DC-DC Converter

The DC‐DC Converter EE290 Fall 2008 Sven Fagerstrom November 25, 2008

ECE Department California State University, Fresno

This document was prepared as a technical writing literature search, Fall 2008. All the technical information and knowledge are disclosed for non‐commercial and academic use only. Any commercial use of the revealed knowledge needs permission from the authors. All rights are reserved by the authors.

SUMMARY Many existing and developing electrical and electronic technologies require voltages of differing levels supplied from a singular available available source voltage such as a battery. In the world of alternating alternating current, changing voltage voltage levels is very simple with the use of transformers. Transformers operate operate with the phenomenon of mutual induction by passing voltage from one winding to another while the magnetic fields expand and collapse due to alternating current. This principal cannot be utilized with with direct current as the voltage level in the primary primary winding remains constant, thereby producing producing a potential of zero on the secondary winding winding of a transformer. Direct current circuits circuits therefore present a different challenge when one DC source voltage is available and another voltage level is required. This challenge is met within within the electrical engineering engineering sub-discipline of power electronics electronics through the design of electronic devices devices referred to as DC to DC converters. converters. DC to DC converters are are devices designed specifically for the purpose of changing DC voltage voltage levels efficiently. This work first examines the DC Chopper, Chopper, a type of DC to DC Converter which can be understood easily due to its simple topology and behavioral characteristics. characteristics. We then examine some of the operating characteristics of DC to DC Converters which are important considerations in design including including Conduction Mode, and Switching Switching Frequency. We then examine three types of practical DC to DC Converters: the Buck Converter, the Boost Converter, and lastly the Buck/Boost Converter. Circuit topologies, brief descriptions of circuit operation, characteristic equations, and circuit behavioral waveforms are included for each design. We then examine examine topics of current current published published literature concerning concerning efficiency. efficiency. The origins of losses are identified identified and descriptive descriptive equations are given. The work concludes with with an examination of current techniques at improving efficiency based on current research.

2

TABLE OF CONTENTS

Table of Contents………………………………………………………………………………………………..

3

List of Figures…………………………………………….………………………………………………………

4

List of Tables…………………………………………………………………………………………………..

5

I.

6

Introduction…………………………………………………………………………………………………..

II. DC to DC Converter Theory and Operation………………………………….…………………………..

6

A. The DC Chopper……………………………….….......................................................................................

6

B. Conduction Mode……………………………………..………....................................................................

7

C. Switching Frequency….…………………………....................................................................................

8

III. Practical DC to DC Converter Design…..……………………………………………………………….

8

A. The Buck Converter………………………………...................................................................................... Buck Converter CCM Equations………………………………………………………………….. Buck Converter DCM Equations…………………………………………………………………..

8 9 9

B. The Boost Converter…………………………………………….................................................................... Boost Converter CCM Equations………………………………………………………………….. Boost Converter DCM Equations…………………………………………………………………..

10 11 12

C. The Buck/Boost Converter ………………………….................................................................................... Buck/Boost Converter Equations…………………………………………………………………..

13 13

IV. Efficiency Improvement Techniques……..……………………………………………………………….

15

A. Examination of Losses………………………………...................................................................................... Load Current Loss……………………………….………………………………………………….. RMS Current Loss……………….………………………………………………………………….. Controller Current Loss……………………………………………………………………………. Switching Current Loss……………………………………………………………………………… Thermal Loss………………………………………………………………………………………….. Loss Examination Summary…………………………………………………………………………

15 16 16 17 17 17 18

18 B. Efficiency Improvement Techniques….….…………...................................................................................... 18 Synchronous Rectification…..…………………………….………………………………………………….. 19 Mode-Hopping……………….…….………………………………………………………………….. 19 Zero-Voltage Switching…….………………………………………………………………………. 19 Variable Frequency……………………………………………………………………………………

V. Conclusion……………………………..……..………………………………………………………………. 20 References……………………………………………………………………………………………………….. 20 3

LIST OF FIGURES

Figure 1: DC Chopper Circuit Topology……………………………………………………………………..

6

Figure 2: DC Chopper Output Voltage Waveform………….…………..………………………………….

7

Figure 3: DC to DC Converter Conduction Modes………………………………………………………….

7

Figure 4: Buck Converter Circuit Topology……..……………….…………………………………………

8

Figure 5: Buck Converter Equivalent Circuit Switch Modes…………………….………………………..…

8

Figure 6: Buck Converter Waveforms…………………………………………………..…………………..

10

Figure 7: Boost Converter Circuit Topology………………………………………….………………………

10

Figure 8: Boost Converter Equivalent Circuit Switch Modes………………………………………………….. 11 Figure 9: Boost Converter Waveforms……………………………………………………………………..

12

Figure 10: Buck/Boost Converter Circuit Topology………………………………………………………..

13

Figure 11: Buck/Boost Converter Equivalent Circuit Switch Modes………………………………………….. 13 Figure 12: Buck/Boost Converter Vo/Vs as a Function of Duty Ratio…………………………………………..14 Figure 13: Buck/Boost Converter Waveforms……..………………………………………………………..

15

Figure 14: Buck/Boost Converter Circuit Topology (Synchronous)………………………………………..

18

Figure 15: Converter Efficiency Comparison, Asynchronous vs. Synchronous……………………………..

18

Figure 16: Mode-Hopping (CCM vs. DCM) Efficiency Comparison………………………………………..

19

Figure 17: Fixed vs. Variable Frequency Loss……………………..………………………………………..

19

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California State University at Fresno The DC-DC Converter

LIST OF TABLES

Table I: DC to DC Converter Source of Losses……………………………….…………….…………………. 16

5


I. INTRODUCTION The DC to DC converter is a device which falls within the electrical engineering sub-discipline of power electronics. The purpose of a DC to DC converter is to provide varying voltage levels different than that supplied. This situation often arises with mobile electronics where a battery of one voltage is supplied, but several on-board systems require different voltage levels. Several techniques of varying efficiency are available which achieve this end. DC to DC converters are used in order to provide the required voltage level efficiently. Various DC to DC converter circuit designs are established providing differing operating characteristics depending on the required application. The basic designs are referred to the Buck Converter, the Boost Converter, and the Buck/Boost Converter. These lower, raise, and lower or raise supply voltage levels, respectively. These designs will be introduced and briefly examined here.

II. DC to DC CONVERTER THEORY AND OPERATION All DC to DC converters, as the name implies, begin with two basic structures: a DC source voltage, intermediate electronics, and a resistive load which requires a different DC voltage supply for power. Various circuit configurations are offered as a means to produce the required output voltage depending on the application.

A.

The DC Chopper

To introduce the concept of a DC to DC converter and how it works, we examine the DC chopper. The DC chopper is the most basic of DC to DC converters and offers a simple and straight-forward circuit topology to illustrate circuit behavior. The DC chopper circuit is created by adding a switch in between the closed circuit of source and load, as shown in Figure 1.

Fig. 1 [2]. DC Chopper Circuit Topology The operation of the DC Chopper is based on the concept of intermittently switching the supply voltage at a certain rate to limit the amount of time that the source voltage is connected to the supply. If this is done within a specific duty cycle, the effective voltage will be lower than the supply voltage. The output voltage of the DC Chopper as a function of the duty ratio D is given in equation (1.1) [2]:

VO = DV I

(1.1)[2]

where

D =

t on

(1.2)[2]

T s

where t on is the on-time of the switch S, and T s is the switching period. The output voltage waveform of the DC Chopper is shown in Figure 2 to illustrate this concept.

6


Fig. 2 [4]. DC Chopper Output Voltage Waveform Note the dashed waveform of Figure 2 shows the average or effective voltage output by the circuit. This output voltage can be calculated by the following equation.

Vo Average =

1 T

∫

T

0

Vo ( t ) dt =

1

( T ∫

ton

0

Vin dt +

∫

T

0

)

0.0 =

t on T

Vin = dVin

(1.3)[4]

The advantage of the DC Chopper is its obvious simplicity. A disadvantage is the creation of harmonics due to the Chopper’s hard switching characteristic which ads a potentially important consideration depending on the application. The major disadvantage of it is that while the switch is off, V o drops to zero and therefore current drops to zero. If current goes to zero the converter by definition is running in Discontinuous Conduction Mode, which is defined in the next section. The forced DCM of the DC Chopper limits its application to applications which do not require continuous output current.

B.

Conduction Mode

In the cases of incandescent lighting, light-emitting diode (LED), and heating element applications, intermittent output current does not affect functionality. In the case of logic circuits, however, interruption of supply current results in a complete loss of functionality. This consideration brings us to the concept of conduction mode. There are two types of conduction modes for DC to DC Converters: Continuous Conduction Mode (CCM), and Discontinuous Conduction Mode (DCM.) The designation is determined based on the load current. If the load current is continuously maintained above zero for the entire period of operation, the converter is operating in CCM [4]. If at any point in the cycle the load current falls to zero, the converter is operating in DCM. Separate sets of equations apply to each mode of operation to describe its behavior [4] p. 2-3. Figure 3 illustrates CCM and DCM operation.

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California State University at resno The DC-DC Con verter

Fig. 3. C to DC Co verter Conduction Modes – CM, DCM espectively [5] pp. 108, 110 Note th t the CCM gr aph of Figure shows Δ I L. his is referre to as current ripple and is efined identically to voltage ri ple: the diff rence betwee the maximu and minim m values of he instantane us output value [7] p.120. Ri ple equations are given for e ach type of converter for Δ I and ΔV simila rly.

C. Switching Freque cy

The concept of switc ing frequency is important hen designing and conside ing the DC to DC Converter. All DC to DC Converters operate in cycles with a certain frequency an period. The switching freq ency f s is the rate at which the switch S of Fi ure 1 is toggl d. As this ha ppens at the s me point in e ch period, we can say that f s=1/Ts. Switching frequency is usually kept above 20kHz so as to avoid th generation o interference i the audio ra ge [4] p. 2-3. S itching frequency also affe ts efficiency and inductor size, to be intr duced and discussed in the pages that follo .

III. P ACTICAL D A.

TO DC CO VERTER DESIGN

The Buck Convert er

The Buck Converter also steps do n the suppli d DC voltag source as t e Chopper does but provides the practical a dvantage of potential CCM operation. T e difference n the topology sees the ad ition of a dio de, an inductor, and a capacitor . Buck Conve ter circuit top ology is show in Figure 4.

Fig. 4 [ ]. Buck Converter Circuit Topology

The Bu k Converter f nctions in tw -modes. Wh n the switch f Figure 4 is N, the input oltage VS for es the diode into the reverse bias region and therefore does not conduct. The circuit effectively be omes that shown in Figure 5 ( ).

Fig. 5.

uck Convert r Equivalent ircuit Modes. (a) Switch O , (b) Switch

8

FF [6]


During the (a) mode of Figure 5, the inductor charges linearly. When the switch is turned off at t = DT, the circuit transfers modes to the equivalent circuit of Figure 5(b) [6] p. 138. At this time the diode becomes forward biased and the energy stored in the inductor will discharge through the load as the diode allows the circuit to “free-wheel.” The capacitor is added to stabilize output voltage whose deviation from average is referred to as ripple [6]. By controlling the duty cycle of the switch S, the effective voltage is lowered below the level of V in or VS [6] p. 137. The concept of CCM/DCM is also applicable to the Buck Converter. As the inductor stores energy and discharges it beyond the point that the supply voltage is disconnected from the circuit, output current will go to zero sometime after the switch S is opened (at t=DT.) The threshold at which the converter goes from CCM to DCM is described by equation (1.4.)

LCCM =

(1 − D ) R (VS − VO ) R =

2 f s

(1.4)[6]

2 f sV S

The equations which describe the behavior of the Buck Converter follow:

observing: λ Buck =

L

(1.5)[6]

LCCM

Buck Converter CCM Equations:

VO = DV S

Current Ripple

(1.6)[6]

Δ I L =



2 λ

⎛

V o ⎜ 1 −

Voltage Ripple C =

ΔV o =



1 − D 8rLf

2 s

V o ⎞

⎟

V s ⎠ 8 LCf s2

⎝

1

=

(1.7)[6]

I O

4λ Rf s r

(1.8)[6]

(1.9)[6]

Buck Converter DCM Equations:

V o =

DV S

Δ I L =

C =

(1.10)[6]

λ 2 λ

(1.11)[6]

I O

( 2 − λ ) 4 Rf s r

ΔVo = rV o

9

2

(1.12)[6] (1.13)[6]


The operational characteristics of the Buck Converter can be seen in the waveforms of Figure 6.

Fig. 6. Buck Converter Waveforms, (a) inductor current, (b) inductor voltage, (c) input current, (d) diode current, capacitor current [6] p. 139.

B.

The Boost Converter

The Boost Converter provides a higher output voltage than input voltage. Boost converter topology can be seen in Figure 7.

Fig. 7 [2]. Boost Converter Circuit Topology

It is interesting to consider and compare that the difference between the Buck Converter and the Boost Converter is the placement of the inductor, diode, and switch. The boost converter also operates in one of two oscillatory

10


modes depending on whether or not switch S is open or closed [13] p. 187. These two modes are effectively illustrated in Figure 8 which follows.

Fig. 8. Boost Converter Equivalent Circuit Modes. (a) Switch ON, (b) Switch OFF [13] p. 187

When switch S is closed, the diode is reversed biased effectively eliminating the Figure 8(b) portion of the circuit and current flows only through the inductor and the switch as shown in Figure 8(a.) This stores energy in the inductor. After a certain time, the switch is turned off and the circuit effectively becomes that shown in Figure 8(b.) The current stored in the inductor then flows through the now forward-biased diode and through the load. Mode (b) ends when the switch is turned on again and the cycle repeats [13]. The concept of CCM/DCM is also applicable to the Boost Converter. determine the CCM/DCM threshold are given in the following equation:

LCCM =

(1 − D )

2

R

The operating characteristics that

(1.14)[13]

2 f s

Equation (1.14) is similar to that of the Buck with the exception being the square is absent on the Buck. Characteristic Boost Equations follow:

Observing: λ Boost =

2 Lf sV o D (1 − D )VR

(1.15)[13]

Boost Converter CCM Equations:

V O =

V S

(1.16)[13]

1 − D

Current Ripple  Δ I L = Voltage Ripple  ΔV o = C =

1 − D 8rLf s2

=

2 λ

I o

Vo D RCf s

1 4λ Rf s r

11

(1.17)[13] (1.18)[13]

(1.19)[13]


Boost Converter DCM Equations:

V o =

V S D

1−

(1.20)[7]

λ

( 2 − λ ) C = 4 Rf s r

2

(1.21)[13]

The behavioral waveforms concerning load voltage, inductor current, switch current, and capacitor current for the Boost Converter can be seen following in Figure 9.

Fig. 9 [2]. Boost Converter Waveforms

12


C. The Buck/Boost Converter

Perhaps the most versatile of the DC to DC Converters examined in this literature search is that of the Buck/Boost Converter. Buck/Boost Converter topology is shown in Figure 10.

Fig. 10 [2]. Buck/Boost Converter Circuit Topology

Fig. 11. Buck/Boost Converter Equivalent Circuit Modes. (a) Switch ON, (b) Switch OFF [13] p. 193 The Buck/Boost is capable of stepping up supply voltage or stepping it down, based on the duty cycle of the switch SS [13] p. 242. When switch SS is “on”, the diode is polarized “off” while the inductor magnetic field charges and current increases linearly. This continues until SS opens whereby the inductor instantly switches polarity and draws current through the load and the diode as the inductor discharges linearly. This results in a negative average voltage across Vo (noting the direction of current flow through the inductor.) The inductor is either in one of two cycles: charging from the source voltage, or discharging through the load. The inductor is therefore effectively disconnected from the load as varies with the duty ratio, as shown by the load connected duty ratio in (1.22). Buck/boost behavioral equations follow. Buck/Boost Converter Equations:

d o = 1 − D , where D  duty ratio

I L max − I L min =

V o L

(1 − D ) T , where T  Period

Vs dT = −Vo (1 − D ) T

(1.22)[13] (1.23)[13] (1.24)[13]

Therefore,

V o Vs

=

D

1− D

13

(1.25)[13]


Also noting,

LCCM Boundary = I o =

I L max

(1 − D )

2

R

(1.26)[13]

2 f s

I L max + I L min

(1 − D )

2

(1.27)[13]

⎡ D dT ⎤ = V in ⎢ + ⎥ 2 ⎢⎣ R (1 − D ) 2 L ⎥⎦

Voltage Ripple 

ΔV o Vo

=

DT RC

Current Ripple  Δ I L =

λ =

= 2 λ

D RCf s

I o

2 Lf sV o D (1 − D )VR

(1.28)[13]

(1.29)[13]

(1.30)[13] (1.31)[13]

In examination of equation (1.25), we note that V o equals Vs when D=0.5 [13] p. 243. This point is therefore the boundary between buck and boost modes. Increasing D will result in an increase in V o, whereas decreasing D will result in a decrease in V o. The duty ratio of D=1 results in a theoretically ideal infinite output voltage. The relationship of equation 1.25 is illustrated in Figure 12 with circuit waveforms following in Figure 13.

Fig. 12. Buck/Boost Converter Vo/Vs as a Function of Duty Ratio

14


Fig. 13 [2]. Buck/Boost Converter Waveforms

IV. EFFICIENCY IMPROVEMENT TECHNIQUES Current Research – Improving DC to DC Converter Efficiency

Current literature available concerning DC to DC converter technology is focused in the area of improving converter efficiency. To examine this closer, we focus on the most versatile of the discussed DC to DC Converter: the Buck/Boost (Figure 10.)

A. Examination of Losses

Ideal circuits are 100% efficient. Non-ideal circuits are not, due to power losses as shown in (1.32.) Therefore, improvement of efficiency is the result of minimization of losses.

Efficiency  η =

Pout Pout + Ploss

=

Pout Pin

(1.32)[1]

Losses of a DC-DC converter can be grouped into several categories as shown in Table 1as proposed by Zhou [1].

15


TABLE I DC-DC Converter Source of Losses

A detailed power analysis by [3] suggests that these losses can be categorized and examined as follows: load current, RMS current, controller current, switching current, and thermal losses.

Load Current Loss

Load current dissipates resistive power losses through the inductor and the switches: 2 PL ESR = I Load ( RL ESR )

(1.33)[3]

PSS = D ( I load ) RSW

(1.34)[3]

2

(

)

2 P LS = (1 − D ) I load RSW

(1.35)[3]

where D is the duty cycle, PSS is source switch power, PLS is load switch power, and RSW is the transistor ON resistance. If a transistor is used instead of the diode (synchronous), then the resistive power dissipated by the diode is:

P D = (1 − D ) I load (VD )

(1.36)[3]

MOSFET body diode dead time losses can be accounted for by:

⎛ t dead ⎞ ⎟ I load (V GS ) ⎝ T ⎠

P BD = 2 ⎜

(1.37)[3]

The dead time is defined as the time between activation of one transistor to the next in sequence, which is required in order to prevent both transistors from being on simultaneously. This loss is therefore applicable only to the synchronous case.

RMS Current Loss

RMS current losses are dissipated through the capacitor, the inductor, and the switches: 2 P ESR = I RMS ( RL ESR + RC ESR )

(

)

2 PSS = D I RMS RSW

16

(1.38)[3] (1.39)[3]


and, 2 P LS = (1 − D ) I RMS RSW

(1.40)[3]

Or for the asynchronous case:

P LS = (1 − D ) I RMSVD

(1.41)[3]

and,

⎛ t dead ⎞ ⎟ I load (V GS ) ⎝ T ⎠

P BD = 2 ⎜

(1.42)[3]

Note: equation (1.42) shows that body diode losses incur only during the time that the MOSFET is not conducting.

Controller Current Loss

Power is dissipated through the gate drive of each transistor as the gate is charged/discharged dissipating quiescent power losses. Gate drive current is independent of load current and can be expressed by:

PQ Controller = I ControllerV in

(1.43)[3]

Parasitic capacitor losses at the gate(s) can be expressed by [3]:

PC Controller = 16 ( C gsV in ) 2

f s

(1.44)[3]

3

Switching Current Loss

Losses occur during switching transitions as voltage and current cross over at the MOSFET. Switching power loss can be expressed by [3]:

⎛ t x ⎞ ⎟ ≈ Iload Vin tx f s ⎝ T ⎠

PSW ≈ I loadVin ⎜

(1.45)[3]

where tx is the total rise and fall time of the transistor. We note that ideal switching efficiency will be when t x=0.

Thermal Loss

If a fan is required to cool components, this power must also be included as loss. As temperature increases, MOSFET ON resistance also increases thereby increasing P SS and PLS [3].

17


Loss Examination Summary

I2R losses are the leading contributor to losses as they increase by the square of load current. Losses are a function of switching frequency at low loading conditions (switching losses.) Synchronous converters incur more conduction losses at low load than asynchronous [3].

B. Efficiency Improvement Techniques

Several methods of efficiency improvement exist. They are Synchronous Rectification, Mode Hopping (CCM/DCM), Zero-Voltage Switching (ZVS), variable switching frequency, and Hybrid (Mode-Hopping and variable frequency) [3]. These techniques are introduced and briefly examined in the following pages.

Synchronous Rectification

In order to eliminate the forward voltage of the rectifying diode, the diode may be replaced with a MOSFET as shown in Figure 14.

Fig. 14 [2]. Buck/Boost Converter Circuit Topology (Synchronous)

Examination of equations (1.35) and (1.36) reveal the benefit of this change. It is noted that the power dissipated in the MOSFET may be higher at high loading conditions as dissipated power is a function of the square of load current for this case, as compared to the unitary exponential of current for the asynchronous case. The use of the MOSFET in lieu of the diode also introduces additional switching losses. Therefore, applications with high switching frequency and high loading conditions prove to be less efficient with synchronous rectification [3]. Figure 4 [10] offers an illustration. This results as the delay of the converter becomes a significant portion of the period (see equation 1.42) thereby dissipating power via the body diodes. Figure 15 reveals the benefit of synchronous vs. asynchronous rectification.

Fig. 15 [10]. Converter Efficiency Comparison, Asynchronous vs. Synchronous In conclusion of synchronous rectification, loading conditions and switching frequency will determine when to use[10]. 18


Mode-Hopping

Mode-Hopping is proposed by [11] is alternating between CCM and DCM modes depending on load current. The mode-hopping analysis determined that optimized efficiency is achieved utilizing synchronous CCM during heavy loads, and asynchronous DMC during light loads as shown in Figure 16 [11].

Fig. 16 [11]. Mode-Hopping (CCM vs. DCM) Efficiency Comparison

Zero Voltage Switching

MOSFET overlap of voltage and current cause switching losses as described by equation (1.45.) Examination of this equation shows the loss goes to zero if there is no overlap in timing of voltage and current. Zero-voltageswitching is achieved by adding a snubber capacitor across the switch forces zero-voltage switching as capacitor voltage cannot change instantaneously. However, the parallel capacitor causes zero-voltage turnoff problems, so a diode is added in series with the capacitor. The diode conducts at turnon permitting slow voltage buildup across the switch, while blocking discharge through the switch during turnoff [13]. A detailed analysis is provided by [12] on ZVS. It is suggested by [3] that it is difficult to ensure proper dead-times for ZVS at all loading conditions.

Variable Frequency

Varying switching frequency f s affects losses by decreasing switching losses according to (1.45.) Figure 6 [12] shows the effect of lowering switching frequency. A detailed analysis is offered by [13] who suggests that lowering switching frequency significantly reduces switching loss at light loads. It is therefore proposed to vary f s according to load below a certain load.

Fig. 17 [13]. Fixed vs. Variable Frequency Loss Of the efficiency improvement techniques examined, the most efficient combination proposed by [3] as a modehopping DC-DC converter employing asynchronous, constant on-time, variable frequency DCM operation for low output currents (up to Io=I p/2) and synchronous, constant frequency CCM operation for high load currents (for Io>I p/2) [3] .

19


V. CONCLUSION DC to DC Converter operation was introduced and discussed with the DC Chopper as well as consideration of Conduction Mode and Switching Frequency. Three practical DC to DC Converter options were introduced and examined briefly including waveforms and characteristic equations. A brief synopsis of current research was given to identify losses, their origin, and current methods for their reduction.

REFERENCES [1] Siyuan Zhou, “Fully Integrated Power-Saving Solutions for DC-DC Converters Targeted for the Mobile, Battery-Powered Applications,” Georgia Tech Analog Consortium Industry Research Review, 2003 [2] Muhammad Rashid, “Power Electronics Handbook,” Academic Press, 2001 [3] M. Gildersleeve, H.P. Forghani-zadeh, and G.A. Rincon-Mora, “A Comprehensive Power Analysis and a Highly Efficient, Mode-Hopping DC-DC Converter,” IEEE Asia-Pacific Conference on ASIC, 2002, pp. 153-156 [4] Timothy L. Skvarenina, “The Power Electronics Handbook,” CRC Press LLC, 2002 [5] Robert W. Erickson, Dragan Maksimovic, “Fundamentals of Power Electronics,” Kluwer Academic Publishers, 2001 [6] Issa Bataresh, “Power Electronic Circuits,” John Wiley & Sons, Inc., 2004 [7] Randall Shaffer, “Fundamentals of Power Electronics with MATLAB,” Career & Professional Group, 2007 [10] O. Djekic and M. Brkovic, “Synchronous Rectifiers vs. Schottky Diodes in a Buck Topology for Low Voltage Applications.” Power Electronics Specialists Conference, Vol. 2, pp. 1974-1980, 1997. [11] A. Prodic and D. Maksimovic, “Digital PWM Controller and Current Esitmator for A Low-Power Switching Converter,” 7th Annual Workshop on Computers in Power Electronics, 2000. [12] J. Stratakos, “High-Efficiency Low-Voltage DC-DC Conversion for Portable Applications,” University of California, Berkeley, Ph.D. Thesis, 1998. [13] Jai P. Agrawal, “Power Electronic Systems Theory and Design,” Prentice Hall, 2001

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DC-DC Converter

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