr, all other thing s heing equa l. AI high frequencies. it become s im-rcasingly puinfulto accept a halving o f gain. IS Such large currents can lead to signiticunr voltage drops along interconnect. [I' Ihese volta ge drops are unconstrained in magnitude and allowed to occur at uncont rolled loc ution s, unanric ipated circ uunpcrurion cun easily IIl:CUr, Including mysterious oscillations (perh:lps arising from the creation of parasitic feedback loo p_" sinc e "ground" may no longcr he le w volts every where) or other deviations from expected behavior. Aguin. the message is 10 be aware th:ll what yuu CUll struct may not always map line -Ill-one with your sim ulatio ns. It is critically important to model rl't'')/hing al RF. and this mcle des seem ingly munda ne Ihings like wire. 19 Using fa(lol'l significantly larger than III is 0111 always wise. Por example. using a ( hoke thai is very much larger Ihao ( ompuled lin Ihis basis coul d result in an inductor with a S( U- reslln~ 1Il freqlk'ncy that is 100 low (i n which case it appears as a capaci tance at the ope rllting freque ncy in\lcad of lI lI. an inJ u(ltlt as d.,'l>iredl. or with excessive resistance (which would le....11\) dec reased dticienc y and pt~,ihl y a heat prutJlcml.

... CHAPTER 13 RF PO W ER A MPLIFIERS

3. 8

J .3V

'25rnA

fiG UR E 13.15. l · W CIa ~s A ampli ~ ef .

X lin,

~

io. -t Q

=:::)

BFL

~

6.4 nil.

(471

Th is value is low e nough thai it is readily prov ided by parasiti c bo ndwire andleadframe inductance. in rure defia nce o f Murp hy's law. Now we need 10 provide bo th a DC bloc king capacitor und an imped ance -transfunning ne twork . T hese functi on s may be combined into on e ci rcu it if a high-pass Lmatch is used. for e xa mple. With that c hoice. the transformation ratio se ts the Q 10 3.4. and the Lmatch cleme nt values become:

RL -.. -,-_ -- 2Jt 1050 3.4 woQ

I

9 •

I C , = -~ '"

woQRs

~

2.3 nil .

J '" JL7 pF. 2" 1O" · 3.4 . .t Q

(481 (491

Th e inductor o f the L-malc h can be combi ned with the lank ind uctor to yield the com pleted circuit ...hown in Figu re 13. 15. A ll o f tbc compo nent values are con..istent w ith rea lization in Ie form. altho ugh some effort is requ ired to build an inductor a~ s mall as 0.6 nil wit h low loss, good acc uracy. and re peatabil ity, In practice, one or both reacti ve elements of the tank would be made adjustable to tunc the amplifier exactly to the desired center freq uency. A n additiona l practica l co nside ration is that we need a way to es tablis h the proper hias conditions. Thi s may be accomplished with the usc of a c urre nt mirror. If we consider the outpu t transistor a... the output half of a mirror the n we can supply, say. J % of the bias c urren t into a nothe r tra nsisto r that is I % the size of the out put tran sistor . Such a biasin g met hod elim inates thermal drift prob le ms associ ated with fixed-voltage gate bia s. If impleme nted in di scret e form. it i... c ritica lly im portant to guara ntee intimate thermal coupling betwee n the two transistors to e nsu re thai they are at the same te mperature. Th e two tra nsistors ca n he connec ted toget her directl y. or through ano ther RF c hoke. a nd the s ignal to be a mplified co upled to the ci rc uit through another DC

13.9 Il: F PA DESIGN EXAMP LES

3'9

3.3V 1l2SmA

RI . = SUQ

FIGU RE 13.16. More complele l -W Claii A cmplifier.

blocking capacitor; see Figure 13. 16. T he power consumed by the biasing circuit can he made suitably small by choos ing a suffi ciently large value of fl . Furthermore. the amplitude of the signal at the com mon gate connection can be augmented by using an inductor in series with the gale drive to provide a resonant boost in amplitude . That is. one may provide some impedance -matching network at the input . In general. practical implementations of power amplifiers would req uire matching networks to mcrcasc power gain.

13 .9. 2 CLASS AB. B, AND C AMPLI FI ER DE SIGN EXAMPLE If we consider only single- ended implementations, then Class AB, B, and C am plifi crs loo k extremely similar, with conduction angle (and therefore biasing details) being the on ly d iffere nce. lienee. the out put network is exactly the same for all three . including the choke and OUIPUI filter values. so a single design example will suffice. For the Class AB amp lifier. the refere nce bias would result in cond uction through lcss than 36(F but more than 18CP . Therefore. the drai n current of the ourput transistorwould have a quiescent value of less than 825 mA. Since the input drive amplitude must then increase 10 give us the same output amp litude as in the Class A case, the gain is obviously smaller than for the Class A amplifier. In the Cla..s B amplifier. the bias is supposed 10 provide a 1 80 ~ conduction angie. In practice. it is impo ssible 10 achieve a co nduction angle o f preci sely this value, so one may well argue thut Class B amp lifi ers exist only in the theoretical world of academia . Hence. all practical "Class R" amplifiers are actually either Class AB or C PAs. Again, the input drive must increase 10 main tain the same output am plitude, and the gain is therefore smaller (by a fucror of about 2 rela tive 10 the Class A amplifier. as alluded to previousl y ). In the Class C amplifier, the com mo n practice in most semicond uctor designs is to use a zero gale bias. crank up the drive amplitude 10 get the desired output power. and

CHA PTE R 13 RF POWER AM PLIF IERS

370

3.3V

H II---,---,--<) vocr

FIGURE 13.17. I-W CIo~5 C a mplifier.

accept whatever conduction angle. gain. and efficiency result. In this case. the refere nce bias tran sistor and associated choke (if used) would be removed. and a resistor (or choke ) wou ld be lied between the gale terminal and ground (see Figure 13.17). Although a MOS resistor is show n here . all ordinary resistor ma y also be used . Either way. it is c hosen 10 prese nt a reasonably high resi stan ce to the coupling capacitor.

13.9.3 CLASS E A M PLIFIER DESIGN EXAMPLE In this (o r in a ny s witc hing ) a mplifier. we me-rel y want to drive the tran sistor hard e nough to net as a switc h. A ny drive in excess of this a mou nt is nor only wasted power , hUI in a bipola r implementatio n may al so se riously degrade the efficiency by driving the tran sistor into dee p sa tura tion. He nce. we compute the maximum drain curre nt required and then adj ust drive co ndi tions on the gate to provide this curre nt. and not much mo re . Aguin. our goal is to supply 1 W into 50 Q with a 3.3.v DC suppl y. Recalling that the ma ximum output powe r is /' = 0.577 . o

V'

ieu , R

(501

we com pute that the desi red load resistance is actually about 6.3 Q. so we need to transform downward by a bit less than for the previous design s. Initially. we'll assume that we will transform to a 5-Q lo ad to acco mm odate various losses; then we'll finish off the design by adding the required im pedan ce transfor mer. First . reca ll the basic topology (shown in Figure 13. 1Xl and the associat ed cquatious :

QR L= - .

(51)

w

C1 =

I w R(::r 2/4

+ I )(::r/2)

I ;:::::. w ( R · 5..w 7 )

(521

13.9 RF PA DES IGN EXAMPLES

371

R

f iGURE 13.18. do ~~ Eompli Fier .

1:

> bnH18

IOAnll 3.8pF

5.8pF

10.

-r-

fiGURE 13.19. More complete d on E amplifier.

C,

'" C' (5.4Q.17)(1+ Q _1.42.0H 2 )•

(53)

where R is 5 Q . Choosing Q = 10 leads us 10 the following values:

L = 8.0 nlt ,

(54 )

C ] = 5.8 pF.

(55)

C, = H pF.

(56)

The required impedance tran sformation may be provided by a simple low-pass L· match whose element values are given by:

t.; ;: : : 2.4 nH,

(57)

c, ;: : :

(5H)

10.6 pF.

Combining the two induct ors in to one leads to the final design shown in Figure 13. 1Y. In practice, the drain ca paci tance of the transistor itself would form part of the 5.8 pf capacitance.

CHAPTER 13 RF POWER AM PLIF IERS

13.10 ADDITION AL DESIGN CONS IDERATIO NS 13 .10 .1 PO W ER · ADDED EFF ICIENCY In the foregoing exa mples. drain efficiency is used to characterize the PAr.. 1I0w· ever. the definition of drain efficiency invo lves on ly the RF output power and the DC inpu t power. Ml it ca n assign a high efficiency 10 a PA thai has no pow er gain. Anot her measure of effic iency has therefore been developed to yiel d a figure of merit that takes power gai n into accou nt . Power-added efficiency (PAE) sim ply replaces RF output pow er with the difference be twee n output and input pow er in the drain efficie ncy equation : PAE sa p out - PIn (59)

Poe

C lea rly. power-added efficiency will alway s be less than the drain efficiency.

13 .10 . 2 PA IN STA BILITY Ampli fiers o f any kind can be un stable with certain co mbination!'> o f load and source impedances, and power amplifiers are no exc e ption. One extre mely import ant problem results from d rain-to -gate coupling (or collec tor-to -base co upling) . As noted in the cha pter o n high -freq uency amplifier design. this coupling can cause the input impedance to have a negat ive real part . In sma ll-...ign al amplifie rs. this problem can he reduced or el iminated entirely by u...ing the various unila teruh zation techniques descri bed earlier . Unfortu nately. these tricks are ge nerally ina ppro pria te for power amplifiers because the requ irement for high efficiency preclude s the use of any technique (suc h as casc oding) that d iminishes supply head room . In general. the problem is usuall y so lved through the brute -force mean s of degradi ng the in put impe dance te.g .. through the usc of a simple res istor acros s the input terminals) to make the feedback less significant. Unfortunately. this action has the side effect of reducing ga in. In general. MO SF ET s - with their larger inherent input impeda nce s - exhibit this stabilit y prob lem to a greater degree than bipolar devices. In any ca se. there is usually a significant stability- gain tradeo ff due 10 the feedback capaci tance. And of course. thoughtfu l layout is required to avoid aug menting the inherent device feedhad capacita nce from an un fortu nate j uxta pos itio n of input and o utput wires.

13 .10 . 3 BRE AKD O W N PHENOMENA

MOS Device~ Downward impedan ce tran sform atio ns were required to deli ver the desired amount of pow er into the output load in all of the design ex amples. Clearly, the tran sformalio n ratio co uld he red uced if a higher power supply voltage were pe rmitted . and thf

13 .10 ADD ITIO N A L DESIGN CO NSIDERATION S

373

reader may reasonably ask why one could not simply demand a high er vo ltage he made availab le. The rea son is thut devices have finite breakdo wn voltages. Furthermore. as Ie technology scales to eve r-smaller dimension s. breakdown voltages tend to diminish as wel l. T hus. increa sing tran sformation ratios are required as devices scale if on e wishes to deliver a certa in fixed amount of pow er to the load. In MOS devices. on e may iden tify four primary limit s In allowable a pplied vo ltages in PAs. These are d rain (or-source) d iode zener breakd own. d rain- source punc hthrough. ti me-dependent d ielectric breakdown (TOD D)• and gate ox ide rup ture. The drain and so urce region s are quite heavily doped to red uce their resistivity. As a consequence. the diodes they form with the substrate have a relative low break down vo ltage. wit h typical values of the order of 10 - 12 V for O.5-/.lm tech nologies. Drain- source puncluhrough is analo gous to base punchrhroug h in bipo lar devices, and occurs when the dra in vo ltage is high enough to cause the depl etio n zone aro und the drain 10 extend all the way to the so urce . effectively eliminating the channel. Current flow then cea ses to he co ntrolled by the gate vo ltage . Pu nc hthroug h problems can be mitigated by usi ng larger channel lengths at the co st of degraded devic e transconductance ; this. in turn . forces the use of a wider device to mai nta in ou tp ut power. Time-depende nt dielectri c breakd own is a co nseq uen ce of ga te oxide dam age by energetic carriers. With the high fields typi cal of mod em short-channel dev ices. it is possible to accelerate carr iers ( primarily elec tron s) to energies sufficient for them to cause the formation o f trups in the o xide . Any charge that gets trapped there the n shifts the device threshold . In NMOS tran sis tors. the thresho ld increases so thut the current obtained for a given gate vo ltage decreases; in PMOS devices, the opposite happens. Time-depe nden t d ielectric breakdown is cum ulative , so it places a limitation on device lifetime . Ty pica lly, TDD B rules are designed with a goa l of no more than 10% degradation in drive current aft er 10 yea rs. As an extre me ly crude rule of thumb. the ratio of gate voltage to oxi de thickness must be kept under appruxi mutel y 0 .5 Vj nm to satisfy this requirement . Recent work sugge sts that T DD B becom es much less of a probl em in extreme ly thin oxides becau se any trapped cha rge is close eno ugh to the gate elec trode (or channel) not to stay trupped. It a ppears that the 5-nm and thinn er ox ide... that arc now becoming commonplace arc relatively free of this problem . As a co nseque nce, the primary limit on allowed gate voltage is imposed by ca tastrophic o xide rupt ure, whic h generally result.. in an irreversible ga te-to -c ha nne l short. A... anoth er crude rule o f thumb, oxide ruptu re occ urs for gate fields exceeding a bout I Vjnm. 111e portion of the gale o xide ncar the drai n freq uently ruptures first in a PA , when the gale is at ibe minimum potential and the drain is at 2Vf)f) (or mo re. depe ndin g on topo logy and load co nd itions). In future tech nologies. it is likely that gate o xide rupture will determine the ma ximum allowable supply voltage in a PA .

CHAPTER 13 RF POW ER AMPLI f iERS

Bi~ r

Devices

Bipo lar transistors have no gate o xide to ru pture, but j unct ion breakdown a nd base punc hthrough im pose importan t limits on allowable supply vo ltages. The collectorbase ju nc tion ca n under go avalanche breakdown in which fields are sufficie ntly high to ca use significa nt hole- elect ron pair ge ne ration a nd mu ltipl ication. In we ll-designed de vice s, this mec hanism impo ses the more serious constraint, although the extremely thin bases that ar e cha racteristic of high-JT de vices ca n oft en ca use base pun chth rough to be important as well. A nother. more subtle. proble m that can plague bipol ar de vices is assoc iated with irre duci ble termi na l indu cta nces that ac t in conce rt w ith large difd t s. When tuming off the devi ce, significa nt base c urre nt ca n Row in the reverse direc tio n until the base c harge is pu m ped ou t. Wh en the base cha rge is gone. the base curre nt abruptly ceases to flow. a nd the large di idt can cause large re verse voltage spikes across base to em iller. Recall that the ba se- emitter j unctio n has a relatively low reve rse breakdown voltage (e.g. 6 - 7 V. although so me power devices exhibit significa ntly larger val ues) , and tha t the da mage from breakdow n de pen ds o n the e ne rgy a nd is cumulative . Specificall y, fi degrades (and the de vice also ge ts no isie r). Hence. gain dec reases, possibly causing incorrect bias. and the spec trum of the output can show an increase in distortion produ c ts as well as a ste adily worse ning noise flou r. In perform ing simulations o f'po wer a mplifiers, it is there for e important to look specifically for this effec t a nd ta ke correc tive action if necessary." Opt ions incl ude clamping diodes connected ac ross the de vice (perhaps integral with the dev ice itse lf to reduct' induc tances betwee n the cl amp di ode a nd the output tran sistor ), or simply reducing L ditd t thro ugh improved layout or better drive contro l. It is possib le ( bu t rare) for a similar phenomenon to occur in MQ S implementat ion s. As the gate drive diminishes during turn-off, the gale ca pac itance drops abrup tly once the gale vo ltage goes be low the threshold . Again. the L d i tdt spike may be la rge e nough to harm the device.

13 .10 . 4 THERM A L RUN AWAY Anothe r problem conce rns therm al effec ts. To ac hieve high power ope ra tion. it is commo n to use paralle led devices. In bipolars, the ba se -e mitte r voltage for a consta nt collector c urre nt has a te mperature coefficie nt of abo ut - 2 mVrc. There fore. i1~ i1 devi ce gets hotte r, it re qui re s less d rive to m aintai n a s peci fied collector current 11m". for a fixed dri ve. the co llecto r c urrent incre ases drama tically as temperature inc reases.

;'ll Again. it is important 10 have a good mOllcl th'lt is bused o n the actua l physical structure.

lJ .10 ADDITI O NAL DESIGN CO NSIDER ATIO NS

375

Now consider what ha ppens in a parallel connection of bipolars if on e device happens to get a little honer than the others , As its tem perat ure increases. the co llec tor current increase s, Th e device ge ts hott er still. stea ls more curre nt. and so o n. Th is thennoelec tric posit ive feed back loop ca n run out of control if the loo p rran smi ssion exceeds unity. resulti ng in rapid device destru ction . To solve the problem, some small res istive degeneration in each tran sistor 's emitter leg is extrem el y helpful. Th is way. as the collec tor curre nt tries to increase in any one device. ns basc -emi uer va llage decreases. offse tting the negat ive Te . and ther mal runaway is avoided . Many manu fact urers integrate such degeneration (ofte n known as ballas/i" g) in to the device struc ture so that none has 10 he added exte rna lly. Even SIl, it is not uncommon ttl observe temperat ure differences of IOGC or more in high-power amplifiers becau se of this po sitive feedback mechanism. Thermal runaway is normally not a problem in Ma S implementatio ns because mobility degradat ion with increasing temperature ca uses drai n curre nt to dim inish , rather than increase, with a fixed gate-source drive. A sub tle exception can occur if feedback control is used to force the gate-source vo ltage to increase with tempe rature to mai ntain a constant drive curre nt. In that case. device losses increase with temperatu re, reviving the possibil ity of a thermal run away scenario. For either bipo lar or ~1a S PAs. it is ofte n prudent to includ e some form of thermal protection to guard again st overload. Fort unatel y, it is trivial in Ie implememutio ns to measure tem peratu re with an on -c hip thermomet er and arrange to red uce device drive according ly.

13 .10 . 5 l A RGE-SIGN AL IMPEDAN CE M ATCHING Although the maximum pow er transfer theo rem is useless in design ing the output ci rcuit of a powe r amplifier. it does have a role in designing the inp ut ci rcuit . Although we have co nsidered Ma S impleme ntations almost exclu sively. it is worth men tioning that driving bipo lar transistors at the large signallevels fou nd in PAs pre se nts a serio us challenge if a decent impedan ce match to the driving source is to he obtaine d. Since the base- cmi uer j unction is. after all. a diode, the input im pedance is highly nonli near. Recognizing this difficulty, manufacturers of bipo lar transistors ofte n specify the input impeda nce at a spec ified power level and freq uency. However. since there is general ly no reliable guide as to how it might change with power level or other operating conditions, and is not eve n guaranteed at any set of cond itions, the designer is left with limited design choices.U The trad itional way of solving this problem is to s wamp OUt the no nlinearity with a s mall -valued resistor con nected fro m base to e mine r. If

21 This despne helpful application s notes from device manufacturers with optimistic titles such as "Syste matic methods make C lass C amplifier design a snap."

CHAPTER 13 RF POWER: AMPLI f iERS

376

the resistor is small e noug h. its resistance dominates the input im ped anc e. In highpower design s. thi s resistor can be as smull as a few ohms or less. so this gives you an idea o f how big: the problem is. In ge neral. bipolar po we r a mplifiers are "twitchie r" than other types beca use o f thi s input nonlinearit y as we ll as ou tput sa tura tio n effects. ~ I O S F ETs presen t d iffer en t challe nges. hut they are usuall y e usie r to man age. Th e bot tom line : lf you wan t to optimize a design. expect to do a fair a moun t of cuta nd-try. Bipol ar Class C a mplifie rs ge nerally require the most iterations; ot her types, fewer. Finally, statutory require me nts on spec tral purity ca nnot always be satisfied with

simple output structures such as the single tanks used in these examples. Add itional filter sections usually have to be cascaded to guarantee accep tably low distortion. Unfortunately. every filter inevitably adds so me loss. In this context. it is important to keep in mind that j ust I dB of attenuation represents a whopping 26% loss. Assiduous atte ntion to managing all so urces of loss is therefore required to keep efficiency high.

13.10 .6 LOA D-PULL CHARACTER IZATIO N OF PA , All o f the examples we've considered so far assumed a purely resistive 50-0 load. Unfortunately. real loads arc rarely purely resistive, exce pt perhaps by acc ident. Antennas in particular hardly ever present their nominal load to a power amplifier, because their impeda nce is influenced by such uncontrolled variables as prox imity 10 other objects (e.g.• a human head in ce ll-phone applications). To explore the effect of a variable load impedance on power delivered. one may systematically vary the real and imagina ry parts of the load impedance and plot contours of co nstant output power in the impedance plane (or. equiva lently. on a Smith chart). The resulting contours are collec tively referred to as a loud-pIIII diagram. The approximate shape (If a load-pull diagram may be derived by continuing to ussumc that the output transistor behaves as perfect controlled -c urre nt source throughout its swing. The derivation that follows is adapted from the classic paper by S. L. Cripps.22 who first applied it to GaA s PAs. Assume that the amplifier ope rates in Class A moue. Then the load resistance is related to the supply voltage and peak drain CUITCnt as follows: H

""

---. _ 2Vf)v

(601

I /) . pk

with an assoc iated output power of

(611

n MA Theory for the Prro i, tinll of GaAs 19!!J, PI'. 22 1-J.

H:~.I

Load-Pull Power Comours," IEEE !tIf T·S lJi.~J/.

13. 10 A DDITIONA L DESIGN CO NSIDERATIO NS

377

Now. if the magnitude of the load imped ance is less than thi s value of resista nce. the output power is limited by the c urren t I n,pl . Th e po wer deli vered to a load in this curren t-limited regime is therefore s imply: (62)

where R I . is the resistive compo nent of the load impedance . The pea k drain vo ltage is the product o f the pea k current and the magn itude of the load impedance :

Vpl = l o.pt. · / Hi

+ xl.

(63)

Substituting for the peak drain c urrent from Eqn. 60 yields (M)

To maintain linear ope ration. the value of Vpl must not exceed 2V/)n . Th is req uirement constra ins the magnitude of the reacti ve load compo nent:

(65) The interpretati on of the preced ing seq ue nce of eq uatio ns is as follows: For load impeda nce magnit udes smalle r tha n R,rp. the peak OUIPUI c urre nt limit s the power: contours of consta nt out put power a re lines of con stan t resistance HI. in the im pedance plane. up 10 the reactance lim it in Eqn . 65 . If the load imped ance magnitude e xceeds Ro[X then the power deli vered is con strained by the supply volta ge . In thi s voltage -swing- Iimited regi me. it is more convenient to con sidcr u load admittance. rather than load impedance. so that the powe r delivered is

PI. =

[ V~n

r

GL •

(66 )

where 0 ,. is the co nducta nce term of the output load admitt ance. Followi ng a meth od a nalogo us to the previou s case. we co mpute the drain current as

;lJ = 2V(){)/ Gl

+ 8;.

(67)

where 8 ,. is the susce pta nce te rm of the out put load admi ttance. Th e maximu m value thai the drain current in Eqn. 67 may have is

(68) Substitut ing Eq n. 68 into Bqn . 67 a nd solving the ineq ua lity yie lds

(69)

378

CHAPTER 13 Rf POWER AMPLIFIERS

The interp retation of the foregoi ng eq uations is that. for load impedance magnitudes larger than R,'f'4 ' contours of constant power are lines of constant conductance G L • up to the susceptance value given by Eqn. 69. The co ntours for the I WO impedance regimes togeth er co mprise the load -pull diagram .

13 .10.7 LOAD · PULL CO N TO UR EXA MPLE To illustrat e the proced ure. let 's construc t a load -pull dia gram for the earlier Class A amplifier ex ample. in whic h the peak volta ge is 6.6 V and the pea k current is 1.65 A. leading 10 a 4 -0 R,>pI ' In order 10 find the locus of all load ad mitta nces ( impedances) that allow us 10 deliver power with in. say. I dB of the optimum design value. we fi rst compute that a I-d B deviation from 4 n corresponds to about 3.2 Q and 5.0 Q . The former value is used in the current-limited regime . and the latter in the voltage -swinglimited regime. In the current-limited regime we follow the 3.2-Q constant-res istance line up to the max imu m allowabl e reactance magnitu de of abo ut 2.6 Q . whereas in the swinglim ited regime we follow the constant co nductance line of 0 .2 S up to the ma ximum allowable susceptance magnitude of 0 .15 S. Rat her than plotti ng the contours in the impedan ce and admittance planes. it is customary to plot the diagram in Smith-c hart form . Since circles in the impedance or admitta nce plane rema in circles in the Smith chart (and lines are co nsidered to be circle s of infin ite radiu s). the finite-length lines of these co nto urs become circular arcs in the Sm ith chart . The co rrespo nd ing dia gram appears as Figu re 13.20. here normali zed to 5 Q and 0 .2 S ( instead o f 50 Q and (J.02 S) in orde r to make the contour big enoug h to see clea rly. The power delivered to a load will therefore he wit hin I d B of the maximum value for all load imped ances lying inside the intersection of two ci rcles: on e of constant resistance (whose value is I d B less than the optim um load resistance) and the other of constant co nductance (w hose value is I d B less tha n the optim um load co nduc tance). Note from this de scription that one need not co mpute the reac tance or susceptance magn itude limits; the intersection of the two circles automatica lly takes care of this computation graphically, Hen ce. con struction of theoretical load-pull diagrams is cons ide rably eas ier than the detai led derivations might imp ly. It should be emphasized that the foregoing developm ent assume s that the transistor behaves a" an ideal. para sitic -free. controlled current source. Device and packag ing narasitics. combined with an external load impedance. co mprise the total effectiv e load for the diagram. In co nstruc ting practical load -pu ll diagra ms, however. nne has knowledge onl y nf the externally impo sed impedance. Hence. load -pu ll contou rs based o n the externa l impeda nce values will ge nera lly be tran slated and rotated relative to the parasitic-free case. Thi s minor incon veni ence may be ex ploited 10 help extract the precise val ue o f these parasitic'> . since one know s that the ce nter of

13.11

379

DE SIGN SUM MA RY

...

.- . .".,• '

'.

"

".

_.. . .... ..,-.... .. ._.. ...,.. .. ,.. ....,....,..- .. " .. .. . . ., ,., -.

~_

,

,

,

.....

" "

"

" '." , ' " , - ',

..

,:O--~-

"

-,

u

"

.,

" ,." '"

FIGURE 13. 2 0 . I -dB load-pull contour (normalized to 5

...

'

.. "

"

"" "

I

. ...

, .,

..., ... ,

,

.

I

, , ,. .

.

nl far Class A amplifier example.

the contour (utter co rrection for the parasitics) must pass through the rea l axis of the Smith chart .

13.11 DESIGN SUMMARY We have seen numerous examples that trade efficiency for linearity and other characteristics. As a co nsequence, one type of power amplifier cannot satisfy all possible requirements.

380

CHA PTER 13 RF POWE R A MPLIfi ERS

Th e va riou s am plifier topologies that have evo lved to accommodate the broad ra nge of requireme nts spa n the ga mut from POOf efficiency a nd high linear ity to high efficiency and low linearity (alas. the high ly efficient- highly linear amplifier has yet 10 he invent ed ). Class A a mplifiers provide the best linearity and WOI'SI efficiency. where as sw itching a mplifiers offer the be st efficiency a nd worst lineari ty. The max imum power tran sfer theorem was see n to he largel y usele ss in the design of the output cir cui t of a I'A . although usefu l in designing the inpu t circu it. Instead. one typically designs to supply a speci fied amount of po wer to the load. and evaluales later ..,. het he r efficie ncy. ga in. linearit y. and robu stness arc acceptable. ite rating as nec essary, Finall y. the need to determi ne se nsitivi ty to real-world load impedance vari ation led to the developme nt of the load-pull diagram. w hic h allows one to assess rapidly how the pow er delivered degrad es as the real a nd imagi nary part s of the load vary.

PROBLEM SET FOR POWER A M PLI FIERS PROBLEM I Co nsider the problem of de termin ing the requ ired de vice w idth for a power amp stage , Specifica lly. suppose that yo u are to design a device fo r use in the l -W C lass A amplifier example in this chapter. Recall tha t the device's maxi mum allowed resistance in the conducting sta te was estimated at around 200 mQ . (a ) De rive a ge neral e xpre ssion for the required width. gi ven an arbitrary resistance specifica tio n. Use the a nalytic al MO SF ET rnodel s fmm C hapt er 3 that take shortc ha nnel effects into accou nt. ( b) Wit h your a nswer to (a) as a guide. es timate the speci fic w idth necessary 10 SUIisfy the 20(J-mQ co nstraint if the ma ximum gate drive is 3.3 V and the threshold voltage is n.7 V. Assume a n effe ctive c ha nnel len gt h of 0 .35 11m. a Co\ of 3.85 mF j m 2 • a mo bili ty of 0 .05 m 2jV_s. and a n E"'l of 4 x 10" Vim . Verify your answer wit h the le vcl- J N MOS device mod el s fro m C hapt e r 3 a nd ite rate if necessary In lind a n acc urate value. (c) If the previou s stage is indu ct ively loaded . the pea k dri ve voltage could potentially dou ble. w hat device wid th wo uld he suffici ent in suc h a case ? (d ) A pproximat ely speaking. how does the req ui red device w idth sc ale wit h supply vo ltage if output po wer and losses du e to transi stor resi stance are to rem ain constnnr? Assume a fixed dev ice tec hnology, Com me nt qu alitativel y on how your answer would c ha nge if the effec tive c han nel le ngth and oxide thickness both scaled with supply vo ltage.

PROBLEM 2 Co nsider a 5OU-JAm x U.5-J1 1Tl (drawn ) tra nsistor used as a power amplifier in wh ich the drain is allowed to sw ing frum ground to 5 V. Using the level-J model s from C hapter 3. plot C gd and C dh as a func tion of drain voltage over this

PROBLEM SET

381

range for a ga le vo ltage of 0 V. 2.5 V, and 5 V. Explain how these variations might affect the performa nce of a power amplifier . PROBLEM 3

(a) Com ple te the " more co mplete" I-W Class A amp lifier example o f Figure 13.16 . Assume that the mirror ratio n is 10 , so that the reference bia s curre nt is 82 .5 mA (neglectin g mirror errors) . Estimate the spec ific width necessary In satisfy the 200 -mQ consuuinr if the maximum gate drive is 6 .6 V and the threshold voltage is 0 .7 V. Assume an e ffective cha nnelleng th 01'0.35 li m , a C... of 3.K5 mF/ m2, a mobility 01'0.05 m 21 V-s, and an E_ of 4 x (00 Vi m. Finally, chouse the inp ut co upling capacitor so that the voltage attenuation acro ss it is less than 0 .3 d B. (b) Simulate the de sign using the level-S models from Chapter 3. Whal input signal amplitude is necessary 10 deliver I W to the load? (c) To what power does the voltage in ( b) correspond? w hat is the power ga in of this amplifier'! PROBLEM 4 Load -pul l experi ments are one important way 10 characterize real power amp lifiers . Th e deriva tion given in the chapter ignores parasitic, uf the device and of the packagin g, so the . . ample co ntour shown applies 10 the impedance of the extemal load combined wit h these parasitic s. Since rea l. pac kaged powe r amplifiers have nonze ro pa rusirics. foad -pull con tours based on the externa l load impeda nce do nul look quit e like the exam ple given.

(a) Exp lore this co nce pt furthe r by rc -dcriving the load -pull contour construction rules if the dev ice and pack aging parasitics may be modeled as a shunt admittance Y = G + j lJ. Hint: Co nverting to an equivalent series im pedance may t'IC helpful for half o f the con tour. (b) As a speci fic exa mp le, suppo se that for the I-W, l -Gll z C lass A PA example the combination of device ami packa ge parasitics may be modeled as shown in Figure 13.2 1. In this model. the capac itance represent s the output capac itance of the device (which , in turn, is modeled as the current source show n). while the inductance represe nts bondwire pamsitics between the package pin and the device itself.

fi GU RE 13 . 21. Simplified

output model.

-

3.'

CHAPTE R 13 RF POWER AMPLIF IER S

Suppose thai C'llII is 2 pF and the bo ndwire indu ctance is 2 nH . Re- draw the load-pull contour. As with the original exa mple. normalize 10 5 Q 10 keep the contour co mfortab ly large. PROBLEM S In the Class C PA exa mple. the zero bias voltage is esta blishe d with a triode -co nnected PET. Here we conside r vary ing the bias 10 values ot her than zero.

(a) Re -design the bias network to allow an external bias source to vary the gate bias from 0 V to increasingly negative values. (h ) Plot th e conduction angle. efficie ncy. voltage gain, and drain supply powe r versus this bias voltage. Co mment on how these parameters trade off among each other as the co nduction ang le vanes. PROBLEM 6 Low distortion ca n be extre me ly importa nt in ce rtain applications. depend ing on the type of modulatio n used . Cl ass A or AD amplifiers are usuall y used in such cases. For the I-W C lass A PA exa mple. assu me tha t the output device is 4 -0101 wide and is built ou t of the technology mod eled with the level-3 parameters given in C hapter 3. (a) Plot the outpu t power as a function of the am plitude of the ga te dri ve voltage . To simplify the simulations you may. if you so desire. drive the transistor with an ideal sinusoidal vo ltage source with an appropriate DC offse t to se t the bias correctly, (b) What is the I-d B output-referred co mpress ion poi nt? (c) Wh at is the voltage ga in under the cond itions in (b) ? Co mpare with the maximum ga in and com ment. PROBLEM 7 output.

Another measure of ampli fier linearit y is the harmoni c conte nt of the

(a) For the circuit of Problem 6. what are the second and third harmonic co mponents of the ou tput at the I-dB ou tpu t compression poi nt? ( b) Repe al part (a). but at hal f the output power. (c) Repeat pa rts (a) and (h) if the output filter induct ance has a Q of H l rnodeled us a resistance in series with the inductance . PROBLEM 8 In FM a pplicatio ns. gain linearit y is relatively unimportant . be t phase linearit y is ex tremely critical bec ause information is encoded as the tim ing of the zero cro ssings . If the phase response is high ly nonlinear. distorti on of the modul ation results. Hence. aside fro m the usua l preoc cupan on s with gain amI enidell\:y.lhc de signer o f FM systems mu st also worry about phase di stortio n. (a) Re -des ign the filter in the I _W C lass C amplifier exa mple for maximally flat time delay (you may wish to rev iew the ma terial un this topic fro m C hapter 8). What Q for the output tank yields this con di tio n?

PROBLEM SET

3.3

(b) Comple te the design by selecting the width of the output device sufficie nt to guarantee a maximum resistance of 200 mO in the co nducting state (with peuk voltage o f 6.6 on the ga te). Use the level-S models from C hap ter 3. (c) Simulate your design with SPICE . driving the output transistor with an idea l square- wave voltage source with an amplitude of 6 .6 V. Over what frequency range does this ampli fier provide a time delay constant to within IOC)f,'! PROB LEM 9 As supply voltages dimi nish. it becomes increasingly difficu lt to de. liver reasonable amo unts of power to a load efficiently, beca use the load impedance must be transfor med to ever-lower values and so tolerable parasitic resistances thus scale at the same time. If an ordinary single-ended stage is used. the requi red device width ca n quickl y grow to absurd values.

On e way to forestall the day of reckoning is to employ a bridged (differen tial) out. put stage in which the load is connected between two amplifiers driven out of phase. That way, the voltage swing across the load doub les. and the power delivered to the load therefore may qua druple for a given supply volta ge. (a) If the width of the ourpur rransistor in a si ngle-ended PA is \Y . how wide should eac h device in a differential PA be made in order to maintain eq ual total on-state losses? (b ) Suppose yo u must deliver I W of power to 3 50 -0 load with a 1.5-V supply, Whal device width would you need to use in a si ngle -ended Class A design to keep the on-state resistance eq ual to 5% o f the load impedance seen by the transistor? Usc the level- J device models from Cha pter J . (c) Combine your answers 10 (a) and ( b) to estimate the width of each device for a differen tial design. PROBLEM 10 C rossover distortion is a prob lem of push-pu ll amplifi ers. many of which are biased as Class AH stages. This problem ex plores some approximate ways to predi ct the crossover distortion of such am plifiers. Assume that an amplifier with crossover distortion produces no output until the magnitude of the input signal exce eds some critica l threshold . Beyond thai threshold, the output follows the input, hut with an offset. Spec ifically. assume thai the amplifier may he modeled as a simple black box whose output, given a sinuso idal input drive. behaves as follows: Vn"l = ViR - e for ViR > s: Vnu l = 0 for IV iRI < f'; and VOUl = Vin + e for ViR < - E: . Assume that 0 < E: < I. What values of s yield a third harmonic component whose amplitude is n . 1%. 1% , and 10% of the fundamental'! PROBL EM 11 In all of the examples given in the chapter. only single-transistor output stages are considered. In ge neral. however, one musr cas cade several stages in

3• •

CHAPTE R 13 I f PO WER AMPLI FIER S

order 10 obt ain sufficie nt overal l power gai n. If linearity is important , it is c ustomary for the early stages 10 be operated as Class A amplifiers and the lasl 10 operate per haps as a Class AB stage (for efficiency ). Expand on the "more complete" ]. W. I-G i lLC lass A amplifier of Figu re 13.16 by precedin g it with enoug h additional ga in stages so that J W is delivered 10 the ourpu t load wit h a l -mW overall inpu t. Perh aps the first ste p in carrying: out the design. then. is to size the ou tpu t tran sistor. Design the imerstage coupling networks so as to maximi ze gain (another cho ice would he 10 maximize linearity. bUI this is an extremel y involved. itera tive process). Use the level-S device mod els from C hapter 3. What is the overall efficie ncy of the amplifier < measured here as the ratio of the powcr deliv ered 10 the load - to the IotaI power supplie d by the DC source to all stages'?

CHAPTER FOURTEEN

FEEDBACK SYSTEMS

14.1 IN TRODUCTION A solid understandi ng o f feedb ack is critica l 10 good circuit design. yet many pmctiei ng engineers have at best a (COUOUS gra...p of this import ant subject. This chapte r is intended as an overview of' the fou ndations of classica l control theory - that is. the study of feedb ack in single -input. single -output. time -invariant. linear co ntinuouslime ... ystems. We' ll sec how to ap ply this knowled ge to the design of oscillators. highly linear broadband amplifiers. and phase -loc ked I(K)PS. amo ng other examples. We'll also sec how (0 extend our des ign intui tion 10 incl ude many non linear systems of practical interest. As usual. we ' ll start with a little history 10 put this subject in ils proper co ntext.

14. 2

A BRIEF HI STORY OF MODERN FEEDBACK

Although application of feed back concepts is very ancient (Og annoy tiger. tige r ea t Og), mathcmaticut trcarments of the subject arc a recent de velopment . Maxwell himself offered the fi rst detailed stability analyses, in a paper 011 the stability of the rings of Salurn (for which he wo n his fi rst mathematical prize). and a later one o n the stability o f speed-c ontrolled steam engines. The first consc ious applica tion of feedback principles in electronics was apparently by rocket pioneer Robe rt Goddard in 191 2. in a vacuum tube oscillator that employe d po sitive feed back .- As far as is known . however. his patent applicat ion was his on ly writing on the subject ( he was sort o f preoc cu pied with that rocketry thing. after all), and his conte mporaries were largely ignorant of his wor k in this field. I

u.s. Pa ll"111it l , JS9,2f)lJ, filed

I AUf' U ~1 19 12, gra nted 2 No vem ber 19 15.

385

.. CHAPfER 1-4 FEEDBACK SYSTE MS

386

'IN ---~

1--- -,--

a

~ vOTJT

f FIGURE 14.1. Positive feedback ampl i~er block d iagra m.

14 . 2.1 ARMSTRO NG AND THE REGENERATIV E AMPLI FIER Armstro ng's 19 15 paper? on vacuum tubes contained the first published expla nation of how positive feedback (regenerat ion) co uld he used to gre atly increase the \'011age ga in of ampli fiers. Although engi neers tod ay have a preju dice aga inst positive feedback , pro gress in electronics in those early ye ars was largel y made possible by Armstrong's regenera tive amplifi er, since there was no other eco nomical way to obtain large amounts of gain from the prim itive (and expensive) vacuum tubes of

me

day.' We can apprecia te the esse ntial features o f Armstron g' s ampli fi er by examining the block diagram of Figu re 14. J. where the quantit y a is known as the forward gain and f is the feedback gain. In our part icular example, a represents the gain of an ordinary (i.e . open-loop) single vacuum tube ampli fi er, while f represents the fraction of the output voltage that is fed had to the amplifier input. Since we have the block diagram . it 's straightforward to derive an expression for the overall gain o f this amplifier. First. recog nize that 0 = V IN

+f

- vovr-

(I)

Next. note that t'otrr = a '

0

=

n . (V IN

+ f ' VOlJf) ·

(2)

Solving for the input-output transfer function yields

A= -

I

-"at -

.

(3)

2 "Some Recent Developments in the Audion Receiver ," Proceedings of the lRf, v. 3, 1915, pp,

215- 41. } The e ffective internal "g",r,," of vacuum tulles t>ack then wa.. on ly on the order o f 5. Hence the gain per ..tage was typ ically qu ite IlIw, requiring many stages if conve ntiona l ttlpolog ie~ were used.

14. 2 A BRIEF HISTORY O F M ODE RN FEEDBACK

387

It is evide nt that any positive value o f af smaller tha n un ity gives us an overall gain A that exceed s a. the "ordinary " ga in o f the vacuu m tulle amplifier. If we make af r..q ual to 0.9 then the overall gain is increased 10 ten limes the ordinary ga in. while an af product of 0.99 gives us a hundred fold gain increase. and so on . In this way. Armstrong was able 10 gel gain from a sing le stage that others co uld obtai n only by cascad ing several. Thi s achievemen t allowed the co nstructio n of relat ively inexpensive. high-gain rece ivers and there fore also enabled dramatic red uction s in transmit ter power because o f the enhanc ed se nsitivity provided by this increased gai n. In short order. the po sitive feedback (rege nerative) amplifier bec ame a nearly universal idiom. and Westingbousc (to whom Arm stron g had assigned pate nt rights ) kep i its legal staff quite busy tryi ng to make sure that only licen sees were using this revolutionary techno logy.

14 . 2 . 2 HA ROLD BLACK AND THE FEE DFO RWA RD A M PLIFIER Althou gh Armstron g's regenerative amplifier pretty much solved the problem o f obtaining large amoun ts of gai n fro m vacuum tube amplifiers . a differen t prob lem preoccu pied the telephone indu stry. In trying to ex tend communicati ons distances, amplifiers were needed to co mpe nsate for transmissio n-line atten uatio n. Us ing amplifiers ava ila ble in tho se ea rly days, d istances of a few hu ndred mi les were routinely achievable and. with great care, perhaps 1000 - 2000 miles was possible. but the q uality was JX)or. After a tremendous amount of wnrk , a crude transcon tinen tal te lephone service was inaugurated in 19 15, with a 68-year-o ld Alexander G raham Bellmaki ng the first call to his former assistant , T hom as Watson, but this feat was mo re of a stunt than a practical achievem en t. The problem wasn 't {Inc of insufficient amp lification; it was trivial to make the signal at the end o f the line qui te loud. Rather , the problem was distortion, Eac h amplifier contributed so me small (say, I % ) distortion. Cascading a hundred of these things guaranteed that what came out didn 't very much resemble what went in. The main "so lution" ut the time was to (try to) guarantee "sma ll-signa l" operation of the amp lifiers. That is. by restrictin g the dynami c range of the signals to a tiny fraction o f the amplifier's overall capability. mo re linear ope ratio n co uld be achieved . Unfortunately, this strategy is quite inefficient since it requ ires the construction of, say, I(X)· W amplifiers to process milliwan sig nals. Becau se o f the arbitrary dis tance betwee n a signal so urce and an amplifier (or possibly between amplifiers ), though . it was d ifficult to guarantee that the inp ut signals were always sufficiently small ro satisfy linearity. And thus was the situation in 1921. when a fresh graduate of Worcester Polytechnic named Harold S. Black. joined the forerun ner of Bell Laboratories. He became

.

388

CHAPTE R 14 FEEDBACK SYSTEMS

{;>>-- - -:;:- - - - - I '------- ~

_?-_A"N+b

b >-''-+ '-''-

(ll«t )

fi GURE 14 . '2. Feedlorwo rd amplifier .

aware of this di stort ion prob le m and devoted much of his spare time to figu ring out a way 10 solve it ." A nd solve it he did. Twice . His first solution involves what is now known as feodforward con~'cl ifJIl. ~ The basic idea is to build two ident ica l amplifier s and usc one a mp lifier to subtract nut the di sto rtio n of the first . To see how this cun he accomplished , co nsid er the block diagram (Figure 14.2) of his feedforwnrd amp lilic r. Notice that there is no feed back at a ll; signals move Hilly fo rward fmm input to o utput . as suggested by the name of this ampli tication technique. Eac h amplifier has a nomi nal gain of A. but may he no nlinear 10 so me degr ee. To distinguish a nonlinear ga in from a perfectly linear on e. we use the symbo l ri. The first amplifier takes the input signal and provid es a nomina l ga in of A , but prod uce s some dis tortio n in the pnl(..ess . Hence, its output is AL'IN plus an error volt age denoted by e. We assume that the amplifier is linear enough that e is small compared wit h the des ired output At'IN. The output o f the first amplifier also feed s a pe rfectl y linear ane nuator. whose gain is If A. The uttenuaror output is then subtrac ted from the inpu t to yield a vo ltage thai is a perfectly scaled version of the distortion . T his pure distortion s ignal feeds another amplifier idcn ticul ro the first one . Beca use we have assumed that the distortion is small in the first place. we expect the sec ond amp lifier to act qui te linearly and thus pro duce an exce llent approxim ation to the original distorti on (i. e.• Ii « F ) . That is. we assume that the error in co mputing the error is itself small. Th e distortion signa l from the seco nd amplifier is subtrac ted from the distorted signal o f the first amplifier to yie ld a fina l outpu t thut has greatly red uced distortion. Anothe r feature is that of redunda ncy, for even if one amplifier fails there still reo mains some output (j ust wi th more distorti on ).

The only one of hi ~ da~\ of new hires to toe pa...~-d over for a 10% pay raise Ihree mo nths aha Marting. Btal:l lll.'arty quitto pursue a career in Nl..i ne~ ... He reco nsidered at the la..1 minute. and decided instead 10 make his marl by solving this crnical prob lem . ~ U.S. Parent ~1 .b86.792 . tiled 3 February 19L'i. granted I} Octobe r 1928.

.I

14.2 A BR IE f HISTORYOf MO DER N fE ED BACK

389

1-- -,-- .. vOUT

a

f FIGUR E 14.3. Negative feedback ompl i ~er block diagram.

Black buill several such am plifiers. hut they proved impractical with the rcc hnology o f his day. He was en co uraged by the pos itive results he obtained .....hen every thing was adjusted right , hut it was virtually impos sible to maintain the tight levels of matching he needed to ma le a feedforward amplifier work well all the time . A goa l o f 0. 1f,l- distortio n. for exa mple. requi res mat chi ng to similar levels, and discrete vacuum tube technology simply co uld not offer this level of matchi ng on a sustainc-d basis. 14. 2 .3

THE NE G ATI VE FEEDBACK A M PLIFI ER

While understandably disappointed with the practical barriers he faced with the feed forward amplifier. the basic notion of measuring and ca nce lling out the offending error terms in the output seemed worthw hile. The practical problem with fccdforward was in using two separate amplifi ers to acco mplish this cance llation. Black began to wonder if one could perform the necessary cance llation with j ust o lle amplifier. That way. he reaso ned. the issue of matching would d isappear. 11 j ust wasn' t d ear how 10do it. Then came the fatefu l da y. On August 2, 1927, while taking the Lack awanna Ferry on the way to work
+" (If -

.

(41

Now make the (If prod uct vel)' muc h larger than unity. In this case,

15)

b H. S. Black, 55--60.

~ I nvent i ng

the Negative Feed back Amplifier," IEEE S/'('c1rum . December 1977, pp.

-

390

CHAPTER 14 FEEDBACK SYSTE MS

A s Black o bse rved. the feed back fac tor f ca ll be implem ent ed with pe rfec tly lincar elements. such as resistive voltage dividers. so that the overall closed -loop behavior is linear even thou gh the amplifier in the block a is not. That is. it doesn't matte r th al li ex hibits a ll so rts of nonlinear be hav ior ay, long as of » t under a ll conditions of interest . The only tradeoff is that the ove rall. dosed-loop gain A is much sma lle r than the fo rward ga in u, Ho wever. if ga in is c heap bUI low distortion isn 't. the n nega tive feedback is a marvelo us solut ion to a very di fficult pro ble m. A s o bvio usly wonderfu l the idea of negative feedback is 10 us today. it was not ut all o bvio us to Blac k's co nte mporaries . It wa s difficu lt to co nvince ot hers that it made sense to work hard to design a hig h-gain amplifier. on ly to redu ce the gai n with feedback . Th e nega tive feedback amplifier rep resented so great a dep arture fru m prevailing practice (remembe r. the Armst rong positive feed back amplifier was then the dominam archi tecture) that it too k " dozen years for the British patent office to issue the patent. III the intervening time. they argued that it could not work , and c ited a lot of prior art to " prover their po int. Black (and AT&T) finally won in the end. but it did take some doing.

14 . 3

A PUZZLE

If you' ve bee n paying attention . you should he a bit co nfused. Suppose one makes in the positive feedback amplifier. Then the math give s us the following resu lt:

"f »

(6)

It wo uld appear that either sign of feed back gives us a linear closed -loo p amplifier. So why do we prefer negati ve feedback ? The math is abso lutely. unassailably correc t. by the way, and the parado x cannot be resol ved within the framework esta blis hed so far. The problem lies in the implicit assumptions that lead to the math; they are not satisfied by physical systems, For the resolution to th is paradox. we now must con sider what happe ns if a and / are no t sca lar q uantities - thai is. if they have some frequency-dependent magn itude and phase. As we sha ll see sho rtly, it turns out that the pos itive feedback amplifier with of » I ca nnot be made stable because all rea l sys tems eventually ex hibit increasing negative phase shift with frequency. If nothi ng else, the finite spe ed of light guarantees that all physica l systems have unconstrai ned negat ive phase shift as the frequen cy increases to infinity. Becau se of the extreme ly important ro le that phase shift plays in determining stabilit y, we will spend a fair amount of time study ing it. Before doin g so, however. lei us ex amine a num ber o f co mmo nly held misconceptions abo ut negati ve feedb ack.

14.4

DESENS ITIV ITY O F NE G ATI VE FEEDBACK SYSTEMS

39 1

14.4 DESENSITIVIT Y OF NEGATIVE FEEDBACK SYSTEMS All sort s of wild claims ubout negative feedback e xist. " It increases b andwidt h"; " it decre ases distortion": " it reduces noise. and remove s unsigh tly facial blemishes," So me of these clai ms ca n be true, hut aren ' t necessari ly[urulamentulus negative feedback . As we' ll sec, there is act ually only on e absolut ely fundamen tal ( but ex traordinari ly important) benefit o f negative feedback systems. and tha t is the desensitivity provided . That is. the overall amplifier possesses an atte nuated sensitivity to changes in the forwa rd gain tI if t lf » I. In ord er to quant ify this notion of desen sinvny, lcr's ca lculate the di fferential change in A that results from a diffe renti al change in a:

=

,,

I

tltl

+ tlf

)

1

= (J

+ tlf>'!

A(

= -;; I

1

+ af

)

(7 )

'

We may rearrange thb expre ssion as follows: 7

(8) This last eq uat ion tells us tha t a given fractional change in A equa ls the frac tiona l change in a, at tenuated C'dcsensitizcd ") by a factor o f I + af. For this rea son, the quantity I + af is often ca lled the desensit iviry of a feedback sys tem." Tim.., if the forwa rd gain varie.. with time, tem pe rature, or inp ut amp litude , then the overall closed -loop gain exhibits smalle r variations since they arc att enu ated by the dcscnsinvity factor. If the factor a] i.. made extreme ly large, then the dc sen sitivit y will he large and variations in A due to changes in a will be greatly suppressed, Let 's perform u similar analysis 10 deduce how variations in the feedback factor affect the close d-loop system:
,, ( , ,

tlf = III

I

+ (If

)

= - (I

,, '

A(

"f )

+ nj) 2 = 7 - I ~ af

'

(9 )

so that. on a norm alized fractional basis, (1 0)

1 Mathemalicians cringe whe never engineers are this cavalier; we don' t worry aboersuch things. I " Return difference" is another term for this q uantity. Th is name deri ves from the observanon that , if we CUI the loop , squirt in a unit signal in one end, and see what dnhhles nut the lither, the difference is I + af.

392

CHAPTE R 14 FEE DBACK SYSTEMS

Here. we sec thai large de scn sit ivit y fac tors d o not he lp us as far as variations in feedhack arc CU IlCCTIl l'tl. In fac t. in the lim it of infinite desen suivi ty, the fract iona l change in A ha s the .\{III/f ' ma gnitud e as the fraction a l c hange in f . This res ult underscores the importa nce o f havin g linea r feed back ne twor ks if o verall d osed -loop linear ope ra tion is the goal (a s o ften . but not always. happen s 10 be the ca se) . For this reason. the feedbac k block is usua lly made of pa ssive e leme nts (commo nly resi stors and capaci tors) ra ther tha n other a mplifiers.

FACI A L BLEMISH ES But how a bout all of those other claims thai a re so com mo nly made about the benefits of negat ive fl.-cdba ck ? Let ' s exa mine them. on e at a time .

BENT CONCEPTION 1: Negative ferdbactc extends ba ndwidth, This ca n he true ( hut co ns ide r an important co untere xa mple, suc h as the Miller efteen. hut it ' s nut nearly a s magical a ... it so unds. If negative feedback were to ecco mphsh th i... bandwidt h ex te ns io n by giving us more ga in at high freque ncies, then there 'd be so me thing to writ e ho me about. But , as we' ll sec in a mo men t , negative feedbac k ex tends ba ndwid th by selectively throwing dlmy gain lJ1 Iowa frequencies. We wi ll dem o nstrate that o ne ma y accompli....h preci se ly the sa me thi ng throu gh the of pure ly open-loo p means. To sec how negative fee d bac k may ext e nd band wid th , let us suppos e tha t the forwar d gain is now not a pure ly sc ala r q ua nt ity hu t is inste ad so me d( S) tha t rolls off wit h s ing le- po le be ha vior. Now. we sa id tha t a s lo ng as tlf had a mag nitude large co m pared with unity. the closed -loop gain wa ... approx ima tely eq ua l to the reciproca l of the feedbac k gai n. We ca n also see that , in the limit of very small u], the closed-loop and for wa rd ga ins conve rge . All entire ly equi valen t de scri ption is: Plo t la(.\")1a nd Il/fl o n the sa me g raph. A good ap pro ximation 10 lA(s) 1ca n be pieced toget her by c hoosi ng the lower of the IWO c urve s. EXl'lwllIti oll: If II/f l is m uc h lo we r th an 1"(.\·) l lhe n this im pl ie s that (/(s )! has a large magn itude. a nd therefore the closed -loo p beha vior is approx ima tely II! (the lower c urve) . If II/II is mu c h hig her tha n 1(/ ( .I' ) I. it mea n" that a (.\.) f has a small magnitude and the c losed-loo p beh avior converge s to II (.I") (sti II the lower c urve ). In the region where 11(.1') and 1/! have sim ilar m agn itude s. we ca n't be sure of what happen s preci se ly. but we c an g uess and cl a im that som e sort o f rea so nable approxima tio n might be obtai ned by cont mumg 10 choose th e lo wer o r ure IWO c urves. Ap ply ing this proc ed ure to o ur sing le- po le ex am ple gc nc nncs Figure 1 ~ .4 {we have used a straig ht-li ne Bode a ppro xi mario r to the act ual s ingle- po le curve). As yo u c an see, the re sponse fonned by concaten aing the lower (If the two c urve s does indeed have a high er corne r freq uen cy than Joe ~ tJ( s) , but nega tive feedback has U
14 .4

393

DESENS ITIVITV OF NEGATIVE f EE DBACK SYS TEMS

Magnitude al s l

Ilf

log llJ

FIGUR E 14.4 . Bandwidth: Als) versus ols}.

' ' -~ -L-: ~ 1, --

FIGURE 14.5 . f eedback system with additive roi se sources.

"0'

accomplished this extension o f bandwidth by reducing gain at lowe r frequ encies. by giving us any more ga in at higher frequencies. Finall y. to see that there is nothing special abo ut negative feed back in the con text of bandwid th extension. co nsider that a ca pacitively loaded resistive divider ca n have its bandwidth extended simply hy placing another resistor in parallel with the capacitor. The bandwidth goe s up. hut the gain goe s down. QED MISGUIDED NOTION 2: Negative feedback reduces noise.

Actually. as we will see whe n we study the issue of noi se in detail. feed back can never prov ide less noise than an otherw ise equ ivalent ope n-loo p amplifier. In fact . the best it ca n do is give you the same noise; wha t's more, in most practical amplifiers. feedba ck typically increases noise. The idea that negative feed back magicall y redu ce s noise ster ns from an incom ptete understand ing of the noise properties of the type of system shown in Figure 14.5. For such a system, the individual tran sfer functions arc:

39'

CHA PTE R 14 FEEDBACK SYSTEMS

FIGURE 14.6 .

Open-loop s~1em with odditive noise soc rces.

1'00.

-

=

Vin

a la 2

I + a,al! '

L'OUI

-= t 'lI l

L'OIII

t',,2

=

-" ~ = t',,)

U I" 2

I

(12)

+ dl tl2 ! '

"2

1 + lIl a 2! I

I II)

+ d11l 2!

'

( Il)

(14)

From these eq ua tio ns. we see tha t the ga in from noi se so urce 1'11' 10 the o ut put is the sa me as tha t from the input 10 the o ut put. Thi s result sho uld be no surp rise : the amphfier cannot distingu ish betwee n the inp ut s ignal and V II ]' as they happe n to enter the sys te m at the sa me poi nt. BUI the gai ns to the o ther two no ise so urces a re smalle r. so one m ight think thai the re's a benefit after a ll. In fact . all this o bservation pro ves is that noise e ntering befo re a ga in stage co ntributes mo re 10 the o utput tha n noi se e nte ring afte r a ga in stage. T his yaw n-ind uci ng resul t has nothing 10 do w ith negative fecdba... k. but on ly with the fac t tha t we happen to have ga in bet wee n two nodes wh e re noi se sig nals could e nter the sys te m. To under sco re the idea that negat ive feedbac k ha s noth ing to do wit h this result, consider the o pen-loo p struc ture uf Figure 14.6. Not e that , w ith the particular choice of K show n, the inp ut-output tra nsfe r fun ctions of the feed back a nd ope n-loo p amplifie rs are ex ac tly the sa me for every inp ut. Thus we sec that feedback offers no ma gi ca l noi se red uctio n beyond what open -loop sys te ms can provi de. Aga in. alt hou g h these properties are not fundame nta l tu negativ e feed back systerns. they may he conve niently ob tain ed through negative feedback . Impedance tran sfor mation. fo r example, ca n be pro vided by o pen - a nd closed -loop syste ms, but feedback implem e ntat ions mig ht be easier to c o nstruc t or adj ust in many instances. In sum mary, dese nsitivu y to the fo rward ga in is the o" ly in herent benefit conIerred by negat ive feedback syste ms. Negat ive feedback may a lso provide ot her benefits ( perha ps more, eve n much more, tha n pract ica lly obtainable throug h o pen-loop mea ns), hut desen sitivity is the only fundamental o ne .

14. 5 STA BILITY O F fEEDBA CK SYSTEMS

14.5

395

STABILITY OF FEEDBACK SYSTEMS

We have see n that use of nega tive feedback allows the closed-loop transfer function A( s ) to approach the reciproca l (If the feedb ack gai n! as (minus) the " loop tran smission" \l a (s )!( s) increases. there by co nferring a benefi t if ! is less subj ect to the vagaries of distortion ami parameter variation than the forward ga in a ( s) . as is o ften the case . As argued ea rlier. this red uction in sensitivity tll ll (.f) is actually the ( 111)' fundamen tal benefit of negative feedb ack; all others can be obtained (although perhaps less co nveniently) through ope n-loop mea ns. Now. large gains are triviall y achieved. so it would. appear that we co uld obtain arbitrarily large desensitivhles without trouble. Unfortunately, we invariab ly d iscover that systems beco me unstable when some loop transm i..ston magn itude is exce eded. And. as luck would have it, the o nset of instab ility frequently occurs with values of loop nun ..mission that are not parti cu larly large, Th us it is ;II.\,/tlbility - rather than the insufficiency of available gain - that usually limit,I the perfo rmance of feet/back system,I. Up to this point. we have discussed instability in rather vague term s. People certainly have some intu itive notions abo ut what is mean t, but we need something a bit more co ncrete to work with if we are to go further. As it happens. there are 2,6 zillion 1o definitions of stability. each with its own subtle nuances. We shall use the bounded -input, bou nded -output ( BIBO) defin ition of stability. whic h states that a system is stable if every bounded input produces a bou nded output. Although we shall not prove il here, a sys tem H ( .f ) is BIBO stable if all nf the poles of H ( s ) are in the oren left half-plane. In order 10 apply this tesr ro our feedba ck system. we must fi nd the poles of A (s ) , that is. the roots of p es ) = I + a( s )f( 5 ). A direct attack u..ing. SOly. a root fi nder is certainly an option. but we 're after the develop ment of dee per design insight than this direct approac h u..ually offers. Furthermo re. explicit polyno mial repr esen tations for U(.I) and f (s ) may not always he available, so we seck alternative methods of determining stability. All of the alternative method s we will examine fo cus on the behavio r of the loo p transmission. The vast simplification that result s CCllIII{If be overemp hasized. Determination of the loo p transmission is usually straightfo rward, whereas that of the closed-loo p transfer function req uires identification o f the forward path (not always trivial. contrary to one 's initial impression) and an add itional mathem atical step (i .e..

9 To find the loop lran ~mi ssi(}n , break the loop [after ~lI i ng all independenl sources 10 their u rn values) , inject a siplal into the break , and take the rauo of " hat co mes back 10 whal you put in, Foe our ca nonical negative fcedb ad, system bl ock diagram, lbe Ioui' Ir.ul!'> ll1issioll is - uf. 10 Aiiasl cou nt, plus or minu s, as reponed by the Bureau of Obsc ure and Generally Useless Suusliu( BOGUS ),

CHAPlE R 14 FEE DBACK SYSTEMS

396

FIGURE 14.7. Disconnected negative

feedbock ~Y5tem . lakin g the ratio o f a ( s) to I + (I( 5 ){( 5 ). Hence. any met hod thai ca n determine sta bility fro m exa mina tio n of the loo p transmission offers a tremendou s saving of labor.

14 . 6

GA I N AND PHA SE MARGIN A S STA BI LITY MEASURES

Co nside r cutting ope n our feedback system (sec Figure 14.7 ), Now imagine supplying a sine wave of MI llie frequ ency to the inverting termin al of the summing ju nction . The sine wave inverts there. then gets mult iplied by the magnitude of a(5 )f(s ) and shifted in phase by the net phase angle o f a( 5)/( 5) . If the magnitude of tl(.~)f( .f ) happe ns 10 be unity at this frequ ency while the net phase of ll ( .I" ) [( S) hap pens to be HID ' , then the out pu t of the f(s ) block is a s ine wave of the same phase und am plitu de as the signal we or iginally supplied. It is conceivab le. then. thut we could dispense with the o riginal input - a sine wave of this frequency mig11, be able 10 persi st if we re -cluse the loop. If the sine wave does survive. it means that we have an output withou t an input . That is. the system is unstable. To determine co nclusive ly whether such a persi ste nt sine wave act ually exists requires the use of the Nyqu ist stabiluy test. However . the derivati on of the Nyquist stability criterion is somewhat involved (although reasonably straightforward 10 apply). and this co mplica tion is enough 10 d iscourage mallYfrom using it. Instead of the Nyquist test. perhaps the stability mea sures mo...l often actuclly used by practic ing engineers are a subset of the Nyquist test. gain margin and phase nwr· gin. These q uantities are easily com puted as follo ws: ( I) Gain margin - Find the frequency OIt whic h the phase shift of a(jw )f(jw) is - I SO"; call this frequ ency W)f ' Then the gain margin is simply

I la (jw)f ) f ( jw,. )1

1151

14,6 GA IN AND PHASE MA RGIN A S STABIlITY M EASURES

397

(2) Plum ' margin - Find the frequen cy at which the magnitude of a (j w )f(jw) is unity. Ca ll this frequency W f ' (he crosso ver fre quency. T hen the phase margi n (in degrees) is simply phase margin = 180"

+ L(ll(jwc)f (jwcH.

( 16 )

As can be inferred from these definitions. gain and phase margin are measure!'. of how clo sel y a ( j w ) f( jw) approaches a magnitude uf unity and a phase shift of 1 80~ . the co nditions that could a llow a persis-tent oscillation. Evidently, these quantities allow us to spe ak abou t the rekuive degree o f stability o f a system since. the larger the margins, the furth er away the system is from unstab le behav ior. Because o f the ease with which ga in and pha se margin are calc ula ted (or obtained from actua l frequ ency response measurements). they are ofte n used in lieu of performing an actua l Nyqu ist test . In fact. mos t engineers o ften dispe nse with a calculation of ga in margin altoge the r and compute onl y the phase margin. However. it should stri ke yo u as remarka ble that the stability of a feedback sys tem could be deter mined by the behavior of the loop transmission at ju st one or two frequencies. M) perh aps yo u wo n' t be surprised to learn that ga in and phase margin are not pe rfectly reliable guides . In (act. there are many pa thological cases (en countered mainly during Ph .D. qualifying ex ams ) in wh ich gain and phase margin fail spec tacula rly. However, it is true that (or mosl commo nly encountered systems. stability ca n he determined rather we ll by these measures. If there is a question as to the applic ability of ga in and phase margin then one must use the Nyquist test, which co nsiders information about a( j w ) f (jtd) at al/ frequenci es. Hence. it can handle the pathological situations that occa sionally arise and that ca nnot be udcq uurely ex amined using only the gain and phase margin. II is important 10 remembe r. then . that the gain and phase margin tests art' only a subse t of the more general Nyq uist test , Thi s important point is freq uent ly overlooke d by practicing engineers. who are often unaware of the limited nature of gain and phase margin as stabi lity measures. This confus ion pe rsists because the stabilit y of the commonest systems happe ns to be well determ ined by gain and phase margin. ami this success enco urages many to make inappro priate generalizat ions. Having pro vided that all-im portant public service an nouncement. we can return to gain and phase margin . It is wort hw hile to point out tha t they arc easily read off of Bode diagra ms (i n fact. it is preci sely this ease that encourages man y designe rs to use gain and phase margin as stability measures); sec Figure 14 .8. Tha t is, expcrimcrually derived dat u may be used to compute ga in and phase margin : no explicit modeling step is neede d. and no transfer functi ons have to be determin ed . Okay. now that we ' ve deri ved a new set of stability criteria that enable us to quantify degre es of relative stability. what values are acceptable in design'! Unfort unately, there arc no universall y correct answers, but we can offer a few guide lines: O ne must choose a ga in margin large enough to accommodate all anticipa ted variations in the

F 398

CHAPTE R 14 FEE DBACK SYS TE MS

IOgIa(;ClI" ,,·" ~)~lt

~

crossover frequency

/

/ "" g.m.

- 18Q'!

•

pi

f iG URE 14. 8 . Gain and phose margin from Bode plo~ .

magnitude of the loop transmission without endangering stability. The more variation in ae I« thai o ne anticipates. the greater the requ ired gain margin. In most cases, a minimum ga in margin of about 3-5 is satisfactory. Simi larly. one must choo se a phase margin large enoug h 10 accommodate all anticip ated variations in the phase shift o f the loo p transmi ssion . Typically. a minimum phase margin of 30"--60' is acceptable. with the lower end of the range generally associated with s ubstantial overs hoo t and ringing in the step response as well as significan t peaki ng in the frequency response. Note tha t this range is quite approximate and will vary acco rding to the details of system com po.. .ition. For example. overshoot might be tolerable in an amplifier but unacceptablei n the landing controls of an aircraft .

14. 7 ROOT·LOCU S TECHNIQUES Gain and phase margin use the behavior of the loop transmission to determine the stability of the clo sed-loop system. We have already noted that a rational transfer function (or even an ana lytical expression o f any type) for the 1(X)p transmission is not need ed to apply the test. However. if one is given a rational transfer function, add itional path... to an...wering the ...tahility que...tion beco me available. As noted earlier, one obvious method is simply to compute explicitly the closedloop transfer function and solve for the roo ts of the denominator po lynomial. Unfortunately. import ant insights rarely emerge from such an exercise (other than thai one should seek an alternative). Fortunately. there exists a method (rather. a collection o f technique..) that allows one to sketch rapidly how the po les of the closed-loop

14.7

ROOT· l O CUS TE CHNI QUES

399

system move as some loop-transmi ss ion para met er (such as DC gain) varies. given only the loo p tran smi ssio n as starting informatio n. To understand hnw on e sketches a root locu s. reca ll that the goa l is to find roo ts of the po lynomi al I + tI ( s)I< s ). Th at is. we want to find the values of s that satisfy p es )

= I

+ ,, ( s ) /

(s)

= 0

=

,,( s)/(s)

= - I.

(1 7)

We may decompo se this complex equation in to se para te co nsuui ms on the magnitude and phase angle o f the loop transmission : I,, ( s ) / ( s )[ ~

1

(1 8)

and LI,, (s )/ (s J! = (2n

+ I) . 180' .

(19 )

Despite thei r simplir.-ity, we can ded uce an amaz ing amo unt ofinfonnation fro m these last two equations. as we shall now St."C.

RULE 1: The locus starts tit th e po les of the loop tran smission , and term inates

011

the zeros (finite or infinile ) of the loop transmis sion.

This ru le deri ves from the magni tud e co nd ition. Suppose we express a (.f) f( s) as kg( s) . where k represents the ga in factor that we are vary ing and g (.f) represe nts all the rest of a (s)I( s) . In this case. the magnit ude con di tion may be stated as

1. (s )1 =

I

t·

(20 J

From Ihis equation. it is app arent tha t the magnitude of g ( s) mu st he extreme ly large for very small value s o f k: (that is. for the stan of the locu s ). li enee. values Of .f thai satisfy this magnitude condition are evidently ncar the poles o f g( s) . Si milarl y, for very large values of k (corresponding to the end of the locus). I.l;'(s) I is quit e small, indicatin g val ues of s ncar the zeros of g (s) .

RULE 2 : If a root-locus hrand l ties 011 the real axis. it resides to tht' left of lin oeltJ number of left hu tf-plan e poles + z.ertJ.f and to the right of 1lI1 odd number of right hulf-plan e poles + Z£'rO.\'. Since we are implicit ly assumi ng that k is a posit ive number in s ketching loc i corrcspendi ng to negat ive Ieedbac k. jhe only way to satisfy the equatio n I + kt<: ( .f) = 0 is for g( s) to be a negat ive num ber. Now. s ince g (s ) is rational by po stulate, we may express it in the followi ng manner: (21)

A()()

CHAPTER 14 FEEDBACK SYSTEMS

Eac h ( r,,· + I ) term contributes a minu s sig n if r .~ is more negati ve than - I. Therefore the sign of g (s) will be negative for values of s more negative than (to the left of) an odd number of left half-plane po les + zeros (where r is positive). including those at the origin. Similarly. g (s ) will be negative forr more positi ve than (10 the right o f) an odd num ber of righi half-plane po les + zeros (w here r is negative). Note that. s ince complex. poles or zeros appear in conj ugate pa irs. their net contribution to the phase at a test point on the rea l axis is zero. and thus they have no effect here. RULE 3 : lf the number of poles exceeds the number of zr tn s by 1"'0 or more. ,ht'n ' he average dis tance of the poles 10 'h e ima ginary' W :i.f is independem of tc ,

Thi s rule deri ves from a property of pol ynomials. Specifically, consider that

(22) We sec that the ratio of the two lead ing coe fficients is the sum o f the root s of L (s ). Note also that the ave rage dist ance of the roots 10 the imagin ary axis is simply this SUIll divid ed by the ord er of the poly no mial (i .e.• the num ber of root s). To usc this observation to derive the rule. let 8(5) = p(J )/q ( .fi). T hen the cherecrcn stic equat ion (after cl earing fractions ) becom es P (s )

=

q (s)

+ kp ( s) .

(23)

From this we observe that . if the order of q( s) exceeds that of p( .f ) by two or more. then the two leadin g coe fficients are indepe ndent (If p (.f) (and there fore of k) and hence the average distance o f the po les to the imaginary axis is a co nstant . RULE 4 : A.f k -;. 00. P - Z brunch es of the IOCUJ hear/off 10 infinitv.osymptoticatty at angles [ with respect to the real axi s) X ; \ '{' lI by

e _ ,(.2:::"_+:,..:..'1)~';-1:R::O:...' n- -

p - z

(24)

In Eqn . 24. fl ranges from 0 to P - Z - 1. P is the number of finite poles of 8(s). and Z is the num ber of finite zeros of g(s) . Th at the angle co ndition leads to this rule is eas ily understood from Figu re 14.9. In the figure. we ass ume tha l .fles, is so far away frorn the po les ami zeros of the loop tran smi ss ion that eac h pole or zero of K (.~ ) forms ap pru ximarely the sa me angle II with respect to .f t....r- Eac h po le then con trib utes a phase ang le of - 0. while each zero co ntributes a phase angle of 0 at 5 k 'I ' Hence. the total phase angle at Sin! is ZO - PO. Now. the angle condition requires that this 10ta 1 phase angle be an odd multiple of 180 > if .f tc " is to be a po le of the cl osed-loop syste m (i .e.• to lie on the Inc us ). Therefore.

• 14 . 7 ROOT· LO CUS TECHNI Q UES

40 1

" I~ n",

Pole s and

tlf g(sl

FIGURE 14 .9. Asymptotic angle rule.

ZfI - PO = (2n

+ I) . 180"

~

,( 2:::,.:-,-,:+,...:.: 1)'-.'-,1:.:8::0-,' €In = -

(25)

Z- p

Recogn izing thai ISO ' i.. the sa me as - 180". we see that this 101". equat ion is in fact equivalent 10 that stared before.

RU LE 5: The asvmptotes of Rille .J tIll intersect tne reut axis til tl point give n by

L Re(poles) - L Re(zcros)

a =

(26)

p- z

To derive this rule. we usc an observation o f Rule 3 - namely. that the ratio of the two leading coe ffi cie nts of a po lynomial is eq ual to the sum of tile r oots. Funh cr note that , since co mplex root s appear in conjugate pairs, the imaginary parts cance l when forming the sum. Hence. we can write our characteristic eq uation as I' ( s )

= 1+

[.\'z + .sZ- 1 L; J Re tzeros) + ... ] ,\,1' + ~.I' - l 1:;=1 Re t pole s) + ...

C1 .

= O.

(27 )

Because we arc interested in the behavior of P ( s ) for large s, we can approximate P (.~ ) by preserving only the firstt wo ter ms of the numera tor and dcnomi muor polyno mials. Performing these truncations and dividing through by the numerato r polynomial yields

o~ I +

.f

P- Z

C.

+ .( 1'-2 -1 [L Re t polc s) - L Retzerosj] + ...

.

(28 )

Clearing fractions. we see that the ratio of the two lead ing coefficients (which is the average distance of the asy mptotes to the j w axis) is a co nstant. lead ing to this rule.

• ,')2

CHAPTER 14 FEEDBA CK SYSTEMS

As an aside, it should be noted thai it is all right for the locus to cross an asymptote. As an add itio nal note. the locus will lie exactl y along an asymptote if the pole -zero pattern happe ns to be perfectly sym metric ab o ut the asymptote ex tende d thro ugh the point a (as given by thi s ru le) .

RU LE 6 : If a real-axis branch of the {OCII S lies between a pair of poles. the locus breaks awayfrom the real a xis somewhere betwee n the poles. Similarly, if a real-axis branch ofthe IOClIS lies between a pair of :;;em s, there will be ' III entr), point between that pair of zeros. Thi s ru le is actua lly a co nse q ue nce of Rule J. since the locu s starts lit the poles of g ( s) and termi nates at the ze ros of g (s ) . li en ee. if a rea l-a xis bran ch lies be tween two po les the n the po les m ust eve ntua lly head off to ze ros, which lie e lsew here by postul ate. And if a rea l-ax is bran ch lies between two zeros (i ncl ud ing o ne at infinity). the n pol es from so me .....here will e nter the real axi s. even tual ly to terminate at the ze ro s. A brea kaway point is found by co mputing the value of 5 bet ween the pule s that m inimi ze s Ig (.I' ) I; sim ilarly. a n e ntry point is the va lue of s between the zeros that

maximizes the value of Ig (s) l. In ge ne ra l. iterated funct io n e va lua tio n (a fancy term for tri a l and e rror) to find the minima /m axi ma is more expedie nt tha n sc tti ng dg (s )/J J eq ual to zero a nd so lvi ng for the root s of the res ulting polyn omial. " since the range o f po ssible valu e s o f 5 is bo unded and kno wn .

RULE 7: The locus fo rms nn initial aflgle O/' witlt respect 10 a complex pole. or an aflgle (}z with respect 10 a comp lex zero: (Jp

L Ll po les] + L L l l.ero~ 1 1 80~ + L Ll pole s l - L zlze ro s] .

= 1 80~ -

ami Oz =

(29)

(JOJ

In Eqn. 30. the sums refer to the a ng les dra wn from a ll o f the pol e s a nd ZCfO!; to the co mple x po le (ze ro ) in questio n. Agai n. the a ngle co nd ition is used to deri ve this ru le. as ca n be see n in Figure 14.10. Here we assume that S leSl is ex tremely clo se to th e po le in que sti on. so that the phase an gle co ntributio n by the o ther pol e s and ze ros to S IC
+L

LI1.ero sl -

L Ll po lcs l.

(311

and this sum must eq ua l ± 1 8 ()~ . leading to the rule. Si milarly. the phase angle contri butio n to S IC" ncar a c o mplex ze ro is sim ply 11 In fact. use nf this fonnaJ method requires finding ,he roots of a polynomial of tkogrec only one less than that of the orig inal. and is therefore rarely worthwhile.

403

14.7 ROOT·lOCUS TECHN IQUE S

Imls)

s• •

•

.- V:"~.,------,

... ...--

-- 'I

... --

....

:

• I{e(sl

f iG URE 14 ,10 . Angle near complex pole.

8z

+L

L(l eros l -

L L( po lesJ,

(321

thus co mpleting the deri vation .

RUL E 8 : If a pa rticular value of s is known to lie

,Ill' loc us. ,11m the value of k necessary to mak e that value of s be a closed -loop poll' location is given b)' Oil

I

k= - - _

(33)

Ig (' ) 1

This rule is simply a restate ment of the magn itude cond ition. The: foregoing eight rules arc by no mean s the only o nes that can be dedu ced from the magnitude and phase cond itions. butt hey should suffice for most pu rposes. 14 . 7. 1

RO OT - LO CU S RULES FOR PO SITIVE FEEDBA C K SYSTEMS

The rules we have developed so far app ly 10 the case of negative feedback systems in which k > O. The same basic construction ideas apply even in the cas e of pos itive feedback. provided we modi fy all the rule s deri ved fro m the phase angle co ndition. For k < 0, the appropriate phase angle co nditio n becomes Lg (s) = n . 360".

(34)

Therefore . all occurrences of ( 2n + I ) . 180C> should be repla ced with 11 • 3 60 ~ . and all as..oci ated rules modifi ed accordingly. II is left as an "exercise for the reader" to perform these modifi cation s.

AOA

CHAPTER 104 f EEDBACK SYSTEMS

14 . 7. 2 ZERO S O F AI,) T he rootlocus tells us o nly how the poles of A(s ) behave. give n the po les and zeros of K (.f ) as start ing information . Since the locus therefo re doe s nut nec essarily tell us a nything abo ut the ze ros of A (s ) . we need 10 do a little ex tra wor k if findi ng lhu-.c closed -loo p ze ros is import a nt. Fortunatel y. thi s task is re lativel y easy. since (35)

Hence the ze ros of A ( .f ) are e vide nt ly the ze ros of tI(s) and the po les of f(s) .

14.8 SUMMARY OF STABILITY CRITER IA We have presented several techn iques fo r determin ing the stabili ty of closed-loop sys tems through exami natio n of loop tra nsm issio n behavior . G ain and phase margin are mo st commonly used in pract ice. but are actually a subse t of the more general Nyqui st lest. Gain and phase margi n (a nd the Nyquist lest) may operate on measured freq uency respon se data: roo t-locu .. techniques req uire a rational transfer function fur a( J ) f ( .~) . All ofthese tests arc relat ively simple to a pply, and allow one to evaluate rapidl y the stability of a feed back system, as we ll as to assess the efficacy of pro posed compe nsation tec hniques. without having to determine di rectl y the actual roots of P ( s ).

14 . 9 MODEL ING FEEDBACK SYSTEMS In our overview of feed back systems so far. we 've identified desensitivity as the only fund amen tal ( bu t extre mely im portan t) benefit conferred by negative feedback. We've see n that the larger the desen sinvity, the grea ter the improvement in linearity that is, the better the redu ction in error. We haven't quantified the notion ti l' error, thou gh . so we need to take care of that little detail now. After all, sta bility co nsiderations construi n the magnitudes of desensinvit y we ca n obtai n..so we ough t to learn how 10 calculate how big the inevitable errors will be . Th is knowled ge will be a useful guide in our efforts at modifying arch itec tures to reduce errors . Before we can evaluate erro rs in feedback systems , hO\~..ever, we need 10 be able to mod el real systems in a fashio n that ullows tract able ana lysis. As we' ll see, this task is often difficult becaus e. contrary to intuition , it is not always possible to provide a I: I map ping between the block s in a mode l and the ci rcuitry of a real system. We 'll also devel op a sci of performance measures that allow us to relate various second-order parameters 10 frequen cy- and time -domain respo nse parameters. Because many feedback syste m.. are dom inated by f irst- or second-o rder dynamics by

14.9 MODE LING FEEDBACK SYS TEMS

.,

"N'

+ "'~_ _

..

40'

v(}l,'T

.,

FIGURE 14. 11. Nooinverting amplifier.

design (ow ing 10 sta bility conside rations ). seco nd-order performance mea..ures have greater general utili ty than one migh t initially recognize.

14 . 9.1 THE TROU BLE W ITH M ODE LIN G f EEDBACK SYS TEMS The noninverti ng op-a mp connectio n is nne of the few examples for which a I: I mapping to our feedback model does ex ist (see Figure 14 . 11). Suppose we choose the forward gain equal to the amplifier gai n. (I

= G.

(36)

and choose the feedback factor equal to the resistive atte nuatio n fact or :

f =

R, HI

+ R2

(37)

For this SCi of model value s. we find that the closed -loop gain is indeed 1/1 in the limit o f infinite loop tra nsmission magnitude : A -

1 - =

( 38 )

f

For that mail er. we find the sa me loop tran smi ssion in bo th the block diagram and the actual amplifi er, so it a ppears that the model pa rameters we chose nrc correct. However. as suggested earlier. this situation is atypical . O ne simple case that males Ibis obvious is the inverting connection shown in Figure 14.1 2. If we insiM on equating the op-a mp gai n G with the forward gai n II of our block diagra m. then we mu..t choose the same feedback factor I as for the noni nverting case if the loop tran smiss..ion s are to be equal. However. with that choice. the closed-loop gain docs nor approach the correc t value as the loop Iran..mission mag nitude a pproac hes infinity, since we know thai inverting amplifiers ideally have a gain give n by

R, R,

11 = - -

(3 9 )

406

CHAPTE R 14 FEEDBACKSYSTEMS

FIGURE 14.12. Inverting amplifier.

while our choices lead 10

I

A _ - =

f

(40)

the same as for the ncmnvertin g case. Part of the problem is simply that our " natural" choice of II = G is wrong. 11Je othe r is thai we need one more degree o f freedom than ou r two -parameter block diagram provides (co ns ide r. for example. the prob lem of gelling a minus sign out of our block diagram). It turns out that there is not necessarily one correct model in general: that is. there are potentially many equivalent models. Operationally speaking. it doesn't matter which of these we use since. by definiti on. equivalent mod el.. . all yield the same answer. A procedure for ge nerating o ne such mod el is as follows: ( 1) select f eq ual 10 the (magnitude of the) reciproc al of the ideal d osed -loop transfer function : (2) selcct « to give the proper loup transm ission with the choice made in step (I ): and (3) add a third block in front of (or after) the rest of the model (0 take ca re of any sign reversal that might be needed. There arc man y other equivalent procedures. of course, hut this one makes use of quantities that arc usuall y easy to di sco ver. For example, the simplification that results from letti ng the loop tran smi ssion magnitude go to infinity usually makes finding the ideal clo sed -loop transfer function a fair ly straightforwa rd affair. Additionally, since the loop transmission itsel f is fo und from cutting the loop , discovering it is also ge nerally simple. Le t ', apply this reci pe to the inverting anmlifier e xample. Fine we select the feedback factor f equal tu R 2/R ), since the ideal clo sed-loop transfer function is - HI / R2. Then. we need to choose a to give us the correc t loo p tran smission:

14.9 MODELING FE EDBACK SYS TEMS

, • +.

f-- - -,-- .

407

' OlIT

FIGURE 14 . 13 . Inverting amplifier blaek diogram.

FIGURE 14. 14. Nanideal inverting amplifier.

Finally, we do need to pro vide one final sign change, so our complete model for the inverting amplifier finally loo ks like that shown in Figure 1-t. 13. Again, this model is not necessarily the only correct possibility (e.g .. consider making bo th a and f negative quantities: o ne may then remove the input negation). But we 're usually happy just to tind one that works.

14.9.2 CLUTC HES A N D LOO P TRAN SMI SSION S We' ve already seen that the loop transmission is an extremely important quantity since it determines stability and dcscnsitivity. In addition, identifying the loop transmission is usually much easier than figuring out the closed-loo p transfer function. making it even more valuable. Although finding the loo p transmission is a trivial matter if we happen 10 have a correct block diagra m for the system, it may be a bit trickier to lind in real systems. The usual problem is how to take load ing effects into account. To sec where we might have a problem , let 's consider an invert ing amplifier that is built with a nonideal op-amp. In this partic ular case, assume that the nonideality involves so me resistance that is connected between the input terminals of the up-amp. The circuit then appears as shown in Figure 14.14. In order- to find the loop trunsmission we suppress all independent sources, so we set the input voltage to zero. Then

• At"


FIGURE 14 .15. Op-ompfol lower.

we have 10 cut the loop. inject it signal lit the cut point . and see what come s back. TIle rat io of the ret urn sig na l 10 the input sig na l is the loop tran smi ssion. If we cui the branc h marked X to the left of th e resistor R . we wil l effec tively (and incorrectly ) e lim inate R fro m the loop trans mission wh en we apply a test voltage to X . To lake the loading effect of R pro per ly into account , we need 10 CUI the loop to the right of R . A nother good c ho ice wo uld be the o utput of the or -amp. Th e ge nera l principle is to find a po int ( if po ssible) that is d rive n by a zero impeda nce. Of that drives into an infinite on e. Thai way. there are no loading-effect issues to co nfound us. Althou gh nor all circuits will automatica lly have such points. it is always possible to generate models that do . For example. co nside r an emitter follower . We can always mode l it as an ideal one with input and output resistances added to acco unt for nun ideal effects found in the orig inal circuit. By using an ideal followe r inside the model. we generate a node whose properties allow us 10 lind the loop transmis sio n of the feedback system of whic h the follower may be a part.

14.10 ERRORS IN FEEDBACK SYSTEMS We have already concluded that the only fund amental property of negative feedback systems is the desensitivity to forward gain variations that they provi de. with greater loop transmission leading to greater dcsensitivit y, Unfortunately. we 've also found that as the loop transm ission magnitude increases. we eventually encounter instability. thus imposing serious co nstraints on the attainable dcscnsitivit y, As a resu lt of having to co mpromise desensitivit y, we expec t the loop 10 produce errors. As a typic al ex ample. conside r a simple vo ltage follower made with an opamp ( Figure 14. 15 ). Assume that the amplifier transfer function is single -pole. and then consider whal hap pens if we app ly a step input. The output will rise with a fi rst-order sha pe. eventually reaching a value clo se to the inp ut. How close? It depends o n the DC gain of the op-a mp. If the inpu t voltage is one volt and the op-amp's DC ga in happens to be 1000. then the output will be one volt to within about a part in a thousand. Viewed another way. a follower is suppose d to . well. follow. So. if the output is to he ncar a volt and the gain is 1000 . the voltage di ff erence at the input has to he 1/ 1000. or I mY. Hence. the erro r between input and output will be about 1 mv. To redu ce this. error we need more gai n. but we can' t always get it owing to stability probl em s.

14.10 ERRORS IN FEEDBACK SYS TEMS

' 09

Altho ugh we can' t reduce the e rror 10 ze ro at all freque ncies beca use of the stability issue, we ca n ofte n obtain zero steudv-state error in respon...e to a ste p input. To reduce the stendy-snne ste p respo nse e rro r to zero. then, we need infin ite gai n only at DC , nu t at all freq ue ncies. As a result , most op-a mps are designed to have huge a mounts of DC gai n (l ike a mill io n or more ) to reduce steady-state error to tiny values . Til so lve the stabili ty problem. they are also ge ne rally des igned to ha ve single-pole behavior ove r ma ny decade s of frequency. Th at is, op-am ps arc ge nerally desig ned to approxim ate Integrators. Ano ther way o f ded ucing the necessity fo r a n inrcgnnion is to recogn ize that , if we want a non zero steady-state outpu t fo r a zero input diffe re nce be tween the np-ump termi nals then we need an integrator, since an inregrutor cun provide any DC outp ut with no inpu t. We can usc thi s obse rvation to dedu ce other usefu l and interesting triv ia . Suppo se, for example, we were interested in a slightly diffe rent prob lem - thai of track ing, with zero steady- state error, a m mp inp ut (a constant veloc ity, if yo u will) . If we now assume that the o utpu t is a ra mp while we have zero inp ut d iffe re nce, the n we need to have two imegnu lon.. in the op-e mp's transfer function (the inpu t ro rbe second integrator mu st be a ste p. and we already kn ow that another integra tion is need ed to give us ze ro steady-stare e rror to a step) . You ca n easily SI.'e tha t three integrations allow us to have ze ro steady-s ta te error in response to a quadra tic ramp (a constant accelera tion ), a nd so on. Is any of this stuff useful? Ahsol utely. For e xa mple, conside r vol tage to be propo rtional to the po sitio n of an objec t we w ish to track . Ze ro ste p respon se e rro r corres po nds to zero po sition error, zcm ramp respo nse correspo nds to zero veloc ity e rror, a nd zero qu adrat ic response corres po nds to zero accele ration error. So, if the obj ec t we're track ing has a consta nt acce leration, we com still trac k it with zero steady-s tate error if we have three integrat ions in G(x) . Yes, we do have to worry a bout stability w ith more than one integrat io n, hut we can fix that up quite ea sily ju st hy adding eno ugh zeros to canc el the phase shift . If P is the numbe r of pol es, we simply have to pro vide P - I zeros well below c ross ove r and we 're all right. Wh en we st udy phase -loc ked loops ( PLLs), we ' ll see that this tracking prob lem reappea rs in the co ntext of F M mod ulation- de mod ulation (as j ust one exa mple) , so we still have to worry abo ut these issues eve n if we're not building an tiairc raft weapo nry,

ER ROR SERIES We' ve see n that it is po ss ible, in principle, 10 elim inate stea dy-snne e rrors if we e mploy e noug h integrations. Howeve r. perfec t integrators are diffic ult to realize in pructice. In any case. we 'd like so me me thod for quantifyi ng errors in ge ne ral.

41 0

CHAPTER 14 FEEDBACK SYSTE MS

On e way is to use an error series, that is, 10 e xpress the e rro r as a po wer series: (42)

If the series converges quickly. we may truncate after a few terms. Fortu nately. if the series doc s 1/01 converge qu ickly. it imp lies that the sys te m is do ing a lousy job of track ing. an d we mig ht as we ll re move th e sys te m . He nce. fo r nearly a ll practical cases. the e rror se ries doe s converge rapid ly. The various erro r coe fficients may be found from the fol lowing eq uation.R (43)

where Vr(.r)jVj( s ) is the input-to-erro r transfer func tion . If you happen 10 forget this formula . there is ano ther way to obtain the error cocfficients. First . find the input-to- error tran sfer funct ion . Then. divide the numeralor pol ynomia l by thai of the de nomina to r 10 o bta in a transfe r fu nc tion in as cending po we rs of ,t. TI1C coe fficient of the S i term in this se ries turn s out to be eq ual to £ i. The vario us error coe fficients ha ve the fo llowing phys ica l Inte rpre tations. The ze ro th-o rde r coe fficie nt is the stea dy -sta te step re sponse e rro r. while the first-order term is the stea dy-sta te delay in re sponse to a ra mp input, a nd so OR. Kno wled ge of the e rror se ries th us a llow s o ne to e stima te ra pidly the error fo r arbitrary inpu t signals. The mo re rigo rou sly minded a mo ng yo u ma y po int o ut that deri va tive s of the input sig nal may no t exist (o r be bounded ) if we a llo w things like ste p inp uts. and therefore we c anno t reall y use o ur e rro r se ries. S trictly spea king, that is true. bUI if we restrict o ur que stion s abo ut erro r 10 times that a re sufficiently re mov ed fro m such disco ntinuities. we ca n stil l use o ur decomposition. W hat is "s ufficie ntly re mo ved'!' A reasonable c rite rio n is 10 wa it a few ti me co nsta nts . 1.\

14.11 FREQUENCY· AND TIME ·DOMAIN CHARACTERISTICS OF FIRST· AND SE COND·ORDER SYSTE MS A n erro r se ries is but on e of many wa ys to characte rize feed back sys te m s. Wc could imagine using o the r me asu re s. such as step res ponse o vershoot . se ttling time , or frequ cncy re spon se peak ing . Depending o n the contex t , so me or a ll of the se parameters could he o f intere st.

12 Sl'C e.g. G. C. Newton, Ir., L. A. Go uld. and 1. F. Kaise r, An alytical [Jl;'si/l1l of U neor f f f'dhud:. Controls. Wiley. New Yor k. 19 57. Append ix C. I J Because high-orde r systems have more than one lime constant. a mute gene rally useful criterion is to wait a senling lime.

14.11

FREQ UENCY- A N D TIM E-DOMA IN CHA RACTER ISTICS

an

In this section, we' ll j ust s imply present a nu mher uf exceed ing ly useful formulas witho ut det ai led derivat ion s. In most instances it should be obvious how to de rive them, hut the tedium invo lved is too great to merit presen tation here. In those cases where the deri vation might not he obvio us, a co mment or two might be added to hel p point the way. We have already asserte d that it sho uld be possible to charac terize mo st feedback systems as systems o f second order at most. Thi s cl ai m derives from the obse rvation that any stable amplifier cannot have more than two (net) po les do minate (he loop tra nsmissio n below cross ove r. li enee, for feedback sy... tern s at least , intimate know ledge of tirst- and second-order characteristics turns out to he sufficie nt for mo st situations of practica l interest. The followi ng form ulas all assum e thai the systems are low-pass with unit DC gain. Therefore, not all of them apply to systems with zeros, for example (you can't hav'e everything, after aliI. With that lill ie warning out of the way, here are form ulas for first - and second-order syste ms.

14.1 1.1

FO RMULA S FOR FIRST-O RDE R LOW-PASS SYS TEMS

Assume that the system trans fe r funct ion is lI ( s )

I = --. TS + I

(44)

For this first-order, low-pass system, we have: t, = r In 9

~

2.2r = 2.2Jwlt,

(45) ( 46)

tp

=

00,

(47) (4 8) (49)

Wp

= 0,

(50)

where the vari ous quantities have the mean ings show n in Figu re 14 .16. Commentary ond Exp lanati ons

Equation 45 - The risetim e defini tion used here is the 10% 10 90 % riscnmc. Equation 46 - Both the step and frequency responses arc mo notonic in a sing lepole system.

CHAPTER 14 FEE DBACK SYSTE MS

\

..." P,-j-- ",

I

0.9

01 time

a) Step Rtspon w Parameters

+

---::;"...,-___.

IIIUw)1 Mp

0.707 -t ------- -+---~

b) Frl'qul'ncy Response Parameters

FIGURE 14 .16 . First- a nd second-order parameters.

Equa tion 4 7 - Becau se the step respo nse is monotonic and asymptotically approach es its final value. there is an infinite wait to see the peak . Equation 48 - An ex po nential settles to wi thin about 2% of final value in four time cons ta nt...

Equation 4 9 - Th e steady -state delay in respon se to a ramp input is equal to the po le lime constant. Equation 50 - The frequency response of a first-ord er system rolls o ff mo notonically from its DC value. Hence. the peak of the freq uency respo nse occurs at zero frequency.

4

14.11 FR EQ UEN CY- AN D TIM E-DOM A IN CHA RACTERIS TICS

14,11.2

413

f ORMUL A S f OR SECO N D -O RDER LO W -PA SS SYSTEMS

Here. assu me a transfer function of the fonn I/( s ) =

" [2 w

ll

, +I + -2(_ W

l

( 51)

n

Then the fo llowin g relationships hold : t,

~ 2. 2r

2.2

=-,

P,, ~I + e xp (

(521

w,

- rri )

~'

vI

(53)

_ (' 2

(5-1)

(55)

(56)

Mp =

I

2i /( - ( ,

.

I

{<

Ji '

(5 7 )

I

15R)

« ..n'

Wh

=

Wn [ I - 2 , '

+ v/ 2

4( '-

+'4,]1/'

=

I

Wn , . 1/.12"

(59)

Commentary and Explanations Equation 52 - The risetim e of a second-order low-pass system is som ewhat dependent o n the damping ratio. In the limit of zero dam ping. the prod uct of bandwid th and risctime CUll he as sma ll as abo ut 1.6. However. fo r any rcasona bly well- da mped syste m. the produ ct will be closer 10 2.2. Equation 5.1- T he pea k of the step response overshoo t ca nnot exceed !(X)% for a second-order low-pass syste m. Equation 54 - The time at which the step respon se pea k ove rshoot occ urs is si m-

ply one ha lf the ringing period . Recall that the ringing frequency is equal to the ima ginary pa ll of the co mplex po le pai r. T he form ula for follows direct ly from the se two facts.

'I'

Equation 55 - J ust as the imagin ary pall of the po le frequ ency contro ls the oscil-

latory pall of the response. the real pan. controls the decay. As in the first-orde r

• CHAPTER 14 f EEDBACK SYSTE MS

' lA

ca.se, it takes abou t four time constants for the envelope to settle to 2% of final value. With the informa tion from Eqn. 54 and Eqn . 55. we ca n also express the eq uation for 1'0 as follows:

P" =

1

+e,p(n) = +e,p(- T~/2 ) . 1_ {2

1

(60)

f r nv

Equation 56 - The steady-state lime delay in respon se to a ramp input is the same as for the first-order case if the dampin g ratio eq uals 0.5, and decreases ;IS the da mping ratio dec reases. appro ach ing zero delay in the limit of zero damping. Equations 5 7 am/58 - The frequency response can exhibit a peak at other than zero freq uency if the da mping ratio is less than 0 .707 . For greater damping ratios. the respon -.e is monotonic and thus ex hibits a peak at DC. For smaller damp ing ratios, the pea k mag nitude asy mptotica lly approac hes infinity in !he limit of zero dam ping. Equation 59 - The - ) -d B frequency equals w" at a damping ratio of 1/ ./2. The band width is a maximum of about 1.55w" in the limit of zero da mping.

14.12 USEFU L RULES OF THUMB No tice that phase margin is conspicuously abscnt from the sci o f equations presented in the prev ious sec tion. To bring phase margin explicitly into the d iscussion requires making a number of limiting ass umptions. beca use in ge nera l there is no unique relationship between. say. pha...e marg in and dam ping ratio. However. out of necessity, stable systems must behave as first- or second-order systems ncar crossover. so we may derive a num ber o f relationships for a second-orde r system and appl y them 10 a much broader class of sy...tents. eve n though they strictly apply o nly to the secondorder system for which they were derived. Specifically. assume in all of the following that we have a two -po le system with purely scalar feedback . Further assume that the two loo p transmi ssion poles are widel y spaced . With these assumpti ons. one may derive the following relationship between damping ratio and phase margin : (6 1)

Thi s cumbersome equatio n may be replaced by a rem arkably sim ple approximation that holds over a restricted ( but use ful) range of phase margins: (621

where ¢J", is the phase margin in degrees in bo th Eqn. 6 1 and Eqn . 62. Thi s relationship is acc urate 10 within abou t 15% for phase margins less than approxi ma tely 7('f .

14 .13 ROO H O CUS EXA MPLES AND COM PENSATION

4 15

Furthermore. it is accurate 10 bette r than 1 0~ fro m abo ut 35° 10 a bit less than 70'0. a range that fortuito usly spa ns the phase marg in targets most oft en encountered in practic e. The damping ratio as estim ated by Eqn . 62 m:lY also be used 10 estimate the step respon se overshoo t:

- n{ ) ::::::: I + exp ( Po =l + exp ( ~ I - {,

- nq,"'

,110'

q,~

),

(631

where the phase margin is aga in ex pressed in degrees. A.. with Ihe express ion for damping ratio. this equation pro vides reason able acc uracy for phase margi ns be low aboul7W . Anothe r rel ationship of considerable utility is that between phase margin and Irequcncy response peaki ng : 1 (64) M,, ::::::: - .- -. Sill

¢m

For our proto type secon d-ordcr syste m. this equation is accurate to within I % up to a phase ma rgin of about 55 °. With these approxima te equalion s. it is a simple maile r III estimate what phase margin is needed 10 satisfy an overs hoot or pea king specific atio n. or 10 estimate phase margin fro m measurements of step res ponse or frequency respo nse . Again. beca use these eq uations apply stric tly to a two -pole system with widel y spaced pole s and scalar feedback . fbey will provid e good estimates for systems Ihal are we ll approxi mated by suc h a two -pole system.

14 .13 ROOT·LO CUS EXAMPL ES AND COMPENS ATIO N The vario us roo t-locus co nstruction rules we' ve deve loped are certai nly not exhaustive, but are more than sufficie nt for the vast majority of loci encountered in practice . We' ll now prese nt a few roo t-loc us examples 10 gai n SOIllC practice with the method . Then we will examine the subject of compe nsatio n. In the pages that fo llow are a number of examples, roughly in order of increasing complexity. In all cases, loci for only pos itive values of k (corres po ndi ng to the mo re commo n case of negative feedback) are shown. As a general rule, it may he useful to employ somethi ng like the following pro cedure. ( I) (2) (3) (4)

First, identify the seg ments o f the real ax is that are pan of the loc us. Compute asy mptotic angles and the rea l-axis Interce pt of those asymptotes. E...timare or calculate break away or entry poi nts. if any. Begin to sketch the locus using additional information . such as the initia l ang le ncar a com plex pole or zero (i f a ppro pria te) or the consta ncy of the ave rage distance 10 the imagin ary axis (i f P > Z + 2) . Remember that every zero is a lenninus for the locu s.

41 .

CHAPTER 14 FEE DBACK SYSTEMS

4

X -X--O--O--K+---~ Re

, f iGURE 14 .17. Root locus with b-enches on real

OX;5

only.

o r C()U ~. this procedure is hardly uniqu e. hut it ofte n suffi ces. Arte r constructing numerou s loci . ce rta in pa tte rns (dare I ca ll them manns?) w ill ofte n recur. and dra wing fool loci will bec o me prog ressively eas ie r.

14 .13 .1

EX AMPLE ; PURELY REA L PO LES A ND ZERO S THAT STAY REA L

We start wit h the sim ple ex am ple ske tched in Figure 14. 17. Th e rules a nd facts used

here are: ( I) real-axis rule; (2) termin i of locus are zeros; (3) asymptote rule (leftmost pol e he ads 10 m inus infinity).

14 .13 . 2

EXA MPL E; TW O POL ES TH AT BECO ME CO M PLEX

This example is a bit more interesting than the last; !'iCC Figure 14.1X. The real-axis ru le tells us that the real-axis segme nt betwee n the po les is pa ri of the loc us. while the asy mpto te rules tell us that the asy mptotes arc at ± 90" a nd intersect the rea l axis ex actly halfway between the pol es. From the break away rule. we can infer that the pu les mu st le ave the real a xis so mewhe re in the mi dd le (e xactly in the middl e. in this case ) of the real-a xis segment. Invo kin g the s ym me try argu ment . the asym ptotes arc ac tu ally ex ac tly a part of the loc us. No te tha t. strictly speaking. a two -po le negative feedback system is never unstable in the HIBO :-.eIlSC. ye t its damping gets prog ress ivel y sma lle r as k inc reases. For suc h a sys te m. the rea l part slays co nsta nt o nce the po les bec om e com ple x. implying

14.13 RO OT· lOCUS EXAM PLE S A N D CO MPENSATIO N

----- ;*""--1~-~----j--- •

•e

FIGURE 14 . 18 . Iwo-pole rcc t lccos.

that the expo nential envelope of the impulse respon se has a co nstant shape. However, the imaginary part increases as Ii. increases, so that more osc illations per unit time occ ur within the expone ntially decaying envelope. Since the ex pone ntial envelope doesn 't change shape as k inc rea ses, the settling time doc s no t change, eith er. Ju st becau se a syste m becom es less slable docs not necessarily imply a degrad ation in settling time. Ob servations and conclusions such as these simply don' t emerge from gain and phase margin ca lculations. illustrau ng the importance of having several ways of assess ing sys tem stability. Additional viewpoi nts accelerate the develop ment of intuition.

14 .13. 3

EXAMPLE : TW O POLE S A N D A ZERO

To stabili ze the previou s system. we may consider the addit ion of a zero. From a phase margin view poin t. we regard the improvement in stability as a consequence ofthe positive phase co ntributed by the zero. An altern ative view, informed by rootlocus co nstructions. is that zeros are auracro rs of poles. so that properly placed ze ros can bend poles away from tbc imaginary axis and toward more highly damped configuratio ns. A spec ific (and frequen tly occurri ng) example is sketched in Figure I·U 9 . Th e locus appears to incl ude a ci rcle ce ntered on the zero. It turn s out to he the correct shape, not just an art efact o f the author ' s laziness. In constructing this locus, make use of the real-axis rule, the breakaway/entr y rule (here, we hnve an entry poi nt between the finite zero shown and an infinite on e), and the asymp tote rule. Those ru les are nor sufficient to ded uce the ci rcular shape shown.

..

CHAPTE R 14 FE EDBACK SYSTE MS

4 16

- - - If----+ 0 - ) E---I--)(+ - - ~

R,

, FIGU RE 14 .19. Root locus of two poles plus zero.

'm .. . /. . :/

.,/ //

...

--~----)(----:*,,~"l<-+--- ~ R,

FIGU RE 14. 20 . Three-pole root locus.

but are enou gh to conclude that the po les eve ntually beco me pu rely rea l again beyond some value of L

14.13 .4

EXA MP LE : SYS TEMS THAT GO UN STAB LE

A simple example of a system that eventually goes unstab le beyond some critical value of l: is Illustrated in Figure 14.20. Use the real-axis. asymptote. and intercept

14.13 ROOT-LOCUS EXAMPLES AND CO M PEN SATIO N

rules to dedu ce this locu s. The asymptot es form ang les of I RO° a nd ± 60" w ith respect to the real ax is. As an a..ide. one ca n es timate the value of ga in a t which the po les c ross the imaginary axi s - as we ll a!'i the corresponding oscillation frequency - by using the asymptotes. A simple trigon om etri c calculation yields the ime rsection of the asy mptotes with the imagin ary axis: this value is a c rude estimate for the o sc illation frequency. Then , plu g this value into the express ion for g( s ) = a (.\') ! (.\·) , compute the magni tude. then take the reciprocal to find the value o f k tha t co rres po nds to this o nse t of in..ta bilit y. If you need exact an ..wers to these que..tions. simply lake g (s ) and find the value of j w where the pha..e become.. e xac tly - 180 '. a nd compute the magnitude of g (s ) at that freq ue nc y. Th at freque ncy is the o scillation freq uency, and the reci procal of the magnitude of g(.~) there is the value of k that j ust results in instability. Thi s pro cedure is equ ivalent 10 de termining the conditions tha t lead to a phase ma rgin o f zero and a gain marg in o f unity. Another potentially use ful ob servation is that we can det ermine the value of k thai corresponds to a speci fied da mpi ng ra tio. Simp ly plot rays of constant da m ping ratio, and com pule the intersection o f the loc us w ith the ra ys. Then. plug that value of s into the ex press ion for g (s). co mpute the magn itud e, and take the recip rocal. Th e result is the requi red value of k ,

14.13 . 5 EX A M PL E, LO CU S W ITH COM PLEX POLE S IN L{ , ) All of our examples so far have had loop tran sm ission pol es and zeros that we re purely real. Let's do so mething a little differe nt and e xe rcise the complex- po le con struc tio n rule. Consider the three-pole locus shown in Figure 14 .21 . He re. we compute the initial angle that the locu s ma kes with the poles hy drawing vectors fro m eac h po le to the other two po les, and adding up the angle co ntributio ns of each. The rea l-axis. asymptote. a nd in terce pt rules tell us what eve ntually hap pen s overall. Again. the asy mptotes the mselves may he used to estimate both the value of ga in tha t ca uses oscillation as well as the oscilla tion frequency. As w ith the previous locus, the asym ptotes are at 180" a nd ± 60' .

14.13 . 6 EXAMPLE , LO CU S WIT H RI GHT HA Lf -PLAN E ZERO IN L{s) In ma ny operationa l am plifiers ( part ic ularly those im plemented in MOS forrnj. there is a significa nt (i .e., rel atively low-freq ue ncy ) rig ht half-plane ze ro. It is easy to get tripped up in const ruc ting the loc us for suc h an a mplifier if we 're as lee p at the wheel.

.. 420

CHAPTER 14 fEE DBACK SYSTE MS

1m

i

, FIGURE 14. 21. Root locus 01system with comple:Jl loop h"onsmiui on poles.

Rem ember the full stateme nt of the real-axis rule : Th e locu s lies to the left of an odd number of left half-plane pole s + zeros, and to the right of an odd number of right hal f-pl ane po les + zeros. Consider Figu re 14 .22. Not ice what happ e ns as we increase the loop transmission magnitude: the pole moves to high er a nd higher frequenci es. Th en , ul some critical value of gain, the sys tem becomes unsta ble as the po le moves from minus infinity to plu s in finit y (yes, this ca n. and docs. happe n at a tinite value of k) . T heil . as the gain increa ses stil l further . the pole asym ptotica lly approac hes the zero (from the right side ), as pol es alway s do. Fr om this locu s, you ca n see wh y right half-pla ne ze ros a re had news from a stability view po int. Again, s ince zeros a re terminallocutions for loci . pole s wi ll eventually move ttl them . and if the ze ros happen to be in the righ t half-pla ne. the syste m will e ve ntually beco me unstab le. It is perhaps initia lly difficult to und e rstand how a zero ca n eve r e ncourage insta bil ity. A fter all. we usc ze ros in lead compe nsators to improve sta bili ty. What' s

14.13 ROOT- LOCUS EXAM PLES AN D COM PENSA TION

421

1m

- - _x- -+- 0--- - .

R<

, FIGURE 14.22. Root locus of system with right half-pkme zero in loop tronsmissiOfi.

differe nt here'! II is len as an exercise for the reade r to show that a right half-plane zero has the same magnitude behav ior as a left half-plane zero. hUI that the phase respo nse has the same gen eral trend as that of a left half-plane pole.

14 .13 .7

EXA M PLE: CO N DITIO N A LLY STA BLE SYSTEM

In all o f the exa mples so far. a system may exhibit instability abo ve some critical value of gain. Thi s be havior is certai nly co mmon. but it is not the only po ssible llchavior. Th ere are systems of practical interest that are unstable both above and be low some range o r gain. In this ex ample, we consider the locu s of such a comlirioi/(/lly stahlc sys tem (sec Figure 14.23). If we co nside r the locu s at relat ively small values of k, we ex pect to sec something like the simple three-pol e locu s. Assume that the pole -zero constellation is such that the com plex pole pair that fo rms doe s cross over to the right half-plane. As the ga in continues to increa se, the attractive influence of the zeros beco mes more promin ent and the co mplex pole s bend back over to the left half-plane. li enee. the sys tem is unstable only for so me finite range of k. Th is type of locu s can result when we attempt 10 build a system that possesses a small steady -state error in tracki ng an input that po ssesses a co nstant acceleration. The ideal of three po les at the origin can' t be realized in ge neral, so they might end up as shown. To stabilize such a loop, two zeros must be added. If the crossove r Irequency is too low, however. the positive phase shift of the zeros doesn' t do any good, and the system ca n be unstable . As the ga in increases abo ve some cri tica l value. the

m

CHAPTER 14 FEE DBACK SYS TEMS

---+-- 0 0 ~-·~~-t-- ~

Re

, FIGURE 14.23 . Root loa n of condi tiona lly stobie 5~tem .

crossover frequency increases enoug h for the zeros to wo rk thei r magic and the S)'Stern bec omes stable aga in. w c'Illater sec that such syste ms may ex hibit instability in practice eve n when our linear analyses tell us thut we have plenty of pha se margi n. The key word is "li near." All rea l sys tems have finite dynam ic ran ge. If we ever saturate an e lement in a cordi-

tionully stable loop then we effectively reduce the gain. since something that saturates doesn ' t provide much of an output change for a given input change . From the shape of our locus. we sec that there is a real dan ger of entering the region of conditional instab ility if a ga in reduction should occur. Furth ermore. any oscil lation associated with this instabil ity migh t be of sufficie nt amp litude to sustai n this saturated condition. and the sys tem may never recover - a tran sient overload might he enough to send it into an oscillat ory mode from which it cannot CSC'lpC.

14.14

SUMMARY OF ROOT· LOCUS TECHNIQUES

We have seen that ju st a handfu l of simple co nstruc tion rules arc sufficient to guess ho w clo sed -loop poles vary as the loo p transmission magn itude varie s. Rem em ber that these rules all deri ve from u magnitude and 'Ingle condition on the loop transmission - namel y. that a value of s ca ll he a clo sed-loop pole location only if the phase of a (s )x (s ) is an odd multiple of I KO' . ami onl y if the magnitude of a (.f) K( s) is unity at the value ofx . Finally. do n' t forge t that the roo t locus only tells us where the clo sed -loop poles are . It is cusy to lose track of this fundamental trut h beca use we employ po les and zeros in co nstructing loci . However. the loo p transmission zeros we use in drawing loc i are not necessaril y the sallie as close d-loo p zeros . If it is necessary to find

14.16 CO M PENS ATIO N THRO UGH GAIN RE DUCTION

423

closed -loop ze ros. it is easy to do so by findi ng the zeros of l/ (.~ ) and uncuncclfcd po les of f( s) . As a pa rting not e. it must be stated tha t not all of the given rules gracefully ta ke into account r ight half-plane po les a nd zeros . beca use the phase behavio r o f po les and ze ros in the righ t half-plane is the opposite of that in the left ha lf-plane . Th e real-axis rule is a n e xce ption. Th e com plex-ang le ru le m ust be mod ified for righi half-plane singularities, a mo ng othe rs. bUI we will not make tho se modifi cation s he re. Just be a hit wary when attempting to draw loci if your loop tran smission contai ns right half-p lane sluff.

14.15 COMPENSATION We have devel oped a number o f methods for eval uati ng the sta bili ty (If feedback sys· terns. For exam ple. roo t-locus tec hn iques allow us to sketc h rapidly how the closed loop pol es vary as we c hange so me parameter of the loo p. but constructing root loci requires a rat ion al ex pression for the loo p transmissio n in order to find the loop tra nsmission po les a nd zeros . Ga in and phase margi n. as well as the Nyqu ist lest . req uire knowledge only of loop tra nsmission gain a nd pha se beha vior, and this informa tio n may be obtai ned experi me ntally, We now shift our focu s away from analyzing stability 10 changing it. As you mig ht ex.JX'CI, we will dra w hea vily from insights thut are im plic it in the vario us
14.16 COMPENSATION THROUGH GAIN REDUCTION Let us consider one implic atio n of pha se margi n as a stability mea sure. If we assum e that the loop tran smi ssion of our uncompe nsated sys tem has an increasingly nega tive phase shift as freq uency incre ases (e.g ., beca use of the prese nce of poles) , then stahili ty could he im proved sim ply by red ucing the crossover freq uency to a value such that the associ ated phase shift is less negative. On e " low-tech" way to effec t suc h a reduction in c ross ove r freq ue ncy is to reduce the loop tran sm ission magnitude by 11 fixed factor a t all freq ue ncies. Si nce using such an anenuato r does not affec t phase behavior, it is triviul tu calcula te the a ttenu a tion factor requ ired to satisfy a given phase margin spcciticutio n. As a speci fic example, consider using a n op-am p that requires trul y ze ro input current and po ssesses zero OUIPUI imped ance. but has the fo llowi ng tra nsfer fu nc tion : G(s)

=

10' (s

+ 1)(I0- 3s + I )

.

(65)


'OUT

FIGURE 14 . 24 . Inverting ampli ~er .

Suppose we lake this op-amp and connect it in an inverting co nfiguratio n with an ideal c losed-loo p ga in of - 99. as shown in Figure 14 .24: we have RI/R z = 99 . With this inform ation. we can readily derive an expression fur the loo p transm ission: - L( s )

=

Hz 105 . G ( s ) = 10- 2 • G ( s ) = R I +R 2 ( 5 + 1)( 10 - ]5 + 1)

(66)

Let 's now com pute the phase margin for this connec tion. First. we find the crossover frequency. In general. u's most co nvenient tu use simple functio n evaluation (fancy nam e for trial and error. guided by a rough Bode pint) . In this case. we can pin down the crossove r freq uency quite acc urately wit hout much computation by ex ploiting a few obse rvations. First. the dominant po le at I rps ca uses a - 20 · d B/dl..'Ciide rol loff until the second pole ill I krp s is reached . Withou t that second pole. the loo p transm ission would have u magnitude of 100 at I krps. T he second po le uccelerutes the roll off ultimatel y to - 40 d B/ decade. so on ly anot her decade beyond I krps takes us from a magnitude of 100 to an extrapolated magn itude of unity. Since crossover is a decade beyo nd the second pu le, we may ass ume that , to a good approximation. the rolloff is - 40 d B/decade there, so that crossove r indeed docs have the following approximate value: (67)

Co mputation of pha se margin is similarly trivial, as crossover occurs well above both loop tnm smis sion pol es. The pule at I rps may be co nsidered to contribute - 9(l ' at 1O ~ rps (the act ual phase shift is only O.()()57 ° shy of - 90" , so little error is invol ved ), while the second pul e also comrib ures nearl y - 90 ", However , a decade beyond a pole we have a resid ual error of 5.7 so here we lind that th e ph ase margin isn' t qu ite tern (bu t it is sma ll). with s uch a s mall ph ase margin. we would expec t large OVCf1>IUXll in the ste p response. large pea king in the frequ ency respo nse. and extreme sensitivity to any addi tiona l negative phase shift Inun unm odelcd pole s that may be lurking in the shadows . In short. the sta bility is very un sat isfact ory. SUpJXI!'C we wanted to achieve a phase margin of atleast 45° by using gain reduction. How could we do it? From inspec tion of the ex pression for loop transmission, 0

,

14 .16 CO M PEN SATIO N THRO UGH GA IN REDU CTIO N

"N

"

' our

FIGURE 14. 2 5 . Inverting amplifier with reduced-gain cornpen!>Otion.

it would appear tha t cha nging the ratio of the feedb ack resistors would he on e posslbility. If we made R 1 smaller. we would reduce the loop tran smi ssion magnitude at all Irequcnclcs. Unfortunately. we would also change the idea l closed -loop gain and. presumably, we 're not perm itted that parti cul ar de gree of freedom. The so lution is to add another resistor. this time across the input terminals of m e op-mnp (see Figu re 14.25 ). Thi s co nnection may be puzzling 10 those who have been taught to treat an up-amp as an ideal element whose input voltage difference is zero (the "virtual ground " co ncep t and all that). It is easy to jump to the incorrec t conclusio n that the add ition of resistance in such a place ca nnot have any effect bec ause the voltage across it is ideally zero. The key word is " ideally:' sinc e we do 1101 have an idea l up-a mp. Consider. for exa mp le. placing a short circuit ucro..s the input terminals of the op-u mp. T he loop tran smission mu st go to very s mall (zero) values in that situation. Look ing at the situation more analytically. no te thai such an additional resistor appears in parallel with Hz as far as the loop rransmission is concerned. but disappears as far as the ideal closed -loo p transfer function is con cerned (t IJi,\' is where we invoke the virtual ground idea). Hence . we ca n effect cha nges in stabilit y witho ut disturbi ng the ideal closed-loo p tran sfer function. Let 's now com pute how mu ch gain reduction we need . Since the phase margin goal is 45 ~ , we need 10 find the frequency at wh ich - [. (.\.) gives ux a phase shift of - 135°, since that will become the new cross over frequency. Again . from inspec tion of the ex pression for the loop tran smission. it should be apparent that the new crossover freq uency should he the frequ ency of the secon d po le, that is, 10 '\ rps, since at that frequency the first pole has contributed essentiall y - 90 " and the seco nd pole another - 45". At the new desired cross over frequency, the uncompensated loop transmission has a magnitude of approximately

IL(j IO' ll =

10'

,J2 " 70.7.

(68 )

IJ I0'+ 11· 1 21

Therefore. this is the factor by whic h we need to redu ce the loop transmission gain.

426

CHAPTE R 14 FEEDBACK SYS TEMS

Th e old loop transm ission ma y he e xpressed as

(691 Th e new loo p tra nsmission if>

R

- L ( 5) = ( Hz

I;

R

+ I ) - ' . G(.i).

(701

whi ch may be rea rra nged as - L ( 5)

=

R ] [ R + ( R, II Rz)

R,

.~

RI

+ Hz

. G( s ) .

(711

where the term in brackets may be considered the co mpensat or's tran sfer function

C(,I. Since we nee d 10 provide a who pping factor of 70 .7 ga in reduction. we need C (s) In howe a value of 1/ 70 .7. Th e refore. we need 10 c hoo se R sma ll e no ugh to give us this atte nuatio n. Becau se ofthe large a tte nua tio n factor. we wo uld ex pec t R io be so sma ll compared with HI and H2 that it sho uld be about 70.7 limes sma lle r than H2. ( 0 a good approximation . A slig htly more rigo rou s calc ula tio n yields a va lue quite close to that estimate : R ::::::: Hz

711.4

(721

In sum mary. the red uced -gai n co m pensato r has taken the sys tem fro m a phase ma rgin of a bo ut 5.7" 10 a phase margin of 45~ by reducin g the loo p transm iss io n by a fuctor of abo ut 70 .7 at a ll frequenc ies. At the sa me lime, c ross over has decreased by a fac to r of 10, fro m a frequency of 104 rps to 10" rps. T he idea l closed-loo p gain remai ns - 99. Th e trad co ffs. of co urse , are a reductio n in band widt h and dcscn sit ivit y, Furth ermor e. fhc reduction in dcsen sitivity occu rs at all frequ enci es. wh er eas stability is detcnnined by just the behavior nca r crosso ver (so say s pha se ma rgin, anywa y). Hence, DC andlow-trcq ucncy dcscnsitivity are apparen tly needle ssl y co mpro m ised by such a simple -minded co mpe nsation sc heme.

14 .17 LAG COM PENSATION If we could red uce crossover by atte nuating the loo p tran sm ission o nly a t higher freq uencie s. we wo uld leave low-frequency desen sitivity unto uc hed while improving sta bilit y. Suc h a se lec tive loop tra nsm ission red uc tio n ca n be accomplis hed by

14 .17 LAG CO MPENSATIO N

R,

'"

R, 'OUT

R C

FIG URE 1-4 . 26 . Inverting amplj ~er wi th lag compensator'.

addin g a capacit or in series wit h the gain reduction resistor of the previous topology. as shown in Figure 14.26 . Tb e c apaci ro r prevcnts the co m pensatio n resi sto r from having any effect at DC and low freque nci es. ..... bile the net wo rk degenerates 10 a sim ple red uced-gain compensator at frequenci es high eno ugh for the ca paci to r to appear as a sho rt . S uch a compensator i.. kno w n as a lag compensator for reasons thai will become clear sho rtly. Once again. the co mpe nsatio n network has 0 0 effec t o n the ideal closed -loo p tran sfer function because it is connec ted across two terminal s th ai have no "ullage d iffe ren ce in the ideal limit of infinite op-ump ga in. To discover the real effec t of this compensa to r, thou gh . let 's deri ve an ex pression for the loo p tran smi ssio n: - L (s)

= [ 8 , II ( 88 ~ 1/ £ )

+I

l

which . afte r so me manipu latio n, " simplifies" to - L (.\·)

=

(731

. G(.,).

s8C + I

1' Cl 8 ( 1 + 8, /8 , ) + 8,] + (1 + 8 ,/8 , 1

I

. G( s ) .

(74)

After furt her manipulati on , this ca n be e xpressed in a so me what more intuitively useful form : - L (.~ )

=

I

.,8C + I

.
II R,)I + I

I

.

R, 8,

+ R,

, G(.~) .

(751

The ter m in braces may he conside red th e transfer funct ion of the co mpe nsa to r, w hile the rest of the eq ua tio n is (minus) the loo p tra nsmissio n of the uncompensa ted syste m. AI DC . the co m pensa tor tra nsfer funct ion is un ity, and the sys te m behaves as in the uncompensa ted ca se . AI very high frequencies, the corn pcns uro r asy mptotically approaches a value of

8 C ( s ) -- cR'""+ --'(""'R'-,--'~ II R, ) '

just as in the red uced -gain co mpe nsator case. as ex pected .

(76)

p


Note that the compe nsator C(.~ ) contains one zero and one pole. As can he seen fro m the full expression for C(.{ ) . the po le is always at a lower frequency than the zero. It is the pole thai ca uses the loo p tran smission magnitud e to decrease (since the magnitude of C ( .{) decreases beyo nd the po le frequency). Unfort unatel y. an unavoid able side effect is the negat ive phase shin thai is assoc iated with the pole. It is this pha se lag that gives this compensator its name . C lea rly. o ne im portant desig n criterio n is to make sure that this Jagging phase shift has been cancelled by the zero 's po...itive phase shifl we ll be low cross over. Otherw ise. phase ma rgin will ac tually degrade ra ther than improve . A simple ( hut not nec essarily optimum) design procedu re is In begin with the red uced -gain co mpensa tor to disco ver the value of R nec essary to force crossover 10 a low enoug h frequency to achieve the speci fied phase margin . For reasons that will become clea r sho rtly, it may be advi sa ble to aim for a ph ase margin abou t five or six degrees larger than you ultima tel y wan t. Then, place the zero a decade below the new desired crossover by choosing 10 RC = - W e .""",

=

10

C = -;;-'-"----

R · w.-._

(77)

With this choice of ze ro location . the pos itive phase shin o f the zero will be about 5.7" shy of its maximu m, wh ile the po le. with its lower frequen cy. has contributed j ust about all of its - 1)0 -' of phase shift . Hence, if the reduced-gain compensa tor is used as a starting po int for lag co mpensa tor de...ign . the phase margin goal should be augme nted by live or six degrees. as staled earlier. A more though tful des ign might requ ire iteration to complete, since the po le and zero locat ion arc both adj ustable param eters. Hen ce, there is no unique sci of R and C that provides u given phase margin . The simplified procedure presen ted here usually suffices , howevcr. 10 provide either a final design or a reasonable initial design from which further optimization may develop . The lug com pens ator pro vides roughly the same crossover (and hence roughly the same clo sed-loop band width) as the reduced -gain compensator, but leaves the low-frequency loop transmission untou ched. Hence. it doesn't degrade desensitivity unnecessarily, and one may obtain all of the associated bene fits. such as reduced steady -stale ste p response error. However. there is one drawhack to the lag co mpensator that de ser ves mention. The compensator employs a zero that is well be low cross over. Furt hermore . since the ideal closed-loo p tran sfer function doc s not contain a zero. our modeli ng proced ure tell s us that the zero mu st come fro m the forward path and that thc zero thus appears in the closed-loo p transfer functi on. Add itionally, fro m roo t-loc us construction rules. we know that zeros are the terminal location s for loc i: they att ract poles. Hence. we ex pect a closed -loop po le close to this cl osed-loo p low -freq uency zero.

14.18 LEAD COM PENSATIO N

Therefore, po les bei ng the natural frequenc ies o f a net wor k , the re will he a slowsettling co mpone nt to transient respon ses.!" The proble m is equivalent 10 co nside ring the effect of imperfec t po le-zero ca nce llation. To ex plore this idea in more detail, le t us exa min e an iso lated pole -zero doublet: D i s) =

ats + I . rs + I

(78 )

To go further, we ' ll need to rec al l the initial- a nd lin al- valuc theorems from Laplace tran..form theory : f(oc) = lim s F(s).

,-"

f( O) = lim sF(s).

(79)

180)

I ---+X

With these for m ulas. it is straightforward to sho w that the initi al value of the step rcsJXmsc of a dou blet is j ust a, while the final val ue is unity. Note that (he initial value is not zero bec ause the high-freq uen cy gai n doe s not go to zero ( in fact . it goes to a) . Now that we kno w that the initia l a nd fina l values a re differe nt, we ne xt have 10 find how we gel from the in itial 10 the final value. f ormall y, one wo uld u..c the inverse Laplace tran sfor m 10 discover thi.. informa tion r igorou..ly. However. we can avoid a lill ie labor by reflect ing once aga in on the meaning of the term " natural frcquency." Evide ntly, the n, the step respo nse evolves expo nentially from its initial 10 final value with a time consta nt eq ual to that of the po le. U Returning 10 our speci fic case of the lag com pensa tor, there is a pole -ze ro do ublet formed by the compe nsating ze ro and its assoc iated cl osed-l oop pole. Becau se the zero is well below c rossove r. the doublet 's po le has a m uch slowe r tim e consta nt than the inverse IO(lp bandwidth . Hence. se ttling to fine accuracy ca n he m uch slowe r than sugges ted by the loo p band width whe n a lag compensator is used.

14.18 LEAD COM PENSATIO N We have see n that phase margi n ca n be im prov ed by redu cin g the magnitu de (If the loo p tran sm ission in order to lower the c rossove r frequ e ncy. The trad eoffs wit h such an approach inclu de a loss of desensitivity and the po ssi bility o f low-freque ncy doublet formation . An alte rnative compensa tion me thod is to alter the phase of the loop tran smission , rathe r than its mugnuu dc. Th at is, we wish to add a po sitive Of It'mlinx phase shift I ~ Here. "s low" means with respect to the cro ssove r frequency. 15 This key fact js e\'ide lllly poorly understood by many. T he po le -zero !\Cparalitln deter mines t h~ rutio of innia l 10 final va lues, while only the pole determires jhe rote at w hich the rcspnn.o,c !>O.'t· rles to Ihe linal value from the initial nne . The zero Ihus has nothing 10 do with Ihe lime enn stalll rhat describes the ~tt ling .

• CHAPTER 14 FEEDBACKSYSTE MS

' 30

c

'

. fi GURE 14. 27 . Inverting a mpl i~ef with lead compenwtor.

ncar crossover 10 improve phase margin. O ne met hod for doi ng so in our op-amp exa mple is illustrated in Figure 14.27. No ll' that we are 110 lon ger maintaining the same ideal closed-loop transfer function. However. as we'I l see. the overal l c losed- loop behavior will ge nerally a pproach the desired ideal more closel y than the redu ced -gain or lag co mpensated systems. First, withou t writ ing any eq uations. let's ce how the add ition of this capacilor ...bould give us a loop transmiss ion zero. A frequ ency increases. the transmission through the capacitor increase s. That's what a zero do cs. so we get a zero, as advertised . If we choose the capaci tor value co rrectly. we can usc the associuted zero to bend the phase shift to mo re positive values and thereb y increase phase margin , There is one dange r, however. A zero provides an increasing magn itude characteristic in add ition to its positive phase shift. Hence. it also pushes out crossover. Therefore. there is the un fortunate po ssibility that a poorl y placed zero will increase cross over so much tha t the po sitive phase shift o f the zero wi ll not o ffset the increased negative phase shift of the uncompen sated system. T he net effect could act ually be a reduc tion in phase margin . so beware of thi s possibility. Okay, now it 's time for an equation or two. Le t 's derive an expression for the loop transm ission for our lead -compen sated system: - L (.f)

=

R, ' G(.\'} , R, + IR, II (l /sel )

(8t)

....-hich may be expressed as

R,

(8 2)

Here, we sec that the loop transm issio n zero is at a lower frequency than the associated po le, the oppos ite of the lag co mpen sa tor. Design ing a lead compensator alm ost alw ays involv es a fair amount of iteration. A few hints may help co nstrain the search space. however. A reason able starting

t.• . 18 LEAD COMPENSA TIO N

4 31

point is to place the zero at the uncompensated syste m's crossover frequ e ncy. Vary the zero loca tion abo ut this frequency a nd find the maximum . If the phase margi n specification ca n be met , you' re finished . Not infrequently, however, you lin d that the phase ma rgin speci fication cannot be reached for a ny val ue of ze ro locat ion . In such cases. a combination of gain redu ction and lead compensation usual ly suffices. Unfortunatel y, w ith two varyi ng paramercrs (gai n red uction a nd zero locat ion), finding an optimu m ca n he a bit involved , and machine computation is de finitely a trem endous hel p. Don 't tum off your brain, though - yo u should always have a rough idea of what the an swe r should be. j ust as a sanity c hec k on the compute r's result s. If the necessa ry gai n reduction is too large, then convert the ga in reduct ion in to a lag net work . The resulting Inul-laN co mpensator the n gives you ma ximu m desen sitivity at DC a nd at high frequ en cies. At thi s po int, yo u ma y be wondering how we can gel away from the do uh let prob lem that afflict s the lag co mpen sator. Th e an swer is twofold . First , recogn ize that the lead zero is located nea r c ros sove r, not well be low it. Hence, any closed-l oop pole that wo uld he associ ated with it wo uld have a time constant consistent wit h the loo p bandwidth . T hat is, any doublet " tail" wo uld sett le ou t at abo ut the same rate as the rise rime. and he nce would he invisible. Thi s observation applies to all lead-compensa ted syste ms . A second reason that ap plies spec ifically to the pa rticu lar up-amp connect ion shown here is that the lead zero doe s not appear in the forward path . Again, to conclude that this must be the case , reco gnize thai the idea l closed-loop transfer function invo lves a po le. He nce, the feedb ac k block in our model m ust supply the zero. T he forward gain block does not ha ve a zero . Sin ce the ze ro a ppears in the feedbac k pa th, it does not show up in the closed-loo p transfer functi on, and thus the re is no closed- loop do uble t. A question that ofte n a rises at this poi nt is why anyone wo uld eve r use anything but a lead compensa tor. Arter all, it can actually provide g rt'afn bandwidth than the uncompen sated sys te m and it ' s free of this doublet problem. Th e a nswe r is tha t bandwidth costs power, a nd sometimes the pri ce is too high . This co nsideration is particularly significant in mechanical sys te ms, w here power requirement s are roughly proport ional to the c ube of bundwidth .!" In large, ind ustrial mac hinery, such a relano nshi p between power a nd band width favors the minimum bandwidth consiste nt with getting the j ob do ne. Even in electronic syste ms, large r bandwidt hs are nOI always desirabl e. Noise is always present, and larger ban dwidth ca n mea n add itional noise . If the bandwidth is

II> Here's a quick hand....a\'ing "derivation" (put on your .....indbreakers, 'cau-e we're going to dll a lor of handwaving): Power or k/ lime = (1 1 )(ineniaiiangular accel~ralion / time ) = 0:11 x f inc.'n ia )(uh/ time = l lW). 1 warned )'00 .

=.. .


in excess of what is ac tuall y needed . then the re is ge nerally a n unnecessa ry degradelion of signa l-to- noise ratio . In many instances. suc h a degradatio n is not tole rable.

14.19

SUMM ARY Of COM PENSATION

We ' ve see n three basic co mpensa tio n techniqu es thai may be used individ ually or in co m binatio n. Bot h the red uced -ga in a nd lag co m pensato rs see k to impro ve phase margin by red uci ng crossover 10 a value where the co rres po nding phase shift isless negative Lhan in the unco mpensated case, the reby inc reasing phase margi n. Th e lag compensa tor improve s o n the simple red uced-gain co mpensator by lea ving untou ched the low-frequency loop tra nsmissio n. bUI introduces a po tentially bothersome doublet thai causes slow settling to high accuracy. Th e lead co mpe nsa to r impro ves phase ma rgin by direc tly improving the phase .~ h i ft of the loop tran smi ssion. In M) doing. the ban dwidth ac tually increases. Purthcrmore. the dou blet problem d isappears because the pol e associated with it is just as fast as the o ve ra ll a mplifier. so an y " ta il" in the res po nse is effec tively ma sked during. the rise ume . The lead co mpe nsato r freq uen tly must be co mbi m..xt with e ither a reduced -gai n o r lag co mpensa tor 10 provide sufficient deg ree s o f freedo m to satisfy a given phase margin speci ficatio n. As a fina l no te o n com pensa tion. it sho uld be MOIled that the type s discussed here do no t co m prise a n e xhaustive list . Additio na lly. eve n thou gh these co mpensators were iIIu.. . trutcd w ith speci fic op-amp ci rc uits. th e fu nda menta l no tions app ly 10 all tit he r feed back syste ms as we ll. He nce. any meth od that red uces the loo p transmissio n magni tude uniformly is a red uced -gain compensato r. anything thut introduces a loo p tra nsm issio n pole -zero pa ir in whic h the po le is at the lowe r freq ue ncy is a lag compensato r. a nd su lin.

PROBLEM SET FO R FE EDB ACK PROBLEM l

O ne import ant benefit of negative feedbac k is the red uct io n of distortion . This pro ble m explo res th is prope rty in mo re qu ant itative deta il. Fo r simplicity, assume that the forwa rd pat h of a uni t feedback syste m has a n input - output transfer c humct c ristic that o ne may expres s as the follo wing cubic po lyn om ial : (P I4.11 Assume that the syste m is o nly we a kly no nlinear. so that the quadratic and cubic terms arc sm all com pa red w ith the first-o rder {desi n.."t1 1tenn . (a) Derive a c ubic po ly nom ial apprc xi rmnion for the overa ll input-output transfer c ha rac te ristic . Verify tha t yo ur eq uation collap.. .es to lluR I !( I + tl uK. ) in the linc ar limit .

PROaL EM SET

433

(bJ By approximately what factor do the qu adratic and cubic terms decrease as the linear loop transmis sio n magnitude increases'? Based on yo ur answer. how do higher-order terms vary as the loo p tran smission magnitude changes? PROB LEM 2 It was noted in the chapter that one may not always lind a I: I mapping between physical clements in a feedback syste m and the ca nonical feedback mod el. furthermore . the ass ignment of tra nsfer functio ns 10 each bind; in the model is not necessari ly unique. Show tha t representing a system in terms of a feedback model can actually he a philoso phical cho ice hy modeli ng the i npu t-outpu t voltage tran sfer characteristic of an ordinary resistive vo ltage divider with a feedback system. PROBL EM 3 Common errors when CUlling the loop to determine loo p tran smission include fa iling to acco unt for various loadi ng effec ts and ignori ng the need to establish correc t DC opera ting point s. Thi s problem ex plores these im porta nt practical issues in more detail. Consider the loop sketched in Figu re l 4.2R. T he com bination of the ideal op-amp and resistor R toget her model a real op- ump thai po ssesses a nonze ro output res istance.

FIGU RE 14. 2 8. Feedback system for loop tra nsmiuion problem.

(a) Sho wn arc two possib le locations for breaking the loop . poi nts A and B. Explain why breaking the loo p at B and dri ving thai po int with a test vo ltage so urce yields the incorrect loop tran smission . (b) Suppose that the or-amp also has an extremely high DC gain. as is typ ical o f many ge neral-purpose op -amps. Assum e that when the syste m is operating as a normal closed-loop feed back syste m. the output com mo n-mode voltage is ncar zero . If V IO is set to zero value w hen determining the loo p transm iss ion. descri be how you would deter mi ne w hat DC value the test gcnenuor should have. Shoul d it he zero. or would a different value be a better choic e'! Explain. PROBLEM 4 II was assert ed in the chapter thut thc root loc us of a lead-compe nsated two-pole system contains a perfect circl e. Show this property forma lly for the case

... 434

CHAPTE R 14 FE EDBACK SYSTEMS

o f a system whose loo p tran smi ssion consists o f two integ rators and a sing le zero whose time constant is Tz• PROBLEM 5 Linear a na lyses are use ful appro xima tions. bUI physica l sys tems exhibit stru ng nonlineariti es if dri ven hard enough . Aside from obvious side effects such as inc reased d isto rtio n. there can be co nseque nce s for stability if the nonlin earity is imbedded within a feedback loop. O ne common no nlinea r phe no me no n is that of saturatio n. in which the o utput c hange s little ( if a t all) beyo nd a give n input ampli tude. To the exte nt that one may thu s interpret saturatio n as a reduction in gain. ex plain the fo llow ing observatio n. A syste m whose loop tra nsm iss io n consists of three integ rators (implemented with sta ndard up-amp circuits ) a nd two coinciden t zeros is adj usted to pro vide a healthy ph ase margin of nearly 60~ . Th e step re sponse fo r sma ll amplitudes is we ll behaved. as woul d be expected for this phase ma rgin . Howe...-er. the step re sponse exhibits a di stinctly frigh ten ing be hav ior o nce th e e xci tatio n is large e noug h to saturate one of the o p-amps. Wo uld you expect the step response rin ging in this reg ime to be of a lo wer o r hig he r freq uency than in the small-sig nal case? Explain yo ur answers in te nus o f bot h a roo t loc us and a Bod e plot o r pha se margin diagr am. Suppose that yo u a re told that the respo nse of a black box appears to be linear and second-order. with a un it DC gain. Measurem en ts reveal that the unit ste p respo nse has a peak va lue of 1.38 a nd that the time for the step respo nse to first pass throu gh unity is 500 ns. PRO BLEM 6

(a ) Assuming that it is second-order. determine the seco nd -o rde r para me ters that can he used to mode l the syste m. ( il ) Using your mod e l of part (a) , estimate the peak va lue of the o utput that would result if the syste m we re e xci ted wit h a unit impu lse. (e) Co mpute the time req uired for the im pulse respo nse to first return to zero. (d) Estimate the frequency res ponse peak ( M p ) a nd - J · d B bandwidth for this system . (c) Estimate the time it takes for the ste p respon se to se ttle to within 2% of final value . (f) Describe a test that yo u wo uld pe rform in o rder to co nfirm that the system is in factlinear to a good approximatio n. PROBLEM 7 Most feedba c k sys tem s we enco unte r a rc low -pa ss in natu re. and it easy 10 ~e t the impression th at stability problem s a rc so lely a property of such syste ms. To break us o ut o f this narro w thin kin g. thi s probl em investigates the stability of higll · pass syste m s. For exa mple, AC · co up led feedback ampli fie rs ca n sometimes ex hibit a n intere sting low-freq ue ncy oscilla tio n known as "moto rboati ng," first observed in vac uum tube audio a mplifiers . S uppo se th ai the loo p tra nsmi ssion for such a n a mplifie r is

PRO BLEM SET

1.( ., )

=

(s

+ 1)(0 . Is + 1)2 '

( 1'14.2)

(a) Sketch the: rootloc us for this amplifier for positive values of l lO · (b) For what range of values of au is the amplifier stable'!

(c) At what freq uency (or frequ enci es ) can this amp lifier osci llate? Your answe r to this question should suggest why the term - mororbocting' is a ppropriate. PROB LEM 8 Phase-loc ked loops are of great utility in com mun ications systems. among othe rs. Although we will take up the detai led study of PL.Ls in a later chapter. it is sufficie nt for now simply 10 accept that a linear second -order model is a good approx imatio n for a ce rta in class of PLLs. Sup pose that such a PLL ha s bee n determined 10 have the followi ng (negative) loop transmi ssion to an excell ent approx -

imation: - [A s )

=

K(n

+ I) 2

(1'14.3)

•

S

(a) Determi ne values for K and

T

that yield a closed-loop po le pair wit h dam ping

ratio of 0 .707 and an w " of 10 Mrp s. (b) Sketc h the mol locus for K > 0 wi th the value of

T

fou nd in part (a ).

(c) What value of K gives us a critically dam ped loo p? (d) Assume that the zero doc s a ppear in the forward path , and assume a uni t closedloop DC gain. Ske tch the freq uency response of the cl osed -loop system for the pole loc urionts) found in part (c ). PROB LEM 9 It is import ant 10 develop a facili ty for modelin g physical circuits as feedback systems. In that spirit, mod e l a conventional textbook source follower with a degeneration resistor as a feed back system. Use a simple, low-frequency, smallsignal model for the MOSF ET , and give expressions for the forward and feedback

gains. PROBL EM 10

Consider a feedback system in which the for ward-pat h transfer func-

lion is given by

K ( ~, 2 - 3s

+ 5)

11'14.4 )

and the feedba ck transfer functio n is ( P I4 .5) The parameter K is a pure scalar q uan tity tha t may be varied . (a) Sketch the roo t Inc us for this system. (b) Identify the values of K , if any. for which the system can become unstable. (c) Wh at value of K will give a damping factor 01' 0 .707 to the dominant po le pai r?

436

CHAPTE R 14 FEEDBACK SYSTEMS

PROBLE M 11

You are given a n operational am plifier thai was designed by SubOp timal Produ ct s, and careful measurem ents revea l that it has a tra nsfe r function give n hy 5 · I O~ IPI4.6) a(.\') = ( .f + 1)( IO - 3s + 1)(1O - 4 .f + I) ' Su ppose it is your misfortune to be given the task of using thi s a mp lifier in a n inverting config uration with an input res istor of 22 kQ a nd a feedback resis tor of 220 kO, so that the ideal closed -loop gain is - 10.

w hat is the phase margin? Sket ch Bod e plots of the loop transm ission for this and subseque nt part s. (h) Suppose you arc now told that we must ha ve a phase margin of 60 . TIle fi rst com pe nsa tion tech nique that comes to mind is a red uced-gain compensator. By what factor is it necessary to red uce the loop tran smission to achieve a 60 ' phase margin? What value o f R cnmp • placed ae ro.... the op-a mp inpu t termi nals, gives us this phase margin? what is the c russever frequ e ncy" (e) The co mpensator in part ( b) th rows aw ay loop uun sm ission OIl all frequencies. but stability is de termi ned ma inly by the be hav ior o f the loop near c rossover. Recognizing this fact . now ame nd yo ur co mpensa tor to recover the lost gain at low freq uencies by designing a lag co mpensator. Feel free to usc this rule of thumb: Place the compensa to r zero a dec ade belo w the crossover frequ ency of part (b). a nd the compe nsa tor po le a decade below that. What is the new phase margin in this ca se'! (d ) The sys te m is stable , and the low-frequency desc nsitivity is restored , but you're unhappy with the ba ndwidt h . You therefore co nsidcr u lead com pensator. after re mem bering tha t it wor ks by add ing: po sitive phase shift to the loo p transmiss ion whe re you need it , instead. of forcing a lowe r crosso ver frequency. So you return to the ori ginal, uncompen sa ted syste m and. add a ca paci tor in parallel with the feedbac k resistor. Wh at value of ca pacito r gives us the maximum phase margin? What is the ma ximum phase margi n'! (c) Suppose it is absolu tely impera tive to achie ve flO' of phase margin . What strategies might yo u e mploy to mod ify the com pensator of part (d) 10 obtain the desired stability? A qualitative an swer will suffice. O ffer at least two suggestions. howeve r. (il ) Deri ve an ex press io n for the loop transmi ssio n.

PROBLEM 12 Consider two feedb ac k syste m... bo th with unit tran sfer function fo r on e system is 0 0(0 .2 s

(l (

while the othe r has

s)

+ I)

= :---,--,,";'7.=::""':-O~;--:-77 Is

+ I )(O.Ols + 1)(0 .2 1., + I)

f.

The fo rward-path

(P I4.7)

PROBLE M SET

{/ (.~ )

=

"0

(, + 1)(0.01' + 1)

,

( PI4.8)

Sketch the roo t-loc us diagrams for both sys tems . Explain why these two sys tems have similar d osed -loop respo nses, PROBLEM 13 Suppose that the ope n-loo p tran sfer function of an automatic ga in control (AGe ) loo p is fo und 10 be

,, (.~) = (O.ls

10 '

+ 1)(10 - 6 $ + 1) 2 '

(P I4.9 1

(a! Determine the gain margin. phase margin. and crossove r frequency for this systern whe n used with unit feedback , (b) What value o f f result !'. in a phase margin o f 4Y '! PROBLEM 14 An up-amp has an ope n-loop transfer function given by

tl(S)

=

rc. n + 1)( 10

5s

+ 1) 2 '

(P I4. 1O)

The op-amp is used in a conventional invert ing configura tion with an ideal gain o f - 10. Determi ne the loo p transm ission gain red uction factor required to provide a phase margin of 45°. Sketch the magnitude and phase of the loop transmission before compensation.

.. CHAPTER FIFTE EN

PHAS E-LO C KED LOOPS

15 . 1 IN TRODUCTIO N Pha se -loc ked loo ps ( PLLs) have become ubiqu itou s in modem com munications syste ms because o f the ir re markable ver sati lity. As o ne important e xam ple. a PLL may be UM:d 10 ge nerate a n output signa l wh ose frequen cy is a prog ram mable , rational multi ple of a fixed inpu t freq uency. The ou tput of suc h / ret/llelle)' syn thesizers may be used as the local osc illator signal in superhet erodyne transceivers. Phase -locked

loops may also be UM:d to pcrfonn frequency mod ulation and demodulation. as wellas to rege nera te the carrier from an input signal in which the carrier has been suppressed. T hei r ve rsatility exte nds to purely di git al sys te ms as we ll. whe re PLLs a rc indispensab le in skew co mpensa tion. clock recove ry. and the ge ne ra tion of clock signals. To und erstand in de tail how PL L~ may pe rfo rm suc h a vas t array of functions. we wilt need to develop lineari zed models of these feed back syste ms. Hut first. of course. we begin w ith a little histo ry to put this subje ct in its proper context. 15 . 2

A SHORT HI STORY OF PLLs

Th e ea rliest de scri ption of what is now know n as a I'LL was provided by H. de Belles-

cizc in 1932. 1 Thi s early wor k offe red an alternat ive a rc hitecture for receiving and dem od ulatin g A M signals. using the dege nerate cas e of a supe rheterody ne receiver in which the interm edi ate frequ e ncy is zero. With this choice. there is no image to reject . and all processing down stream of the frequ ency conve rsion takes place in the uudio range , To functi on correc tly, howe ver. the /ltJll roJ YI/(' o r d i rect-convers ion rece iver requi res a local osci llator ( LO) whose frequ ency is prrciselv the same as that of the incoming carrier. Furthermore. the local oscilla tor mu st he in phase with the incoming

I "La Reception Synchrone," L 'U nd(' [ {(,cinque. v. II. June 1932. pp. 230- .«). 438

15.2

A SHO RT HI STORY O F PLL I

439

carrier for maximum output. If the phase relationship is uncont rolled. the gain could be a... small as zero (as in the case where the LO happens to be in quadrature with the carrier). or vary in some irritatin g manner. De Bellescize described a way to solve this problem by providing a local oscillator whose phase is locked to that o f the carrier. For various reaso ns. the homodync receiver did not displace the ordinary superheterodyne rece iver. which had come to domin ate the radio market by about 1930. However. there has rece ntly bee n a renewal of interest in the homodyne architecture because its relaxed filtering requirements possibly improve amenability to integratio n.I The next PLL-like circuit to appear was used in televisions for over three decades. In standard broadcast television . two sawtooth generators provide the vertical and horizontal deflection C'sweep") signals. To allow the rece iver to synchronize the sweep signals with those at the studio, timing pulses arc transmitted along with the audio and video signals. To perform synchronization in older sets, the TV 's sweep oscillators were adj usted to free -run at a somewhat lower freq uency than the actual transmilled sweep rate. In a technique known as inj ection locking. " the timing pulses caused the sawtooth oscilla tors to terminate eac h cycle prematurely. thereby effecting synchronization. As long as the received signal had relatively lillie noise. the synchronizn no n worked we ll. However. as signal-to- no ise ratio degraded. synchronization suffered either as timing pulses disappeared or as noise was misinterpreted as timing pulses. In the days when such circ uits were the nonn, every TV set had to have vertical and horizonlal " bold" co ntrols to allow the co nsumer to fiddle with the free -runni ng frequ ency and, therefore. the quality of the lock achieved. Improper adj ustment ca used vertical rolling or horizon tal " tearing" of the picture. In modem TV s. true PLLs are used to extract the synchronizing informatio n robustly even when the sig nal-to -noise ratio has degraded severely. As a result , vertical and horizontal hold adjustments thank full y have all but di sappeared . The next wide application of II l' LLbke circuit was also in televisions. When various co lor television systems were being considered in the late I940 s and early 1950s, the Federal Co mmunications Commission (FCC) impo sed a requirement of

2 How ever, the humodyne requ ires excep aonal from-e nd linearity and is intole rant of OC offsets. Furthermore, since the RF and LO freque ncies are me same, LO leakage had. OUI of the antenna is a proolem Additionally, this La leakage ca n sneak hack inlu the from end, whe re il mixes with the LO wilh some random phase, resulting in a varying DC otfr.ellhal can he seve ral orders o f magnitude larger than the RF signal. ~ prohl ems are perha~ a~ difficuh ro solve as the fillering proble m, and are considered in gre..ter de tail in Chapter UI. J The circad ian rilythms of huma ns provide anuthe r eumple o f injecliun kld.ing. In the absence of a synchroni zing signal from ue sun (lif oeer periodic light source ), a "day" fUf most peop le is ouoond25-27 hours . Noe that the free -runnin g frequency is again some ....hat lower than the lucked frequency.

CHAPTER 15 PHASE -l OCKE D LOOPS

co mpat ibility with the existing black-a nd-w hite standard, and furth er dec reed that the co lor television signal could not req uire any additiona l ba ndw idth . Since monochrome telev ision had bee n develo ped wit hout loo king forwa rd to a colorful future. it was decided ly nontrivial to satis fy the se constraining requiremen ts. In particular. it seemed impo ssible to squr.r.eze a color T V signal into the same spectru m as a monochrome signal without degrading something. The breakthrou gh was in recog nizing that the 30 -Hz frame rare of television results in a co mbhke - rather than co ntinuous - spectrum, with pea ks spaced 30 HI.apart . Color informatio n co uld then be shoe horned in between these pea ks with out requiring add itional bandw idth. To acc omplis h this remarkable feat. the added color informati on is modulated on a subcarrier of approx imately 3.58 M H z :~ The subcarricr frequency is carefully chosen so that the sideba nds of the chroma signal fall precis ely mid way between the spectral peaks of the monoc hrom e sig nal. The combined mono chrome (also known as the "luminance' or "brig htness" signal) and chro ma signals subsequently modu late the fina l carrier that is ultimat ely tran smitt ed . The U.S. version of thi s sche me is known as NTSC (for National Te levision Standard s Co mmittee). Color information is encoded as a vec tor whose phase with respect to the subcarTier determines the huc and whose magnitud e det erm ines the am plitude ("sa turation") of the co lor. The receiv er must therefore extract or rege nerate the subcarricr quite accurately to pre serve thc W phase refe rence: otherwise. the reprod uced colors will not match tho se tra nsmitted . To enable this phase-loc kin g. the video sig nal includes a " burst" of a numhcr of cycles (NTSC spcciticntio ns dictat e a minimu m of 8) o f a 3.58-MHz re fere nce oscillation transmitted during the retrace of the C RT' s elec tro n beam as it returns to the left side of the sc ree n. Thi s burs t s ignal feeds a circuit inside the rece iver whose job is to regenerate a con tinuo us 3.58· M ll z subcarrie r that is phase -locked to this burst. Since the burst is not appl ied con tinuo usly. the receiver' s oscillator mu...( free-run du ring the scan across a line . To preven t co lor . .hitt s. the phase of this regenerated subca mc r must not drift , Earl y impleme ntations did not always accomplish this goal successfully. leadi ng some wags to dub NTSC "n ever twice the same co lor: " Europe (with the exception of France ") chose 10 adopt a similar chroma sc heme, but addresse d the phase drift problem by alternating the polarity of thc reference every line. This way. phase drifts tend to average out to zero ove r two succes sive lines.

4 If you rea lly wamm k now. I~ exact freq uency i:. 3.5795..15 Mll z. deri ved from the ..l.5-MHl ropKing betwee n Ihe video and audIO earner frequ cocies 111t11lil' lied hy 4 ' 5/ :'J72. , In fact . Ihe very earhes t suchcircuns dispe nsed with ano.......i llalnr altogether. lnsread.rbe kim signal merel y excited a high-(J reson ator (a quartz crystal) , and the resulting ringing was used as l he ~. erated subcamcr. Th e ringi ng had 10 pe1'\i ~1 fl)l'ov er 200 cycks wit hou t excessive decay . (J)eny! 6 The Fre och w illi" lele\'isinn system is klllMn as SECAM. for Seq uenue l Coo leu r a\'CC Ml!TrIOIl't. In this system. lum inance and ehrominance infllml alil'n are sent serially in lime and m'tlfls1TUl;tN in the receiver .

1.5 .3

Input Signal

LI NEARI ZE D Pl l MODELS

Phase Detec tor

vco

44 \

I--~-_ Ou tput

fiGURE 15.1. Phcse-locked b:>p o rchitecture .

reducin g til' eliminating perceived color shifts. Th us was born the phase -alternatin g line ( PAL) sys tem. Anot her ea rly applica tion of Pl. L-like circuits was in stereo Fl\.I radio . Again . to preserve bac kward co mpatibility. the stereo inform ation is encoded on a subcurriel'. this time at 38 kll z. Treating the monaura l signal as the sum o f a I.: ft and right channel (and bandlim ited to 15 kil l ). stereo broadcast is enabled by mod ulating the subcarn er with the difference betwee n the left and right channels. Th is L-R difference signal is encoded as a double-sideband, suppressed-carrier ( DSB-SC) signal. The rec eiver then regenerates the 38- kl lz subearrier. and recovers the indi vidualleft and right signals through simple addition and subtrac tion of the L+ R monau ral and L-R di fference signals. To simplify receiver design. the trun... milled signal includes a low-amplitude /,iff,! signal at precisely half the subcarrie r freq uency. which i.. then doub led at the receiver and used to demod ulate the L- R signal. As we "Il see shortly. a PLL can eas ily perform rhis frequency-dou bling functio n even without a pilot, but for the circuit.. . o f 1960 , it wa s a tremen dous help. Early Pl. Ls were mainly of the injec tion-loc ked variety because the cost of a complete, te xtbook I'LL was too great for most co nsumer applications. Exct.. pl for a few exotic situations, such as satellite co mmunicat ions and scie ntific in... trume ntulion. such " pure" Pl.Ls didn 't exist in significant numbers untilthe 1970 s, when Ie technology had advanced enough 10 provide a true PLL fo r stereo FM demodulat ion. Since then. the PLL has become commonplace, found in systems ranging from the mund ane 10 the highly specialized . From the forego ing, it should be clear that phase locking e na bles a rich variety of applications. With that back ground, w e now consider a description of "textbook" PLLs. 15 .3

LINEARIZED PLL MODEL S

The basic PLL architecture is shown in Figure 15. 1. and is seen to co nsist of a phase detector and a voltage controlled oscillato r ( VeO). The phase detector co mpares the phase of an inco ming reference signal with that of the VCO, and produces an output that is so me function of the phase d ifference. The VCO simply genenn es a signal whose frequency is so me funct ion of the contro l voltage.

CHAPTE R 15 PHASE-l O CKED lO O PS

" 2

Om~

-

H ( s)

ve tr l

K -s o

f iGURE 15. 2. Linearized PU model.

The ge ne ra l idea is th at the ou tpu t of the ph ase detector drives th e veo freq uency in a direction that reduces the ph ase difference ; that is. it 's a negative feedback system . Once the loop ac hieve s loc k . the phase of the input refer ence and veo output sig nals idea lly have a fixed phase relati o nsh ip (most c o mmo nly ()" o r 9
15 .3 . 1 FIRST· O RDER PLl TIle simple st PLL is o ne in whic h the fu ncti o n I/ (s ) is simply a sca lar ga in (call it K". wit h un its ofvolls pe r rad ia n). Becau se the loo p tra nsm ission the n po sse sses just

1.5 .3

LINE A RIZED PL L M O DElS

a single pole. this type of loo p is known as a first-order I'LL. Aside fro m simplici ty. its main attribute is the ease with whic h large phase margins are obtained . Offsetting those positive attributes is an impo rtant shortcoming. however : bandwidth and steady-state phase error are stro ngly co upled in this type of loo p. Because one generally wa nts the stead y-state pha.s e error to be zero. independent o f bandwidth . first-order loops are infrequently used. We may use our linea r PLL mood to eval uate qu anti tatively the limitations of a fi rst-order loop . Speci fically. the input-output phase tran sfer function is read ily derived:

-

KoKo s + Ko K o

The closed-loop bandwidth is therefore

To verify that the ba ndw idth and phase error are linked . let' s nnw derive the input to-error transfer function: tP~(s)

tP,n(s )

s S

(3)

+ KoKo

If we assume that the input signal is a co nstant -freq uency sinusoid of freq uency then the phase ramps Iincar ly with time at a rare of w, rad ians per second. Th us. the Laplace -do main representation of the input signal is

Wi .

(4)

so that

(5) The steudy- stntc error with a constant frequency input is therefore •

Wi

Wi

RoKD

Wh

lim S¢ t(!>') = - ,- - = . ..... 0

.

(6)

The steady-state phase error is thu s simply the ratio of the input frequ ency to the loop bandwidth ; a one -radian phase error results whe n the loop bandwidth equals the input frequ ency. A small steady-s tate phase error therefore requires a large loop bandwidth; the two parameters are tightly linked . as asserted ear lier. An intuitive way to arrive q ualitative ly at this result is to recognize that, in ge neral. a nonzero voltage is required 10 drive the veo to the correct frequ ency. Since the control voltage deri ves fro m the outpu t of the phase detector. there must be a no nzero phase error . To prod uce a given control voltage with a s maller phase error requ ires an increase in the gain that relates the co ntro l voltage to the phase detector output.

• CHAPTER IS PHASE- LOCKED LOOPS

ljI;n

~ -

Kll("t zS+ [J

,

Y~ lrI

K" -s

ljIOUl

FIGURE 15.3 . Model of second-order Pll.

Becau se an increase in ga in raise s the loop transm ission uni fon n ly a t all frequencies, a bandwidth incre ase necessaril y accompanies a red uction in pha se error. To produce zero phase error, we req uire a n e le me nt that can generate a n arbitrary v e o contro l voltage fro m a ze ro phase det ector o utpu t , i mply ing the need for an infi nite gain. To deco uple the stea dy-state e rror from the bandwidth . however, this ele ment IlCL'dS to have infinite ga in o nly a t DC . rathe r than a t a ll freq uencies. An integrator has the prescribed cha rac teri stics. a nd its usc leads 10 a sec o nd-o rde r loop.

15 .3 . 2 SECO N D·ORDER PLL Th e mod el for a seco nd- o rder PLL is show n in Figu re 15.3. Th e ljO ~ negat ive phase ~h i ft contributed by the added integrator has to be offse t by the po sitive phase shift of a lo op -stabilizing zero. A.. with an y ot her feedback syste m compensau..-d in this manner. the ze ro ..houl d he placed well be low the crossove r freq uency to obtain acce prable phase margin. In th is mode l. the co nsta nt K /) has the units of volls per second beca use of the ext ra integration. Also thanks to the added integ ratio n. the loo p bandwidth may be adj usted ind epe nde ntly of the steady -sta te phase e rro r (w hich is ze ro he re ). as is clear from study ing: the loo p tran smi ssio n ma gnitude be hav ior g ra phed in Figure 15A . The sta bility o f thi s loop can he e xplo red w ith the roo t-locus diagram o f Fig ure 15.5. As the loo p tra nsm issio n magni tude increases ( by inc reasing Kf) K o ). the loo p become progressively bett er dam ped because a n incr ease in crossover frequ ency a llow s more of the ze ro 's positive ph ase shift to offse t the negat ive phase shift of the pole s. For very large loop tran smi ssio ns. o ne closed -loo p pole e nds up a t nea rly the frequency of the zero. whi le the o ther po le heads fo r infinitely large frequ ency, In this PLL imple me ntat ion. the loo p -stabilizin g ze ro come s fro m the fo rward path: hence. this ze rn a lso sho ws up in the closed -loop transfe r functi on . 11 is stra ightfo rward to sho w th at the phase tran sfer fun ct io n is

fru m whic h we determine that

15.3

LINEA RIZED Pll MOD EL S

log 11.11

'J -2 -I

,I

log tI,

FIGURE 15.4 . Loop tra nsmission of second-order Pll .

1m

z

FIGURE 15. 5 . Root locos of secood-order PLl.

(.0"

= j KnKo

(R)

and W nTz _t~,j ,-,-k~l~'K =o ' = - 2- = 2 .

(C)l

Furthermore, the crossover frequency for the loop may he expressed as

W .,

UJn ' = [ 2w;

2

W

n

+ W n '4I ( W~ ) ' + 1] '/' •

(I())

which simpli f ies considerably if the crossover frequency is well above the zero frequency. as it often is: ( II )

.....

CHA PTE R 15 PHASE · LOCKED LOOPS

II)

= Iltz

-.

fi GURE 15 .6 . Closed-loop pho se trans fer function for PU (Bode approximation) .

Both Figu re l 5A and Eqn . 10 ..how that the crossover frequ ency always exceeds w". whic h - from Figu re 15.4 and Eqn . 8 - is the extrapolated cru ....o ver frequencyof the loo p with no zero. Fina lly. it s hould be clear that increasing the zero's time constant impro...es the damping. glven a fixed w". Thu s. the band width and stability of a sccond-ordc r loop may be adj u..ted ,IS de sired wh ile preserving a zero ..teady-srae phase error. Jitter Peaking in Sec:ond
From the root locu s for this loop. we see that the zero is to the right of its associated (clo..ed -loop) pole at larger dampi ng ratio s. li enee. the clo sed -loop frequency response initiall y exceed.... unity until the zero 's e ffe ct is cance lled by the pules (set Figure 15.6). We M."C that there is a rise with frequ ency. starting at the zero location. then a "al· tenmg ca used by the first pule. As see n in the figure. the phase tran sfer function ha~ a magnitu de grea ter than unity above the zero freque ncy unt il the ..eco nd pole introdu ces a sufficient rollo ff. There is thus a band of freq uencies. bounded roughly by the Lew location and the second po le. where the magnit ude of the tran sfer functioo exceeds unit y. The implic atio n o f this peaking is thai if there is any modu lation ( intended ororherwise} on the input with ...pcctral co mpo nents within that ce rtain frequency band. the output mod ulation will have ,I pha se excursion thut exceeds the excursion on the input. Unfortunate ly. we can sec from the locus tha t such peak ing is an inherent property of such loop" as lon g as the zero is contributed by the forward path. Tberefore. if this pea king is 10 be kept to a minim um . we requ ire large loop transmic..ion.. to keep the first pol e as clo ...c to the zero as possible ." Alt hough this ji tter peaking

7 The peaking problem is compounded. of course, if there art' I'LLs in cascade. The overall peaking of a cascade CUll actually be large enough to cuusc downstream I'LLs tn lose lock. While Ihil

15 . 4 SO ME N OI SE PROPE RTIE S O F Pll$

FIGU RE 15.7. Li nearized PlL rnodeI

447

wi'" ocise infXl l.

may be totall y eliminated by using a voltage -controlle d delay elemen t 10 provide the loop zero in the feed bac k path . it is satisfactory in mo st RF application s simply to cboose sufficiently large da mping ratios."

15 .4 SOME NOI SE PROPER TIE S Of Pl l . 15 .4 .1 REJE CTIO N OF VCO DISTU RBANCES Aside fro m the response 10 the desired input, it is also important in practical systems to evaluate the respo nse 10 noise ( particularly in l'Cs, where noise generated by other parts of the chip can couple into the PLL). We therefore now exa mine how the cla ssic PLL behaves in response to no ise on the input of the PI.1. and on the co ntrol line of the Yeo. The linear model of Figu re 15.7 considers noise as additive at the control port of the veo, It is a relatively straightforward exercise to show that the noise -to -phase error tran sfer function for this system is

,\K o

¢£ - VN =

.f 2

+ sTzKo K o + Kn K o .

(1 21

Assuming a unit step- function noi se input, inverse Laplace -transforming g ives us the resulting phase error as a function of time : - ¢l (I)

=

D. Wj

~

W"y( 2 _ I

inh (w" v~ exp( - ( WIlt) Sin (" . - In.

(1 31

where 6.w l is the initial frequency error du e 10 the step-function disturbance VN • The damping ratio ( of the closed-loop poles is still given by issue is aimo' l never o f conce rn 1tJ RF designers. il can be a significant problem in some dig ital networks (e.g., loke n rings ). I T. Lee and J. Bulzacchelli, MA ISSMHz Clock Reco very Delay- and Phase -Locked Loop,' IEEE J. Sf/lid ,S'll lt' Cirrlliu. Dece mber 1992.

CHA PTER IS PHASE-LOCK ED l OO PS

( 14)

and the natu ral freque ncy

w~

re mai ns

j

w" =

( 15)

KfJ K o .

Th e ma ximum phase e rror is g iven by the fo llowin g intuitively obvious equation: - q,c. ma~ =

ilow,

~

W"V, ~ -

I

(

ex p -

(

· sinh tanh - I

,

~

v r;2 - 1

R-=! ) l;

,

ta nh

_I

R-=! ) r;

( 16)

ami occurs at a lime t max

=

I

~

w"vr; 2- I

tanh

_I

(17)

Now. while the utility of the fore goi ng eq uations may be so mewha t elus ive owing thei r c um bersome nat ure. these ex press ions simplify considera bly in the limit of high damping:

10

(18)

I "", ~ ;:::::

2 1n 2r;

---,
(19)

where W e is the cross ove r freq uen cy of the loo p. We 1I0 W have a cha nce of deve lo ping so me insig ht fro m these equation s. We see that the m aximum phase error is ap prox ima tely the ra tio of the initial veo freq uency shirt 10 the loo p c rossover freque ncy (w hich is a pproxi mately the closed- loop bandwid th). T his relationship may be unde rstood intuitively by recalling tha t phase is the integral of frequ e ncy. so that any departure fro m the cor rect frequency causes some integ ra tio n of phase e rro r, a nd tha t thi s accumulation of phase e rror persists for a time on the orde r of the reciproca l loop bandwidth . De pending on the context. thi s vary ing phase em u is calk'd j iut'T or phase noise. fro m the foregoing equa tion s. it is cle ar that minimi zing jit ter or pha se noise caused by power supply (or whatever) noise requires muxirnizing loo p bandwidth and millim izing the initi al shift in the veo freque ncy. Unfo rtunately. a rbitrari ly large loop bandwidth s arc not possible becau se all practical feed bac k syste ms ultimately suffer phase ma rgin degradatio n from a varie ty o f so urces. such as ( poss ibly) poor ly modeled para sit ic ele me nts. Additional ly. many PLL s a rc sa mpled- data sys tems (phase e rror measurem ents are made a t discr ete inter val s). and thi s natu re imp oses further

1.5 .4 SO ME NOISE PROPE RTIE S O F PUs

.49

b ound s on the crossover freq uen cy if stability is not to he compro m ised. T hese co nside ratio ns. in conjunc tio n w ith the need to abso rb co mponent to lerances and dri fts with te m perature a nd supply volt agc. typica lly force the usc of loop bandwidths thai are o nly a sma ll frac tion (c.g.• < 10% ) of the clock frequ e ncy to g uara ntee ucceprable worst -case phase margin s. At this point . a numerical exa mple ma y be useful in und erscoring the magni tude of the problem . S uppose that . for so me vulu e of ste p disturbance o n the pow er supplie... ~w, i!'> 2 ~ of W ( amt"f ' Further assum e that the c rossover freq uen cy W I ' of the loo p i.. ulso 2<,;, o f W earn C\" In th is case. the ma ximum ph ase e m ir i s o ne radian. or abou t 630 ps. w ith a ca rrier input o f 250 M Hz (a peri od of ~ n s). Thi s ma gnitude of jitter (over 15% ) is ge nera lly intole rable . Viewed from an IC comm unicatio ns systems perspec tive. th e sensitivi ty of PLL.. to exte rna l a nd internal noise so urces ma kes it ex tre me ly difficult In me rge the m with digital c ircu itry (whic h te nds 10 ge nerate large a mo unts of noi se w ith a ll that switc hing going o n) wit ho ut co mprom ising the spectra l purity of the Pl.L o utput. From the eq uations. we we that the se nsitivity to external noise is minimized if we m inim ize the y eO gain and maximize the loo p bandwidth . Minimizin g this noise se nsitivity remains one of the most sig nifica nt c halle nge.. in the qu est for furt her imcgrauon of RF c irc uitry with di git a l d e ments.

15 .4 . 2 REJE CTION O F N O ISE O N IN PUT We' ve just see n thai ma ximizing the ban d width o f the Pl.I . helps to minimize the in100 dee p making a system fa..tcr means tha t it recovers more quick ly from e rro rs. what ever the so urce. However, ther e is a po tential d rawback to maxim izing the ban dw idth . abov e and beyond the stability issue . A s the loop bandwidth increa ses. the loop gels better at track ing the input. If the in put is no ise -free (o r at least less noi sy than the PLL' s own yeO), then ther e is a net improvem e nt overa ll. However, if tile input signa l is noisier than the PLL' s YeO, then the hig h-ba ndw idth loo p will fait hfu lly reproduce this input no ise at the ou tp ut. lie nee. there is a tradeoff between se nsitivity to noi se on the input 10 the loo p (a co nside ratio n that favo rs s ma ller loo p ba nd wid th s), and se ns itivity to noi se that dis tur bs the YCO frequ ency (a co nside ratio n thai we ' ve see n favors larger loop ban dwi dt hs) . In ge nera l. tuned osc illators (e.g., l.C or crysta l-base d ) arc inhe re nt ly less (oft e n much less ) noisy. at a give n power le ve l. than rela xati on osc illa to rs (suc h as ring or RC pha se -shift osci llators) . lienee, if the refere nce input to the PI.L is ..uppl ied fro m a tuned osci llator whe n the yeO is based on a relaxat ion osci llator topolog y, larger bandwidths are favored . If. instead. the situa tio n is rever sed (a rarer occurrence ) a nd tlu ence of disturbances thai a lte r the YCO frequency. This Insight is nor

CHAPTER 1.5 PHA SE-LOCKED LO O PS

A costlll -

-

} - - - ~ A R(cos
-

Bcos(o.>t +t )

FIGURE 15 . 8 . Multiplier 0' pho se detector.

a relaxation oscillator supplies the reference to a crystal oscillator- bused PLL, then smaller loop bandwidths will generally be favored.

15 .5 PHASE DETECTORS We 've taken a look att he classical phase -locked loop at the block diagram level. with a particular focus on thc linear behavior of a second-order loop in lock . We now considcra few implementation details 10 sec how real PLLs arc built and how they behave. In this section. we' H examine several representative phase detectors. In subsequent sections. we will examine one or two types of voltage -co ntrolled oscillators to help develop a feel for how practical loop s look , although we will defer a detailed discussion of oscillators to yet a later chapter.

15 . 5 .1 THE A NA LOG MULTIPLIER A S A PHA SE DETECTOR

In PLLs thai have sine-wave inputs and sine- wave veos. the most common phase detector by far is the multiplier. often implemented with a Gilbert -type topology. For an ideal multiplier. it isn' t too d ifficult to derive the input- o utput relationship, a\ shown in Figure 15.8. Using some trigonometr ic identities, we fi nd that the output of the multiplier may be expressed as A H cos w I cos(wt

+ 4»

AB

= 2

' cos(4) ) - cos (2wt

+ 4» ).

(20)

Note that the output of the multiplier consists of a DC term and a double -frequency term. For phase detecto r operation, we are interested only in the DC term. Hence. the average output o f the phase detector is ( A H coswt cos (evl

+ 4») =

AB

2

[cos 4> l.

(21)

We see that the phase delector gain "constant" is a function of the phase angle and is given by

15•.5 PHASE DETECTORS

451

FIGURE 15. 9. Multiplier pbcse detector outpvt ven us phose d iHerence . d

AH

K lJ = d¢ (Y"UI) = - T

.

[sJn(¢ )].

(22 )

If we plot the average output as a function of phase angle. we get something that looks roughly as sho wn in Figure 15.9. Notice that the output is period ic. Furt her note that the phase detector gain consta nt is zero whe n the ph ase difference is zero, and is greates t whe n the input ph ase d ifference is 90 °. Hence, to maximize the useful phase detection ran ge, the loo p should be arranged to lock to a phase difference of 90" . For this reason, a multi plier is o ften called a quadrature phase det ector. When the loop is locked in quadrat ure. the phase detector has an incre mental ga in constant given by :

AH

= -2

( 23)

In what follow s, we will glibly ignore minus sig ns. The reason for this neglect is that a loop may se rvo 10 either a 90" or -9
15.5 .2 THE CO M M UTA TIN G M U LTIPLIER AS A PHA SE D ETECTO R

In the pre vious section. we assumed tha t both inputs to the loop were sinusoidal. However. one or bo th of these inpu ts may be we ll approximated by a square wave

A52

CHA PTER IS PHASE· LO CKED lO O PS

AnIS\OlI -

-

-

B ~g n l cos( ux + 4') 1

FIGURE 15.10. Multiplier wirh ooe squa re-wave input.

in ma ny COI 'iC .'i of practical interest. so lei us now modi fy our result s 10 accommodate a single ..qua re -wave input. In thi s ca se. we haw the situation depicted in Figure 15.10. where "sgn" is the sig num func tion defi ned as sgn(x) = I

sgn( x)

= - I

if .r >

n.

if .r < O.

Nnw. recall thai a sq ua re wave of'a mplit ude B has a funda men tal component whose a mplitude is ~1l11t . If we assume thai we care abou t only the Iundam ental co mponent of the sq uare wa ve. then the avera ge o utput o f (he mul tiplie r is

4 AB n 2

2 sr

( V0IJ1) = - - Icos(,p)) = - A Hlcosl ¢ )I .

(26)

The corresponding phase detecto r gain is similarly j ust 4j 1t times as large as in the purely sinusoidal case: 2 A lJ

=-- -

(27)

Altho ugh the ex pres sions fo r the phase detec tor o utput ami ga in arc qu ite similar to those fo r the purel y sinusoidal cas e, there is an important qualitative di fference between these two detectors. Because the sq uare wa ve co nsists of more than ju st the fund am ent al co m po nent , the loop ca n actu ally lock on to harm onics or subharmonics of the input frequ ency. Co nsider, for e xa mple, the case where the IJ sq uare- wave input is al prec isel y o ne third the frequency of the sinusoidal input frequency, Now, sq uare waves') consist of odd harm o nic s. a nd the th ird harmon ic willthen he at the sa llie freq uency as the inpu t sine wave. Th ose two signa ls w ill pro vide a DC output from the mu ltip lier.

'I

Wc are implid tly avsuming that the square w i1\'e~ are IIf :'ill'l dUl y cycle. As)'mmctrical sqUaft ..... a\·~ w iIl atso comnn even as .....e ll as odd harmonic components, pro viding an "op pon unuy" to lod ; to even multiples of the inco ming reference in addition tu odd multiples.

IS .S

PHASE DETE CTOR S

453

Because the spectru m of a sq uare wave dro ps off a.. l/f . IO the average o utp ut ge ts progressively sma lle r as we attempt 10 lock 10 hi gher and highe r har mo nic s. Th e atten da nt reductio n in phase detec tor gai n con stant thu s makes il more difficult to achie ve o r maintain lock at the highe r harmoni cs. bur this issu e must be add ressed in all pract ica l loo ps thai use this type o f detector. Sometimes harm o nic locking is dcsirublc. and so me time... il isn' t, If it isn' t, then the veo frequ ency ran ge usually has to he restri cted (o r acquisition care fully managed) in ord er (0 pre v en t ha rmo nic lockin g. Another oh..er vation worth making is tha t mult iplic ation o r a signa l hy a peri od ic signum functio n is cq uivale nt to Inverting the phase o f th e s ig nal periodi c ally. l ien ee, a multiplier used this way ca n be re placed by switches (also kno wn as "commute10rv:· by a na logy wi th a compone nt of rotat ing machines ). Bec a use switc hes a rc easier to impleme nt in so me technologies (such as C MOS) than are Gi lbert multipliers. this observation ca n lead di rectl y to simplified circuitry. Even if Gilbert-type multipliers a rc U M.'t1 . they are o flen dri ven with larg e e nou gh signals o n one port thut they beha....e as polarit y switc hes 10 a good a pprox ima tio n. 15 .5 . 3

THE EXCLU SIVE - OR GATE A S A PHA SE DETECTOR

If we now dri ve an analog multipli er with sq uare waves on hath input s, we could unalyzc the sit uatio n by using the Fourier ser ies fo r eac h of the inputs, multipl yi ng the m togeth er . and so forth . Howe ver . it turn s OUI th at a nalyzing thi s particu lar situa tio n in the lime domain is much ea sier. so that 's what we ' H do . Th e rea de r is welcome (i ndeed. e nco uraged ) to ex plore the uhemative met hod a nd perform the analysis in the frequency domain as a recreational exercise . In this case. the two sq uare-wave inputs produce the o utput sho wn in Figure 15.11. As we c hange the input phase differe nce. the ou tput tukcs the form o f a squa re wave of ....ary ing duty cycle, with a 50 % d uty cycl e correspo nd ing to a qu adratu re relation ship bet wee n the inputs. Si nce the dut y cycle is in fac t proportionulro the input phase differ ence. we ca n readil y produce a plot ( Figure 15.1 2l of (he average o utput as a function of the input pha se differ ence. '!lIC ph ase det ector consta nt is a consta nt in thi s instance, and is eq ual to

,

Kn = ~ A B. n

(28)

We see tha i thi s val ue is the sam e us fur the pure ly sinuso ida l case about quadrature. 10 Here' s anot her fun piece o f trivia .....ith ..... hic h to amaze party gue,ts: In ge neral. the spectru m of a lioignal will decay alio I/r . \10 here " ilio the num ber o f dcriv anves of the sig nal req uired III yield an imp ulse. Hence, the spec trum of an ideal sine wave halio an innnucly fa.st roltoff (lioinee no num ber IIf deri vatives ever yields an impulse], thai of an impulse doe sn't mil uff (since II = 0). lhal of a square .....ave mlls nlTas Ilf , that nf a triangle wave a,\ 1// 2, and Ml lin.

-CHAPTER 15 PHASE-LOCKED LOOPS

A'-j----,

S

u AS

FIGURE 1S.11. Multiplier inputs and output.

AS

FI GURE 15.12. Multiplier charac teristic with I'wo square-wove inputs.

A~

in the case with one square-wave input , this phase detector allows the loop to

lock 10 various harmonics of the inpu t. Aga in. depe ndi ng on the application. this propert y may or may not be desirable. If we examine the waveforms for this detector more closely. we see that they have precisely the same ...ha pe as would be obtai ned from using a d igit al excl usive-Ok gale, the only difference be ing DC offse ts on the inp uts and outpu ts. as well as an

15.6

SEQ U EN TIA L PHAS E D ETECTORS

455

Von

f iG URE 15.13. Characteristic of XOR a s quodrcture phose delector.

inversion here or there. Hence. an XO R may be co nsidered an overdriven analog multiplier . For the special case where the inputs and output are logic levels that swing between gro und and some supply m ilage VOD (as in C MOS). the phase detector output has an average value that behaves as graphed in Figure 15. I3. The co rres ponding phase detec tor ga in is then VDD

Ko = - -.

n

(29)

Because o f the case with whic h they arc implemented . and because of their compatibility with other digital circuitry. XO R phase detectors are frequen tly fou nd in simple ic PLLs. 15 .6

5EQ UEN TIA l PHA SE DET ECTORS

Loops that usc mult iplie r-bused phase detector s lock to a qua drature phase rela tionship between the inputs to the phase detector. However. there are many practic al instances (de Bellc scizc's homod yne AM detector is one example) where a zat) phase difference is the desired condition in lock . Additionally. the phase detector co nstants at the meta stable and desired equilib rium points have the same magnitude. resulting in pote ntially long residence times in the meta stable state. perhaps dela ying the acquisition of lock . Sequc r uinl phase detectors can provide a zero (or perhaps UIO ' ) phase difference in loc k. and also have vastly different gain constants for the metastable and stable equilibrium poi nts. Additiona lly. some sequential phase detectors ha....e an output that is proportionul to the phase erro r ove r a span that exceeds 27l radians. Seq uential detectors do possess so me disad van tages. Since they ope rate o nly on transitions, they tend to be qui te scnsitive to missing edges (althnugh there are modifi cations thut ca n red uce this sensitivity ); this is in contrast to multipliers. whic h look

.,.

CHAPTER 15 PHASE·LOC KED lOOPS

InpUI A

Input B (q :.• \'<"' 0)

1'1; Output _

n n n rL - - - - ' '----------.J

FIGURE 15 .14 . SRRip ·Rop pha~

'-----~

detector waveforms.

a t the whole wave form. Fu rtherm ore. a not he r conseq ue nce of their edge-triggered 1I,IlUfe is that they in trod uce a sampling operation into the loo p. As we will see later, sampling inherently adds something s imilar to a time delay into the loop transmission. The associat ed increasin g negative pha se shift w ith incr easin g frequ ency imposes an uppe r bo und Oil the allowa ble crossover frequencies thai is ofte n substa ntially more

restrictive than if a different phase detector were used. 15 . 6 .1 THE SR FLIP-flO PA S A PHA SE DETECTO R T he simplest sequential phase dete ct or is the set- reset (SR) flip-flop . Here. a transilion (say. a po sit ive -goi ng o ne ) 011 OIlCinput sets the flip -flop. while
_ Von .

n lJ -

--.

2rr

we have assumed the typical case for a C MOS imple menta tion . where the outpul swi ng s fro m rail to rail. From examination o f Figure 15. 15. we can see that ma ximi zing the phasedetecta ran ge here req uires c hoosi ng a n equilibrium phase di fferen ce of I HO" . fu rtherm ore, the gain a t the metastable po int is extreme ly high (idea lly. infinite ). su that the likelihood of the loop resid ing there is muc h smalle r than for, say, the XO R detector. The forego ing ana lysis assumes im plici tly tha t the Hip-flop respon ds equally fa.~t to the se t a nd reset inpu ts. An y speed di ffe rence results in a static phase error, a\ some thi ng other tha n a I!U)" pha se differe nce is necessa ry to yield an average output

15.6

SEQ UEN TIA L PH ASE DETECTORS

457

""-+ -If-+ -If-+ -IL-+ - - ., FIGURE 15 ,15 . Charoctefi!ttic of SR Rip-Rep as phase detector.

Sel

f iGURE 15 .16 . Texlbook 51/: Rip-Rop implemenfrction.

of Vf)IJ / 2 if the set and reset operations take place at different speeds. Consider, for example. the classic textbook SR flip-flop with cross-coupled NOR gates (Figure 15.16). Note that this circuit has faster respon se to the reset than to the set input, and is therefore not the preferred implementation for those applications in which a small static phase error is important. 15.6 . 2

SEQ UEN TIA L DETECTORS W ITH EXTEN DED RAN GE

Sometimes, a (I . (ruther than 180 ·) degree phase difference in loc k is absolutely necessary. In such cases, the SR flip -flop is usually nOI a suitable phase detec tor.II Furthermore. it is often desirable to exlend the phase detection range to span more than one period (say, tw o].

II Of cou rse , an inverter may he aJ,k-d 1<) 0"", of the inp uts III cancel nom ina lly the 1110 > phase:: rel at ion.~ h ip . How ever, the mvcrter dela y now ad ds directly to the ph ase d iffere nce . In lOu m.:- applications, the a"sn d aloo pha'i.t' em ir is of no co nseq ue nce , hUI in many il is a 'i.t' rious problem. particularl y at high frequ encies.

AS8

CHAPTER 15 PHA SE· LO CKED l O OPS

10 ic " I"

0

~

U

0

~

D

V

R

FIGURE 15.17. PheW!

de!edof with extended range.

v

,.

•

FIGU RE 1S . 18 . Chorocteri ~lic of extended-ra nge pho~ del9clor .

A widely used circuit thai possesses both of these attributes consists of two D flip -flops and a reset gate; see Figu re 15.17. The designat ion s R and V stand for " reference" and "veo," while U and D stand for " up" and "down," terms that will mean something shortly.

For this circuit, the up and down outputs have an average difference that behaves us shown in Figu re 15.18. Note that the input range now spans 4;r radians, with a cons tant phase detector gain o f KfJ =

~l;).

w hich is the sam e as for the SR Hip-flop. It should he cl ear from Figure 15.l 8 1hal a If' loc k po int should be chose n to maxim ize the lock range. For those who arc interested in building circuits wit h this type of phase detector. it should be no ted that this phase detector is function ally equivalent to that used in the 4044. except for some logical inversions.

15.6 SEQ UEN TIA L PHA SE DETECTORS

459

15 . 6 . 3 PHA SE DETEC TO RS V ERSUS FRE QUEN CY DETEC TORS In many applications. it is important (o r a t least useful ) to have some informatio n about the magnitude of any!requellcy diffe rence be twee n the two inp uts. Such information co uld he used to aid acq uisition, fo r exam ple. Whereas multiplier -based phase dete c tors ca nnot prov ide such infor mat ion . sequential pha se detectors can. Consider the extended-ra nge phase de tec tor of the previous sec tio n. If the frequ ency of the veo exceeds that of the refe rence the n the U output will have a high dut y cycle. because it is se t by a rising edge o f the higher -frequ ency veo but isn' t cl eare d until there is anothe r rising edge on the lower -frequ ency refere nce . He nce, not o nly doe s this type of phase detec tor pro vide a large and linea r pha se de tection range. it also provides a signal that is ind icative of the sign and magnitu de of the frequency erro r. These a ttributes accoun t fo r this detector's e normo us popul arit y. De tec tors with thi s frequency disc riminatio n pro pert y are known co llec tively as pha se-frequency detectors. 11 should he me ntioned that rhis de tec tor doe s have some pro blems, however. Being a sequential det ector. it is se nsitive to missing edges. Here. it would mi sinterp ret a missin g edge as a freque ncy e rror a nd the loop wo uld he dri ven to "correct" this error. Addition ally, the slope of the phase detector cha racteristic nea r zcru phase e rror ma y ac tually be some what d ifferent fro m wha t is shown. bec a use bot h the U and D outputs are narro w slivers in the vicinity of the lock point. Since all rea l circuits have limited speed. the nonze ro risetimes will ca use a depa rture fro m the ideal linear shape show n. as the areas of the slivers will no lon ger have a linear rel ation ship to the input time ( phase) differences . In so me systems, thi s problem is so lved by intenti on ally introd ucin g a DC offse t into the loop so that the phase detec tor output must be nonzero to ac hieve lock . By biasing the balanced conditio n away from the detect or 's ce nter, the no nli nca ritie s ca n be greatly suppress ed. Unfort unately. thi s re medy is ina ppropriate for app licat ions that require small error, since the add ed offset trunslutcs into a static phase e rror.

15.6 .4 O THER SEQ U EN TIA L PHA SE DETECTOR S As we'v e see n, seq uential phase detect ors can exhibit a great se nsitiv ity to missin g pulses. O f course , this behavior ca n be modi fied sig nificantly, and a grea t deal of crea tive effo rt ha s bee n direc ted at dev ising ways to mitig ate this problem. One simple struregy is to have the veo outp ut cause flip -flops in the phase detec tor to togg le ra ther than reset. Th is way, mis.. . ing input pul ses cause no error (on average) a nd undesirable loo p behavior is grea tly minimized , as lon g as the loo p filter ca n re mo ve the rip ple on the cont rolli ne thai results fro m the toggl ing.

'60

CHAPTER 15

PHA SE-LO C KED l O O PS

Ph,l>c Detector O utput

FIGURE 15. 19. Hogge'~ phose detedof.

Another strategy is to recog nize tha t we'd like to implem ent a "do -noth ing" slate in the e vent of missin g input pu lses. II turn :'> o ut that it is po ssible In provide just such a slate. Furthermore. lTlany phase detectors of this type ca n he used tn recover tbe carrier (cloc k} from certain types of di gital da ta strea ms. Th e first ofthcsc " trista te" detecto rs we'I l co nsider i..d ue 10 I Ioggc l 2 (even though it does n' t q uite sol ve the prob lem ); sec Figu re 15. 19. Th i.. ci rc uit direct ly compares the phases o f the dela yed data and the dock in the followi ng mann er. After a change in the sta le of the dela yed da ta . the 0 input a nd Q output of Dctype flip-flop U, arc no longe r eq ua l. cau sing the o utp ut of XO R ga te VI to go hig h. Th e output of VI re mai ns hig h unt il the ne xt risin g edg e of the cl ock . at which time the delayed data' s new state is cloc ked throug h V., . eliminat ing the inequ alit y betw ee n the 0 and Q lines of V.l. At the sa me time, XO R gate V2 ra ise s its o utput high because the D a nd Q lines of V4 a re no w uneq ua l. Th e o utput of V2 re main s hig h until the ne xt falli ng edge of the cloc k . a t whic h lime the del ayed dat a' s new stale is cloc ked thro ug h V4 . If we as sume that the clo ck has a 50% dut y cycle. then V 2 's output is a positive pulse with a wid th eq ua l to ha lf th e c loc k period for eac h data transition. The output of VI is a lso it posit ive pu lse for eac h data tran sitio n. hut its w idth depends on the ph ase erro r between the del ayed da ta and the cloc k: its wid th equals half a dud period when the dela yed da ta and the cloc k ar c optimally aligned . li enee. the phase erro r ca n he obtained by co mparing the widths of the pu lses o ut of VI and VI . Figure 15.20 and Figu re 15.21 arc timin g di agra ms fo r this detecto r with the delayed data and clod optimall y ali gned (i n thi s case, with the fal ling edge ot tbe clock) a nd with da til uhcud (If the cluc k, respecti vel y. In the forme r case. the output of the phase de tec tor has ze ro ave rage va lue. a nd the re is no ne t cha nge in the loup

12 C. R . It ll~~e. MA Se tr-Correcnng Clod : Recovery Circuit: ' J. U /(h/Ht"'e Te(·h",,/,,[(.\ ·. v. LT·}.00 6. l<,IllS. pp. 1312- 14.

15.6 SEQ UEN TIA L PHASE DETE CTORS

L-_IIL-_ Cluck

Q OUlf"'1 " r UJ

Q OulP1i1 of U4 Oulput ofU I

Ou'put of

L'~

I'ba.~ 111<1« I' lf ( kupu l

Outpul or IA" ", Ink J rat' l!"

FIGURE 15 . 20 . waveforms For Hogge's detector in lock.

IL-_II'-----_ Clo,:k

Q Output of U3

Q Out put or U4

,--__11,--__

_ _IIL-_I

Ou tpul llf U I

Chlll'UI or

U~

Phase ( kt ~.·t (lr OUlput OU1l'Ul of L' " 1fl hUCj: ralllr

fiGURE 15 . 2 1. Waveforms b- Hogge's detector with dota input

ahead of dock.

461

• 462

CHAPJE R 15 PHASE-LOCKE D LOOPS

FIGURE 15. 2 2. Cha racteristic of Hogge's pha se detector.

integrator 's output; in the latter case, the output of the phase detector has a positive average value. As a result. the loop integrator 's output exhibits a net increase. Conversely, if the delayed data were behind the dock. the phase detector's output would. have a negative average value. and the loop integrator 's output would ex hibit a net decrease. Plott ing the phase detector 's average output (assuming maximum data transition density) as a function of phase error yield s the familiar sawtoo th characteristic exhibited in Figure 15.22. Consistent with Figure 15.20.lhc phase detector's average output equals zero when the phase error between the delayed data and the clock is zero. One noteworthy feature of this pbase detector is that the decision-making circuit is an integral componen t of the phase detector (for the output of flip-flop V} is the retuned data ). However. this detector does suffer from a sensitivity to the data iransitio n density, Since each triangular pulse on the output of the loo p integrator has positive net area (sec Figure 15.20 ), the presence or absence of such a pulse affects the average output of the loop integrator. The (data-dependent) jitt er thus introduced is often large enough to he objectionable . The phase detector shown in Figure 15.23 greatly reduces this problem by replacing the triangular correction pulses (which have net area . even when the delayed data and clock are properly aligned) with " rriwaves," whose net area is zero when clock and data are aligned. As in Hogge's detector. the width o f VI's output is depe ndent o n the phase error betwee n the dela yed data and the clock . while the output of U2 and U} are al.....ays half a clock cycle wide (assuming Ihat the clock possesses a 50% duty cycle), The phase error can thus be obtained by cu mparing the variable-width pulse from VI with the fixed-width pulses from U2 and U3. Note that the pulses out of VI and V} are weighted by I. while the pulse nul of U 2 is weighted by - 2.

1.5 .7

LO O P fi LTER S AND CHARGE PUMP S

463

Phase

Detector O utpu l

( 'I

Del aved

D.",

~

O' r O' us Clock --.J r

FIGU RE 15. 23 . Triwove phose deledor.

Figure 15.24 is the liming diagram for the triwave detector with the delayed data and the clock optimally aligned. Note thai each data transition initiates a three sectioned transient (the triwave) on the output of the loop integra tor, and thai this triwave has zero area . Therefore, its presence or absence does not change the average output of the loop integrator. Hence, the triwave detector exhibits a much reduced sensitivity to data transition density. However, the rriwave detector is somewhat more sensitive to duty-cycle d isto rtion in the clock signal than is Hogge's implementation, owing to the unequa l weightings used. Th is sensitivity to duty cycle can be restored to that of Hogge's implementation with the simple modification shown in Figure 15.25. The modified m wave detector uses two distinct down-integration intervals cloc ked on opposite edges of the clock , rather than a single down-integration of twice the strength clocked on a single clock edge. As a consequence. duty-cycle effects are attenuated. 15.7

LOOP FILTERS AND CHARGE PUMPS

So far, we've examined the behavior o f PLLs using a linear model. as well as a number of ways to implement phase detectors. We now consider how to implement the rest of the loo p. w c'tl take a look at various types of loop filters and survey a couple ofcommon techniques fur realizing veos. We'll wrap up hy going through an actual example to illustrate a typical design procedu re. 15.7.1

LO OP FILTER S

Recall that we generally want to have zero phase erro r in lock. Now. the veo requires some control voltage to produce an output of the desired frequency, To provide

• CHAP TER. 15

I

Drlayed 0<01.

PHA SE·LOCK ED LO O PS

'-------

Cl
QOull""of U4

Q O Ul",,1 nf

LJlL-_ ri. •cLJlL_

us

Q OUII"" " r Ub

Out pul " , L:l

Outp ut " I U2

()1I11'1I1 " f U.1

f'ho1-e 1)c1C"',,,r Outpul

0

Oulput 0' Lo" ", I n'c~l'"

fiGURE 15. 2 4 . waveforms of triwove detector in lock.

Ph ~

Detector Output

,---- -----+.

~.'------

Dela cd Data

a ook

FIGURE 15. 25 • .Modified triwove

detector.

15.7

lOO P FILTER S AND CHARG E PUMPS

R,

465

C

R,

Ph", De lec tor

f-iW--'-- - -f-'\

veo

fi GURE 15.26. PLL with typica l loop ~he(.

lhi ~

control voltage with a zero output from the phase de tector (and hence zero phase error), the loop filter mu st provide an integration. Then. to ensu re loop stabili ty. the loop filter must also prov ide a zero . A classic architecture tha t satisfies the se requ irement s appears as Figur e 15.26. It should be easy to dedu ce the ge neral properties of the loop filter without reso rting toequations. (O kay, may be it 's eas ier after yo u've done it once or twice so that yo u already know the answer. bUI work with me here.I AI very low freq uen cies. the capacitor '!'> impedance dominates the op -amp's feedbad.• so the loo p filter behaves as an integra tor. As the freque ncy increases. tho ugh . the capacitive reac tance decreases and eve ntua lly equals the series resistance R 2. Beyond that freq uency. the ca pacitive reac tance becomes incre asingly negligible compared with H2 • and the gain ultimately flattens out to simply - Rd HI. Stating these observations anothe r way. we have a po le at the origin and a zero whose time constant i.. H2C . Furt hermore, the value of HI can be adjusted to provide whatever loop transmission magnitude we'd like. so the op -amp ci rcu it provides us with the desired loo p filter transfer function. Before go ing. further, it should be mentioned that PL.!.s need not include an active loop filter of the type shown. In the simplest case, a passive He network co uld be used to connect the phase detector with the Yeo. However, the static phase error will then no t he zero, and the loo p bandw idt h will be coupled (i nversely) with the static phase error. Because of these limitations. such a simple loop filler is used only in noncritical applications. The ci rcuit of the figure is commonly used in discrete implcmcnuulons. hut a differenr (altho ugh funct iona lly equivalent) approach is used in most le s. T he reason is that it is not necessary to build an entire op-amp 10 obtai n the desired loo p filter transfer function. A considerable red uction in co mp lexity and area (not to me ntion power co nsumption) ca n be obtained by using an eleme nt that is less ge neral-purpose than an op -amp. A popu lar alterna tive 10 the op -amp loo p filter is the use of a cha rge Putt/I'. working in tandem with an He net work . Here. the phase detector contro ls one or more current sources . and the He network provides the necessary loop dynamics.

CHAPTE R IS

Pump Up

PHA SE · LO CKED LOO PS

._. _..••.••••. _ ...

r'vco is hue")

,r

Pump Down (- veO is carly" ,

...

- - -r - - - . .

Vco nlml

R

Figure 15 .27 sho ws how a c harge pu mp provides the necessary loop filte r action. Here. the phase detector is ass umed ( 0 provide a digita l " pump up " or " pump down" signa l. If the ph ase detector det ermines that the yeo ou tput is lagging the inpn refe rence. it activates the top curre nt so urce, depositin g c harge o nto the ca pacitor ( pumping up ). If the y eO is ahead . the bott om curre nt source is activated. withdrawing charge from the capacitor ( pumping down ), lf there were no resistor then we wo uld have a pur e integra tio n. As usual. the series res isto r pro vides the necessa ry loo p -stabi lizin g ze ro by for cin g the hig h-frequency asy mpto tic impedance to a no nzero value. Sin ce sw itc hed c urre nt sou rces a re easily im plemented wi th a very sm all number of tran sistors, the c barge- purnp approach a llow s th e synthesis of the desired loop fi lter witho ut the co m plex ity, area . a nd power co ns umptio n of a textboo k up-amp. The natu re of the co ntrol ulso meshes nice ly wit h the many existing digital phase detecto rs (e .g., seque ntia l phase de tectors).

Con,"" Une Ripple ond Higher-Order Poles On e i.ssue th at needs to be addressed is that of ripple on th e co ntrol line. As a conseq ue nce of the loo p -stabili zing ze ro. there ca n be sig nifica nt high-freq uency content on the co ntrol line th at d rives the Yeo. Th is " hash" can co me fro m the higher-order mi xing products in a multiplier -type of detecto r ( i.e., ess entially th e dou ble frequency te rm ) o r fro m the mu ltip le -order prod uct s fro m a c harge -pump-de tec tor combinetion. Th e resulta nt mod ula tio n of the yeO freq uen cy may be unacceptable in many applicat ions. Reducin g the loop band width he lps, of course . but other desig n co nsiderations may impose a lower bou nd o n the ba nd wid th . He nce. we need to see k alternative' method s. Luc kily, the ripp le co mpo ne nts are at a higher freq uency th an the output of the Yeo. and the loop cros so ver mu st he lower still. Thu s, we may add o ne or more po les to filter the ripp le wit hout serio usly degrading the phase ma rgin of the loop. S uc h high-orde r poles ca n red uce the ripple (and a ttenda nt undes ired FM or jitter on the y e O o utput) by d ramatic amo unts.

15 .7

LO OP FILTE RS AND CHA RGE PUMPS

Ta ble 15. 1. Higher-order poles \ 'US US

ripple reduction Ripple

" I

0.25 0. 14

2 3

,

Pump Up

11.10

0.08

Pump Down ...

7."

13.4 I_U 14.1

_ ..

("veo is lase" ( ~VCO

auenruanon

flU ,

f---,-,r- " .-

is c.. rly")

V ."Onllnl

R

FIGURE 15 . 2 8 . Cha rge- pump loop filter wi th one extra pole.

Determi ning how ma ny pules sho uld he added - und where these pol es sho uld he placed - is so me what Involved. Since the de rivation itsel f is uncnligbtcn ing, we will simply pre se nt the re sul t tha i one or two higher-order poles ca n a ttenuate the ripple by about an order o f m agnitude or more if they are placed around a factor of ~ -7 above crossovcr.U More preci sel y. we ca n offer Table 15. J. e xtrac ted from da ta from the refere nce in footno te 13. In constructi ng this tabl e , it was assumed tha t a ll of the additional poles are coincident. From exami ning the table e ntries . we can sec that the addition al utrenuatio n of ripple beyo nd two higher-order po les is modest, whi le the dan ger of degra d ing pha se margin increa ses. In practice , frequently just o ne suc h additional pol e is used , and it is read ily prov ided a s shown in Figu re 15.28. The ex tra ca pac ito r provide s co ntin ued uuenuation a t suff iciently hig h frequen cy. If we assum e that the zero is far bel ow c rossover. then the seco nd pol e has a tim e constant of RCu ln. to a good a pproxi matio n. Mo re r igorously, the second pole frequency is g ive n by

Il

J. Bulzacchelfi. "A Delay-Locked Loop for Chwk Recovcry and Data Synchrunizatiun," Master's thesis. Massachu'-t'ILs InstilulCof Technoloi:Y, 1991.1.

. ..

CHAPTER 15 PHASE· LOCKED LOOP S

R,

e

R,

Phase Detector

'-,""'--'-- - -1" I

veo

FIGURE 15. 29 . Clcssic loop ~her with exira pole .

Th us. the effec tive capac ita nce fac ing the res istance is j ust the se ries co mbinat ion of the two capacita nces. T he classic np -a mp loo p filte r may he sim ila rly mod ified by placi ng an extra capacito r in parali c I wit h the serie s He feed back ne twor k to pro vide the e xtra pole, as sho wn in Fig ure 15.29.

15 .7.2

vco.

Although we will lake up the detailed study of oscillators OI l a later lime. we ClIa mine he re one co m mo n arc hitec ture for rea lizing y eOs in integra ted circuits: the currcm-s tarved ring oscillator. in which the effective propagation delay of each inverter in the ring is controlled by a current source .

Ring OKilloton Ring oscillators are an ex tre me ly popular id io m. since they de rive from digital-like bu ilding bloch . Wc 'lI sec later that , co mpared to tuned oscillators (i.e., those that explicitly use high - Q reson ato rs), they have substantially infe rior phase noise perfo rtnunce for a given le ve l of po wer co ns um ptio n. For many app licat ions, however, thei r relatively la rge tuning ra nge and simplicity are stro ng e no ugh attrib utes to make them att ractive . Th e co ntro llable ring osc illato r de rives fro m the uncontrolled ring oscilla tor and co nsists sim ply o f " inverte rs in a rin g. w here 1/ is odd.':' In its s implest form. it appears as shown in Figure 15.30. In th e simples t a na lysis of suc h an o scill ato r. it is assumed that eac h inverter can be c harac te rized by a propagat ion de lay Tpd . No stable DC point e xi sts, and a logic 1 ~ Ring osctltarors in whieh the individ ual stages ore differe ntial allow the use o f an even numberor

stages. The necessary inversioncan he ubraincd simply hy reversing one dittc renua l pair of s ignal~.

15. 7 lOO P FILTER S AN D CHA RGE PU M PS

II

469

inverters ; n odd

FIGURE I S.30 . Ringoscillotor,

FIGURE 15. 31. Simplecurrenl-storved

CMOS inverter_ level propagates aro und the loo p. ex perie nci ng o ne net inve rsion cac h tra ver sa 1. Th e oscillatio n period is therefo re simply twice the 100a i propagation de lay:

I f (O>J;;

=

")

_n ·

T. .

(3 3 )

pd

Now. to co nvert th is thin g. into a co ntrollable oscillnto r. fhc pr opagation de lay see ms the most natural q ua ntity to adj ust. One ca n ima gin e a g reat man y speci fic method s for adjustin g the dela y, hut they all boil down to e ither c hang ing the load (e.g., by vary ing the effective a mo unt of c upacinm ce see n by eac h invert er o utput) o r varying the c urre nt dri ve of the inve rters. One cheesy wuy to acco m plish the latt er is sho wn in Figure 15.3 1, whe re a PMO S curre nt mirror pro vides a limited, varia ble pul l-up c urre nt to the C MOS invcrt cr.P By adj usting this c urre nt, the effective propagation delay of the inverter c an he adjusted. a lterin g the osci llatio n frequen cy in the process. Th e ring osci llator (and its many var ia nts) requires o nly c le me nts that are norma lly available in o rdinary d igita l process tec hnology, T hisauribute. co mbi ned w ith its fundament a l s implici ty. has ma de the ring oscillato r a nea r-ubiq uito us topo logy in ICs.

15 Olwiulisly. an NMOS mirror coo kt also be used 10 consiram the pull-dow n current . hUl lhis WtlllloJ

mi lia Ihe cheese level.

470

CHAPTER 1.5

PHA SE-LO CK ED LO OP S

15 .8 Pll DESIGN EXAMPLES Now tha t we've studied the basics of phase -locked loops, it 's time to consider a few design examples. The particular examples we' ll study use a co mmercia lly available I'LL chip, the 4046. It is an inexpensive (..... $0.25-$ 1) C MOS device that contains two phase detectors (one XOR and on e seq uentia l phase detector) and a Yeo. We will co nsider the design of a PLL with each of the phase det ectors and a couple of loop filters. The 4046 is a relatively slow device, with a maximu m osci llation frcqu encyofonly abo ut I Mil l or so. Still, the design procedure we'll follow is generally applicable 10 PLL" w hose output frequ ency is much higher, so what follows isn't a purely academ ic exercise. In any eve nt, the device remains useful for many a pplicatio ns even tod ay, and it is ce rta inly an exceptionally inexpen sive PLL lUtoriallab-on -a-chip.

15.8. 1 CHARACTERISTICS OF THE 40 46 CMOS Pl l Phase Detector I The chip co ntains two phase detectors ( PDs). O ne, know n 3S " phase detector I ,~ is a s imple XO R gate. Rcc311 from the sec tion on phase detectors that an XOR has a gain co nsta nt gi ven by Voo KD = V/rad. (].I I

"

Th rou ghout these design exa mples. we will use a po wer supply voltage of 5 V, so the spec ific num erica l value for our design s will be

Voo K[) = ::::: 1.59 V/rad.

" Phese Detector II The chip's ot her phase detector (or "comparator" ) is a sequential phase detector that operates only on the positive edges of the input signals. It has two distinct regions of be havior depe nding on which inp ut is ahea d. If the signa l input edge precedes the veo feedback edge hy up to one period. then the output of the phase detector is sci high (that is, to Vnn ) hy the signal edge and sent into a high-impedance state by the feed back edge. ( We' ll sec momentarily why it can he advantageo us to have this high-impedance state.) If the signal inpu t edge lags the veo output by up 10 o ne period. then the output is set low (to ground) by the veo edge ..nd sent into a high-imped ance state by the signal input ed ge. A nd that's all there is to this phase det ector. The high-im ped ance state allow s o ne to redu ce (he amo unt o f ripple on the control line whe n in the loc ked state. Hen ce, the amount of unintended phase and frequency

1.5 . 8

PLL D ESIGN EXA MP LES

471

. 2n

FI GURE 15.32 . Characteristic of phaloe detector II .

modulation of the y e O during lock can be much smaller than whe n other detectors are used . It should also he cl ear thai a PLL that uses this seq uential phase detector forces a zero ph ase difference in lock . in co ntrast with the qu adrature condition that results with an XO R det ector. The other bit of information we need in orde r to carry OUI a design is the ph ase detector gain constant. Unfortunately, this particular detec tor doe s not have a particularly well -defined KI) because the output voltage in the high-impedance state depend s on ex ternal elements. rather than on the phase error alone. A good solution to this prob lem is to remove the uncert ai nty by forcing the output voltage to Vno/ 2 during the high -impedance co nd itio n (e.g.• with a simple resistive d ivider). Wit h this modification , Ko can he det ermined . For phase errors of less than one pe riod (signal inpu t leading), the average output voltage will be linearly pro po rtional to the phase erro r. The minimum output is Von / 2 for zero phase error, and is a maxim um value of VD!) with a 2JT phase em u . The minimum output is determined by the added resistive divider , while the maximum output is simply controlled by the supply voltage . Similarly, in the cnse of a lagging input signal, the avera ge outpu t voltage will he Voo / 2 for zero phase erro r and zero volts for a 2JT phase erro r. Hence. the ph ase detector characte ristic look s as shown in Figure 15.32. After solving Schr odi ngcr's equation with approp ria te boundary co nd itions, we find that the slope of the line is VDn

Kn = -

4"

Y/ rad .

(36)

For our assumed Voo of 5 Y. the phase detector ga in is a ppro ximately 0 .40 Y/ rad.

veo Charocterisrics The y eO UM-oJ in the 40 46 is reminiscent of the emitter-co upled mult ivibrator used in many bipol ar Yeas. Here. an ex terna l capacitor is alternately charged o ne way,

CHAPTER IS PHASE· LOCKED LOOPS

then the ot her. by a c urre nt source. A simple differen tial comparato r sw itches the p0la riry of the c urrent so urce when the ca pac itor voltage exceeds some trip point The feedback pol arity is c hose n 10 keep the circ uit in oscillatio n. The main y eO ou tpu t is a squa re wave, deriv ed from one output of the differential co mparator. An app roximation to a tria ngle wave is also a vai lable across the ca paci tor te r mi nals. The trian gle-wolve signa l is useful if a sine -wave output is desired. since eithe r a filte rin g networ k O f no nlinear wave shuper cun he used 10 convert the tria ngle wave into so me se mbla nce of a s ine wave . Frequency control is provided through adj ustme nt of the capacitive charging cur-

rent. Both the center frequency and y e O gain can be adjusted independently b)' choosing two external resistors. One resistor. R2 • sets the charging current{and hence the VCO frequen cy) in the absence of an input. thus biasing the output frequencyversus-cornrol vo ltage curve. The ot her resistor. R I • sets the transconductance of a common-source stage and therefore adjusts the VCO gain. Conspicuously absent from the data shee ts. however. is an explicit formula for rela ting the VCO frequency 10 the various external component values. A quasiempirical (and highly approximate) for mula thai provides this crucia l bit of information is as follows : 16

(31)

From this formula. the VCO gain constant is easily determined by taking the derivative with respect to control voltage :

2 R,C

K o :::::: - -

rad / ~/V .

(lSI

MiKellany Notice thai the phase detector gains are function s of the supply voltage. Additionally, the VCO freque ncy is a function of V/J/J as well. Hence. if the supply voltage varies then so will the loop dynam ics. for example. If power supply variations (including noise) arc not In influence IlXlP behavior. il is necessary 10 provide a well-regulated and well-filtered supply. As a convenience. the 40-l6 includes a 5.2-V (plus or minus about 15%) ze ner diode that may be used for this purpose. The -lO-lf:l also includes a simple source follower to buffer the control voltage. This feature is handy for those a pplicatio ns in which Ihe PLL is being used as an FM

I~

Th is formula is rhc result uf measurements nil only one particular device with a ~·V power SU~ ply. Your mileage may vury, especially if you usc resistance values below about 50- IOUkG (the yeO control function gets quite nonlinear at higher CIl1Tt'nl S ). Cavratnerdus.

1S.8

c

VUI.l

' in -

Pha-,< Detec tor II

1

473

PLl DE SIGN EXAM PLES

·T~ .J, ., ., veo

. . ' nul

c,

FIGURE 15.33 . Pllwithpho!>e de led od l.

demod ulato r. for example . The demodulated signal is equal to the veo contro l vo ltage, so that a buffe red version of this con trol signal is co nvenie nt for dri ving ex ternal

circuitry, Finally, the chip inclu des an "inhibi t" co ntro l signal that shuts off the osc illa tor and so urce foll ower to reduce chip dissipation to thc lOO-jlW range (e ven less if the signal input is a con sta nt logic level).

15.8 .2

DESIGN EXAMPLE S AT LA ST

Sec::ond-order PU with Possive

Ie loop Filtet" ond PO II

We know that active filte rs can provide superior performan ce. part icularly with regard to steady -sta te error. However. there are some application s for which fully passive filters arc adeq uate. and thu s for which acti ve filters would simply consume addi tional area and power . Suppose we usc phase detector II and a simple RC low-pass loo p niter (w ithout a loop-stabili zin g zero) . Design a circuit to meetthe foll owing speci fications : cross over freque ncy: I kmd / scc: phase margin: 45 ~ ; center frequency : 20 kHz . Solution : First , we recogni ze that the high-im pedance charnctcrisric of this phase delector requ ires the usc of the resistive di vider, as menti oned earlier. Then. to provide

the ability to dri ve an arbitrary RC net work. it is advisable to add a buffe r. Hence. the PLI. appears as show n in Figur e 15.33. The value of R is nor parti cu larly critical, but should he large enoug h 10 uvcl d excessive loading of the phase detector 's wimpy outpu ts. Values on the orde r of tens of kilohms are acce ptable . Note that the loo p tra nsmission may he wr it ten as

po

CHAPTER 15 PHASE-lOCKED l O O PS

Th e phase margin speci fica tio n requires us to choose the pole frequency of the loop filte r eq ual 10 the desired crossover freque ncy. since we do not have a loop -stabilizing zero . Havin g ma de that c hoice, we adj ust the veo ga in throu g h se lectio n of RIC. Finally. we choo se R2 to satisfy the ce nter frequ ency specification. Carrying out these steps. while being mindful lhat resistance values should be no lower tha n abou t 50 kn to validate the qu asie mpirtca l veo eq uatio n. yie lds the following sequence of computations. half-truths. and outrig ht lies. (I ) As sta ted ea rlier, the phase margi n spec ificatio n requires a loo p filter time con-

stant of 1 ms. Somewhat arbitrarily choose H.I = 100 kQ . so that C l = 0 .01 J1. F. Both values happen 10 correspo nd to standard component values. (2) Because the crossove r frequency must be I krps while R3CI and the phase detector gain con stant are both known, K o must be chosen to yield the desired crossover frequency: II. (jwc)1 = K /) ·

1 M ",2

'

K

,0

10- rps

= I

==:::}

RjC = 0.582 ms.

(40)

Arbitrarily choose the capacitor eq ua l to a standard value, 0.00 1 lifo: so that the required resistance is 582 kQ (not quite a standard value. but close to 560 I n , which is). Just for reference. the corresponding veo gain constant is about 3.56 k'P ' / V. (3) Now select R2 to yield the desired center frequency (here defined as the veo frequency that results with a control voltage of VoD / 2) with the veo capacitor chosen in step (2). From the semiempirtcal veo formula , we lind that R2 should be approximately 67.3 kfl (the closest standard value is 68 kfl) . Because of variability from device to device, it is advisable to make R2 variable over some range if the veo center frequency must be accurately set. That co mpletes the design. With the parameters as chosen, let us compute the veo tuning range, the steadystate phase error throughout this range, and the lock range (something we haven't explicitly discussed before). The luck range is defined here as the range over which we may vary the input frequency before the loop loses lock, For the frequency tuning range, we again use the Veo form ula . With the values we've chosen, the veo can tune about 1 kHz abo ve and below the center frequency. Th is range sets an upper bound on the overall PLL frequency range. Because of the passive loo p filter, the static phase error will not be zero in general since a nonzero phase detector output is required to provide a nonzero veo control voltage.!" Now, if we assume that the veo gain constant is, well. co m fa llt , we can 11 Here. zero control volta ge- is interpreted a., a dc"ialiun from the ce nter value of V DD / 2.

475

15.8 PLL DESIGN EXA M PLE S

compute precisely how much control voltage change is req uired to adju st the Irequency over the ran ge computed in ste p ( I). lfthe correspo nding phase erro r exceed s the ±21f span of the ph ase detect or, then the loop will be unable to maintain lock oyer the entire ± I-kHz freq uency range . The voltage necessary to move the o ut put frequency is fou nd from Ko , and is related to the phase detector gain constan t and the phase erro r as foll ows: ~.vclrl

t;w = - .- = KnfPCrTOf '

(4 1)

Ao

Using ou r compo nent values, the ph ase error is pred icted to be abou t 4.4 rad at I kHz off of center freq uency. Actual measurements revea l that, at the lower frequ ency limit (I kHz be low ce nter) , the phase error is 4.3 rad. T heory triumphs ( here, anyway). At I kHz above cente r, tho ugh , the measured pha se error is actually about 5.9 rad. The reason for this rather significant discre pancy is that the v eo frequ ency isn' t quite linear ly related to the co ntrol vo ltage at higher cont rol voltages. It turn s out that a larger than ex pec ted co ntrol voltage is req uired to reach the upper freq uency limit. Hence, a larger phase det ector output is req uired and so a larger co rrespo nding phase error results. Since angles of bo th 4 .3 rad and 5.9 rad are still within the phase detector ' s linear range, however, it is the y eO's lim ited tun ing range - rathe r than the phase detector's chara cteri stics - that de termines the overall PLL' s lock range in this particular case.

Second-order PU with Passive Ite Loop Filter and PO I It is instruc tive to re -do the pre vious design with the XO R phase detector repl acin g

the sequential phase detector. Becau se the XO R has four times the ga in of PD II , the value of K o mu st be adj usted dow nward by this facto r to maintain the crossover frequency. We may adjust K o by increasing HI to fou r times its previous value. In order to maintain a 20 -kHz ce nter frequ ency, H2 must be adj usted as well (dow nward). Becau se the XO R does not have a high-impedance output state, the resist ive divider and buffer may be eliminated . Once these changes have bee n made, the locked loop displays dyn am ics tha t arc similar 10 those observed with the previou s design . However . the v e o mod ifica tion s alter the y eO tun ing range and, therefore, the co rrespond ing phase error: ~Vetrl

~w

= -

Ko

= KnfP.,mll

~w

=}

fPermr = - - - . KoKn

(42)

Because HI has been changed upward, the veo tun ing range has decreased to a foun h of its previous value, whi le the prod uct of phase detector gai n and y eO gain remains unchan ged . Now, the XO R is linear over on ly a fo un h of the pha se error span o f the sequential phase de tector. Hence, for a given crossover frequ ency and

.. CHAPI ER 15 PHA SE· LOC KED LOOPS

476

Ph a o;e

1(3

Detector f-.'W--'---~ II

veo

FIGUR E 15. 34 . PlLwith a ctive loop filter (defective).

dam ping, use of the XO R phase detector ca n ca use the loo p In possess a narrower loc k runge. II is left as an exerci se for the reader 10 carry out actual numerical calculations10 veri fy these assertions. ( In this case . it turn s out that the veo tuning range is still the limiting fact or. but just barcly.) As a few final notes o n the usc of the XOR . it sho uld be men tioned that this type of detector is se nsitive 10 the du ty cycle of the inp ut signals. The ideal triangular character istic of the XO R pha se detect or is obtai ned o nly whe n bot h inp uts possess a 50% duty cycle. If there arc any asymmetries. the average output will no longer reach both supply rails at the ex tremes of phase error. The sequentia l phase detector is an edge -trigge red device and so does nor sutl er this duty- cycle se nsitivity. Ano ther important note is to reiterate thai the XOR phase detector allows locking onto harmonic s of the input . s ince the ac tion of the XOR is equivalent to multiplying two sine waves together. The rich harm o nic co ntent of square wave s prov ides many opportunities for a coin cide nce in frequency be tween com po nent s of the input and veo output. permitting loc k to occ ur. If harmonic locki ng is undesirable. U ~ of an XOR phase detector may cause some problem s.

Second-order PU with Active RC Loop Filter a nd POII Now let's consider re placi ng the simple pa ssive Re loo p litter with an active fi ller. Let this filter pro vide a pole at the origin to drive the steady -stale phase error to zero. Add itiona lly. assume that we want to achieve precisel y the same cross over frequency and phase margi n
15 .8

Pll DESIGN EXAMPLE S

477

vco

FIGURE 15. 35 . PU with active loop filter (fixed).

positive feedback loop. To fix this problem. we must prov ide an additional invers ion in the co ntro l line. The re is ano the r prob lem wi th the circuit: The op- ump's noninven lng terminal is grounded . The implicatio n is that the output of the lo op filter ca n never integrate up. since the minim um output of the phase detect or is ground . To fix this last ( known) problem . we need to connec t the noninvert ing terminal to Vnn / 2. as shown in Fig ure 15.35. Now we ca n set about determi ning the various co mponent values . First, not e that our loop tran smission is (4 3)

To achieve a 45° phase marg in. the zero must be placed at cros sover. since the two poles at the ori gin contribute a total phase shift of - ISO". Hence. R4, C I must equa l I ms. Choosing values with the same mod eratel y constrained arbi trariness as in the passive filter case . we let R4, = 100 kQ . so tha t the value of C 1 is O.OIIIF. Next. note that the loop transmission magnitude is contro lled by bot h R) and K o • so we would have an undcrcon srraincd problem if achieving a speci fied crossover frequency were the on ly co nsideratio n. Since there is a requirement 011 the lock range of the loop. however. there is an additional constraint that allow s us to fix ho th R) and K o . Spe cifically, the control vo ltage has an effect on veo frequency only from about 1.2 V to 5 V. acco rd ing to the empirical formul a . III The ce nter of this voltage range is 3. 1 V. lint the 2.5 V implicitly ass umed. If we continue to usc 2.5 V as our definition of ce nter. tho ugh . the loc k range will not be symmetrical abo ut 20 kil l . As there is no specification a bout a symmetrica l lock runge. we will remai n consistent in our use of 2.5 V as the contro l voltage that co rrespond s to the center frequ ency of the yeo . With that cho ice. the lowe r frequency limit is sma ller than the higher one. To satisfy our 10 -kH z spec ification. we must he a ble to change the VCO freq uency by

18 The control vcltage tcrrn is no( allowed to take on a negative value in the formula .

CHA PTE R 15

PHASE -lOCKED LO O PS

10 kllz (or more) with the co ntro l voltage at its minimum value o f 1.2 Y, corresponding to a deviatio n of 1.3 V from the center. Hence , we require Ko >

21f . JO kHz 3 ::::: 4 Jl x J04 rpsjV. I. V

Maintaining a center freq uency of 20 kf-lz wit h this followi ng cho ices for the three veo co mponents:

C = 0.001 "F.

R, = 42krl .

(44)

veo gain constant leads to the

R, = 130 kn .

Here. the closes t standa rd (10 % tolerance) resistor:" for HI and Hl are 39 1:11 and 120 kQ . respectivel y. Finally. having determ ined everything else. the cross over frequ ency requirement fixes the value o f the op-amp input resistor: (45)

Therefore. HJ = 2.8 MQ (2.7 meg is the closest standard value ), and the design is co mplete. Note tha t. for this design. it is defi nitely the VCO tuning range and not the phase detector characteristics thai determines Ihe lock range. wuh a loop filter that provide s an integration . any steady-stare YCO contro l voltage can be obtained with zero phase error. Therefo re. the phase detector cha racteristics are irrelevant with respect to the steady-state loc k range.

15.9 SUMMARY The design examples presented arc representative o f typical practice, allhough they arc a very tiny subclass o f a vast universe of possible PLL ••pplicutions. In Section 16.7 we will enco unter on e more application of PLI.s, in frequency synthesis, since synthesizers arc exceedingly impo rtant bui lding block s for modem RF communications gear.

PROB LEM SET FOR PHA SE -LO CKED LOOPS 1 5 . ~ 6 . Assume that the phase detector is a simple C MOS XO R whose logic levels are ground and Vf)fJ . Further assume that bo th the input to the loo p and the veo output are square waves that swing bet ween ground and Vnn . Finally. assume that the YCO has a perfectly linear relatio nship betw een control voltage and output freq uency of 10 M tlz per volt. Polarities are such tha t an increase in cuntrol voltage ca uses an increase in YCO frequency.

PROBLEM 1 Consider the PLL shown in Figure

PROBLE M SET

Phase Uelecto r

"

veo

f-~_ . 41oul

,4 co ntrol voltage

f iGURE 15.36 .

Second-on:ler PLL.

(a) First suppose that the loop has been in lock forever. and that the input signal frequency has remained co nstant for that time. Sketch the input signals to the XOR (as functions of lime ). (b) Derive expressions for the loop transmission and 4JuuI / 4Jin. (c) Initially assume N2 = 0 and Nt = 100 n . What value of C gives us a loop crosso ver frequency of 100 kHz? What is the phase margin? Assume that the op-amp is ideal. (d) With the value of C Irom pan (c). allow nonzero values of R2 10 provide a phase margin of 450 while preservi ng a 100 -kHz cro ssover frequency, (e) Now suppose thai a frequency divider of factor N is inserted into the feedback path. with the component values of part (d), what is the largest N that can be tolerated without shrinking phase margin below I-t ? Do not co nsider "divider delay" in the phase margin calc ulations. D

PROBLE M 2 Derive the transfer funct ion for a charge -pump phase detector. As-

sume perfect switches, a pump current of I . and a pump capacitor of C . How does your answer change when a resistor is inserted in series with the capacitor? PROBLEM 3 When an exclusive-O R gate is used as a phase detector. harmonic loc kin g becomes possible. If both the input and yeO signals are sq uare waves with

precisely 50% duty cycle, then only odd harmonic locking is allowed. (a) Show that, if the input signal has a duty ratio /) other than 50%, locking to even harmonics is possible. Hint: Find the Fourier series representation for a non-50% duty cycle sq uare wave. (b) Provide an explicit expression for the loop transmission of such a first-order PLL as a function of harmonic number. Normalize your answer in tenns of the loop transmission for the fundamental. (c) Based o n your answer to (b ), what conclusions may you draw about the ease or diffi culty of loc king un to harmonics? PROBLEM 4 Statements such as ··PUs using XORs lock in quadrature," or '·PLLs

using (certain) sequential phase detectors lock in phase" are incomplete because the

480

CHA PTER IS

PHA SE·LO CKED l O O PS

loc k po int is actually a function of mo re than simply th e type of phase detect or used. Offsets, either random or purposefully introd uced . may also change the lock point (sec Fig ure 15.37). Here. assume that the loo p filte r co nta ins an imc grarion with zero o ffset .

,,

1\l(JI'f'SI: , '

'If' - Y

FIGURE 15. 3 7. PU with offset.

(a) What is the lock point of this loo p if the phase detector. here represented as the three -input sum ming nod e. is an XO R? (b) S uppo se tha t the q,otf'W:l is related to an o ffset vo ltage th roug h the phase detector ga in c.'o nsta nt. Ko . Wh at offset voltage co rresponds to a phase error (offset) of 0 . 1 nul jf the phase detec tor is a simple C MOS XO R with a Vno of 3.3 V1 (e) On e way fo r offse ts to be introd uced is thro ugh vari ation in bias voltages. such as in the ci rc uit of Fig ure 15.38. If the phase detecto r itse lf has no offset, express the loc k po int of this PLL as a func tion of VR1....S • For simplicity, you may assume tha t the phase de tector c ha rac teristic is sym me trica l a bout ze ro. such as the ex tended-range phase -and -frequ ency detecto r discu ssed in the c hapter. R,

C

R,

yeo

¢I,n-

l

control voltage

FIGURE 15 . 3 8. Second-o rder Pll wi th offset.

PROBLEM 5 In real I'LL circ uits. of co urse . the o utput of the y eO is used by somethi ng e lse. As a co nseq uence . it is freq uent ly necessary to buffer the y eO signal. However. a ll rea l buffers are noi sy. so eve n if the y eO ou tput pru pcr is quire clean (spectra lly speaking), it is po ssible for buffer noise to negat e the co nsiderable design effo rt ex pended in m aking the veo itse lf rel at ive ly no ise -free. There a re two c hoice s fo r ho w to c lose the loop if a buffer is ex plici tly considered. O ne is to close the loo p a ro und the bu ffe r. and the o the r is to cl ose the loo p without the buffe r. To unde rstand the noise co nseq ue nce s of these two cho ices. model lhe

48 '

PRO BLE M SET

buffer as an ideal un it-gai n block w hose output is corrupted by add itive noise. For simplici ty. you may assume that the added no ise is white. (a ) First co nside r the case depicted in Figure 15.39 . For simplici ty. assum e that the input signa l to the PLL is co mpletely noise -free. Furthe rmo re. ass ume thai the veo is als o noise -free. G iven these two assumptions. sketch the expected output spec tru m. VNO l st

vco

BUFFER

FIGURE 15.39. PU with noisy buffer outside of loop.

(b) Now consider enclosing the buffe r (a nd its noise source) inside the PLL. as shown in Figure 15.40. With the same assumptio ns as in part (a). what is the spec trum of the output now'! Take into account the effec t of the loop filter o n the resulti ng spectru m. VNOISE

·'~,------H,------H LPF

vco

BUFFER

r0r.o~

FIGURE 15.40 . PU with noisy buffer inside of loop.

(c ) Given your an swers 10 the previo us part s of this question. whic h architecture leads to a cl eaner o utput spec trum'! Would your answer change if the input slgnal to the PLL were not perfect ly noiseless? If so, how? In all ordi nary PLL s, the voltage dri ving the veo consists of a DC value (co rrespond ing to the correc t average frequency of the PLL output), on which rides som e ripple co mponent.

PROBLEM 6

(a) Assu me an excl usive-O R gate is used as the phase det ect or, Sketch the o utpu t of the phase detect or as a function of time, assuming squa re -wave inputs when the PLL is lock ed. (b) If you add itionally assume thut the veo has a perfectly linear frequency-versusco ntrol voltage characteris tic. then the sketch yo u drew in part (a) is in fact a plot of frequency versus time. C lear ly, the output spectrum cannot he spectrally pure.

482

CHAPTE R 15 PHA SE· LO CKED LO O PS

Us ing whatevcr sim ulatio n tools are at your disposal . pl ot the out put spectrum of a perfec t sinuso idal YCO d rive n by such a con tro l voltage waveform. (c ) Com me nt on the effe c tive ness of filtering the co ntro l voltage to redu ce the ripple. Ca n yo u provide arbitra rily large a ttenuation of this ripp le? If not . why not? Assume that. for a pa rticular fi rst-ord er loop. the y eO has a gain consta nt K o of 20n 7T f\.t rps /V, K[) is 0.8 V/rad. a nd the oscilla tion freque ncy f~ = 50n ~t l lz . Sketc h the contro l vol tag e at the o utput of the ph ase de tect o r if the input

PROBLEM 7

freq uency ju mps fro m 500 MH z to 650 MHz. PROBLEM 8 In man y PLL -hased frequen cy sy nthesi zers. a freque ncy divid er is used in the feedback path . allowing the generatio n of output frequ encies that art

mu ltip les of an input reference freq uency. Th ese divide rs are almost always

COIl'

structed out of digita l logic e lemen ts that ca n introduce noi se into the loop. as shown in Figure 15A 1.

vco

L -_ _-{ +

fi GURE 15 .4 1. Pl.lwith noisy d ivide ·by-N in feedl:xxk path.

Sketc h a n ude plo t of the transfer fun ct ion ¢ "ul /¢ no;-e a nd di scu ss how the bandwid th of the loop ca n alter the effec t of the divider's noi se. You may as sume that the loo p fille r conta ins a n integ rat ion and a loop-stabilizing ze ro : the yeO is modeled as a perfect integ rato r. PROBLEM 9

Co nside r the sec o nd-o rde r pha se-locked loo p used to ge nerate a 10 Hz ou tp ut signa l ( Figure 15.42). No ise coupled into the I'LL by o ther c ircuitry is rnodeled us additi ve at the co ntro l port of the Yeo. The yeO has a gain co nstant Ko

of 400 Mrpsl

v. VNO, ISE

vco

FIGURE 15 .4 2 . Second-order Pl.l with no ise injection .

PRO BLE M SET

483

The input reference signal is supplied by a crystal oscilla tor whose outpu t specuum is exceptionally free of noise. The design prob lem is 10 se lect loop constants mal guarantee reasonable immunit y 10 Vnoi'C" (a) Determine the effective phase detector gain K n and loop-stabilizing zero time consta nt f = to produce a maximum output phase error o f o ne radian in respo nse 10 a IOO-mV step o n Vnoi'C" (b) How would your answer to part (a ) change ift hc maximum tolerable phase error were 0. 1 rad? PROBL EM 10 Practical veos generate imperfect signals. Suppose we mod el one such impe rfectio n. noise, :IS a voltage added to the OUlp UI of the YeO; see Figure

15.43.

Ll'f

vco

. . 4J()lJT

fiGURE 15••13 . PU with noisy veo.

In this mode l, then. the veo is assumed to produce a perfectly pure output signal rluu is co rrupted by the add itive noise voltage shown. Assume further that the input signal to the loop is also noi seless. Derive an expression for the output spectrum in terms of the transfer charac teristics of the loop e lements. You may assume that the spectrum of Vnoi~ is white. What does your equation say abo ut how one should select loop bandwidth?

CHAPTER SIXTEEN

OSCILLATORS AND SYNTHESIZERS

16.1 INTRODU CTION Given the effort ex pended in avoi ding instability in mo st feedback syste ms. it would see m trivial to construc t oscillators. Howeve r. s imply ge ne rating some periodic outpu t is not sufficie nt for modern high-performan ce RF receivers a nd tra nsmitters. Issues of spec tral purity a nd am plitude stability IlI US! be addressed . In this c hapter. We consider seve ral aspects (If osci llator des ign. Firs t. we show why purely linear osc illators are a pract ical impossib ility. We the n present a linea rization technique that uses describin g functions to dev elop insight into how nonhnearities affect osci llato r pe rfo rma nce . with a partic ular e mphas is o n predict ing the amplitude (I f oscil lation. A survey of resonator techn ol ogi es is incl uded. a nd we ,,1:-;0 re visit PLLs. this time in the co ntex t of frequency synthesizers. We concl ude this c hapter w ith a survey of oscillator arc hitec tures. Th e important issue of phase no ise is considered in detail in Chapter 17.

16.2

THE PROBLEM WITH PURElY LINEAR OSCILLATORS

In negat ive fee dback sys tems. we aim fo r large po sitive pha se margins to avoid instability. To make an osci llator. then . it might see m that all we have to do is ~hooc for zero or nega tive phase margin s. Let's examine thi s not ion more carefully, using the root locus sketc hed in Figure I ft. I. Th is loc us rec urs fr equently in osci llator design beca use it applies to a two-pole bandpass reso nator with feedback . As seen in the locu s. the closed -loo p pules lie exactly on the im agina ry axis for some part icul ar value of loop tran smi ssion magnitude. Th e co rres po nd ing impulse response is the refore a sinusoid thai neither decays nor grows with time, a nd it would see m that we have a n osc illator. • 84

16 .3 DESCRIBIN G FUN CTIO N S

FIGURE 16 . 1. Root locus

'"

for oscilbtor excmple.

There are a co uple o f practical difficu lties with this scenario, however. First, the amplitude of the oscillation depends on the magn itude of the impulse (i t is a linear system. after all) . Th is behavio r is ge nerally undesirable; in nearl y all cases. we want the osc illator to produ ce a co nstant amp litude output that is independent of initial cond itions. Another prob lem is that if the closed -loop pol es don't lie pred st'ly o n the imaginary ax is, then oscillations will either grow or decay expo nentially with time. These problems are inherent in any pu rely linear approach 10 oscillator design . The so lution to these pro blems therefore lies in a purposeful exploitation uf non linear effe cts: !IIIp ractical osc illators depend on nontinearuies. To understand j ust how nonlincaritics ca n be be neficia l in this co ntext, and to develop intuition useful for both analysis and design, we now consider the subject of describing fun ctions.

16 . 3 DE SCRIBIN G FUNCTION S We have see n that linear descriptions of systems often suffice, even if those systems are nonlinear. For example, the incremen tal model of a bipo lar transistor arises from a linearization of the device' s inherent exponential tran sfer characteristic. As long as excitations are "s ufficiently small," the assumption of linear behavior is we ll satis fied, An alternative to linearizing an input-out put tran sfer characteris tic is to perform the linearization in the f reqllt'IIl'y domain . Specifica lly, co nside r exc iting a nonlin ear system with a sinusoid of some particu lar frequency and amplit ude. The output will generally consist of a nu mber of sinusoids of vario us frequencies and amplitudes . A linear description of the sys tem can be obtained by disca rding all output components exce pt the one whose frequency matches that of the input. The co llection o f all possible input-output phase shifts and amplitude ratios for the surviving co mponent comprise!"> the describing function for the non linearit y. If the output spec trum is

".

CHAPTER 16 OSC ILLATO RS A N D SYN THESIZ ERS

dom inat ed by the fund a mental compo nent , results ob tained with a desc ribing functio n approximat ion will be reaso nably accu rate. To validate furt her our subseque nt analyses. we will also impose the following restric tion Oil the nonlinearit ies: they must ge nerate no subharmonics of the input (DC is a subharmonic). T he reaso n for this restriction will become d ea r mome ntarily. For RF syste ms. this req uire me nt is perhaps nol as restric tive as it initially appears. because bandpass filter s can ofte n be used to elim inate subharmo nic a nd harmonic components. As a spec ific exa mple of generating a desc ribin g function. conside r a n ideal comparutor who se output de pen ds on the inpu t as fo llow s: (I)

If we drive such a comparator with a sine wave of so me frequency wand amplitude E . then the outp ut wi ll he a sq uare wa ve of the sa me frequ ency but of a con stant amplitud e B . independent of the input amplitude. Furthermore. the zero cros sings of the inpu t a nd output will coi ncide (so there is no phase shift). Hence. the output can he e xpressed as the following Fourier se ries : VOUl

_ 4 8 ~ sin w nt -

L.-

-

Jr

I

.

1/

odd.

(21

1/

Preservi ng only the funda me nta l term (n = I ) a nd ta king the ratio of output to input yields the describ ing functio n for the com parato r: G l )( E )

48 = -. rr£

(3)

Since there is no phase shift or frequ ency depend ence in this parti cular case. the describing function depen ds only on the input am plitude . Note that thc descri bin g fu nction for the com parator shows that the effec tive gain is inversely proportionalus the driv e am plitud e. in contrast with a pur ely linear syste m in which the gain is indep endent of dri ve a mplitude . We shall soo n see that this inverse gain be hav ior can be extre mely useful in provid ing nega tive feedb ac k to stabilize the a mp litude.

16 . 3 .1 A BRIEF CATA LOG O F DES CRIBIN G FUN CTION S Havin g show n how one goe s abo ut genera ting describing fu nc tion s. we now present a small list . w itho ut derivation. of describing func tion s for some commo nly e ncountc red nonlincarities.! 1 Sec e.g. the excellent boo k by J. K. Roberge. Opa ulillnu! Amp/ifias. Wiley. New Yor k, 1975.

487

16 .3 DESCR IRING FUNCTIONS

, FIGURE 16 .2. Tran~fer cha racteris tic for saturating ampli ~eI".

~ slope "' K

V.

FIGURE 16.3 . Tram.fer chorodefislic ampri ~er with crossover distortion.

for

For a saturating amplifi er (Figure 16.2) we have G

lJ(

E ) -

jf E < EM .

K

1 (2 K/;rr)( sin- 1 R + R .J I _ R 2 )

if E > EM.

where R = EM / E. For an amplifier with crossove r distortion ( Figure 16.3). the describing function is

if E < EM . if E > EM . Finally. for a Schmitt trigger we have Gp (E ) = 4Bj JrE .

L(- si n- I HI

for R < I (see Figure 16 .4 ). In this last example. the value (If R must be less than unity beca use otherwi se the Schmitt never triggers. in whic h case the comparator output would be only a OC value of eithe r B or - 8 .

-I

4"

CHAPTER 16 OSC Ill ATO RS AN D SYN THESIZERS

v~

II

v.. EM

FIGURE 16 . 4 . Tron ~Fer charadefi slic

for Schmitt trigger .

II j " import ant to no te that desc ribin g funct ion s themselves are linear eve n though the functio ns thai they describe may be nonlinear (got that'!). Hence. superposi tion holds: the describing funct ion for a sum of non linearit ics is eq ual to the sum of the indi vid ual describing functions. Thi s property is extreme ly useful fur deri ving desc ribing funct io ns for non linearities not included in the short catalog presented here.

16.3 .2 DESCR IBIN G fUN CTION S f O R MO S A N D BIPO LAR TRAN SISTO RS Although the foregoing collection of describing functions is ex tremely useful, perhaps more relevant to the RF oscillator design problem arc describing funct ions for O Il C - and two -transistor circuits. since the high frequencies that characte rize RF operation arc difficu lt to ge nerate with many transi stors in a loop. To illu strate a ge neral approach , consider the circu it in Figur e 16.5. The capacitor is assumed large enough to be have as a short at frequency w . and the transistor is idea l. We wi ll be using this circuit in tuned osci llators. so the bandpass act ion prov ided by the tank guarantees that describing functio n analysis will yiel d accurate results. Before emba rking on a dct uiled derivation of the large-signal (i.e. , describing funclion) tran scon du ctance. let ' s anticipate the qua lita tive outlines of the resu lt. As the amplitude VI increases, the source voltage V:, is pulled to higher values. reaching a m ;llillllllll roughly whe n the inp ut doe ". Soon after the g:lle d rive h",ad o; ha c !.: dow n, ward fro m the pea k, the transistor cuts off as the input vo ltage falls faster than the curre nt source can discharge the ca paci tor. Beca use the curren t source d ischa rge s the capacitor between cycles. the gate -source j unction aga in forward-biases when the inp ut return s to near its peak value, result ing in a pulse o f drain current. The cycle repe ats, so the drai n curre nt consis ts o f periodic pu lses.

16 .3

DESCRI8 1NG FUNCTIO NS

' 89

• '0

Vss FIGURE 16 . 5 . Large -signa l tronscond uctor.

v,

' ",

FIGURE 16 . 6 . Hypotneticol source a nd gale voltage, and dra in current, for large input voltage.

Remarkabl y, we do not need to kno w any more about the detailed shape of drain current in order to derive quantit ati vely the large -signalt ransconductance in the limit of large drive am plitud es. The only relevant fact is that (he current pulses consist of relatively narrow slivers in that limit. as in the hypo thet ical plots o f gate voltage. source voltage. and drain current shown in Figure 16.6.2 Whatever the current waveform, KCL dem ands that its average value eq ual /BIAS. Th at is. 2 The word " hypothetical" i ~ here a euphemism for " wrong." How ever, eve n though the detailed waveforms shown are not strictly correct. the results and insights IlNairM."lJ are. In particular, thi" picture allow s us to understand why the describing fun,,:tion u anscond uctance for large drive amplitudes is essentially the same for bipolan. and MOSFl:: i s I buth lung- and short -channel), as well as for JFETs and vacuum t uhi:'~ .

' 90

CHAPTE R 16 OSC lltATORS AND SYNTHESIZERS

11'

(if)) = T

0

if)( t)

dt =

(4)

ItU AS .

Nnw. the fundamental co mpo ne nt of the dra in c urre nt has an amp litud e give n by

~ lT io (I) cos wt dt .

t, =

T

(5)

0

Although we may not know the detailed functional form of i o (t ). we do know that it co ns ist.. of narrow pulse s in the lim it of large d rive amp litudes . Furt hermore , these curre nt pul ses occ ur ro ugh ly whe n the input is a max imum . so thut the cosine may he ap proxi mated there by unity fo r the sho rt d uration o f the pu lse . The n.

21'

I, = -

T o

21'

io (t )coswt d, ::::: -

T

0

;o (l) d l = 2I aIAs .

(6)

That is. the a mplitude of the funda me nta l co mpone nt hi approxima tel y twice the bias curre nt, again in the lim it of larg e VI . The mag nit ude of the de scribing fun ction is there fore

(7) 11i..importan t to no te tha t the foregoing derivation doc s not depend o n detailed transistor c harac te ristics at any step alo ng the wa y, Becau se no de vice -spec ific a ssumption s are used . Eq n. 7 is qui te ge nera l and applies 10 MOSF ET s ( bo th lon g- a nd shortc hannel ), a s we ll as to bipo lar'S, J F ET s, GOlAs f\IESFbIS, and e ven vac uum tubes, In de rivin g Eqn. 7, we have as sumed that the d rive am plit ude , VI , is " large ," To qu a ntify this noti on , le t us co mpute the G m / 8m rati o for lon g- and sho rt-c hannel M OS F ET s and bipo lar device s . For lon g-ch annel dev ice s, the ra tio of 8m to d rain curre nt

1 1I1 AS

ma y be writte n as

2

8m

( 8)

V, '

VI/.,

I IH AS

so that Gm 8m

=

Vx. -

VI

(9)

V,

Ev ide ntly, " la rge " VI is de fined relative to ( VR• - V, ) for lon g-c hann el MO SFETs. Repeating thi s ex e rcise for sho rt- cha nnel device s yic h.h 3

s-

- -= I BIAS

2 VgJ

-

V,

E..a,L

+ (Vx.,

(10)

V, ) '

3 Here. we have used the appro~ i mate , analytic model for shon -channet MOSFETs imrod uced in Chapter 3.

. 91

16 . 3 DESCRIBING FUN CTIO N S

\-0-- - -

Appm \ jmatio n

v, FIGURE 16 . 7. G m/gmversus V, .

.....hich , in the limit of very short channels. converges to a value precisely half that of the long-charme! case. Thu s. .:. ( v"'':' ' ;;- ",,:, V'C): < _ G_. <

= 2.:. (v ",';' ;--" ":'V'C):

V,

g",

VI

(I I)

Finally. for bipo lar devices, we have

g. { 8 IAS

so that

1

= VT '

G",

gm =

2VT

v;-

(1 2 )

( 13)

In bipolar devices, large VI is therefore defined relative to the therm al voltage . Although the equation for G... is valid o nly for large VI. practical oscillators usually satisfy this conditio n and so the restriction is much less co nstraining than one might think . We will also sec in the next chapter that large VI is highly desirable for reducing phase noise. so one may argue that all well-designed oscillators automatically satisfy the cond itions necessary to validate the approximations used. Nonetheless. it is important to recog nize that Gm can never exceed !:m. so o ne must he careful not to misapply formulas such as Eqn. 13. To underscore this po int, Figure 16.7 shows. in an approximate way, the actual behavior of G... / g", co ntrasted with the behavior as predicted by Eqn. 13. Although this equation applies strictly to the bipolar case. the overall behavior shown in Figure 16.7 holds generally. Having presented numero us describing functions. we now consider two examples to illustrate how to use them to analyze oscillators. 16. 3. 3

EXAMP LE I : FUNCTION G EN ERATOR

To get good results with describing functions. it's important to satisfy the conditions used to derive them. That is. the circuit must be sufficiently low-pa...s or bandpass in

CHAPTER 16 OSC ill ATORS AND SYNTHESIZE RS

-

.

~

. .,

fi,l

. .,

B .. V ,n

f- ..

- l Is

f-

I

FIGURE 16 .8 . Functioo geoerotof core .

natu re 10 provide a near-sinusoidal drive 10 the nonlinearity. If this condition i.. not well satisfied , the resu lts of descri bing function analy sis can not he expected 10 be accura te. Fortunate ly. many systems of practi cal interest do satisfy this req uirement, and describi ng functions can be used succes sfully in ot her than purely academic settings. Oscill ators for use in communications systems are partic ularly amenable to describing function ana lysis because high· Q reson ators are often present , forci ng nearly sinusoidal drives. To illustrate the utility of describi ng functions, let 's analyze an osci llator two different ways. As a specifi c exa mple, co ns ide r the ci rcuit of Figure 16 .8, wh ich is used as the basis for many labo ratory functi on ge nerators [i.e., the thin gs that generate sine, square, and tnu nglc waves. all in on e instrume nt ). As see n in the figure. the oscillator consists of a Schmi tt trigger and an inverti ng integrator. 111e outpu t of the Schmitt is a square wave of amplitude H. while the outpu t of the integrator is a triangle wave of amplitude EM . Specifically, the output waveforms arc as depic ted in Figure 16 .9. From this direct analysi s in the time domain . we sec that the per iod of oscillation is simply ( 14)

Now thai we know the exact answe r, lei 's sec if we ca n ob tai n a reaso nable pred iction of the amplitude and frequency of osc illa tion from a describing funct ion ana lysis. Befo re p rt>l:e e Jinl,; fu rthe r. we note th a t s o me e rror iii: to be cx(\"c tc d , s i nc e t he w nvc -

form at the input to the no nlinearity (the Schmitt trigger ) is not a particularly good upproxirnation to a sinuso id . Neve rthele ss, a triang le wave's spectrum fall s off as I j w 2 • Ml per haps the analysis won ' t be completel y worthle ss. To analyze this oscillator with descri bing functio ns. conside r Figu re 16. IO's model for thc loo p. Wc have ex plicitly assigned the inversion its ow n sepa rate block in order

16. 3 DESCRIBING FUNCTION S

493

-E M

R -R

FIGU RE 16 .9. Functio n generator waveforms.

~.

-

GOm )

I I FIGU RE 16 .10 . Describing

-I

.

lis

I I

function kxlp model for oKillator .

to make the model co nsistent with the negative feedb ack-Ioop block diagram s we have used previou sly. Recall that a necessary [ but no t sufficient) condition for o..cillation is a unit loop trans mission magnitude and a zero phase margin . Fur this system. thai req uire ment trans lates 10 I G,, (E ) -. = - 1. ( 15)

J'" We can express this requirement a little differently: -~

[cu

- I

- I

G n(E)

41J/rrE (L[ - sin- I RJ)

( 16)

This last equation suggests a grap hical tech nique for disco vering po.... ible oscillation frequencies and umphru dcs: Plot the behavior of the purely linear stuff in the gain phase plane. and similarly plot the negative rec iprocal of the de..cu bing fu nctio n .~ If any intersec tion s occur. the corres po nding frequencie s and ampl itude s are possible oscillatory sta les .

~ Recallthat the magnitude of the reciprocal Is just the reciprocal uf the magnitude . Similarly. the

phase of lhe reciprocal

i ~ j U~1

the algd' raic inverse of the original phase.

CHA PTE R 16 O SCILLATORS A ND SYNTHESIZ ER S

,

- IKlt

FIGUII:E 16 . 11.

-90"

Gain phose plon for cscillcroe exomple.

For the sa ke of simplicity. let B = EM = I . Th en. applying this n..cipe to our speci fic exa mp le. we ob tai n plot s in Ihe gain phase plane tsee Figu re 16.11). There is only one intersection.I and it occ urs at an ampli tude E = I and at a frequ ency lJ) = ~/Jr rps. corres po nding to a period of oscillation o f

T....: =

rr ' : :;: 4.9 s. "2

(1 7)

The exact ana lysis performed earlier shows that the actual amplitude is in fact unity for these values , w hile the oscilla tio n period is 4 s. Th is level of agreement is fairly reasonable considering thai a trian gle wave, rat herthan a s inuso id, dr ives the nonlinearity. We can glean so me important Insight s from this type o f analysis. If the goal is to achieve high spec tral purity. then it is desirable for the no nlinea rity to be relatively "soft:' Furthermore. the linear clements in the loop should he low-pa ss (or per haps ba ndpass) in nature to atten uate the dis tortion products thai arc ge nerated by the non linearity. Fina lly, one should usc the inpu t to the nonlinearity as the output signal, since it will have the lowest distortion. As an add itio nal bo nus. sat isfyi ng all of these requirement s also guarantees conditions that favo r highly accurate describing function analyses. As a dosing 1I0te on this example , it should he men tioned that the sine-wave output in co mmercial functi on generators is obtained by nonlinear waves hapi ng of the triangle wave output, as d iscu ssed in an earlier chapter. The maximum operating frequ ency for discrete implementat ions of such generators is typically around JO-50 Ml lz , and is limited primarily by the qu ality of the square wave at higher frequencies. At higher frequenci es. general-purpose function generators are re placed

j

Strictly speaking, we have a tangency rather than an mrersecnon. Neven beless, we see that the resuns o r the ..k~ri bi n g Iuecuon anal y~i s are in reasonable agreement with the exact analysis.

16 .3 DESCRIBIN G FUNCTIO N S

. 95

V l lt )

R \'-'1..'"

C, V

FIGURE

16.12 . Colpitts oscillator

(bia sing details not shown).

by special-purpose ci rcuits. such as high -speed sq uare -wave o r pul se generators and tuned si ne -wave osci llators, that use di fferent architect ures optimiz ed for eac h par ticu lar type o f out put waveform .

16 . 3. 4 EXAM PLE 2 , COlPITTS O SCILLATOR Relaxatio n oscilla tors such as the function generator are rarely used in high-pcrformunce tran sceivers because they generate signa ls of inadequ ate spec tral purity. Much more co mmon are tuned osc illators. primarily for rea sons that we may appreciate only after study ing the subject of phase noise. For now. s imply acce pt as an axiom the superiority of tuned osci llators . Our prese nt focu s, then. is the use of describi ng functions to predict the output amplitude of a typic al tuned osci llator. such as the Co lpitts circuit shown in Figure 16. 12. We shall see in this cha pter that a varie ty of osci llators differing in trivial details are name d for their inventors. In kee ping with standard practice. we will reta in this naming conventio n. hut the reader is advised to foc us on ope rat ing princi ples ruther than nomencl ature. The basic recipe fo r these osci llators is simple: Combine a resonator with an active device. The dist ingui sh ing fe ature of a Co lpitts oscillator is the ca pucitively tapped resonator, with positive feedback pro vided by the active device to make oscillation s possible. In Figure 16. 12. the resistance R represen ts the total loadi ng due to finite tank Q. tran sistor output resistance. and whatever is driven by the oscill ator ( presumably the osci llator' s output is used somewhere). The curre nt source is frequently replaced by an ordinary resistor in practical implem entations. but is used here to simplify (marginally) the analysis. Fro m our describing functi on derivation . we know that the transistor may he characteri zed by a large -signal tran sconductance G... . For the sake of s implicity, we wi ll ignore all dynamic elemen ts of the tran sistor. as we ll as all parasitic resistances, altho ugh an accurate analysis ought to take these into account. The transistor also

496

CHAPTER 16 O SCILLA TORS AN D SYN THESIZERS

Gm = 2 J/l1.45""' J

VI H, = IIG",

R

---- ·~ G"' VI :.~~~;~~~;;'

f-_-.J V,

?'

model

FI GURE 16 .13. Describing Fvnction model of Colpitts ~i llalof .

L

c"

fi GURE 16.14 . SimpliJiedmodel of Colpilh oscilla tor.

has a large -sign al source-gate resistance. of co urse , which m ust be modeled as well. Tak ing a c ue fro m describi ng fu nc tion s. it see ms reaso nable 10 define this re sistance

as the ratio o f the fundamental component o f source current to the source -gate voltage. We ' ve actu ally a lready found this ratio; it is sim ply JIG",. Thus, we may model the osc illator with the ci rcui t of Fig ure 16. 13 . Tn simplify the ana lysis further. first reflectt he input resistance H; acros s the main ta nk te rm inals by trea ting the ca paci tive divider as an ideal tra nsfor mer (sec Chapter 4), so rbat we end up with a simple RLC ta nk e mbedded within a po sitive feedback loo p. No te that the re sulting c ircuit has ze ro pha se ma rgin
a nd

C ,C 2 Cet! = -;;-7-0,C j + C,

(18)

I

J LC,, '

(19)

16 .3 DESCRIBING f UNCTIO NS

497

Similarly. R rlj is the parall el com bina tion oft he or iginal tank resistance Rand thc reflected large -signal input resistance o f the tran sistor:"

Req

'"

1 R II " 'G- . II · ' tn

(20 )

where 1/ is the capacitive voltage divide factor.

12 1) The amplitude VI is simply the amplitude capacitive divide factor . so we may write

Vla n1

V,ank

of the tank voltage multiplied by the

V,

~ -

.

n

Now we have col lected enoug h eq uations 10 ge t the job don e. The amplitude of the tank voltage at resonance is simply the product o f the curren t source ampli tude and the net tank resistance :

which ultim atel y simplifies to (24)

Thus. the amplitude of oscillation is d irect ly proporti onal to the bias current and the effective tank resistance. The loadi ng of the tank by the transistor 's input resistance is taken into account by the ( I - 1/ ) factor and is therefore controllable by choice of the capacitive divide ratio. Since R also co ntrols Q. it is usually made as large a!'> possible. and adj ustment of ' BIAS is consequently the main method of defining the amplitude. As a specific numerical exa mple. co nsider the -- 60- M Hz oscillator ci rcuit that is sketched in Figu re 16. 15. For the particular element values shown. the capaci tive divide factor n is abou t 0 . 155.7 The expected oscillation ampl itude (VWl k ) is therefo re

6 In thi s and related eq uations. the reaso n for tbe ~app ro, i1l1aldy equals" syrn bol is thar we arc treat ing the capacitive divider as an idea l imped ance transformer. As mc nuoned in C hapter -t. the appruximanon is good as lon g as the in- circuit Q is large. J In prac tice, il is gent'rally true lhal best pha -.e n<.lise pe rformance lends 10 occ ur for a C2/ C I rollin nf ab out -t. correspon ding 10 n = 0.2. Th is rule uf Ihumb ca n be put on a more rigorous meore nc al basis by making usc of the time -varying theory d iscu sse d in Ch ap ter 17.

' 98

CHA PTER 16

OSC ILLATORS A ND SYN THESIZ ERS

Voo

R5nu V'""k 3JpF V

ISOpF

FIGURE 16 .1.5. CoIpitn

O5oCillalof example.

abo ut I A V. Measureme nts mad e on a bipolar ve rsion of this circuit reveal an amplitud e of 1.3 V. in good agree me nt with theoretical prediction s. II is important to und er score agai n that thi s result is largely independent of the type of act ive device used 10 build the osci llator. Th e prediction was mad e (or a MOSFET design and experimentall y verified with a bipolar devi ce .

Startup, Second-Order Effe.m, a nd POthologiM In the preceding analysis. nothing spec ific was men tiom."'. t! about co ndition s for guarantee ing the startup of osc illatio ns. From the ge neral roo t locu s of Figure 16 .1. howeve r, it should he clear that a necessary condition is a grea te r-than-unity value of small-signal loop transmi ssion. To eva luate whethe r startup migh t be a problem. one should se t the transcond uctance eq ual to its small- sig nal val ue (a n appropriate c ho ice. as the circuit is ce rta inly in the small-s ignal reg ime be fore osci llations have started] and compute the loo p transmission magnitude. If it docs not exceed unity then the oscillato r wi ll not start up. To fix this pro ble m. adj ust some combi nation of bia s current , device size, and tapping ra tio . In the ca se of the example ju st co nsidered, let us ident ify the minimum acceptable transcon d uctance for guarantee ing startup. That minimum 1(m . toge ther w ith the g iven value of bias curre nt. defines the width of the de vice. We use the model in Figu re 16.16. T he a mp litude of the voltage acro ss the tank at resonance is (25) w hic h reduce s to the fo llowing express io n for th e mi nimu m trunsconduc tauc c :

8m >

2 ' H (II - II )

(26 )

With " = 0 .155 a nd H = 850 n , the a bsolute mini m um acceptab le transconducta nce wo rks out to app roxima tel y 9 mS . Howeve r. not e tha t me rely having e nough

16 . 3 DESCRIBING FUN CTIO N S

L

fiGURE 16.16. Startup

model of

Cotpilt!. oscillator.

transconductance to ach ieve net unit loop gain with no oscillation is nor sufficientto make a good oscillator. Addi tionally. the desc ribing function is accu rate only in the limit of large amplitudes. and therefo re on ly if the small-signal transcondu ctance is substantially larger than the large -signal value. A rea..onable cho ice for a tin-t-cut design is to select K... to be fi ve times the minimum acceptable value. Hence. we will design for a 45-mS small-signal transconductance. To est imate the necessary device width. initially assume that the gale overdri ve is small enough that the device co nforms to square -law behavior. Then we may use Eqn. 8 to estimate the ove rdrive as fo llows: (27)

This overdrive is indeed small compared with typical values of E..., L (e.g.• 1- 2 V). so we will continue to ass ume operation in the long-channel regime. Solving for WIL in this regime then yields a value of abou t 6fXXl for typical values of mobi lity and C Ol ' For a D.5-11m channel length . then. the width should he roughly J IXX) It Ill . which is quite large. Thi s large width is a co nsequence of the low bias current; a higher hias current would permit the usc of a substantially smaller device. Aside fro m the neglect of startup conditions in the for egoing development. several other simplifying assumpti ons were invoked to reduce clutter in the derivations. Transistor parasitics were ignored. for ex.ample. We now consider how to modi fy the analysis to take these into account. The gate - drain and drai n-hulk capac itances appear in para llel with the tank. ami a first-order correction for their effec t simply involves a reduction in the explicit capacitance added externally to keep the osc illation frequency constant. However. these capacitances arc nonlinear. so dist ortion may be unsatisfactorily high if they COIll· prise a significant fraction o f the total tank capacitance. Temperature dritt properti es may also be affected . The source-gate and source- bulk capac itances appear directly in parallel with C 2 • and the same comments apply as for the other device ca pacitances.

500

CHA PTER 16 OSC ILLATORS AND SYNTHESIZE RS

One must also worry about the ou tput resistance of the trans istor. for it loads the tank as well. Many hig h-speed tran sistors have low Early vo ltages (e.g.. 10- 20 V or less ), so this load ing ca n be sig nifica nt a t limes. In se rious case s, cascodi ng (or some equivale nt rem edy) may be necessary 10 mitigate this pro ble m. In other insta nces. Ihis loading mere ly needs 10 be ta ke n into account to pred ic t the a mplitude more accuratel y. As a final comme nt on the issue o f amplitude. il mu st he e m phasized that the re is al ways the possibility of amp litude instabi lity. since feedback control of the amplitulle is fundamen tally involv ed. That is, instead of stay ing constant, the amplitude may vary in so me manner (e.g.• q uasi-sinusoidally ). Th is type o f behavior is known as "squcgging" and is the bane of oscillator designers. Recognizing that feedback is involved is the key to devel opi ng an intuition about what remedies might be applied. In parti cu lar, reducing the driv e amplitud e VI by reducing the factor " (fur a fixed bias curre nt) is useful to eliminate sq ucgg mg . The main drawbacks of this remedy are that the phase noise worse ns (see Chapter 17) and that the small -signal loo p ga in drops, possibly enda ngering startup.

16 . 4

RESONATOR S

T he previou s describing function exa mple analyzed a tuned oscillator. Since tuned circuits inherentl y perform a bandpass filtering funct ion , d istortion produ cts and noise are a tten ua ted relative to the fundamental compo ne nt. Not surprising, then. is that the performan ce of these circuits is int im ately linked to the qu ality of available resonators. Before proceeding onward to a detai led disc ussion of oscillator circ uitry, then . we first survey a number of reson ator technologies.

RE SON ATOR TECHN OL O GIE S Quarter-Wave Resonators

Aside from the famili ar and venerable RLC tan k ci rcuit, there arc many ways to make resonat ors. At high frequ encies. it becom es increa singly difficult tu obtain adequate (] fro m lum ped resonators because requ ired component values arc often impractical to reali ze. On e alternat ive is to use a distributed resonator such as a qu art er-wave piece of tra nsm ission line . To suggest w hy suc h a c ho ice may be se ns ible. recall thut Q is pro po rtional to the rat io of energy stored to energy dissipa ted. Some distributed sIrUCturcs store energy in a volume. while diss ipa tio n is due mainl y to surface effects Ie.g .• s kin effec t) within Ihe vo lume . Hence. the vo lume/surface -area ratio is impo rtant in determining Q. Thi s ratio may be made quite large fur ce rtai n distributed structures, so high Q values are possible.

16 .4 RE SONATOR S

so,

The required physical dimen sion s ge nerally favor practi cal reali zation in discrete form in the UHF band a nd above. As a n example. the free -space wave length at .30 MH z is one mete r, so that a quarter-wave resonato r would be about ten inches . If the resonator is filled with a dielec tric material othe r than air . the dim ensions shrink a.. the sq uare root of the relative di elec tric consnmr. Dime nsions become compat ible with ICs at mid-gigahert z freq ue ncie s. At 3 G Hz. for e xamp le , the tree -space quarte r-wa vel engt h is about one inch . Wit h diel ec tric materials that are co mmo nly a vailable. qua rte r-wa ve IC reso nators of abou t ha lf an inch or so are po ss ible . If we terminate suc h a tran smi ssion line in a short . the inpu t impedance is ideall y an open ci rc uit (l imited by the Q of the line). At frequencie s be low the resonant Irequcncy, the line appea rs as an inductor; above reso nance. it a ppears as a ca pacitor. Hence. for small disp lacement s about the reso nant cond ition. the line a ppears very much like a pa ralle l RI.C net work . There is an importa nt differen ce, howe ver. bet ween a shor ted line a nd a lumped RL C reso nator: the line appea rs as an infinite (o r at least large ) impedance ut all odd multiples o f the funda mental reso nance. Somcumes. this pe riodic behavior is desired . but it can al so result in oscillation simulta neou sly on m ultiple frequ encies. or a chaotic hopping fro m o ne mode to ano ther. Addi tio nal tun ed cleme nts may be required to suppress osc illation on unwa nted modes. The osci llators in most ce llular telephones use off-c hip qu arter-wave reso nators in which a pie zoel ec tric materi al. such as bariu m titana te. is UM.'d as the die lec tric. Th e high dielec tric consta nt of such mat erial s allows the rea lization of physica lly sm all resonators that possess exce llent Q -values (e.g .• 20,(XK)). It is virtual ly impos sib le to obtai n such high Q -values using ordinary lu mped el ements. Quartz Crystals

The most co mmo n non- R LC res ona tor is made of qu art z . Th e re ma rkable properlies and potential of quartz for use in the radio art were first seriously appreciat ed in the 1920 ,.. particularly hy W. G . Ca dy of Bell Laboratorie s. Qu art z is a piezoelectric materia l. and thus exhibits a reciprocal tran sduction betwee n mechan ical strain and electri c c harge. When a voltage is applied across a s fuh nf qu artz, the cry sta l phys ically deforms. Whe n a mechanical strain is applied. c harges appear across the cry stal. Most practical qu art z cry sta ls used a t rad io frequ encie s" employ a bulk shear vi· brational mode (see Figure 16.17). In this mode.fhe resonant freque ncy is inversel y proportional to the thick ness of the slab. according to the roug h formula" in the figure (51 units ass umed ).

8 The n yslals used in dig ital watches emp loy a tors ional mode o f \lihralion to allow resonance al a low frequency (32.768 U b i in a small sbe . 9 The form ula oeglcc rs l~ influence of lhe oeer dimension...

soz

CHAPTER 16

OS CI LLA TO RS AND SYN THESIZ ERS

C-.V

,J

r-

' 6711"

FIGURE 16 . 17. lIl u~trotion 01bulk

sheer mode"

Rs

FIGURE 16.18 . Symbol and model

"" """"I. Even though quartz does not exhibit a part icularly large piezoelectric effect. it has other properties that make it extremely valuable for use in RF circuitry. Chief among them is the exceptional stability (both electrical and mechanical) of the material. Furthermore. it is possible to obtain crystals with very low temperature coe fficients by cutting the quartz at certain angles.!'' Additionally. the transduction is virtually lossless. and Q values arc in the range o f 10 4 to 10". II An electrical mode l for a q uartz resonator is shown in Figure 16.18. The capacitance Co represents the parallel plate capacitance associa ted with the contacts and the lead wires. while emand L m represent the mechanical energy storage. Resistance Rs accounts for the nonzero lossiness that all real systems must exhibit. To a very crude approximation. the resistance of well-made crystals is inversely proport ional to the resonant frequency, and generally follows a 5 x

1 0~

Rs '" ----,,----

/.,

(28)

relationship. This formul a is a semiempirical o ne. and should be used only if measurements aren't available.'?

'0 II is aIM) ptlssible to obtain controlled. nonzero temperature ctld lk icnts. This pro perty has been II

II

exploued 10 make temperature -to -frequency transd uce rs that function attem peratures too e xtreme Iur ord inary electronic circuits. At lowe r frequencies. damping by air lowe rs Q significantly. The higher Q values correspond 10 crystals mou nted inside a vacuum. This form ula strictly applies only to "AT- ("lJ t ~ cry....als ope raung in the fundamental mode .

16 .4

RESONATORS

503

The values of C", and L", can he co mputed if R s . Q. and the resonant frequ ency are given. In ge ne....II. beca use of the extnsordinarily high Q -values that qu artz crystals possess. the effec tive induct anc e value will be surprising ly high . while the series ca pacitance value will be vanishingly small. For example. a I-MH z crystal with a Q of IOj has an effective inductance of about eight he nries (no type here. that ' s reall y eight henries ) and a Cm of abo ut 3.2 fF (again. no typo ). It is apparent tha t crystals offe r significant ad vantage s over lu mped LC realiz ation s. where such element values are unattainable for all practical purposes. Above a bout 2{)- 30 M Hz. the requ ired slab thickness becomes impractical ly small. For exa mple. a 100-M Hz fundamental mod e cry stal would be only a bout 17-Jl m thick . However. crystals of reasonable thickness can still be used if higher vibrationa! modes are used . The bo undary condition s are such thai only odd overtones arc allowed. Beca use of a vari ety of eff ects, the ove rto nes are nOI exac tly integ er mu ltiples of the fundamental (bu t they' re close. off by 0 .1% or so in the high direclion). Th ird and fiflh overtone crys tals arc fairly co mmon. bUI seventh or even ninth overtone oscillators are occasionally encountered . However, as the ove rto ne orde r increases. so doe s the d ifficulty of guaranteeing oscillation on only the desired mode. As another extremely crude rule of thumb. the effective series resistance grows as the square of the overton e mod e. Hence.

«, ~

5

X

8

10

2

10 N .

(29 )

where 10 is here interpreted as the frequ ency of the N ih ove rtone. Becau se the ove rtones are not at exact integer mu ltiple s o f the fundamental mode. the crystal must be cui to the co rrec t frequ ency at the desired ove rtone. Well-cuI overtone crys tals JXlssess Q values similar to those of fundamental -mod e crys tals . Quart z crystal fabri cation tec hnology is an ex tremely advanced art . Crys tals with resonant freq uen cies guaranteed within 50 ppm are routinely avai lable. and substantially better performance ca n he obtained . although at higher cos t. The ge neral chemical inertness of quart z guarantees excellent sta bility over tim e. and a judicious choice of cui in conj unctio n with passive or active temperature compensation and/or co ntrol can lead to tem perature coe fficients of well under I ppmr C. For these reaso ns. quart z oscillators are nearly ubiquitous in communications equipment and instrumentation (not to mention the lowly wr istwatch . where a one-mi nute drift in a month corres po nds 10 an em u of on ly about 20 ppm ).

Surface Acoustic Wove ISAWI Devices Because quartz crys tals operate in bulk vibrational mod es. high-frequency operation requires exceedingly thin slabs . A I·G Hz fundamenta l-mode quartz crys tal wou ld have a thick ness of only a bout 1.7 Ji m. for example. Aside from obvious fabrication difficulties. thin slabs break easily if the electric al exci tation is 100 great. Because of

so,

CHA PTE R 16

OSCI LLATO RS AN D SYN TH ESIZ ERS

thei r hig h Q. it is easy 10 deve lop large a mplitude vibrations wit h ve ry modest e lectrica l d rive. Eve n befor e ou trig ht fractu re occ urs, the e xtre me bending resul ts in a host of ge ne rally undesired nonlinear behavior. O ne way to evade these limitat ions is 10 e m ploy surface. ra ther tha n bulk. acoustic waves. If the mate ria l suppo rts suc h surface modes then the eff ec tive thickness ca n be much sma ller tha n the physica1thickness. a llow ing hig h reso na nt freq uenci es 10 be o bta ined with crysta ls of prac tica l dime nsions. Lithium niobate ( LiN bO .\) is a pie zoel ectric mute rial thut suppo rts surface acoustic wave s with lill ie loss. and has bee n used ex te nsively to make reso nators a nd filters at freq uencies practically unt ouchab le by qu art z. Control o f frequ enc y to qua rtz-crysta l acc uracy is not ye t ob tainable at low cost. unfortunately. but performance is adequate to sa tisfy such high -volumc.Iow-cost applicati on s as au tomati c gurage -door c peners . wh ic h typically work around 250-300 MU z. as well as fron t-end f ilt ers for cellular telephones. Sadly. neither quart z nor lithi um niobate is compatible with ord inary Ie fab rication processes. Also disappointi ng is the lac k of a ny pie zoel ectric activity in silicon. Hence, no inhe ren tly hig:h-Q resona tor ca n he made with laye rs normally fo und in ICs.

16.5

A CATA LO G OF TUN ED OSCillATORS

Th e re seems to be no limit to the number of ways to combi ne a reso na tor with a tra nsistor or two 10 make an oscill ator. as will become evident shortly. In the examples that fo llow. only the mo st minim al ex planations arc usually pro vided . perhaps lea ving the render in do ubt as to wh ich topo logy is "best." It is generally true that , w ith sufficient diligen ce and care, j ust about any of these topo logies ca n he made to perform wel l enough for a given applica tio n. When we cons ide r the issue of pha se noise in the ne xt chapter. more rational se lec tion cr iteria wi ll become evident.

16 . 5 .1

BA SIC

ic FEE DBACK OSC ILLATO RS

T he basic ingred ient s in these oscillato rs arc simple: one tran sistor plus a reso nator. Many of the oscillators are nam ed after the fellows who first ca rne up w ith the topolog ies hut . as wc 'lI sec , a more or less unified desc riptio n of these designs is possible. Ignoring biasin g detai ls. the basic topologies arc as follow s. As menti oned in the earlier exam ple. a ca pacitive voltage di vide r off of the ta nk provides feedb ac k to an ampli fie r in a Co lpitts osci llato r ( Figure 1().19 ). Notice that the feedback is positive; you should ske tc h the roo t loc us 10 convince yourse lf that this thin g ca n osci llate. Th e locus will sugges t why positive feed bac k to polo gies are ge nerally favored wit h bandpass structures . In alternative ve rs ions of the Co lpitts. the feedback is from so urce back to the gale ruther than fm m drain 10 source. Th at is. the tran sisto r may be con nected eithe r as

16 . 5 A CATALOG O F TUN ED O SCILLATO RS

505

FIGURE 16 .19. CoIpith oscillolor lbiosing not shownl.

lr

c--

,;. FIG URE 16 . 20 . Hortley oscillalor (b iasing slill not shown) .

a source followe r or as a co mmon-source amplifier. Either way, there is net po sitive feedback, The Hartley oscillator ( Figure 16.20) is essentially identical to the Co lpitts, hut uses a tapped inductor for feedbac k instead of a tapped capaci tor. The Hartley osci llator has its origins in the very early days of radio. when tapped inductors were readily nvailable . It is much less common today. One could also usc a tapped resistor. in principle, hut that particular configuration doesn't seem to have a name attached to it. The Clapp oscillator (Figure 16.2 1) is a modified Colpitts oscillator. with a series LC replacing the lone inductor, The Clapp oscillator is actually ju st a Colpills oscillator with an addi tional tap on the capacitive divider chain. as is evident in the re -drawn schematic of Figure 16.22. The extra tap allows the voltage swing across the inductor (and d ivider chain) to exceed considerably that of either the drain or source, and therefore to exceed the supply und even device breakdown voltages. The larger signal energy helps ove rcome the effect o f various noise processes (specifically, phase noise, as discussed in Chapter 17) and so improves spectral purity. Of these topologies, the Colpit ts is almost certainly the most commonly encountered. lis requirement of tapped capacitors is co mpatible with Ie implementations, although the inductor is generally not (of course - you can' t have everything). One

CHAPTE R 16 OSCILLATORS AND SYNTHES IZERS

'=r='

,7

~

i

I:!=

FI GURE 16. 2 1. Clopposcil lctor (bia sing still not shown ).

=c=' L-

"

~

7

FIGURE 16.22 . Re ·drawn Clapp oscilla tor.

ot he r import a nt rea so n for the popularit y of the Colpitis co nfig uration is th ai it is capeble of excellent phase noi se performan ce, as we ' ll sec . Ano ther oscillator idiom actu ally ow es its ex iste nce 10 the instabi lity o f so me tu ned amplifiers. Recull that it is possible for a co mmo n-sou rce amplifi er to have a negalive input ad m ittance if it operates w ith a tuned load below the reson ant frequ e ncy of the loud (so that it looks inductivc j.U Thi s negative resistance ca n be used to overco me the loss in another resonant ci rcuit to produce oscillu tlons: sec Fig ure 16.23. The T ITO oscillator uses a M iller-effect coupling ca paci tor . In man y designs (particularly at very high frequ encies), an explicit coupling capaci tor is unnecessary; the device's inherent feedback capacitance is sufficient to pro vide the desired negative res istance. T his obse rvation underscores the difficulty of using tuned ci rcuits in bo th the input and output ci rcuits o f nonunilatcral amplifiers at high frequencies. Because of its pai r of tuned circui ts, the T IT O osci llator is theoreticall y ca pable of prod uci ng sig nals with good spec tral purity. However. its need of two ind uctors makes this topo logy unattractive for Ie implem entati on . An add itio nal strike against it is the need for ca reful tun ing of two reson ators if prope r ope ration is to be obtained.

lJ

Sali!>fying Ihis condiram is not sufficient. however .

16.5

A CATALOG O F TUNED OSCILLATORS

507

FIG URE 16.23. Tuned input-tvned output (TIlO) oscillator {bios details incomplete}.

FIGURE 16. 2 4 . Colpitts cry5ta loscillator.

16.5 . 2

C RYSTA L OSC I LLA TO R PO TPOU RRI

Many crystal oscillators arc recognizab ly derived from l.C co unterparts. In Figure 16.24. for example. the crystal is used in its series resonant mode (where it appears as a low resistance) 10 close the feedback loop only at the desired frequency. The inductance across the crystal is frequently (b ut not always) needed in practical designs to prevent unwanted off-frequency o scillation s due to feedbac k provided by the crystal's parallel capacitance (Co). The inductance resonates out this capacitor, so that on ly the series Rl. C arm of the crystal controls the feedback . A variation on this theme is shown in Figure 16.25. In this particulurcon figuration. the capacitive divider off the tan k provides the feedb ack as in a classic LC Co lpitts. However . the crystal grounds the gate only at the series resonant frequency of the crys tal, permitting the loop to have suffi cie nt gain to sustain osc illations at that frequency only. Th is topo logy is useful if one term inal of the crystal must be grou nded .

508

CHAPTE R 16 O SCILLATORS AND SYN THES IZERS

f i GURE 16.2 5.

Modified

Colpilt$ cl)"$,kJl oscilla tor.

FIGURE 16. 26 . Pierce c~tol

oscillator.

Yet anot her topo logy is the Pierce oscilla tor, sketched in Figu re 16 .26 . In this oscillato r. assume that the ca pacitors mod el tran sistor und stra y parasitics. so thai the tran sistor itself is idea l. G ive n this assu mpt ion . the on ly wa y 10 satisfy the zeropha se margin criterion is for the oscillat ion frequency to occu r a bit abov e the series resonance of the crystal. Thai is, the crystal mu st look inductive ut thc oscillation frequency. Thi s property co nfers the advantage thai no external inductance is therefore required for this oscillator to function (the RF cho ke may he repl aced by a large valued resistor or current source). Hence, it is more amenable to integra tion than a Co lpitis, for example, particu larly at low frequen cies. T hat the crystal mu st loo k ind ucti ve can be argued as foll ows. If we art: to have a phase margin of zero . and the transistor 's tran sco nducta nce already pro vides a 180" phase shift. then the passive elements mu st supply the ot her 180" . T here' s no way for a two -po le Ht: ne twork to pro vide l ~ rF (close . hur c!o" e doe >; n', cHunll. "0 Ihe crystal mu st look inductive. Because the output frequ ency of a Pierce oscillator therefore docs not coincide with the series resonance of the cry stal, o ne must use a cry stal tha t has been cur to oscilla te at the desired freq uency with CI specified load capacitance ( in this case , the value of the two capac itors in series).

16 .6 N EGA TIVE RESISTA NCE OSC ILLATORS

509

- KJ,

K/s

FIG URE 16 . 2 7. Ouodroture O$Cillator bleek diogmm

As a final note on Ihe Pie rce oscillator. ir happens 10 form the ba sis of many "digi tal" osc illa tors . A n ord inary Cf\.10S invert er. for example. ca n act as the gain clement if biased to its linear region (e.g.. with a large feedback resistor). Just add the appro pria rc amount of input and out put capacitance. toss in the cry stal from input to output. and chances are very good that yo u' ll have an oscillator. Gener ally one or two stages of buffering (with mo re inverters. of course) are necessary In obtai n fu ll C MOS swi ng s and also 10 isolate load cha nge s from the oscillator core.

16. 5 .3

O THER OSC ILLATOR CO N FIG URATIO NS

In so me a pplicatio ns. it is desirable 10 ha ve two outputs in quadrature. On e oscillalor architec ture that naturally provides quadrature outputs (atleast in principle) uses a pair of integrat ors in a feedback loo p (see Figure 16.27). From the magni tude co ndition. we can dedu ce thai the frequency of oscillation is W.>M:

= K.

( 3U)

Thus . luning may be effec ted by varying the integrator ga in. Further more. the desired quadrature relationshi p is obtained across any of the integra tors. In pra ctice. unmodclcd dynamics cause a departu re fro m idea l be havior. Consider. for ex ample. the effect of additi on al po les on the root locu s. Rathe r than co nsisting of a purel y ima ginary pa ir. the loc us w ith additiona l po les brea ks awa y from the imaginary nxis . Furt hermore. these unmod eled parasitic» tend 10 he rathe r un reliable. so that allowi ng the osci llation frequency 10 de pend on the m is undesirable. Despit e these obstacles. howeve r. I-OBz qu ad ratu re oscillato rs with reaso nable q uad ratu re phase (erro r under 0.5") have been reported.'!

16 .6

NEGATI V E RE SI STAN CE O SCILLATORS

A perfect ly lossle ss resonant ci rcuit is very near ly an oscillator. burlossle ss elc mc ms arc difficult to realize . Overcoming Ihe e nergy loss implied by lhe Ii nile Q uf practic al

101 R. Dunca n el al., "A I G il l (Ju.adralurcSinusoidalOsc' illalllf:· I£ £ £ Cle e Oigl'.l I . IW5. pp. 91-4.

F 5 10

CHAPTER. 16 OSCill ATORS AND SYNTHES IZERS

z, Z ,n ----(A - DR

" FIGURE 16.2 8 . Generalized impedonce converter.

resonat ors with the energy-supplying action of active elements is one poten tially attractive way to build practical oscillators. as in the T IT O ex ample. The foregoing description is quite general. and covers both feedback and openloop topologies. Among the former is a classic textbook circu it. the negative impeda nce co nverter ( NIC ). The NIC ca n be realized with a simple op -amp circuit that em ploys bot h positive and negati ve feedback . Speci fica lly. co nsider the co nfiguration of Figure 16 .28. If ideal up-amp be havior is assumed . it is ea sy 10 show that the input impedance is related 10 the feedback impedance as

z., =

21

I - A"

(3 1)

If the closed -loo p gain A is set equal to prec isely 2. then the inp ut impedance will be the algebra ic inverse o f the feedba ck impedan ce. If the feedb ack impedance is in tum chose n to be a pure po sitive resistance. then the input impedance will be a purely negat ive resistance. This negative resistance may be used to offse t the positive resistance o f all pra ctical resonator s to produce an oscillator , As usual, the inherent nonlin earitics of all rea l active devices provide amplitude limi ting. and describ ing functions can be used to estimate the oscillation amplitude. if desired . Describin g functions may also be used to verify that the oscillator will, in

fact. oscillate. As a specific exa mple. consider the oscillator shown in Figur e 16.29. In order to guarantee oscillation. we require the net resista nce across the tank to be negative. Th us. we must eotis fy the fo llowing inequality : (32)

The nonlinea rity tha t most typi cally limit s the amplitude at low frequencies is the finite o utput swing o f all rea l amplifiers. Since there is a gain o f2 from the tank to the

16 . 6 N EGATIV E RES ISTANCE OSCILLATORS

511

R,

R, R R

FIGURE 16 .29. Negative resistance o!>Cillator.

T

FIGU RE 16 . 30 . Canonical RF negotive resistance {biasing

not r-hownl.

op-amp output. the signal across the lank will generally limit to a value somewhat greater than half the supply. corresponding to period ic saturation of the amplifier output, AI higher frequencies. it is possible for the finite stew rate o f rbc amplifier to control the amplitude ( partially, if nor tot ally). In general, this situation is undesirable because the phase lag associ ated with slew limit ing can cause a shirt in osci llation frequency. In extreme cases, the amplitude control provided by slew limiting (or almost lilly other kind of amplitude limit ing) can he unstable. and squcgging can occur, Finally. the various oscillator configurations presented earlier (e.g ., Colpitis, Pierce, etc.) may them sel ves be viewed as negative resistance oscillators. A more practical negative resistance is eas ily obtained by exploiting yet another "parasitic" effect: Inductance in the gate circuit of a common-gate device can cause a negative resistance to appear at the so urce terminal: sec Figure 16.30. A straightforward analysis reveals that , if C ( d is neglected. then Z in has a negative real part for freq uencies greater than the resonant frequency of the inductor and of C(~ . For frequencies much larger than that resonant frequency hut much smaller than W T , the real part o f Z ,n is app roximately

r '"

CH A PTER 16

O SCI LLATO RS A N D SYN THESIZE RS

L

f iG URE 16 . 31. Simple differentiol negative resiskJoce osci llator.

R

ln :::::

w1L -

-

-

wr

W

= - -

wT

IZLI·

(33)

The ease with which this circuit provides a negative resis tance accounts for its populari ty. However. it should be obvious thai this case also underscore s the impo rtance of minim izing parasitic ga te inductance when it negative resistance is 1I0 t desired. A circuit that has become a freq uently rec urri ng idio m in recen t years uses a crossco upled di fferen tial pair to synthesi ze the negat ive resistance. " II is len as an exercise for the reader " to analyze this circuit. which is pictured in Figure 16 .3 1. As will he shown in the next chapter, spectral pu rity improves if the signal amplitudes arc maximized (be cau se this increases the signal- to- noise ratio) . In man y oscil lator s. such as the ci rcuit of Figure 16.3 1. the allowable signal amplitudes are constrained by the available supply voltage or breakdown voltage conside rations. Because it is the energy in the tank that co mprises the "s ignal," one co uld take a cue from the C lapp oscillator and employ tapped resonators ttl allow peak tan k voltages that exceed the devic e breakdown limits or supply volta ge, as in the negative resistance oscillat or I ~ of Figure 16.32. 1h The differential connec tion might make it a bit difficult to sec that this ci rcuit indeed employs a tapped reson ato r, so consider a simplified half-ci rcuit (Pi gu re 16 .33). lnt his simplified half-circu it. the tran sistors arc replaced by a negative resistor, and

13 J. Crauinc kx and M . Steyacrt . "A C MOS 1.8Ci1 b Low-Ph ase-Noise Vo ltage ·Colllro lloo Oscutetor with Prescele r," ISSCC Digol of Technical Pupa s. February 199 .'1, pp. 2M- 7. "The mductc rs are bondw ires surched across the die . It> II is bestto han ' a tail curren t sou rce 10 con strain the swing. 001 we omit this de lail in Ihe inte rest simplici ty.

or

16 .6 N EGATIVE RES ISTAN CE OSCi ll ATORS

513

Ypo

R

R

c,I

0000 L2

I

C2

LI

FIGURE 16 . 3 2 . Negative resi~lance

osciJla1cM" with

mod; ~ ed lank (~j mpfi~ed) .

-R

')( C1 . C 21 - C I + C2

FIG URE 16 . 3 3 . Simplj ~ed

half·circuit of nega tive resistcnce oscillalof.

the positive resistors are not shown at all. Furthermore. the twn ca pacitors are rcplaced by their series equivalent, while the j unction oft he two inductors corresponds to the drain conne r.-tion of the original circuit. It should he clear that the swing across the equivalent capacitance (or across 1.2) can eXI.-eed the supply vo ltage (and even the transistor breakdown voltage) because of the tapped configuration, so that this oscillator is the philosoph ical cousin of the ChlPP confi gUnition. Useful out put may be obtained either through a buffer interposed between the oscillato r core and load, or through a capacitive voltage divider to avoid spoiling resona tor Q. As a consequence of the large energy stored in the tank with either a single- or double-tapped resonator, this topology is capable of exce llent phase noise performance, as will he appreciated after the next chapter. Tuning of this (and all tither L C ) oscillators may he accomplished by realizing all or part of C J or C 2 as a variable capacitor (such as the junction capacitor formed with a p+ diffusion in an n-well j. and tuning its effective ca pacitance with an uppropri ure bias control voltage. Since C MOS junction capacitors have relative ly poor Q. it is advisable 10 use only as much j unction ca pacitance as nec essary tn achieve

,.. 514

CHAPTER 16 OSCILLATORS A ND SYNTHESIZERS

~ R

I nMO

"

Vet rl

L2

CI

R

" C2

O?~O 11

l

1r =

~

2S

1l

7 f iGURE 16 . 3 4 . Voltoge ' cOl1trolleel negative resista nce o K illotor {limplifieell ,

the desired lu ning range . In practice. tuning ranges arc frequen tly limited to below 5-10% if excess ive degradation of phase noise is to be avoided . A simple. but il lustrative, example of a voltage -controlled oscillator using this method is shown in Figure 16.34 . As a final comment on negative resistan ce osci llators, it should he clear that many ( if not all) oscillators may he co nsidered as negati ve resista nce oscillators since. from the point of view of the tank . the active clemen ts cancel the loss due 10 finite Q {If the resonators. Hence. whether ( 0 ca ll an oscillator a " negative resistance" type is actu all y more a philosophical dec ision than anything fundamental.

16.7

FREQUEN CY SYNTHESIS

Oscillators built with high - Q crystals exhibit the best spec tral purity. but can' t be tuned ove r a range of more than se veral hundred part s per mi llion or so . Since mo st trun scel vers mu st operate at a number of different freq uencies that span a considerably larger range than that, one simple way to accommodate this lack of tuning capability is to use a se parate resonator for each freq uen cy, Clearl y. this straightforward approach is pract ical only if the number of required frequenci es is small. Instead , virtually all modern gear uses so me form of freq uency synthes is. in whic h a single quartz-con troll ed oscillator is co mbined with a PL L and some digit al elemen ts to provide a multitude of ou tput frequenci es that are traceable to that highly ~t:l "" ';> ~fercnee .

In the idea l ",a "e , t he n, o n", ",a n o b ta in a w iJo; " I'",rati ,,/; r~u", "",)'

range anti high stability from line o scillator. Before underta king a detailed investigati on of vario us synthesizers. however. we need to d igress briefly to exa mine an issue that strongly influences architectural choices. A freq uency divid er is used in all of the synthes izers we shall study. and it is im portant 10 model properly its effect on I(XIP stabili ty,

16 .7 FREQUENCY SYNTHES IS

515

T IT ··· FIGUR E 16.35. Somple-ond-~action.

16. 7.1 DIVIDER " DEl AY" Occasionally, one encounters the term "divider delay" in the literature o n PLL synrhesizers. in the co ntext o f eva luati ng loop smhility, We 'll Si..'C mom entaril y that the phen omenon is somewhat inaccurately named , hUI there is indeed a stability issue associ ated with the presence of dividers in the loo p transmi ssion. The use of a frequency d ivider ge nerally impli es that the phase detector i.... d igital in nat ure." As a conseq uence. knowledge abo ut phase error is availab le to the loo p on ly at disc rete instants. Th at is. the loo p is a sampled -data system, If a di vider is prese nt. the loo p samples the phase error less frequen tly than migh t be implied by the veo frequency. To model the PLL correc tly. then, we need to account properly for this sampled nature. In orde r to develop the necessa ry insight, consider a process in which a contin uous time function is sampled and held period ically; see Figure 16.35. The sam ple-andho ld (S/ H ) operation shown introd uces a ph ase lag into the proc ess. Mathem atic s need not he invoked to concl ude that this is so; simply "eye ball" the sa mpled-andheld waveform and co nside r its tim e relationship with the ori ginal co ntinuous-lime waveform . You should be able to convince yourself tha i the best fit occurs when yo u slide the original waveform to the righ t hy abo ut half of a sample time. More for mally, the " hold" part of the S/ H opera tion can he model ed by an element whose impulse respon se is a rectangle of unit area and T-sccond du ration, as seen in Figure 16.36. This element . kno wn formally as a zero -order hold (Z O Hl . has a transfe r function given by 1M

17 Although there are excep nons te.g., sc bharmonic mjection-Jocked no;cilJalun;). we will limit 1m' present di S("u ~s ion to the more com mon implementations. tK If you Il.uuld IiLe a 4ukk derivauon of the transfer function. nore that the impulse response of the zero-order hold is lhe same as that of the difference nf t .....o integrations tone del ayed in time). Tha t should tIC' e nough of a hint.

516

CHApt ER 16 O SCIll ATORS AND SYNT HESIZE RS

hll)

T

FIGURE 16 . 3 6 . Impulse respcose of zer-o -order hold.

'--=-I _

1/( s) ~

(' - s f

sT

Th e ma g nitu de of the tra nsfer func tion i... . sin C1J (T/ 2) II/i) w) 1 = w( T/ 2) .

(35)

LI I/( j w ») = - w ( T/ 2).

(3 6)

while the phase is simply

Note thai the phase is linearly proportional 10 frequ ency. and is the same as thai of a pure lime del ay of 1'/ 2 seconds. It is from thi s ph ase behav ior that the term "divide r delay" deri ves. Ho we ve r. since the magnitude term is nor conslant with freq ue ncy. the term "d e lay" is nor exactly correct.!" Now let 's apply this informat io n 10 the spec ific exa mple of a PLL with a frequency divider in the loop trans mission. From the expressio n for phase shift. we can see the de le ter ious effec t on loo p sta biluy that di vider s can int rod uce. A s the divi de mod ulus increases , the sampling pe riod T incre ases (assu min g a fixed veo output freq uency). TIIC lidded negative ph ase shift thus becomes increa singly worse , deg rading phase margin . A s a co nseq ue nce , loop c rosso ver mu st he forced to a freq uency tha t is low co mpa red w ith I I T in o rder to avoid these effects . Since the sa mpling rate is de termined by the o utput of the dividers a nd he nce of the freq uency at which phase co m parisons are made, rath er than th e o utput of the Yeo. a hig h div isilln factor ca n result in a se vere constrain t o n loop band width . with all of the atte nda nt negative implications for settli ng speed and noi se pe rfo rm am..e . It is therefore co m mo n 10 c hoose loop c rosso ver freque ncies that are about a ten th of the phase comparison rate.

19 The magnitude is dust:' tu unity. however. for w T < 1.

16 , 7

..

5 17

FREQ UEN CY SYN TH ESIS

+N

,"

I' LL

'-----

+M

... -

FIGURE 16.37. Clonic PLL frequency ,ynthesizer .

16. 7. 2 SYN THES IZERS W ITH STATIC MO DULI Having developed an understanding o f the constraints imposed by the presence of a frequency divider in the loop transmission, we now tum to an exuminution (If various synthesizer topologies. The simplest PLL frequency synthesizer uses one reference oscillator and two freq uency d ividers, a" shown in Figure 16.37. The loop forces the veo to a frequency that makes the inputs to the PLL equal in frequency. li enee. we may write f re f

f OUl

/i- M"'

(37)

so that

/.>Ul =

M N . f'ef'

(38 )

Thus. by varying the divide mod uli M and N, any rational multiple of the input reference frequency can he generated. The long-term stability of the output u hat is, the average frequency) is every hit as good as that of the refe rence. hut stability in the shortest term ( phase noise) depends on the net divide mod ulus as well as on the propert ies o f the PLL's veo and loop dynamic s.j" Note thatthe output frequency can be incremented in steps of f rd I N, and that this frequency represents the rate at which phase detection is performed in the PLL, Stability considerations as well as the need to suppress control- voltage ripple force the usc of loop bandwidt hs that are small comp ared with IrefIN. To obtain the maximum benefi t of the ( presumed) low noise reference, however. we would like the PLL to track that low noise reference over as wide a bandwidth as possible. Additionally, a

20 Within the I'LL's !Ullp bandwidth, the ou tput phase noise will be AIIN ume s that (If the reference o-cill.nor. since a phase division necessarily acco mpanies a frequency d ivisio n. O Ulsidc (If the I'LL bandwidth, fl-edhack is inerfecnve. and the OUlput phase noise will mcrcfore he thai of the PLL's own veo.

~

I

518

CHAPTER 16 OSCI ll ATORS AND SYNTHESIZE RS

I'LL

L FIGURE 16.38 .

+M

,-

+P

r---.

f'ltJl

Modified PU frequency synthesizer .

hig h loo p band width w ill speed settling after a cha nge in modulu s. Th ese conflicting req uire ment s have led to the developme nt o f a ltern at ive a rchitec tures. On e si mple modi fica tio n that is occ asio nally used is dep icted in Figu re 16.38. For

this synthesizer. we may write

M f""" = N P . f re f.

(39 )

The minimum output frequency increment is evidently Jeer/NI' , but the loop compares ph ases at f ret/N , o r P tim es as fast ;IS the pre vious archit ec ture. Th is modifi cation therefore imp ro ves the loo p band wid th co nstraint by a facto r of P , at the cos t of requ iring the PLL 10 osci llate P limes faster a nd the + M counter to ru n tha t much faster as well. Yet another mod ification is the irueger-w synthesize r of Figure 16.39. In this widely used synthesizer. the divider logic consists of two counters and a dual-modulus prescalcr (divide r). One counter, called the channel-spaci ng (or "sw allow") counter is made programmable to enable channel selection. The other counter, which we' ll call the frame counter (also known as the program counter). is usually fixed and determines the total number o f prescaler cycle s thai co mprise the following operation: The prescaler initially divides by N + I until the channel-spaci ng co unter overflows. then divides by N unti l the frame counter o verflows: the prescaler modulus is reset to N + I. and the cycle repeats. If S is the maximum value of the ehanncl-...pacing counter and F is the maximum value of the frame counter. then the presculer d ivides the veo output by N + I for S cycles. and by F - S for N cycles, before repeating. The effective overall divide mod ulus M is therefore M

~

(N

+ 1)5 + If' -

5 )N = N F

+ 5,

(4Q)

16.7 FREOUENCY SYN THESIS

..

-

519

PI-I.

Divider Lo gie

-

fiGU RE 16. 39. Integer-N freq uency synthesizer.

f. r

---i

PLL

+

NIN+I

t

Dither comrol

f iGURE 16.40 . Block diagram for frequency synthesizer with dithered modulus.

The output frequency increment is thus equal to the reference fn.c. quency, This architecture is the most popular way of implementing the basic block diagram of Figure 16.37. and gets its name from the fact that the output frequency is an integer multiple of the refere nce frequency.

16. 7. 3 SYNTHESIZERS W ITH DITHERING M ODU LI In the synthesizers studied so far. the desired channel spacing directly constrains the loop bandwidth. An approach that eases this problem is to dither betwee n two divide moduli to generate channel spacings that are smaller than the reference frequency; this is shown in Figure 16.40. As an illustrat ion of the basic idea . consider that dividing alternately by. say. 4 and then 5 with a 50% dut y ratio is equivalent to dividing by 4.5 on average. Chang ing the percentage o f time spent on anyone mod ulus thus changes the effective (average) mod ulus. so that the synthesized output can he incremented by frequency steps smaller than the input reference frequency.

520

CHA PTE R 16 OSCI LL ATORS A ND SYN THESIZERS

Ther e a re many strategies for switching be tween two moduli thai yield the sa me average modulus. of cou rse . and not all of them are equally desirable since the inS1(l tlltllU'OIH freq uen cy is also of import ance. Th e most com mo n stra tegy is used by the fructional- Y synthesizer, in which o ne d ivide s the y eO o utput by (Inc mod ulus (ca ll it N + I ) e very K y e O cy cles. a nd by the ot her (N) for the rest of the time . T he average d ivide factor is thus

so that

N,. =(N + llG )+N( I - ~) =N + ~ . J...

=

N,.1., ~ (N+ ~ )1.,.

(4 1)

(4 21

We sec that the resolut ion is de term ined hy K , so thai the minimu m frequ ency incre me nt can he much .1"/1111ller tha n the refe rence frequ ency. However. unlike the other synthesizers studied so far. the phase detector ope rates with inpu ts whose frequ ency is milch higher than the mini mum increment (i n fact . the pha se det ector is driven with sig nal:..of frequ ency f rrf) . thus providi ng a muc h-desired dec ou plin g betwee n synthesize r freq ue nc y resolution and I'LL sa mpling frequency. To ill ustrate the opera tion of this a rchitec ture in greate r det ai l. conside r the problem of ge nerating a freque ncy of 27 . 135 M ll z with a reference input o f 100 klt z. The integral modulus N therefore equals 271. while the fractio nal pan ( 1/ K ) equals 0 .35. Thus. we wish to d ivide by 272 ( = N + I ) for 35 out of every 100 veo cyc les (fo r example j. nnd by 27 1 (= N) for the othe r 65 cyc les. Of the many possible stra tegies fur impleme nting this des ired beh avior. the most commo n ( hut not nec essaril y optimum) one is to increm ent an acc umulator by the fractional pan o f the modulus ( here. 0 .35) every cycle . Each lime the accumulator over flows ( here defined as equalling or exceeding uni ty), the divi de mod ulus is set to N +1. Th e resid ue after overtlow is preserved . and the loup co ntinues to operate as befurl'. It should be appa rent that the reso lutio n is set by the size of the acc umulator. a nd is equal to the refe rence frequ e ncy divided by the to ta l ucc um uhuo r Sill'. In our exam ple with a l OO-kllz refe rence, a 5-d igit BCD (hinary-cod ed decimal ) accumu la tor wou ld allow us to sy nthes ize output ste ps as small as I li z. Th ere is o ne other property of fracuonal- w sy nthe sizers that needs to be men tio ned . Because the loo p operates by periodically switc hing bet wee n two divide mod uli. there is necessar ily a periodic modulatio n of the control voltage ami hence of the veo out put freq uency. Th e refore. even tho ugh the ou tput frequ e ncy j " correct on average. it ma y not be on a n instantaneous basis . and the output spec trum therefor e conta ins sidebands . Furtherm ore. the size a nd loc ati on of the sidebands depe nd on the partic ular modul i , IS well as on loop paramete rs. In practical loo ps of this kind , co mpensation fo r thi." modulation is usual ly necessal)'. Th is co mpensa tion is ena bled by the fuct that the mod ulation is de rerministic -.

16 . 1 FR EQU ENCY SYNTHESIS

FIGURE 16 .4 1.

521

Ofhet synthesizer loop.

we know in advance.' w hat the control-line ripp le will he . Hence. a co mpensa ting control ..o hage variation may he injected to offset the undesired mod ulation. In pract ice. this techniqu e {so metime s called API. for analog p hase interpolations is capable of providing between 20 dB and ~o dB suppression of the sidebands. Achieving the highe r levels of suppres sion (a nd beyond) req uires intimate know ledge of the co ntrul cbaracteri sucs of the Yeo. incl udin g temperature and supply voltage effects, so det ails vary considera bly from design 10 design.?' A n altern ative to this type of cancella tion is to eliminate the periodi c co ntrol ..u llage ripple altoge ther by employi ng a more sophisticated st rategy for switching betwee n the I W O mod uli. Fo r example. one might rando mize this switching 10 decrease the amplitudes of spurious spectral co mponents at the expense of increasing the noise floo r. A powerful impro vement on thut strategy is to use delta -sigma techniques 10 distribu te the noise IIommiform/y.21 If the spectrum is shaped to push the noise nul to frequencies far fro m the carrier. subsequent filtering can read ily remove the no ise. The loop itself takes cafe of noise near the carrie r. sn the overall ou tput can possess exce ptional spectral purity.

16 . 7. 4 CO M BINATIO N SYN THESIZERS Another a pproac h is 10 co mbine the outputs of two or more synthesi zers. The addition al degree of freedom thus pro vided can ease some of the performance tradeoff's. but at the expe nse o f increased complexity and power consum ptio n. The most common expression of this idea is to mi x the output of a fixed frequ ency source wit h that of a variable one. T he offset synthesize r ( Figure 16.4 1) is one architecture that implements that parti cular cho ice .

l J See. for example. V. Mmnasscwnsch, Frt''1III'n L")" Syll/he .~i=er.~. 3nJ ed .• Wiley. New Ynrl.. 11)l!7. n The da.\Sic paper on Ihis architecture is by T. Riley et al.• "Sigma-Ue lla Mod ulanon in Pracuon alN Frequency Sy nthesis ... IEEE J. Solid -StUll' Ci'n~jl$, v. 28. May 1993. pp. 553- 1). The tenll'i. "delta-sigm a" and "sig ma-de lta" are frequently used imerchangeahly, 001 the former nomenclalure " as used by the inventors (If the concept.

CHAPTER 16 OSC ILLATO RS A ND SYN THESIZ ER S

With this architecture. the loop does not servo to an equality of output and reference frequencies. because the add itional intermediate mixing offsets the equilibrium point. Without an intermedia te mixing. note that the balance point would corre spo nd to a zero frequency output from the low-pass filter that fo llows the (first and only) mixer. In the o ffse tloop. then, the balance point corresponds 10 a zero frequency output from the final low-pass filter . Anned with this observation, it is a straightforward matter to dete rmine the relationship betwee n fOOl and frdThe low-pass filters selectively eliminate the sum frequency components arising from the mixing ope rations. Hence. we may write fl =

fOOl -

f~t .

(4 3)

h = Iv -: foff~ = f fJIJ' - f~f - f"ff~ '

(44)

Selling [z equa l 10 zero and solving for the output freque ncy yield... (45)

Thus. the output frequency is the sum of the two input frequencies. An important advantage of this approac h is that the output frequency is not a multiplied vers ion of a reference. Hence. the phase noise similarly undergoes no multiplication. making it substantially eas ier to produce a low-phase noise output signal. A rela ted result is that any phase or frequency mod ulation on either of the Iwo input signals is directly transferred to the output without scaling by a multipl icative factor. As a consequence of these attributes. the offset synthesizer has found wide use in transmitt ers for F M / PM systems. There are other techniques for combining two frequencies to produ ce a third. For exam ple. one might usc two complete PLLs and combine the outputs with a mixer. To select out the sum rather than the difference (or vice versa). one would conventionally usc a filter. One may also ease the fi lter 's burden by using a single-sideband mixer (also known as a complex mixer) to reduce the magnitud e of the undesired compouent.U However. such loops are rarely used in IC imp lementations becau se of the difficu lty of preventing two PLLs fro m interacting with eac h other. A common prob lem is for the two loo ps to lock to each other (or to attem pt to) parasitically through s ubs tru tc

~ uu p l i ll g

o r incomplete rever...e b llla lilJlI tlu uugh

a lllpl i tk r ~

a nd

other circui try. T hese problems are sufficie ntly difficult to solve that such dual-loop synthes izers are rarel y used at present.

23 We !ihall explore more fully the uses o f SS B mixers in Chapter t8.

16.7

523

FRE QU ENCY SYN THESIS

-~

Ace 1--";..

[~

co~
DAC

-

•! «r... i

Lf

dcgh l

FIGURE 16.4 2. Direct digital frequency synthesize r.

16 .7. 5 DI RE CT DIG ITAL SYNTHESIS There arc so me applications that require the ability to cha nge frequencies at a relatively high rate. Examples. include frequency-hopped spread-s pectrum systems. in which the carrier frequency changes in a pseud orandom pauem ." Co nventional syr nbesizers may he hard-pressed to provide the fast settling req uired. so alternative means have been developed. The fastest-settling sy nthes izers are open-loop systems and thereby evade the constraints imposed by the stability considerations o f feed back systems (such as PLLs). One extremely agile type of sy nthes izer employs direct digital synthesis ( DDS). The basic block d iagra m o f such a synthesizer is shown in Figure 16.42. T his synthesizer consists of an acc umulator. a read-only memory (RO M) loo kup table (w ith integral output register). and a d igital-to -analog con verter ( DAC ). T he accumulato r accepts a frequency command signal fi ne as an input . and increments its output by this amount every clock cycle. The output therefore increases linearly until an overfl ow occ urs and the cycle repeats. The output ¢ thus follows a sawtoo th pattern . A useful insight is that phase is the integral of frequency. so the output of the accumulator is analogous to the integral of the frequency input command. The frequency of the resulting sawtooth pattern is then a function of the cloc k frequency. accumulato r word length. and input command. T he phase output of the accumulator then dri ves the address lines of a ROM cosine lookup table thai converts the digital phase values into digital amplitude val ues .2~ Pinally, a DAC converts those values into analog outputs. Generally. a filter follows the DAC to improve spectral purity 10 acceptable levels.

l~

This strategy is partin Jliirly useful in avo iding detection and jamming in military scenarios, fur which it was fir"il developed . because the m.utting spec trum luuls very much like white noise. 2S With II Iittte additionatlog.ic. one can easily reduce tbe amoum of ROM required by 751J.. since one quadrant's worth of values i~ readity reused 10 reccesuuc t an entire period.

524

CHAPTER 16 OSCILLATORS AND SYN THESIZERS

The frequency ca n be changed rap idly (w ith a latency of on ly a couple clock cycles ), and in a phase -conti nuo us man ner, simply by chang ing the value of /inc ' Furthcrmorc. modulation o f bo th frequen cy and phase are trivially obtained by addi ng the modulatio n directly in the digi tal domain to hoc or 4>. respectively. Finally. eve n amp litude mod ulat io n can be add ed by using a mult iplying DAC ( ~I DAC). in which the analog output is the produ ct of an ana log in put ( here . the amplitude modu lation) and the d igit al input from the ROM .26 The chief prob lem with this type of synthes ize r is thai the spectral purity is mar kedl y inferi or 10 thai of the Pl.Lbased appro ach es con sidered curlie r. The number of bus in the DAC set o ne hound on the spec tral purity ( l'f'')' loosc:1y speaking, the carrie r-to -spurious ratio is abo ut 6 dB per bit), while the num ber of RO M poi nts per cycle det ermi ne the loca tion o f the harmo nic co mponents (wi th ajudicious choice o f the " poin ts. the first significant harmonic can be made to occ ur at " - I times the fundamenta l), Since the clock will necessarily run much faster than the output frequency ultimately generated , these types of synthesizers prod uce signal s whose frequ encies arc a considerably sma ller fract io n of a given technology' s ultim ate speed than VCO / PLL -hased synthes izers. Freq uently, the output of a DDS is upcon verted through mi xing wi th the o utput of a Pl.Lbased synthesize r (or used as one of the inpu ts in an offset synthesizer) 10 effect a co mpromise betwee n the two .

16.8

SUMMARY

We have examined how the amplitude of oscillation can be sta bilized thro ugh nonlinear mean s. and extended feedb ack concepts to incl ude a particular type of lineurize d non linearity. A rmed with describin g functions and knowledge of the rest of the clements in a loo p transmission, both oscillation frequ ency and amp litude ca n he det erm ined , We looked at a variety of oscillators of both open -loop and feedb ack topologie s. The Col pitts and Han ley oscillators usc ta pped tank s to provide po sitive feedback , wherea s the T ITO oscillator employs the negat ive resistance that a tuned amplifier wit h Mil ler feedback can prov ide. Th e C lapp osci llator uses an ex tra tap to allow resonator swings that exceed the supply voltage. and so perm its signal energy to do minate noise. Crystal oscillator versio ns of L C osci llators were also described . Since a quartz crystal behav es much like an L C resonator wi th extraord inarily high Q , it permi ts the reali zation o f oscilla to rs wi th e xce lle nt spe ctra l puri ty a nd low power eon,;um p -

lion , The Colpitts co nfiguration osci llates at thc series resonant freq uency o f the

ll> O ne may also perfor m the amplitude mod ulation in the digital do main simpty by multiplyillg the ROM ou tput with a dig ital represe ntation o f the desired am plnudc mod ulunon before driving

the nAc.

PROBLEM SET

525

crys tal and thus requires an l.C lank . The Pierce oscillator operates at a frequency whe re the crystal loo ks inductive. and therefore req uires no ex ternal inductance, T he o ff-reso nant operatio n. however. forces the use of crys tals that have been cut specifica lly for a particu lar load capacitance. A random sampling of other oscilfutors was also provided. including a quadra ture oscillator using two integrators in a feedback loop, as well as several negative rcsisranee lJ~iIIa tlJrs , Again. tapped resonators were ..ee n to be benefi cia l for improving phase noi se, Fina lly, a number of frequ ency synthesi ze rs were examined. Stab ility co nsiderations force loop crossover frequencies well below the phase comparison frequency; phase noise conside ratio ns favor large loop bandwidths . Because the output frequency increment is tightly coupled to the phase co mparison frequency in simple architectures. it is d ifficult to synthesize frequenc ies with fine increments wh ile additionally co nferri ng tot he output the good phase noise o f the refer ence, The fractionalN synthesizer dcco uplcs the frequency increm ent from the phase comparison rate. allowing the usc of greater loop bandwidths. However. while phase noise is therefore improved, various ...purious components can be ge nerated ow ing to ripp le o n the co ntro l voltage . Su ppressio n o f these spurious tones is possib le eithe r by ca nce llation of the ripple (since it is deterministic in the case of the classical fractional-Y arc hitecture) or through the use of randomization or noise shaping of the spectrum.

PROBLEM SET FOR OSCILLATORS PROBLEM 1 Conside r the Co lpin s osc illa tor o f Figure 16 .43.

l.og

f' v.....

FIGURE 16.4 3 . Colpills oscil1alof

example for Problem 1.

(a) Ass ume that the inductor has a finite Q. Derive an expression for the minimum Q of the inductor such that the circuit j ust satisfies the co nditions for osci llation . Express your answer in terms of the small-signal transcondu ctance o f the transistor .

526

CHA PTER 16

OSC ILLA TO RS A ND SYN THESI ZERS

(h ) Nnw assume that the inductor has a Q of 10. If n = C 1 / (C I + C 2 ) , provide an express ion for the minim um n co nsistent with j ust sntisfy mg the co nditions for oscillation. (e ) Expla in why these min ima exist. PROBLE M 2 Th is problem examines the startup question from a somewhat di ff erent pcrspec..uvc than the previou s prob lem. Co nside r the Col pitts osci llator shown in Figure 16.44 . v oo=)V

tOlD

~I~ = 0 .065 mZI V., 101 = 9n m

C0 1

v

100p F

=3.IWmF/m 2

Em,= " )(

fl

IO Vlm

FIGURE 16 . 44 . Colpitts o:r.cillotor exomple for Problem 2.

(a) C alculate the minimum W/ L necessary for a startup loop ga in 0( 2. Assume longchannel operation and neglect tran sis tor capacitances, body effec t. and channellength mod ulati on . ( b) Repeat (41) bUI assume operation deep into the short-c hannel regime. (c ) Ca lculate the gate-so urce capac itance in bot h cases and calc ulate the im pedance o f these capac itances at I GH z. PROBL EM 3 In the C lapp oscillator show n in Figure 16.4". calculate the osci llation frequ ency and the amplitude of osci llation at the gate. as well as across the inductor. Assume long-channel operat ion . and neglect all transistor parnsitics.

J.1 n = O.06Sm 2/Y .s I.$V __M-r---,--~ 1

t", = 9nm C", = 3.84 ml'/ m 2 Een, = 4x lOeVi m

120pF

FIGURE 16 . 4 5. Clopp o5oCillofor.

Also plot the sha pe of the drain cu rre nt wave form. Wh at is the pea k drain current ?

527

PROBLE M SET

PROBLE M 4 (a) In the differential oscillator of Figure 16.46. ca lculate the oscillation frequency and amplitude acro ss the LC tank. Assume an inductor Q of 10 and ignore all transistor pcrasuics exce pt C6 • • Voo-3V

~. = O.06Sm 2/ V_s '.... = 9nm

,

C.... = 3.84mF/ m

FIGU Il:E 16." 6 . Negative resistance oscilla tor.

(b) What is the amplitude -limiting mech anism ? (c) Find the minimum supply voltage that guarantees startup. (d) Estimate the power dissipation of this oscillator. What limits if ! Is this mech unism reliable? (e) Is it possible to usc a series tank instead of a para llel tank here? Explain. PROBL EM 5 Suppose the capacitor in the previous problem is replac ed with a series HC combination. Assume that the resistor can vary from small to large values. (a) Derive an equivalent para llel Re network. fo r this combination. What is the Q of the tank if the IO-nll inductor is perfect? (b) Plot the frequency of oscillat ion and tank Q as a function of R . (c) Using your answer to ( b), what are the advantages and disadvantages of using this resistance variation as a tuning method ? PRO BLE M 6 The text's derivation of oscillation frequency and amp litude for the Colpitis configuration glibly ignores any possible phase shih assoc iated with Gm • Amend those derivations if the large -signal tran sconductance actually has the following form : (1)16 .1) G - G ..- j w T(J '"

-

.I ",o ~

•

where G",o is the phase -free large -sign al transconductance given in the text.

CHAPTE R 16 OSCILLATORS A N D SYNTHESIZERS

Treating the mixers in an offse t synthes izer as ideal multipliers. and the osci llutor input s as perfect sinusoid s. deri ve an ex press ion for the loop tran smi ssion. Call the two filter transfer funct ion s " t ( .f ) and " 2( .f ) . Comm ent on how or if the input frequ encies constrai n the loop bandwidth .

PROBLEM 7

As menti oned in the text. an alternative to off se t synthes izers is simply to take two oscillators and co mbine their ou tputs with a mixer. T he mixer' s output is then filtered to yield ei ther the sum or d ifference freq uency, wh ichever is desired. Q ualitatively co mpare these two methods of synthes is. PROBLEM 8

To ea se the dem and s on filtering in the type of ..ymhes izer descri bed in the previous problem. a more complica ted mixer is somet imes used to pro vide suppression of the undesired term prio r to filterin g. A syr nbesizer emplo ying such a mixer a ppears as Figu re 16.4 7. PROBLEM 9

f"

.. r'lUl fZ<) -

-

FIGURE 16.4 7. Combination synthesizer w ith complex mixer.

The subsc ripts I and Q refe r to " in-phase" and "quadrature:' respectiv ely. and indicutc that there are two sig nals with a 90 " phase differen ce bet ween them at each frequ ency. Wh at is the output frequ ency? The co mplex mixer of the previou s problem ca n mix two frequ encies and inherently produ ce only the sum nr differen ce freque ncy with no addi tional filtering, at least in principle. T he gains of the two paths must match . and the I and Q signals mu st be in perfect quadrature in order to cancel the undesired term . Deri..-e an ex plicit ex pressi on for the ratio of undesired to desi red output amplitude if there is on ly a ga in mismatch in the two paths. Ex press yo ur answer in terms of s, where I + e i ~ the gain ratio. PROBLEM 10

PRO BLEM SET

PROBLEM 11

529

The offset synthes izer relies on filtering to select the differe nce co mpo nent out of the various mixing operations. The filter bandwidths constrain the settling time of the loop. however. To ease this constraint. co nsider using the co mplex mixer of the previous two prob lems. Sketch a block diagram of an offset synthesize r that uses two sets of co mplex mixers. Comment on the filtering require ments relative to the classical offset synthes izer.

CHAPTER SEVENTEEN

PHASE NOISE

17.1

IN TRODU CTION

We asse rted in the previous cha pter that tuned oscillators produ ce outputs with higher spectral purity tha n relaxation oscillators . O ne stra ightforward reason is simply that a high -Q reso nator attenuates spectral compo nents removed fro m the ce nter frequency. As a consequence, d istortion is suppressed . and the waveform of a welldesigned tuned oscillator is typically sinusoidal 10 an excellent approximat ion . Aside from suppressing distortion products. a resonator also attenuates spectral co mpone nts co ntributed by source s such as the thermal noise associ ated with finite resonator Q. or by the active elemenl(s) present in all oscillators . While noise from such sources can perturb both amplitude and phase, the amplitu de -li miting mech anism inherently present in all practical oscillators atte nuates amplitude fluctua tions, so we will focu s primari ly on phase noise. To place this subj ect in its proper co ntext, we first ident ify so me fundamental tradeoffs among key parameters. These inclu de power dissipation, osci llation frequ ency, reson ator Q , and noise. Afte r study ing these tmdeoffs qualitatively in a hypot hetica l idea l osci llator , we consi de r qu antitatively how various noise processes corrupt the output spectrum of real oscillators.

17.2

GENERAL CONSIDERATIONS

Tn high light some important issues in a very approximate and general way, consider single HL C ban dpa ...s reso nator Ilsed in an nvcil lamr. The rt'sn natn r is conn ected to an active e lement that has the remarkable (and unrea lizable) property that it co ntribut es no noise of its ow n; sec Figure 17. 1. The noi seless magic bo x supplies j ust enough energy to the tan k 10 compensate for the d issipa tion by the lank' s resistance, thereby leading 10 a co nstant-amplitude oscillation . Although the magic box is noiseless, the tank resistance is not. 11

530

17. 2 G EN ERAL CO NSIDERATIO NS

531

FIGURE 17. 1. "Perfectly efficient- RiC oscilla tor.

The signal e nergy stored in me tank is simply (I)

so that the mean-square signa l (carrier) voltage is

(2) where we have assumed a sinusoidal waveform. By postulate. the o nly so urce of noise is the tank resistance. The total mean-square noise voltage is found in the usual way. by integrating the resistor' s thermal noise over the noise bandwidth of the RL C filter (see Chapter 10):

-

Z(j) V ' = 4kTR l ~ I - I ' df " 0 R

~

kT. 4 kTR · - I = -tRC C

(3)

Taking the ratio of mean-square voltages . we find that the noise-to-signal ratio is equal to the ratio of thermal energy to the stored signal energy: (4 )

Thus. we want In maximi ze the energy of the desired signal (currier) relative to the thermal energy. All other factors held constant. one therefore needs In use the maximum possible signal levels if the noise-to-carrier ratio is to be minimized. To bring power consumption and resonator Q explicitly into the discussion. recall from the chap ter on passive RLC networks that Q is most generally defined us proport ional to the energy stored. divided by the energy dissipated:

(5) Hence. we may write N

S

(6)

CHAPTE R 11 PHASE NO ISE

532

The power cons umed by a perfec tly efficient oscillator ! is simply eq ual to Pd"••• the amo unt necessary to compensate exactly for the dissipation in the tank to maintain a constan t oscillat ion am plit ude. For such an oscillator-' the ca rrier-Io -noise ratio is direc tly propo rtional to the prod uct o f reson ator Q and the pow er co nsumed . and inversely proporti onal to the freq uency of oscillation. Of particu lar significance for low-power operation , then . is the observation that the power required to build an oscillator with a given 11.'\'('1of noise can be made inversely proportional to Q. Eve n though rea l oscillators differ in detail from the one just analyzed . the perfornumcc limits defined by the analysis constitute a reference thai is useful for rating the perfor mance o f real oscillators. Furtherm ore. im po rtant design insights gained from the analysis remain generally correct for rea l osc illators . For exa mple. hig her re...onator Q usually leads to better noise performance for a given level of power consumption, or allows lower power consumption at a given level of noise performance. Also. u....ing the maximum practical signal amplitudes tends to maximize the ca rrier-to- noise rati o. As d iscussed in the previous chapter, the C lapp oscillator configuration increases the signal energy by using a tapped resonator. as docs a more recent C MOS imple-

rncntation.J 17. 3

DETAILED CO NS I DERA TIO N S: PHASE NOISE

The insigh ts conferred by the foregoin g ana lysis are certainly useful. hut hard ly constitute a quantitative theory of phase noise , For exa mple, we have alluded vaguely to the corruption of the output spectrum by various noise processes, but nothing presented so far allows us to calculate an actual output spec trum. We will sec shortly that computing the out put spectrum of real oscillators involves some subtle conside rations. For thut reason . we ' ll begin with the idea l oscillator shown in Figure 17, I. since co mputing its output spectrum is relatively straightforwant - provided we a...sume linearity and time invariunce . We will later revisit these a...sumptio ns. and reco nsider O UT ca lculation, in orde r to develop a co mple tel y general and quantitative model of phase noise. PHA SE N OI SE O f AN ID EAL O SCILLATO R

Assume that the output in Figure 17. 1 is the voltage ucross the tunk . We continue to ussume that the on ly source of noise is the white therm al noise of the tank resistance. I Pracncal ol.d ll.ltlll1i usually violate many o f the a ~~u m p' i (ln s made in getting [0 [ hi~ poin t, hence [he wease l term"perfe ctly e fficient." An example of an " imperfect ly e fficient" oscillator might he tine in which II I-Q re~i sttlr is lied acro ss the power supply. C learly, such an osc ill.ator will CQIl sume much more power than the hypothclkaJ "e fficient" oscillator convidcred in this secnon. 2 J, Cranmckx and M . Stcyaen , "A C MOS l .llGHt Low-Phase -Noise Voltage .Contftllled Oscillator with t' rescelcr." ISS CC 1Ji8f'H of Trrh nicn l Papers. February 1lJ95. pp . 266 -7.

17.3

533

DETAILED CO N SIDERATIO N S· PHASE NO ISE

which we muy represent in Norton form as a current so urce mean-square spectral density of

'1 ~ = 4 t TG tJ. f

UI.TOSS

the tank with U

.

where G is the reciprocal of the tank resistance . Th is current noise becomes voltage noise when multiplied by the impedance of the tank. We must he a little bit careful, however. in co mputing the impedance seen by the current so urce beca use it is necessarily modified by the magic box. Si nce. by postulate. the circuit osci llates with a co nstant amplitude. the magic noiseless box must present an overall effective negative resistance that precisely cancels the resistance of the tank . l ienee. the net effe ctive impedance seen by the shunt noise current so urce is simply that of a perfectly lossless L C resonator. For small displacement s li.w frum the center frequency wo. the impedance of an LC tank may 11c approximated by Z (wo + li.w) ::::: I

wo L 2(li.w j wu)

~==--

(8)

We may write the impedance in a form more useful for our purposes hy noting that the tank Q may be expressed as)

R

1 woG I.

(9)

Q ~ - = -- .

woL

Solving Eq n. 9 for L and substituting into Eqn. Kyields IZ (Wll

+ li.w )1 ~ - I

Wll

(10)

. - -.

G 2Qli.w

All we'v e done is removed explicit dependence on the inductance. and replaced it with Q and the conductance G. Next. we multip ly the spectral density of the mean-square noise current by the squared magnitude of the tank impedance to obtain the spectral density of the meansquare noise voltage (V still equals I Z . even for noise ):

W )' = 4 kTR (-W " - )'

' '1 21 2 = 4 kTG ( -1 . - -.,'" = -;""tJ. f

tJ. f

G

2QtJ.w

2QtJ. w

( 11)

It is clear that the spectral density of the noise is no longer Independent of frequency, owing to the filtering action of the tank. and in fact increases without bou nd as the

) II is im portan t tu ntlle lhal thiro Q is tha i o f the lank in Iscla uon.

534

CHA PTER 17 PHA SE N O ISE

frequency approaches woo Note also that an increase in lank Q reduces the noise, when all other parameters are held co nsta nt. Now, as mentioned previously, thermal no ise ca uses fluct uations in both ampli-

tude and phase, and Eqn. II accounts for both. However. all practical oscillators employ some Conn of amplitude limit ing. as noted previou s ly. Consequently. amplitude variati on s in rea l osc illators are atten uated and phase fluctuation s dom inate. The eq uipart ition theorem of thermodynamic s allows us 10 asse rt that. in the abse nce of amplitude limiting, noise energy splits evenly between amplitude and phase fluctu ation s. so tha t suppressio n o f amplitude varia tio ns leaves us with ha lf the noi se given

by Eqn. II . Add itionall y. we are o ften more interested in how large this noise is relative to the carrie r. rather tha n its absolute value. so it is trad ition al to norm alize the mean -square noise voltage de nsity to the mean-square carrier voltage, and report the ratio in decibe ls. Performing this normaliz ation yields the followi ng equation for ph ase noise:

[V;/6/ ]

[2kT ( -, -Wo - ) ' ] .

L (.1. w ) = 10 · log --=- = 10· Jog - _ . v2. POI'

".

(12)

_Q.1.w

The units of Eq n. 12 are co mmonly expressed as deci bel s below the carrie r per hertz. or d Bc/ l lz. at some offse t frequ ency .1.w from the carrie r freq uency wooHence. one might speak of a I -G Hz oscillator's phase noise as "- 11 0 d Bc/ Hz at a 100-kHz offset." Purists may co mplain that the " per hertz" actually applies to the argu ment of the log , rather than to the log itse lf. but it is common usage. Equation 12 shows that the phase noise at a given o lTset im proves as bo th the carrier power and Q increase, as pred ict ed by the general analysis of Sec tion 17.2. It also sho ws that the noise varies as the inverse square of the frequency offse t. These de pendencies should not be surprising. Increasing the signal po wer improves the ratio simply because the the r mal noise is fixed ; increasing Q or the offset improves the ratio qu adr aticall y bec ause the tank 's impedance fall s olf as I/Q.1. w . and the square of the no ise voltage is proport ional to the square of the impedance. The behavior of actua l oscilla tors conforms a ppro xima tely to these expec tations. but there are sc me signific ant differences be twee n the spectru m predi cted by Eq n. 12 and what one typically measures in pract ice. Whi le real spec tra do po ssess a region whe re the observed den sity is proportion al to t j (.1. w )2, the magnitudes arc typically quite a hit larger than predicted by Eqn . 12. mainly because there arc impo rtant noise so urces in practic al osci llutors ot her tha n tank lo ss . The mo st sig nific a nt of the se is the noise assoc iated with the active clements inside any physical implementation of a " mag ic bo x." Furth ermore. measured spectra eventually tlanen out for large frequency offsets. rather than continuing to drop as the square of .1.w. Such a noise floo r may be due 10 the noise assoc iated with any active element... (such as buffers) placed between the tank and the outside world. or it ca n even reflect limitat ion s in

17.3 DETA il ED CO NSIDERATIO NS: PHASE N OI SE

"'. 2Q

FIGURE 17.2 . Phase noise s,pectnJm according to Leeson, venus Eqn . 12.

the measurement instrumentation itsel f. Eve n if the output were ta ken directl y fro m the ta nk. any resi stance in se ries with eithe r the inductor or capacitor would im pose a bo und on the amou nt o f filte ring provided by the tan k at large frequency o ffsets. thu s produ cin g a noise floor. Finally, it is always fou nd that the phase noi se spec trum possesses a 1/ (6 w )3 region at sufficiently small offsets. In an effort to account for all three discrepancies. Leeso n" has propo sed an ad hoc mod ification to Eq n. 12: L (6 w )

= 10 . log

I [ ( )'] ( 2 Fk T -.-. P" g

1+

Wn

2 - -Q6 w

.

1+

6w I )

If

16wI

)I .

(1 3)

Leeson 's modifications to Eqn . 12 incl ude the introduction of OJ factor F to acc ount for the increased no ise in the I j (6 w )2 region, an addit ive factor of unity (in side the brackets ) to acco unt for the no ise 1100r, and a mult iplicat ive factor
~

D. B. Leeson. "A Simple Model of Feedback Oscillator Noise Spec trum," PmC'. I EEE, v. 54. r-ehruary 1966. pp. 329- 30.

CHA PTER 17

536

PHA SE N O ISE

FIGURE 17. 3 . LC oscillator

excited by current pulse.

Th e model imp lied by the spec trum of Eqn. 12 or Eqn . 13 suggests that increasing bo th the Q and signal amplitude are the only wa ys to reduce phase noise. the sa me insights developed back in Section 17.2. Unfo rtu nate ly. not hi ng in Lee so n's model guide s us in the com puta tion or reduction of F . a nd we have already noted tha i 6Wl lf l is an e mpiric al factor as we ll. Sin ce we ca n' t qu ite make qu antit ative predictio ns abo ut phase noi se from Eqn . J 2 or even Eqn . 13. we must conclude thai nul all of the assum ptions used in the deriva tions arc valid. desp ite their appare nt reaso nabl eness. To de velop a quantitative theory, the n. we need to rea ssess and correc t these ass umptions.

17. 4

THE HAJIMIRI MODEL : A TIME-VARYIN G PHASE NOISE THEORY

Th e de rivation in Section 17.3 assu mes linearit y and time invcri ance. T his lime in variance ex te nds 10 the noise so urces themsel ves. where we have implicitly assumed .~la lim/(l r;ty. meaning thai the descriptive measu res we usc to c haracterize no ise (c.g.• spec tral den sity) arc presumed 10 be time -in variant as well. Although nonlinearity is clea rly a funda mental pro perty of all rea1osci llators. we have already noted that its prima ry effec t is to re move (o r a t lcust attenuate) amplitude noise in the process of provid ing amp litude limit ing (and otherwise definin g the shape of the osci llation wavefo rm). Phase fluctuatio ns then domi nate. and since noi se is typ icall y substa ntially smalle r in magnitude than the carr ie r, linearity wo uld ap pear to he a reasonable assumpt ion a s far as the notse-to-ptiase transfe r f unction is concerned. so we wi ll continue to invo ke it. Time invariancc (a nd its noise counterp art. stationarity), on the othe r hand, is less obvio usly defe nsib le. In fac t. it is a trivial ma tter to dem onstrate (hat oscillators arc fu ndam e ntally time-varying syste ms. Recognizing this truth is on e key to de velopi ng a correct theory of phase no ise . To show tha t time invariance fails to hold. co ns ide r exp licitly how a n impu lse of c urre nt affec ts the wavefo rm of an ideal I.C osci llator (see Figu re 17.3). We assu me that the system is osc illating with so me consta nt a mplit ude until the impulse occurs . We cons ide r how the syste m responds to a n im pulse injected at two d iffer en t times. as gra phed in Figure 17A . If the impulse happen s 10 coincide with a vo ltage ma xim um (as in the upper plot). the a m plitude inc reases a bruptly by a n amount 6 V = 6 QI C. bUI bec ause the

17.4

THE HAJ IM IRI MODEL: A TlM E·VA RYIN G PHASE N O ISE THE O RY

537

V' '''I

FIGU RE 17 . 4 . W aveforms for irnp4.llse excitation

of lC osci llator .

respo nse to the impulse superposes exactly in phase with the pre -existing oscillation. ,ht' timing of'h e zem crossings dot's fl01 chan ge. On the other hand. an impulse injeered at any ot her time d isplaces the zero cross ings. as seen in the lower plot . Unlike amplitude perturbations. which may be uuenuated by the amplitude- limiting mechanisms present in real osc illators. no analogous restorat ive mecbani..m exl..ts for phase . Hence. an impulsive input produces a step in phase. so thai integration is an inherent property of the impulse -to -phase transfer function . Beca use the phase displacement depends on when the impulse is applied. the system is d early time-varying. and the shortcomings oft he I.TI (linear time -invariant) noise
h¢(t , r ) = :....:c=-' u (t - r),

( 14)

qmax

where tj mal is the maximum charge d isplacement across the capacitor and 1I(t ) is the unit step. The functio n f (x ) is called the impu lse sensit ivity fu nction n SF). and is a dimension less. frequency- and amplitude-independent function thai is periodic in 211" . As its name sugges ts. it encodes informatio n abo ut the sensitivity o f the system to an impulse injected at phase W oT. In our example of the L C oscillator, red has its maximum value ncar the zero crossings of the osc illation. and a zero value at maxima of the osc illation waveform. In general. it is most practical to determi ne r (.~) through simulation. but one may also use one of the fonnal methods outlined in (the

CHAPTER 17 PHASE NO ISE

S38

[(wI)

FI GUR E 17.5 . Example impulse semilivity fu nction for LC oscilla tor.

appendi x 10 ) the pape r by Hajimiri and Lee :~ To develop a fee l for typ ical shapes of ISFs. co nside r two repre se ntative ex amples, on e for an LC osc illa tor ( Figure 17.5) and another for a ring oscillator (Fig ure 17.6). Note that . in the case of the LC osci llator. the ISF is approx ima tely proportional to the derivalive uf thc oscillation waveform itse lf. a relationship that hol ds generally to zeroth orde r even for othe r types of oscilla tors. as in Figure 17.6 . O nce the ISF has bee n det erm ined (by wha te ver means). we may co mpu te the ex cess phase through use of the superpos ition integral (reme mber, we're still assuming lineari ty): 4'(1)

=

I

N

h q, (/ . r) i (r) d r = - J

qmu

- 00

J'

r (w ot )i ( t ) d r .

(1 5 )

- 00

To cast this eq uation in a mor e pra cticall y useful form. note that the ISF is period ic and therefore expres sible as a Fourier series: r
2Co + L~ c.. cos(nwoT + 8..),

."

( 16)

where the coefficients r.. are rea l and where 8.. is the phase of the nth harmonic of the ISF. We will ignore 0" in all that follow s because we will be assum ing that noise co mponents are uncorrela ted . so that their relative phase is irre levant.

j

A . Hajimiri and T. Lee , "A General Theory of Phase Noio;e in Electrical Oscillators," IEEE 1. So lid -StUll' CirO.li u, Pebruary 199 8.

17.4

THE HA JIMI RI MODEl : A TlME·VA IlYIN G PHA SE N O ISE THEORY

539

O wl )

FIGURE 17. 6 . Example impul!>e !>en ~itivi ly function for ring oscillator.

The reason for this decomposition is thai , like many functions associated with physical phenomen a , the series typically converges rapid ly. so that it is often well approximated by just the first few (e .g., two ) term s of the series . Substituting the Fourier expansion into Eqn. 15. and exchanging summation and integration. one obtains: ¢( t)

[c l' i( r) tlr + L:>'"l' i(r )cos (nwor ) tlr ].

= -I- ~ q mn 2

00

-00

,, =I

( 17)

- 'Xl

T his equation allows us to compute the excess phase caused by an arbitrary noi se current injected into the system. once the Fourier coe fficients of the ISF have been determ ined . T he fact that the system is not LTi leaves open the possibility that signals (noise) injected into the system at some frequency may produce spectral compo nents at a different freq uency. To demonstrate this propert y ex plicitly, consider the injection of a sinusoida l current whose frequency is near an integer mult iple m of the oscillation frequency, so that (1 8) i(1) = /", cos [(mwo + 6wlt 1. where 6w « w . Substituting Eqn. 18 into Eqn. 15 yields ¢ (t)

= - '- [ co q lJ\;il.

2

I'

i(r ) d t

» oc

+ ~ CII i~{I", cos[(mwo + 6 W)tJ)cOS(nWOf) d r ).

(19 )

SAO

CHAPTER 17

PHA SE N OI SE

We can simplify Eqn. 19 co nside rab ly to e xtract so me intuit ive value fro m it. Because the integ rals may he interp reted as proportional to a " running ave rage" of the integrunds. there is a negligible contribution by term s other than when" = m . so Eqn. II} co llapses to ( 20)

The spectrum o f q,(I) is therefore see n 10 consist of two eq ual sideband s at ± ~ w . even though the injec tion occ urs near some integer mll/lip/e of wo. Thi s frequency conversion (or "fold ing") . which ca nnot occ ur in an LTI system. is fundamental to und erstand ing the e volutio n of noi se in a n osci llato r. Unfortunately. we're nor q uite done: EqR. 20 allows us to figure out the spectrum of 41(1), but we ultimately want to find the spec trum of the output voltage of the oscillator. which is nor qui te the same thing. The two quantiti es are, however. linked thro ugh thc actual output waveform. To illustrate what we mean hy this linkage, co nside r a speci fic case where the output may be app ro ximated a.s a sinusoid. so that l'out(/) = co.s lw ol + 1'(/») . Th is equ ation may be co nsidered a phase -to -voltage converter; it takes phase as an input , and produ ces fro m it the output voltage. Thi s conversion is fu ndam e ntally nonlinear because it involves the phase mod ulation of OJ sinusoid . Performi ng this phase -to -voltage co nversion. we find that the single -tone injection leading to Eqn. 20 results in two eq ual-power sidebands symmetrically d isposed abo ut the ca rrier: PSBd 6 w ) = 10 . log [ I,., e,., (21) 4qmu 6 w

I

It's ju st a short hop from here to the general case of a white noise source:

'" L00 • ] " [ -

~f

PSHc( 6 w ) = 10 · log

c2

• - 0

2 - 2 . 4qmax 6 w

(22)

Equa tion 22 implies both up - and dow nconversion of noise into the noise ncar the carrier, as illustrated in Figure 17.7. This figure summarizes what the foregoing equations tell us: Com ponents of noise near integer multiples of the carrier frequency get co nverted into no ise ncar the carrier itself. Noise ncar DC gets upconvened, weighted by coefficie nt e lh so I/ f device noise becom es Iff 3 noise near the ca rrier: noise near the ca rrier ..rays there. weighted hy CI : and white no ise near higher integer multip les of the carrier undergoes downconvers ion. turning into noise in the J/f 2 region . Note that the l /f 2 shape results from the integration implied by the step change in phase ca used by an impulsive noise input . Since an integration (even a time-varyin g one) gives a white voltage or current spectru m a I/f character. the power spectra l density will have a l/f 2 shape.

11.4

THE HAJIM IRI MO DEL: A TIMEN ARYIN G PHA SE NOI SE THEORY

541

N(w)

++--..I,,-f-,L----'+ ,L---\:-f-,L----• •

o.

J - '--"=------+-___+__

s• •

~

...I

I

" OJ

""

f iGURE 17.7. Evolution of noise.

II is clear from Figure 17.7 that minimizing the various coe fficients ell (b y minimizing the ISF ) will minimize the phase noise. To underscore this point quantitatively. we may use Purseval' s theorem to write

(23) so that the spectrum in the

1// 2 region may he ex pressed as

' 2 Ii! [

2 -" r nns

L( li w ) = 10 · log

·2

2qmu liw

] '

(24)

where r rm, is the rills value of the ISF. All other factor s held eq ual. reducing r nn' will reduce the phase noise at all frequencies. Equation 22 (or Eqn. 24) is the rigorous equation for the 1// 2 region. and is one key resu lt o f the linear time -varying ( LT V) model. Note that . unlike the Leeson model. no empirical curve-fitting parameters arc present in Eqn. 24. Among other attributes, Eqn. 24 allows us to study quantitatively the upconversion of 1// noise into close-in phase noise. Noise ncar the carrier is particularly impo rtant in communication systems with narrow channel spacings. In fact. the allowable

'"

CHAP TER 17

PHA SE N O ISE

channel spaci ngs are frequ ently co nstra ined by the achievable pha se noise. Unfortunate ly. it is not po ssible 10 pred ict cl ose -in phase noise with LTI models. so various (and unreli able) rules o f thumb have evolved 10 guide designers. These rules of thum b are nor needed if the LT V model is UM.- d. Specifica lly, assume tha i the curre nt noise behaves as follows in the I/! region: -.,' " ,l lf

~

'n'

=

w ill

(25)

tiw '

where w ill is the II [ co mer freq uency. Substitutio n into Eqn . 22 gives us

'.

L ( tl.w ) = 10 . log [

which descri bes the phase noise in the

z

'"i 'fiCO 8Qmu6w

] 2 '

IIf 3 region .

W ill

•

.1.w

( 26)

The l/ f 3 comer frequency is

then

Co af"rm,

z ~ will ' D.w l// J = w ill ' - -,-

( -Co ) ' ,

(27)

Cl

fro m which we see thai the IIf 3 phase noise come r is nor nece ssari ly the sa me as the II! device /circuit no ise come r. In fact. since Co is the DC value of the ISF, there is a possibility of reducing by large f actors the Iff 3 phase no ise co mer. The ISF is a function of the waveform. and hence po tentially under the control of the designer. This result is not anticipated by LT I appro aches, and is on e o f the mll!'> t powerful insig hts conferred by this LT V model. Hence. Eqn. 22 and Eq n. 27 ta ken together are the important results of the LTV mod el. Anoth er extremely powerful insight conce rns the intfucnce of cyclostationary noise sources . In most osci llators. the noise sources are not well mode led as station ary. A typ ical example is the no minally white collector shot noi se (or MO SFET drain curre nt noise). which may vary because the collector curre nts vary pe riod ica lly with (he osci llating wavefor m. The LTV mod el is able to accom modate a cyclostarionar y w hite noise source with ease, since we may treat such a source as the produ ct of a stationary white noi se source and a pe riod ic function : ;,,(t )

= ;"0(1) . a(w ot) .

(28 )

Here, ;,,0 is a statio nary white noise source who se peak value is equal to thaI o r the cyclostatio nary source. and a ( x) is a periodic funct io n with a pea k value of unity. Substituting this into Eq n. 15 allows us to treat cyclosratio nary noise as a stationary noise source, provided we define an effective ISF as follows:

r..ff ( x )

= F'( x } · a (.t ).

( 29)

17.4 THE HA JIMI RI M ODEL: A TIM E-VA RYIN G PHASE NO ISE THEORY

543

R

v

c,

c,

FIGU RE 17. 8 . CoIpilh oscillator (biosing detail s not shown) .

Th us. none of the preced ing concl usions chan ges as long as f rff is used in all of the equation s. Having identified the factors that influence oscillator noise. we're now in a position to articulate the req uirements that must be satisfied to make a go od oscillator. First. note that an active device is always necessary to co mpensare for tank loss. Of the many possible ways that this active e lement could return energy to the tank , this energy should be delivered all 01 O1lc e, at the peak of the tank voltage. where the ISF has its minimum value. In an ideal oscil lator. then. the transistor would deliver an impulse of current at the peak and then go into a coma each cycle. The extent to which real oscillators approximate this behavior determines the quality of their phase noise properties. Let us re-examine the Colpitts oscillator of the previous chapter. now that we have develope d these insights (see Figure 17.8). Recall that the wavefor ms for this oscillator appear approximately as shown in Figure 17.9. Note that the drain current fl ows only during a short interval coincident with the peak of the tank voltage. Hence. a Colpitts oscillator comes close to behaving ideally. Its correspo nding exce llent phase noise properties accou nt fo r the popularity of this confi guration. It is impor tant to underscore that the foregoing descript ion presumes that the amplitude of the source- gate voltage is large enough to cause narrow drain current spikes. As one adj usts the feedback factor from zero to unity. the current spikes narrow. Offsetting the noise -suppressing value of that reduction in conduction angle is the increasing pro minence of loading by the source impedance. A minimum phase noise co ndition thus exists for a particular value of tapping ratio. Over the years. oscillator designers have converged on various rules of thumh for choosing tapping ratios (e.g.• a 4 : I capacitance ratio in the tank). but until the develop ment of the LT V theory there was no rational basis for these rules. As an exa mple of a circuit that does not well approximate ideal behavior. consider a ring oscillator. First. the " resonator" Q is poor; in fact. it is unity. since the

CHAPTER 17 PHA SE N OI SE

"tllnk

v,.

+ --\,-----+---\-+---\- +---\- - - .. t

FIGU RE 17.9. Approximate il"lCTemel1tol tonk voltage and drain current for Colpitts osc illator .

energy store d in the node capacitance s is reset (disc harged ) every cycle. Hence. if the resonator of a Colpitts.. oscillator is analog ous to a fine wine glass. the reson ato r of a ring oscillator correspo nd!'. to a lum p of clay. Next . ene rgy is restored to the resonator duri ng the ed ges. rather than the vo ltage maxim a . These factors account for the well-know n terrible phase noise performa nce o f ring oscillators . As a co nsequcnce. ring oscillators are foun d only in the most non crit ical app lica tions. or inside wide band PLL s that clean up the spec trum.

17.5

SUMMARY

We have examined the all-impo rta nt issue o f phase noise. The insights gained from studying LTlmod els arc simple and intuitively satisfying: One needs to ma ximize signal amp litude and resonator Q. An additional, implicit ill-sight is that the phase shifts aro und the loo p generally must be arranged so that oscilla tio n occurs at or very nca r the center frequency of the resonato r. Thi s way. there is a maximu m atte nuation by the resonato r of off-ce nte r spectral components."

I>

Th is requ irement i ~ onen stated somewhat d ilTerently in o lder works bused un tll< Leeson mode l. In that viewpo int. one wants to force oscilla tion at a frequency where the phase -versus-frequency curve of the lonp l....lIlsmission has the maximu m slope . In the case of a bandpaxs reso nator. that po int happens to corre spond precisely to me center freq uency o f the resomnor. Ho w ever . this equivalency is not general (conside r e.g . the add ition o f non minimum phase eleme nts in the lotlp transmi""io nl. and the LTV model is correc t.

PROBLEM SET

545

Deeper insig hts pro vided by the LTV mod el a re that the resonator e nergy should be restored impu lsivel y at the voltage maximum. instead of evenly throu ghout a cycle. and that the DC value of the e ffective ISF shou ld be mad e a" close 10 zero as possib le to suppress the upconversloo of III noi se into close -in phase noise .

PROBLEM SE T FO R PHA SE N O ISE PROBLEM 1 Calc ula te the phase imp ulse respon se for a volta ge -d riven serie s RL C net work . PROBLEM 2 In eve ry pructica l osci llator, the tank is not the only source of phase shift. Hence, the ac tua l oscillation freq uency may d iffe r somewha t from the resonant frequ ency o f the lank, for example. Usin g the time-varying mod el. e xp lain why the osci llator 's phase noise ca n degrade if such off-frequency oscillations occ ur. PROB LEM 3 Assume Ihal the srcady-snu e output a mplit ude o f the following oscillator is I V. Calc ulate the phase noise in dBel Hz at an o ffset of 100 kHz from the carrier for the signal coming out o f the ideal comparator (sec Figure 17.10 ).

L'r

+

c

FIGU RE 17,10 . Osc illa tor w ith compa ro tor.

=

Assume tha t L. 25 nil . L 2 = 100 nl-l, M = 10 nil. a nd C assume tha i the noise c urre nt is

i;. = 4kTGefft1l , whe re I/ Geff = 50 Q . The te mperatu re of the circ uit is J CXI K.

= 100 pF. Further ( P I7. 1)

546

CHAPTER. 17 PH ASE NOIS E

PRO BLEM 4 Consider the C MOS Colpitts oscilla tor shown in Figure 17.11 . Vpp=3V

IlIkD

1ll0pF

FIGURE 17.11. CoIpitf$ o ~illotor example for Problem .4 .

(a ) Calculate V". and Vd • before oscillation starts. Calc ulate the steady-state tank oscillation amplitude and freq uency. You may assume that the drain current co nsists of narro w pulses. Assume operation deep in the shim -channel regime and ignore all transistor capacitances. (b) Inductors used in practical osci llators typically have relatively low Q. Assume that an eq uivalent parallel resistance of 10 kfl mode ls energy losses in the inductor, as well as loading effects caused by a subsequent stage (not shown). Cal culate the phase noise in the 1/1 2 region due 10 this resistance, assuming that the transistor and current source are noiseless. Assume also that the ca pacitors arc lo ssless. It may help 10 point OUI that the resistance in question introduces a stationary noise source . PROBLEM S For the Colpitts oscillator of the previou s problem, find the DC value of VN• when the oscillator is in the steady slate. Also find the conduction angle 2¢ , assuming for simplicity that the drain current waveform consists of triangu lar pulses (sec Figure 17.1 2). Hint: Keep in mind that the average drain current must equal the bias current, I HIA.S.

'"' - +---"---'--- ------'----"---- ~" FI GURE 17.12 .

I

A~~umed drain curren t wave form

lor Problem 5 .

547

PRO aLEM SET

Explain qualit atively how the phase noise in the 1/f 2 region depends on the conducnon angle.

Assume a noiseless resistor in parallel with an ot herwise lossless tank . and calculate the phase noise due to drain curre nt noise for the Colpitis oscillator of Problem -to Note that the drain curre nt noise cannot be treated as stationary.

PROBLEM 6

(a) Calc ulate (1" 2(0) and the effec tive value for r rm" assuming a pu rely sinusoidal n (9 ) . Note thai the drain curre nt noise in any MOS de vice may be expressed as ( P17.2) Ass ume that the drai n current waveform co nsists of the tips of a sinusoidal waveform, as shown in Figure 17.13.

I

RIAS

" .•••••••..•. -.'

••

FIGURE 17.13 . A.!.sumed dra in current waveform for Problem 6 .

(b ) Use the calc ulated

r nn,

(effective) to ca lculate the phase noise du e to NM OS

drain curre nt noise. (c) Calc ulate F . the filling parameter fro m the Leeson model. Find the voltage change. 6 V1 • across the tank caused by a current im pulse of area 8 £12 injected at time t = r in to the middl e node of a capacitiv el y ta pped

PROBLEM 7

lank , as shown in Figure 17. 14.

L

FIGURE 17.14. Topped lank a nd noise source .

.

548

CHAPTE R 17 PHA SE NO ISE

(a ) Findthe equiva lent c harge 6 qcq injec ted in par allel with the ind uc tor that would result in Ihe sa me volta ge cha nge . Exp ress yo ur result in term s of the ca paci tive di vide ra tion = CI / (C r + C 2). ( b) Find the power spec trum of that eq uivalent noise so urce. e xpressed in term s of the pow er spec tru m of the orig inal noise c urre nt so urce in pa rallel with C2 • PROBLEM 8 Suppose that the tail c urre nt so urce of the oscillator o f Figu re J 7. 15 is act ua lly imp le me nted wit h a J.3- kO resistor. Model the noise due to the resistor as an equivale nt noi se c urre nt source in parallel with the indu c tor. Assume tha t this noise is do minant ove r all othe r sources and calcula te the pha se noi se result ing from thi s resistor 's noise.

V(JO"'3V

200nH ~

IOkO

f"v"",

1.5 V

-1

~ 100pF

V

f iG URE 17.15. Colpitts

o!oCillob" exomple for

Problem B.

PROBLEM 9 For the Colpitis osci llator of Pro ble m 4. suppos e the N MO S transistor po ssesses a II! norse corne r of 200 kHz (this value IS not atypic al. unfortunately). Ca lcula te the 11! 3 come r frequ ency of the phase noise. Hint: Calc ulate en using

Co

= -1

1,2'r cff((J)

](0

d O = -1

1," r Un

· a (O) d O

(1' 17.3)

](0

a nd assume a trian gular drain curre nt waveform a.s in Pro blem 5. PROBLEM 10 Re -con sid er the high -level model for an H L C oscillator (see Figure 17.16 ). Note tha t the active eleme nts thut kee p the sys te m in oscillation a re no longer cons ide red noise less. In partic ula r. mod e l the " mag ic bo x" ex plicitly as possessing an equivalent inpu t noise c urre nt and noise voltage . For s implicity. assu me that these noise so urce s
PROBLEM SET

FIGURE 17.16.

549

RiC oscillator.

(a) Derive an expression for the phase noise spectrum assuming LTI behavior . (b) Re-derive this expression using the LTV model. assuming that the oscillation waveform is sinusoidal . Further assume that the noise sources are stationary, (c) Compare your answers and provide an explicit expression for F. the Leeson fitting parameter.

CHAPTER EIGHTEEN

ARCHITECTURE S

18 .1 I NTRODUCTION Bec ause of its high performa nce. the superheterody ne is the only basic architecture presently in use for bo th receivers and transmitters. One should not then infer. however. thai all rec ei vers and transmitters are therefore topologically iden tical. for there are many varia tion... on a basic theme. For example. we will see thai it may be desirable to use more than one intermediate frequency to aid the rejection of certain signals. feuding to a question of how many IFs there should be. and what freq uencies they should have. Answering those q uestions is known as fre quency plannin g. and converging on an acceptable frequ ency plan generally involves a substantial amount of iteration .

An important co nstraint is thai on-c hip energy storage clements generally consume significant die area , Furthermore. they tend not to scale gracefully (i f at all) as technology improves. Hence. the " ideal" integrated arc hitecture should require the minimum number of energy storage clements. and there are continuing efforts even to eliminate the need for high-quality filters through architectural means. Complete success has been elusive, though , and one must accept that the desired performance freq uently may be achieved only if cxtemal tiltcrs arc used, It is not too much of an exaggeration to assert that architectures arc esse ntially determ ined by available filter technology, Once a basic architec ture and its associated frequency plan have been chosen, other key considera tions incl ude how best to distribute the huge power gain (typica lly 120140 lin for receivers) atmmg the various stages. becau se import ant factors such as system noise. stability. and linearity arc all strong functions of gain distribution. The great d iversity of existing arc hitectures re fl ects the inability of any single one to satisfy all requirements of interest. So. after considering so me universal system issues, what follows is a sampling of several rep resentative receiver and transmitter architectures. along with a discussion of their attributes and limitations.

18.2 DY NAMIC RANGE

551

FIGURE 18 . 1. Cascaded systems for noise ~gure computation .

18.2 DYNAMIC RANGE Dynam ic range is on e of the most basic sys tem conside rations. and we have alread y ide ntified the Iwo para met ers. inte rcept and noise figure. thai hou nd it. Howe ver, we need 10 exte nd our understandin g to how the dynamic ra nge of a cascade of system s depend s on the intercepts and noise figures of the individual subsys tems. We now de velop the relevant co mbina tion rules in this sec tion.

18.2 .1 NOI SE FIGUR E O F CASCADED SYS TEMS Th e overall noise figu re of a cascade of syste ms de pends on both the ind ividual noise figures as well as their gains. Th e dependency on the gain results fro m the fac t tha t. once the signal has been am plified. the noise of subsequent stages is less importan t. As a result. sys te m noise figure te nd s to be dominated by the noise performance of the first couple of stages in a receiver. How the ind ividual noise figu res co mbine to yield the overall noise figu re is complicated by the variety of impeda nce le vel s typicall y found in the syste m. To deve lop an equation for the syste m noi se figure. consider the bind. diagram of Figure 18. 1, where eac h F" is a noise fac tor a nd each G " is a power gain (specifically. the availabl e gain. the ga in obtai ned with a matched load ), Since noise factor de pends on source resistance. on e must compute the ind ividual noise ligures relati ve to the output impedance of the preceding stage to keep the calc ulat ion ho nest. Thi s issue a rises less frequ ent ly in discrete designs, where impedance level s a rc often sta ndardized, hut req uires careful atte ntion in Ie implem entat ion s. No ise fact or may he expressed in seve ral ways. hut one form that is particul arl y useful for our task at hand is (I )

where R, is a (possibly fictitious) resistance that accou nts for the observed noise in excess of that due to R I ' Th e quantity N r is thu s a n excess no ise powe r ratio. equal tof· - 1. Reflecting this pow er rati o bad: to the input of the precedi ng stage simply invo lves a d ivision by the available power gai n o f the previous sta ge. Reflecting the excess

552

CHAPTER 18 ARCHITECTU RE S

noise cont r ibutio n of a give n stage all the way back to the input thu s requ ires division by the total ava ilable gain between thai stage and the overall input. Th e total noise fac tor is the sum of these individu al co ntributions. and is therefore gi ven by F = I whic h s implifies

+ F, -

1+

F2 - I - I - + -fl'-+ ... +

G]

l2

GG

FN - J . G"

n:,., .1

(2)

10

(3 )

II is d ear tha i the syste m noi se figure is in fuct dominated by the no ise performance o f the first few gain stages . Hence. in trying to ac hieve a good noise figu re. mo st of the design effort will generally focu s on the first few stages. It should also be not ed that the preceding eq uations mu st be u..cd carefully if any of the stages is 3 mixe r. because noi se at a give n IF can result from the tra nslation of noise fru m two diffe re nt freq uencies. These equ al ions will a pply a~ long ax the noise figure o f the stages preceding a mixer is co mputed by exa mining the noise at both signal and image frequencies.

18.2 .2

LINE ARITY O F CA SCA D ED SYSTEMS

T he ot her figure of merit tha t hou nds system dynamic range is the intercept poi nt . Even thou gh we have disc ussed only third-ord er interce pts. it should be mention ed that there arc also instances in which the second-orde r intercept is a relevant linearity measure. A notabl e ex ample is the degenerate case of a superheterod yne in whic h the IF is zero . We will study such direct-conversion receivers in greater detail shortly; we bring up the subj ect now merel y to call attention to the fact that both second- and third-order (a nd possibly higher-order) intercep ts may he usefu l measures of system linearity. A difficulty in developi ng the desired equation is that the distortion produ cts of (J il l' sta ge combine with those of a later stage in ways that depe nd on thei r relative phases. Hence, there is no simple, fixed rel ation ship bet ween the ind ividual and overall interce pts. However , it is possible to deri ve a con servative (worst- case ) estimate by ass uming that the amp litude s of the distortion prod ucts add directly. Th is choice in turn makes it most natural to express the gains as vo ltage rati os, in contrast with the usc of power ga ins in the expressio n for system no ise figure. Thi s is sketched in Figu re 18.2. where e uch AI'" is a voltage gain and each II VM" is an M th-order input intercept voltage . To faci litate the deri vations. we use Vd M ." to denote the M lh-order ( intermod ula tinn ) distortion produ ct
18. 2 DYNA MIC RAN GE

FIGU RE 18.2 .

553

Cascaded systems for input intercept cokulotion.

the input of that stage. Further note that. from the definition of an input intercept. the input-referred Mth -order 1M distortion prod uct may be written as

V"

(4 )

Vd M = II VMM r '

Let us carry out the derivation for the specific case of the third-order intercept. and for a cascade of ju st two stages. The third-order 1M at the output of the first stage is

v dl.1

=

A ~IVJ

(5)

I1V32 ' I

The third-order 1M voltage at the output of the seco nd stage is the sum of two cumponcnts. One is simply a scaled version of the distortion produced by the fi rst Mage. and the other is the d istortion produced by the second stage. Add ing these directly together yields the following pessimistic estimate: ( 6)

The input-referred third-order distortion is found by dividing through by the total gain: (7)

Substituting Eqn. 4 and Bqn. 5 into Eqn. 7 yields ( 8)

This last equation confirms that the later stages bear a greater burden because of the gain that precedes them. We can also see that the reciproc al IIV3 of a given stage. normalized by the total gain up to the output of that stage. cont ributes to the overall reciproca l input-referred intercept in root-sum-squared fashion. Although Eqn. 8 applies strictly to a two -stage cascade. it is readily extended to an arbitrary numhcr of stages as follows: I II V32 = 1,,1

I

" I ,HnA;.;j.

II V3~J + L J=2

I1 V32

J , .. I

(9 )

• 554

CHAPTER 18 ARCHITECTURES

L

RFA mp! Prc!'>eIC'Clor

Tuning contr ol -

•

- >'

FIGURE 18.3 . 5lJpe~rodyne receiver block diagram.

On e ma y follow a similar procedure to determine the overall input-referred interce pt for di ..tortion products of any order. Having deriv ed ex press ions for the noise figure a nd intercept of a n arbitrary cas-

cade of systems. we may now tum 10 an examination of receiver and transmitter arc hitectures .

18.2 .3 THE SINGLE-CO NV ERSION RECEIVER Insigh t into importa nt desig n issues. a nd an unde rstandi ng of why alternative archilectu res were developed. may be ob tai ned by study ing the sta ndard superhe te rod yne bloc k diag ra m introduced in previou s cha pters : see Figure 18.3. To d istinguish this ba sic architecture from more ela bora te ones thai em ploy more than one mixer a nd intermed iate freq uen cy, this one is known as a .5;/I~/e -Cmf\'e r.f io" supe rheterod yne receiver, To appreciate some of the design tradeoffs invol ved, con side r first the image rejecfi rm problem. ca used hy the fact that there are two inpu t freque nci es tha t ca n prod uce

a n IF of a give n freq uency, Suppose, as is common. thai the differe nce freque ncy co mpo nent ou t of the mi xe r is chosen as the IE As a speci fic example. suppose the IF is 70 MH z and we des ire to tune in an RF signal at 8{X) Mll z . With a corres ponding (ass umed) local osc illator frequ ency of 870 M Hz, an RP signal at 940 MH z would also prod uce a n IF signal at 70 M il l , Th is undesired signal is know n as the image signal. Most typi cally, a filte r at the fron t-end ( know n as a preselector or image -rej ect filte r) is used to a ttenuate greatly the image signal prio r to mi xing. Note thut the image sig nal is displaced from the desired frequ ency by twice the IF. To ma ke it easier to fille r the image. it is the refore generally desirable 10 c hoose a relutively high IE To allow the use of a fixed filter, the IF should be high e noug h so thai images never fall in band . In the case of ordi nary AM radi o, wh ich spans 530 kHz to 16 10 kt t z, o ne wo uld no rmall y wa nt to choose an II' of at least (16 10 - 530)/2 kHz, or a boul540 kll z. Unfo rtu nately. 455 kHz e volved as the IF a nd has rema ined

t 8 , 2 DYN AM IC RANG E

(a nd w ill probab ly forever re mai n) the norm. ' Th e preselec tor therefore can' t be (a nd isn' t) a fixed filler in A M rudies thai UM: the histor ically conventional freq ue ncy plan. In such recei vers . the prcselec tor filler tracks the LO to suppress the image. Whereas a high IF is favored 10 rela x the require ments on front-e nd filte ring, a low IF is prefe rred to reduce dema nds on the IF am plifie r and filter. Th e resu lting frequency pla n is partl y a consequence o f balancing these factors agai nst one a nother. Ano the r part of frequency planning involves the c ho ice of LO freq ue ncies since. once aga in, the re are two possib le values that prod uce a given IF fro m a given RF. Norm ally, it is de sirable to choose a n LO ra nge that corres po nds 10 the RF ra nge plu s the desired IF. rat he r tha n the RF ra nge minus the desired IF. Alt hough both c ho ices are valid. the former c boice. known as high-side injection. reduce s the ratio of maxim um to minimum LO frequency requi red , easing oscillator design . Continuing with our A M radio exa mple, our LO choices would he one that spans 75 kHz (= 5JO- 455 kHz) to 1155 kHz ( = 1610 - 455 kll z), or one that spans 985 kHz to 2065 kll z. Th e former cho ice requ ires a tuning range in e xcess of 15 : I. while the latter c hoice requires only about a 2. 1: I tunin g ra nge . It is muc h eas ier to des ign an osci lla tor to span a 2. 1: I frequency ran ge , part icularl y if the frequ ency variation is accomplished by varying j ust a sing le capacitance in a tank (w hic h is the most common meth od ). Since the resonant freque ncy of a ta nk is pro portional to the inverse square roo t of the LC produ ct. the capaci ta nce must vary ove r a ra nge that is the square of the fre quency range . Obtaini ng a (2.1)2 ran ge in capacitance is much easier to ac hieve than a ( 15)2 ra tio, either mechanicall y or elec tronically. For this reason , high-sid e injection is the nea rly universa l cho ice.

18. 2 .4

UPCON VERSION

Th e primary mot ivation for c hoosing the difference freq uency component out of the mixer is to perform a downconve rsion from RF to a lowe r-frequ ency IF. Th e impli cit assum ption is that such a freq uenc y lowe ring ma kes it easier to rea lize high -q ual ity IF filter s a nd to obtain the requisite gain, As a conseq uence, the vast majority of supe rheterod yne recei vers are of this type . In many cases, howeve r. other design conside rations may influence the frequency plan . One option is to choose an IF that is actually hiXher than the RF. Such a choice greatly reduce s the image reject ion problem a nd there fore relaxes front-end filteri ng requ irements considerably. Another significa nt ben efit is a reduction in the fractional tu ning ra nge req uired of the LO. Thus. if either o f these co nside rations is im port ant. a n upconvers io n arc hitec ture ma y be prefe rred .

I FM broadcast radio is in better shape. It spurts the 88- lOH-Mli /. band (in the United Slalcs),llml the typicallf is 10 .7 MH t , Sll that no image of aleglrirnarc F ~t radio signa l coincides with anothe r lcgitimalc FM radio signal.

CHAPTE R 18 A RCHITECTURES

556

Le t's carry nut a design exercise to see how upconversion might be applied to the design of a n A M recei ver (perhaps. eve n, to pe rmit a full y integrated sol ution ). Ruther than c ho osi ng the traditiona l IF of 455 k jl z , suppo sc we select. some what arbitraril y. an IF of 5 M ll z. To cove r the e ntire AM ba nd. we might wa nt the LO 10 tUIIC fro m 5.530 M i ll to 6 .6 10 J\.H l z. for a n La luning rati o of just 1.2: I . Constructing an oscillator thai tunes over a 20% range is relatively easy, With this choice of IF, the image frequency is 10 M Uz away fro m the desired RF signal. I I is easy 10 build a fixed filter that m ils off abo ve 1610 kl l z 10 provide e xce lle nt unenuation by 10.530 MHz, so the image rejectio n problem largel y d isappears . We M.'C that upco nversion eases demands on the performance o f both the local oscilla tor and the preselector. The price paid, of course. is that channel selection has to occur at a higher freq uency (the IF), and this places greater demands o n elem ents in the IF chain. Furthermo re, the receiver is now susceptible to interference from sources within the relatively large bandwidth of the front end. so linearity requirements bec ome more severe ( perhaps much more so ). Choo sing In use upconvervion therefore depen d!'. on w hether the increa sed susceptibility to interferers is acceptable in exchange for improved image rej ection and simplified LO design. 18 . 2 .5

DUAL CO NV ERSION

We ' ve no ted that a low intermediate frequency is demanding of presclector and LO performance. but places no great de mands o n IF filtering. We 've also noted that a high IF increases the difficult y of realizing the channel-selective IF filters. but mitigares the image rejection problem and therefore rela xes requirements o n the preselector. A high II-' also happens to simplify LO design owing 10 the reduction in required oscillator tuning range. A dual -conversion receiver uses two IFs in an effort to com bine the benefits of both downconvcrsion and upconversion architectures hy using both techniques together. In a dual-conversion superheterodyne receiver. a fi rst mixer prod uces a high IF to take care oft he image rejection issue, while a second mixer and low IF ease the channel selection problem. S\JIllC receivers employ a third IF tn provide even greater fl exibility in the tradeoff. Most modem high-performance receivers implemented in d iscrete technolog ies usc a dual-conversion architecture. with the frequency plan deter mined in large part by what high-quality fil ters arc available at reaso nable cost. 18 . 2 .6

THE IM AG E-REJECT RE CEIV ER

The image -reject receiver employs a complex mixer that exploits the relationship betwee n the desired signal and the undesired image to cancel the image d uring the mixing process. Thus. no preselecto r filter is needed (at least in principle ), and design of the overall architecture can proceed with d iminished co ncern fur the image problem.

557

18. 2 DYNAM IC RA NGE

'X )"

LPF

~ ~Ntput CYIF

RF ln

1

LPF

~

o'

FIGURE 18.4 . Image -re ject mixer.

LPF

IF

RF lnpu!

( N l puI

~1

LPF

~

I, FIGU RE 18 . 5 . Weaver architecture.

In particul ar, a rel ati vel y low IF may be chose n to ease the requi rement s on IF filtering, analog-to- digital conversion. ami sub sequent baseband proc essing . See Figure 18.4. To perform this miracle, the RF signal feeds two mixers. Two LO signals, in quadrature, drive the other mixer ports. The desired difference freq uency co mpo nent is the same in both mi xer outputs, but the image term s arc in quadrature. In the cl assic implementation show n. a constant-ga in, freq uency -independen t 90~ phase shifter allow s simple addi tio n of the two signals to cance l the und esired im age terms. Becau se it is somewhat di fficult to build bro adb and quadrature phase shifters. the alternative architec ture (due to Weavcr 2) show n in Figure 18.5 is often attractive. In

2 D. K. Weaver. "A Third Method of Generano n and Derccuon of Single-Sideband Signals," Pn« , IR£ . 1Jel.-cmher 19 56. pp. 1703- 5.

5"

CHAPTE R 18 ARCHITECTUR ES

the Weaver arch itec ture. a pair of qu adrature mixing operat ions el iminates the need for the phase shifte rs. As long as the two pairs of La s ignals arc truly in qu adrature. a nd the gains in the two paths are pe rfec tly matc hed . there will be perfec t cancellation of the unwanted image .

Effect of Ga in and Phose Erron Complete image rejec tio n in the preceding arc hitectures depends on perfect phase qu adratu re a nd perfect gain mat chi ng. Of CU UrM\ it is more real istic 10 assume that nei the r cond ition is satisfied in prac tice . Th e ul tima te rejection of the ima ge the n depends on the level s of matchi ng achieved. To quantify the mat ch ing requireme nt. first co nsider explicitly the effect of ga in e rrors in the block diagram o f Figure ISA. Specificall y, but withou t loss o f generality, let the I and Q signals mod el all the e rro r as follows: 1 = B COS(WLO t) ,

( 10)

Q = A sin (w w f ).

(II )

If the RF input signal is COS (WRFf) . the n the output of the I mixer is (12)

a nd the outp ut of the Q mixer is

' RFt + w to r I - sm( . wRFf 2"AIsm(w

WLo f ) I.

( 13)

The low-pass filters re move the sum frequency compo nent , while the 90~ pha se shifte r converts (minus ) sine to cos ine . T he sens itivity o fthe difference frequ en cy compo ne nt in Eq n. 13 10 the sign of the argument is ultimat el y the source of this mixe r 's image -rej ection pro perty, since an RF signal above the LO frequ ency by an amount equa l to the IF produces an IF signal with a sign oppos ite to that prod uced by mi xing do wn the image . Phase shifting , and then sum ming with a cos ine -mixed ver sion of the La . resu lts in a reinforce me nt of the desired co mponent a nd a simulta neous suppress ion of the image. For an RF signal above the La, the overall output may he exp ressed as ~ [ A COS(WIFt)

+ B cos(U)Jpt)J :

(14)

the image RF signal produ ces an outpu t gi ven by ( 15) Let us defi ne the image-rejection ratio (IRR) as the power ratio of the desi red ou tput to the undesired output. Th e n

18 .2 DYNA M IC RAN GE

B]' = [ II +- BfA BIA]' = [ II -+ (I(1 ++ e' )]' "" 4, ; ,.,n= [ A+ A _ B ) e

IRR

559

(1 6)

in the la.. .1 approxi mation. it is assumed that the ga in error £ is much sma ller than uni ty. As we shall sec sho rtly, the gain error of any pract ical image -reject mixer must satisfy this ineq uality in order to meet realistic performance specif ications. Hence. thc approximation is guaranteed to be valid in all cases of practical interest. O ne may cart)' out an analogou s derivation a...suming perfect gain matching but impe rfec t 90" phase shift. In that case. the IR R du e to a departure tJ.q, (in rad ians) from quadr ature may be expressed as

4 IR R plla.-e = 1 + -ttcot tJ.¢ )2 :'::::: (6¢)2'

(17 )

Again assuming small ga in and phase errors. the net IR R may be computed using the sum of the erro rs given hy Eqn. 16 and Eqn. 17:

IRR101 ""

4 (6q,) 2

+£2.

(1 8)

It is quite d ifficult to ac hieve much better than about a 0 . 1% ga in error and a I" phase error. parti cularly at high frequ encies. without some form of ca libration. These numbers correspond to an IR R of abo ut 4 1 d B. More typically. image rejec tion is rarely significantly better than about 35 dB. Because most receivers require much larger image rejec tion (e.g.• SO-d B). the image-reject architecture alone rarel y provides sufficient rejec tion. Auroc alibration ca n hel p close the gap. but it is unlikely to pro vide an additional 40 dB of image reject ion. li enee. additio nal filteri ng is usuall y req uired . Aside from that limitatio n. an additiona l drawback of the image-reject mixer is thai mixers are noisy and can be power-hu ngry. so addi ng ano ther o ne can be unattractive from the perspec tive of power consumption and dynamic range. Quadrature Generators

The image -reject mixer req uires the ability to generate qu adrature phase shifts. Altho ugh it isn' t difficult to prov ide a 90 ° shift over a broad frequenc y range. it is much harder to main tain a constant amplitude respo nse at the same time. In fact. no finite net work can provide both a constant phase shift and a constant gain magnitude over an infinite freq uency range. so on e mu st settle for approx imations thai work well only over some limited range . The HC- CN network shown in Figure 18.6 is a popular method for generating quadrature signals over a narrow frequency band. The phase shift is 9(jO at all frequencies. hut the magnitude response is not constant, T he phase shift of the I branc h is zero at DC. and asymptotica lly heads down toward - 9

CHA PTE R 18

ARCHI TECTURES

~ Q

R

fiG U RE 18.6 . Passive analog

quadratu re generator.

starts off at + 9()" at DC, and heads downward toward zero with the same functional shape as the I branch . Hence . eve n thou gh the phase shift o f each HC branch is not co nstant. the 90' differe nce between the m is. Unfortunately. the amplitude of eac h output cha nges dram atically with frequency, since the I bra nch is low-pass in nature whereas the Q branc h is a high -pass filter. The amplitude s o f the two outputs are eq ual only at the pole frequency:

1 w= HC '

(I9)

There is, of co urse. a 3-d B atte nuation at this freq uen cy, so on e must accep t this signiticantloss when using this network . Furthe rmore . this networ k is thermally noi sy, and its co ntribution to o verall noise figure must be kept under control. The design of this q uadrature generator is straightforward in princip le : the resistor and capacitor val ues are chose n to set the po le frequ ency eq ual to the operating frequency. In doing so, the effec t o f parasitic loading mu st he taken into acco unt. Even so, an amplitude mismatch is all but inevitable becau se of compo nent to lerance s. If nece ssary. o ne may restore amp litude equa lity with a variable gain amplifier in an AGe loo p or with a lim iter. C lear ly. this type of co mpe nsation is effective only in generating one frequ ency ut time. suc h as in produ cin g a quadrature LO. In such single- frequency applica tions. it may be appropri ate to consider alternative quad rat ure generation methods that autom ati cally provide equa l-amplitude outputs. Q uadrature oscillators . consisting of IWo integrators in a feedback loop . can inherently produ ce equa l-amplitude sine wave s in quad rature (sec Cha pter 16), In those instances whe re square- wave signals arc acce ptable , one may use digi1:11ci rcuits 10 e liminate the :Implitude matching proble m . Ring (l-:cill;l!nrl:, in which d ifferential gain stages are used ( in a two · or four-stage ring, for ex ample). ca n also produce exce llent quadrature outpu ts. Such an oscillator must he embedded within a PLL in orde r to set the frequency. Ot her digital options that do not require a PLL involve clocked circuits (see Figure 18.7 ). The first circuit prod uces qu ad rature ou tputs whose frequ ency is one fourth

18 .2 DYN A M IC RAN GE

56 1

FIGURE 18.7. Digilol quodrature generators.

that of the c loc]c. so the clock frequency may have to be exceedingly high . One attribute o f the ci rcuit. though . is that it is insen sitive to the duty cycle of the inco ming clock . since state changes occur only o n the rising edge. The output of the second ci rcuit is at one half the clock frequ ency, so this circuit demands less of the ci rcuitry. However, since slate changes occur on both rising and fallin g edges of the clock. the Qualit y of the qu ad rature outputs is now sensitive to the du ty cycle of the clock . If the clock and its inverse are ge nerated with the crude method show n. furth er degradation results from the propa gation delay of the inverter. Various othe r phase -shiftin g met hods have evolved to handle those cases where one must pro vide a quadrature relationship to a relatively broadband signa l. A COIll mo n exam ple is the passive He network shown in Figure 18.8. The geometric mean of the R C frequencies is chose n equal to the desired center frequency. Th e detai led se lection of the ind ividual stage HC time coust an rs to satisfy an arbitra ry ga in con stanc y constraint is quit e involved . bUI an extremely crude ru le of thumb often suffices for a first-cu t design. Each stage pro vides reasonably constant gain (w ithin about 0.2 dB) ove r roughly a 10% band width . so the requ ired number of stages is determin ed by the band width over which a constant gain is required . The two -stage design shown can pro vide relatively constant gain over appro ximately a ± 20% bandwidth abo ut the ce nter frequ ency by stagg ering the two time constants. For ex ample. to design for a I-GHz ce nter frequency. one might choose HIC I to correspo nd 10 abo ut l)OO Mllz . while R 2C2 might be selected to correspo nd to abo ut 1.1 GlI z.J A signiticenr di sad vantage of this network. however. is its unenuation and

norse. J

This netll.,(n is evioenuy due ttl M. J. GingelL "Single Sideband Modulation Using Selluence Asymmetric Pol yphase Networks," Electricui Communication . y.48. 1973. pp. 2 1-5. Pracucel

• 562

CHAPTER. 18 ARCHITECTURE S

FIGURE 18. 8 . Two-stage broocb:md quadrature generator .

18.2 .7

DIRE CT CO N VERS IO N

Th e upconvers ion and dual -con ve rsion receive r arc hitect ure s sec k 10 solve the image rejectio n prob lem by using a high IF 10 disp lace the image enough 10 allow a simple tiller 10 provide the necessary rejection. An alternativ e c hoice is 10 use a dow neonversion arc hitec ture in which (he intermed iate frequ e ncy is ze ro. S ince a signal and its image a re se para ted by tw ice the IF, a zero IF impli es thai the desired sig nal is its own image. T herefore. goe s the argu me nt , there is no image 10 rejec t a nd hence no need for a front-end image -rejec t filter: fro m-e nd filte ring requireme nts t1lU S become

especially easy to satisfy," Furthermore. with a zero IF. all subsequent baseband processin g can take place at the lowest po ssib le freq uency. For example, relatively low-frequ ency a nalog-to -digital conveters (A De s) and digital -signal processin g engines can implem ent the channel filter, as well as perform dem od ulation and a ncillary hou se kee ping fu ncti ons . Hen ce. no exte rnal filter s are required (in princi ple, anywa y). Furthermore, the fle xibil ity inhere nt in digital approaches opens the po ssibility fo r a " unive rsal" receiver , nne that can accommodate man y differe nt standards with one piece of hardwar e; see Figure 18.9. The mo st general dir ect-conversion recei ver req uires two mi xers a nd LO s. The reason is that the phase of the LO wit h respect 10 the incoming RF signal is importa m . If the phases :Ire coi ncid e nt (or a nticoincidc nt), the dc mod ulotcd ~ i s n a l is of

e",amplcs that Iunctinn over a decade freq uency range may aIM) be fou nd in vanou s edmons of the ANNL Handbook / or Radio Amfllt'urs. fm m abou t 198 1 10 199 2. ~ II is nonelhe less ~ene ra lly good prac tice to use some type of filler in any ca.se - to avoid. for example, oo l-of·band interferers over loading the fronl end .

18. 2 DYNAMIC RANGE

563

RF Input

FIG URE 18. 9. Direct-c0rtYer5ion receiver.

maximum strength . If the phase relationship happens to be a q uadrature one. the demod ulated signal is zero. In order to accommodate arb itrary phase relationship s between the RF and LO signals. then. a mixer must be augmented with another o ne driven by an LO in quadrature with the first one. By combining the outputs of these two mixers. correct demodulation is possible with arbitrary input phase. Mismatch in the I and Q paths is not nearly as serious here as in an image reject architect ure because - rather than attempting to reject the image - this architecture actually exploits the " image." The on ly effect of mismatches is some distort ion. With all of these attributes. it seems as if the d irect-convers ion receiver has no peer. especially for amenab ility to IC imp lementation. Indeed. the ubiquitous pager is often a direct-conversion receiver. The simple architecture (and simple sig naling scheme) thus appears to permit a highly integrated. low-cost realization. However, there arc several serious draw backs to direct conversion. and these impediment.s have thus far stymied efforts to use this architecture for more sophisticated applications. despite a considerable number of earnest attempts. Among these problems is an unfortun ate. extreme sensitivity to DC offsets (bo th interna l a." well externally induced) and II! noise : with a zero IF. offsets and II! noise represent error components within the same band as the desired signal. Co nsider . for example. the problem of detecting a 1O -J1V input signal (this value is nor atypicnll. Offsets (and II[ noise) that are small compared with this value are not eas ily achieved. Hence . noise fi gures tend to be rather pOOT. and it is easy for o ffsets to dominate the output and overload subsequent stages. The only way that pagers have managed to evade these problems to some degree is through the use of relatively unsophisticated two tone signaling. The resulting spectrum has little DC energy. so simple AC coupling largely solves the problem there.

564

CHAPTE R 18 ARCHITE CTURES

Unfortunate ly. straightforward AC coupling is nor a panacea . A low cutoff frequency is necessa ry if the mod ulat ion hap pens to have significant spec tral co mpo nents at low frequencie s. A low cutoff freq uency in turn forces the U ~ of rela tively large capac itors. poss ibly requiring the use o f off-c hip element s. A m\)TC serious difficulty is that such a network recovers slowly from overload. For exa mple. a HX)· fb CUIotT freq uency impl ies se ttli ng times on the order of 10 ms. Such slow recovery cou ld ca use the receiver 10 drop bits. Whi le so me of these problem:". may be addre ssed at the system and pro tocol level by manda ting the usc o f mod ulation met hod s thai minimize low-frequency spec tral energy. thai approach is clearly pra ct ical only in new systems. where mod ulation sc he mes have not already bee n cod ified in a standard . Other met hod s for allev iating the probl ems with ca paci tive coupling include the usc of active offset cance llation. Time -division multiple -access (T D MA ) syste ms, fo r example, inherently provide users with intervals of tim e in which they are doin g nothin g. O ffsets may he measured and removed during these period ic id le times.! An alte rnative is to use two sets of mixers so that , at any given mom ent . one is being used and the other is having its offse ts cance lled . T he two se ts of mixers exc hange places periodica lly. An import ant co nside ration in all o f these proposed methods is that the offset cance llation ca paci tors must he large enough for k TjC noise to be negligib le. Frequently, the requ ired capa ci tances are a good fraction o f a nanofarad . so the die area consumed can ofte n he unatt ractive. Another d ifficulty is intolerance of front-end no nlinearity. Any even-o rder distortion produ ces a DC offset that is signal-depe ndent. and thu s represe nts a nother " no ise" term . Prom-e nd LNAs. for example. mu st therefore be des ig ned ttl have exce ptionally high II P 2 (more relevant for this architec ture than II PJ , althoug h it's useful to have both numbers). Thi s req uiremen t usuall y forces a sig nifica nt increase in fro nt-end power d issipation since . all ot her factors hel d co nstant. ele vating bias leve ls improves linearity. Furthermore. diff erentia l struc ture s are alm ost certainly ,I necessity in the front end becau se their symmetry reduces eve n-order distortion. However. use o f differential circuits involves a doubling o f pow er consumption. Yet another problem is that of La radi ation. Since the LO is the same frequ ency as that of the RF inpu t signal. LO energy ca n find its way to the antenna and radia te, causing interfe rence to other receivers. Worse yet. the La ca n ca use interference to it.. . OWl! recei ver. Depe ndin g Oil the phase rela tionship be tween the LO compo nent a ppeari ng at the RF po rt of the mi xer and that at the norma l LO po rt. yet another DC " noise" com ponen t w ill 3ppe 3r in the base band s ignal as a result of the mi'lin g action. Since the LO puwer is ge nerally stronger than the RF s ign al (pe rhaps by many orders of magnitude ), this self-rectification of LO energy is a significant pro blem ~ J. Seve nhans et al., "An Integrated Si Bipolar Rf Transceiver fOf a Zero IF 9(XlMUz GSM Digital

Mllhile Radio mnl-End of a Il and Portable Radio: ' Proc. IEEE ClCC. May 199t , pp. 7.7. 1_4.

18.3

SUBSAMPUN G

565

indeed . Extrao rdinary isolati on must he achieved to preven t the DC offset from do minati ng the output of the mixer. Furth erm ore. the amount of LO radiation that docs leak hack into the fron t end often finds it!'. way back thro ugh a path that may be quit e sensitive to facto rs such a!'. the pro ximity of the anten na 10 nearby objects. Finally. leakage d uring tran smit o f the PA output back to the LO can cause pulling o f the LO due 10 parasitic coupling. Thi s feedback ca n cause the generation o f a whole host of highly obje ctionable spurs . In summary. the dire ct-co nversio n recei ver thus needs an exce ptionally linear LNA . two exceptionally linear mixers. two l Os Ioperuring at or ncar the RF. which may he a relatively high frequ ency). a met hod for obtaining a qu adrature relationship be twee n the two LO signals. extraord inary isola tion of the energy from these two LOs. ami a method for achieving sub microvoh offsets and Ilf noise. T hese requ irements are difficult 10 satis fy simultaneously, O ne other point deserves ment ion. The direct-conversion receiver ge ts much of its gain at one frequency. O n a power basi s. this might he as high as 10 12 or even higher. Thus. it becomes critically import ant to avoid any parasitic feedback loo p.. in order to avoid oscillation. Becau se o f the large ga in. ex traordinary input-output isolation is required to guara ntee a parasitic loop tn.ansmission we ll be low unity. Again. it can he challenging to provide this level of isola tion. Research into solv ing these prob lems continues. with progress in so me areas. hut success remains elusive.

18.3

SUBSAMPLIN G

There has bee n a fair amount o f activity recent ly on the develop ment o f subsampling urchirccturcs. This class of receive rs exploits the large ratio o f ca rrier freq uen cy to bandwidth that characterize s almost all RF links. Satisfying the Nyq uist sampling criterion requires only that we sample at twice the signal band widt h, not at twice the carrier. So. in principl e. we can sample the RF directly, 001with a muc h lower sampling frequency than the RF (sec C bupter 12). T he fly in the oi ntmen t here is that all of the noi se within the bandwid th of the front end fold s into the subsam plcd baseband . To avoid this pro blem, one would need RF filters whose band width s were impract ically narrow. If we had such filters in the first place. we would structure receivers very differently indeed . Because of this unfortunate noise -foldin g prope rty, subsampling architectures typically exhibit terrible noi se figu res (e.g.. 30 dB !). As a consequence. designers of such rece ivers are forced instead to direc t the reader's attention to lineari ty. The ca reful reade r will note. however. that co mparable "linearity" may be obtained simply by preceding an ord inary architecture with an attenuator, as mentioned in the co ntext of certain mixers (e.g.• subsampling and pote ntiometri c mixers). Th is eq uivalency should leave one with some skepticism abo ut the general utility of subsampling. Hence. although research continues. succes s ag ain remai ns elusive.

566

CHAPTER 18 ARCHITECTURE S

. To PA

BPF

FIGURE 18.10 .

Superheterodyne tron smitte...

BPF

FIGURE 18 .11. Direct-conversion

" To PA

tran ~m jlter .

18.4 TRAN SMITTER ARCHITE CTURES As one might suspect. tran smi tter architectures are ge nerally the inverses of correspo nding receiver architectures. For exa mple, a traditi on al ordi nary superheterody ne (now ofte n simply shortened to " heterody ne") transm itter is s ketched in Figure 18.10. The heterod yne transmitt er allows modul ation at a low frequency In he Ir:tn s la h:· tl up to RF in steps , but req uires filters that may be difficult to impleme nt in integrated form . To reduce the filtering requireme nts. o ne may em ploy direct (upjconversion. as shown in Figure 18. 11. As mentioned previously. feed back from the PA can perturb the LO (since they are at the same frequency) if this arc hitecture if, implemented

18. 5 O SCILL ATOR STABILI TY

567

To PA BPF ..

1-

F] FIGU RE 18 . 12 . Trommiltef with offset frequency 5yn~zer.

literally.H uge red uctions in La pulling by the PA ca n be achieved if an offset sy nrheslzer is used instead o f an La ope rating directl y at the desired RF ca rrier frequency; see Figure 18.12. In this architecture, the output carrier frequ ency is not the same as the modu latio n ca rrie r frequ ency Fl , reducing gre atly the pulling prob lems discu ssed ea rlier. Furrherm ore. any mod ulation on the input is transferred direc tly to the ca rrier without any scaling. Synthesizer loo ps that employ freq uency dividers in the feed back path multip ly any phase noise at the reference inp ut by (he divide factor N . Such noise enhancement can be highly objectionable. so the o ffset sy nthesize r 's lack of any such multi plication is a potentially significant advantage .

18. 5

O SCILLATOR STA 81l1TY

An issue that requires some discu ssion is that of La freq uency stability, In the AMPS analog cellular teleph one system, for example, the channel spaci ng is 30 klt z while the carrier frequency is appro ximately 900 Mll z . Since the cha nne l spacing is therefore roughly 30 ppm of the carrier, the 1.0 frequency must be controlled to within. say, 3 ppm if the error is to be an acceptably small fraction of the channel spaci ng. Needless 10 say, it is rather challenging to achieve - let alone sustain over variations in temperat ure and supply vo ltage - this level of accuracy. Stable voltage regulators are routinely used to so lve the voltage sensitivity problem, but eliminating the effect of temperature variation remains troubleso me. O ne straightforward solution is to enclose the oscillator in a thermostaticall y controlled environme nt. Such "ovenizcd'' osci llators may exhibit excellent stability, hut generally consume 10 0 much power for most portable applications.

568

CHAPTE R 18 ARCHITE CTURE S

An ulternufivc tech nique e xploits the repeatab ility of a crys tal' s drift wit h temperatu re. T he act ual te mperat ure is continuously measur ed . d igitized with an ADC . lhcn fed lo a calibration ROM . The output of the ROM drives a DAC. which in tum dri ves a varac tor to compc n...ate for the drift ." Thi s open-loop correction me thod is w idel y used in cellula r telephon es. a nd typicall y pro vides about an order-of-magnitude improvcrncnt in net drift ove r a temperature range of we 10 4()"C. A n.-. cenr app roach is 10 e m ploy closed-loop method s in whic h a di gital sig nal processor( DSP) continuous ly es timates the freq uency em ir by exa mining featu res o f the downconvcrt cd s ignal (typica lly at baseband) a nd retunes the osci llator 10 minimize the e rror," Thi s tec hnique el iminates the the rmomete r. ADC . ami ROM . However. blind estimation meth od s are not always robu st , so bes t performan ce results if provisions are ex plic itly made in the protocol to accommodate thi s type o f closed-loop frequency contro l. 18.6

SUMMA RY

Th is c hapte r has bui lt on the foundati on laid by the pre viou s c hapters 10 provide the basic information nee ded to construct receivers a nd tran smit te rs. We' ve see n thai the se arch for a full y integrat ed rece iver red uces to the quest for a n architec ture that does nor req uire an e xternal filter. Roth the zero -If ( i.e .• direct -co nve rsion) and low-IF (e.g.. image -rej ect) recei vers are po te ntial candidates. but the lack of co nvinc ing exislence proofs unde rscores the need fo r continued researc h . Linearit y requirem e nts of the form er a re almost ludic rously stringe nt. while the achievable image rejection of the lutte r fa lls far short of value s typically needed by most syste ms . Pe rhaps more so than a t any other level, the tradeoff's at the urchit ccturullc vel are parti cularly seriou s and e xisting solutions unsati sfying.

PROB LEM SET FOR A RC HITEC TU RES PROBL EM 1 De rive Bqn . 17, the image rejection ratio due to impe rfect quadra ture. PROBLEM 2 (a ) Design a sim ple HC- CR quadrature ge nera tor for a I-G lJ z ce nter frequency. First se lec t the ca paci ta nce so that the kTj C noi se is 1.6 x 10 - 11 V 2. a nd then determine the nec essa ry resistance from the ce nte r frequency specifica tio n, Is this resistance value reasonable'! Explai n.

In Icss so phisticated version s, the " ROf\.1" and " DAC" are simply com parators that switch capacitors into and oe t of the c ncuu . 1 In ,',OntC sys te ms. the "features" examined ca n include a !OCt paucm spec itically transm itted 10 facil itate this t)'J1(' orcorrecnon. It

n

569

PROBLEM SET

(b ) Su ppose we may ign ore any po ssible practi cal difficulties assoc iated with the rcsiste r value . However . we d iscover that the ca pacitors in this technology ha ve a bottom -pl ate capacitanc e to subsu ure that is 30 % of the main ca paci ta nce. Co mpute the gain and ph ase error at I GH z res ulting from the bo nom-plate proble m . You m3Yassume that the substrate is at ground po tential and is a superconductor. (e ) Estimate the ima ge rejection ratio if all the errors in an im age-reject architecture co me fro m this phase shift network. Derive an expression for the IR R of Ihe image -reject mixer without making any simplifying assumptions. Verify that your express ion red uces to the ones given in the text in the limit of small erro rs.

PROBLEM 3

The low-pa ss filters in the image-reject mixer would a ppear to he superfluou s because even the sum frequ ency co mponen ts are theoret icall y rejected by the architecture. Exp lain why the filter is nonetheless important in practical mi xers of this type . PROBLEM 4

The text allud ed 10 the usefulness o f the o ffset frequ ency synthesizer for usc in me direc t co nversio n receiver. Consider the transmitter architec ture show n in Figure 18.13.

PROBLEM 5

Q.

1• ,1 1

'"

F,

FIGURE 18.13. Transmitter with offset Frequency synthesizer.

(al For this transmitt er arc hitect ure. how is the output carr ier freq uency. Fi , relat ed to F I and 1-3 if lI(s ) is a low-pass filler? (hi Repe at part (a) if lI ( s ) is a high -pass filler . (e) Disc uss the advantages and d isadvantage s of the two choices.

570

CHAPTE R 18 ARCHITE CTURE S

PROBLEM 6 Assum e a cubic nonlinearity and derive a relat ionship between II P 2 and IIP3 in term s of the power series coe fficients. PROBLEM 7 Followin g a development ana logo us to the one in the text , derive an ex pression for the input st'wnd -orde r (voltage) interc ep t for a cascade of systems in which individual gains and intercepts are known .

l

CHAPTE R NINETEEN

RF CIRCUITS THROUGH THE AGES

19.1 INTRODUCTION As menti oned in C hapter I. there just isn't enoug h room in a standard engineeri ng curriculum 10 incl ude much material of a historical ruuure. In any case, goes one cornmon argume nt , looking backw ards is a waste of time. part icu larly since co nstraints on ci rcu it design in the Ie era arc conside rably di fferent from wha t they once were. T herefore , co ntinues this reasoning. solutio ns that worked well in some past age are irrelevant now. Th ai argume nt might be valid. bUI perha ps authors shouldn't prej udge the issue. Thi s final chapter therefore clo ses the bonk by presenting a tiny (and nonuniform ) sampling of circuits that represent import..nr milestones in RF (actually. radio ) design. Thi s selection is somewhat skewed . of course, because it reflects the author' s biases: co nspic uo usly abse nt. for exa mple, is the remarkable story of radar, ma inly because it has been so we ll documen ted elsewhere. Ci rcuits developed specifically for television are not incl uded either. Rather, the foc us is more on early consumer radio circuits, since the literature on this subject is scat tered and scarce. There is thus a heav y representation by circ uits of Armstrong, because he laid the foundations of modern communications ci rcuitry.

19.2

ARMSTRONG

Armstrong was the first to publi sh a co herent explanation o f vac uum tube operation. In "Some Recent Developments in the Aud ion Recei ver ," pu blished in 191 5, he offers a plot of the I- V charac teristics of the triode. som ething that the quantitatively

I IRE Pwct't'fllnR.l. v. 3. 191 5, pp. 215-4 7. II should be noted thaI he had first published triode characteristic s some.....hat earlier, in an ank le in Hlect rira t WorM ( 12 Dec em ber 1914, pp. 1149-52).

u

571

________.

• 572

CHA PTER 19

RF CIRCU ITS TH ROU G H THE AG ES

n.

c,

FIGURE 19. 1. Grid-look detector.

impaired de Forest had never bo there d to prescm .! In this landmark paper, Annstrong follow s up on the circuit implica tions of those characteristics to explain the operation of an AM demod ulator and the first pos itive -feedback amplifie r. and also 10 show how the latter may also functio n as, an oscilla tor and heterodyning element (mixer). We now study these ea rly achieveme nts of the 24- year-old inventor.

19. 2 .1 THE G RID· LEA K AM DEMO DULATO R In this demodu lator. re- draw n with mod em symbo ls as Figu re 19 . 1. a process called "grid-leak detection" is used.' For those unfamiliar with vacuum lube operation , it is per haps helpful to regard the triode as a depletion-mode n-ch ennel ju nction FET, with the- cathode . grid. and plate ana logou s to source. gate. and drain. respectively.' In " normal" oper ation. the grid potential is negat ive. so that negligi ble grid curre nt llows. However. ju st as with such a FET. a small positi ve vo ltage o n the gri d forwardbias es a diode, causing grid curre nt to flow. The grid-leak det ector uses this d iode to perform the demod ula tion. Although Arm stron g did not invent the cir cuit. its principles of operati on were shrouded in mystery until publi cation of his pa per.

2 De Forest' s conrcmpcnmeous paper . " The Aud ion _ Detector and Amplifie r" (/RE: t'rocrcdings, 111 1-1. pr o I5-J (1 ), stands in remarkable cuntrasttn An nMrong' s. Nowhe re in the 20 + page paper can one rind any quanntauve informa tion UOc.lU1 how one uses a triod e Audion. although there are a ( OUI'Ie nf hilllt!sn l1lC phntographs of equipment. Typical o f the intel lect ual qualily of the writing are stalemenls sucf as "Posnrvc Of negati ve c harges unpre ssed on the grid cuuse a diminution in [the plale ] ... current," along with the seemingly incompatible as~n i"n thai the triode Audion c\hibits •• . . . freedom from . . . distonio n." Armstrong explained h"w the forme r ctluld toe true. hul ntll al lhe same time as the latter. De Forest saw no iSMlC that needed further disl:uss inn. .1 In rallio par lance. "de tec tion" and "demod ulation" are usually symmY llltius. 4 Indeed. vacuum usbe drc uits translate quite grace fu jly imo W ET form with very few nKKlilicalions other than supply voltages .

573

19. 2 A RMSTRO NG

B-

FIGURE 19. 2 . f irst regenerative ompl i ~ er (adopted from Arm ~trongl .

In this circ uit. the sig nal fro m the ante nna is tran sformer-coupled to a resonant LC filter. which performs channel selection. A co upling capaci tor. C 2 • connects this tuned RF s ignal to the grid. A pos itive -go ing input s ignal forw ard-biases the grid cathode diode. caus ing grid current to flow and charging C2 toward the pea k input level; C2 then discharges s lowly (relative to the carrier frequ en cy). us a co nsequence of grid curre nt and of leakage in the capacitor (represe nted by R). unt il the next po sitive -going in put pulse occ urs . The voltage across C 2 (as well as the average grid voltage) therefore fo llows the modu lation enve lope. The triode then gains up this demod ulated signal ju st as in any other co mmo n-cathode amplifier. leaving a healthy a udio sig nal in the headphones. As another trivia no te. " 8+" is the standard notation for the pla te supply. Why " 8 "1 < bec ause "A" is reserved for the filament supply. Th ere is also a " C" s upply. for grid bias in tho se ci rcu its that use it.

19.2 .2

THE RE GE NERATIVE A M PLIf i ER A ND D ETECTOR

In the same paper. Arm stron g presents the first positive feedb ack am plifie r. with which one could obtai n much mor e gain fro m a single stage than had previously bee n po ssib le. His ampli fier is shown in Figure 19 .2. again re -draw n with modem symbo ls. Here. co upled indu ctors L 2 and L ) provide the positive feedback . The plate current is sampled through L ) . w hile L 2 pro vides the feed back to the grid circ uit . Th e amoun t o f feedback may be adju sted thro ugh vario us means (such as vary ing the relative spaci ng and orie ntation o f L 2 and L j ) to contro l the gain (a nd . simultaneo usly. the Q) . Wh en the sys tem is on the verge of instability, the overall gain can be consideru bly larger than without regenera tion. Furthermore. this circuit can simultaneously demodu late through grid-leak de tect ion.

574

CHAPTER. 19 RF CIRC UITS THRO UG H THE AG ES

In the sa me paper, Armstrong desc ribes alterna tive met hod s for achi eving regeneration. A mo ng these is the usc of an inductive pla te load impedan ce. acting in concert with feedback throu gh the plate- grid capac itance. He was thus the Iirst to recognize. and then explo it. the destabi lizi ng pro perties of this co mbination.

19.2 .3 OSC ILLATOR A N D MI XER The ci rcuit of Figure 19.2 ca n also he used as an osc illator simply by overcouph ng L 2 and L j . Such a co mpact generator of continuous waves wa s co nside red so mewhat of a miracle at the time because spark a pparatus. arc o scillators. and large rotating ma -

chines were then the main sources of RF energy. Furthermore. mixing action lakes place betwee n the osc illation and any RF input sig nal beca use of the nonlinearit y of the triode. Thus. demodulation ca n occ ur throu gh homod yne detec tion. with a co nversion ga in roughly proportional to the amplitude o f the oscil lation (within limits ). If the oscillatio n frequency differs from the RF. the mixing action lead s to heterodyne co nversion. and is part icu larly useful for detecting C W code signa ls. Working with heterodyne converters undo ub tedl y set the stage for Armstro ng' s invention of the supe rhete rody ne two or three years later.

19.2.4 THE SUPERREG EN ERATIV E A M PLIFIER Whil e fooling aro und with the regenerative amplifier. A rrnstm ng discovered superregenerative umplificati on.! In his paper of 1922. he describe s the basic principles and then proceed s to show a number of ci rcuits for reali zing supcrregcnerutive amplifiers. In all o f these examples. the quench oscillator is se par ure fro m the ma in amplifier. as shown in Figure 19 ,3. In this ci rcuit . VI is a regenerative oscillator /detector. with enoug h feedback fro m L l 10 L to ensure osci llations when V2 is removed from the circuit ." Device V2 is driven by a low-frequ ency oscillator and
~ " Some Rece nt Dcvetopmenb on the Rege nerative Receiver," IRE Pmct't'Jing_~ . 1922. 6 T he slightly d ifferent feet.lback connection :r.how n here was by far the most popular way of achieving regeec rauon.

.._roo- - -

19. 3 THE " A LL -A M ERICA N" 5·TUBE SUPER HET

575

R+

FI GURE 19. 3 . Eariy superregeneratiYe receiver

(wit+, sepcrcte quench oscillator).

Later. engineers discovered that a se parate quench osc illator is nor strictly nee essary, although be lief per form ance is possible if one is ava ilable . The inte rmit tent osc illa tions characteristic of a supe rregenemtive am plifier can he produced in a standard Armstrong oscillator with ce rta in choices of grid-leak ele ment values? II is poss ible for so much negative bias to build up from the oscillation s that the rube actually cuts 011' for a time unt il the grid -leak resistor disc harges the grid- leak capaci tor. Hence. the same circuit as show n in Figure 19.2 ca n quench itsel t. Jcudi ng to its usc as a superrcgencranve amplifier and detector (thro ugh nonlinear action. as usual). As should he ev ide nt by now. this is an exce ptionally versatile ci rcu it. Modem replication s using n-channc t JF ET s function similarly well, and are excellent hobby circuits for weekend experime ntation.

19.3 THE " Al l· A MERICAN " 5·TU BE SUPERHET Armstron g' s superhete rody ne underwent considerable refinement during the 1920 s and 19.10 s. Unp recedented ease of opera tio n conferred by the sing le requi red tuning control. coupled with circuit impro vements and cos t reductions mad e po ssible by better vacuum tubes. mad e the supe rhet the dominant architect ure by 19.10 . For over two decad es, the standard low-end consumer AM table radio used a complement of five vacuu m tubes (with only fo ur in the actual signal path), as shown in

7 Actually, Ann..tmn g himself describes uus effect in hi.. 191 5 paper. hut in Ihe co ntes t uf con dinons III avoid if one wants to con ..IJllCI good osc ittaors.

576

CHAPTER. 19

RF CI RCU ITS THRO UG H THE A G ES

B+

V~ I ::t:::.., V)

RI

ell

To antennacoil L _ _L!R"! Vol.

R6

AC l n, ~""" from V3

-21

R7~-V R8 B+

f iG URE 19.4 . "All -Amencc n ~ 5·tube superhet (typicol Khemotic).

the schematic of Figure 19.4.M The f il ament voltage s sum to about 120 V, allowi ng a direct connection to the AC line and so eliminating the need for a costly power transformer." The chassis also is actually connec ted to nne side of the AC line . so thai miswiring can inad vertentl y lead 10 the chassis being connected to the hot side of AC power . Hen ce, an isolati on transform er should always be used when servici ng these units. else touchin g the chassis could ca use your eye ba lls to coun ter-rotate '1 160 Hz . Th is circuit evide ntly originated in an a pplica tions note publ ished in the I940 s by RCA . who developed (and naturally wanted to sell) the tubes used in the circuit. I have nor been able to locate a primary source, however, so this conjecture remains to he verified ,

MEarl y WOi ions of this circ uu used the larger oc tal 125 A7, 125 K7, 12SQ 7, 5flL6, and 35 7.5 tulles in place of the miniature 12BE6, 12BA6, 12AV6. 5OC5, and 3SW,f, respe ctive ly, " The pow er rec nficr tube' s filament has a tap so that a 6.3-V pilothgfu ("ptlwe r lin" inoJi{'atur ) may he htXl!<{"d up to it. Th e tap also teeds the anode o f the rectitier, prov id mg a bit o f m.i ~tance betwee n the AC pow er line anoJ the:' B+ .-.upply in orde r to limit peak curre nt.

19. 3 THE " A LL-A M ERICA N" 5 -TUBE SUPERHET

577

Thc fi rst tube CVd. a 12BE6 pcntagrid converter. acts as both loca l oscilla tor and mixer. This "uutodyne" circuit is an improvement over the Ar mstrong oscillator/m ixer ill that interna l cascodmg within the tube ( provided by incrementally grounded grids 2 und 4 ) reduces coupling between the oscillator and RF port s, making independent tuning eas ier 10 achicve.!" In the aurodyne circuit, the local oscil lator is an echo of Armstrong's original regenerative circuit. T he ca thode current couples back to the first grid through transformer Tv. who se tuned secondary controls the oscillation frequency and thus the channel selected . as in all superhets. Simultaneo us tunin g of a simple bandpass filter at the RF input port aids ima ge rejectio n. Th e tuning capacitors for both circ uits are mec hanically linked {vganged"), so thai the consumer on ly has 10 turn one knob to change freq uency, Grid 2 is incrementall y grou nded and acts as a Faraday shield to isolate the oscillator and RF circuits . as d iscussed earlier. The RF signal feed s grid 3. and nonlinear interaction within the tube performs the mixing action. Grids 4 and 5 are incrernentally grounded ; they remove the Miller effect and suppress secondary electron em ission. res pectively. Note the absence of an RF amplifying stage . Strictly speaking. one is not needed for ordinary broadcast AM radio because atmospheric noise in the 1· 1\.1111 band is much larger than the e lectronic noise generated by any practical front -end circuit. Hence. an RF amplifier would not impro ve the overall noise performance of a rece iver in this band. Indee d. its gain would be somewhat of a liabilit y. as large signals co uld Ihen overload the front end. The OUlpUI of the first stage is coupled through a double -resonant IF bandpass tilter to a single IF amplifier, a 12BA6 ( V2) . operating at 455 kil l.. The 12BA6 is a pentodc and thus behaves muc h like a cascod e. allowing one to use fi tters on both the input and output ports without worry ing abo ut det uning or Instability from feedback. Demodu lation and audio am plifi catio n lake place in VJ. a 12AV6. which contains two diodes and a triode within one glass envelope. The diod es!' are used in an ordinary envelope detector (rea rranged slightly to acco mmoda te a grounded ca thode). and the triode amplifies the demodulated audio . Because there are 110 tuned circuits connected to the input and output port s. the Miller feedback of an ordinary triode is acceptable. The demodulated output is sent to two destinations. One is the output power amplifier. V4 . The other is an additiona l low-pass filter, the o utput of which is the average of the demod ulated o utput . Thi s signal is used to control automatica lly the gains o f

10 Sume puriSls reserve "a utod yne" for circuits in which the lube has signals applied In on ly tint" grid. I am not one o f the m. 11 A common variant grounds one o f the d iodes. while annther uses both in separate audio and AGe tuncuon s.

578

CHAP TER. 19

RF CI RCU ITS THRO UG H THE A G ES

the uutod yne and IF am pli fier as a function of rece ived sig nal stre ngth . T he g reate r the demodula ted o utput. the mor e negat ive the bias fed back In those stages. red ucing the ir ga in. Th is a uto ma tic ga in contro l (AGe) o r a uto matic volume co ntrol (AVC )12 thus red uces potentially jarri ng variations in output amp litude as one runes acro ss the dia l.

sues

bea m-power tube used in a Class A aud io co nfigura tion. Device V4 is a Transformer coupli ng provides the necessa ry impeda nce transformation to deliver ro ugh ly a walt of a ud io into the spea ke r. Th e power rec tifier used to generate the B+ pla te supply for the o ther lubes is a 35W4 ( V j ) . In so me later ve rs ions. this tube was replaced by a se m ico nd ucto r rec tifier. With minor varia tio ns, the All -Am erican was wid e ly copied. and clones could be found a ll o ver the wo rld. On ce it ca ug ht o n. hig h ma nufac turing volumes drove do wn the cos t of these five part icular tube type s, so anyone design ing a new radio intended fo r a cost-sensit ive applicati on tended to U 5e th e sa me tube s, and thus used sim ilar circui ts. Vari ation s a mo ng different versions we re rea lly qu ite slight (e .g.• sma ll resistor o r ca paci to r value differ ences. absence or presen ce o f ca thode resistor bypass ca pacitors. etc.).

19.4 TH E REGENCY TR - I TRANSISTOR RAD IO Th e first po rtab le transistor radio became ava ilable in lime for Christ ma s in 1954 and wa s the result of an effo rt by a yo ung Te xas Instrument s (T i l to c reate a ma ss market fo r transisto rs. Up to this time, the o nly co m me rcia l use for tra nsistor s had bee n in heari ng aids. A s the fathe r of the project , Patrick Haggert y, later noted. the a d ram at ic acco m plish ment by Ius wo uld ) aw aken pote ntia l user s idea was that to the 1':11,:1 that we were read y, will ing, a nd abl e to supply Itra nsistors l," 13 TI arr anged a deul w ith a sm all co mpany ca lled IDEA (I nd ustria l Deve lopme nt Engineering Associa tes) to perform so me cost-red uci ng modi fic ation s of Tf's first-pass circ uit ( princi pa lly designed by Paul D. Davis a nd Roger We bster) a nd then to ma nufac ture the radios through IDEA's Regency Division . Th e tas k wa s cha lle nging , as no o ne had much ex pertise w ith tra nsistor s yet . To make a tough jo b evenmore diffi cult . the ge rma nium tran sisto rs then ava ilable we re q uite poor by tod ay's sta ndards (fr s of o nly a few megah ertz at best, a nd f3 s of 10- 20), and thei r cos t wa s high . Co mpounding those difficulties was the lac k of m iniat ure com pon e nts. IDEA eve ntua lly had to s ubco ntrac t o ut the manufa ct ure of so me of the ca pacitor s to a unive rsity profe sso r who se t up a little s ide business. Th e JX}(l f q uali ty o f the sma ll spea ke rs a lso

12 AVe ""'3.\ ye t anot he r mvenuon of the ines tim able Harold Wh eeler. lJ

"The fasc inau ng story o f how the Iirsl ua nstsronzed portable rad io came 10 be is to ld in detail by Mic hae l Wolff in IEEE Spectrum, Dece mbe r 1985 . pp. 6-1- 70 .

n

•

---.J~

19.4

579

THE REG ENCY TR -I TRA N SISTOR RAD IO

+22.~V~

C17J:

c ro

" '

~

C l4

R2

cz

01

0"

fi GURE 19.5 . The Regency TR-) portable Iran~ stor rcdic.

gave the enginee rs infinit e grief. as did numerous.. othe r de sign and production difficulties . It was quite a struggle 10 cra m all of the ci rcuitry into a case small enough to fit in a (la rge) shirt pocke t. Cost was a big problem a... well. and the expensive transistors dominated it. It was determined early on that no more than four transistors co uld he used or IDEA would not he ab le to make a profit at the targeted sale price of $49 .95 {ut a time when an All-America n j -tuhcr co uld he purchased for abo ut S I5). T he four transistors accounted for about half the materials costs. As seen in Figur e 19.5. four transistors were enough , In this ci rcuit, the first transistor, QI, function s as an autod ync converter. A common-base oscillator configuration is used . with transformer coupling between collector and emit ter circ uits pro viding the feedbac k necessary for osci llation. The incomin g RF signal is tuned using a mechani sm ca lled "absorption," evidently 14 developed by the German company Telefunken aro und World war 1. In this tech nique. an LC tank is co upled to the inpu t circuit so that it shorts out (absorbs) signals at all freq uencies other tha n the resonant freq uency of the tank . The RF signal can pass to the base of Q l only when this shorting disappears . at the tank's resonant frequ ency. The non linearity of the base -emitter d iode provides the mixi ng action .

I~

"Funken" means "spark ," gi\'ing you an idea of when the company came into being.

_

580

CHA PTER 19

RF CIRCU ITS THROUG H TH E AGES

li enee. in addi tion to the local osc illator signal. the collector CUITem also has a compo nent at the sum a nd diffe rence hete rodyne ter ms. T he d ifference signal is the n fed 10 the tir51 1Fa mplifier. Q 2. through a n LC ba ndpass tiller tuned 10 the IF 0( 262 kll z, T he unusually low IF allows the low-fr tra nsistors 10 provide usefu l a mou nts of gain. bUI ex ace rba tes a n already bad image rejection pro blem. IS Th e variable ca pac itor in the a bsorptive L C fro nt-end lan k is ga nged w ith the La va ria ble capaci tor. The degree of image rejection achieved he re is best described as adequate. Th e second IF a mp lifier. Qj , is co nnected in a ma nner esse ntial ly ide ntical 10 Q 2' Th e large CIJ values ( probably a bout 30-50 pF) arc partia lly ca ncelled hy pos itive feedba ck throu gh C ln and Cu . in an homage to the neutmdyne circ uit of the 1920 s. lfl Dem odu lation is performed with a standard e nve lope detector, foll owed by a single stage o f aud io a mplificat ion. Transformers couple signals into the detector and out of the a ud io a mplifie r. Automatic gain contro l actio n is pro vided in a famili ar mann er: the demodulated audio is further R C~filtered ( here by R II a nd C 9 ) , and the result ing negative -pol arity feedback sign al controls the gai n of only the first l f stage by varying its bias. Th e huge success of the T R-I had importan t co nseq uences beyond establishing TI us a leader in the se miconductor busmess.! ? O f particular significance is tha t I B ~1 quickly aband oned developmen t of new vac uu m tu be computers. w ith Thomas Watson. J r. rea so ning tha t if tra nsistor s were mature e nough to show up in high . volume co n...umer gear, they were ready for pri me time. As he later told the story. e very time on e of his subordina tes expressed doubt about transistors. he 'd give him a T R-I . and that usually se ttled the arg ume nt. I II

19.5 THREE ·TRA N SISTOR TOY CB WALKI E·TALKI E Anot her e ngi neer at Texas Instruments. Jerry Norris. was respo nsible for develop ing the first toy walkie-talkie , in 1962. 19 Thi s widely copied and ingenious circui t uses a I ~ This IF is not without precede nt , however. Ear ly vacuum tube supc rhcts used a wi de variety of Ifl value s, with 175. 26 2,455. As me ntioned in C hapter l . Har ofd Whee ler invented neu tralization whil e wor ki ng for Hazeltine. 11 Tu as lnvn u mc ms had another major triumph in 1954. with the production of the llr:.! silicon Iran sislnrs. (Jo rdon Te al, who had joined TI fro m Bell Laboratories, te lls the slory of bei ng the last speaker at a confere nce whe re eve ryon e woo we nt before him predic ted that il wou ld take a few more years In produce silico n tra nsistors, III,' wowed the audience oy pulli ng a few silico n de vices ou t IIf his pocket and anoounci ng their availability. Th e othe r speaL:crli. were partiall y ng t u, lhou gh : T I had a mon opo ly lin silicon tran sist ors for Iour more yea rs, For more un this fasc inating story, se e Tears reminiscenc es in IEEE Tram . Electron De vices. Jul y 1976. (The enure issue is devoted 10 histori es lo ld by lhe pri ncipals fhemselves.} 1M Wo lll op. cil . 19 S{'(' J. Norri s, " Th ree-Transistor CB Tran sce iver," EIer.'I",nic.1 World . Nove mbe r 1%2. pp. 3~9.

·.., 19. .5 THREE ·TR AN SISTOR TOYCB WA LKIE-TA LKIE

581

RI3 -/IN

J

~

A

C IS

c1 T

R7

14

R

"

-9V

RI2

RIO

R

CI

CI1

R'

fiGURE 19.6. Three-Iransislor CB walkie-talkie {fI Rswitch shown in receive model.

sing le -transistor superregenerative detec tor, followed by two stages o f audio amplification in rece ive mod e (sec Figure 19 .6 ). When transmitting. the supcrregcncrative stage becomes a stable crystal-controlled 27-f\.1Hz oscillator, amplitude -mod ulated by an aud io amplifier built out of the other two transistors. The speaker doubles as a micropho ne in this mode .

19. 5 .1

RE CEI VE MODE

Transistor QI, a J()()·Mllz /T 2N2 l8 9. doe s all the RF work in this circuit. TIle anten na is physically shorter than ideal, so loading coil L3 helps " lengthen" it, electrically spcaking.j'' The inco ming RF signal is co upled through a transfor mer to the collector circuit of QI ' Ca pacitor Cs provides positive feedba ck from collector to emitter in a Colpitts oscillator configuration. As oscillations build up ex ponentially from whatever initial conditions are established by the inpu t signal. the emi tter-base

20 An elec trically shon d ipo le has a net capac itive reactance. so a series intlucuuK"t.' permits benc r power tra nsfer . To improve mdiation (and, by rec iprocuy, sen~i tivity l, a decem grou nd plane is es sential. O ne ca n increase gre
582

CHA PTE R 19

RF CI RCUITS THROUGH THE AGES

diod e ca uses the e mitter vo ltage swing 10 be asym met rical. with negative -go ing signa ls experiencing a higher imped ance than positive -go ing ones . One con sequence of this asymmetry is thai the average emitte r voltage head s 10 more nega tive values. Capaci tor C.1 charges up to this avera ge voltage through RF choke L s . Resistor R tJ discha rges C3. out at a relatively slow rate (the C2- HS net work has such a large lime consta nt that the voltage across it ma y be conside red constan t) . Th e DC value of the err uner voltage there fore heads so far downward that the oscillation actuall y terminates until CJ d isc harges sufficiently 10 restart oscillatio ns. resulting in an intermittcnt osci llation W hOM~ quench freq uency is too high to be audible. Whil e this self-q uenching action hard ly result s in optimum superregeneratio n (i n part icu lar. selectivity suffers mighti ly). it permits a one -trans istor ci rcuit to perform remarkab ly well. Anot her consequence of the asymmetric al emitter volta ge is that the tran sistor amplifies the modul ated RF signal asymmetrical ly, Hence. the collec tor curre nt contains a co mponen t roughly proport ional to the modulation itself A low-pas.. filter co nsisting of C9 • L ... and C IO removes the RF com ponent. pa..s ing only the modulat ion to the two -transistor aud io amp lifier .

19. 5. 2 TR AN SMIT MODE Although Q l acts as a self-quenched L C oscillator in the rece ive mod e, FCC regulation s requi re much more frequ ency stability and acc uracy during tran...mi t than can be provided by a simple L C network." Hence. a quartz cry stal is used to control the frequ en cy o f o sci llatio n in the transm it mode. Capacitor C 1 • which bypa sses the base to ground in the rece ive mode. acts as a coupling ca pacitor to gro unded crys tal XTAL du ring tran smit. Th e cr ystal shorts the base to ground only at its series resonance. and therefore pe rmits osci llation s only at that frequ en cy. Resistor R" is shorted ou t during tran smit to prevent intermittent osc illations . An audio sig nal derived fro m the two-tran sistor audio amplifier varie s the effective co llec tor supply voltage of the oscilla tor. thereby effecting amplitude mod ula tion. T he speaker function.s as a micro phone in this mode. and the amp lified audio signal is available at the co llector of QJ. Because of tran sformer T2 ' s co nnection to the battery. the coll ector voltage o f Q J co nsists o f a DC component on which the audio modulation is superimpos ed. Th is DC + AC signal supplies me co llec tor voltage to Q l through by pass capaci tor C 1 and indu ctor L 1, which here behaves simply a... a short. The osci lla tor ampli tude is rou ghly proportional to the collec tor supply volta ge. MJ the varyi ng supply voltage amplitude -modulate s the carri er. Althou gh the distortion

21 Channels are spaced IO·k1h apart in the 27-Mfll band. so freq ue ncies must be accurate to better than ahuut .50ppm. tnj uiring the usc of crystal-controlled escilla tors.

19. 5 TH REE-TRAN SI STOR TOY CB W A LKIE-TA LKIE

58J

from this process hardly meets the standards of high-fideli ty audio. it is certainly adequate for vo ice co mmunications ami most definitely adequate for a toy. Because thi..simple circuit provides such large gain with so few transistors. it dominates the low-end walkie-talkie market . having been copied and modified countless time.. by toy manufacturers. Every supcrrege ncrauve walkie-talkie P who ..e circ uit I've traced has j u..t one transistor doing all of the RF work. j ust as in Norris's original version . As with the All-American fi -tuber, variations amo ng different manufacturers are relativel y minor. The relativel y expensive and bulky audio coupling transform er is not used in mode m implementations since transistors now prov ide enough ga in without u. The choke L 4 is also generally omitted. ordinary He fillers being adequare. Other than these simple modifications. other superregenerative walkie-talkie circ uits are quite similar. ~ II is trivial 10 detennine wberber a sU pl.'1TC~encI1l1 i ..e circuit b used. e\ ('n withoul tracing the sche ma tic. gu perregencranve receivers have a rauctlu~ hiss unlcss a reaso nahly strung signal is

llcing received. Thai is. noise is always pre-e m. and. diminishes in ampliludc only as the signal strenglh increases. This hIl'haviur is quilC' different from Ihal of a superbererodync. for e xample. where the had ,gn Nnd no ise is roughly constant in amplitude afld irklepcoJe nl of the received signal strength.

J

INDEX

AC poIIoer. fur tilamenl luhe.14 ~;ti~c in,Juo;lof"l." noi -e ~i<"lI of. 21i9

Adltf. R. B., ad milLuli.'C

I~ ~

<.-onT latiun llllm inalll.'C. 1511, 275 of intinile.imal Cllpa<'"ila....-e. 110 opl imurn ..........-.: adrninaocc. lSlI. 2M of paralld R LC t;ml . Kto- 7, 90 of tapped "apa<'"i tur ~nal''', ~, 100 w minan,;e relaliun. relko;t ;ull <.' oeflicicnl relation 10.

270

Advan,'ed M,>t>ile Phone Se rvke, W AF \ igna l... 17 AIILen. Il ugh , 11. 22

AlC'un.kN'n. F.m~l F. W., 7 AlcllanJen;.tl ll a h.. malor , II "AlI. Am•.m ean" S-Iube' ' uPl'rhet (rad io), devd opme m

or, 32. S7S-1i. SIB alt..rnalor ta:lmulugy. hi.hlf)' uf. 22

aluminum inlcrconncd. Jt> A~t

or.

band.. idlh 21 FM sig nal conversion il\lu. 2H AM hru",ka,l , ti r'\ hy Fcsscll
AM !>walk".' hmll! , 1.'1 AM hro:l,k a'l redro ncmrulizmiun for. l ll.'l srarukrrd II' fur, 3 10- 17 A M ucHludulalllr

Arm,t wo g', u p!aoali" n of, 512 envelope d cloclur "" . Il l , 112

American Naliunal SI
f«dflll""'Md amplific:n.. .1K6-o,l gain of. I IlO bigh-f req ueoc y Iype . 178-222 in\t-rtirl g amplifu:n. 85. 4C1h. 4C17 R.llTOYo'band Iype. IN negal iH~ f«dt-a,J . ;lIlll' liticrs. ) )N... "l) neu lnl liLllliufl uf.1tJJ-6 noi", pe rformanc e u f. 268 ntJlli n\,ening amplili~ 405 posinve f«<1Nck amp hti...rs. .1M shllnl alld se no:lI peak ing .. r. 11l8--Q , hunl· pealed. IIl4 , hUnl-!ioeries type. I QI -7 so lid· ""1<' type, 20 -- 1 su perrcge neralive type. 2 11- 13 IlIned lyped . Il,l9-20 J two -port l'landw i,lth ...nhanc... mcnt of. 1117- o,l l un ilal eralization of, 20 J-6 wi lh vo llag e-..... nsil i\e cal.....ilallce.IIJ arnp litU A M tahl... radio. ...arly model ("AII· Amc ric,m" ). 32.

515- 11. Sll3 anal og circuits, 36 Ana log Devic es, 2M analog- lo- dig ilal conven ers (AIX:s ). 562. Soil amireso nan r freq ucrll'y. "I' parallel Hi.C lank . 111 arc hitC<.'l ures, SSO- 7U d irec t con ve..,.ion in . Sh2-S for dual-conversiun rec..ivers, SSt> in d ynam ic range. 551-05 for ima ge -reject rec eiv...r . 550 - 61 prob lem sel 1'1".. 56!l-1 0 fo r , ing le -cur",cn ion receiver, 554 -5 fo r . ubsampling . 50s fl". transminers, SM - 7 for up,:(Invcnion. 555 - f1 an: techno log y hislUl)' of. 22 ror indu'itriaJ illumina tio ll. 7

585

58.

IN D EX

Arm\l mllg. Edwin ll " .. ani . l , 12. 13--15, 17, 19, 2 1- 2. 2 11. 2 12, )111I , Jil l), .N Il grid- leak AM
or

alll' nu.ll io n l"t1Q,tanl. of lranslIU...siu n lillC'S. ' 42-3

;oudinn'\.. a, de Fur..... vec ue m lube, _ 11- 13. 23. 27,

m

au,Ji" ph o n IC,," Jl<....l"r ," IQ\um f'l iun in. 3411 aoo llury sy,te m, of humans. h

autoo)'1IC cin:u il , or "AII-Amrrican" 5-fuhe superf1.. t.

vn

aUI"IT\;IIH;: It" i.. c<"'lml (Af'oC' l. in ·'AIl. Amencan" 5· lube oupfflM:l. 578 l u!'>tn;Ihc 'l>lu ~ cuntrol (AVe). in "A II-A maican" 5·lul'<' 'll pcrtlrt . 5711

aU l1c "'''\c p:....-n. 2.... ~· .. ·g al .. !>illS e ffect .

77, 240

ha tla\t ing. 375 ""ndgap f'l'kn'''k-n

da....ic. 229-37 ok.. gn cumplc for. 1.'0 -3 halklga p rdn-en.:c principle. 229 Noogap "u''''ge n:fCll'nc.~. in C MOS tech ool~y. 221. 2.1.\- 5 N OOI""5 tillers. for inl...n erer...:... redu cnon. 7 Nnd", idrh channe l capdd ly related In. 124 ~l i mali(1n u f. sO" hdnd"..i"'lh estim atio n fr dou lller ...n han~'('nk.·nl u f. IQ7- Q negative fl'Cdllacl and. 3Q2-J for nui "" . 2.th q ualily fa,:lo' and, 'JI) ri ....l ime dllll ,le idy. Itl7- 74 risenme relauon s with , 171-.1, 175 \h ri nka ~c of, 17,1. 207 -Q of , hunl-se ri,·s amphfic rs, 195- 7 T-~· .. i1 ... nh'mc...mem of. I'JI) /.l."rn~ a,s e nhallcel'" of, 1711, 17')...9 1 ha nd""i"'lh- dcla Ylra ,k o lT. 1119 hamf widlh estimntion. 1..\1>- 77 open-circuit tune e(>rl ~ lml l ' (OCTs) melhod , 1..\ 7- tl1. 1Il.l pm llle lll Nel fOT. 174 -7 , h n rt '~' in: u i l l i rnc Clln, lanl.s (SC TS) InCrhnd , IIl I _1l lIa rde... n, ]ulln. 112 barrencr. Il a a llje ~. Car l. 1"11- 9 1I;ll1je~ fr "'u ubler . IQII. 217 .....am-forming e leclw,le. 3 1-2 Bell. A lC'~ andt'r Graham. JIl1 Bell Lollu nu urie5. 2 1. (,2. 247 . J1I7. 50 1. 5110 l>1a5 el n·Ulh . supply· intlcJlC'ntlc nr. 225--7 biasi ng. 223--42 prul>lem 1oC't fur . 2.17- 42 tlid' " ..." ,. in '" w -noi-e dml'litiCT. 294

Big B an~. e.:l1l,...5 of. 2411. -' I I hih ncar Irand oTmalion , 135 bipo lar ci~u i .. bia., ing ba-ed un. 223 C MOS vers ion of. IWI- 9 bipolar common -emill...r amph lier• .lOti tlipolar . 1l2. 75--1l biJllllar tec hnoluglC'll. I" ,ly ...muters for , 35 bipoler lran, i, ror mixer , c",w~i(ln ,am uf . 311l " ipolar rr~s.i ..."l'1i. lln:-ak duw n phe nomena in. 31 .de....:n l>ing fUlldjOl1~ f..r . JIl II-QI Earl y effec t in. llicl " lMli se in. !.'iJ gu ld doping of. 255 ;n':",lJlenlal mudd of. J 85 noir.e mude l fur . 1M thermal runa.. ay in. 375 biII line ..s. 119 Bode: plu. ... 170. 3Q1. 3Q7. 3911. J 24. 436.446

n

......y effect . "r hao:k·~art l>1 u body-dTect cotfficKrn.llJ Bolu l1l.ilnn·s run' la nl . 244. 2J5 bo" d w ire m"'lIcton.. 34. 5 2. flO 1ItN:'. J. C.•5 boo nded-mp ul ...."' rwkd-.. Ulpul ( HIHO , de lini.ion. o f MaMily. 395 Branly cohere r, 3 Branly. F..douard. -' Bran ain. John . (,2 Braun, I~rd inand . 5 hli dging capacuaece. 1119. 1110-1 hri~ bl ne,s Sign;l!' in n ,l<>r lcle~ i~ion. J.JI I Brirh n Marco ni. 10 Bmkaw c....1. 22'1. 2.10. 232 Bmll<'nian 1Il" li"n. noi..... a".... 243 Burns. Ken. 22 hlll"'l n"i..........r I" 'f'Cnm noise Cad y. W. G.. 50l capal'ilance eq uation s tor• 41). 51. 51- II " f imcrcun nect. ..\ 1- 1 of MOSFET s.1l 7_70 cepacinve dtl1en" ralion, .1211 capaci tol'l'l, 37- 41 ca rbon re, isl<>rs . flicker n,,;\e in. 2:'1 3 ca rboru ndu m detec to r s, Il. 9. I I . 20 Car~nn. Jobn R .• 2 1 Ca rt.. si"" f.... dhnc ~ . It>l ca...:ade " mphliers , 155. 1:'1. 1:'111. I1l5. 1tlto- l o bantlwidlh ,hri nh~e in, 201_9 noise temperature of. 2tlO opti mum gai n pe r \Ia ge in. :!ll'l--I I .. ilh si n~ r... tuned I........ 20..\ cascade sys lems de lay in. 161- 9 for imcrcept ca kulaliun, 55.1

.n.

-'J.

INDEX

linearil y 0(, 55 2-4 fur nnlSC'· fis ure co m putati
m

ca....,omp r jrcuu, J21 cath..des oiL,i c principles of, 2.' -5 o nce -coa ted. 24, 2.'i ci1ll.. hi'oler . ..... detec10r pan . 5. (). 20 ct'Uular lC kphune~ cha nne l r.pao: ing in. 567 FM by, 344 lithium niub ate n iter use in. 504 ('halllld capa.:ily, bandwillth related Ill, 124 cha nnel lcngths mnd ulation of. 77-'8. 11-I ill ~hort-" hanlK' l MOSdnk('ll. 75-6 chan""I·~ins cou nl"', in integer-N syllthe, iu r.

u""

'"

charge pump, as alterna tive f" r op-a mp I.." p filter , 465-6 , 479 cirellJ ian rfl}'lhms. o( hu man s. ...~ inj«1i..n I,d ;,ing,

."

(ireuil tbigll, ecuntJrtlics of. I ci rcular ca pac itance, coerec ucn fao.1on (or frins ing in.

"

Clapp nsci llatllr , 505, SOli, 5 12, 5 13. 524, 5J2 amphfie rs . 346 - 9, 364. 3110. J 8 1, 3'82, 3);4.

Cla'~ A

'"

des.illll eUlllple for• •'t>5-9 drai n cfticieocy of. 347 load-pu ll charac leriu linn of, :n 6--8 mud ulation of. 362- 3 Cia' " All amplifiers . 354 d<"sign u ample for, .169 _70 mtJtiulalinn o f, 36 2- 3 n a' ii B amp llhers. J 49-5 I. 3M t10Kisn eu m ple for , 369--70 drain c llieicncy of. 350 mcdulutinn of. 362- 3 Cia'" C amplifie rs. 35 1- 4, 3ti5, 311 2 de~ign cxa mple for , 309-70 drain efficiency o f. JS4 ilentiom of, J7 6 lTl ipolar Iran , isturs in, 22 5 dindo:", in, 225 di'lribu led amplifier U'C in. 2 16

o ......

' 87

implan lc:.J dill u, iun, in. 23 1 inferi ori ly ufo 183 in\~rs in. 469. 509 ju....1ion cap~:ilon in. 513 mixers, 324 -5. 33 1. 332 noise fi~ UfC~ of, 278 pha..e ,k lel:lllrS in, 4:'i1i re,i'lor opliuns in. 34, 35 RF inlegrd led circ uil dni~. 134 swit.:hes in•.119, J 3 1. 335 , 453 cOiu ial cabte. f'O"'c r· ha ndli nll capac ity of. 14 1- 2 C. ""hron . 8 .1 _, Ili7 coh crcr Ilranl y's, .1 repl...."l:lllt:nl of. 8 col nrtd n isioo . pu'· hl e crrcuu in. 4Jo,I- 41) Col pilh ' T) '>lal ' l<'>CiUal
'"

des;:rihing func lion analysis of, 49 5- 5110 slart up 1l1tl tkl of, 49' J '••a,'d onn, for. S44 Colum hia Univmll-StlUree (CS ) am phf'<"l", 154 indul 'lively de ge nerat ed. 28 2 'It'ulralitalillfl of. 2ll5. 206 with ., hunt inpul re..isltlr . 2711, .'05 wilh si nllk tuned ItW , 200-1 UTO·peakd. 187 com municati tlns syucms, n\Cillator.l for . 49 2 comm uta ung mulnpliers. as phase: detectors . 45 1- 3 com parator dest..ri binll furlt:tinn (nr , 4116 ...:illator wi th. 545 compe nsa ling carec oaece. in nanrody ne ampl iti<"l", 16 cornpens.at'un n . >t ·locu~ eump les and. 4 15- 22, -02 thro ugh ga in red ucuon. 423-·6 compleme ntary metal-uvide silic" n proce' sc N, .,'~ .. C MO S pmces >cs complementary 10 ab..ohne le mpt"rah ue tCTATl. 224 . 225 , 2211 ,229 compulers use of ,...,twtln an alysis, 146 v:..:uum tul>es in. 581l conductors three at.Ij",,·ent wires in. 45-7 wire-()\ 'er-sroond plane. 4 1- 3 wire IoaIldwic hed in, 44 - 5 CllIl'>lanl·cn~cklf'C' amphfiers. 344 ,.16 2 cnn'>lant-S.. hi a.~, 235-1 , 2J8. 241 co n,tant· l lirles. 118 l h.. C" nlin",,,,,,' WIH'f (Aitk en) . 22 c
588

ctos.."v~ r fn:tjlll' IICy. J'f7. 4 25, 4 29 . 44 5, 448 cm.......q uad MOSFET. 326. ) 40 LTO,;..-qUd\l Irans<:ooouL' lor, 327-11 CRT5. W) dc,i gnali" n, fur, .n cryslao.lyne ~~jvo: , . of u-v, 20-1

cry.tal a, q Udl11 n: "' lRJlur in m n
lcmperatun: ·indu....ed dri ft of. 5t>l1

-rte ('ry >.la llkll'l,:lo, HI Doug la.. I. 22 cry , tal
his.,1I'Y or. 22

1n\"C'nl ion u f. 4 - 5 cry,."l d iode. IS

cry"'al u.ciliaIOl'l<. .'I07~9. S14, 514 cry'lal rildio. II J t arly n,,~1 of. S. 6 CU m'h l m;m ", prohkm ba-ed on. Il4 C'UloIT f rrtjUl"nc)'. of lumped hne~ 118-9 ""'1\.":5. of TM,j ~. 542

cyd.l\l..lt llll iU)'

DAC. ~rr di ~ i taI·lo-l nalog coov e rter

D.rrlln l(ltlll pair, ,,, Ir dou l'ok-r. 198 . 2 17 D-",js. Paul D.. 578 DC lra n, mi..., itlll. of p.......er , III ;l. Lee. 7.8. 1O~ 1 4, 17. 22. 25. 27 De N~ lri" d...audion. 9, 12. 2.1. 57 1- 2 ....Ia)'. in ca...:ade , y'l..-m... 167-119 dt'lla-s iglTlilla :hn iljUC. for lIuii'C di..u-ihution . 52 1 cen...d lllali..n in ~A II- A lTK'rkan" 5 ' lu~ MJprnx-l. 577 in t'arly lram i, tur rad io. 580 d...m..dulal"r, a~ ,lrtt'l;t' lf. 337 tkp.... li.....·m" de
40H

oJ,.'lc.:tnrs, " ar ly p;lICn1 s for. :'i DIHL, ." '1' drain-irutt": tt1 hamer Illwering differenlial pair, a, fr douhler, 198 digitul uC\kc . r ohcrc r a" 4 digil'll el.'Cln lllk"" IC pn l<''' '''' ' for. 34 digilal signal pn ..·e' " '' ( DS P)• .'I Ml digiful sysl..ms, hit ,'m,rs in. 182 digiwl·I.. ·mlalnl> eonvertcr ( P AC ), 523, 524. 5611 dij:ilal watche s, q uanl crys tals in, .'50 I, .503 diod e -ring m i ~ e ,.,. , :H 7_4() di,l ."" hn"I" IlY, 22' noi.'C m odel for. 2M struclure uf, 23 V_I t'haraclr nstio;5 o r. 27 diIl:<.1 ·co nvt'f" iun Il:<.-ei\ et'i. :'i52 .Sidto Nllld inr..n rutiun in. 31 1 dil"t"Cl·cu n\-ef'li.1O Iransm lller. % d ll~s:1 digiral syn l~\i/er. 523-4

INDEX

displacement current. invrrn i"n of. 2 di" OI1i"n, fmm am illilicn , 3117 di.sl" n i' lO products. L1 nil for. 14 1 distrihulcd amp lilit'n. 19 1. 214 .. l tI d i...rihuled sy"... ms. 114 - 33 lumped syslems link to. 11b--17 problem set for. 131-3 d lthning mo l
... ,

doob le-sidchand 00.., ti, urt' I USO NF l. 3 10- 11 dou hk 4 appcd ",""lOal'''' . 1OJ--4 douhlel problem ... in I..g cornpo:n.~li"n . 419 , 4 .1 1 nou ,lo1.... A .• 22 drain CUfl't'nl upm;~ion ror. 67 in MOSF ET lTiode "" ion, 65--7 in !oit ur~ titlO , 65- 7 dra m currenl nui-e. 2411-9 drai n diode leht'T hrcakdt""'·n. in MOS de\·icn. 373 drai n dlic imcy. "I' p Mn am phlim<. 34K. 3~. 3.'i4 dram·induced bani" ..... ·ni ng I DIBL). 76. 236 cu mulari\'o: dft'd.s of. 85 dra in modu lalion. in p lW('f amplihrn. 363 drain n"i .., . gate noi "C' com: lali" n with . 272. 274 drain-llOUK-epunchthn-ugh . in MOS tk\ i."n. 373 driv ing-poi nt impcdan<-e. 1211 of itera ted "' rut.111mt. 117- 19 dual,cul... o:~ion receivers , archll~_ ru", fnr. 5 ~ "duck te~I " \~iOll. of Ot."Cam '~ nu tJl'. 11f! DUh..'cod y, H..nry lIam .. 1O Cha-e. 5. I I Ea rly err....1 . in bipolar tran'i' turs. 77. ~ I(J Earl .\I R"J'" Wal'<' rf Phillirs ), 22 Early mllaj:t'. 229 F..di'o
n r....,,,rs

p

....

INDEX

Europe , T V chr oma ..:he me of, W I-I El't''' ;'' /f Tt'It'lI rurn, 7 e_ d usi\'e -O R gale , 'IS fIha 'C' de lco;-lnr, 4.'i3-5. 479 Farad ay shid d . .10 , 57,.'i77 FllI'lIday'. law. 115 1M FUlht-, /1 H" J,/ , Ilk f ure sil . 22 Ir d" u hler hand wi lh c:'Tlha........1TIr111 \'o ilh, 197- 9 Ballp lype, 198--9,2 17 Dart lnglnn pair 11-\. 197. 2 17 diffl.'f1"nlial pai r " s. 197 F.:.kra l Cvmmunieali,,,,,, Com miui,,,,, ( R'C ). 4.19--40 ~al Te\q>hnno: and Tek1!raph Co., 12 " "'j>dhn A"'f'Iiu Principil'f I R..sen'4.4rt I. 1(,7 f«dhdo:l< . y" e lm., 3S~37 with aoJd'tJ\'e noj'C ....>UI'l.'e'!I. 393 oompen....tinn in. 41 5--22, 413 t:4 - 5 Iheml al nl1i .... in, 2411--5 0 F FT alguri1hms. 2<,1 11, 325 tield -.. tTec·t lran.s i slo'~ • .fl'j> fille,., lI i ll., in~ " f. 237 ti""Htrd..r I" w-pass .~ Y' le m ~ . formu las for. 4 11- 12 first-nrder ph ase -loc ked IlK'p S. 442- 4 fi,.,. l-pa " design. inc remental Illodel of, 154 [~ clll i ng. John AmhTtlse, 10. I I [·lemi ng va lve, 10. I I. 23 flick er noise, 252 -3 in j urwt ions. 255 in MO SFEn , 25 4 ~5 in resisltlr•• 253-4

1''''\

FM

or;

tmnd widlh 21 Il.'~al hall ie " v..r, 2 1- 2 rh ase lrneamy imJKK'lan"'e in. ) 112 slupe dc nKKlulalinn of, 20 F M hroadca st rad iu. siltnal uf, 5~5 I' M / PM sy~lcm ... .. ff'Cl ~y nllK:si lA:'r use in. ~22 F M r;l(J iu. ..~ nler fl'aj """"ies of, 20 1 404h C MOS pha se -lud".."tJ I'K""" U.se in design ..u mr le-<. 470 ---ll

5'9

Fuu rie, cuml"menr_, uf sil:na l_, 12J. 18 2 Fo u rier Irans forms, It>lI-<,l. 3D. 45 .1, 411t.• .'i311, 53<,1 - fo ur ....ven. Mpalelll (117771I . .. f MaK o ni. 1 f,.....·Iis, fl't' §ynliK:>;i/t"f'> frin~ e cOll«1ion. f" r circ ular npacitance• .'ilt Fu fler , Leonard. 1 fuocti,,,,, ~cncr-.ll' ''', 49 1 _~ GaA ~ IOI.- hnolugy. distrihu led amrli 'iCT use in. 2 1t. ga in. prm isi" tl by surnrq:cncr.uor, III gai n----b.ond", idlh prod......, IS , 2 12 linear ga in- h.md \'o ilf Mla.. .M2 11 ~ain---dc lay mtd..",ffs, eumpks of. 2 1J ! ain matlin, "'OO1ptJtuion of. J% J ain phase plllll<'. of O\oI:i ll,ilur , 4'1.\-4 Jain reducnoe. cclfflpen'illion Ihmu ll:h , 423-6 , ak "'-"«It ...... 5, 6 ~aml lyl. 5l'1' prupall:a1iuncn".. .unl

r -p1anc. 135 ~arage-d< ..". ope"""

filhium niuh.ale re_al,'" u...... in. 504 supe rrege ncral ur u.'OC' in. 20 galt capaci....wc. of MOS tlev kes, IIJ galt capa<:i l'>r. 4(J n.... In C Mo.<; pr............, J<,I ga le currenl. from hoi e]a'ln>l15, 76 gale noise, 249-50. 25 I d rain noi.... CCItTl: I.llit"" wi lli. 212 . 214 gate ",·erdriv e, 74 delinilinn of. 2M tale oxide rupture . in MOS de vices. 3n Gaussian d iSlrill' Jli'lfl . 244, 25 1, 25J Gauss', law, 115 Gen eral EI<'<: lrie Co .• 7, 24 ge rmanium di" ,k . 5 gem lanium lransislor". early lyrc ~ 57 11 gig ahe rtz range. mic rowave d i,K1... in, D7 gig a pre fix. in unils. 141 nnten. Barr ic. J i ll Gilbe rt m ixers. 3 1'). 329. 3.1 1, 4.~(). 4 ~ J Iinearily «r. 324 noi sc figure of, 32.1 - 4 Gilbert multi-tan h arrunge r ncnt. 32 \1, .14 ,' Ginz ton, E. 1.., 2 14 GI" hal l'm ili" ning S ys l~rn (!iI'S) , fre... llJ ~n ..y u",d in. 2113 II,. cel l. 32\1 u.K1dar d. Ro be r1 , 12, J K.'i gold do ping, of lliJKllar lra nsish.... . 25~ golden rarin (gn hlen sec uon I. I III Gra hc l. A ., 1(,7 Gray, P. E.• 1M GI't'r, Arm " r" ng·. ..xplanali' lfl of, 512 gyralor. 2611

u',

590

INDEX

" a, geny. !>-.urick. 5711 lI aj imiri mUdd. fn,. rha-e noi.'
half--e<:li'>Ils for line tt'nn miltion of. I l ll - .lU m ·deri~ed.

130- 1

han nnn ic t1i.' lun ion. 2%. 2~

lIarllcy n••c illat"" ~~. 524 lI v elline, Louis. 16, 17.205,5110 Il eavi' idc. Oh \Tl". 2. 124 hc-pl.lde " in ' -ilnJUm tulle... J2

Hcrtz , He Inrich . 1- 2. 22 tlC'lc",dy ne pn nd pk . I... .mIl U<"Akll , W. R., 214 higb-\k,'ke -count c ircuit. I high-f""'!uC' ,. ;UllJ'lIfid" lk\i gn. 1711--222 prohk m ' ror, 217-22 high-fmjUl:ocy I'l"fnrmall«', hgu m; of merit for.

70-2 high-sid<'io}<'l;-I".... 1.0 f""'luency and, :'iSS high-speN r"II"lIrml, heha\ ior of. 2111--1\1 1I,,"e '5 ph;a.\C lk1ector ,-klO- ) hoIk..., l;J>ind iooUOl--ton.. 4<1 homodyl1C' m :ei"t'B, " 55

It,,,al 'N: ill'''' lf r",. 439 o«ond.. lI'dtT intm.-ep in. 195 IISPICE oof'VraJe. 325 h)'hlid f'UoIrt1l"1crl1. 138

hyPffill>ropl ju....1i..... . 411

Ie romp"nc nl,.. .l4-41, 571 prohlr m \(' 1 for. :'i1l- 6 1 In n"i... in. .... 7, 44 9 Ie induct"",, 34 1"C'....... 'Ol'

IC miun . 3H Ie ~,i'l'''''', 34 IF. ".... inlcnnCtliah,' f~ uenc)' (I F) sig na l image L'(JIld Ul.1a nce• .'I1 ima,!!l'-rcje.c1 titte r . o r ' in,!!le ·co n\'ers io n recti.'er , .'1.'1 4 image rejc.·li,JIl prollkm , ' in,!! le ·cUI1\'ersio n I'l.'t:t'iver and• .'1 .'14 irnage ·rcj"di on prulller n, ." lUli'JIl "r. .'in2 irna lle ·rcj.'Cli' m ratin (lRR l. de tin rtion o r, .'i.'i1l--9• .'\n9 imege -reje ct rcce,wr arch iln:l lln: rur , .'I.'\n- li l , .'In~ ga in 'tlld pha .... errors in. 55~-9 quadrature g..nerators of• .'I .~9- n l image . igna l o r mix(" ' , 3 1U "r , ing le _convers io n receiv er• .'1.'1 4 ' M'lrrir d'l Gen il'." (de I'orcsn, 22 itTI f'<'<.l"'''; e, co nvene r ror• .'1 10 mod ificati"n of. lin o r HI .C nclwurk , 1110 , Ian..lard i, ed (5ll or 7.'1 ,>l1I1.s ). 14 1- 4 lran"hmllatll>n o r. 2 111 Iran, ;enl ( plll.-e) type , 12ll impedance ~ ipr..canon , 2n~ imf'<'<.la ncre Inm,fnrm ali" n mndel . 2&1) impu l,< resp' >n'<. ITlI. m..n" of. 16 7_8 implll,< ...n, iliv;l)' fun.·linn IIS F). 54 1• .'1 ".'1 or I.C '''CillaIOr • .'137 • .'\.18 incandc ....'C'nt li,hl hu ll!. 9. 23

Inde re ndenl Rad io Man ufac lun:rs Associanon (IR M A). 17 inductance form ulas. 49, .'I.'\-7 ind uelive T-tlt'IW"rL. as n u.te l for ';' lUpled i ndUL' t'~.

,..

induelOl'!l. J 7_6 1 d ie area of. 3J los "itlt'''' of. " 1- 2 in p..ssrve Rl. C llt'!Vl'urIc".lln Ind uslrial Dt'\-ek'l'tnenl En,!!II'..e ring A""lCia'e~ I IDEA )• .'I71l--9 injeclion ·lock ed (If;(,"illal on. • .'1 1.'1 injecliorl locLing. 4.19 input impedance. in veceum tulle > ;and MOSFETs. 280. 2111 input ---<1U lJ'Ul impcd.mces , of . h unl - lIoCrin amplJli en. 19.'\- 7 inpu l -OUlpul ...... i"'..ncCll. o( ~hunt~ arnphtien.

193-.'1 InMilule of Ek..1rical and EkctmnM:5 Eng.no:en lIEEE ).1 41 integer - '" 5)'nlne.;7lT • .'1 19 inlegra led ci ru>;I.......... Ie ;nl~ p. >int. third ,ll'def I IPJ ). 3 12, J42 N ima linn of. 297-JOO. 3 24 in single -.....ge LNA ... JOI • .'\()2 inlrrronned. J 4. 37. 311. I J7 ca pac ilance of. 4 1_7 . 11lJ in power amplinen, .1" 7 problem" on• .'I". no imerfcrrnce rrduclin" of. 6. 7, 14

from ~UI'ClTq;"IICTllI."". I "

intr rmt'dialc fne4 in"..n; ng amp lifier. 406, 407, 424 with la, cornp.. nsaln r, 427 w ith lead compen sator, 4 ,\() with rcd uL'ed-gllin .' (.ml"' nsatio n. 42.'1 io n implant ation, source -d rain ditlu vions defined b)',

zw,

as

[SF. ,<<'e impul.... se nsitivity fu nction iterat ed stru ctu re.•• dri vin g-point impedance

"r. 117- 19

Jasbc rg. J. H.. 2 14

JFETs. t>4 de scribing functiun tran ....on d uct alicC for . 411~. 4'Xl j iller noise, 4411, 4nn j iller rea king. in ,<;C(;oncl..o n "' r flhase -lnd t't1 t" o J's. 446-7 Juhnso n. J. D,. 243 John "' JIl nn i'l', s.... lhermal noi.-e juoc liu n ca pac ilan,·e. 40- 1 j uacnon capaciton• .'I9- hll in o 'loC ilidlor IUlllng. 5 13 j Unc1io ns. 20 llkl cr lM:>i se in. 2.'1.'1

INDE X

"," nnoo y,1 . E . 3011 K irchtlufr ~ cUrTenl law ( Ke L ), 115, 116, 347, 4119 KirchhutT' s laws, 114 Kin:h huff 's voltag e law (K VL l, 115. 116, 229 ladder nClworb, 117, 118. 175 inpu t impedancc o f, 127 lag compcnwion dn""t.:k of, ·UII in fee.Jhad l:sY\l:crm, 426-9 l ..ilplao:c transfoem, 429. 44 7 larg c -sign al pt-'l'fOfffi;ln~"e. in lo.... -noi.c amplili cn.. 295 -301 laicralllu ~ capaci lur, .111- 9 Laue r, Mariu s. 17 I .C ci rcui l, .1l9 in cry>tal rad i"s, S, 7 u.c 10 cnange loca l....cillalOr freq ue......-ics. 30lI L C t...ollalor impuloc 5Cn~ilivily func tioon, 537, nil wa~'dnnns for , 531 lead CCll1lpenl.alion , in feedbac k l y,lems. 429-32 LEO. discuvery of, I. 9, 14 GaN hlue type, 9 , 20 LIX,o n mudd, fur phase noise. B5, 53 6, 54 1. 547 Levy, Lucien, 30 11 Lewi.., Tom. 22 lighl , freq ue ncy o f, 247 lighlly duped drain I LDDl struct ure. in MOS devices.

7. lilM:'nfdd , Juliull, 62

nrear-e nverope oper..lion , of JlO""cr amp li lier.;, :u4 lineari ly assu mption 3Illi in luw-nuise al1 lpliticrs , 295--301 lim,'arita liull, in freq uency do main . 4115 linear lime -invariant ( Ll' ll noisc ana lysis. 537 , 5.19, 542, 544 Ii Ilt:'M lime -\'ary ing ( LT V) ootse mode l, 54 1. 542. 544 .

or.

'"

b"luid l'wrellCf, II Iilhium niuha le. use in ~'\()nalor.; ;and lillen.. 504 [...mllich anemJII, 104- 5 L-malch circuil(l ). 94 -5. 109. 1% , 3Nl rr -matc h circui t as cascade of, 9fl LM.l 09 5-V regUlalOr rc. 227 LNA , ,' cc low-noi ..: amplifier local ll>CiIlalor ( LO I amplitude of. 309 f~uellt;y choice for , 5S5 Iwmonics of, 332 . 34 1 f,.. homodyne receivee . 4}1l L C wml'linatiun 10 cnang<: fmjUoroc y of. 30lI oo i.c tig ure and IP} of. 334 po....... er tran sfer fro m. 3 10 radiation frum, 564 RF signal mix ing wit h, 320 signa l rower of, 3 13, 3 16. 3 17 LOO~e , Oliver, 3 , 7 Luetlner. E.• 22 lo" LaIs, 32 Ioog-cnannellimi l .leI;hncOOgy sca ling in. 72

•

59 1

luup tillcfll in phase - Ioc k~d loop... 463-8 . 4 7 ~t> rippl e components in, 4fl1'l~8 lou ps, induc tanc e ur. 56 lou p tran , mi" iuns, clut ches and. 40 7-8 Lo..ev, O leg, I , 20 - 1. 22 Losev cry~lad)'ne receiver. 2 1 Ios~y lnn'mi,~ ion line" lumped rnook' i f... , 119-22 kM--frntu.e......"Y ~ai n, 0( andw idlt\ of. 247 LT V, .",'1' linear lime -varying ( LTV) no ise mode l lunun eace sign al. in color tele vision. 44i1 lumped mudd . fur lus, y Irllnsm issi"n line, 119-20 lum ped RL C rnook'l. of lram mij,j;ion· llIlc lIe~ mclll_ 119 lumP"'! ~y\l:ems, 114 - 33 charack'lislic impedance of. 88.89 cutu ff frrt.lue......"Y o f, 128-9 t1iSlr lhutC'l.i sysle rns h nklo. ll b- I1 , 13 1 lenn inaliun of, 129-30 magic hox . in phase no ise, 5.~ O, 54 ~ , 54~ magne sium getter, in Vlll'u um lube. 25 Manha lllln slwrs lep jogs. 47 M.m.-o ni , G ugliel mo. 2, 4. 5. " 22 Ma.,qchu!i('(h Inslil ule uf TedlnoIogy ( Min 141 m.uimum pcw er IImsfer lhrorem, 93-4, 344 , 375 Max""eII. James C lerk, 1-2. 385 Maxwell pIl'gram. 46. 47 MlU "" c ll '~ eq uanons. 2. 116 in diffe ren tia l for m, 114- 15 vector c..kul u, ap plied Ill , 124 McCand less, W., II m-de rtved balf- secrion s, I :\tj- I Mf'ij,...F" k"e rna I"qualiun and met hod . 43 MESH:.' s, ....'SClibing fune lions for, 4~1 meta l inlen:urmet. a, small ~sistl ...,.16 Meca-Suf''Aart' CoqluraIion, 325 "A Melhud f,... the Dct enuinalion of the Transfer fU llCliull of Electne ucC ircuits" (Coc hrun and GrallC'h . 167 micfohcnrit s, illduclance !livcn in. 55 micrustrip lines, 133 microwa ve ci rcuits and sys tems, 134 microwave diude s, for radar, 337 micn ......ave radrati.." , in uni' en.e, 2411 rnili lary application s, o f direct digual sy nlhcs i.leT, 523

I

5"

iNDEX

Milk... e ffec t. 2lJ, JO, 1.55, 156, 17(" 192. 195. 1'J6. 202. 2t~ . 2lJl. 392. 506. mim 'r l!aJ~im"mo;l"r. ill fle ming valve. 10. I I mirrors, in PMOS. 226_7. 236, 2311 m ix" K Jim-.n coovtT'>i" " gain o f, J 10

sn

di.d' ·ting mh.,.... 337- 40 double·hala....-ed mi1ef'l. 322- 9. 331-5 fundal11l:nlal~ inle~J11

o f, ) U9- IJ

pli nl J

or. 324

i...>l"rion and, JII- l ) tiIK ar, nonlllled, ~y 'lem, b . J 14 hll('m ly J'llflIl11I:lel'1l of . 3 11- 13 IlIIeari u li' lII led m iqUC'll for . 325-9

muluplicr-b.a......,j rnil W<, J IS-J5 n"i'l' in. 272

, hul noise in. 25 1 thermal n"i se in, 248-511 lri"dc reg inn in. 65, 66 two -port nolS('!"' ramcl en ur, 273-7 M as lran. i,I...., use a' a ITsisl...., 35- h " mOlnm. >aling. luw -frrq u.:ncy .'"'Ci llalion a" 434, M

0 5 Mlh-onler f inl('nn, . lulaliun J di'lurtion p....Iu<1. 552 -3 mulliplit' r-ha"t'd mi \c~. 3 13--35 mullipllCn co m mulalinll mu llipllt'n;, 45 1- 3 a., r h'lose lk1o:clUR , 4~ muh i-Wlh IrdnS('()rlojocl,..., 329. J43 mu tual condocloUlCe, ~f'<' Ir:Ifl"coooUl.unc('

J->knlitlfTll'uil: mil Cn. 329--J I pfutolcm """ f••, 34H--J RF lran...:" nd oclon for . 32 1 . i" glc:·haJan.:n! mil tn. J 1 ~22

naml'WhanoJ amph licn. 1'N Natiuna l S<'mic onducl' ... Cnrp.• 227 Nation.ll Tck....i,t.J(l Slantlato.1, c.....rm ju ee (msc,_

,"!lIafl:·la'" m l l Cf . l iS- IS

n-ch annel junction F ~T, f>.1 n-c har me l MOSF ET. 64 . b5 fll:llau,'e ro:cdhack, for m ""lNJnoJ illnph lten, 19 1 llC'~ali"c r~ k sys k ms deSC'n' lt" 'ity of. J9 1-5 di...'<.mnectc'd, oWn noise ami . 393-4 ne~ati"c impedance tUnv('ner t NIC ), 5 10 lICi!dlive ~~i~tal'lC'e lI'lC i1Ial' I("'S . :11»--14 IIl.'twurl. ('n~rgy. of par-d id RL C tank , KII nctllrali u li'-J(l. o f arnph li,.".. 20 .l -6 .....u lmdyne .. mpliher . 11'I N....· }f,n Ht'ro/d . 7 Tht' Nt'''', Y" n TI", ,,s. JIl l,l N1C. ~"" nc~ali'e iml'l'd a....'C' con verte r nichmm~ ~~i'lon;, 3n N icl !oCn '~ C
"llNrnphng mixon. n5-7 !hm:' IXlf1 mi..er , J 14

' ......·p..n milCf. 3 14 mil.ng fun,:t iun . definit ion for . 333 mol>itity degradation, in MO S ,x,,'ices. 79 "lOS capad l'>r•.W MOS dr\- it:n

In'al dt_ n phcn..mena in. 372- 3. 374 de!;cril>j ng hm,:.;...., for. 41\8-9 1 hi. ..")' o f. 62- 3 m" b,lil y dc~rdllaliun in. 7'1 m'iso: mood [nr . 2M ('r"l'r.lliun in w(';ok ill\,t'T'oinn, 72-3 phy~io .. r. 1'I2-85 pruhlt-m w i ["r. III - 5 RF apr lit'ali"n, uf, t « "I.I b<.lra' c CUrTt' nl in, 76 l('mlll'ralu!'C \ ar-illliurl in. 7K thertuul runa .. ay in , 375 Ihl'o: \ I",ld refucuon in, 71'1 ,rans;llilnc c fTecl" in , 79- 110 velo city s aturation e1fCt 'I' "n. 75~n M OS I'ET~

al'l'Ultlulali" Il-ln"d e type. 3':1 bac k-gate hia, in, 77- 11 b andwidth e.• tinunhm r" r, 152 c" p"c il1lllcc, "r. h7-70 crtl"-'l uad. 32n, 341 J describin g rU ll\Cl iurl ~ fur, 4119, 4911, 498 di"d e-connt'Clcd , 227 dra in current noi'>C in, 542 Hid er nube in. 254 -5 gale structure of. 2110 1111'111 irnped.ul\Ce in, 211l1. 211 1 in' lahilily " r. 372 long-channeltype. 72. 3 16 mi\ ing hy, _UO ope ralion in weak invcni"n. 72- 3 f'dra,ilil: t'i1I>aci lilnl'C's o f, 2 17 phy , it~ «, 64 - 72 prino:iplc5 0[. t>3 in , hort-o:hilnno:l re~i me . 73-7

..,

mimlf'. 4n'l

n"i .e in, 255 transcon duc tance in,lI l up w"rd thres hold shills in, 77 NMOS lran ,i sh'rs , dev ice thre, h" ld nf, 373 Nobe l Prize winners . 5, 63, 2411 Noe, J . U.• 2 14 n"i ~ . 24 J-. 7 1 bandwid th for , 24l> culculauons of. 2(,0 - 3 cyclnslalionilry s( l\lrt'C~ " r, 542 from DC devicc. , 5M defin ilion or. 243 urain current noio;.c . 248-1,1 evo lution "f. 54 1 nid .CT I" 'i sc. 2~2-3 i!alC noise. 24l)-5 0 inplll inro phase-loc ked lu.'f'I', 447_50 j ille r noise, 44 K limils " f opl im il.alinn nf. 2tJ1) hoca r lime -..-ary ini! model [,..., 5-& 1 neg ative ret.xl "a.:k and, J 9.1--4 n" i.e perl"orma.,,:e. 264 - 5

OQ

INDEX

paramet ers for, 272 plia.'C' "" i\C... .... pha\C flui'" p" f'I.'um " ,,is<'', 255-250 p ower-coesuain•.-..J ''l''imilOll i,," prtlh k m 'ICt f'l{. 2M _7] pru,;e..,;i n~

cka" h""'" 10

a~"id.

or. 2lW-S 25h

,h,,' "o i..... 2:'Wl-2, S42 therma l noise.". 24 J-'~lI •.'ill ' ....o -port ItleOry ur. 2..~-flO

unil

r..... 14 1

of. W) rw.1i S<" (1ICIo, . 26 1 u pcon~ e", ion

Jcn'auunof, 2..~ Hl ",, is<' fi, ure ( NF l. 2hO. 26 1. 291 of ca.sc.s...... y-.ten.... 267. 551-2 compulau nn uf. 2n 7. J ill..1B. 55 1 " f G ilhn1

mj ~Cf'•.12J-.4

~i n, k-~idchll.oo

VI. "'ouhk -~idclwkt. JIO - II Iloi...: modd_. 2M-6 noioc IC'mpcnlllre. 2Hl . 2M ",';'C"'o·pha.o;e-('ffUf U":o.n.. fcr fur",:t,un, -l4 7 n,," )' rr. i,t i'-e 1ldV.,.rl . 261. 2"2 nonim erung amplilio:n..-$Q~ 1lI...1">e'ari,..,liun.
Nnm \ . Jnry. SIlO. 5113 Non..n etju ivalml mud... . n"i\C in. 245- 6

NOf1on form, H3 nu,-j' lor. ~ . pu of vacuum IU ~ e...llUlinn , .n Ny
' 93

Colpnts osc tlknor. 4'15-.51111. !\04_ n. 5 I I, 52-1, :'i25-(,. 54 n. !l47 cl)"stal o..cillal......, .507-9. 5 14. 524 dependen ce on no ntmeannes. -I!I.'I de'oCrihing fuoc linn ana lysis of. 492 - .' fUOC linn ge neralnr . 4' 11-.5 Harl ley O!\l.·illalnr • .5I1S. 524 linear type, 4114 - 5 1Iel:1I1ive resi"'aoce O'iCi llal'lf , .511'1-1 4 . 5 27 nn i", fm m. 4_.19 osc illalinn amplil utk in. 4ll.4 O!oCillal inn fretjuency in. 5.10 osc itlalion stert up in. 47-8 superreg ene ranve amplifier P. 19 TITO osc illa" lf. :'iOb- 7 1Wk"d type. :'\l»--9 osc illosc opev ph..'\(' di"',>rti"n in. 1112 prohe of. IU. I II~ vacuu m 1 1I~ d l~trihuled ampl' hers in. 2 15 tllllpUl ("OmPfC"-~inn pnin l . .112 OlIlput lhi nJ-ordt... inl,'Tl'e"J'l PUlllII Ol P3 1. 312 o. erlap capacilall('e!iO. r.II m .iok'S. !rap formali" n in. 373 paralle l plate cerecuors. 37-4 7 para lk l re"'",an,:~, III' p;!l"alkl Il I.C I.nl . 87 para llel re"'lI1anl lanl Circllli . f'l"..h k rn ,\fl. ~9 parallel R LC lanl . 11(,-9 adminance "f.Il6- 7 paralk l ll l.C tan l ba nd lO.'idlh of. 'X) l'harao:.'eri~lic impedance .. f. 11M Ilt"lw, rl en"rgy .. r. Mil q ualuy fa,,-'ur of. 1111_9 para mete rs. fur SPIC E lcvel- J mn,k'1s, 1I2 parllllldric cunWrtl'l1i (Il',ramelri,' alilplifiers ). 3 10.

3311 para ~ili c l·al"'c i laIKe .~

and i n
pe....'l'a nt·e . 2/l. 2M phaw -altc maungli ...· j PALlsy. It..I1l. 4-11 plla.se de lt'Clnr ana log mulliplit"r a~. 450- 1 ('Jldu.si~e·OR gale as.. 4~ J-..~ . 47 9 ffC
.

594

INDEX

rha~ tk lre l." ' ("<"'t.) lI"U!e'~ 1'1la-e de lOX!"", -1hO- 3

in pllase ·I<,,:ked loop cin·"i h . -1-1 1, 442 '1uaoJralUI\' tyJl'l.'. -1 ~1. 453 o"'"'!Ul: nlial typc. -1 5~63 XO k pha.'C: tktect.lf. 4.55••H~6. 4711 "Moe det«tor CUlIs llln' , 453. -156 pOa\C' detector Fain. -lID_ I . -ISS pha-e .... le.; l""'. tri"'l ve de tector. 46 2. 463. -164

phiII>C di'.un ion. ~ cauced by, 1112 phase·k-;led 10l~ t P1J.A). 409, -lJS-IIJ archuro;lul\' of. 44 1 hiMc>ry of. 4 ~ I

litlC'Mircd. 44 1- 7 koop li1t~ in. -1M-II nui-e input into. 44 7

nOl ' Y. -111 1- 2 prohkm ~ for. -171W13 quaru -conln>lkd O'Cilllllor ('Om h,onl ...ilh, 5 14 11-" .... mpled- dota 'y\.lClfI5. .u8 ~ ...>nln"tyre . 435. ~ -7 . ·H~ &' _y nl~iHnl. SI S. 5 16. 5 17. 5 111. 522 . 524 u-e ..' f M dernodulalun. 471-3 II.....

or. 43K, 4 )'1

veo (IUtput in. 4*l pha.'"l: marg in. rompulatiun of. 424 pha.... ooi.'"l:• .14K. 4&4. 530-49 etjuatioo!l fur, 534 lIaj lnliri mo..ud for, SJ li-l4

PtlU l,..,nTlkker. 14

pm", DC lransmi" inn of. 10 dis, i!",linn nf. 530 handlL'tl oy l;'Oill ial cable, 14 1 published gain fij:ure.. fnr, 316 unus fnr. 140 --1 pnw er amp lifiers. 344--84 hreaL.d",".-n phell4. J69.-.10 CIa•., C lImplifie.... 351-4. 362-3 •.lM. J6'J....70 CIa" 0 amplifim . 355--7, 365 nl.... E mlphfien. 357....9 . 362.... 3. 365. 370--1 CIa F ampltlier'l..359--63. 365 des ign eum ples f,.. . 365-11

design \limIT1&}' for , 3"1'J4lO general mode l f,",~. 345 inM..l>ility of. 312 Iall:e - ~ignal impedano:e TnoilLiling in, 375-6 Ioad--pull charao..,erilalion of. 376 load·pull CQIllour rumple of. 37~9

mndulatioo of••'62-3 ptJWCT· ltdded e fficiency of.

312

problem roet for . 3110......

pul'oCWKllh modulatIon of• •'64 puslt-pull amphfiers, 3~ I. 356. 3113 1ilenna.1fUna.. ay in. 374 - 5

of ideal o...: illal..... 532- 6 U-Ml model ror. 535, .536, 54 1. ~ 7 linear limr -im m anl analysi. o f. ~lJ7. 539

pnw er-comtrained opll mit.atinn of lII. i'o(', 2114--.8

pruhkm set fur. ~·t,-<,I IM ..r)' of. 532-f>

p'>Wn gain . at I,.... f""'l ucncie... <,1 3- 104 power RF O\Cillahln., in\'enlinn uf. 13

pha'oC Ele..:I" ," ic. N,V.. 2611 1'I1I 1I1P'>. V.. 22 ph'''l'o"r,de, ignaiion rur, H ph<~ocloctric rfflX1 . " f Einstein. 2(J Pkkanl. Greenleaf Whiuier. 4. 6 I'ien'e eryslal oSl:i l1;llllr, ...01l ~<,I . 51 1. n5 x -rnatch circuit. 95--8. lO5. 107 with transformed rij:ht-han,t Lsecnon . 97 pinched-off channels, 67 pink nob e , u .. flicker noise

Pl
P" is.oll' . nj uation. 26 po k -I.<'nl dou ble!, l lln pol y re' 1S1ms. J' polysilicnll re, i' lor, proh lcm (111. nO...I

p' >fICom 1II1i 'o(', 255...6 p'''ili\"e fcctlhock amplifieD. 3116, 390 p'''ilive fC<"t.lhock .•yslems, root-k..·..s rules for. 403-4 p" tcntiolnelric mil en;, 329--31 PtM.JI'ICn, Val.....mar. 7. 14

pow er tnm' fer , 116

ma \ imum . 9 3.... 4 . 106

pre.....Iec·lnr. of ~i n!! Ie ·''t,nvcl'lii'lI1 rece i..er, 554 problem "'Is

for architectures. ~1U for handwidth c. timation, 174_7 fot "ia ,i n~ and vol l a~c n:fen:n,'c'. 231--42

fur dist ribu ted system•• 131.... 3 for fecdlla.:k . y, t,'ms. 432- 1 r Ot

high -frequen cy amplifie r d" . ij:n, 211... 22

Ie compo nent" 5 11~"' 1 for low ·noj 'e eurpiiticrs. 30-1 ....7 for mixer s. 3-1()-3 for MOS dC \'i,''' ' . 11 1--5 for noise. 2M ....7 t for ()~iIt ll tOI'li. 525....<,1 for passive RL C network•. 10"'- 13 fur phase-locked 1'~'I'" -1711 _11.1 fill" phase noise. 545--9 ror I" ....er amphtiers. 3110- 4 for Smith eholrt and S·l'aflllllclen. 144 propagalll>l1 Const;(f1t (y ). 11'J relationship tu line paramelel'll. 122-4 fm

proporuo nal 10 all' n lute te mper ature ( PlAT), 228.

229. 230 pulse impeda nce. 1m pulsewjd th modulaliun I PW M., of p'......er amph lierl;,

3f>4• .165 P'tJpin, Michael. 124

595

INDEX

P npin cu iIs, 124 push--pull amplifier,;

crossov.., di!J.OI1 ion in. 3113 pOII.a capa hihly of. 35 1. 3~

Q , J~~ qualily factor quad ralllre ~ ..neraIOf'll. for ima~e - rejecl rec.. i~",.. 559-6 1,562 lIu adr~l ure oscillators. 51)1}, 525, 5hO qu adra lure plla...e dt-I~'lur. 45 J. 479--80 qUilhly fllClOl'( Ql, so. 9S. 211S of panlkl RLC tilnl . R8--9 ringin, and, OJ()..I uf la!'P"d n~ i,lJI' re...", altll'. 1110 quantum e tTL'CI, in carhoru ndulll LEo'; , 20 quart er-wave reson mors . 500 -1 'lUaTIlcry'lal, fabricaliun uf. So.1 as re..."'l1on. , 50 I , )'m btll :and modt-I for. 501 quench o..t.'illaltll'. in early sU ~i!enenu;ve receiver,

'"

Qu inn, P"al , 321 radar , microw ave diotks for. 331 radio. hiMt>l)'of. 1- 33 r.odioal.'1 i~ity, d;'Io('t)Ytf)' of_ 10 radio a"'nllMlITIy, , i
radiocirt"lliIS.1S-111 Radio Corptll'alion of America ( RCAI. IS. 19. 2 1. 3 1. 33 , 212, S16 radiu el'~' lrtm ic" dc~e l"pmem " in, 15 radio ·frequeocy IR F I ~'amers, . ine·wa.'e lype . 7 rad io- freq uency I RF ) , i, nal, .II , IS r.tiote le,nphy, 8. 22 tnlmilior! In DtliOlC lepht",y. 9 radiOlC lepht"'y.9, 19 radiu s of I yral ioo_ 1m

randum lelegraph sil(nals I RTSl, J U P"flCum ooi<;e HC low-pa ss tiue r, , tl'p respun <;e in, 17 J reacta nce rube. 269 read-on ly-lDC"mtlf}' IROMllt....up table. 523. S24. 5t>K receiver

",im.cohf:,ror. 3 d lrn:l -con ,en.iOll Iype:. 56 2- 3 ~ uni ~"rNa l.~ 56 2 reflection ~,,,.. fficienl , 126. 121, 134- 5 ad millaoce reJal;un h>, 270 rofl..~ circu i l s. 1 7-111 R..getl(}' TR- I ~i"'or radio. hi, lory of, 5711-80 regenenui,e ampI; fier/d..let,:I.,.-loKilIa ll,.-. in.enli"n of. I). 17. 18. 3116. 573-4. 5n regencralive radios. de FOI'l." ,l'5 sale of, 17 re~eneral i\'e re~'e; . er, 13, 18 resis tance s, ur shunl-se ries amplifi..rs. 193- 5 rC"sis,on , 34 -7 !Iicler Ik.ise in, ~3-4 !ayou l of. 36- 7 lherm al noi'C' of, 2-45-6. 265 resonance , branch cum pts ... . 119 re...",aPI circuits, am plificr function in. 107 re",,,, anl fre'l ....ncy, of parallel RL C ta nk. K7 reson ant voltage, magn ifica tion of, 9 1

resonators, 5(X)- 4 q uarte r-wave ~"'''';t\lIl'S, 51Ml- 1 q uartl- cry""'l ro,o,,,,IOfI. 501 S AW de.,ices . 503-4 RETMA . 33 RF circuits anolsyslem s h;,">I)' 57 1- II.l

»r.

noi.... in, 25 1

superrcgenerators in, 2 J2- 13 UP;ls [n. 140--- 1

RF recuners. 10 RF . i, na)s, 17 RF uan..t:ondu..1.lIl>. for mi _cn. 32 1

ring O'I\.'illal' lI' impu l..e sc::nsit i~ ily furo.:lit'" f"r . 5311, S39 uo;c::s nf, 544 ripple components . in I' K'p fillen;. 4M

ri;,clime b.alkiwitlm. relatitlllS wuh , 17 1-3. 175, 2 13 ofn\CMIe ~y!>lems. l~7 1

ri'>CIimea
Rontgen. Wilhelm Kunrad, III n){)I - I'''l;u. t~h n i'l ues

com pen!Wllion and , 4 15-22. 432 for fc::eJhad , ysl.. m~ 39R-UJ3 f,,, p""nive fted~k . y" em , 403-4 w mrnary of. 422- 3 Ro..en'>larl . S.. 161 Rou nJ, Hen ry J.. 9, 14, 20 Salura i eq uation and melhoJ , 42. 43, 4.~ , 46 , 47 \.ample·amI-huld K lion. of synthe. iu l'i. 5 15 !>aITlple·and-hoId cirt:Uil. 270 S.am u rr, Davtd, U . 19, 22 u ld lite wmmunicl1i.",s. ptw;e ·lockc::d k...ps in, 441 ~atura t; ng am plifier. der.cllhmg fuoct;'''' fur. 4tH saturation. definhion s for MOS and l'> ifKllar devices,

''''

!WIllIraliun curre nl. in shor\ 'l'hanncl MO SF l:Ts S.AW. surface llCou \.fic WI"e (S AW ) devices s.a"'1u'lIh l!C'Iltf'
s,.,

"""Ucring parameters, JU S-parameten Scbnun bigger. 4117. 48 8. 49 2 Sc::ho.lIlky, Walter. N 3, 250. 252 . 30Jl Schouky diode, 5

Searle , C . L .. 1M se.:ond-ordc::r ItlW ' pass syslem•• formu lilli f.lI'. 4 13- 14 ~~>rder plllI....-It...· ktd I.- 'fl". 444- 7. 4 79 in des ign eu.mples . 446- 7. 47S-ll jille r pea kinl! in, 446-7

sec"UIldo. numhe r in year, 253

self-biased cir cuits, Man -up ncl""ork in. 226. n self-res onant pruperti es. uf ,p iral indul'lurs, 5 1

7

»

* •

596

INDEX

. ...If-reslo ring
't' mio;.IIld ul" Or do.'IL'.:tor,lirsl pnem for. :'i ...-quelltial pha.'IC' ,lelcc lul'!I. 4~~3 wilh ute lllkt.l range. 4.H - 8 S~u"nlid ('..u k ur a'"CC M'
•h"r1 ci",,,illl, Io:" " i",u ing pun" in, 139 ""lft-o;in:u'l ti~ ctln-unts ,SCrsl......,u.lII .:curill.')' of. 11),4 a,h-" nl;JgelI (If, 174 for Noo...idth e>t imalion. 161 - 6 ~1n..1 '''fll

or. 1b4 -b

, hot nui-e. 250-2 •hunt ao..ImJlta.....'C. 127 ..nun! caracllaoce, l.'l.Iualion for , 50 lIhunt-peal.n:I .unl'hfien. 179- 9 1 , hunt-pe.ikn:l RLC n~ wor" . 211 , hunt peak ing, del.ign t u mplt for . IIB .... 4 .J1unt-litrit.amplilitr.>. 19 1_7. 30t. ha nd ll.ith of. 19S- 7 tktaik'd ..... i~n uf. 193- 7 input -o utput improam:C'l' nf. 195- 7 input....nOlput rt, i"alll'C'l' " r. 193-5 1,..."-frtqu(,lJll' ~ain of IOJ3- 5 nn iw figure " f. 27K. 279 si hrornt "", i' l" n<•.1n si, han , 20-- 1 soo rct'-co upled aml' liht"J' . wuh single nmed klall.

••

." un·e - t1rain diffu. iuns vre' i.lon frum. 35

source f" lInwCrll. cal>iro: ilivc ly I"aded. 282 Sn"iet Uni"n. 20 S-para mc ters. 134 . 0 11_411 prubl e m set rnr . 14.. ....5 . J>ark-[MI'
'.3

S PICE . 23 9 IP3 " \l imalio n u.sing. 2911 mis er simulali" n using. 324 u..... for bandwKll h est imalinn. 14OJ. 152 . 155. 157. 1511. 159. I flO. 175 S PICE l...u~I-3 mnoc [", KO- 1. 2 17. 3KO p;u'ilm..-Ie., r..... 112 SPICE M Q.'W ET m" ,ltl•• so . prral intluct.. n . 47- 52 sl'Orioo .... rlU' , iiJi . J'p nu imalio.", of luop IOl'Oh • 45 1- 3.454 'Ithaviur . 5(1l1 r.la bility. BIBO dt linilio lfl of. 395 . ta nd rng W~\ ... ' alio (S WR). 132 ~lali<: modu li. 5ymhe- il l'n w rlh. 5 17- 19 stationanlY. in n" i'C c1'Mnc1rri1.olIiun. 536 .I~ fM radio. ......se- kx ked loop c in:urt, m.

ran,...

....

,

"Subhi~lolie~

of tht Lighl·Emi tting Diudc-" t Loe bner l. 22 suh"" mpJing. an'hrtrrtutn fo... 56 5 . Ubs tr..l...current . frum h" l d cc lrun ... 76 . urcmu lTer, ror OCr ca l<:ulalion. 177 superheter odyne AM radro. Il.ilh ht'plu,k . 32 . up..-rhc1e rod yllt rece ivcr. 4.111, "]9 blu,:k 0, 5ht1 ~uJ'Crrtgcneratoe. snv..nlio n 01. 111-2(1. 2 1 I sup<.Tregenera tor aml'[ili....., 13 I. 2 11- 13. 2 1II Arm'I" >rt g'S ime nti'>rl uf, 574 -5 inl erferen..-c Irorn. 2 12 . upen"Cg... ncrator d... h:c ltlr. U \C in tny walkie -talkie s. 20,511 1. :'i83 ~ u rprt."or gri d. 3 I surfuce acou stic wave {SAW ) devices. us res"nalnr. ,

501-4 . wa [low courser, in inlcger- N ' yrrtlrcsi/.cr, 5 [ II sweep n....lllurors . U \C in ldevisinn . 439 switches. C MOS tc..-hno [o8Y, .1 [9, n r. 335 sy nthc" i/.cn<.51 4....24 cmnbin" tinn .y mhesilcl'lI, 52 1-2. 5 211 dire ct digila [ .y nt he ~ i le r. 52 1- 2 wi lh di1hering m,,,luli , 5 [9-2 1 d ivider "delay" in. 5 1S- 16 inleger· N .y m hc ~ i le ,. 5 17- 11,1 vlT"'1.\ ynlhcsiu r. 521 . 5 29 pha.se ·lod e lank ...... i.lance. a. noise

"''''TU:. 53 1. 532- 3

4 597

INDEX

tmn Nl.·'lndll~' ''1I"

l"malum. in lll!hl hult..... 23 larf"'''' ~ara,:i lur re sonator , illS- b. 107

CroSHjll;ll,1type. 327-11 lal};e ·" i~ nal ty pe , 41111, ·W ) lran 'oCunlinen lal lele phlllle sc rvice, i nau~ u(;lh"n IIf,

us nnpedauce lIlal,"hi lllt nelwllr k , '1'~ lll l lapped inJul:!n r ma..:h d rcuit, 102- 3. lOll T· ~"il,

twmdwi

in pa."iw RL. ncl"url ... llt.

Teal. G.II'doIfl, 51lll Tchl'Oll j ~. 1'111. 21S

quarter-w ave type. 117

Td dunl.en,5N

It'lcpho.>ne repeaser amplifier. 12 k lephnn c [nn.'mi ....;on R"du.:li"n in. 124

.ek,-j, ion ampl iti

lill('~. ~ignal· ...i-.pt'niOfl

for. I N

001 'le'l_inn ... W.......I ('.my amplifier- for. 3J ll:" in-tww.lwil h ampli fier< fur.::! I) milcn In UHF [li nen (nr . .\.17 pha'l'- Iod.ed Ie.. '!" in, ".N

-,nI.... , .... ~

H

3K7 transformers

nlli .... in. 243 :i l1al"".. in. ".N

a. hladho~ f~lI"eno.',,), I I~

Telqcn. R Il. H . 2M tcm(lC'raturc. tlQ-dlip ~

,i\ar;lo.'1m>.tic impedull:" of. lIlI, 11'1 cutoff fl"C
rem.·nl of. J75 lcnmnalinn'...flran'nll ,itlll l "~ ... 125 - 7

Te-.la. N,I.,,1a. 7. 9 1 Tc»la cn i l. 91 l.:tn.lr. ,·haral: I..ri' I...~ uf. ,\Ii Tcu.• ln-t rume nl.• lTh. 57&, 5KO

Thaden y. U . 22 thenllal I"I<>i...:. 2·U-~ I. SoU epi no i'l' 111', 29 1 in MOSFET... 2411-.50 ~5 , 245 Th um 'Ofl, J. 1. ~ lhuri" tw tu n~st ..n. in ti l
1lk'\'.. nin re, i, talll.-':,

2~8-':l,

.'no.

JlH lhrce 'J" >r1 mi l " "'. J I4 Ih,..., I1< ,loJ I"t'
.n.'

''' '

time- oJo main NinliJI" lmN, simulatio n .. f'mixer IP] with . 324 -5 time invari
token rings, pea king prnb!em in, 44 7 tuys rad io-cornmlled. cuberer u..... in, 4 , R sUJ't'Jll'ge ....,r alllr use in, 19- 20 ...... "/",, walki" ' lal kie, lrack -;\oo ·h<.1J suh""mplinll' mix er, .U t. tran ...,.. ntIu~' talll.·e C'\juations fur . 2'1 or .J1,o.II1·{ hanoe l MOSFETI, JlX) temperalure dfec1S un . In

UP"'dnJ aoo "'.wn", ard 1),J't'... 94 Iran sienl impcth ncc . 120 Ir.lI1,i , lor .lln pliners. OC'n r' lI". 15 1-2 tr.1I1, islur I"lldio, hi' llll)' o f. 57l1-1lO lran , i, tt>r.i. H heRavjor of, t.2-85 ins'cnlion of, I velo,..ily 1\.illUr;lli"n effect 011. 7'5-6 transit ume effect •. in MO S LInl ot:. 142- 3

archuectu rcs for, 5M -7 of Hert z . 2 supe rrel:e llt.'r alill' I"<'<.-.:i,'..r a,. 1'1 triode, , tru<.1ure and ch.ir.":teri '>lic, nl , 211. 571 triode vacuum tulle , lJ, n, 27

llF hmi tatiu ns uf, 29-3(1 inc'rem,"" lal nl<"lcl rOf, 2\1 tnvenuon fi t, II tri state d<'le : I<""'. 4hO mwavc pfla..... detector. 4lJ2, 4t' J . 4f>.l tuned amplif ers, 1lJ<)-20-'. 2 111 luneoJ input -tu neoJ output ,,,<:ill ah'r. ,,','" TIT O osclllatnr tuned radio -frt'q uc'n~'Y (T RFl rt'CeivN . 15-1 t. lu ngsk n, in Iig hl hu lMs. 24 two -diode mixers..HIl two -port bandwidth enhall.e ", c nl. ,, / ampliti,'r" 11\7 - Ql tw,,- purl m i~ cr" . J 14. -' 15- 111 two -port nu lsc theo ry, 25t>_t>U two -pon va r iable tk tinilion , . 1.111 t"..,- I'lIle thiroJ-mdcr inler~'c l'l , of nmer Iin...drity, ]12 1)ne, Gerald, I I " uhrav iulet caUst rup he:' 247 un ildterdlil alin n. of amplihcl""l, ltH- t> U.S . A rmy S igna l CUtp'!. 14 - umve rsa r' C'\judtinm . fu r impn\.lnl·c tran,formallo n.

94.95

5.' Ullsilidc kd 1'"ly. no_,j, .ivily of, 35 IIp<:l>nve,,ion anchilroatka._' rad lO'l, 205, 575-8 ... amplifier.!, J~7 Ni..ic informat ion nn. 22- 33

de Fnd uetance for. -189, 490

oo,t lnJ'n.,n! of, 9-1J. 22. 2-1 3 di"li hulal ampl ifier 11'>(' ...ilh, 21S Slid numher in. 32 input imp.-dJoco: in. 2RO. 211 l a~ o:.cillalon. 311.'1 V_I cll.lrao:lm~ics ufoll-.U Van lkr 7.J.c.o1. A., 2411 ' anlI.1'"". 40, 41. 3 10 jU ' K1ftca['olClh"" a., • .'i9--tIO v.. k ity .....11lr,IIion, effect 00 tnnsiJ>lno. 75 "Tf)' large 'loCaie i'llt p:alion. JU Vr.s l technologir.

V_I { ha:rac.1m'tif;"

of MOSH , \ . 83 of U'"llum ' uhes. 2>-33 \idro amplihcn, shUll1f'C'al mll: derivatiOl1 from, IS) VLSJ citl:uil~. f"I,,~i\'C' com~nl, of. 86 VLS l loxhnologiclo. " 'llll:e --drai n diffusions in. 35 v"I"'80:. of d" llb. 2lJ...4 \'{Ih.~ ccll1ln>lIro O'oCillalon I Vco,, ). 4M-- 9 Ila~ 111'1 !l"1;Ualinn O'ICilJalor topo logy. 449 in 4Q.lt, C MOS
.....ulk i..·l" lkies (tuy ) devel" l' m'>lIt or. 5110_3 rC"cive nu"te of. 511 1_2

INDEX

.~ Upe m: gc rlC ra lor use in. 20. 58 1 Iran, mil m<>l1 ( Mu.....Ul. 115 Weaver arch il~u re> 557. 558 We"'Ier. Roger. 5711 weUR'lIi.'lt>r.l. 35 W..stingh.. use. If> Whee ...... Uaru ld. If>. 205. 5711 Whe'<'1..r ·! form ula . Sf> " When Tuhe.~ 'kal Crysta"l; Early Ra•.h o o...,lccl<>nM {Thad ..enIyl. 22 ...hile "..i se. 523. 5-lO \\-"hiu ier.lnn n GrunJe.lf• .5 Widlar . Bon. 221 Wib " n . Ruhat . 247-8 ...ing e lectrooJe, 11. 25 ... 'i ll''''!§ communicarion . M'm :unj', invef1(itJn of. 4 Wi",I, J, ll"'rfd. 9 ...ire k>up. ind ucb nce: " t .!of> WolI""t" n " ire. 8, II Wood ', metal. in tktn.' tn. 5 W~ .....,. PoIy1cchni e Instilule. 3117 World War I, 7. 14. 15.21. 519 World War II . J.l7

"f.

I

XOR pfla-e tktec1 455. 478 uSC' in de~i,n amp le. 475-6 X-ray... d i"-CO\'ery of. 10 X-ray tuhn. 24

year. second, in. 253 Yuan f..rm ula and II1('lhod . 4 1_2. 4.1 ze ner diode, 227

'(T(. damp ing. of f«d had . §yMelrui. 4 14 ' .em-order hnld (ZO Il). tran.f..r fu"'"tioo or. 5 15- 1f> zel\>-pt.'a l etl amp li';"r. 2/ 9_20 zero-po w..r eecein n . 11.1 Uln,

a., hand..... idlh cn ha'Kerl< , 178. J 79- 9 1 in runt · lnt:u. tec hniq ues, 41 5. 4 1n _17, 41 <)....23 . 429. 43 1 zij,!lag wi re ..te,-tn"le, in tn,,,1<... I I zfncite cry.la l dind es. 2fl Z- planc map•. 135

I

Lee - The Design of CMOS RF Integrated Circuits

Recommend Documents