Microwave Electronics GIOVANNI GHIONE, MARCO PIROLA March 15, 2013
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ii
Contents
1
A syst system em intr introd oduc ucti tion on to micr microowave elec electr tron onic icss
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Passi assive ve ele elemen ments and and circ circui uitt lay layout out
2.1
Transmission lines 2.1.1 Transmission line theory 2.1.1.1 More on series losses 2.1.1.2 More on parallel losses 2.1.2 Parameters of quasi-TEM lines 2.1.3 Working with transmission lines: the Smith chart 2.2 Plana lanarr tran transm smiissio ssion n line liness in micr microowave int integra egratted circ ircuits uits 2.2.1 The coaxial cable 2.2.2 The microstrip 2.2.2.1 Analysis formulae 2.2.2.2 Design formulae 2.2.3 The coplanar waveguide 2.2.3.1 Analysis formulae 2.2.3.2 Coupling and radiation losses in planar lines 2.3 Lumped parameter components 2.3.1 Inductors 2.3.1.1 Strip and loop inductors 2.3.1.2 Spiral inductors 2.3.1.3 Inductance of bonding wires 2.3.2 Capacitors 2.3.3 Resistors 2.3.4 Chip inductors, resistors and capacitors 2.4 Layout of planar hybrid and integrated circuits 2.4.1 Some layout-connected issues 2.4.1.1 Connecting series and parallel elements 2.4.1.2 The stub 2.4.1.3 Active element mounting 2.4.1.4 Planar line discontinuities 2.4.2 Hybrid layout
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Contents
2.4.3 Integrated layout 2.5 Microwave circuit packaging 2.6 Questions and problems
60 65 68
71 3.1 Representations of linear two-ports 71 3.2 The scattering parameters 74 3.2.1 Power waves 74 3.2.2 Power wave n-port model 76 3.2.3 Power wave equivalent circuit - definition and evaluation 79 3.2.4 Solving a network in terms of power waves 82 3.2.5 Properties of the S-matrix: power, reciprocity, reactivity 83 3.3 Generator-load power transfer 85 3.4 Power transfer in loaded two-ports 87 3.5 Gains of loaded two-ports 91 3.5.1 Maximum gain and maximum power transfer 91 3.5.2 Operational gain 93 3.5.3 Available power gain 96 3.5.4 Transducer gain 98 3.5.5 Is power matching always possible? 98 3.6 Stability 99 3.6.1 Analysis of stability conditions 101 3.6.2 Unconditional stability necessary and sufficient conditions 106 3.6.3 Proof of stability criteria 107 3.6.3.1 Output stability criterion 107 3.6.3.2 Input stability criterion 109 3.6.3.3 Input and output stability 110 3.7 One-parameter stability criteria 112 3.7.1 Proof of the single parameter criterium 112 3.8 Two-port stability and power matching 114 3.8.1 Power matching and maximum gain: can it be always realized? 114 3.8.2 Managing conditional stability 119 3.8.3 Stability circles and constant gain contours 120 3.8.4 Unilateral two-port 122 3.9 Examples 123 3.9.1 Stability and gains at constant frequency 123 3.9.2 Stability and gains as a function of frequency 125 3.10 Questions and problems 126
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Power gain and stability of a loaded two-port
4
Directional couplers and power dividers
4.1 Coupled quasi-TEM lines 4.1.1 Analysis of symmetrical coupled lines 4.1.2 Coupled planar lines
132 132 133 138
Contents
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4.1.2.1 Coupled microstrips 4.2 The directional coupler 4.3 The two-conductor coupled line coupler 4.3.1 Frequency behaviour of the synchronous coupler 4.3.2 Effect of velocity mismatch and compensation techniques 4.4 Multiconductor line couplers 4.4.1 The Lange coupler 4.5 Interference couplers 4.5.1 Branch-line coupler 4.5.2 Lumped-parameter directional couplers 4.5.3 The hybrid ring 4.6 Power combiners and dividers 4.6.1 Wilkinson distributed dividers 4.6.2 Wilkinson lumped dividers 4.7 Conclusions 4.8 Questions and problems
139 141 144 150 152 155 161 164 165 170 175 176 177 181 183 183
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Active microwave devices and device models
186
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Noise and noise models
187
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LLinear amplifiers
188
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Power amplifiers
189
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1
A system introduction to microwave electronics
1
2
Passive elements and circuit layout
2.1
Transmission lines Transmission lines (TLXs) are the key distributed element model in microwave circuits, acting as signal transducers between circuit components but also being the basis for a number of passive distributed elements, such as couplers, filters and matching sections. In hybrid and monolithic microwave circuits typical guiding structures are the so-called TEM or quasi-TEM lines, characterized by broadband behaviour (almost constant propagation parameters from DC up) and by the absence of cutoff frequency (as it is found in rectangular or circular waveguides). From a theoretical standpoint, a set of N metal conductors plus a ground plane supports N TEM or quasi-TEM propagation modes. TEM (Transverse ElectroMagnetic) modes are characterized by and electric field and magnetic field are transverse, i.e. orthogonal to the line axis (i.e. no longitudinal field components exist along the line axis and the propagation direction). A purely TEM mode exists in theory only in a lossless line with homogeneous cross-section, but in practice a quasi-TEM mode (with small longitudinal components) is supported by lossy lines and lines with a non homogeneous cross section, e.g. a microstrip line where the cross section is partly filled by a dielectric and partly by air. If N = 1 we talk about simple lines, in N > 1 we have coupled or multiconductor lines. TEM and quasi-TEM lines may also support upper propagation modes with a cutoff frequency; however those modes have to be avoided because they contribute to radiation losses and cannot be exploited to useful purposes.
2.1.1
Transmission line theory Transmission line theory is a convenient model for 1D wave propagation. Two parallel ideal conductors (one is the active or signal conductor, the other the return or ground conductor) surrounded by a homogeneous, lossless medium, support a Transverse Electromagnetic (TEM) propagation mode in which both the electric and the magnetic fields lie in the line cross section and are orthogonal to the line axis and wave propagation direction, see Fig. 2.1, (a). In such a TEM TXL the electric field can be rigorously derived from a potential function satisfy, in the line cross-section, the Laplace equation. The transverse electric
2
3
2.1 Transmission lines
Figure 2.1 Example
of TEM transmission line (a) and equivalent circuit of a line cell of length dz in the lossless (b) and lossy (c) cases.
potential is uniquely determined by the conductor potentials, or, assuming one as the reference, by the signal line potential v(z, t) with z parallel to the line axis and propagation direction. In the same conditions, the transverse magnetic field is related to the total current i(z, t) flowing in the signal conductor. From the Maxwell equation, v and i can be rigorously shown to satisfy the partial differential equation system (called the telegraphers’ equations ): ∂ ∂ i(z, t) = v(z, t) (2.1) ∂z ∂t ∂ ∂ v(z, t) = i(z, t) (2.2) ∂z ∂t where is the per-unit-length (p.u.l.) line inductance, the p.u.l. line capacitance. The telegraphers’ equations are compatible with the voltage and current Kirchhoff equations applied to the lumped equivalent circuit of a (lossless) line cell of infinitesimal length, see Fig.2.1 (b). The p.u.l. parameters have a straightforward meaning, i.e. they correspond to the total series inductance of unit length cell and to the total capacitance between the two conductors in a unit length cell. In real lines some series conduction losses are associated to the line conductor and some parallel or shunt losses are associated to the dissipation mechanisms in the dielectric substrate; an additional series resistance and parallel conductance can be incorporated in the model as discussed further on to account for losses. The telegraphers’ equations admit, in the lossless cases, a general solution in terms of forward (V + , I + ) or backward (V − , I − ) propagating waves:
−C −L
L
C
v(z, t) = V ± (z
∓ vf t) i(z, t) = I ± (z ∓ vf t)
4
Passive elements and circuit layout
substitution into system (2.1), (2.2): ∂ I ± (z ∂z ∂ V ± (z ∂z
∓ vf t) = −C ∂t∂ V ±(z ∓ vf t) → I ± = ±C vf V ± I ± (z ∓ vf t) → V ± = ±Lvf I ± , ∓ vf t) = −L ∂ ∂t and elimination of the current or voltage unknown (I ± = ±C vf V ± = LC vf 2 I ± ) allows to conclude that the (phase) velocity propagation is given by: vf =
√ 1LC ,
while the voltage and current waveforms are related by the characteristic impedance Z 0 (also denoted as Z ∞ or Z c ):
L
V ± = ±
I ± → V ± = ±Z 0 I ± , Z 0 = C
L
C.
A lossless TXL therefore supports undistorted wave propagation. Time harmonic voltages or currents of frequency f and angular frequency ω = 2πf yield propagating waves of the form:
√
ω v(z, t) = V ± (z ∓ vf t) = 2Re V ± exp jωt ∓ j z
√ 2 Re
V ± exp ( jωt
i(z, t) = I ± (z 1 Z 0
√ ∓ vf ±t) =
√ ∓ jβz)±
2Re I exp(jωt
2Re V exp (jωt
∓ jβz)
,
vf
∓ jβz)
=
=
where V ± is a complex proportionality constant to be determined through the initial and boundary conditions,1 while: ω β = = ω vf
√ LC
is the propagation constant of the line. The time periodic waveform with period T = 1/f is also periodic in space with spatial periodicity corresponding to the guided wavelength λg such as: β =
2π λg
→ λg = vf f = nλeff 0 ,
√
where neff = eff is the line effective refractive index, eff is the line effective (relative) permittivity. The circuit model can be extended to account for losses by introducing a series p.u.l. resistance (associated to ohmic losses in the conductors) and a parallel p.u.l. conductance (associated to the dielectric losses in the surrounding medium), see Fig. 2.1, (c). In fact, series losses cause small longitudinal field components, thus making the field distributions slightly different from the ideal TEM
R
1
The
G
√ 2 factor is introduce to normalize V ± to the voltage effective rather than peak value.
2.1 Transmission lines
5
pattern; however, the TXL model can be heuristically extended also to cases in which the cross section is inhomogeneous and therefore the structure supports a so-called quasi-TEM mode. Quasi-TEM propagation can be approximately modeled as TXL with frequency-dependent propagation parameters. Both in the TEM and quasi-TEM cases the operating bandwidth is wide, ranging from DC to an upper frequency limit associated to the onset of high-order modes or sometimes to limitations related to line losses, and the frequency dispersion of the propagation parameters (due to modal dispersion in the quasi-TEM case but also to ohmic losses) is low, at least in the high-frequency range. Undistorted propagation is typical of lossless TXLs where the signal phase velocity is frequency independent. For lossy lines the telegraphers’ equations can be modified, by inspection of the related equivalent circuit, as: ∂ ∂ i(z, t) = v(z, t) v(z, t) (2.3) ∂z ∂t ∂ ∂ v(z, t) = i(z, t) i(z, t). (2.4) ∂z ∂t In this case propagation is not undistorted any longer and the simple propagation solution outlined so far is not generally valid. The lossy case can be conveniently addressed in the frequency domain, i.e. for time-harmonic v and i. We generally assume in this case that the complex time-domain signal has the form:
−C −L
v(z, t) = i(z, t) =
√ √ 2Re
−G −R
V ± (z, ω)exp(jωt)
2Re I ± (z, ω)exp(jωt) ,
V ± (z, ω) and I ± (z, ω) are the space-dependent phasors associated to v and i (in the general case V (z, ω) and I (z, ω)), such as: V ± (z, ω) = V ± exp ( αz
∓ ∓ jβz) = V ± exp (∓γz) V ± (z, ω) V ± I ± (z, ω) = I ± exp (∓αz ∓ jβz) = = exp(∓γz) , Z 0 Z 0
(2.5) (2.6)
where α is the line attenuation, γ = α + jβ is the complex propagation constant , Z 0 is the (now possibly complex) characteristic impedance, and V ± a constant to be determined from initial and boundary conditions. For time-harmonic signals, system (2.3), (2.4) become: ∂ V (z, ω) = (jω + )I (z, ω) (2.7) ∂z ∂ I (z, ω) = (jω + )V (z, ω). (2.8) ∂z Substituting from (2.5) and (2.6) we obtain, for the complex propagation constant γ and for the complex characteristic impedance Z 0 :
− L R − C G
L R L R
α + jβ = γ = V ± (z, ω) = I ± (z, ω)
±
(jω +
jω + jω +
)(jω + )
C G
C G ≡ ±Z 0.
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Passive elements and circuit layout
The dispersive behaviour of the line is thus apparent; as a matter of fact, the very p.u.l. parameters are frequency-dependent because of the presence of losses. The real part of the complex propagation constant γ = α + jβ , α, can be further splitted (in the so-called high-frequency regime, see further on) into a conductor attenuation αc and a dielectric attenuation αd ; the imaginary part is the real propagation constant, β = ω/vf , vf phase velocity. The propagation constant β is measured in rad/m, while the attenuation is expressed in Np/m (neper per meter), or. more commonly, in dB/m or dB/cm (decibel per meter or per centimeter); the units are related as: α dB/m = 8.6859α
|
α dB/cm = 0.086859α
|
(2.9)
(2.10)
where α is in natural units, i.e. in Np/m. In fact, given a forward propagating voltage V + (z), we have, in the presence of attenuation:
|V +(z + L)| = |V +(z)| exp(−αL) where α is in Np/m and L in m. In dB we obtain:
V + (z) V + (z + L)
dB
V + (z) = 20 log10 + = 20 log10 exp(αL) = V (z + L) = αL 20log10 e = 8.6859αL = α dB/m L.
×
|
Electromagnetic theory shows that at high frequency the current density is not uniform over the conductor cross section (as in DC) but, rather, electric and magnetic fields only penetrate the line conductors down to an average thickness called the skin penetration depth δ : δ =
2 = µσω
1 , πµσf
(2.11)
where µ µ0 = 4π 10−7 H/m is the metal permittivity (we assume conductors to be non-magnetic). If δ is much smaller than the conductor thickness the current flow is limited to a thin surface sheet having sheet impedance: 2
≈
×
1+j Z s (ω) = R s + jX s = = (1 + j) σδ
ωµ . 2σ
(2.12)
Thus, the high-frequency p.u.l. resistance follows the law:
R(f ) ≈ R(f 0)
f , f 0
while the high-frequency p.u.l. inductance can be split into two contributions, the external inductance ex (related to the magnetic energy stored in the dielectric surrounding the line), and the frequency-dependent) internal inductance in
L
2
L
The sheet impedance is the impedance of a square piece of conductor; it is often expressed in Ohm/square.
2.1 Transmission lines
7
Frequency behaviour of the line resistance and inductance from the DC to the high-frequency (skin effect) regime. Figure 2.2
(related to the magnetic energy stored within the conductors); since the corresponding reactance X in (f ) X in (f 0 ) f /f 0 one has:
≈
L(f ) = Lex + Lin(f ) ≈ Lex + Lin(f 0)
f 0 f f →∞
≈ Lex.
At high frequency, therefore, the total inductance can be approximated by the external contribution. The behaviour of the p.u.l. resistance and inductance of a lossy line as a function of frequency are shown in Fig. 2.2; the high-frequency regime it typically the one in which microwave circuits operate. A short discussion follows on the relationship between the surface impedance Z s and the per-unit-length parameters. From the definition, Z s is the impedance of a metal patch of width w and of length l = w. For a conductor of periphery p and length l = p the total input impedance will be: Z =
Z l = Z p = (Rl + jX l) ≡ Z s;
it follows that the per unit length impedance of the conductor
Z = Z ps .
Z is:
For example, for a circular wire of radius r and for a strip of width w and thickness t we have, respectively: Z s Z s , Z strip = . Z wire = 2πr 2 (w + t)
(2.13)
For different reasons, also the p.u.l. conductance will be frequency dependent; in fact, this is associated to the complex permittivity of the surrounding
8
Passive elements and circuit layout
dielectrics = j” = (1 j tan θ), where tan θ θ is the (typically small, 10−2 10−4 ) dielectric loss tangent . Suppose in fact to consider a parallel-plate capacitor of area A and electrode spacing h; the capacitor impedance will be:
−
−
−
≈
A A A = jω + ω” = jωC + G(ω). h h h The result can be generalized to a transmission line with transversally homogeneous (or also inhomogeneous) lossy dielectrics, where in general: Y = jω
G (f ) ≈ f f 0 G (f 0), i.e. the line conductance linearly increases with frequency. Materials characterized by heavy conductor losses (like doped semiconductors) have, on the other hand, frequency-independent conductivity, leading to a frequency-independent line conductance. In a lossy line the propagation parameters γ and Z 0 are real at DC and very low frequency:
≈ √ RG R, Z 0 ≈ G
α + jβ
since in this case the line works as a resistive distributed attenuator. In an intermediate frequency range jω + jω while jω + in most lines, since typically series losses prevail over parallel losses. The line performances are therefore dominated by the p.u.l. resistance and capacitance ( RC regime ), with parameters:
C G≈ C
L R≈R
√ ≈ 1√ +2j ωCR 1 − j R Z 0 ≈ √ . ωC 2
α + jβ
In the RC regime the line is strongly dispersive and the characteristic impedance complex. The RC model is adequate e.g. for digital interconnects in Si integrated circuits. Finally, in the high-frequency regime jω and jω ; the imaginary part of Z 0 can be neglected and the complex propagation constant can be approximated as:
CG
Z 0
LR
L
≈ Z 0l = C R (f ) + G (f ) Z 0 + jω√ LC = αc (f ) + αd (f ) + jβ l , γ = α + jβ ≈ 2Z 0 2 √ where Z 0l is the impedance of the lossless line, αc ∝ f and αd ∝ f are the conductor and dielectric attenuation, respectively (usually αc αd in the RF and microwave range), and β l is the propagation constant of the lossless line. Therefore, in the high-frequency regime a wideband signal (e.g. a passband pulse)
2.1 Transmission lines
9
propagates almost undistorted, apart from the signal attenuation. The onset of the high-frequency regime depends on line parameters; integrated structures with micron-scale dimensions can operate in the RC range for frequencies as high as a few GHz. Moreover, the impact of losses is related to the length of the TXL; in short structures signal distortion can be modest even though the line operates under very broadband excitation. While the low-frequency or RG range is of little interest for microwaves, the transition between the RG and the LC behaviour critically depends on the line parameters and often occurs within the microwave frequency range, particularly in monolithic microwave circuits with small features. An example of behaviour is discussed in the Example 2.1.
Example 2.1:
• A transmission line has 50 Ω high-frequency impedance, effective permittivity equal to 6, conductor attenuation of 0.5 dB/cm, dielectric attenuation of 0.01 dB/cm at f 0 = 1 GHz. Suppose that the two attenuations do not depend on frequency. Evaluate the line parameters in the RG, RC and LC regime, specifying the frequency ranges of validity.
◦ For the sake of definiteness, suppose that at f = 1 the line is already in the 0
high-frequency regime; in the LC approximation we have: Z 0 =
L C /
√ √ vf = 1/ LC = 3 × 108 / eff
i.e.: 1/ = 50
C
√ × 3 × 108/√ eff −→ C = 6/
and thus:
150
× 108
= 1.633
× 10−10 F/m
L = C Z 02 = 4.0825 × 10−7 H/m The attenuations in the high-frequency approximation yield:
R ≈ 2Z 0
αc
≈ G 2Z 0
αd
i.e., since αc = 0.5 dB/cm = 1/0.086859 = 5.75 Np/m; αd = 0.01 dB/cm = 0.115 Np/m:
R = 2Z 0αc = 100 × 5.75 = 575 Ω/m G = 2αd/Z 0 = 2 × 0.115/50 = 0.0046
S/m.
Let us verify that the line actually is in the high-frequency regime at 1 GHz; for this we require:
L R → 6.28 × 109 × 4.0825 × 10−7 = 2564 575 2πf 0 C G → 6.28 × 109 × 1.633 × 10−10 = 1.025 0.0046
2πf 0
10
Passive elements and circuit layout
Figure 2.3 Frequency
behaviour of attenuation and propagation constant from
Example 2.1.
and both conditions are verified; the line parameters therefore are:
L = 4.0825 × 10−7 H/m C = 1.633 × 10−10 F/m R = 575 Ω/m G = 0.0046 S/m. The frequency behaviour of the propagation constant and attenuation are shown in Fig. 2.3. The low- intermediate- and high-frequency regimes are clearly visible both on the attenuation and on the characteristic impedance, shown in Fig. 2.4; in particular, in the intermediate frequency range the real and imaginary parts of the impedance are approximately the same. The high-frequency attenuation is constant because we have neglected the frequency dependence typical of skin losses and parallel dielectric losses.
2.1.1.1
More on series losses To justify the skin-effect analysis of losses, consider a metal slab of conductivity σ and thickness t, carrying, on a width w, a total current I at frequency f . Supposing that the system is transversally homogeneous (see Fig. 2.5), the input impedance of the metal slab can be evaluated as follows. In DC conditions the current density is uniform in the slab cross section and J z = I /(wt) = σ z (independent of x). At frequency f , the electric field phasor E z satisfies the wave
E
11
2.1 Transmission lines
Figure 2.4 Frequency
behavour of characteristic impedance from Example 2.1.
equation. Being µ the magnetic permittivity, = σ/(jω) the complex dielectric permittivity of the metal, dominated by the imaginary part (i.e. by the conductivity), E z satisfies, from the Maxwell equations, the wave equation:
−
d2 E z = jωµσE z . dx2 As a boundary condition, we assume that on the conductor ( x = t/2) the field is known and equal to E 0 . The symmetry of the problem suggests to write the solution as:
±
E z (x) = A cosh
1+j x δ
where A is a constant to be determined and δ is the skin penetration depth, see (2.11). Imposing the surface field we have:
1+j x δ E z (x) = E 0 . 1+j cosh t 2δ cosh
From the Maxwell equations we obtain that the magnetic field is directed along y and has the value:
H y =
1 dE z 1 = jωµ dx 1+j
1+j x 2σ δ E 0 . 1+j ωµ cosh t 2δ
sinh
12
Passive elements and circuit layout
Thus, the contour integral of the magnetic field along γ in Fig. 2.5 equals the total current flowing in a strip of width w, I : I =
·
H dl = wH y (t/2)
γ
1 = 2wH y (t/2) = 2w 1+j
− wH y (−t/2) =
2σ E 0 tanh ωµ
1+j t . 2δ
The potential drop at the strip surface over a length L is V = E 0 L; it follows that the impedance of a slab of width w and length L is: V L Z = = (1 + j) I 2w Notice that for ω
ωµ coth 2σ
1+j t . 2δ
→ 0, δ → ∞ and, since: 1+j 2δ 1 2 1 coth t ≈ = 2δ 1+j t 1+j t
(2.14)
2 µσω
we obtain: L lim Z = (1 + j) ω →0 2w i.e. the DC resistance. For ω Z (ω)
≈
× ωµ 2σ
2 1 1+j t
2 L = µσω wtσ
→ ∞, δ → 0 leading to the approximation:
L (1 + j) 2w
ωµ L 1 = (1 + j) . 2σ 2w σδ
At high frequency (see Fig. 2.6) the current is concentrated on the upper and lower surface of the strip and the penetration of the current occurs on a depth of the order of δ t; this justifies the name skin effect . The total impedance is in that case the parallel of the upper and lower surface impedance; for L = w we recover the definition the metal surface impedance, given by:
1+j Z s (f ) = Rs (f ) + jX s (f ) = = (1 + j) σδ
2πf µ , σ
see (2.12). The surface resistance and reactance have the same magnitude and they are proportional to f . An example of the frequency behaviour of the impedance of a metal strip conductor is shown Fig. 2.7. The internal reactance (inductive) linearly grows at low frequency and becomes, in the skin effect region, equal to the resistance. The resistance equals the DC resistance up to a frequency at which the strip thickness becomes comparable with δ . In the skin-effect region the conductor resistance becomes, from (2.13):
√
R =
Rs 1 = . 2 (w + t) 2δσ (w + t)
For a good conductor δ is of the order of a few µm at frequencies of the order of a few GHz. The frequency behaviour of R s and δ is shown in Fig. 2.8 for several
2.1 Transmission lines
13
Figure 2.5 Evaluating
the impedance of a metal layer of thickness t , infinitely thick in the y direction; we only consider a section of width w . All fields are invariant vs. y and z .
Figure 2.6 Frequency
behaviour of the current distribution (magnitude of J z normalized vs. the surface value) with a copper metal layer, thickness t = 10 µm, conductivity σ = 4 107 S/m.
×
values of conductivity; the maximum value σ = 1 108 S/m is slightly larger to the typical value of good conductors (for copper σ 6 107 S/m). In the planar microwave technology we exploit composite metal layers obtained first through a thin sputtered or evaporated adhesion layer (0.1-0.2 µm) followed by sputtered gold or Al (up to 500 nm). Larger thicknesses (from a few µm to 1520 µm) can be obtained by gold electroplating. For such thicknesses the transition
× ≈ ×
14
Passive elements and circuit layout
Frequency behaviour of the impedance of a copper layer with w = L , σ = 4 10 S/m, thickness t = 10 µm. Figure 2.7
×
7
between the RC range and the high-frequency regime (with fully developed skin effect) takes place in the microwave range. 2.1.1.2
More on parallel losses Parallel losses are accounted for by the p.u.l. conductance and are associated to dielectric losses. The dielectric response of a material, leading to a relative permittivity larger than one, is caused by the interaction between the EM wave and microscopic mechanisms occurring the material itself. The main interaction with the EM field are with dipolar molecules (e.g. water), atoms and electrons. Each interaction is characterized by a low-pass behaviour: for low frequencies the interaction is active and provides a contribution to the dielectric response, at high frequency the interacting agent is not able to follow any more the time variations of the field and the contribution to the dielectric response vanishes. However the interaction is also affected by losses that are maximized around the transition frequency. The transition frequencies of many molecular interactions fall in the microwave range and anyway atomic and electronic contributions are active (such contributions disappear in the UV range mainly). We therefore have losses basically proportional to the frequency (each cycle leads to a loss of energy, so the dissipated power increases with the number of cycles per unit time). This can be described by a complex dielectric constant:
G
rc = r
− j2 = r (1 − j tan δ (f ))
15
2.1 Transmission lines
Figure 2.8 Frequency
behaviour of surface resistance and skin-effect depth for different values of conductivity.
where δ (generally a weak function of frequency) is the dielectric loss angle (nothing to do with the skin penetration depth!). The dielectric conductivity will therefore be: σ = ω2 0 = ω r 0 tan δ. In a TEM (or transversally homogeneous) line the p.u.l. parallel admittance can be evaluated simply as:
Y = jωC = jωrc Ca = jωr Ca + ωr tan δ Ca; where a is the capacitance p.u.l. in air (i.e. with rc = 1). the second (real) term is a conductance, yielding:
C
G = ωr tan δ Ca = σ Ca /0.
(2.15)
Notice that is the dielectric medium has large conduction losses (rather than dielectric losses), like in many semiconductors like Si, the conductance simply is proportional to the conductivity and is frequency independent . If the line has inhomogeneous cross section (like in a quasi-TEM line) evaluating is slightly more involved. Consider for instance a microstrip line with a lossy substrate and an air (lossless) overlayer; we can split the total capacitance p.u.l. as = air + sub = air + rc sub,a where sub,a is the substrate
G
C C
C
C
C
C
16
Passive elements and circuit layout
Material r tan δ
Allumina 9.8 10−3
Quartz 3.78 10−4
Teflon 2 10−4
Beryl ox. 6 10−3
GaAs 12.9 10−3
InP 12.4 10−3
Si 11.9 10−2
Table 2.1. Characteristics of some dielectric substrates fo hybrid and integrated circuits.
Figure 2.9 Examples
of non-TEM, TEM, quasi-TEM waveguides.
capacitance in air. The p.u.l. admittance is therefore:
Y = jωC = jωCair + jωrc Csub,a = = jωCair + jωr Csub,a + ω r tan δ Csub,a , leading to:
G = ω r tan δ Csub,a = σ Csub,a /0. with the same frequency behaviour. The remarks made so far hold for most of the low-loss dielectric of semiconductor substrates, see Table 2.1; for Si conduction losses can be high, and for this reason Si circuits exploit oxide layers to screen the top conductors from the substrate.
2.1.2
Parameters of quasi-TEM lines In planar microwave circuits transmission lines made by one signal conductor supported by a dielectric substrate and a ground plane are the most common technological solution. Such lines have an inhomogeneous cross section and therefore support a quasi-TEM mode. Examples of non-TEM (rectangular waveguide), quasi-TEM (microstrips) and TEM (striplines) guiding structure are shown in Fig. 2.9. Non-TEM waveguides have a simply connected metal cross-section and therefore do not support DC conduction, while both TEM and quasi-TEM lines allow for DC conduction through two separated metal conductors. The main difference between the TEM and quasi-TEM case is the fact that in the latter the phase velocity and characteristic impedance are a weak function of frequency. The frequency dispersion of quasi-TEM parameters is significant for the phase velocity (or effective permittivity) and can be properly taken into account in the operating frequency range; above a certain frequency (that increases with decreasing line dimensions and substrate thickness) higher-order modes can appear leading to radiation losses. Neglecting in the first approximation losses and confining in any case to the high-frequency approximation we have that the quasi-TEM line characterization
2.1 Transmission lines
17
is based on evaluating and . First of all, we show that in a TEM line (homogeneous cross section, relative permittivity r ), does not depend on the dielectric permittivity. In fact, let us denote with the inductance with dielectrics and a the inductance in air; the phase velocity with dielectrics coincides with the phase velocity in the medium, i.e.:
L
C
L
L
L
vf =
√ 1LC = √ c0 r
where c0 is the velocity of light in air; on the other hand, the phase velocity of the line in which r = 1 (line in air or in vacuo) will be:
√ L1 C ; (2.16) a a however, C = Ca r , and therefore L = La , i.e. the inductance with dielectrics is c0 =
the inductance in air. The same result holds for quasi-TEM lines; in fact, the inductance is related to the magnetic field, that, from a quasi-static standpoint, is not influenced by the presence of dielectrics. In a quasi-TEM line the p.u.l. inductance is a function of the p.u.l. capacitance in air; in fact from (2.16) we obtain:
L = La = c21Ca .
(2.17)
0
Therefore we can express the characteristic impedance and phase velocity as a function of the capacitances in air and with dielectric as follows:
L
1 Z 0a √ = √ C c0 CC a r 1 Ca = √ c0 vf = = c 0 r LC C
Z 0 =
=
(2.18)
(2.19)
where Z 0a is the impedance in air. In the quasi-TEM case we can introduce an effective permittivity eff such as:
C = eff Ca;
√
we also define the effective refractive index neff = eff . Thus, in a quasi-TEM line: 1 Z 0a Z 0 = = (2.20) c0 a eff eff c0 vf = (2.21) eff
C √ √
√
and, furthermore:
√ λ0eff √ β = β 0 eff .
λg =
(2.22)
(2.23)
18
Passive elements and circuit layout
Figure 2.10 Parallel
plate quasi-TEM line with inhomogeneous dielectric.
A simple example of evaluation of the effective permittivity of a quasi-TEM line is discussed in Example 2.2.
Example 2.2:
• Consider a quasi-TEM line made by two parallel metal conductors; the dielectric in not homogeneous as shown in Fig. 2.10. Evaluate the effective permittivity and impedance of the line supposing that the field lines are orthogonal to the conductors.
◦ An elementary line section with length dz is made of two parallel capacitors
of size h, W , dz, one in air and the other with a dielectric constant r . The total capacitance p.u.l. will be:
C
C 1 = = dz dz
W dz W dz 0 + 0 r h h
=
W (1 + r )0 h
while: 20 . Ca = W h The effective permittivity is the ratio between the capacitance with dielectrics and the capacitance in air; we obtain: eff = For the impedance: 1
C = 1 + r . 2 Ca
h √ Z 0 = = 0 µ0 × × √ c0 Ca eff 2W 0 where
2 = 120π 1 + r
2 h 1 + r 2W
µ0 /0 = 120π is the characteristic impedance of vacuum.
In general, in a planar line on a dielectric substrate, the effective dielectric constant has values between 1 (the air constant) and the dielectric constant of
19
2.1 Transmission lines
Figure 2.11 Behaviour
of the effective permittivity of a quasi-TEM line as a function
of frequency.
the substrate. A non-quasi static analysis permits to find out that in a quasiTEM line the effective permittivity is frequency dependent according to the typical behavior shown in Fig. 2.11. The effective permittivity grows slowly with frequency from the quasi-static value eff (0); the increase becomes fast after the inflection frequency f infl which also corresponds (approximately) to the cutoff frequency of the first higher-order mode of the structure. Higher-order modes are mainly guided by the dielectric substrate and therefore lead to power leakage and radiation; for this reason the inflection frequency limits the useful operation range of the structure. For very high frequencies the effective permittivity tends to the substrate permittivity r . The useful frequency range of the line is below the inflection frequency. The behaviour can be approximated through empirical expressions, e.g.: eff (f )
≈
α eff (0) +
α α r eff (0) 1 + (f infl/f )β
−
1/α
(2.24)
where α and β are fitting parameters, of the order of 1. The characteristic impedance does not exactly follow the behaviour of the effective permittivity; however, its variation with respect to the quasi-static condition is less important and we will always use the low-frequency value for it.
2.1.3
Working with transmission lines: the Smith chart Transmission lines can be exploited as circuit elements using the line solution in terms of forward and backward waves. A circuit including transmission lines can be easily shown to be amenable to a well posed solution since each line has
20
Passive elements and circuit layout
Figure 2.12 Input
impedance of a loaded transmission line.
two unknowns (the forward and backward voltages) and two relationships are imposed on the line input and output, depending on the loading conditions. As a first example consider a line of length l closed on a load Z L (Fig.2.12); the voltage and current phasors can be written as a superposition of forward and backward waves as: V (z) = V + exp( γz) + V − exp (γz) V + V − I (z) = exp( γz) exp(γz) , Z 0 Z 0
− − −
with boundary condition (the line current is directed towards increasing z ): V (l) = Z L I (l)
→ V + exp(−γl) + V − exp (γl) = Z Z L0
i.e.:
V + exp( γl)
− − V − exp (γl)
V − = ΓL exp ( 2γl) , V +
−
where: ΓL =
Z L Z 0 zL 1 = Z L + Z 0 zL + 1
−
−
is the so-called load reflection coefficient with respect to Z 0 and z = Z/Z 0 is the normalized impedance . The line input impedance will be: Z i = Z (0) =
V (0) V + + V − 1 + Γ L exp ( 2γl) = Z 0 + = Z . 0 I (0) V V − 1 ΓL exp ( 2γl)
−
−
− −
Expanding the load reflection coefficient and expressing the exponential in terms of hyperbolic functions we obtain: Z i = Z 0
Z L cosh (γl) + Z 0 sinh (γl) . Z L sinh (γl) + Z 0 cosh (γl)
Notice that for l , Z i Z 0 independent on the load. For a lossless line, however, the input impedance is periodic vs. the line length, with periodicity
→ ∞ →
,
2.1 Transmission lines
21
λg /2 (due to the tan function): Z i = Z 0
Z L + jZ 0 tan (βl) . Z 0 + jZ L tan (βl)
(2.25)
The input impedance of a lossless line of infinite length does not converge therefore to the characteristic impedance. This is of course purely theoretical, since no lossless line exist and an infinitely long line would never get into a steady state condition in a finite time (thus a reflected signal would appear after an infinitely long time). We will consider now some particular and useful cases: if the load is a short (Z L = 0) or an open (Y L = 0) we have: Z i (Z L = 0) = jZ 0 tan (βl) Z i (Y L = 0) = jZ 0 cot (βl) ;
−
a reactive load is therefore obtained, alternatively inductive and capacitive according to the value of the line electrical angle φ = βl = 2πl/λg . For a lossy line we have: Z i (Z L = 0) = Z 0 tanh(γl) Z i (Y L = 0) =
−Z 0 coth(γl);
It can be readily shown by inspection than the input impedance of a shorted lossy line for l 0 is Z i jω l + l, while the input impedance of a short line in open circuit is Z i (jω l + l)−1 . For a quarter-wave line (l = λ g /4):
→
≈ L R ≈ C G Z i = Z 0
Z L sinh(αl) + Z 0 cosh(αl) Z 0 sinh(αl) + Z L cosh(αl)
while for a half-wavelength line (l = λ g /2): Z i = Z 0
Z L cosh(αl) + Z 0 sinh(αl) . Z 0 cosh(αl) + Z L sinh(αl)
For a lossless line we have on the other hand: Z i = Z 0
Z L 0 + Z 0 1 Z 2 = 0 Z 0 0 + Z L 1 Z L
· ·
· ·
(a quarter-wave line, yielding the so-called frequency transformer); Z i = Z 0
Z L 1 + Z 0 0 = Z L Z 0 1 + Z L 0
· ·
· ·
(half-wavelength line). In the above treatment we made use of the reflection coefficient simply as a notational shortcut. However, this is able to yield a simpler and possibly more significant picture of the line. As already remarked, the forward and backward wave amplitudes uniquely determine the voltages and currents on the whole line. We often prefer to identify the two amplitudes by assigning e.g. V + (i.e. V 0+ ) and the ratio Γ(z) = V − (z)/V + (z), the reflection coefficient at section z.
22
Passive elements and circuit layout
The variation of the reflection coefficient with position is immediately found; we immediately have: V − (z) V − (0)exp(jβz) Γ(z) = + = + = Γ(0)exp (2jβz); V (z) V (0)exp( jβz)
−
(2.26)
in other words Γ(z) is periodic along the line with a periodicity of λg /2 (voltages and currents have a periodicity λg ). The reflection coefficient is known everywhere if it is known in one section of the line. The evolution of Γ( z) as a complex number is simple, the corresponding phasor rotates with constant magnitude in the complex plane with periodicity λ g /2. In the presence of losses the magnitude changes as well, in fact: Γ(z) =
V − (z) V − (0)exp(αz + jβz) = = Γ(0)exp(2αz + 2jβz). V + (z) V + (0)exp( αz jβz)
− −
The impedance or admittance seen from a section of the line can be immediately identified as follows: V (z) V + (z) + V − (z) V + (z) + V − (z) 1 + Γ(z) = + = Z = Z (2.27) ∞ ∞ I (z) I (z) + I − (z) V + (z) V − (z) 1 Γ(z) I (z) I + (z) + I − (z) 1 V + (z) V − (z) 1 1 Γ(z) Y (z) = = + = = (2.28) + − − Z (z) V (z) + V (z) Z ∞ V (z) + V (z) Z ∞ 1 + Γ(z) Z (z) =
− −
− −
with inverse formulae: Z (z) Z ∞ z(z) 1 = Z (z) + Z ∞ z(z) + 1 Y ∞ Y (z) 1 y(z) Γ(z) = = Y ∞ + Y (z) 1 + y(z)
−
Γ(z) =
−
−
−
(2.29)
(2.30)
−1 and z(z) = Z (z)/Z ∞ and y(z) = Y (z)/Y ∞ are the normalized where Y ∞ = Z ∞ impedances (admittances). Let us review the input impedance problem in terms of the reflection coefficient. A line with length l is closed on Z L ; we want to evaluate the input impedance in z = 0. Assume for simplicity that the line is lossless. We have: Γ(l) = from which: Γ(0) = Thus: 1+ Z (0) = Z ∞ as found in (2.25).
1
−
Z L Z ∞ Z L + Z ∞
−
Z L Z ∞ exp( 2jβl). Z L + Z ∞
−
−
Z L Z ∞ exp( 2jβl) Z L + jZ ∞ tan βl Z L + Z ∞ = Z ∞ Z L Z ∞ Z ∞ + jZ L tan βl exp( 2jβl) Z L + Z ∞
− −
− −
(2.31)
2.2 Planar transmission lines in microwave integrated circuits
23
The relation (2.31) between the normalized impedance and the reflection coefficient: z 1 Γ= (2.32) z + 1
−
is an analytical mapping from the complex plane z to the complex plane Γ, with the following properties:
the angle between two intersecting curves in z plane is preserved in the Γ plane, i.e. the mapping is conformal ; circles or straight lines in plane z are transformed in circles or straight lines in plane Γ; constant resistance (z = r + jx) lines transform into circles in Γ plane with center on the real axis; constant reactance lines transform into circles in Γ plane going through the origin; the half plane Re(z) > 0 is transformed into the circle Γ 1; purely reactive impedance are transformed into the unit circle Γ = 1.
| |
| | ≤
Some important points of the Γ plane are as follows. For z = 1 (reference impedance) Γ = 0 (center of the reflection coefficient complex plane). Short and open circuits correspond to Γ = 1, respectively. Reactive impedances yield:
∓
Γ=
jx 1 jx + 1
− → |Γ| = 1;
in particular, inductive impedances have reflection coefficients in the upper Γ plane, capacitive impedances in the lower Γ plane. The above remarks are summarized in Fig. 2.13. Notice that from its definition Γ ΓV , the voltage reflection coefficient; the current reflection coefficient trivially is Γ I = ΓV . Such idea have a graphical representation in the so-called Smith chart (from his inventor, Phillip Hagar Smith, 1905-1987), see Fig. 2.14. The Smith chart reproduces a number of circles corresponding to impedances with constant real or imaginary parts in the Γ plane, and can be used both to identify the reflection coefficient corresponding to a certain impedance, and to carry out graphical computations exploiting the fact that along a (lossless) transmission line the reflection coefficient rotates with constant magnitude. The Smith chart as a design tool has been superseded by CAD tools but its use in the graphical representation of parameters amenable to reflection coefficients (including the scattering parameters corresponding to reflectances) is widespread both in the instrumentation and in the CAD tools themselves.
≡
2.2
−
Planar transmission lines in microwave integrated circuits Fig. 2.15 shows some TEM, quasi-TEM and non-TEM microwave waveguides. Apart from the slot line , that may be exploited in antenna transitions, hybrid and
2.2 Planar transmission lines in microwave integrated circuits
25
monolithic microwave integrated circuits are based on quasi-TEM or TEM lines. Closed TEM structures like the stripline are exploited in particular applications where radiation losses are important or to obtain high-directivity directional couplers; other shielded lines like the finline are important at millimeter waves where low metal losses are difficult to obtain. However, most of the integrated microwave circuits are based on the microstrip or the coplanar waveguide (CPW). CPWs have propagation characteristics almost independent on the substrate thickness and are popular at millimeter wave frequencies, although their layout is less compact than for the microstrip. Suspended microstrips are sometimes exploited in hybrid circuits at millimeter waves. However the vast majority of planar microwave circuits is based on the microstrip. The design of monolithic microwave integrated circuits (MMIC’s) has quite naturally followed the guidelines derived from hybrid integrated circuits. Also in the choice of transmissive media, the solutions commonly adopted in MMIC’s are those already successfully implemented in hybrid circuits, such as using quasiTEM transmission lines rather than non-TEM waveguides. Among the advantages of quasi-TEM transmission lines are their intrinsic wideband behaviour, as opposed to the more dispersive nature of non-TEM media, and the easy ground definition they allow. A disadvantage of the quasi-TEM lines, which limits their use in the millimeter wave range, are their heavy ohmic losses due to the use of strip conductors. Since the semiconductor layer which makes the substrate of a MMIC has two sides, a quasi-TEM transmission line (i.e. a line made of two separate conductors, a ground plane and a “hot” strip) can either use both sides of the substrate, or lie entirely on the top side. The first case corresponds to the microstrip approach; the second case to the coplanar or uniplanar approach (coplanar waveguide, CPW). It should be pointed out that the microstrip and coplanar approach are not simply a different choice of transmission media, but also entail a markedly different circuit design philosophy. Moreover, those two approaches to MMIC’s are by no means the only conceivable ones. Indeed, the monolithic approach allows multilayer circuits to be made, in which transmission lines and discrete components are located on the top and bottom sides of the substrate, but also stacked structures separated by dielectric passivation layers are implemented. As an example, an microstrip patch array can be printed on the top surface of the substrate, while feed lines coupled to patches are located either on the bottom side or under a passivating dielectric layer. A multilayer design strategy is today rather challenging since it requires a strict control of coupling between different lines and components, which can be achieved in turn only by accurate 3D electromagnetic modeling. Today, the microstrip approach is by far the most popular in MMIC’s. Actually, the microstrip on GaAs integrated circuits substrates of thickness ranging from 50 µm to 400 µm is up to approximately 40-50 GHz a transmission line with reasonably low losses and dispersion. The practical impedance range is wide enough as to cover most practical applications; series element connection is very
26
Passive elements and circuit layout layout
easy, whereas parallel connection is more troublesome, owing to the absence of a topmost ground plane. Suitable techniques exist to circumvent this problem, such as the use of topmost wrap-around ground planes or via holes making the bottom plane accessible from the top of the substrate; however, the first solution imposes constraints on circuit layout, while the second one increases the IC process complexity. complexity. Although the use of coplanar or uniplanar circuits is limited, the main advantages offered by coplanar lines can be listed as follows:
The performances of CPW’s are comparable, and sometimes even better, than those of the microstrip in terms of guided wavelength, losses, dispersion, and impedance range. The CPW allows easy series and shunt element connection. The CPW impedance is almost insensitive to substrate thickness. Active elements can be easily connected since they are intrinsically coplanar. The coupling between neighbouring lines is reduced owing to the presence of grounded surfaces lying in between. On-wafer measurements through coplanar probes are easier and more direct than in microstrip circuits.
Moreover, the coplanar approach allows greater flexibility in the use of mixed structures structures and transitions transitions to slot lines, coupled slot lines etc. which which can be profitably exploited in some applications (e.g. mixers, balancing units). However, coplanar lines also have disadvantages that more or less confine their use to low-power applications:
2.2 2.2.1
The power handling capabilities of uniplanar circuits are unsatisfactory, due to the absence of a lower heat sink and to the need of making the substrate rather thick. Introducing a lower ground planes partly overcomes the problem, but also creates troubles due to spurious coupling with parasitic modes. To suppress a parasitic slot-like mode, the ground planes have to be connected together by means air bridges, thereby making circuit realization more complex.
The coax coaxia iall cab cable The coaxial cable has a particular role in microwave systems and in the instrumentation, although it is obviously not amenable to integration. In can be realized either in rigid or flexible form. It has comparatively low attenuation and high immunity to electromagnetic disturbances (it is a shielded structure). Let us call a call a and and b the b the inner and outer conductor radii; the p.u.l. capacitance can be expressed as: 55 55..556 556r C = log(b/a log(b/a))
pF/ pF/m,
2.2 Planar transmission lines in microwave integrated circuits
27
Figure 2.15 Waveguides
and transmission lines in microwave circuits.
and the p.u.l. inductance: inductance: b 200 log L = 200 a
nH/ nH/m.
The characteristic impedance is: 60 Z 0 = log r
√
b a
Ω,
while the effective permittivity is the dielectric permittivity. The dielectric and conductor (we assume a copper conductor, for a different one the attenuation scales according to the square root of resistivity; in many cases however the inner and outer conductors are different, e.g. and inner copper wire and an outer aluminium jacket) attenuations are, respectively: tan δ dB/ dB/m λ0 10−5 (a + b) r ab log(b/a log(b/a))
√
αd = 27 27..3 r αc =
9.5
×
√
f GHz dB/m. GHz dB/
The useful frequency range of a coaxial cable is limited by the cutoff frequency of the first higher-order propagation mode, corresponding to the cutoff wavelength:
√
λc = π r (a (a + b)
28
Passive elements and circuit layout layout
where the cutoff frequency is f = c 0 /λc .
Example 2.3:
• Consider a copper coaxial cable with a Teflon (( = 2) dielectric. Find the ratio r
b/a corresponding to the minimum conductor losses and evaluate the resulting impedance. Dimension the cable so that the maximum operating frequency is 50 GHz.
◦ We have:
1+b/a 1+b/a 1+x 1+x = = f ( f (x), ∝ log(b/a log(b/a)) log x
αc
x > 1
df ( f (x) 1 = dx log x
1+x 1+x 1 x log x − (1+x (1+x) = =0 − (log x)2 x x (log x)2
corresponding to x to x = 3.5911. Thus: 60 Z 0 = log r
√
b a
=
60 √ log log 3.5911 = 54. 54.24 Ω. 2
We then have: c0 c0 c0 f max = = max = λc π r (a (a + b) πa r (1+b/a (1+b/a))
√
√
→
c0 3 × 108 √ (1+3..5911) = 0.0 .294 mm = √ πf max (1+b/a)) π · 50 × 109 2 (1+3 max r (1+b/a b = 3.5911 · a = 3.5911 · 0.294 = 1. 1.06 mm. Thus the coax outer diameter is 2. 2 .12 mm while the inner diameter is ≈ 0.6 mm. The conductor attenuation at 50 GHz is: √ 9.5 × 10−5 (a + b) r a =
αc =
ab log(b/a log(b/a))
f =
√ √
9.5 10−5 (1. (1.06+0 06+0..294) 10−3 2 50 = 3. 3.223 dB/m. 1.06 10−3 0.294 10−3 log log (3. (3.5911)
× ×
2.2 2.2.2
·
×
× ·
The micr micro ostri strip p The microstrip microstrip (see Fig. 2.16 for the cross section) section) is a quasi-TEM quasi-TEM transmission transmission line due to the inhomogeneous cross section. Analysis and design formulae are presented in the following sections for the line parameters; take into account that most CAD tools for microwave circuit design have embedded microstrip line calculators.
2.2.2. 2.2.2.1 1
Analys Analysis is formu formulae lae Today all CAD tools have built-in analysis formulae for the microstrip parameters; approximate expressions are given here:
29
2.2 Planar transmission lines in microwave integrated circuits
Figure 2.16 Microstrip
cross section.
Impedance:
Z 0 =
60 8h W log + , eff W 4h 120π W W + 1.393 + 0.667 log + 1.444 eff h h
√
√
W 1 h −1 W , > 1 h
≤
where the effective strip width W accounts for the strip thickness t: W h
=
W 1.25t 4πW + 1 + log , h πh t W 1.25t 2h + 1 + log , h πh t
W 1 h 2π W 1 > h 2π
≤
Effective permittivity:
−
1 + r r 1 W eff = + F 2 2 h
−
r 1 t 4.6 h
−
h W
where:
W F h
=
12h 1+ W
−
+ 0.04 1
12h 1+ W
−1/2
W h
2
,
W 1 h W > 1 h
≤
Dielectric attenuation:
r αd = 27.3 eff
√
−1/2
eff 1 r 1
− −
tan δ dB/m λ0
Conductor attenuation in dB/m:
αc = where:
Λ=
Rs 32 (W 2 Λ, hZ 0 32 + (W 2 Rs Z 0 eff W 0.667W /h 10−5 + h h W /h + 1.444
−
1.38
6.1
×
W h W Λ, h
h 1.25t 1.25 4πW W 1+ 1+ + log , W πW π t h h 1.25t 1.25 2h W 1+ 1 + log , W πh π t h
−
≤ 2π1 ≥ 2π1
≤1 ≥1
30
Passive elements and circuit layout
Figure 2.17 The
microstrip characteristic impedance vs. W /h for different substrate
permittivities.
Rs = ωµ/2σ = (σδ )−1 is the surface resistance. Dispersion of effective permittivity:
eff (f )
√ − √ ≈
r eff + − 1 + 4F 1.5
2
√ eff
where 4F −1.5 implicitly defines the inflection frequency. In fact:
√ − 1
4h r F = λ0
W 0.5 + 1 + 2 log 1 + h
2
where f = c 0 /λ0 ; thus F = kf where:
√ − 1
4h r k = c0
W 0.5 + 1 + 2 log 1 + h
2
and f = f infl when 4F −1.5 = 1 i.e. when F = 2 4/3 = kf infl ; thus f infl = 24/3 /k. Fig. 2.17 and 2.18 show examples of the microstrip parameters (impedance and refractive index) as a function of the W/h ratio for different substrate permittivities (GaAs, r = 13; allumina, 10; Teflon, 2.5). The minimum W/h is suggested by technological constraints (strips cannot be narrower than 10-20 µm) while the maximum is related to the onset of transversal resonances. Fig. 2.19 shows an example of the metal and substrate losses for a microstrip on allumina, substrate thickness 0.5 mm. At 1 GHz the metal losses prevail, but, due to the different frequency behaviour, dielectric losses can be important
31
2.2 Planar transmission lines in microwave integrated circuits
Figure 2.18 The
microstrip effective refractive index vs. W /h for different substrate
permittivities.
at millimeter waves. The metal losses decrease with the strip width, i.e. are important for high impedance (narrow) lines. 2.2.2.2
Design formulae The design formulae yield the needed W/h ratio to obtain, with a substrate with given permittivity, a certain characteristic impedance Z 0 . A classical set of approximate design formulae was developed by Wheeler:
For Z 0
≥ 44 − 2r Ω: W = h
exp(B) 8
Z 0 B = 60
−
1 4exp(B)
r + 1 1 r 1 + 2 2 r + 1
−
−1
− 2r Ω: W 2 2 r − 1 = (d − 1) − log(2d − 1) + h π π πr
(2.33)
0.2416 0.4516 + r
(2.34)
For Z 0 < 44
60π 2 d = Z 0 r
√
log(d
− 1) + 0.293 −
0.517 r (2.35) (2.36)
The effective permittivity can be derived by a set of simplified expressions:
32
Passive elements and circuit layout
Figure 2.19 Behaviour
of the dielectric and conductor attenuation for a microstrip as a
function of W /h.
for W/h
≤ 1: r + 1 r 1 eff = + 2 2
−
for W/h
12h 1+ W
−1/2
−
+ 0.04 1
W h
2
≥ 1: r + 1 r 1 eff = + 2 2
−
12h 1+ W
−1/2
.
A finite strip correction (that cannot be obviously included directly in the synthesis) is as follows:
for W/h
for W/h
≥ 1/2π:
W W t = + h h πh
2h 1 + log t
≤ 1/2π: W e W t = + h h πh
4πW 1 + log . t
Finally, the synthesis formulae are the approximate inverse of the following analysis formulae:
33
2.2 Planar transmission lines in microwave integrated circuits
for W /h
≤ 1: Z 0 =
for W /h
60 log eff
√
8h W + W 4h
≥ 1: 120π W W Z 0 = 1.393 + + 0.667 log 1.444 + eff h h
√
−1
.
Example 2.4:
• Design a 50 Ω microstrip using the substrates (a) CER-10- 0250 and (b) TLY-
5-0620, data in Table 2.2. Assume gold metallizations, conductivity σ = 4.1 107 S/m and thickness t = 7 µm.. Plot the frequency behaviour of the effective permittivity and conductor and dielectric attenuation in the two cases.
×
◦ Using the Wheeler formulae we have in case (a) = 9.5 from which Z = 0
r
50 > 44
− 2 × 9.5; thus we use: Z 0 B = 60
r + 1 1 r 1 + 2 2 r + 1
−
0.2416 0.4516 + r
= 2.1025
i.e.: W = h
exp(B) 8
−
1 4exp(B)
−1
= 1.0073
yielding W = 1.0073 0.63 = 0.63 mm. In case (b) r = 2.2 and Z 0 > 44 2r ; we obtain B = 1.16 from which W = 3.12h = 3.12 1.57 = 4.9 mm. The lowfrequency effective permittivities result, respectively, (a) eff = 6.41, (b) eff = 1.87. Let us evaluate now dispersion; in case (a) the k coefficient appearing in F = kf is:
×
√ − 1
4h r k = c0
−
×
W 0.5 + 1 + 2 log 1 + h
2
= 1.517
× 10−10
while in case (b) k = 3.4823 10−10 . The inflection frequencies f inf = 24/3 /k are in case (a) 24/3 / 1.517 10−10 = 16.6 GHz, in case (b) 24/3 / 3.4823 10−10 = 7.23 GHz. The second substrate, being thicker, is more dispersive. In fact the frequency behaviour given by:
×
×
eff (f )
×
√ − √ ≈
r eff + 1 + 4(kf )−1.5
√ eff
2
shown in Fig. 2.20, confirm that case (b) has a lower inflection frequency, but case (a) exhibits a larger absolute variation of the effective permittivity. Concerning attenuation, we have (a) tan δ = 0.0035 and (b) tan δ = 0.0009; thus at 1 GHz the attenuations are (a) αd = 0.0076 dB/cm, αc = 0.0166 dB/cm; for case (b) αd = 9.57 10−4 dB/cm, αc =0.0027 dB/cm. Line (b) has lower conductor losses because it is wider. The behaviour of losses vs. frequency is
×
34
Passive elements and circuit layout
Figure 2.20 Frequency
behaviour of effective pernittivity, lines in Example 2.4.
Figure 2.21 Frequency
behaviour of attenuation, lines in Example 2.4.
reported in Fig. 2.21; metal losses prevail at low frequency but dielectric losses become important at high frequency
2.2 Planar transmission lines in microwave integrated circuits
r 9.5 10.0 9.8 2.20 2.20 2.20 2.33 2.33 2.50 2.50 2.55 2.55 2.55 2.55 2.55 2.95 2.95 3.20 3.20 3.00 3.00 3.00 3.50 3.50 3.50 3.50
h, mm 0.63 1.57 1.27 1.57 0.78 0.51 1.57 0.51 1.52 0.76 1.52 0.76 0.51 1.52 0.76 0.38 0.25 1.57 0.78 1.57 0.78 0.51 1.52 0.76 0.51 0.25
t, µm 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35
ρ/ρAu 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118 0.7118
35
tan δ Name .0035 CER-10-0250 .0035 CER-10-0620 .0035 CER-10-0500 .0009 TLY-5-0620 .0009 TLY-5-0310 .0009 TLY-5-0200 .0009 TLY-3-0620 .0009 TLY-3-0200 .0019 TLX-9-0600 .0019 TLX-9-0300 .0019 TLX-8-0600 .0019 TLX-8-0300 .0019 TLX-8-0200 .0006 TLT-8-0600 .0006 TLT-8-0300 .0028 TLE-95-0150 .0028 TLE-95-0100 .0030 TLC-32-0620 .0030 TLC-32-0310 .0030 TLC-30-0620 .0030 TLC-30-0310 .0030 TLC-30-0200 .0025 RF-35-0600 .0025 RF-35-0300 .0025 RF-35-0200 .0025 RF-35-0100
Table 2.2. Characteristics of Taconic commercial substrates, h is the substrate thickness,
t the thickness of the lower ground (and of the upper ground if present), ρ the metal resistivity, tan δ the substrate loss angle.
In conclusion, a 50 Ω microstrip line on a heavy substrate (allumina) with r = 10 requires W = h while on GaAs the same impedance level can be achieved for W/h 0.7. The microstrip ohmic losses decrease for increasing strip width and are large for high-impedance, narrow-strip lines. The impedance ranges achievable in practice on allumina substrates are from 20 to 120 Ω approximately.
≈
2.2.3
The coplanar waveguide Ideal coplanar waveguides (Fig. 2.22) on an infinitely thick substrate are characterized by eff = (r + 1)/2, independent of geometry and line impedance; more-
36
Passive elements and circuit layout
Figure 2.22 Coplanar
waveguide (CPW): simple, with lower ground plane, with finite-extent lateral grounds.
over, the line impedance only depends on the ratio between the slot width and the strip width, or, equivalently, on the ratio a/b, where a = w/2, b = s + a. Notice that 2b is the overall lateral extent of the line. The property eff = (r + 1)/2 is partly lost in practical lines on non-ideal (finite-thickness, Fig. 2.22) substrates; moreover, in this case the line impedance also becomes sensitive to the substrate thickness. As a rule of thumb, the substrate should be at least as thick as the overall lateral extent of the line, i.e. h > 2b to make the influence of h on Z 0 almost negligible. For a given h, this conversely imposes a limitation on the maximum line dimensions: e.g. for h = 300 µm one must have b < 150 µm. Although no limitation occurs to the impedance range, which only depends on the shape ratio a/b, thin substrates require small lines which in turn are affected by heavy ohmic losses. A further cause of non-ideal behaviour is the finite extent wgp = c b of the lateral ground planes. As a consequence, the line impedance increases and eff slightly decreases. In practical circuits, the overall lateral line extent should be kept as small as possible, provided that no spurious coupling arises between neighbouring lines and that the impedance level of the line is
−
2.2 Planar transmission lines in microwave integrated circuits
37
not seriously affected. A reasonable compromise, which can be actually observed in most practical coplanar circuits, is to have c 3b at least. A last variety of coplanar waveguide is the so-called conductor-backed CPW. There seemingly are many reasons to advocate the use of conductor backing in uniplanar circuits made on semiconductor substrates: GaAs substrates have poor mechanical properties; in particular, thin substrates (e.g. h < 100 100 µ µm) m) are very brittle. Thus, coplanar circuits should be made on rather thick substrates, which, however, have poor thermal properties (high thermal resistance seen by active devices). Conductor backing allows thinner substrates to be used, and permits to connect the back of the circuit to a suitable heat sink. Unfortunately, conductor backing also has some disadvantages. Apart from problems connected to spurious coupling with parasitic modes, conductor backing lowers the impedance level of the line and makes it again dependent on h. In other words, conductor backing leads to a mixed coplanar-microstrip structure. If the aim is to obtain a coplanar rather than a microstrip behaviour, the substrate should be approximately as thick as in the case where no backing is present, i.e. one should have h > 2b 2 b. Now, if we come to the actual substrate thickness needed to allow reasonable power dissipation even to medium-power devices, one finds that very low h are h are needed, e.g. h 30 60 µm. If such a low thickness can be locally obtained through substrate thinning and connection to a heat sink, making a whole conductor-backed coplanar circuit on such a thin substrate does not seem a good policy, since the requirement h > 2b 2 b leads to extremely small lines with unacceptable ohmic losses.
≈
≈ −
2.2.3. 2.2.3.1 1
Analy Analysis sis formu formulae lae QuasiQuasi-sta static tic expres expressio sions ns for the line line parame parameter terss have have been deriv derived ed throug through h approximate conformal mapping techniques, and are accurate if the substrate thickness is not small with respect to the line width 2 b. Although the exact asymptoti asymptoticc limit is obtained obtained for h 0, it is advisable to confine the use of these expressions to h > b/2. b/2. In all following formulae, K (k) is the complete elliptic integral of the first kind , of argument k, while k = 1 k 2 . The ratio K (k )/K (k ) can be accurately approximated as follows:
→
√ −
K (k ) K (k ) K (k ) K (k )
≈
1 log π
≈ π1 log
√ k − √ k √ 1 + k √ 2 1 − k 1+ 2 1
≤ k2 < 1
,
0.5
,
0 < k2
≤ 0.5
(2.37) (2.38)
One has for the characteristic impedance ( Z 0 ) and effective permittivity ( (eff ):
38
Passive elements and circuit layout layout
Coplanar waveguide (CPW) on infinitely thick substrate: 30 30π π K (k ) Z 0 = eff K (k ) r + 1 eff = 2 k = a/b
√
(2.39)
(2.40)
(2.41)
Coplanar waveguide with finite-thickness substrate:
30 30π π K (k ) √ eff K (k ) r − 1 K (k ) K (k1 ) eff = 1 + Z 0 =
(2.42)
K (k ) K (k1 )
2
(2.43)
where: k = a/b sinh(πa/ sinh(πa/22h) k1 = sinh(πb/ sinh(πb/22h)
(2.44)
(2.45)
Conductor-backed coplanar waveguide: Z 0 =
60 60π π √ eff K (k )
1
(2.46)
K (k2 ) + K (k ) K (k2 ) K (k ) K (k2 ) + r K (k ) K (k2 ) eff = K (k ) K (k2 ) + K (k ) K (k2 )
(2.47)
where: k = a/b tanh(πa/ tanh(πa/22h) k2 = tanh(πb/ tanh(πb/22h)
(2.48)
(2.49)
Coplanar waveguide with finite-thickness substrate and finite-extent ground planes:
30 30π π K (k3 ) √ eff K (k3 ) r − 1 K (k3 ) K (k4 ) eff = 1 + Z 0 =
2
K (k3 ) K (k4 )
(2.50)
(2.51)
where: a k3 = b
− 1 1
−
(b/c) b/c)2 (a/c) a/c)2
sinh(πa/ sinh(πa/22h) k4 = sinh(πb/ sinh(πb/22h)
− 1 1
−
(sinh(πb/ (sinh(πb/22h)/ sinh(πc/ sinh(πc/22h))2 (sinh(πa/ (sinh(πa/22h)/ sinh(πc/ sinh(πc/22h))2
(2.52)
(2.53)
39
2.2 Planar transmission lines in microwave integrated circuits
Concerning the frequency dispersion of the effective permittivity, an analytical expression of the frequency-dependent behaviour of the effective permittivity for a coplanar waveguide with finite substrate is:
eff (f ) f ) =
where: A = exp
0.54
eff (0) +
√ r −
eff (0) 1 + A(f /f T E )−1.8
log(2 (2a/h 0 .015[log(2a/h 015[log(2a/h)] )]2 a/h)) + 0. − 0.64 log
0.86 log(2 log(2a/h) a/h) + 0. 0 .540[log(2a/h 540[log(2a/h)] )]2 −√ = c /(4h (4h − 1) 0.43
f T E
0
r
(2.54)
log(2a/ log(2a/((b
− a))+
(2.55) (2.56)
The conductor and dielectric attenuation can be expressed in dB per unit length as: 8.68 68R Rs (f ) f ) eff αc = 480πK 480 πK (k )K (k 2 ) 1 8πa(1 πa (1 k ) 1 8πb(1 πb(1 k) π + log + π + log (2.57) a t(1 + k ) b t(1 + k ) tan δ r K (k1 ) K (k ) αd = 27 27..83 (2.58) λ0 2 eff K (k1 ) K (k )
√
×
−
−
√
Fig. 2.23 and Fig. 2.24 show show the behaviour of the characte characteristi risticc impedance and attenuation of a CPW on allumina substrate as a function of the shape ratio 0 < 0 < a/b < 1. < 1. Since increasing a the capacitance increases, the impedance decreases; on the other hand losses are maximum for a for a 0 (large strip resistance due to the narrow strip) and for a b (the impedance vanishes in the limit and therefore α therefore α c 1/Z 0 diverges). The minimum loss occurs around a/b around a/b 0.5, that also corresponds to a 50 Ω impedance on allumina.
∝
2.2.3. 2.2.3.2 2
→
→
≈
Coupli Coupling ng and radiati radiation on losses losses in planar planar lines lines Parasitic coupling on uniform lines can occur either because the quasi-TEM field of the line couples with other quasi-TEM fields (line-to-line coupling) or because coupling occurs with surface waves or free-space radiation. Coupling between quasi-TEM modes and other guided or radiated waves is significant only in the presence of phase velocity synchronism; if this condition occurs, circuit operation is severely deteriorated owing to power conversion to spurious modes or radiation. A different sort of coupling can occur in the presence of line discontinuities. In such cases, higher-order line modes are excited and the related power can easily convert into surface waves or free-space radiation. Such effects modify the circuit behaviour of line discontinuities such as impedance steps, open circuits etc. and can be approximately modelled by means of concentrated radiation conductances. In what follows, we shall try to collect some ideas on spurious coupling mechanisms so as to give design criteria on this fairly complex matter.
40
Passive elements and circuit layout
Characteristic impedance of a CPW on allumina vs. the aspect ratio a/b . We have 2b = 600 µ m and the substrate is infinitely thick. Figure 2.23
Characteristic impedance of a CPW on allumina vs. the aspect ratio a/b . We have 2b = 600 µ m and the substrate is infinitely thick. The metal strips are gold with thickness t = 5 µ m; the dielectric loss angle is tan δ = 0.001; frequency is 10 GHz. Figure 2.24
2.2 Planar transmission lines in microwave integrated circuits
41
Coupling with spurious modes is an important issue in coplanar design. Actually, coplanar waveguides also support a variety of quasi-TEM or non-TEM (slotlike) modes which have to be suppressed as much as possible. Namely:
All coplanar lines support a parasitic slot-like mode which is odd with respect to the central conductor. Such a coupled-slot wave is a true quasi-TEM mode when the lateral ground planes have finite extent and can be excited both by discontinuities and by synchronous coupling. In order to suppress this mode, the lateral ground planes have to be connected together at short intervals (less than λ g /4) by means of airbridges. Conductor-backed coplanar lines with finite-extent lateral ground planes also support a microstrip-like mode in which all strips have the same potential. Although synchronous coupling with this mode is impossible due to the large difference of effective permittivities, the microstrip-like mode can be excited at discontinuities (typically, at short circuits). Although mode conversion at short circuits is not dramatic from a quantitative point of view, suppressing this spurious mode is practically impossible – one should have to connect the upper and lower ground planes through via-holes and wrap-arounds, which is precisely what one wants to avoid doing by the use of coplanar waveguides.
Both the microstrip and the coplanar waveguide show the possibility of spurious couplings with surface waves supported by the (grounded) dielectric substrate. As shown in Fig. 2.25 to every guiding structure a possible parasitic waveguide is associated, with upper or lower ground planes, or both as in the CPW with lower ground plane. The spurious coupling can assume two forms:
synchronous coupling, when the spurious and the original mode travel with the same phase velocity; asynchronous coupling, that may arise only in the presence of discontinuities.
A grounded dielectric slab carries TE or TM waves vs. the propagation direction. In the TE case the electric field is parallel to the ground plane, in the TM case orthogonal. This second field topology is more compatible with the microstrip field topology, while the CPW has both horizontal and vertical field components. There are an infinite set of surface waves TE n and TMn , n = 0, 1, 2...,with cutoff frequencies: c0 n f cTMn = = 2n f cTE (2.59) 2h r 1 c0 (2n + 1) f cTEn = = (2n + 1) f cTE (2.60) 4h r 1
√ − √ −
·
0
·
0
i.e. the fundamental TM0 mode has zero cutoff frequency. The same remark holds for the fundamental mode in a metallized dielectric layers shielded, at a distance H , by a second metal plane (Fig. 2.25). The dispersion relationship of
42
Passive elements and circuit layout
Figure 2.25 Waveguiding
quasi-TEM structures and associated surface wave dielectric
waveguides.
the TM0 mode for this structure can be shown to be: 1 r
−√ √ − −
−
r eff tanh(2c0 f (H h) eff 1) = eff 1 tan(2c0 f h r eff )
−
(2.61)
where c 0 is the velocity of light in vacuo. A dielectric slab metallized both side carries a TEM mode (requiring however a potential difference between the two planes) and TE and TM modes similar to the single-side metallized slab. Fig. 2.26 shows the dispersion curves for several waveguiding structure on a GaAs substrate (r = 13) with h = 300 µm. We report a 50 Ω microstrip effective permittivity, a microstrip permittivity in the limit of large impedance, and of two coplanar waveguides with 50 Ω and different transverse dimensions. For surface waves, we show the dispersion curve of the fundamental mode of a metallized dielectric slab with a cover H = 1.5 mm above the latter; we also show the TM 1 mode of a dielectric slab with lower and upper metal planes. Concerning coupling with microstrip or coplanar lines, the following remarks hold:
Synchronous coupling with the TM0 mode in a microstrip is possible only for very high impedance lines; on the other hand synchronous coupling is possible in coplanar lines, with a synchronous frequency f s given by: f s =
c0 tan−1 (r ) πh
2(r
− 1)
43
2.3 Lumped parameter components
Figure 2.26 Dispersion
curves for quasi-TEM modes and surface waves..
that, for heavy substrates like allumina, simplifies to: f s
106 [mm] r
≈ h
GHz.
Synchronous coupling with the TE0 mode is possible but takes place at a higher frequency f s > f s : f s =
√ − 1
3c0 4h 2(r
− 1) .
Finally, coplanar lines with a lower ground plane ha synchronous coupling with the TM1 mode of the dielectric waveguide with upper and lower ground planes; such a coupling occurs at the frequency: c0 f ss = h (r 1)
−
Spurious surface waves can be controlled by inserting dissipative materials in critical points of the circuit.
2.3
Lumped parameter components Lumped parameter components (resistors, capacitors and also, at RF and beyond, inductors) are exploited in planar circuits to implement several functions: bias and stabilization resistances, DC blocks (capacitors) or RF blocks )inductors), feedback resistances, lumped parameter couplers, power dividers,
44
Passive elements and circuit layout
matching sections. Although lumped components can be monolithically integrated, components with large values often have to be inserted in hybrid form exploiting discrete elements (often called chip capacitors, resistors and inductors since they are realized on a dielectric chip). Integrated resistors and capacitors have a compact layout, while integrated inductors are large and with a poor quality factor. In principle, lumped elements can be obtained from short transmission lines. In fact, the input impedance of a short line closed on Z L can be written, taking into account that tanh x x for small argument, in the form:
≈
Z in = Z 0
Z L + Z 0 γ l . Z 0 + Z L γ l
Taking into account the expression of the characteristic impedance and complex propagation constant: Z 0 =
R G
+ jω , + jω
L C
γ =
we have: Z in =
(
+ jω )( + jω )
L G
C
Z L + ( + jω )l ; 1 + Z L ( + jω )l
R G
therefore, for a shorted line: Z in =
R
L C
Rl + jωLl = R + jωL
which synthesizes, for a low-loss line, and inductor. For a short line in open we have: Y in = l + jω l = G + jωC
G
C
typically corresponding to a capacitor. The value obtained this way are however small. A summary of the parameters of several classes of lumped elements is shown in Table 2.3. For a reactive element the quality factor is defined as the ratio between the resistive and reactive component as follows: R ωC = ωL G Taking into account that RF inductors are in air (no magnetic losses) and that the high frequency resistance is proportional to f we have that the inductor Q decreases at high frequency like f −1/2 . On the other hand, since for a capacitor the main loss mechanism are dielectric losses for which G = ωC tan δ we have that for a capacitor Q 1/δ . High Q resonators cannon typically obtained with lumped elements but require external components (quartz resonators, surface wave acoustic components). Q =
√
≈
45
2.3 Lumped parameter components
Figure 2.27 (a)
strip inductor, (b) loop inductor (horsehoe) (c) spiral inductor and related equivalent circuits.
2.3.1
Inductors Microwave inductors are in air, due to the fact that alla magnetic materials (also ferroelectric like ferrites) are restricted to operation below 1 GHz. Microwave integrated inductors are limited by ohmic losses, by an upper operating frequency connected to the resonance from the parasitic capacitance, and by a range of values limited to 10-50 nH, due to the large size. Large inductors have a large parasitic capacitance and therefore a limited operating frequency range.
2.3.1.1
Strip and loop inductors Small values of inductance (up to 2 nH approx.) can be obtained through high impedance lines (Fig. 2.27 (a)) or through loop inductors (Fig. 2.27 (b)). For the strip inductor we have the following model (lengths are in mm):
−
2πl Lstrip = 2 · 10−1 l log
W W Rstrip = 1.4 + 0.217 log 5t
W 1+ nH 2πl Rs l Ω 2(W + t)
(2.62) (2.63)
46
Passive elements and circuit layout
Type
Valuse
Q (10 GHz)
Material
0.01-0.5 nH 0.5-10 nH
30-60 20-40
Gold
INDUCTORS High Z 0 lines Spiral inductors CAPACITORS Microstrip gap Interdigitated MIM (parallel plate)
0.001-0.005 pF 0.01-0.5 pF 0.1-100 pF
RESISTORS This film Monolithic
5 Ω - 1 kΩ 5 Ω - 1 kΩ
≈ 50 ≈ 50 ≈ 25
Si3 N4 ,SiO2 , Polymide Ta2 O5 NiCr, TaN GaAs implanted
Table 2.3. Parameters of lumped elements.
where R s is the surface resistance of the metallization, t the metal thickness. For the loop inductor we have the so-called Grover formula (dimensions are in mm):
− 8πa W
Lloop = 1.257a log
Rloop = 1.4 + 0.217 log
2
W 5t
nH
Rs πa Ω. W + t
(2.64)
(2.65)
The behaviour of the strip and loop inductance (with l = 2πa) is shown in Fig. 2.28 for different values of W . Notice that, for the same length, the loop inductance is lower than the strip inductance.
Example 2.5:
• Evaluate the inductance of a strip inductor of length l = 0.5 mm, W = 50
µm and t = 5 µm, on a 300 µm GaAs substrate (r = 13). Compare the input impedance with the input impedance of a microstrip line having the same dimensions as a function of frequency, neglecting the effect of losses; find the resonant frequency of the inductor. (all lengths in mm): ◦ For the strip inductor we use the model L = 2 10−1 l log
·
2πl W
− 1 + W πl
= 0.32 nH;
the input impedance is Z i = jωL. We can model the same inductor with a microstrip model, using the highfrequency expression of the characteristic parameters. Since h/W = 300/50 = 6 < 2π, we exploit the width correction: 2h W W 1.25t = + 1 + log = 0.205 h h πh t
47
2.3 Lumped parameter components
Figure 2.28 Inductance
of a strip and loop inductor vs. the total length, for different values of the strip width W .
i.e. W = 0.205h = 0.0615 mm; then we exploit the narrow strip formula for the effective permittivity:
12h F = 1 + W
−1/2
−
+ 0.04 1
W h
2
= 0.144
i.e.: 1 + r r 1 r 1 t eff = + F 2 2 4.6 h
− − −
h = 7.76. W
The impedance and impedance in air are: 60 8h W Z 0 = log + 0.25 = 78.9 Ω eff W 4h 8h W Z 00 = 60log + 0.25 = 220 Ω W 4h
√
yielding the total inductance lZ 00 /c0 = 0.36 nH, in fair agreement with the value obtained through the concentrated model. The propagation constant is: β =
2πf eff = j5.83 3 108
×
√
× 10−8f
leading to an input impedance of the shorted line: Z i = jZ 0 tan(βl).
48
Passive elements and circuit layout
Figure 2.29 Behaviour
vs. frequency of an inductor reactance according to the lumped and microstrip model (Example 2.5).
The magnitude of the reactance evaluated from the lumped and the microstrip model is shown in Fig. 2.29; the microstrip model yields a resonance around 55 GHz. The limit l < λ g /20 would confine the frequency range of the inductor to frequencies belwo 11 GHz.
2.3.1.2
Spiral inductors Larger inductances can be obtained through spiral inductors, see Fig. 2.27 (c), where the spiral shape can be square, circular, octagonal. Integrated spiral inductors require air bridges, see Fig. 2.30. Approximate formulae exist for the circular spiral inductor inductance, parasitic resistance and parasitic capacitance: a2 n2 Do + Di K g nH, a = , 8a + 11c 4 −1.7 πanR S s Rsp = 1 + 0.333 1 + W W Lsp = 39.37
C 3 = 3.5 10−2 Do + 0.06 pF
·
c =
Do
− Di 2
(2.66) (2.67) (2.68)
where Do and Di are in mm. The parameters are defined in Fig. 2.27, (c); n is the number of turns. We approximately have: Do
− Di ≈ nW + (n − 1)S 2
2.3 Lumped parameter components
49
Figure 2.30 Rectangular
spiral inductor with air bridges.
from which: n
≈
1 Do Di + S . W + S 2
−
A factor K g takes into account the effect of the lower ground plane; thus we have: L = K g L0 where L0 = L sp is the inductance without the ground plane. An approximation holding for W/h > 0.05 where W is the strip width is: K g
≈ 0.57 − 0.145log
W h
(2.69)
where h is the substrate thickness. For W/h < 0.05 the effect of the ground plane and negligible and K g = 1. For the rectangular spiral inductor we have the approximate model: Lrsp
≈ 0.85
√
An5/3 K g nH
·
(2.70)
where A is the inductor total area, n the number of turns.
Example 2.6:
• We want to implement a family of spiral inductor withe external diameter
Do = 1 mm, W = 50 µm, S = W, varying the number of turns n, with t = 5 µm. Evaluate the inductance that can be obtained on a 300 µm substrate varying the number of turns, with a constant external diameter Do , the quality factor, and the resonant frequency. Metal conductors are made of gold.
50
Passive elements and circuit layout
◦ We have, with constant D : o
Do
− Di ≈ nW + (n − 1)S = (2n − 1) W
2 i.e. for the internal diameter:
Di = D o
− 2 (2n − 1) W.
The maximum turn number corresponds to D i = 0, i.e.: 1 Do + = 5.5 2 4W
n =
≈5
We then have: 1 Do + Di = [Do (2n 1) W ] = 0.525 0.05n mm 4 2 Do Di c = = (2n 1) W = 0.1n 0.05 mm. 2 The substrate correction factor is: a =
−
−
K g = 0.57
−
−
−
−
−
W 0.145 log h
= 0.57
− 0.145 log
50 300
= 0.83
while the gold surface resistance is: Rs =
2πf µ = 0.098 f GHz Ω 2σ
we then obtain for the inductor parameters, with W = 0.05 mm: a2 n2 (0.525 0.05n)2 n2 nH L = 39.37 K g = 32.7 8a + 11c 3.65 + 0.7n R = 6. 46 10−2 (21 2.0n) n f GHz Ω
−
×
C 3 = 0.095 pF
−
For the quality factor: QL =
2πf L = 79.5 R
− n) n × (10.5 3.65 + 0.7n
while the inductor resonant frequency is:
√ −
1 25.5 73.0 + 14n f 0 = = (21 2n) n 2π LC 3
√
f GHz
GHz.
The inductance, quality factor and resonance frequency that can be obtained for n = 1...5 are reported in Table 2.4. Notice that the resonant frequencies are optimistic since the feedback capacitance is evaluated only in an approximate way.
2.3 Lumped parameter components
n L (nH) f 0 (GHz) QL @ 1 GHz
1 1.70 12.55 173.6
2 4.68 7.55 267.6
3 7.20 6.09 311.1
4 8.57 5.58 320.4
51
5 8.65 5.56 305.8
Table 2.4. Inductance, quality factor and resonance frequency from Example 2.6.
2.3.1.3
Inductance of bonding wires Bonding wires are exploited in hybrid circuits within the circuit and to connect integrated circuits to external ports. Instead of circular wires, ribbons can be exploited with the advantage of a lower inductance. The resistance and inductance of a wire of diameter d and length l (in mm) are:
l d
L0,wire = 0.20 log R =
+ 0.386 l nH
Rs l . πd
(2.71)
(2.72)
A ground plane correction is:
√
Lfilo = 0.2l log
4h + log d
+ l +
l2 + d2 /4
l +
l2 + 4h
l +
4h2 l2
−
l +
d2
4l2
+
− 2 hl + 2ld
nH
(2.73)
where h is the distance between the wire and the ground plane. such an inductance is not negligible; e.g. a 100 µm wire has an inductance of 500 pH/mm, while the inductance grows to around 800 pH/mm for a diameter of 25 µm.
2.3.2
Capacitors Capacitors can be realized through passive structures or through junctions (Schottky or pn). Passive capacitors can be microstrip patches, interdigitated capacitors, MIM capacitors. Microstrip patches or gaps have a low capacitance, while interdigitated capacitors have a capacitance of the order of 0.5 pF/mm 2 . Larger values can be obtained (up to around 30 pF) with MIM (Metal Insulator Metal) capacitors, see Fig. 2.31 (b), for which the parallel plate formula holds: Wl d where W l is tha capacitor area, the absolute dielectric constant of the dielectric, d the dielectric thickness. The parallel conductance is related to dielectric losses. For example a MIM capacitor with a 1 µm silicon oxide layer (relative permittivity 4) has a specific capacitance around 35 pF/mm 2 . C =
52
Passive elements and circuit layout
Figure 2.31 MIM
Dielectric
SiO SiO2 Si3 N4 Ta2 O5 Al2 O3 Polymide
and interdigitated capacitors.
C a , d = .2µm 275-325 175-230 300-400 1000-1200 350-400 30-40
Q + ++ ++ = = –
θ 100-500 50-100 20-40 10-150 400-600 400-500
F cv – = + = + +
F cq = = + + + =
Deposition technology
Evaporation Evaporation, sputtering Sputtering, CVD Sputtering, anodization Sputtering, CVD Spinning
Table 2.5. Dielectrics for MIM capacitors. θ is the temperature coefficient in ppm/ o C. The
polymide figure refers to a thickness of 1 µ m.
Two common figure of merits for dielectrics exploited in realizing capacitors are the product between the capacitance and the breakdown voltage: F cv = C a V b = 0 r E b F V/m2
(2.74)
·
typically in the range (8 itance - capacitor Q:
− 30) × 103 pF-V/mm2, and the product specific capacF cq = C a / tan δ d F/m2
(2.75)
where C a is the capacitance per unit surface, V b the breakdown voltage, E b the breakdown electric field, δ d the loss angle. Typical values are E b = 1 2 MV/cm , r = 4 20 per la costante dielettrica, tan δ d = 10−1 10−3 . A summary of some relevant dielectrics is reported in Table 2.5. The tolerance of MIM capacitors is limited by the ability to control the dielectric thickness accurately.
−
−
−
2.3 Lumped parameter components
53
Figure 2.32 Integrated microwave resistors.
Material
Cr Ta Ti TaN NiCr GaAs
Rs , Ω/ 10-20 30-200 10-100 250-300 40-100 100-1200
θ 3000 100-500 2500 150-300 200 3000
Accuracy
Stability
Deposition
= = = ++ ++ –
= ++ = = ++ ++
Evaporation, sputtering Sputtering Evaporation, sputtering Sputtering Evaporation, sputtering Implantation, epitaxy
Table 2.6. Resistive materials for resistors. θ is the temperature coefficient in ppm/ o C.
2.3.3
Resistors Planar integrated resistors can be obtained either deposing a thin film on a dielectric substrate (thin film resistors) or through semiconductor resistive films (mesa resistors) or doping a semi-insulating substrate (implanted resistors), see Fig. 2.32; a summary of the relevant materials is shown in Table 2.6.
54
Passive elements and circuit layout
Figure 2.33 Examples
of discrete RF lumped components: thin film chip resistor for surface mount; chip capacitor; chip inductor; ultrabroadband conical inductor.
The input impedance can be evaluate through a short (shorted) RC line model:
R CR ≈ −
≈
1 R = 1 C jω 1 + 13 jωC R + jω R 3 where R = l, C = l is the parasitic capacitance. The series resistance can be evaluated in the DC or skin effect range according to the operating frequency, the resistor thickness and material. Z in =
C
R
2.3.4
tanh
jω
l
R 1
1 jωC R 3
C
Chip inductors, resistors and capacitors Discrete lumped components can be externally inserted in hybrid integrated circuits (i.e. circuits where the substrate is dielectric, and the active semiconductor devices are not monolithically integrated), usually as surface-mount chip resistors, inductors or capacitors. Chip resistors are obtained by deposing a resistive thin film over a dielectric (e.g. ceramic) chip. Wrap-around or flip-chip contacts are then added, allowing for surface mounting (SM) on a microstrip or coplanar circuit. An example of such structures (shown bottom up) can be found in Fig. 2.33; the side size of the component often is well below 1 mm. The resistance of chip resistors typically ranges from a few Ω to 10 kΩ. Chip capacitors can be obtained by deposit-
2.4 Layout of planar hybrid and integrated circuits
55
ing a dielectric layer (e.g. SiO 2 ) on a conductor or semiconductor (e.g. Si); the dielectric layer is then coated with metal so as to define the external contacts, which can be surface mounted through flip-chip (i.e. by connecting the component upside down), see Fig. 2.33. The capacitance of chip capacitors typically ranges from a few pF to 1 µF. While chip resistors can be properly manufactured so as to achieve spectacular bandwidths (e.g. from DC to millimeter waves), thus making it possible to provide ultrabroadband matched terminations, broadband inductors are difficult to obtain, due to the increase of losses with frequency and to the upper limitation related to the LC resonant frequency. The quality factor of RF and microwave inductors typically peaks in a very narrow band, with maximum values well below 102 . An example of RF and microwave chip inductor is shown in Fig. 2.33; achievable inductance values typically decrease with increasing operating frequency and are limited to 500 nF approximately, with maximum operating frequencies below 10 GHz. However, ultrabroadband bias Ts typically for instrumentation require broadband inductors as RF blocks. Conical inductors (Fig. 2.33) are a particular technology allowing for very broadband behaviour, due to a strong reduction of the parasitic capacitance and to the scaling invariance of the design.
2.4
Layout of planar hybrid and integrated circuits High-speed electronic integrated circuits (ICs) can be implemented through two complementary approaches, the hybrid IC and the monolithic IC. Integrated circuits operating in the microwave range (i.e. up to 30-40 GHz or 40 Gbps) are often denoted as (Monolithic) Microwave Integrated Circuits, (M)MICs. In the hybrid approach, the circuit is realized on a dielectric substrate, integrating all distributed components and, possibly, some lumped components (which may, however, also be inserted as discrete lumped elements through wire bonding or surface mount techniques). In the hybrid approach, the semiconductor active elements are inserted as lumped components and connected again through wirebonding or surface mount. On the other hand, monolithic circuits integrate, on a semiconductor substrate, all active and passive elements. While hybrid circuits often exploit, at least for narrowband applications, distributed components based on transmission line approaches, in monolithic circuits the lumped approach is preferred, owing to the possibility of reducing the circuit size (lumped components are much smaller than the guided wavelength, while distributed elements have, as already recalled, characteristic sizes of the order of λ g /4 at centerband). Monolithic integrated circuits can be based on GaAs, InP or Si substrates and may exploit, as active elements, FETs or bipolars (typically HBTs). According to the transmission medium used, we may have microstrip or coplanar integrated circuits. Microstrip circuits are more compact in size due to the lower ground plane, but require a precise control of the dielectric thickness, while
56
Passive elements and circuit layout
Figure 2.34 Coplanar
probes for on-wafer measurement of a three-port (mixer) with coplanar finite-ground layout.
coplanar circuits can be preferred at very high frequency (mm waves). Microstrip circuits in fact only allow for straightforward connection of series elements (parallel elements require to reach the lower ground plane, often by etching a hole in the substrate, the so-called via hole ) while coplanar circuit allow for the connection of both series and parallel elements. Finally, the on-wafer high-frequency characterization requires to connect the integrated circuit to the measurement setup through coplanar probes ; to this purpose, coplanar ground planes must be made available (e.g. through via holes) at the circuit input and output. An example of coplanar probing of a three-port component, a passive mixer implemented in finite-ground coplanar technology, is shown in Fig. 2.34; the probes are coaxial cables with a coplanar tip transferring by pressure contact the ground planes (lateral) on the lateral ground plane coplanar pads of the circuit, while the center conductor is the signal conductor. The finite-ground coplanar waveguides allow for reducing the lateral size of the lines leading to the device input to the active element (a planar diode). A qualitative example of a hybrid or monolithic integrated circuit in the microstrip or coplanar technology can be introduced as a simple, single-stage open-loop amplifier with two lumped bias Ts and input and output matching section. Fig. 2.35 shows a simplified schematic of the single-stage amplifier, with an input matching section, two bias circuits connected to the active element, and an output matching section. The purpose of the matching sections is to transform
57
2.4 Layout of planar hybrid and integrated circuits
Figure 2.35 Open-loop
single-stage amplifier schematic.
the load impedance (typically 50 Ohm) in the optimum impedance that must be seen at the amplifier input and output port according to a maximum gain, maximum power or minimum noise criterion. The bias circuits, also called bias Ts, are a combination of a DC block (the capacitor) and an RF block (the inductor) whose aim is to separate the paths of the RF and DC currents in such a way that the RF circuit is not loaded by the DC supply and the RF load is isolated from the DC bias. Active devices require in principle two bias sources, although with proper bias schemes these can be reduced to one. Fig. 2.36 presents two possible circuit implementations, with distributed matching sections (a), typically (but not necessarily) hybrid, or with lumped matching sections (b), usually monolithic. Fig. 2.36 (a) also shows the equivalent circuit of two microstrip to coaxial connectors, modeled through a low-pass filter. The generator and load are external and connected though coaxial connector and coaxial to microstrip transitions.
2.4.1
Some layout-connected issues Before discussing details on the hybrid and monolithic layout, let us discuss some layout-connected problems.
2.4.1.1
Connecting series and parallel elements Series and parallel connections can be made, see Fig. 2.37. In microstrip circuits the series connection is easy while the parallel one requires via holes (particularly critical in integrated circuits) or wrap-arounds , see Fig. 2.38; the use of bonding wires is inconvenient due to the parasitic inductance. In coplanar lines (Fig. 2.39) both the series and parallel insertion is possible; symmetric parallel elements are preferred at high frequency.
2.4.1.2
The stub Stubs are short lines exploited to synthesize reactive elements; open and shorted stubs are easy in coplanar circuits, in microstrip circuits the stub in short is a problem due to the need to connect to the ground plane, see Fig. 2.40. Open
58
Passive elements and circuit layout
Figure 2.36 Simplified
circuit of oper-loop amplifier with distributed (a) or lumped (b)
matching sections.
Figure 2.37 Series
and parallel insertion of concentrated elements.
circuit stubs can be trimmed through the use of small metal patches that can be connected at the end of the stub. 2.4.1.3
Active element mounting Active elements can be mounted in chip or in package . In the first case bonding wires or ribbons are needed, in the second case high-frequency packages are often of the flatpack kind, see Fig. 2.41; for symmetry the package has two source or emitter contacts besides the input (gate or base) and the output (drain or collector) terminals.
2.4.1.4
Planar line discontinuities In the design of microstrip and coplanar circuits layout features are introduced that introduce additional parasitics with respect to the ideal optimized schematic made of transmission lines and ideal lumped parameter circuits. For instance, a
2.4 Layout of planar hybrid and integrated circuits
Figure 2.38 Series
and parallel insertion of microstrip elements.
Figure 2.39 Series
and parallel insertion of coplanar elements.
Figure 2.40 Example
of microstrip layout with stubs and discontinuities.
Figure 2.41 Flatpack
package active element.
59
60
Passive elements and circuit layout
microstrip stub connected to a line introduces in the layout a T-junction that is the source of capacitive and inductive parasitics; the same happens when a microstrip bend causes an additional parasitic capacitance to be introduced in correspondence to the bend (minimizing the capacitance is possible by cham fering the bend). As a further example, an open circuit stub has an additional fringing capacitance towards ground that be compensated for by adjusting the length of the stub. Finally, the microstrip gap cannot be strictly speaking considered a discontinuity since it is exploited to implement a small series capacitance; microstrip gaps are e.g. common in the implementation of microstrip filters. Such layout-induced parasitics are collectively called microstrip (or coplanar) discontinuities. Examples of microstrip discontinuities are shown in Fig. 2.40 while a set of discontinuities and related models is shown in Fig. 2.42. In general, the circuit optimization is carried out by working with ideal elements, the layout is extracted and then, from the layout, an augmented netlist including discontinuity models is generated than can be exploited in order to verify the design and further trim the circuit.
2.4.2
Hybrid layout Fig. 2.43 shows a simplified hybrid microstrip implementation of the single-stage amplifier, in which the input and output matching sections have been separately realized on two different ceramic substrates. The active device is introduced in packaged form and exploits as the ground plane (and also as the heat sink) a ridge in the metal package. Bias Ts are implemented using as series inductors the parasitic wire bonding inductance; chip capacitors connected to the package as the ground are also part of the bias T. The microstrip lines are connected to the exterior of the circuit through coaxial connectors, see Fig. 2.44. In the coplanar layout ground planes are located on the same (upper) airdielectric interface as the signal conductors. The area occupation of the ground planes can be often reduced by using finite-width grounds. However, since the ground planes are connected to a potential reference only on the periphery of the circuit, in each section of the coplanar line the left and right ground planes may actually have (locally) a different potential. Such a potential difference may imply the excitation of a parasitic slot mode that may be suppressed by using bonding wire in the cross section with a spacing small with respect to the wavelength (e.g. λg /8), as shown in Fig. 2.45. Coplanar hybrid circuits are however uncommon, the coplanar solution being usually implemented in MMICs. In Fig. 2.44 we also show a coplanar-coaxial connector.
2.4.3
Integrated layout In the monolithic layout no external elements can be integrated within the circuit (although lumped element may be connected externally); as a qualitative example, the monolithic implementation of the single-stage amplifier already described
2.4 Layout of planar hybrid and integrated circuits
Figure 2.42 Microstrip discontinuities and equivalent circuits.
Figure 2.43 Example
of microstrip hybrid layout.
61
62
Passive elements and circuit layout
Figure 2.44 Coax-microstrip
and coax-coplanar connector.
Figure 2.45 Example
of coplanar hybrid layout.
in shown (microstrip form) in Fig. 2.46; lumped input and output matching sections are exploited. Via holes are used quite liberally to provide local grounding, besides the ground pads needed for the input and output coplanar connectors; the circuit is shown as unpackaged. Finally, Fig. 2.47 is a coplanar waveguide monolithic implementation exploiting distributed matching sections. Due to the typically small size of MMICs, such a solution is realistic only if the frequency is high enough to make distributed elements compact, e.g. for millimeter wave operation. While coplanar waveguides easily allow for open- and short-circuit line stubs (i.e. short pieces of transmission lines for the implementation of the distributed matching sections), the layout is globally less compact, and ground planes have to be connected together by airbridges (rather than bonding wires, as in the hybrid implementation) to suppress spurious modes where the two ground planes are at different potential. Both in the microstrip and in the coplanar layout a source air bridge is used in the active component. The active device layout has been kept the same in the microstrip and coplanar version, although the difference in operation frequency (microwave vs. millimeter wave) also has an impact on the FET layout (e.g. on the length of the gate fingers, which is decreasing with increasing frequency).
2.4 Layout of planar hybrid and integrated circuits
63
Figure 2.46 Qualitative
example of microstrip integrated implementation of single-stage amplifier with DC bias.
Figure 2.47 Qualitative
example of coplanar integrated implementation of single-stage amplifier with DC bias.
An example of vintage coplanar MMIC (a front-end single-stage low-noise amplifier for TV satellite using a MESFET as the active element) developed by CISE (Milan) in 1982 is shown in Fig. 2.48; this is the first example of coplanar MMIC designed in Italy. An overview of active and passive elements for a microstrip MMIC is shown in Fig. 2.49; the circuit shown does not implement any useful function but is meant to be just a catalog of components. The ion-implanted MESFET active element has been replaced during the last 15 years by the HEMT as a FET(Field Effect Transistor) of choice. The computer aided design of MMIC is today a well developed technology, although analog circuit CAD has a degree of automatic design that is far less developed than the digital circuit CAD. Microwave CAD tools make use of a database including libraries of element models (passive and active), a circuit
64
Passive elements and circuit layout
Figure 2.48 Example
of early coplanar MMIC: a TV satellite front-end amplifier in coplanar MESFET technology, CISE, Milan, 1982.
Figure 2.49 Layout
elements of a monolithic integrated circuit sectioned in correspondence of a via hole.
simulator (small signal, large signal steady state, large signal time-domain, often noise), optimization tools, often a layout generator. The circuit is described by a low-level ASCII format (like a netlist) specifying in the minimal case the network connectivity and the element characteristics (value, library, associated layout files etc.). The designer assembles, typically with a graphical interface, a circuit interconnecting elements and making use of element libraries. Optimization with respect to some prescribed design goal is then made, using however in the first phase of design ideal element with a minimal parasitic set. In fact, the use of full models with complex topologies makes optimization critical, not only because
2.5 Microwave circuit packaging
Figure 2.50 Multistage
MMIC amplifier: schematic with ideal elements.
Figure 2.51 Multistage
MMIC amplifier: layout.
65
the element count is too large, but also because the element parasitics are correlated with each other. An example of low-noise three-stage MMIC schematic in shown in Fig. 2.50; the first stage with inductive source feedback is a typical low-noise solution; the second and third stage are resistive feedback amplifiers needed to achieved a reasonable overall gain. From the netlist the layout can be obtained, see Fig. 2.51; from the layout many CAD tools can derive an augmented schematic with parasitic elements that may be used in order to check the circuit performances and perform a further (limited) optimization and tuning, see Fig. 2.52.
2.5
Microwave circuit packaging Both hybrid and monolithic integrated circuits are typically packaged (in a metal or dielectric enclosure) and connected to other subsystems through electrical connectors. The circuit package is an important part of the microwave circuit, also
66
Passive elements and circuit layout
Figure 2.52 Multistage
MMIC amplifier: extended schematic with real elements.
in terms of cost; it should protect the circuit mechanically, offer electromagnetic (EM) shielding, protect the circuit from chemicals and allow for heat dissipation. Packages can be hermetic (sealed, sometimes filled with inert gases to avoid oxigen contamination through leakage), or open. The two most common approaches are the metal package (often alluminium or brass) and the dielectric pacakge. The metal package is a high-Q solution that often exhibits internal resonances that may be suppressed by locating dissipative media (e.g. layers of carbon loaded foam) in critical positions. Dielectri pacakges are low- Q and therefore resonances are less dangerous; often they have a MIM multilayered structure to improve EM shielding. Metal packages can be tailored (i.e. fabricated in the lab) while ceramic packages typically are standard, off-the-shelf products. MMICs can also be mounted package-free within a hybrid circuit, see Fig. 2.53; the package shown is metal, while a ceramic package with a kind of flatpack compensated connector is shown in Fig. 2.54. Package connectors make use of microstrip or flatpack transitions that are compatible with interconnecting planar lines; an example of compensated microstrip transition for a ceramic package is shown in Fig. 2.55. Metallic packages and system-level modules typically exploit coaxial transitions, see Fig. 2.56 and Fig. 2.57). A few coaxial to microstrip transitions ordered by increasing frequency operating range are shown in Fig. 2.56. Soldered connectors can be used up to a few GHz, while at higher frequency wire bonding or ribbon bonding (having lower parasitic p.u.l. inductance) are exploited. High-frequency connectors operating, e.g. at 40 GHz and beyond are not based on wire or ribbon bonding, but rather on contact connectors. 3 3
High-frequency coaxial connectors are denotes by conventional names, some of them referring to the frequency band they were initially meant to cover. Thus we have the K connectors (up to 40 GHz), the V connectors (up to 60 GHz) and the so-called W1 connectors (Anritsu name, 1 mm radius) up to 110 GHz, which currently is the highest frequency exploited in standard instrumentation. The connector size decreases with increasing frequency.
67
2.5 Microwave circuit packaging
Figure 2.53 Hybrid
MMIC mounting in a metal package with coaxial connectors.
Figure 2.54 Ceramic
package with flatpack connectors
Thermal and thermo-mechanical problems are a last area that heavily influences the packaging issue. The thermal performance of a package can be defined by its thermal resistance , defined as the ratio between the temperature rise of the circuit and the dissipated power. Complex cooling techniques are often needed
68
Passive elements and circuit layout
Figure 2.55 A
flatpack compensated microstrip transition for ceramic packages.
Figure 2.56 Coaxial-microstrip transitions.
in power modules to keep the circuit temperature to an acceptable level. Heating during circuit operation also may cause mechanical problems due to the different expansion coefficients of the materials involved in the package, the circuit and the soldering materials. Synthetic materials made by dispersion of metal (e.g. copper or tungsten) powders in an epoxy matrix can be manufactured with the aim to equalize the expansion coefficient of semiconductors like GaAs. 4
2.6
Questions and problems 1. Q Explain the difference between a TEM and a quasi-TEM transmission line. 4
The expansion coefficients of allumina and GaAs are of the order of 6-7 10− 6/Co at ambient temperature, the copper expansion coefficient is about 3 times larger, the one of tungsten lower (5 10− 6/Co ). Copper-tungsten alloys can be obtained that are able to have the same expansion coefficient as the substrates.
×
×
2.6 Questions and problems
Figure 2.57 Transitions
69
between 3/7 coax or SMA connector and microstrip on
allumina substrate.
2. Q A lossy transmission line has per-unit-length parameters , , , . Express the characteristic impedance and complex propagation constant of the line in terms of the parameters for the general case and in the high-frequency approximation. Identify, in the high-frequency approximation, the propagation constant and the attenuation. 3. P A lossless quasi-TEM line has a 50 Ω impedance and an effective permittivity eff = 2. Evaluate the per-unit-length parameters , . Compute the guided wavelength at 10 GHz. 4. P A lossy quasi-TEM line has a 50 Ω impedance.The dielectric attenuation is 0.1 dB/cm while the conductor attenuation is 1 dB/cm at 1 GHz. Evaluate the per-unit-length parameters , . Estimate their values and the resulting dielectric and conductor attenuation at 10 GHz. Assuming an effective permittivity eff = 7, evaluate the total loss over 1 guided wavelength at 10 GHz. 5. P The conductivity of a 2 µm thick conductor is σ = 1 105 S/m. Evaluate the frequency at which the skin-effect penetration depth is equal to the conductor thickness. 6. P A lossless transmission line with 50 Ω characteristic impedance and 5 mm guided wavelength is closed on Z L = 50 + j50 Ω. Compute the input impedance for a 2.5 and 1.25 mm long line. 7. Q A lossless line is infinitely long. Is the input impedance always equal to the characteristic impedance? Explain. 8. Q A quasi-TEM line has a per-unit-length capacitance of 5 pF/mm and an in-vacuo capacitance of 2 pF/mm. What is the effective permittivity? 9. Q Sketch the cross section of a microstrip and of a coplanar waveguide. 10. Q A microstrip on 0.5 mm thick allumina substrate has a strip width of 0.5 mm. What is (approximately) the characteristic impedance? 11. Q Sketch the attenuation of a microstrip and of a coplanar waveguide as a function of the strip width.
LCRG
L C
R G
×
70
Passive elements and circuit layout layout
12. Q Sketch the behaviour of the attenuation of a transmission line as a function of frequency. 13. Q Sketch a strip, a loop and a spiral inductor. What usually limits the frequency range on which integrated RF and microwave inductors can operate? 14. Q List some possible uses of inductors in integrated RF circuits. 15. Q Sketch an interdigitated and a MIM capacitor. 16. P In P In a MIM capacitor the dielectric is 100 nm thick, width permittivity equal to 2. What is the capacitance per mm 2 area? 17. Q What are chip inductors, capacitors and resistors? are they used in hybrid or integrated implementations? 18. Q What is a coaxial-to-microstrip transition? 19. Q What are the main differences between a coplanar and a microstrip circuit layout?
3
Power gain and stabilit stabilityy of a loaded loaded two-port
3.1 3.1
Rep Represe resent ntat atio ions ns of linea linearr two-po o-port rtss Consider an electronic subsystem (including or not active elements such as transistors) interacting with the rest of the circuit with two electrical ports whose instantaneous electrical state is given by the current entering the port ( ik (t)) and by the voltage between the pole were the current enters and the second pole of the port (v ( vk (t)), see Fig. 3.1. Such an element is denoted as a two-port; if the two-port includes only linear elements we define it as a linear two port. The two-port may or may not include indepenent voltage or current sources; in the first case we call it autonomous , in the second case non autonomous . A non-autonomous two-port can include dependent sources, in that case the twoport has zero open-circuit voltages or short-circuit currents, but is able anyway to provide voltage or current amplification or power gain. A two-port able to provide gain to the input signal is often called active ; it typically includes active elements such as transistors. On the other hand, a passive two-port is made of passive elements, lumped (as resistors, capacitors, inductors, transformers etc.) or distributed (as simple or coupled transmission lines). Active Active device device (e.g. (e.g. transistor transistors) s) operating operating in small-signa small-signall conditions conditions can be modelled as linear two-ports. In fact, in this case the transistor characteristics have been linearized around a DC bias point and the signal generators have an amplitude compatible with the linear approximation of the transistor characteristics. An active device small-signal model is the basis for the design of linear amplifiers, including maximum gain and low-noise amplifiers; throughout this chapter, we will consider linear, active two-ports that can be interpreted as transistors operating in small-signal conditions. In a linear linear circui circuitt superpos superpositi ition on applie appliess and theref therefore ore the analys analysis is can be carried carried out considerin consideringg a steady-sta steady-state, te, single single frequency frequency sinusoida sinusoidall excitation excitation.. Such Such a frequencyfrequency-domain domain analysis analysis is also consistent consistent with the fact that many RF electronic systems actually are narrowband. Given a linear two-port with current phasors I 1 and I 2 and voltage phasors V 1 and V 2 the constitutive equation set of the two port is made by two equations expressing a linear relationship between two independent variables and two dependent variables. In a set of four variables we have six possible way to select a pair of dependent and a pair of independent variables, thus obtaining six differ-
71
72
Power Power gain and stability stability of a loaded two-port two-port
Figure 3.1 Two-port
and its relevant electrical parameters.
ent ways to write the constitutive relationship set or in other words six possible representations of the two port, namely:
The current-driven current-driven or series representation where I 1 and I 2 are the independent variables and V and V 1 and V and V 2 the dependent variables, the model reads: V 1 = Z 11 11 I 1 + Z 12 12 I 2 + V 01 01 V 2 = Z 21 21 I 1 + Z 22 22 I 2 + V 02 02
where Z ij ij are the elements of the impedance matrix (Ω) and V 0i are the open-circuit voltages at port i, i , zero for a non-autonomous two-port; The voltage-driven or parallel representation representation where V 1 and V 2 are the independent variables and I and I 1 and I and I 2 the dependent variables; the model reads: I 1 = Y 11 11 V 1 + Y 12 12 V 2 + I 01 01 I 2 = Y 21 21 V 1 + Y 22 22 V 2 + I 02 02
where Y ij ij are the elements of the admittance matrix and I 0i are the shortcircuit currents at port i, zero for a non-autonomous two-port; the model is particularly suited to represent field-effect transistors in small-signal commonsource operation since it describes through the transimpedance Y 21 21 the effect of the gate voltage on the drain current; The hybrid-I representation where I 1 and V 2 are the independent variables and V and V 1 and I and I 2 the dependent variables; the model reads: H 1 V 1 = H 11 11 I 1 + H 12 12 V 2 + V 01 H 1 I 2 = H 21 21 I 1 + H 22 22 V 2 + I 02
where H where H ij ij are the elements of the hybrid-I matrix (the diagonal elements are impedances impedances and admittanc admittances, es, respectiv respectively ely,, the non-diagona non-diagonall elements elements pure H 1 numbers), V 01 is the open-circuit open-circuit voltag voltagee at port 1 when port 2 is shorted shorted H 1 and I and I 02 is the short-circuit short-circuit current current at port 2 in the same loading conditions conditions;; again, both are zero for a non-autonomous two-port; this representation is well suited to model bipolar transistors in small-signal common-emitter operation since the parameter H parameter H 21 21 is the current gain between the base and the collector currents;
3.1 Representations of linear two-ports
73
The hybrid-II representation where V 1 and I 2 are the independent variables and I 1 and V 2 the dependent variables, the model is the inverse of the hybrid-I, we omit the relevant equations since this model is not particularly important; The so-called transmission-I and transmission-II models that exploit as independent variables the variables at port 2 (I 2 and V 2 ) or 1 (I 1 and V 1 ) and the other set as dependent variables; it is sometimes exploited in evaluating the representation of two-ports in cascade but we will omit details.
There are many reasons why the above representations, whose describing variables are voltages or current, are not particularly popular in RF or microwave circuits:
At RF and microwaves, total voltages and currents are difficult to measure through conventional instruments, and even the definition of these quantities may be questionable in some cases; for instance, in a non-TEM waveguide circuit currents and voltages in the conventional sense do not exist at all; In the measurement of conventional (impedance, admittance, hybrid) twoport parameters, short and open circuits are required as loads. However, they are difficult to implement at RF over a broad band of frequencies, so that a wideband characterization of a component becomes difficult. Most RF transistors cannot be measured in short or open-circuit conditions because they are unstable with such reactive loads.
In fact, the evaluation of the impedance parameters does not necessarily require open-circuit conditions to be imposed at the two ports; any loading conditions can be in fact exploited to derive any set of paramers, see Example 3.1; however, of course the problem remains of measuring current or voltages.
Example 3.1:
• Suppose that a non-autonomous two-port is loaded with an input and output
generators with open circuit voltages E 1 and E 2 and internal impedances Z 1 and Z 2 ; show that the impedance parameters can be derived by measuring I 1 and I 2 in two conditions: first we set E 2 = 0 and measure I 11 and I 21 ; then we set E 1 = 0 and measure I 12 and I 22 .
◦ We have four relationships:
V 11 = E 1 V 21 =
− Z 1I 11 = Z 11I 11 + Z 12I 21
−Z 2I 21 = Z 21I 11 + Z 22I 21 V 12 = −Z 1 I 12 = Z 11 I 12 + Z 12 I 22 V 22 = E 2 − Z 2 I 22 = Z 21 I 12 + Z 22 I 22
74
Power gain and stability of a loaded two-port
therefore we obtain two linear systems: I 11 Z 11 + I 21 Z 12 = E 1 I 12 Z 11 + I 22 Z 12 =
− Z 1I 11
−Z 1I 12
and: I 11 Z 21 + I 21 Z 22 =
−Z 2I 21 I 12 Z 21 + I 22 Z 22 = E 2 − Z 2 I 22 from which the four impedance parameters can be evaluated.
To finally overcome the problems associated with the definition of the conventional parameters a different representation technique was devised, that exploits the measurement of progressive and regressive waves (called power waves) in the presence of a matched (resistive) load conditions. The approach is derived, from a physical standpoint, from transmission line theory, but equally applies to lumped-parameter two-ports. The representative small-signal parameters are denoted as scattering parameters or S -parameters. For the sake of generality we will introduce the subject by assuming that the structure is an n-port.
3.2
The scattering parameters
3.2.1
Power waves Consider a linear n-port (i.e. a component with n pairs of poles), see Fig. 3.2. Notice that for n > 2 the impedance and admittance representations still exist while all other representatons introduced for a two-port have to be extended and re-defined. In the general case we will confine ourselves, therefore, to the series or parallel representations. The state of the n-port is determined by the set of current and voltage phasors at port k, V k and I k . Let us associate to port k the so-called normalization impedance R 0k that in principle is arbitrary (provided it has positive real part); however, we will for the sake of simplicity assume it to be real in what follows (and call it normalization resistance). Then, let us introduce for port k the power waves a k and b k as a linear combination of V k and I k :
V k + R0k I k 2 R0k V k R0k I k bk = . 2 R0k ak =
√ −√
(3.1)
√ R1 (R0k ak + R0k bk ) 0k 1 = √ (ak − bk ) . R
(3.2)
Inverting system (3.1) we have:
V k = I k
0k
75
3.2 The scattering parameters
From a physical standpoint power waves ak and bk can be traced back to the theory of transmission lines; in fact, in a line with characteristic impedance Z ∞ two (forward and backward) waves propagate with voltages (V + , V − ) and currents (I + , I − ), related as:
V + = Z ∞ I + V − = Z ∞ I − .
−
(3.3)
while the total voltage and current are obtained by superposition as:
V = V + + V − I = I + + I − .
(3.4)
The power flowing on the line finally is: P = (V I ∗ ) = V + 2 /Z ∞
| |
− |V −|2/Z ∞
By comparing (3.4) and (3.2) we can readily associate (assuming that R0k power waves to the normalized forward and backward voltages:
≡ Z ∞),
ak = V k+ / R0k , bk = V k− / R0k ; while the power entering port k (or, in the analogy, flowing on the line) is: P k = (V k I k∗ ) = V k+ 2 /Z ∞
(3.5) | | − |V k− |2/Z ∞ = |ak |2 − |bk |2; thus ak is related to√ the incident power, bk to the reflected power, both having the dimension of a W, from which the name “power waves”. Notice that the definition of power waves is independent from whether propagation actually takes place - they can be defined also for a lumped-parameter circuits.
Figure 3.2 Linear n -port.
From (3.1), if V k = R0k I k , then ak = 0. This happens if port k is loaded by the normalization resistance (resistance matching or matching, that does not imply power matching, i.e. maximum power transfer between the port and the load). In such conditions we also have:
−
bk = V k /
R0k .
(3.6)
76
Power gain and stability of a loaded two-port
3.2.2
Power wave n -port model For the sake of generality let us consider an autonomous n-port, where V 0k = 0 and I 0k = 0. Define the vector of port voltages and currents:
V =
V 1 V 2 . . V n
, I =
I 1 I 2 . . I n
.
For a linear n-port we have the impedance or series representation: V = Z I + V 0
(3.7)
where Z is the impedance matrix and V 0 the open-circuit voltage vector; similarly we have the parallel representation: I = Y V + I 0
(3.8)
where Y is the admittance matrix and I 0 is the short-circuit current vector. The two representations are not necessarily defined since one of the two matrices may be singular. Let a and b be the power wave vectors and let R0 the diagonal matrix of normalization resistances
a =
a1 a2 . ak . an
, b =
b1 b2 . bk . bn
, R0 =
R01 0 . . . 0 0 R02 0 . . 0 . . . . . . . . 0 R 0k 0 . . . . . . . 0 0 . . 0 R 0n
.
The power waves a and b are related by a linear relationship that can be identified as follows. Eqs.(3.2) can be written in matrix form as:
1/2
V = R 0 (a + b) −1/2 (a b) . I = R 0
(3.9)
−
1/2
−1/2
Since R0 is diagonal, also R0 and R0 are diagonal (the function of a diagonal matrix is the diagonal matrix of the functions of the diagonal elements). Substituting (3.9) in (3.7) we obtain: 1/2
R0
−1/2 (a − b) + V
(a + b) = ZR 0
0
that is:
−1/2 ZR−1/2 + I )−1 (R−1/2 ZR−1/2 0 0 0 − − − / 1/2 1/2 1 2 − +(R + I ) 1R V . ZR
b = (R0
0
0
0
0
− I )a
(3.10)
77
3.2 The scattering parameters
Figure 3.3 Connecting
two n -ports.
The power wave consistutive relatioship of the n-port therefore is: onde di potenza: b = S a + b0 .
(3.11)
where the scattering matrix S is defined as: S
≡ (R−0 1/2 ZR−0 1/2 + I )−1(R−0 1/2 ZR−0 1/2 − I ) = −1/2 ZR−1/2 − I )(R−1/2 ZR−1/2 + I )−1 = = (R0 0 0 0 − 1/2 1/2 − 1 = R 0 (Z + R0 ) (Z − R0 )R0 = 1/2 1 2 − / = R 0 (Z − R0 )(Z + R0 )−1 R0 .
(3.12)
The above equations in (3.12) are all equivalent since functions of the same matrix commute. The vector of forward wave generators b 0 is then obtained as:
≡ (R−0 1/2 ZR−0 1/2 + I )−1R−0 1/2 V 0 = R10/2 (Z + R0 )−1V 0 .
b0
(3.13)
If n = 1 the n-port reduces to a bipole or one-port; we thus obtain: S = and: b0 =
Z R0 Z + R0
−
√ R
0
Z + R0
V 0
(3.14)
(3.15)
in other words, S is the reflection coefficien of the impedance Z with respect to the normalization resistance. The normalization resistance matrix R0 is arbitrary (provided it is not singular); however, in many cases all normalization resistances are chosen as equal (R0 = R 0 I ) and often R0 = 50 Ω as a default. If the normalization resistance R0 = 1/G0 is uniform for all ports we obtain the following simpler relation: S = (Z
− R0I)(Z + R0I)−1 = (G0I − Y)(G0I + Y)−1.
(3.16)
78
Power gain and stability of a loaded two-port
Figure 3.4 Connecting
two n -ports in terms of scattering parameters.
S 11 =
(z11 1) (z22 + 1) (z11 + 1) (z22 + 1)
S 21 =
2z21 (z11 + 1) (z22 + 1)
S 11 =
(1 y11 ) (1 + y22 ) + y12y21 (1 + y11 ) (1 + y22 ) y12y21
S 12 =
S 21 =
−2y21
S 22 =
(1 + y11 ) (1 y22 ) + y12 y21 (1 + y11 ) (1 + y22 ) y12 y21
S 12 =
2h12 (h11 + 1) (h22 + 1)
S 22 =
(1 + h11 ) (1 h22 ) + h12 h21 (h11 + 1) (h22 + 1) h12 h21
S 11 =
S 21 =
−
−
− z12z21 − z12z21 − z12z21 −
(1 + y11 ) (1 + y22 )
− y12y21
(h11 1) (h22 + 1) (h11 + 1) (h22 + 1)
− h12h21 − h12h21
−
−2h21
(h11 + 1) (h22 + 1)
− h12h21
S 12 =
2z12 (z11 + 1) (z22 + 1)
S 22 =
(z11 + 1) (z22 1) (z11 + 1) (z22 + 1)
− z12z21
− − z12z21 − z12z21
−2y12
(1 + y11 ) (1 + y22 )
−
−
− y12y21 −
− h12h21 −
Table 3.1. Conversion between the Z , Y and H and the scattering parameters for a two-
port with normalization resistance R 0 at both ports. We have z ij = Z ij /R0 , yij = Y ij R0 , h11 = H 11 /R0 , h22 = H 22 R0 , h12 = H 12, h21 = H 21 .
Tables 3.1 and 3.2 report the conversion formulae between scattering, admittance and impedance parameters for a two-port having normalization resistance R0 at both ports. We denote with lowercase symbols the admittance impedance or hybrid parameters normalized vs. the normalization resistance or conductance.
3.2 The scattering parameters
z11 =
z21 =
(1 + S 11 ) (1 (1 S 11 ) (1
−
(1
−
− S 22) + S 12S 21 − S 22) − S 12S 21
2S 21 S 11 ) (1 S 22 )
−
− S 12S 21
z12 =
z22 =
2S 12 S 11 ) (1 S 22 )
(1
−
(1 (1
− S 11) (1 + S 22) + S 12S 21 − S 11) (1 − S 22) − S 12S 21
−
(1 S 11 ) (1 + S 22 ) + S 12 S 21 (1 + S 11 ) (1 + S 22 ) S 12 S 21
y12 =
y21 =
−2S 21
y22 =
(1 + S 11 ) (1 S 22 ) + S 12 S 21 (1 + S 11 ) (1 + S 22 ) S 12 S 21 2S 12 S 11 ) (1 + S 22 ) + S 12 S 21
−
(1 + S 11 ) (1 + S 22 )
− S 12S 21
h11 =
(1 + S 11 ) (1 + S 22 ) S 12 S 21 (1 S 11 ) (1 + S 22 ) + S 12 S 21
h12 =
h21 =
−2S 21
h22 =
−
−
(1
− S 11) (1 + S 22) + S 12S 21
−2S 12
− S 12S 21
y11 =
−
(1 + S 11 ) (1 + S 22 )
−
79
− S 12S 21 −
(1
−
(1 (1
− S 11) (1 − S 22) − S 12S 21 − S 11) (1 + S 22) + S 12S 21
Table 3.2. Conversion between the Z , Y and H and the scattering parameters for a two-
port with normalization resistance R 0 at both ports. We have z ij = Z ij /R0 , yij = Y ij R0 , h11 = H 11 /R0 , h22 = H 22 R0 , h12 = H 12, h21 = H 21 .
3.2.3
Power wave equivalent circuit - definition and evaluation From the series and parallel representations (3.7) and (3.8) we can derive the equivalent circuit (Fig. 3.5). We can do similarly with the analytical power wave representation (3.11). In order to do this we need to introduce two new components, the forward wave generator and the backward wave generator ) shown together with its constitutive relationships: in Fig. 3.6. Fig. 3.7 shows the equivalent power wave representation that can be derived directly by inspection. Eq. (3.12) and (3.13) enable to evaluate S and the forward wave generators b0 from Z and the open-circuit voltage vector V 0 . Similar relations exist for the admittance matrix and the short-circuit currents. S and b0 can be, however, directly evaluated from their definition (this also suggests a measurement technique for these parameters). From (3.11) we obtain b = b 0 when a = 0, i.e. when all ports are closed on their normalization resistances. The elements of b0 derive from the total port voltages; from (3.6) we
80
Power gain and stability of a loaded two-port
Figure 3.5 Series
and parallel equivalent circuit of an n -port.
Figure 3.6 Forward
(left) and backward (right) wave generators.
obtain: b0i =
√ V Ri
.
(3.17)
0i
Suppose now to set b0 = 0 by turning off all independent internal sources (thus making the n-port non-autonomous). From (3.11) we have b = S a. For the elements of S we have: bi S ij = aj
(3.18) ak =0 k=j
∀
Condition ak = 0 k = j is achieved by closing all ports apart from the j-the one on the corresponding normalization resistance, and feeding with a real gen-
∀
3.2 The scattering parameters
81
Figure 3.7 Power wave equivalent circuit.
erator (with an internal impedance that can be conveniently chosen as the port normalization resistance) the j -th port, see Fig. 3.8. The diagonal element S ii is immediately derived from (3.18) as the reflection coefficient at port i when all other ports are closed on their normalization resistance, i.e.: S ii
Z i − R0i G0i − Y i = ≡ Γi = abii = Z G0i + Y i i + R0i
(3.19)
where G0i = 1/R0i is the normalization conductance of port i and Y i is the input admittance of the one-port obtained by closing all ports but the i-th one on the normalization resistances. The out-of-diagonal elements of the scattering matrix are transmission coefficients. To identify them consider the circuit in Fig. 3.8; the power wave bG coming out of the generator connected to port j is derived from (3.15) setting Z = R 0 :
bG = V 0j /2 R0j ; but the same wave enters port j , thus:
aj = b G = V 0j /2 R0j . We also have from (3.6): bi
|a =0∀k=j = √ V Ri k
(3.20)
(3.21) 0i
finally then: bi S ij i= j = a j
|
V i =2 V 0j j ak =0∀k =
R0j . R0i
(3.22)
82
Power gain and stability of a loaded two-port
Figure 3.8 Evaluating
the out-of-diagonal elements of the scattering matrix.
To evaluate the out-of-diagonal terms of S we simply need to compute or measure voltage ratios according to circuit theory.
3.2.4
Solving a network in terms of power waves Consider a network deriving from interconnecting an arbitrary number of mports. To each m-port we can associate m sides with 2m unknowns, the port voltages and the current entering (and exiting) each port. Suppose the total number of ports is k (interconnected together) with a total number of unknowns 2k (voltages and currents). We can reformulate the problem by stating that the total number of unknowns is 2k, each port hosting a forward and backward power waves. The question is of course whether the Kirchoff voltage and current laws plus the consistitutive relationships translate into a well-posed set in terms of power waves. Consider first what happens when connecting two ports belonging to two different n-ports, see Fig. 3.3. From the Kirchhoff voltage and current laws we obtain:
V ja = V ib I ja = I ib .
−
(3.23)
Since for any pair of connected ports we can generate two such equations, the total number of topological equations is 2k/2 = k. Since for each port we obtain one constitutive equation, the problem is well posed, since we have k topological plus k constitutive equations in terms of port voltages and currents. Coming to power wawes, any couple of connected ports implies four unknowns, two forward and two backward waves, see Fig. 3.4. The total number of unknowns is again 2k. The constitutive relations based on scattering parameters (3.11) yield k relations, but k topological relationships can be obtained by expressing (3.23) in terms of power waves; from the definition of the power waves at ports i and
83
3.2 The scattering parameters
j connected together we immediately have:
√ − − − − − R0ja (aja + bja ) =
(aja
bja )/ R0ja =
R0ib (aib + bib )
(aib
bib
(3.24)
√ )/ R
0ib
that is, solving:
R0ja R0ib R0ja + R0ib 2 R0ja R0ib 2 R0ja R0ib
aja
=
R0ja + R0ib R0ja R0ib 2 R0ja R0ib 2 R0ja R0ib
bja
aib
.
(3.25)
bib
If the normalization resistances are the same for ports i and j , (3.25) reduces to: aja = b ib bja = a ib
(3.26)
i.e. power waves are continuous across the interconnecting port. Generally speaking therefore each couple of connected ports yields two topological relationships, yielding in total k relations that added to the constitutive ones finally yield 2k relations, equal to the number of unknowns. Therefore the problem is well posed.
3.2.5
Properties of the S-matrix: power, reciprocity, reactivity From (3.5) the net power entering port k can be expressed in terms of power waves; the total power P tot entering the n-port will therefore be: n
P tot =
n
| P k =
k =1
( ak
k=1
|2 − |bk |2) = aT a∗ − bT b∗ ,
(3.27)
where T denotes the transpose. If the circuit is non-autonomous (no internal independent sources) b = S a; substituting in (3.27) we obtain: P tot = aT (I
− ST S∗)a∗ .
(3.28)
For a reactive n-port P tot = 0 independent on the excitation, this can only be obtained by imposing S T S∗ I = 0; thus for a reactive n-port:
−
S−1 = S ∗T
(3.29)
i.e. the scattering matrix is hermitian (the inverse equals the complex conjugate of the transposed). Reciprocity characterizes most networks made of passive components (although some microwave passive components including magnetic materials, such as circulators, are non-reciprocal); in terms of the impedance matrix the reciprocity condition reads: Z = Z T
(3.30)
84
Power gain and stability of a loaded two-port
but, from (3.12), we obtain: Z = R 1/2 (I
− S)−1(I + S)R1/2
(3.31)
and therefore, taking into account that R1/2 is diagonal and R1/2 = (R1/2 )T , we obtain: ZT = R 1/2 (I + ST )(I
− ST )−1R1/2 .
(3.32)
Substituting (3.31) and (3.32) into (3.30) we obtain: (I
− S)−1(I + S) = ( I + ST )(I − ST )−1
i.e.: (I + ST )−1 (I
− S)−1(I + S)(I − ST ) = I
in other words: (I + S)(I
− ST ) = (I − S)(I + ST )
and finally: S = S T .
(3.33)
Formally therefore the reciprocity condition for the scattering matrix coincides with the conditions for Z and Y, i.e. S is symmetric. Additionally, for a reciprocal and reactive n-port we finally have: S−1 = S ∗ .
(3.34)
Example 3.2:
• In a reactive and reciprocal two-port, make the relations b etween the scattering parameters from (3.34) explicit. ◦ From (3.34) we obtain: SS∗ = I
i.e., developing the product:
|S 11|2 + |S 12|2 = 1
∗ + S 12 S ∗ = 0 S 11 S 12 22 ∗ + S 22 S ∗ = 0 S 12 S 11 12
|S 22|2 + |S 12|2 = 1 thus S 11 and S 22 have the same magnitude (note that the second and third equation are equivalent). This lead to a relationship between phases: φ11
− φ12 = −φ22 + φ12 + nπ,
3.3 Generator-load power transfer
85
Figure 3.9 Evaluating
the power transfer on a load.
with n odd, i.e.: φ11 + φ22 = 2φ12 + nπ.
3.3
Generator-load power transfer Consider the real generator connected to a load in Fig. 3.9; the power absorbed by the load impedance (Z L = R L + jX L ), P L , is: P L =
(V L I L∗ ) = |V 0|2 |Z G +RLZ L|2 .
(3.35)
The maximum power trasfer (maximum power on the load) occurs in power matching conditions, i.e. when:
∗ . Z L = Z G
(3.36)
(see Example 3.3); the maximum load power, also called generator available power , is: P av =
|V 0|2 . 4RG
(3.37)
Example 3.3:
• Obtain (3.36) and (3.37). ◦ The maximum of P vs. R
and X L corresponds to a zero of the partial derivatives of P L vs. the two variables, i.e. to conditions: L
L
2 L + RG ) =0 | |2 |Z G + Z L||Z G− +2RZ LL(R 4 |
∂P L = V 0 ∂R L ∂P L = V 0 ∂X L
L + X G ) =0 | |2 −2R|Z LG(X + Z L |4
86
Power gain and stability of a loaded two-port
Figure 3.10 Evaluating
the power on a load through power waves.
from the second equation we find X L = X G ; substituting in the first we obtain RL = RG and therefore the load corresponding to the maximum power transfer from a generator with internal impedance Z G is:
−
∗ . Z L = Z G
(3.38)
Substituting we find the generator maximum or available power: P av =
|V 0|2 .
4RG
(3.39)
The same result can be obtained by describing the circuit in terms of power waves. The circuit in Fig. 3.9 results from the connection of two one-ports, the real generator and the load with impedance Z L . Suppose for simplicity that the normalization resistance is the same for both one-ports; the circuit can be represented as in Fig. 3.10. The scattering matrices of the generator and load are the reflection coefficients Γ G and ΓL , respectively, and the forward wave generator b 0 are expressed from (3.19) and (3.13) by:
ΓG =
Z G R0 Z G + R0
ΓL =
Z L R0 Z L + R0
−
b0 = V 0
−
√ R
0
Z G + R0
(3.40)
.
The power on the load can be evaluated by taking into account that the circuit in Fig. 3.10 implies two topological relationships in terms of power wave continuity and two consistutive relations (generator and load):
aL aG bG bL
= b G = b L = b 0 + Γ G aG = ΓL aL
87
3.4 Power transfer in loaded two-ports
solving, the power wave on the load are:
aL =
1
−
b0 ΓG ΓL
.
b0 ΓL bL = 1 ΓG ΓL
−
Thus, from (3.5) the power on the load is: P L = aL
2
| | − |b L |
2
1 ΓL 2 . 1 ΓG ΓL 2
| | − |ΓL| ) = |b0| | −− | | | 2
2
= aL (1
2
(3.41)
∗ ); the maximum power (or generP L is maximum for ΓG = Γ∗L (i.e. for Z L = Z G ator available power) is:
| |2 1 − |1ΓG|2 = |V 0|2 |Z G +R0R0|2 1 − |1ΓG|2
P av = b0
(3.42)
that is equivalent to (3.37). From (3.41) and (3.42) we obtain: P L = P av
(1
− |ΓG|2)(1 − |ΓL|2) . |1 − ΓGΓL |2
(3.43)
Notice that for ΓL = 0 we do not have maximum power transfer; in fact in that case we obtain from (3.43): P L = P av (1
− |ΓG|2) ≤ P av
and P L = P av only if ΓG = 0, implying that both the load and the source impedances coincide with the normalization resistance.
3.4
Power transfer in loaded two-ports In the circuit in Fig. 3.11 we want to evaluate the load power P L as a function of the S-parameters of the rwo-port and of the load and generator reflection coefficients ΓL and ΓG . We assume the same normalization resistance at all ports. The power wave continuity and constitutive relationships read:
topological relations:
constitutive relations:
a1 b1 b2 a2
= b G = a G = a L = b L
bG = b 0 + Γ G aG b1 = S 11 a1 + S 12 a2 b2 = S 21 a1 + S 22 a2 bL = ΓL aL .
88
Power gain and stability of a loaded two-port
Figure 3.11 Power
transfer between generator and load through a two-port.
Eliminating aG , bG , aL and bL from the power wave continuity equations we obtain the reduced system:
with solution:
−−
1 S 11 S 21 0
−ΓG 1 0 0
− −
0 S 12 S 22 1
0 0 1 ΓL
−
a1 b1 a2 b2
= b 0
1 0 0 0
(3.44)
1
− S 22ΓL (1 − S 11 ΓG )(1 − S 22 ΓL ) − S 12 S 21 ΓG ΓL S 12 S 21 ΓL + S 11 (1 − S 22 ΓL ) b1 = a G = b 0 = (1 − S 11 ΓG )(1 − S 22 ΓL ) − S 12 S 21 ΓG ΓL S 11 − ∆S ΓL = b 0 (1 − S 11 ΓG )(1 − S 22 ΓL ) − S 12 S 21 ΓG ΓL a1 = b G = b 0
a2 = b L = b 0 b2 = a L = b 0
(1
− S 11ΓG)(1 −
(1
− S 11ΓG)(1 −
ΓL S 21 S 22 ΓL ) S 21 S 22 ΓL )
(3.45a)
(3.45b) (3.45c)
− S 12S 21ΓGΓL
(3.45d)
− S 12S 21ΓGΓL
where ∆S is the determinant of the S-matrix. From (3.45) we can derive a number of parameters:
The input reflection coefficient of the loaded two-port: Γin =
b1 S 12 S 21 ΓL S 11 ∆S ΓL = S 11 + = . a1 1 S 22ΓL 1 S 22 ΓL
−
− −
(3.46)
Two-port input power, can be expressed in a direct way: 2 2 22 ΓL | − |S 11 − ∆S ΓL | | |2 − |b1|2 = |b0|2 |(1 − S |111−ΓGS )(1 − S 22ΓL ) − S 12S 21ΓGΓL|2 ,
P in = a1
(3.47)
89
3.4 Power transfer in loaded two-ports
or by exploiting the input reflection coefficient; in that case the analysis reduces to a one-port (the generator) loaded by Γ in and therefore: a1 = b 0
1
from which:
−
1 ΓG Γin 1 Γin 2 . 1 ΓG Γin 2
| | − |Γin| ) = |b0| | −− | | | 2
2
P in = a1 (1
2
(3.48)
Expressions (3.47) and (3.48) are of course equivalent but they will be conveniently used in what follows. Two-port output equivalent circuit. Since the two-port is closed at the input by a generator, its equivalent circuit at the output port will be the one of a non-autonomous structure, whose power wave equivalent circuit corresponds to the following constitutive equation: b2 = b 0 + Γ out a2 . We have that b 2 = b 0 when port 2 is loaded by the normalization resistance, i.e. when ΓL = 0. In this case we have: b0 = b 0
1
−
S 21 . S 11 ΓG
(3.49)
The output reflection coefficient is derived by symmetry exchanging ports 1 and 2 and ΓL with ΓG : Γout = S 22 +
S 12 S 21 ΓG S 22 ∆S ΓG = . 1 S 11 ΓG 1 S 11 ΓG
−
− −
(3.50)
Power on the load. Can be expressed in several equivalent ways, either
directly: S 21 |2 (1 − |ΓL |2 ) | P L = |aL | − |bL | = |b0 | |(1 − S 11ΓG)(1 − S 22ΓL ) − S 12S 21ΓGΓL|2 2
2
2
(3.51)
or by means of the equivalent circuit at port 2, cfr. Fig. 3.12:
|ΓL|2 = |b0|2 |S 21|2(1 − |ΓL |2) . | |2 |1 1−−Γout ΓL |2 |1 − ΓLΓout|2|1 − S 11ΓG|2
P L = b0
(3.52)
or, finally, with reference to the input reflection coefficient (see Example 3.4): 2 2 | |2 |1 − |ΓS G21Γ| in(1|2−|1 |−ΓLS |22)ΓL |2 .
P L = b0
Example 3.4:
• Demonstrate (3.53).
(3.53)
90
Power gain and stability of a loaded two-port
Figure 3.12 Loaded
two-port: equivalent circuit at port 1 (above), equivalent circuit at
port 2 (below).
◦ From the expression of a : L
aL = b 0 we have, collecting (1 (3.46):
(1
− S 11ΓG)(1 −
S 21 S 22 ΓL )
− S 12S 21ΓGΓL
− S 22ΓL) and taking into account the expression of Γin aL = b 0
from which, finally:
(1
−
S 21 S 22 ΓL )(1
− ΓGΓin)
2 2 | | − |ΓL|2) = |b0|2 |1 − |ΓS G21Γ| in(1|2−|1 |−ΓLS |22)ΓL |2
P L = aL 2 (1
which corresponds to (3.53).
Example 3.5:
• Derive the inverses of (3.46) and (3.50). ◦ From (3.46) we obtain: (Γin
− S 11)(1 − S 22ΓL) = S 12S 21ΓL,
3.5 Gains of loaded two-ports
91
that is, developing: Γin
− Γin S 22ΓL − S 11 + S 11S 22ΓL = S 12S 21ΓL,
from which, collecting ΓL : (S 11 S 22
− S 12S 21 − Γin S 22)ΓL = (∆S − ΓinS 22)ΓL = S 11 − Γin.
Therefore, deriving ΓL : ΓL =
S 11 Γin ∆S S 22 Γin
− −
(3.54)
and, exchanging port 1 and 2 and the generator with the load: ΓG =
3.5
S 22 Γout . ∆S S 11 Γout
− −
(3.55)
Gains of loaded two-ports Inserting a two-port between a generator and a load changes the power exchange between them. We can quantify this effect by a set of power gains expressing by ratios of power- and generator-referred powers: 1. Operational gain Gop : the ratio between the power on the load P L and the power P in entering the input port of the two-port, P in . As will be explicitly shown, the operational gain depends on Γ L but does not depend from ΓG ; changing ΓG in fact we change the input available power and the input matching condition, thus the input power, but, in the same way, also the output power; 2. Available power gain Gav : the ratio between the input (generator) available power P av,in and the available power at the output port of the two-port P av,L . It depends on ΓG but does not depend on Γ L ; in fact, the output available power is the power on the load when the load is power matched to the output port of the two-port, independent of the actual value of Γ L ; 3. Transducer gain Gt : the ratio between the power on the load P L and the input available power P av,in; it depends both on ΓG and on ΓL .
3.5.1
Maximum gain and maximum power transfer The maximum power transfer between a generator and a load connected through a two-port occurs when two conditions are met, i.e. the input power is the generator (input) available power and the power on the load is the output available power. This condition of maximum power transfer implies simultaneous power (or conjugate) impedance matching at the two ports. While the maximum power transfer implies maximum gain, the opposite is true only for the transducer gain.
92
Power gain and stability of a loaded two-port
Figure 3.13 Block
diagram describing the flow from the generator available power to the load power and the definitions of operational gain (above), available power gain (center), transducer gain (below).
A maximum in the operational gain and in the available power gain imply maximum power transfer only if a second condition is met, corresponding to the input or output matching, respectively. However, it is quite clear that the maxima of all gains coincide. In fact we can write: P L Gt,M = P av,in
M
M
≡ P P M L
av,in
M where we denote as P LM and P av,in the power on the load and the input available power in maximum power transfer conditions and Gt,M is the maximum transducer gain, obtained by properly selecting the generator and load reflectances. M However when maximum power trasfer is achieved P LM = P av,L because if the output is power matched (see Fig.3.13, middle) then the load power coincides with the output available power. Therefore we have: M P av,L P LM Gt,M = M = M P av,in P av,in
≡ Gav,M
i.e. the maximum avalaible power gain coincides with the maximum transducer M gain; notice that P av,in derives from an optimization with respect to the source M impedance while the condition P LM = P av,L is obtained through an additional constraint on the load impedance. Similarly, in maximum power transfer conditions, the input power coincides with the source available power, implying conjugate matching at the input by properly selecting the source reflectance. We
93
3.5 Gains of loaded two-ports
thus have: Gt,M =
P LM P LM = M M P av,in P in
≡ Gop,M
M where P in is the input power in maximum power transfer conditions. Thus, also the maximum of the operational gain (obtained by properly selecting the load reflectance) coincides with the maximum transducer gain, but the maximum power transfer condition additionally requires the conjugate input matching to be achieved (see Fig.3.13, top). In conclusion, while we have:
Gt,M = Gop,M = Gav,M optimization of the whole chain leading to P av,in to P L (Fig.3.13, bottom) by properly selecting the input and output reflectances lead to maximum power transfer, while optimization of the chain leading from P in to P L by properly selecting the load reflectance (maximum operational gain) or optimization of the chain leading from P av,in to P av,L by selecting the source reflectance (maximum available gain) do lead to maximum gain but not to maximum power transfer unless a second condition is met on the output or input power matching, respectively. For historical reasons however the maximum gain is often referred to as MAG, i.e. the maximum available gain. Achieving simultaneous power matching at the input and output ports is not, however, always possible. Such a condition may be implemented only if the twoport is unconditionally stable, i.e. stable for any load and generator impedances having positive real part. If this is not the case the two-port is potentially unstable and the maximum gain is infinity. The potentially unstable condition was regarded in the early stages of the development of electronics as a useful tool to boost the amplifier gain, and was even artificially caused through positive feedback; however, in current design the stable gain of transistors is large enough to suggest a design strategy based on the stabilization of active devices within the design bandwidth. Out-of-band stabilization is mandatory anyway to suppress (typically low-frequency) spurious oscillations.
3.5.2
Operational gain The expression for the operational gain is obtained from (3.47) and (3.51): Gop = i.e., developing:
P L = S 21 P in
2
| |2 |1 − S 22ΓL1|2−−|Γ|S L1|1 − ∆S ΓL|2
(3.56a)
2 | |2 1 − |S 11|2 + |ΓL |2(|S 22|2 1−−|∆|ΓS L|2|) + 2(ΓL (S ∗ ∆S − S 22)) . 11
Gop = S 21
(3.56b)
94
Power gain and stability of a loaded two-port
Note from (3.56) that the operational gain is a real function of the complex variable ΓL ; as shown in Example 3.6 the constant gain curves are circles in the ΓL plane. Moreover, the operational gain vanishes on the unit circle of the Γ L Smith chart, i.e. for ΓL = 1; in such a case the load is reactive and the average (or active) power dissipated by it must vanish.
| |
Example 3.6:
• Show that the constant operational gain curves are circles in the Γ find their center and radius. ◦ Reworking (3.56) we obtain:
L plane
and
|ΓL|2(Gop (|S 22|2 − |∆S |2) + |S 21|2)+ ∗ ∆S − S 22 )) = |S 21 |2 − Gop (1 − |S 11 |2 ) Gop 2(ΓL (S 11
from which:
|ΓL
|− 2
2
∗ ∆S ) Gop (S 22 S 11 ΓL Gop ( S 22 2 ∆S 2 ) + S 21
− | | −| |
| |2
=
|S 21|2 − Gop (1 − |S 11|2) . Gop (|S 22 |2 − |∆S |2 ) + |S 21 |2
(3.57)
Summing the last relation to both terms:
∗ ∆S ) Gop (S 22 S 11 α = Gop ( S 22 2 ∆S 2 ) + S 21
− | | −| |
|
Eq. (3.57) can be rewritten as:
|
2
S 21 |2 − Gop (1 − |S 11 |2 ) | |ΓL − C | = Gop (|S 22|2 − |∆S |2) + |S 21|2 + α 2
where: C =
∗ ∆S ) Gop (S 22 S 11 Gop ( S 22 2 ∆S 2 ) + S 21
− | | −| |
| |2
∗
=
(3.58)
2
(3.59)
∗ S 11 ∆∗ ) Gop (S 22 S Gop ( S 22 2 ∆S 2 ) + S 21
− | | −| |
| |2 . (3.60)
Comparing the last expression with the equation of a circle in the Z plane, with center Z C and radius R,
|Z − Z C |2 = R2 . we derive that if: R2 =
|S 21|2 − Gop (1 − |S 11|2) + α ≥ 0 Gop (|S 22 |2 − |∆S |2 ) + |S 21 |2
(3.61)
then the constant operational gain curves are circles with center C and radius R. Developing and taking into account the relation:
|S 22 − S 11∗ ∆S |2 = |S 22|2 + |S 11|2|∆S |2 − 2(S 22S 11∆∗S ) = |S 22 |2 + |S 11 |2 |∆S |2 − |S 11 |2 |S 22 |2 − |∆S |2 + |S 21 |2 |S 12 |2 = (|S 22 |2 − |∆S |2 )(1 − |S 11 |2 ) + |S 21 |2 |S 12 |2
3.5 Gains of loaded two-ports
95
Figure 3.14 Constant
operational gain circles in plane Γ L .
we obtain for the radius the expression: R = S 21
| |
|
|2 − 2K |S 21||S 12|Gop + |S 12|2G2op |Gop(|S 22|2 − |∆S |2) + |S 21|2|
S 21
(3.62)
where we have introduced the real parameter K : K =
1
− |S 11|2 − |S 22|2 + |∆S |2 2|S 21 ||S 12 |
(3.63)
called the Linville coefficient. From (3.60) we also find that varying Gop the centers of the circles lie on a straight line with slope given by:
∗ − ∆∗ S 11 ) (S 22 ∗ S 22 − S 11 ∆S ) = ∗ (S − ∆∗ S 11) .
arg(S ∗
22
(3.64)
S
In conclusion, if (3.61) is verified, the constant gain curves in the plane Γ L are circles whose centers lie on a straight line.
Starting from the expression of the radius (3.62), whose denominator is positive anyway, we can understand under which conditions constant gain circles exist in the above form. The term under square root in the numerator should also be positive if the radius is real; taking into account that this term is a second order polynomial in Gop with positive coefficient of the second-order term, we
96
Power gain and stability of a loaded two-port
find that the polynomial is positive for values of Gop larger or smaller than the two roots of the polynomial, i.e. for:
−
S 21 Gop > (K + S 12 S 21 Gop < (K S 12
K 2
− 1)
K 2
− 1).
(3.65)
(3.66)
Since the operational gain is real we also obtain the condition K 1. Notice that if we had K 1 the operational gain range in which the radius exists would correspond to negative gain, a condition that will be shown to be inacceptable for reasons connected to stability; therefore the condition:
≤ −
| | ≥
K
≥ 1
should hold. We usually draw the constant (operational) gain circles in the Γ L Smith chart, as shown in Fig. 3.14. As discussed further on, if the two port is unconditionally stable (cfr. Sec. 3.6) Gop has a maximum within the ΓL Smith chart, corresponding to (3.66), while the minimum corresponding to (3.65) falls outside the Smith chart and has no interest. The maximum operational gain derived by setting R = 0 is given by: GopMAX
−
S 21 = (K S 12
K 2
− 1) .
(3.67)
Substituting in (3.60) we derive the corresponding optimum value of the load reflectance Γ Lopt : ΓLopt =
B2
−
− |
B22 4 C 2 2C 2
|2
(3.68)
where:
| |2 − |S 11|2 − |∆S |2 ∗ . C 2 = S 22 − ∆S S 11
B2 = 1 + S 22
(3.69)
(3.70)
If the two port is unconditionally stable ΓLopt is within the Smith chart and ensures power matching at the output port. Notice finally that for K = 1 the maximum gain becomes S 21 /S 12 , also called (for reasons to be explained later) Maximum Stable Gain or MSG.
|
3.5.3
|
Available power gain The available power gain is the ratio between the load available power ( P av,L ) and the input available power (3.42):
| |2 1 − |1ΓG|2
P av,in = b0
(3.71)
97
3.5 Gains of loaded two-ports
while the output available power is equal to the load power when Γ L = Γ∗out . In such conditions (3.52) becomes:
| |2|S 21|2 (1 − |Γout|2)1|1 − S 11ΓG|2
P av,L = b0
(3.72)
(3.73a)
from which, substituting (3.50), we obtain: Gav =
2
P av,L = S 21 P av,in
| |2 |1 − S 11ΓG1|2−−|Γ|S G2|2 − ∆S ΓG|2
i.e.:
ΓG |2 − | Gav = |S 21 | ∗ ∆S − S 11 )) . 1 − |S 22 |2 + |ΓG |2 (|S 11 |2 − |∆S |2 ) + 2 (ΓG (S 22 1
2
(3.73b) Gav only depends on ΓG . If we compare Gav / S 21 to Gop / S 21 we immediately notice that the two terms correspond each other by replacing the generator with the load and exchanging port 1 with port 2. It follows immediately that the constant level curves of Gav in the plane ΓG will be again circles with centers lying on a straight line; if again the Linville coefficients satisfies K > 1 the radius and centers of the circles can be derived from (3.60) and (3.62) by exchanging the indices 1 and 2. Moreover, for an unconditionally stable two-port the available power gain shows a maximum G avMAX that coincides with the operational gain maximum in (3.67). In fact we obtain G av / S 21 2 from G op / S 21 2 by exchanging ports 1 and 2, but the maximum of G op / S 21 2 , G opMAX / S 21 2 , is invariant with respect to such exchange; thus GavMAX / S 21 2 = GopMAX / S 21 2 and therefore GavMAX = GopMAX , i.e.:
| |
2
| |
| | | | | |
GavMAX
| | | | | |
− − ≡ − − | |
S 21 = (K S 12
K 2
2
1)
GopMAX .
(3.74)
The optimum ΓG leading to maximum available gain is: ΓGopt =
B1
B12 4 C 1 2C 1
2
(3.75)
where:
| |2 − |S 22|2 − |∆S |2 ∗ . C 1 = S 11 − ∆S S 22
B1 = 1 + S 11
(3.76)
(3.77)
B1 , C 1 and ΓGopt can be derived from B 2 , C 2 and ΓLopt by exchanging ports.
98
Power gain and stability of a loaded two-port
3.5.4
Transducer gain The transducer gain G t is the ratio between the power on the load and the input available power. From (3.71) and (3.51) we immediately obtain: Gt =
− |ΓL|2)(1 − |ΓG|2) | |2 |(1 − ΓLS 2(12)(1 − ΓGS 11) − S 12S 21ΓGΓL|2 .
P L = S 21 P av,in
(3.78)
As already noticed, the transducer gain depends both on Γ G and on ΓL . Let us define now a new parameter, the unilateral transducer gain G u . This is the transducer gain of a two-port having S 12 = 0; we call this a unilateral two port since is only shows forward action from port 1 to port 2 and no reverse internal feedback from port 2 to port 1. As we will show later, a unilateral two-port is (apart from some fancy cases) unconditionally stable. Real high-frequency transistors are often almost unilateral. Finally, the unilateral approximation makes the treatment of power matching very easy. We will therefore start the discussion with the unilateral assumption. If S 12 = 0 we obtain from (3.78): Gu = Gt
|S
12
2 − |ΓG|2) . | |2 |1(1−−ΓL|ΓS L22| |2)(1 |1 − ΓGS 11|2
=0 = S 21
(3.79)
In this case Gu is clearly maximum when conjugate matching is simultaneously achieved at both ports:
∗ ΓG = S 11 ∗ ΓL = S 22
(3.80)
with maximum unilateral gain (MUG): Gumax =
|S 21|2 . (1 − |S 11 |2 )(1 − |S 22 |2 )
(3.81)
In the general case of a non-unilateral two-port the maximum transducer gain corresponds to the simultaneous conjugate matching at both ports (assuming this is feasible), i.e. to the coupled equations:
3.5.5
ΓG = Γ∗in (ΓL ) . ΓL = Γ∗out (ΓG )
(3.82)
Is power matching always possible? As discussed in Sec. 3.6, simultaneous power matching is possible when the twoport is unconditionally stable. In fact:
If the two-port is unconditionally stable simultaneous power matching at the two ports is possible, the input power is the source available power, the power on the load is the output available power, and all gains (operational, available, transducer) a maximum with the same value given by (3.67). If the two-port is potentially unstable simultaneous power matching at the two ports is not possibile and there is a set of general and load impedances
99
3.6 Stability
Gop =
Gav =
Gt =
2
| | − |S | + |Γ | (|S | 1−−|∆|Γ | |) + 2(Γ
P L = S 21 2 P in 1
L
2
11
2
22
2
S
2
L
∗ ∆S (S 11
− S
P L = S 21 P av,in
))
G
22
2
G
2
11
2
S
2
(1 ΓL 2 )(1 ΓG 2 ) ΓG S 11 ) S 12 S 21 ΓG ΓL L S 22 )(1
2
22
2
| | − |S | + |Γ | (|S | 1−−|∆|Γ | |) + 2(Γ
P av,L = S 21 2 P av,in 1
MSG =
L
−| | −
| | |(1 − Γ
−| | −
G
∗ ∆S (S 22
− S
11
))
2
|
S 21
S 12
2
MUG =
|S | (1 − |S | )(1 − |S | ) 21
11
2
22
2
Table 3.3. Summary of gain definitions for a loaded two-port.
(with positive real part) for which the gain tends to infinity, thus leading to the onset of oscillations. In practice, since the transistor gain decreases with frequency, active devices are more prone to be potentially unstable at low frequency. While out-of-band stabilization is mandatory to avoid spurious lowfrequency oscillations, a design with an in-band potentially unstable device is possible, provided that the terminating impedances are chosen so as to be far enough away from the potentially unstable termination set. However, while this is easier in hybrid design, integrated design must allow for technological fluctuations, and therefore in-band stabilization is the preferred choice. A summary of the definitions of gains of a loaded two-port is provided in Table 3.3.
3.6
Stability The stability issue is important in the design of amplifiers, mixers and oscillators (a mixer can be seen like an amplifier also providing frequency conversion). Typically we want stable, non self-oscillating behaviour from amplifiers and mixers. On the other hand, instability is sought in the design of oscillators: in the linear approximations an unstable circuit generates oscillations with infinite amplitude, but of course in practice the amplitude is limited by nonlinear saturation effects that are present in all active devices.
100
Power gain and stability of a loaded two-port
A two-port loaded with generator and load impedances with positive real part (we call those passive or “physically realizable”) is unconditionally stable if the input impedance has positive real part for any value of the load impedance and the output impedance has positive real part for any value of the generator impedance. The same condition can be expressed through reflectances: for any value of ΓL (with ΓL < 1) we have Γin < 1, and for any value of Γ G (with ΓG < 1) we have Γout < 1. Since the two-port parameters are frequency dependent, stability depends on the operating frequency. Moreover stability is a global property of a circuit, unless this is made of subcircuits that are isolated with respect to each other. On the other hand, we say that a two-port is conditionally stable or potentially unstable if there is a set of passive impedances at port 1 or 2 such as the output or input reflection coefficient of the two-port has magnitude larger than one. Notice that the fact that e.g. Γin > 1 does not automatically imply that the circuit will oscillate, since the oscillation condition is in fact Γ in ΓG = 1 and we can conveniently select ΓG such as Γin ΓG < 1; however such a circuit does not properly behave as an amplifier, as discussed in Example 3.7. The only proper choice is therefore to select, in a potentially unstable two-port, a set of termination such as the input and output reflectances have magnitude less than one. As already stressed, for an unconditionally stable two-port a well defined maximum power transfer condition exists corresponding to conjugate matching at both ports. For a potentially unstable devices in the operating bandwidth, on the other hand, the linear gain is theoretically unbounded, and the choice of terminations should be made so as to ensure stability, a large enough gain, but also a termination not too close to the instability boundary, to avoid self-oscillations induced by process variations. An important result is that unconditional stability can be detected by a set of simple equivalent tests to be made on the two-port scattering parameters. If the two-port is potentially unstable graphical tools like the stability circles can be exploited in order to make sure that the terminations ensure stable behaviour with a good enough margin.
| |
| | | |
| |
| |
|
|
Example 3.7:
• Suppose a two-port is loaded at port 2 with a reflectance Γ
L such
as Γin > 1 (or Z in and Y in have negative real part). Can we stabilize the circuit with a proper choice of Γ G ? In that case is the circuit working as an amplifier from the generator to the load?
| |
◦ Consider a sinusoidal voltage generator with open circuit voltage E and pasG
sive internal impedance Z G , loaded with an impedance Z in . The instability condition corresponds to an infinite current in the load at an angular frequency
101
3.6 Stability
ω0 : I L =
E G Z L + Z in
→∞
i.e. to RL (ω0 ) + Rin (ω0 ) = 0, X L (ω0 ) + X in (ω0 ) = 0. In other words the total loop impedance is zero and the input resistance (negative) compensates for the generator resistance. A dynamic analysis of the system (that can be carried out postulating around ω 0 an equivalent circuit including reactive elements, e.g. lumped capacitors and inductors) shows that the system is stable (i.e. the pulse response decays for t ) if RL (ω0 ) + Rin (ω0 ) > 0, unstable (the pulse response diverges) for RL (ω0 ) + Rin (ω0 ) < 0. Similarly we can show that the stability conditions implies (in a parallel representation) G L (ω0 ) + Gin (ω0 ) > 0. Therefore we can make a circuit stable by properly selecting the generator resistance or conductance so as to enforce the stability of the series or parallel circuit. Even in this case, however, the circuit does not operate like e conventional amplifier. Since we have assumed that Γin > 1, from (3.41) we obtain for the input power in the two-port:
→∞
| |
2
| |2 |11−−Γ|GΓinΓin| |2 < 0
P in = b0G
where b0G is associated to the forward wave generator at port 1. This means that the generator power is reflected back by the two-port (with a possibile amplification) and dissipated in the generator resistance. This circuit therefore does not operate like a conventional amplifier in which power is amplified from port 1 to port 2, but rather as a reflection amplifier where power is reflected back by an active element with amplification and dissipated on the generator resistance, acting in this case as a load.
3.6.1
Analysis of stability conditions Let us consider again (3.44); the system admits a nonzero solution when b 0 = 0 only if the system matrix determinant is zero, i.e.:
−−
1 S 11 S 21 0
−ΓG 1 0 0
− −
0 S 12 S 22 1
0 0 = (1 1 ΓL
−
− S 11ΓG)(1 − S 22ΓL) − S 12S 21ΓGΓL = 0 .
Taking into account (3.46) and (3.50) the condition can be written in one of the two following ways: (1
− S 11ΓG)(1 − ΓLΓout) = 0 (1 − S 22 ΓL )(1 − ΓG Γin ) = 0 .
(3.83a)
(3.83b)
102
Power gain and stability of a loaded two-port
Therefore, at least one of the following conditions should be met: S 11 ΓG = 1
(3.84a)
S 22 ΓL = 1
(3.84b)
ΓL Γout = 1
(3.84c)
ΓG Γin = 1 .
(3.84d)
We immediately remark that a two-port having S 11 1 or S 22 1 has little practical interest, since they could not be safely measured when terminated on the normalization impedances (however this condition could occur in real devices, thus requiring e.g. a change of the normalization impedance from e.g. 50 Ω to a lower value, as it happens in large periphery power transistors). Unless stated differently, we will therefore assume that S 11 < 1 and S 22 < 1; therefore, if we take into account that Γ L and ΓG refer to passive terminations and have magnitude lower than 1, the first two equations of (3.84) are never satisfied. We therefore reduce to the analysis of the behaviour of Γ G Γin or ΓL Γout as a function of ΓL and ΓG . Note that the product Γ G Γin is the amplification of a wave experiencing a double reflection at port 1 (generator and input reflectance); similarly for the product ΓL Γout at port 2. Now, if ΓL Γout = 1 or ΓL Γout = 1 this implies linear instability, since in the presence of an independent wave generator an incident wave, after a loop, will superimpose in phase and with the same amplitude to itself, thus building up a response tending to infinity. This condition is never met if Γin ΓG < 1 or Γout ΓL < 1, since in such cases multiple reflections lead to a response converging for t . However, ΓG and ΓL are passive, therefore the two-port is unconditionally stable if:
| |≥
| |
|
|
|
| |≥
| |
|
→∞
for any passive ΓL , Γin < 1; or for any passive ΓG , Γout < 1.
| | | |
Using the Smith chart and taking into account that the transformation between ΓL (Γ G ) and Γin (Γ out ) is a conformal mapping between complex planes (also called linear fractional transformation or M¨obius transformation) transforming circles into circles, we have the following interpretation: 1. The circles ΓL < 1 ( ΓG < 1) are transformed into an image in the plane Γin (Γout ) consisting in a circle plus its interior or exterior, see Fig. 3.15; if the image falls within the Smith chart, we have unconditional stability (Fig. 3.15, above); if it partially falls outside the Smith chart, we have potential instability (Fig. 3.15, below). 2. Alternatively, the circle Γin < 1 ( Γout < 1) has as a counterimage a circle of the ΓL (Γ G ) plane and its interior or exterior. If the counterimage includes the whole ΓL (ΓG ) Smith chart, we have unconditional stability because certainly all ΓL (ΓG ) within the unit circle will yield a Γ in (Γout ) within the unit circle, see Fig. 3.17. If on the other hand the counterimage does not include the whole ΓL (ΓG ) Smith chart, we hve potential instability, see Fig. 3.16.
| |
| |
| |
|
|
103
3.6 Stability
Figure 3.15 Example
of unconditional stability (above) and conditional stability (below) in the input (output) reflection coefficient plane.
The counterimage of Γout = 1 in the plane Γ G is denoted as the input stability circle , whereas with output stability circle we denote the counterimage of Γin = 1 in plane ΓL .1 As shown in Example 3.8, the center and the radius of the output and input stability circles (plane Γ L and ΓG , respectively) are given by:
|
|
| |
∗ S 11 ∆∗S S 22 ∆S 2 S 22 2 S 12 S 21
− | | −| | | | RLC = |∆S |2 − |S 22|2 ΓLC =
1
(3.85a) (3.85b)
Notice that a stability circle should perhaps called more properly a stability circumference the image of the stable region is not necessarily a circumference and its interior, i.e. a circle, but may be a circumference and its exterior.
104
Power gain and stability of a loaded two-port
Figure 3.16 Output
stability circle: example of unconditional stability.
Figure 3.17 Input
stability circle: example of conditional stability.
and by:
∗ S 22 ∆∗S S 11 ∆S 2 S 11 2 S 12 S 21
− (3.86a) | | −| | | | . RGC = (3.86b) |∆S |2 − |S 11|2 Supposing that |S 11 | < 1 and |S 22 | < 1, we can immediately understand whether ΓGC =
the stable region identified by the stability circle corresponds to the region internal or external to the circle. In fact, the origin of the Γ G (ΓL ) plane corresponds in the Γin (Γout ) plane to the point S 11 (S 22 ), that we have assumed to lie within the unit circle (the ratio is that the two-port closed on the normalization resistances should be stable, otherwise the measurement itself of the scattering parameters would be impossible) Therefore the output (input) stability circle is the region of the plane ΓL (ΓG ) delimited by the circumferences described by (3.85) and (3.86) including the origin . According to whether the stability circle is internal or external to the Smith chart we can have the six cases shown in Fig. 3.18.
105
3.6 Stability
Example 3.8:
• Demonstrate Eq. (3.85) and (3.86). ◦ The relation (3.46) yielding Γ as a function of Γ is a linear fractional transin
L
formation between complex variables of the kind: w =
az + b cz + d
(3.87)
that transforms che circles of z plane into circles of w plane. The unit circle in w plane will therefore correspond to the condition:
and thus:
az + b cz + d
2
(az + b)(a∗ z ∗ + b∗ ) = =1 (cz + d)(c∗ z ∗ + d∗ )
∗ − cd∗ ) ∗ ∗ 2 2 ∗ (a b − c d) = |d| − |b| + z |z|2 + z (ab |a|2 − |c|2 |a|2 − |c|2 |a|2 − |c|2
(3.88)
.
(3.89)
Eq. (3.89) is the equation of a circle in the z plane, as it is clear if we sum and substract the factor
|c∗d − a∗b|2 , (|a|2 − |c|2 )2 to the left-hand side of (3.89), that becomes: c∗ d a2
a∗ b c2
− − | | −| | z
2
∗ ∗ 2 | | − |c|2 + (||ca|d2 −− a|c|b2|)2 .
= a2
(3.90)
From (3.90) we immediately obtain that the center C and radius R of the afore mentioned circle are given by:
− a∗b | | − | c |2 ad − cb R = |a|2 − |c|2 C =
c∗ d a2
From (3.46), comparing with (3.87) we obtain:
(3.91)
.
a = ∆S b = S 11 c = S 22 d = 1
− −
that, after substitution into (3.91), yield the first two equations (3.85). If we exchange ΓL with ΓG and Γin with Γout we similarly obtain Eqs. (3.86).
106
Power gain and stability of a loaded two-port
Figure 3.18 Stability
cases (a) e (b): unconditional stability; (c) (d) (e) (f): conditional stability. The Smith chart refers to Γ G (output stability circles) or Γ L (input stability circles).
3.6.2
Unconditional stability necessary and sufficient conditions Suppose a two-port is stable when closed by its normalization resistances (i.e. S 11 < 1 and S 22 < 1). A set of necessary and sufficient conditions for unconditional stability is given by the necessary (but not sufficient) condition on the
| |
| |
107
3.6 Stability
Linville or stability coefficent K : K =
1
− |S 22|2 − |S 11|2 + |∆S |2 > 1 2 |S 21 S 12 |
(3.92)
together with one of the following conditions:
|S 12S 21| < 1 − |S 11|2 |S 12S 21| < 1 − |S 22|2 |∆S | < 1
(3.93) (3.94) (3.95)
The most popular set probably is K > 1, ∆S < 1. Most CAD tools show the frequency behaviour of K ; if K < 1 the two port is potentially unstable, but also in regions where K > 1 potential instability can arise if the determinant of the scattering matrix is larger than one, see Example 3.9.
Example 3.9:
• Show and example where a two-port with K > 1 is potentially unstable. ◦ Consider a two-portwith the scattering matrix: 0.5292 j0.6643 0.1375 5.3756 + j2.9848 0.5918
S =
−
−
− j0.1346 − j0.5800
.
The Linville coefficient is K = 1.2787 > 1; however, the two-port is not unconditionally stable. For example, close port 2 with the passive load Γ L = 0.3762 + j0.5264; we have Γin = 1.0464 j0.4481 with magnitude larger than one. This is confirmed by the fact that ∆S = 1.8528 > 1, contrarily to what requested by condition (3.95).
−
− | |
The stability criteria are demonstrated in Sec. 3.6.3.
3.6.3
Proof of stability criteria
3.6.3.1
Output stability criterion The output stability circle is the region of plane Γ L delimited by the circumference with center Γ LC and radius RLC , see (3.85), including the origin. Two cases are possible: 1. ΓL = 0 lies outside the circumference, and the stable region is external to it (Fig.3.19). We have unconditional stability if the circumference defined by (3.85) lies completely outside the Smith chart, i.e. when ΓLC > 1 + RLC , implying a fortiori ΓLC 2 > RLC 2 . Substituting the relevant expressions
|
|
|
|
|
|
108
Power gain and stability of a loaded two-port
Figure 3.19 Case
(1): the stable region in the load reflection coefficient is external to the output stability circle.
(3.85) the previous inequality becomes:
|S 11∆∗S − S 22∗ |2 = |S 12S 21|2 + 1 − |S 11|2 |S 22|2 − |∆S |2
> S 12 S 21
|
|2
(3.96)
implying in turn:
− | | | 1
S 11
2
S 22
|2 − |∆S |2
> 0
that is satisfied only if S 22 > ∆S . This result allows to correctly choose the sign of the denominator of RLC , eliminating the absolute value, see (3.85). We start again from ΓLC > 1 + RLC , taking the square of both members and substituting the value of R LC we obtain:
| | | |
|
|
|S 11∆∗ − S ∗
22
S
|
2
>
and, exploiting Eq. (3.96):
|S 12S 21|
2
− | | |
+ 1
S 11
2
2
S 22
2
2
| − |∆S | + |S 12S 21
| |
K =
| − |∆S
2
>
S 22
2
|
2
2
| − |∆S | + |S 12S 21
|
2
. (3.97) This relation can be rewritten by using the Linville coefficient K defined in (3.63): 1
S 22
|
− |S 22|2 − |S 11|2 + |∆S |2 > 1 . 2 |S 21 S 12 |
(3.98)
We have therefore shown that the output unconditional stability implies K > 1; thus, this condition is necessary .
3.6 Stability
109
2. ΓL = 0 is internal to the circle, and therefore the stable region lies within the circumference (Fig. 3.20). Thus, we have unconditional stability if the circumference defined in (3.86) is completely internal to the Smith chart, i.e. 2 if ΓLC < RLC 1. This implies a fortiori ΓLC 2 < RLC . With analogy to case (1), e desume that this time the following condition should be verified:
|
|
−
|
|
|S 22| < |∆S | ; therefore condition |ΓLC | < RLC − 1 is equivalent to the pair: |ΓLC |2 < (RLC − 1)2 .
RLC
(3.99)
> 1
Developing the first of the (3.99) we have:
|
∆S 2
| − |S 22|2 − |S 12S 21|
2
− | | |
> S 12 S 21 2 + 1
|
|
S 11
2
S 22
|2 − |∆S |2
from which we get back condition K > 1. To impose the second of the (3.99) we start from the expression of R LC ; since ∆S > S 22 , we have:
| | | | |S 12S 21| > |∆S |2 − |S 22|2 .
Substituing this expression in the one imposing K > 1 with the explicit expression of K we obtain:
− | | − | 1
that is: 1
S 11
2
S 22
|2 − |∆S |2
> 2 S 12 S 21
− |S 11|2 > 2 |S 12S 21| − |∆S |2 − |S 22|2
and finally: 1
− |S 11|2 > |S 12S 21| .
|
|
> S 12 S 21
|
| (3.100)
Thus Eq. (3.100) must be verified together with condition K > 1. In case (1) the condition is implicitly satisfied if K > 1. Conditions K > 1 and 1 S 11 2 > S 12 S 21 can be easily shown to be not only necessary, but also sufficient, we only have to run backwards the above demonstration line.
−| |
3.6.3.2
|
|
Input stability criterion By exchanging port 1 and port 2 and the input and output reflection coefficients we immediately obtain:
1
K > 1 S 22 2 > S 12 S 21 .
− | | |
|
110
Power gain and stability of a loaded two-port
Case (2): the stable region of the load reflection coefficient is internal to the input stability circle. Figure 3.20
3.6.3.3
Input and output stability Since stability is global, the input and output stability criteria should be equivalent. Assembling the stability conditions obtained in the previous sections we obtain that a necessary and sufficient set for the unconditional stability of a two-port (with S 11 < 1 and S 22 < 1) turns out to be:
| |
| |
K>1
(3.101) 2
|S 12S 21| < 1 − |S 11| |S 12S 21| < 1 − |S 22|2 .
(3.102)
(3.103)
If K > 1 we can show that (3.102) implies (3.103) and viceversa (see Example 3.10).
Example 3.10:
• Show that set (3.101), (3.102) implies set (3.101), (3.103) and viceversa. ◦ Suppose that (3.102) is verified and that K > 1. In this case, we have incon-
ditional output stability, i.e. for every Γ L with magnitude < 1 whe have that Γin has magnitude < 1. Since S 11 < 1, ΓL = 0 (in correspondence of which Γin = S 11 ) certainly falls within the unit circle. This implies that the image circumference ΓL = 1 in plane Γin must have radius < 1 (if it were not so there would be loads that make potentially unstable the two-port at the input). Proceeding as discussed in Example 3.6, with application to (3.54), we obtain that
| |
| |
111
3.6 Stability
the image circle Γin ( ΓL = 1) has radius:
| |
Rin =
|S 12S 21| . |1 − |S 22|2|
(3.104)
This radius is < 1 if the following condition holds:
|S 12S 21| < |1 − |S 22|2| but we also have S 22 < 1; therefore it follows:
| |
|S 12S 21| < 1 − |S 22|2 i.e. (3.103) holds. Similarly, we show that if (3.101) holds, Eq. (3.103) implies (3.102).
A further condition, alternative to (3.103) or to (3.102) is obtained by summing these two last relationships:
|S 12S 21| < 1 − 12 |S 11|2 − 12 |S 22|2 . Taking into account that: ∆S = S 11 S 22
|
− S 21S 12| < |S 11S 22| + |S 21S 12|
and exploiting the previous equation we find:
|∆S | < |S 11S 22| + 1 − 21 |S 11|2 − 21 |S 22|2 = 1 = 1 − ( |S 11 | − |S 22 |)2 < 1 . 2 If Eq. (3.102) and (3.103) hold, then |∆S | < 1; it follows that if a two-port is
unconditionally stable we also have:
K > 1 ∆S < 1.
| |
(3.105)
Inversely, if ∆S < 1 and K > 1, we should have:
| |
−| | −| |
2 S 21 S 12 < 1
|
|
S 22
2
+ 1
S 11
2
from which we obtain that at least one of the two expressions must be true:
|S 21S 12| < 1 − |S 11|2 |S 21S 12| < 1 − |S 22|2 but, since if K > 1 they imply each other, both must be true. The set (3.105) can be therefore used to test the unconditional stability of the two-port.
112
Power gain and stability of a loaded two-port
3.7
One-parameter stability criteria The classical stability criterion, as summarized above, was, and still is, widely exploited in CAD tools; it is, however, somewhat inconvenient, since the simple condition K > 1 is necessary but not sufficient to ensure stability. In 1992 Edwards and Sinksky [4] proved that a condition on a single parameter is sufficient to assess unconditional stability; namely, they showed that the two port unconditional stability conditions can be put into biunique correspondence with one of the following: 1 S 11 2 ∗ ∆ + S 12 S 21 > 1 S 11 1 S 22 2 ∗ ∆ + S 12 S 21 > 1 S 22
−| | |S 22 − | | −| | µ2 = |S 11 − | | µ1 =
3.7.1
(3.106)
|
(3.107)
|
Proof of the single parameter criterium In this section we will establish a relationship between µ 1 and the classical stability parameters K and B1 . We start from the definition of µ1 , which can be written as: 2
|S 22 − S 11∗ ∆| = 1 − µ|S 111| − |S 12S 21|
(3.108)
Since:
|S 22 − S 11∗ ∆|2 = (|S 22|2 − |∆|2)(1 − |S 11|2) + |S 12|2|S 21|2
(3.109)
squaring both terms of (3.108), subtracting on both sides the factor S 12 S 21 2 , and using (3.109) we have:
|
|
|2 − |∆|2 1 − |S 11|2 = 2 2 1 − |S 11 |2 − 2|S ||S | 1 − |S 11|
S 22
=
|
µ21
21
12
µ1
and therefore: 2
|S 22| − |∆|
2
=
1
− |S 11|2 − 2 |S 21||S 12| µ21
µ1
(3.110)
Since:
|S 22|2 − |∆|2 = −|S 21||S 12|(2K − B1) after some rearrangement, (3.110) gives: B1 µ21
− µ21 + 2K − B1 = 0
(3.111)
113
3.7 One-parameter stability criteria
This equation has always real solutions since its discriminant is positive:
∗ ∆|2 S 22 − S 11 | 1 − B1 (2K − B1 ) = |S 12|2|S 21|2 > 0 moreover, rearranging (3.108) one obtains: b1 s22 s∗11 ∆ = + 1 > 1 µ1 s12 s21
| − | | |
thus implying that only the solution of (3.111) with the plus sign is valid: B1
µ1 = 1+
− 1
(3.112)
2KB1 +
B12
A direct contour plot of µ1 as a function of K and B1 from (3.112) shows that the region with K > 1 and B 1 > 1 where µ 1 is real, corresponds to µ 1 > 1, and viceversa (i.e. µ1 < 1 anywhere else) [4]. (The same conclusions can obviously shown to hold for the set µ 2 , K and b 2 .) A straightforward algebraic proof of the above equivalence is now provided. We will now give a simple demonstration of the equivalence, which has been shown to hold through graphical inspection, of the two conditions K > 1 and B1 > 1 with (3.106). We first notice that, from (3.106), it is trivial to see that µ1 < 1 implies S 11 > 1 and viceversa so that, when the two-port cannot be unconditionally stable because S 11 > 1, the parameter µ1 correctly predicts conditional stability. We can therefore focus on the case S 11 < 1 or µ 1 > 0. We now prove that µ1 implies K > 1 and B1 > 1. In fact, condition µ1 > 1 can be expressed as:
| |
| |
| |
|S 22 − S 11∗ ∆| < 1 − |S 11|2 − |S 12S 21| = |S 21||S 12|(B1 − 1)
(3.113)
from which we immediately find that B 1 > 1. Then, squaring (3.113), and using (3.109), we find that:
|2 − |∆|2 1 − |S 11|2 + |S 12S 21|2 < 2 < 1 − |S 11 |2 − 2 |S 12 S 21 | 1 − |S 11 |2 + |S 12 S 21 |2 Since |S 11 | < 1 and therefore 1 − |S 11 |2 > 0, it follows that: |S 22|2 − |∆|2 < 1 − |S 11|2 − 2 |S 12S 21|
|
S 22
(3.114)
(3.115)
(3.116)
and therefore: 1
− |S 11|2 − |S 22|2 + |∆|2 > 2 |S 12S 21|
i.e., from the definition of the Rollet parameter, K > 1. By reversing the above proof, we show now that B1 > 1 and K > 1 implies µ1 > 1. Starting from (3.116), we obtain (3.115) and then (3.114) by multiplying (3.115) by the factor 1 S 11 2 (positive because B 1 > 1), and adding S 12 S 21 2
−| |
|
|
114
Power gain and stability of a loaded two-port
to both sides. Taking the square root of the two terms, and using (3.109), we have:
|S 22 − S 11∗ ∆| ≤ 1 − |S 11|2 − |S 12S 21| but, since B1 > 1, we have 1 − |S 11 |2 − |S 12 S 21 | > 0 and finally obtain (3.113),
which is equivalent to condition µ 1 > 1. By means of the same procedure, we obtain that µ 2 > 1 implies K > 1, b 2 > 1 and viceversa . The reciprocal implication of the conditions µ1 > 1 and µ2 > 1 can be stated equivalently by showing that the condition K > 1, B 1 > 1 implies b2 > 1. This demonstration has already been carried out in [4], but a simpler proof will be given here, which also has the advantage of immediately showing that if K > 1 conditions B 1 > 1 and b2 > 1 mutually imply each other. We start by evaluating the product ( B1 1)(b2 1); a direct computation shows that:
−
−
(B1
− 1)(b2 − 1) = 1 − |S 11 |2 − |S 22 |2 + |S 11 |2 |S 22 |2 + |S 21 |2 |S 12 |2 = + |S 12|2|S 21|2 2 |S 21 ||S 12 | − |S 21 ||S 12 | |S 11 |2 + |S 22 |2 = − |S 12|2|S 21|2 1 − |S 11 |2 − |S 22 |2 + |∆|2 − 2|S 21 ||S 12 | = + |S 12|2|S 21|2 ∗ S ∗ ) S 21 ||S 12 | |S 11 |2 + |S 22 |2 + 2 (S 11 S 22 S 21 | 12 + = 2 2 |S 12| |S 21| ∗ √ S 21S 12 2 ∗ S ∗ + S 22 S 11 S 21 K − 1 12 =2 |S 21||S 12| + |S 12|2|S 21|2 This implies that, if K > 1, (B1 − 1)(b2 − 1) > 0, i.e. B 1 and b 2 are both either
larger or smaller than unity. Therefore condition K > 1, B 1 > 1 implies b2 > 1, and thus µ 1 > 1 implies µ 2 > 1.
3.8
Two-port stability and power matching
3.8.1
Power matching and maximum gain: can it be always realized? To maximize the power transfer between port 1 and 2 we must impose simultaneous matching at the two ports:
Γin = Γ∗L . Γout = Γ∗G
(3.117)
3.8 Two-port stability and power matching
115
Replacing the expressions of Γ in (3.46) and Γout (3.50) we obtain the following nonlinear system:
S 11 ∆S ΓL 1 S 22 ΓL ∗ ∆∗ Γ∗ . S 22 ∗ S S ΓL = Γout = ∗ 1 S 11 Γ∗G Γ∗G = Γin =
− − − −
Substituting the second equation into the first one we obtain a second order equation in ΓG yielding its optimum value; similarly we can obtain a secondorder equation for the optimum ΓL . Solving we obtain: ΓGopt ΓLopt
± − | | ± − | |
1 = B1 2C 1 1 = B2 2C 2
B12
4 C 1
2
B22
4 C 2
2
(3.118)
where the coefficients B 1 C 1 B2 e C 2 aleady defined in (3.76), (3.77), (3.69) and (3.70) are conveniently reported again here: (3.119) − |S 22|2 + |S 11|2 − |∆S |2 2 2 2 B2 = 1 + |S 22 | − |S 11 | − |∆S | (3.120) ∗ C 1 = S 11 − ∆S S 22 (3.121) ∗ C 2 = S 22 − ∆S S 11 . (3.122) The choice between signs + and − should grant that the optimum reflection B1 = 1
coefficients have magnitude less than one. As shown in the Example 3.11, this is possible if and only if the two-port is unconditionally stable. In that case the optimum reflection coefficient are expressed by: ΓGopt ΓLopt
− − | | − − | |
1 = B1 2C 1 1 = B2 2C 2
B12
4 C 1
2
B22
4 C 2
2
.
(3.123)
(3.124)
and the maximum gain (Example 3.12) is given by the expression, already introduced: GMA X =
|S 21| (K − |S 12|
K 2
− 1) .
(3.125)
This corresponds to the maxima of the transducer, available, and operating gain, as already stated for simultaneous power matching the three conditions coincide.
Example 3.11:
• Show that the two-port simultaneous matching is possible only if the two-port is unconditionally stable.
116
Power gain and stability of a loaded two-port
stable two-port K > 1, implying: ◦ In an unconditionally 1
− |S 22|2 − |S 11|2 + |∆S |2
2
> 4 S 21 S 12
|
|2
i.e. summing and subtracting the term:
− | | | | − | | | | − | | | |
4 1 we obtain:
−| 1
S 22 2 + S 11
| | |2 − |∆S 2
S 22
2
2
S 11
2
∆S 2
> 4 S 21 S 12 2 + 4 1
S 22
2
S 11
|2 − |∆S |2
. (3.126)
Taking into account (3.96) rewritten exchanging port 1 and 2:
|S 22∆∗S − S 11∗ |2 = |S 12S 21|2 + 1 − |S 22|2 |S 11|2 − |∆S |2 we obtain that(3.126) is equivalent to:
−| 1
S 22 2 + S 11
| | |2 − |∆S |2
2
− S 11∗ |2 .
> 4 S 22 ∆∗S
|
The tow members of the above equation clearly are always positive. Taking the square root we obtain:
|B1| = 2 |C 1 |
− | 1
S 22 2 + S 11
| | |2 − |∆S |2 ∗| 2|S 22 ∆∗S − S 11
Exchanging again ports we obtain that if K > 1 then:
> 1 .
|B2| > 1 . 2 |C 2 |
(3.127)
(3.128)
We will now show that if the two-port is unconditionally stable then B1 > 0; notice that if this occurs the stability circle lies completely outside the Smith chart, or covers it completely (see cases (a) and (b) of Fig. 3.18). In the first case we certainly have S 11 > ∆S and therefore 2 S 11 2 ∆S 2 > 0; summing this equation to (3.92) we immediately obtain B1 > 0. In the second case S 11 < ∆S and the stability circle has radius larger than one; thus, from the second equation in (3.86), we obtain ∆S 2 S 11 2 < S 12 S 21 . Since in the case of unconditional stability ((3.94) holds, 1 S 22 2 > S 12 S 21 , and we obtain:
| | | |
|
| −| |
| | −| | | | −| | | | |∆S |2 − |S 11|2 < 1 − |S 22|2
and therefore: B1 = 1
− |S 22|2 − |∆S |2 + |S 11|2 > 0 .
| | | |
117
3.8 Two-port stability and power matching
Similarly we can show that unconditional stability implies B2 > 0. Taking into account of the results obtained so far we can write:
ΓGopt
B1 = 2 C 1
|
ΓLopt =
B2 2 C 2
| | | ± − | | 1
| |
1
1 1
±
4 C 1 2 B12
− 4 BC 22
2
.
2
From those formulae, taking into account (3.127) and (3.128), we have that if the two reflectances have magnitude less than one we need to select the solutions with minus sign in the formulae; also the choice of the sign in (3.118) is forced and the optimum reflectances maximizing gain are: ΓGopt ΓLopt
− − | | − − | |
1 = B1 2C 1 1 = B2 2C 2
B12
4 C 1
2
B22
4 C 2
2
.
(3.129)
(3.130)
Therefore if the two-port is unconditionally stable the optimum terminations exist and are uniquely defined.
As a byproduct of the previous example, we can identify a last set of necessary and sufficient stability conditions as the set K > 1 together with one of the conditions B 1 > 0, B2 > 0.
Example 3.12:
• Derive the expression of the maximum gain when simultaneous power matching at the two ports is achieved. ◦ We substitute in G the reflectances Γ = Γ and Γ = Γ , thus obtaint
L
Lopt
G
Gopt
ing the maximum transducer gain G tMAX (see (3.53), (3.71) and (3.117)): 2 | |2 (1 − |ΓG1 −|2|)Γ|1L− S | 22ΓL |2 opt
GtMAX = S 21
opt
opt
Exploiting (3.46) we have: ΓGopt =
S 11 ∆S ΓLopt ∗ 1 S 22 ΓLopt
− −
and therefore: 1
2 2 − |ΓG |2 = |1 − S 22ΓL|1 −| S −22|ΓS L11 −|2∆S ΓL | . opt
opt
opt
opt
(3.131)
118
Power gain and stability of a loaded two-port
Using this last expression (3.131) becomes: 2 | |2 |1 − S 22ΓL 1 |−2 −|ΓL|S 11| − ∆S ΓL |2 = 1 − |ΓL |2 = |S 21 |2 1 − |S 11 |2 + |ΓL |2 (|S 22 |2 − |∆S |2 ) − 2(ΓL opt
GtMAX = S 21
opt
opt
| |2 N D
opt
opt
opt
= S 21
C 2 )
(3.132) where C 2 was defined in (3.70). We will now express the different terms of (3.132) as a function of K . From (3.96) and (3.70) we have:
|C 2|2 = |S 12S 21|2 + 1 − |S 11|2 |S 22|2 − |∆S |2
(3.133)
while from the first of (3.69) and from (3.63) we easily obtain the following relationships: 1 1
− |S 11|2 = 2K |S 21S 12| + |S 22|2 − |∆S |2 − |S 11|2 = B2 − (|S 22|2 − |∆S |2) .
Summing and subtracting we obtain:
− |S 11|2 = K |S 21S 12| + B22 |S 22|2 − |∆S |2 = B22 − K |S 21S 12| 1
(3.134)
(3.135)
|
(3.136)
that, substituted into (3.133), yield: 2
2
B2 + 2
|C 2| = |S 12S 21| + K |S 21S 12| B2 = 2 − |S 12 S 21 |2 (K 2 − 1) . 4
B2 2
− K |S 21S 12
From (3.124), noticing that from (3.136) we obtain: B22 4
− |C 2|2 = |S 12S 21|2(K 2 − 1)
we get: ΓLopt =
B2
− 2|S 12S 21|√ K 2 − 1
(3.137)
2C 2
from which, since the numerator of the right-hand side is real, we also have that also ΓLopt C 2 is real and in particular: 2 (ΓLopt C 2 ) = 2ΓLopt C 2 = B2
− 2|S 12S 21|
K 2
− 1 .
(3.138)
119
3.8 Two-port stability and power matching
Moreover, using (3.136) and (3.137) we have: 2
1
− |ΓL |
2
=
opt
|C 2|2 − B42 + B2|S 12S 21|
= 2 S 12 S 21
|
=
|
K 2
−
B2 1 22 B2 4
K 2
|C 2|2
− 1 − |S 12S 21|2(K 2 − 1)
− |S 12S 21| K 2 − 1 − |S 12S 21|2(K 2 − 1)
|√ K 2 − 1 . B2 + |S 12S 21 | K 2 − 1 2 2 S 12 S 21
|
=
(3.139)
Using (3.134), (3.135), (3.138) and (3.139) the denominator D of (3.132) can be rewritten as: B 2 B2 D = K S 12 S 21 + + ΓLopt 2 ( K S 12 S 21 )+ 2 2 B2 + 2 S 12 S 21 K 2 1 =
−
= (1
|
|
|
| | √ | −
− |
|
− |ΓL |2)(K |S 12S 21| − B22 ) + 2|S 12S 21| opt
K 2
− |ΓL |2)(K |S 12S 21| − B22 + B22 + |S 12S 21| √ = |S 12 S 21 |(1 − |ΓL |2 )(K + K 2 − 1) . = (1
opt
−1=
K 2
− 1) =
opt
After substituting this expression into (3.132) we finally obtain an explicit expression for the maximum transducer gain, identical to (3.67) and (3.74): GtMAX =
3.8.2
|S 21| |S 12|
1 |S 21| (K − √ = |S | K + K 2 − 1 12
K 2
− 1) .
(3.140)
Managing conditional stability If the two-port is not unconditionally stable we should identify the regions of plane ΓG and ΓL granting values of Γout and Γin within the Smith chart, i.e. the stability of the two-port with a given set of loads. Tee region sought for clearly are given by the intersection of the stability circles with the Smith chart. With reference e.g. to Γ G we should:
identify the circumference limiting the input stability circle; decide if the stability circle is the external or internal region to the circumference; remember that the stable region includes the origin of the Γ G plane; identify the intersection of the above region with the Smith chart.
120
Power gain and stability of a loaded two-port
We similarly proceed to find the region of the Γ L plane corresponding to stable behaviour. In the case of conditional stability, the conditions corresponding to conjugate matching at both ports and maximum gain do not exist any more; in certain conditions, in fact the gain has a minimum within the stable region, and tends to infinity on the boundary between the stable and the unstable region. A parameter called MSG (Maximum Stable Gain) is often introduced, this corresponds to the maximum gain for a two-port where K = 1
S 21 GMS G = . S 12
(3.141)
Since K depends on frequency and is typically (in transistors) smaller than one at low frequency and larger than one (but asymptotically tending to one) at high frequency, there is one (or in some cases more than one) frequency in wich K = 1 exactly. When plotting the MAG as a function of frequency the MSG is usually shown in the regions where the device is not unconditionally stable (and, as a consequence, the MAG is not defined).
3.8.3
Stability circles and constant gain contours There is a close relationship between the stability circles and the constant gain contours of a loaded two-port. In particular:
the constant operational gain contours in plane Γ L are related to the output stability circle ( Γin = 1 circles in plane ΓL ); the constant available power gain contours in plane Γ G are related to the input stability circle ( Γout = 1 circles in plane ΓG );
| |
|
|
For the sake of definiteness, let us refer to the level curves of the available power gain in plane ΓG . For an unconditionally stable two-port (Figure 3.21) the gain has a maximum within the Smith chart and is singular outside it (i.e. for active terminations) on the boundary stability circle, that is completely outside the unit circle. In the previous case K > 1; if K = 1 we are in a limiting condition, the stability circle is tangent to the unit circle of the Smith chart and the constant gain level curves are tangent to the same tangent point (Fig. 3.22). In this situation the maximum gain occurs in the limit on the tangent point and corresponds to the MSG. Finally, if the two-port is conditionally stable the gain goes to infinity in the part of the stability circumference internal to the Smith chart (Fig. 3.23). In theory, this gain amplification could be exploited in circuit design but the choice of a load too close to the unstable region is dangerous due to possible technological fluctuations that may lead the circuit to oscillate. The load and generator should be therefore chosen by allowing enough stability margin. The choice of terminations could be in practice always influenced by other specifications different
3.8 Two-port stability and power matching
121
Figure 3.21 Constant
gain level curves and input stability circle for an unconditionally
stable two-port.
Figure 3.22 Constant
gain level curves and input stability circle for a two-port in the
limit of stability.
from the maximum small-signal gain, i.e. specifications on noise or (as in power amplifier) on the maximum power; such cases will be discussed in the relevant chapters.
122
Power gain and stability of a loaded two-port
Figure 3.23 Constant
gain level curves and input stability circle for a two-port that is
conditionally stable.
3.8.4
Unilateral two-port In a unilateral device the internal feedback between port 2 and port 1 is zero, i.e. S 12 = 0. In many semiconductor trasistors this condition is almost verified, and some devices can be (as a first approximation) considered as unilateral. This can be quantified by the unilaterality index U defined as: U =
|S 11S 12S 21S 22| . (1 − |S 11 |2 )(1 − |S 22 |2 )
For an exactly unilateral device U = 0. We can show that the ratio between the MAG and the maximum unilateral gain (MUG, see (3.142)) satisfies the inequality: (1 + U )−2 < MAG/MUG < (1
− U )−2
i.e., for small U , the error introduced by assuming the device unilateral is of the order of 4U . We should however stress that the unilateral approximation neglects stability problems; in fact, for a device with S 11 < 1 and S 22 < 1 the unilateral approximation is always unconditionally stable (the Linville parameter K tends to infinity in this case).Therefore the unilateral approximation is meaningful only if the original device is unconditionally stable. For unilateral device the conjugate matching at both ports simplifies since Γin = S 11 , Γout = S 22 ; therefore we can simply impose the two separate condi-
| |
| |
123
3.9 Examples
|S 11| 1 2 3 4 5 6 7 8 9
0.2 0.75 1.05 0.5 0.95 0.69 0.1 1.2 0.1
ph(S 11 ), degrees 20 -60 20 0 -22 -123 0 0 0
|S 12| 0.05 0.3 0.05 0.025 0.04 0.11 0 0 0
ph(S 12 ), degrees 120 70 120 180 80 48 0 0 0
|S 21| 3 6 3 2 3.5 1.29 0 0 0
ph(S 21 ), degrees 30 90 40 0 165 78 0 0 0
|S 22| 0.5 0.5 0.5 0.1 0.61 0.52 0.3 0.3 1.3
ph(S 22 ), degrees -50 60 -50 0 -13 -77 0 0 0
Table 3.4. Scattering parameters of the analiyzed two-ports.
tions:
∗ ΓG = S 11 at the input, and
∗ ΓL = S 22 at the output. Similar simplifications occur in the gain expression; as already seen the maximum available gain of a unilateral device reads: Gumax
S 21 |2 | = . (1 − |S 11 |2 )(1 − |S 22 |2 )
(3.142)
called MUG (Maximum Unilateral Gain).
3.9
Examples
3.9.1
Stability and gains at constant frequency In this section we analyze some two-ports whose scattering parameters are known, see Table 3.4. Out of the nine cases considered, the last three are unidirectional devices. Examples include cases deliberately not common in practice or anomalous. Consider to begin with the first six examples of non-unidirectional two-ports. Table 3.5 shows the values of the center (modulus and phase) and the radius of the input and output stability circles. Note that in case 4, ∆S = S 22 and the radius of the output stability circle is infinity. Table 3.6 shows the values of K and ∆S for the cases in which there is unconditional stability (denoted by ST, the remaining UNST). We also report the optimal value of the source and load terminations and the value of maximum gain. The following comments apply:
| | | |
| |
124
Power gain and stability of a loaded two-port
|ΓGC | ph(ΓGC ), 1 2 3 4 5 6
3.33 0.10 1.10 2.04 1.07 1.37
degrees 160 107 -19 0 30 127
RSC
|ΓLC |
6.70 0.44 0.23 0.21 0.24 0.34
2.40 0.26 1.02 undef. 3.45 1.74
ph(ΓLC ), degrees 50 -36 42 undef. 71 86
RLC 0.80 0.41 0.74 inf 3.12 0.69
Table 3.5. Values of the center and radius of the stability circles for the non-unidirectional
two-ports in Table 3.4.
1 2 3 4 5 6
K
|∆ |
2.57 1.34 0.34 7.50 0.19 1.12
0.249 2.156 0.673 0.1 0.572 0.254
S
Tipo ST UNST UNST ST UNST ST
|Γ
ph(ΓGopt ), degrees 0.10 -20 undef. undef. undef. undef. 0.50 0 undef. undef. 0.88 127 Gopt
|
|Γ
ph(ΓLopt ), degrees 0.48 50 undef. undef. undef. undef. 0.07 0 undef. undef. 0.82 86 Lopt
GMAX , dB 10.8 undef. undef. 7.3 undef. 8.6
|
Table 3.6. Coefficients for the calculation of the stability for the bi-directional two-ports;
Examples 1, 4, 6 are unconditionally stable; examples 2, 3, 5 conditionally stable.
If (case 1) K > 1 and ∆S < 1, the device is unconditionally stable. In cases 2 and 3 we have potential instability because at least one of the stability conditions is violated. In case 3 in particular we also have S 11 > 1. The parameters of the maximum gain and simultaneous power matching are therefore not defined. In case 4 we have unconditional stability, since K > 1 and ∆S < 1. In case 5 we have potential instability because K < 1, the gain and power matching parameters are not defined. Finally, in case 6 K > 1 and ∆S < 1, thus the two-port is unconditionally stable.
| |
| |
| |
| |
Examples 7–9 refer to unidirectional two-ports, always unconditionally stable if S 22 < 1 and S 11 < 1. Therefore, the example 7 is unconditionally stable, examples 8 and 9 potentially unstable. Note that in the latter two cases we have K < 1, and obviously we always have ∆S = S 11 S 22 < 1 since S 11 < 1 and S 22 < 1. Note that in Example 7, the device has no gain, ie its MAG is, in dB, equal to infty and 0 in natural units (shown in the table).
| |
| |
| |
| | |
−
|
| |
125
3.9 Examples
K 7 8 9
+
∞ −∞ −∞
|∆ |
Type
0.03 0.36 0.13
ST UN U NST UN U NST
S
|Γ
ph(ΓGopt ), degrees 0.1 0 indef. indef. indef. indef. Gopt
|Γ
ph(ΓLopt ), degrees 0.3 0 indef. indef. indef. indef.
|
Lopt
|
GMAX
0 indef. indef.
Table 3.7. Coefficients for the stability evaluation of unidirectional two-ports in Table 3.4.
f , f , GHz 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000
|S 1111|
0.949 0.821 0.648 0.512 0.472 0.464 0.441 0.411 0.454 0.551
φ11 -29.8 -59.8 -94.2 -133.0 -165.2 176.0 158.2 127.5 91.4 66.6
|S 2121|
4.825 4.531 4.092 3.516 3.025 2.714 2.505 2.321 2.093 1.836
φ21 151.1 123.8 97.6 73.9 54.7 38.4 22.1 4.0 -15.1 -34.5
|S 1212|
0.038 0.070 0.092 0.102 0.108 0.118 0.134 0.151 0.168 0.181
φ12 72.1 56.0 41.4 30.5 25.3 23.7 20.2 15.0 7.0 -2.8
|S 2222|
0.781 0.696 0.600 0.518 0.444 0.367 0.302 0.281 0.300 0.328
φ22 -14.4 -28.9 -42.4 -51.8 -57.8 -65.4 -80.8 -105.9 -134.2 -169.8
Table 3.8. Scattering parameters from 1 to 10 GHz of an active microwave device; the
phase is in degrees.
3.9. 3.9.2 2
Stab Stabil ilit ityy and and gain gainss as a fun funct ctio ion n of freq freque uenc ncyy Table 3.8 shows the frequency behaviour of a real device (MESFET NEC) measured at intervals of 1 GHz. Although the device is almost unilateral (i.e. S 12 12 is small), the stability parameters (K ( K and and ∆S ) evaluated as a function of frequency (Fig. 3.24 ) clearly shows that the device is potentially unstable for frequencies below 5 GHz. Such potential instability at low frequency is typical of devices operating in the microwave field, and is related to the decrease of S 21 21 with frequency. In Figure 3.25 we show (in semi-logarithmic scale) the maximum, unilateral and maximum stable gains (MAG, MUG, MSG), and the squared modulus of S 21 21 , which represents the operational gain when the device is closed on its reference impedances. Notice that the MAG coincides with the MSG at the limit frequency corresponding to the transition between unconditional and conditional stability; beyond that frequency the MAG is undefined and the MSG is shown instead. Also note that, in the frequency range in which the device is stable, the MSG is greater than the MAG; this was expected since the MSG refers to a device brought to the edge of instability. The unilateral gain MUG is always quite different from the MAG, showing that the device is not, actually, unilateral. Finally, the gain on the reference impedances, which do not correspond, in the stable frequency
| |
126
Power Power gain and stability stability of a loaded two-port two-port
Figure 3.24 Frequency
K and ∆S for the device in Table 3.8. The behaviour of K potentially unstable region is shown in gray.
range, to the optimum condition, is always lower than the MAG, as expected. In the unstable region the maximum gain is obviously infinite, so the MSG should be understood as a figure of merit of the device, not the maximum gain that may be actually achieved. Suppose now that we modify the device so as to make it more unilateral, for example by dividing S 12 12 by 10. The new device thus obtained, while retaining all other parameters unchanged, presents different characteristics. The region of instability moves to lower frequency, see Fig. 3.26, while the unilateral gain comes very close to the MAG, see Fig. 3.27. Although in this case the device is, at high frequency, virtually unilateral, we cannot neglect its potential instability at low frequency, which can give rise to spurious oscillations at frequencies much lower than the operating band. Therefore, it is seldom convenient to assume that an active device is unilateral; on the other hand, it is often necessary to stabilize devices at low-frequency, as discussed in the chapter devoted to linear amplifiers.
3.10 3.10
Ques Questi tion onss and and probl roblem emss 1. P The P The open-circuit voltage of a real generator is V 0 = 10 V and the internal impedance is Z is Z G = 50 + j + j50 50 Ω. What is the generator available power? What is the load impedance yielding power match to the generator?
127
3.10 Questions and problems
Frequency requency b ehaviour ehaviour of the gain of the devices devices in Table 3.8. The unstable unstable region is in gray. Notice that in that region the MAG is not defined.
Figure 3.25
Figure 3.26 Frequency
K and ∆S for the device in Table 3.8, but with behaviour of K S 12 12 made 10 times smaller. The potentially unstable region is in grey.
128
Power gain and stability of a loaded two-port
Figure 3.27 Frequency
behaviour of the gain of the devices in Table 3.8, with S 12 modified. The unstable region is in gray. Notice that in that region the MAG is not defined.
2. P Suppose the normalization impedance is 50 Ω. Locate on a Smith chart (approximately) the following impedances: Z L1 = 50 Ω; Z L2 = 50 + j50 Ω; Z L3 = 50 j50 Ω; Z L4 = 100 Ω; Z L5 = 25 Ω; Z L6 = 0 Ω; Z L7 = Ω. 3. P A transmission line with length equal to λg /4 is loaded with a 100 Ω impedance. The characteristic impedance is 50 Ω. Locate on the Smith chart the load impedance and input impedance. What is Z i ? This structure is called a quarter-wavelength transformer . 4. P A resistive two-port has the following impedance matrix:
−
∞
Z = R
21 12
Sketch a possible structure (implementing the above impedance matrix and evaluate the scattering matrix (assume the normalization impedance R0 = R) . 5. P A reactive two-port has the following impedance matrix: Z = jX
21 12
Evaluate the scattering matrix assuming R 0 = X and check that the properties of the S-matrix of a lossless two-port are verified.
3.10 Questions and problems
129
6. P A real generator has internal impedance Z G = 50 j50 Ω and open circuit voltage V 0 = 10 V. Assuming R0 = 50 Ω derive the power wave equivalent circuit (ΓG and b 0 ). 7. P A load exhibits a reflection coefficient Γ equal in magnitude to -10 dB. What part (in %) of the incident power is reflected? 8. P A real generator with Γ G = 0.2 and b0 = 1 W1/2 is connected to a load with ΓL = 0.5. Evaluate the power delivered to the load and the maximum available power of the generator. 9. P A loaded two-port has the following characteristics: P in = 10 mW; P av,in = 20 mW; P L = 100 mW; P av,out = 300 mW. Evaluate the two-port gains G op , Gav , Gt . 10. P A two-port has the following scattering matrix (R0 = 50 Ω):
−
0 0 10 0
S =
Evaluate the two-port MAG. Is the two-port unilateral? 11. P A two-port has the following scattering matrix (R0 = 50 Ω): S =
12.
13. 14. 15.
0.1 0.01 10 0.1
Compute the input and output reflection coefficients when the two-port is loaded on 100 Ω. Q Maximum power transfer between generator and load through a two-port implies simultaneous power matching at both ports. Is this condition always possible? P A two-port has K = 2, S 21 = 15(1 + j) and S 12 = 0.1. Evaluate the twoport MAG and MSG. Assume the two-port is unconditionally stable. Q A two-port has K = 2.5, ∆S = 1.5. Is the two-port unconditionally stable? P Discuss the stability (according to the one- and two-parameter criteria) of the two-port with scattering matrix:
| |
j0.1 10 S = 0.1 0.1
.
Suppose now to exchange ports 1 and 2, the new scattering matrix becomes: S =
0.1 0.1 10 j0.1
.
Does the 2-port stability change? 16. Q Discuss the stability (according to the one- and two-parameter criteria) of the unilateral two-port with scattering matrix: S =
j1.1 0 5 0.1
17. Q Is a unilateral device always unconditionally stable?
130
Power gain and stability of a loaded two-port
18. Q Suppose a device in unconditionally stable above f 0 and potentially unstable below f 0 . Qualitatively sketch the behaviour of the device MAG and MSG as a function of frequency. 19. Q Consider two passive two-ports, one reactive (lossless), the other resistive (lossy). What kind of property do we expect from their stability factors?
References
[1] [2]
[3] [4]
[5]
[6] [7] [8] [9] [10] [11] [12] [13]
W. H. Ku, ”Unilateral Gain and Stability Criterion of Active Two-Ports in Terms of Scattering Parameters”, Proceedings of IEEE , vol. 54, pp. 1966-1967, November 1996. D. Woods, ”Reappraisal of the Unconditional Stability Criteria for Active 2-Port Networks in Terms of S Parameter”, IEEE Transactions on Circuits and Systems , vol. 23, No. 2, pp. 73-81, February 1976. E. F. Bolinder, ”Survey of Some Properties of Linear Network”, IRE Transactions on Circuit Theory , pp. 70-78, September 1957. M. Lee Edwards and J. H. Sinksky, ”A New Criterion for Linear 2-Port Stability Using a Single Geometrically Derived Parameter”, IEEE Transactions on Microwave Theory and Techniques , vol. 40, No. 12, pp. 2303-2311, December 1992. G. Lombardi, B. Neri, ”Criteria for the Evaluation of Unconditional Stability of Microwave Linear Two-Ports: a Critical Review and New Proof”, IEEE Transactions on Microwave Theory and Techniques , vol. 47, No. 6, pp. 746-751, June 1999. R. P. Meys, ”Review and Discussion of Stability Criteria for Linear 2-Ports”, IEEE Transactions on Circuits and Systems , vol. 37, No. 11, pp. 1450-1452, November 1990. K. Kurokawa, ”Power Waves and the Scattering Matrix”, IEEE Transactions on Microwave Theory and Techniques , vol. 13, No. 3, pp. 194-202, March 1965. T. T. Ha, ”Solid State Microwave Amplifier Design”, New York Wiley , Appendix 1-4, pp.285-296, 1981. G. Gonzales, ”Microwave Amplifier Design”, New Englewood Cliff , 1984. J. M. Rollett, ”Stability and Power-Gain Invariants of Linear Twoports”, IRE Transactions on Circuit Theory , vol. 9, pp. 29-32, March 1962. J. M. Mason, ”Power Gain in Feedback Amplifiers”, Transactions of the IRE , vol. 1, pp. 20-25, June 1964. J. M. Mason, ”Some Properties of Three-Terminal Devices”, IRE Transactions on Circuit Theory , vol. 4, pp. 330-332, December 1957. G. E. Bodway, ”Two Port Power Flow Analysis Using Generalized Scattering Parameters”, The Microwave Journal , pp. 61-69, May 1967.
131
4
Directional couplers and power dividers
4.1
Coupled quasi-TEM lines We define coupled transmission lines multiconductor lines formed by N conductors plus a ground plane. Multiconductor lines have different applications in microwave and high-speed logical circuits:
As multiconductor buses, suitable to transmit of N parallel digital signals (high-speed digital circuits), in which case of course the coupling between lines is unwanted (i.e. translates into crosstalk between the lines); as coupled structures used in many distributed analog components, such as directional couplers, coupled line filters, and so forth.
In analog circuit components coupled lines are usually made of two or four conductors; common are for example two- or four-conductor coupled microstrips (see Fig. 4.1) and the coupled coplanar lines (see Fig. 4.2).
Figure 4.1 Two
or four-conductor coupled microstrips.
Figure 4.2 Coupled
coplanar lines.
132
4.1 Coupled quasi-TEM lines
133
Figure 4.3 Scheme
4.1.1
of coupled lines and elementary equivalent circuit.
Analysis of symmetrical coupled lines In general, a multiconductor line with N conductors plus ground supports N TEM or quasi-TEM propagation modes. The analysis will be limited here to twoconductor lines (plus ground), operating in sinusoidal steady state. Therefore, voltages and currents will be indicated by their associated phasors. Two coupled lines have voltages V 1 (z) and V 2 (z) and currents I 1 (z) and I 2 (z), as shown in Fig. 4.3. Voltages and currents can be combined into a vector of tensions V and currents I such that: V =
V 1 V 2
,
I =
I 1 I 2
.
The line is defined, per unit length, by specific parameters that describe not only the inductive and capacitive phenomena typical of a single line (per unit length inductance and capacitance) but also the capacitive coupling (mutual capacitance) or the inductive coupling (mutual inductance) between neighbouring lines. From now on, we assume for simplicity (and because this is the most significant case) that the two lines are geometrically (and electrically) symmetric, and that losses are negligible. Starting from the per-unit-length equivalent circuit we can describe the capacitive part (a π capacitor circuit, see Fig. 4.3) with a per-unit-length capacitance matrix: C =
C
−Cm −Cm C0 0
(4.1)
similarly the inductive part is described by a per-unit-length inductance matrix: L =
L L 0
m
Lm L0
.
(4.2)
134
Directional couplers and power dividers
The p.u.l. capacitance matrix relates the vector of the charges q (per unit lenght) induced in the two lines to the line voltages V as: q = C V while the inductance matrix (p.u.l.) relates the vector of the p.u.l. magnetic fluxes Φ to the vector of the currents flowing in the lines, I , as: φ = L I. The voltages and currents satisfy the generalized telegraphers’ equation: dV (z) = jω LI (4.3) dz dI (z) = jω CV (4.4) dz from which, eliminating e.g. the current from the second into the first equation, we obtain: d2 V (z) = ω 2 LCV . dz 2 We look for exponential solutions for the complex voltage phasors under the form of propagating waves:
−
V (z) = V 0 exp( jβz)
−
where V 0 = (V 01 , V 02 )T is a constant vector. Substituting we obtain the linear homogeneous system: β 2 I
− ω2LC
V 0 = 0.
To obtain a nontrivial (zero) solution we have to set the system determinant to zero; this allows to derive the values of the propagation constant β . For simplicity let us set: LC =
L L C 0
m
Lm L0
−Cm −Cm C0 0
L C −L C =
−L0Cm + Lm C0 −L0Cm + Lm C0 L0C0 − Lm Cm 0 0
m m
=
AB BA
Substituting and setting the determinant to zero we obtain: β 2 = ω 2 (A
± B).
Thus β can assume four values, two with positive sign and two with negative sign, implying forward and backward propagating waves, respectively; the absolute value of the propagation constants is however in general different, implying that two propagation modes can exist with different phase velocity. Developing the previous equations and substituting we obtain the two values: β 12 = ω 2 (
L0 + Lm )(C0 − Cm) β 22 = ω 2 (L0 − Lm )(C0 + Cm ).
(4.5)
(4.6)
.
4.1 Coupled quasi-TEM lines
135
Before trying a physical interpretation of the two propagation modes, let us eliminate the two p.u.l. inductances by introducing, as done for a single line, the in air or in vacuo capacitances. In fact, for an in vacuo line or set of coupled lines all propagation modes have phase velocity equal to the velocity of light in vacuo, c 0 . Thus, for a line in air we have: ω2 = ω 2 ( c20 ω2 β 22a = 2 = ω 2 ( c0 β 12a =
L0 + Lm)(C0a − Cma )
(4.7)
L0 − Lm)(C0a + Cma ).
(4.8)
As for a single line, the inductances are not affected by the presence of a dielectric. Deriving the p.u.l. inductances as a function of the in-air capacitances we finally obtain for the propagation constants of the coupled line β 1 and β 2 (the double sign refers to the forward and backward modes) where:
±
ω β 1 = c0 ω β 2 = c0
C − C CC − CC 0
0a
0 +
m
±
(4.9)
ma m
C0a + Cma
(4.10)
We can finally express the propagation constants in terms of a proper effective permittivity as follows: ω β 1 = eff1 (4.11) c0 ω β 2 = eff2 . (4.12) c0
√ √
where:
C0 − Cm C0a − Cma C0 + Cm eff2 = C0a + Cma eff1 =
(4.13) (4.14)
To derive a physical interpretation of the two propagation modes we can derive from the telegraphers’ equation the solution eigenvectors. Since the determinant of the system is zero the two equations are linearly dependent; substituting in the first equation of the syste the first value of β 2 we obtain for the eigenvector V 0 the relationship: V 01 = V 02 while for the second value of β 2 we obtain: V 01 =
−V 02.
In the first propagation mode the potential of the two lines is equal section by section; this mode is called the even mode and the electric field topology is symmetrical, see Fig. 4.4 for a coupled microstrip line. In the second propagation
136
Directional couplers and power dividers
Figure 4.4 Field
topologies for the even and odd modes in coupled microstrips.
modes the line potentials are equal but opposite section by section; we call this an odd propagation mode, with an antisymmetrical electric field pattern as shwon in Fig. 4.4. For the even mode we can therefore set V 01 = V 02 = V 0 , while for the odd mode V 01 = V 0 , V 02 = V 0 . To simplify the notation let us introduce the concept of even and odd-mode per unit lenght capacitances. Those are defined as the capacitances towards ground of one line when the two lines have the same potential or opposite potentials, respectively. With reference to Fig. 4.5, we obtain that the capacitance of a single line towards the ground is, for the two modes:
−
Ce = C0 − Cm Co = C0 + Cm.
(4.15)
(4.16)
Note that we always have:
Co ≥ Ce
(4.17)
since m 0; the equality hold only if the strips are not coupled any more, i.e. their distance tends to infinity implying m 0. The propagation constants of the two modes can be therefore identified in terms of the even and odd mode capacitances. We obtain:
C ≥
C →
ω β 1 = β e = c0 ω β 2 = β o = c0
C CC
e
(4.18)
ea o
Coa
(4.19)
137
4.1 Coupled quasi-TEM lines
Figure 4.5
Definition of the even and odd mode capacitances in a two-conductor line.
from which the even and odd mode effective permittivities result as:
Ce C pa Co . eff d = Cda eff p =
(4.20)
(4.21)
In a purely TEM coupled line the effective permittivities of the two modes are equal and equal to the medium permittivity. This does not happen in general to a quasi-TEM line with a inhomogeneous cross section, since the field patterns of the two modes are different. Substituting the voltage solutions in the telegraphers’ equations we find that for both modes the forward wave current is proportional to the forward wave voltage V 0 . The proportionality factor can be interpreted as the even (odd) mode characteristic admittance, i.e. the inverse of the even (odd) mode characteristic impedance Z 0e (Z 0o ). These turn out to be expressed as: 1 Z 0ae √ = √ eff e c0 Ce Cae 1 Z 0ao Z 0o = √ = √ o c C C Z 0e =
0
o ao
(4.22) (4.23)
eff
where the index a denotes in-air quantities. The two characteristic impedances follow the inequality: Z 0e
≥ Z 0 ≥ Z 0o
(4.24)
where Z 0 is the impedance of an isolated line and the equality sign only holds if the disstance bewtween the coupled lines tends to infinity. 1 No general relationship exists instead for the even and odd mode effective permittivities. Increasing the spacing between lines the even mode and the odd mode impedances asymptotically tend to the isolated line impedance from above and below, respectively, see Fig. 4.6. 1
In fact we have o 0 0 a0 e but also ao a0 ae from which o ao e ad , from which, taking into account the definition of impedances, we immediately have (4.24).
C ≥ C ≥ C
C ≥ C ≥ C
C C ≥ C C ≥ C C
138
Directional couplers and power dividers
Figure 4.6 Behaviour
of the even and odd mode characteristic impedances as a function of the line spacing.
Figure 4.7 Some
4.1.2
examples of coupled microstrips, coplanar lines and striplines.
Coupled planar lines Some examples of coupled planar lines are shown in Fig. 4.7. When exploited in the realization of directional couplers such structures should allow for a large difference between the even and odd mode characteristic impedances (deriving from a strong coupling C , see the definition in (4.55)) while the even and odd mode phase velocities should be (ideally) equal. In practice a two-conductor microstrip or coplanar line do not allow for either of the two conditions, and are therefore suited to fabricate low-coupling couplers only. Strong coupling can be achieved by broadside-coupled striplines or by properly exploited multiconductor
139
4.1 Coupled quasi-TEM lines
lines (see e.g. the so-called Lange coupler); while the copled stripline is a full TEM structure the microstrip-based couplers are not ideal from the stanpoint of having the same even and odd mode phase velocity. 4.1.2.1
Coupled microstrips Two-conductor coupled microstrips are a textbook example of coupled quasiTEM line, although in practice such a structure is limited to low coupling values, for technological reasons related to the minimum separation value that can be achieved (of the order of 50-10 µm). Let us call h the substrate thickness of dielectric constant r , W the strip width, S the slot width, and further define the normalized strip and slot widths: u = W/h
(4.25)
g = S/h
(4.26)
The effective permittivity of the even and odd modes can be obtined by empirical approximations as follows: r + 1 r 1 10 ae (v)be (r ) eff e = 1+ 2 2 v r + 1 eff o = + ao (u, r ) eff exp( co g do ) + eff 2
− −
−
−
(4.27)
−
(4.28)
while for the impedances we have: Z 0e = Z 0
eff eff e
1
√
(4.29) Z 0 eff Q4 1 377 eff 1 Z 0o = Z 0 . (4.30) Z 0 eff Q10 eff o 1 377 The previous formulae exploit the following parameters of the isolated line of width W : r + 1 r 1 eff = + 2 2
−
r + 1 r 1 eff = + 2 2 moreover:
−
−
12 1+ u
12 1+ u
8 u + u 4
−
−1/2
√
+ 0.04(1
−1/2
u
− u)2
u
≤1
(4.31)
≥1
Z 0 =
√ 60eff log
Z 0 =
120π 1 √ eff 1.393 + u + 0.667 log (1.444 + u)
u
(4.32)
≤1
(4.33) u
≥ 1.
(4.34)
140
Directional couplers and power dividers
The following parameters are also exploited: 20 + g 2 v = u + g exp( g) 10 + g 2
−
2
− − −
v v4 + 52 1 1 v ae (v) = 1 + log + log 1 + 49 v 4 + 0.432 18.7 18.1
3
r 0.9 0.053 be (r ) = 0.564 r + 3 r + 1 [1 exp( 0.179u)] ao (u, r ) = 0.7287 eff 2 0.747r bo (r ) = 0.15 + r cd = b d (r ) [bd (r ) 0.207] exp( 0.414u)
−
−
−
do = 0.539 + 0.694 exp( 0.562u)
−
−
and the fitting function set: Q1 = 0.8695u0.194 Q2 = 1 + 0.7519g + 0.189g 2.31
8.4 g
Q3 = 0.1975 + 16.6 +
6
−0.387
1 g10 + log 241 1 + (g/3.4)10
2Q1 1 Q Q2 exp( g)u + [2 exp( g)] u−Q 0.638 Q5 = 1.794 + 1.14log 1 + g + 0.517g 2.43 1 g 10 1 Q6 = 0.2305 + log + log 1 + 0.598g1.154 10 281.3 5.1 1 + (g/5.8) Q4 =
−
Q7 =
−
3
−
10 + 190g 2 1 + 82.3g 3
Q8 = exp
− − − − 6.5
Q2 Q4
g 0.15
0.95log g
Q9 = log Q7 Q8 + Q10 =
3
5
1 16.5
Q5 exp Q6 u−Q log u Q2 9
An example of the behaviour of even and odd mode characteristic impedances and of the related permittivities is shown in Fig. 4.8 and Fig. 4.9. Note that the even and odd mode impedances tend, for large values of the ratio S/h, to the value proper of the isolated microstrip. On the global behaviour of the even and odd mode impedances the following remarks hold:
141
4.2 The directional coupler
Behavior of even and o ddo mode characteristic impedances as a function of the ratio W /h for coupled microstrips on a GaAs substrate, h = 300 mu m. Figure 4.8
For growing W/h both impedances decrease because the capacitance towards ground of both modes increases; For increasing S/h the odd mode capacitance decreases and therefore the odd mode impedance increases. At the same time the even mod capacitance increases and therefore the even mode impedance decreases.
The behavior of the effective permettivity is less straightforward. The even mode impedance in coupled microstrips is higher because most field lines go through the substrate in the even mode. The odd mode permittivity is close (for high coupling) to (r + 1)/2 because the field lines are approximately distributed in an equal way in the substrate and in the air. Furthermore:
4.2
With increasing of W/h the field is increasingly confined in the substrate and both permittivities grow; With decreasing by S/h the odd mode field is increasingly concentrated in the slot between the two strips, and the odd mode permittivity decreases approaching ( epsilonr + 1)/ 2, and is little affected by S/h.
The directional coupler The directional coupler is four-port network, which has in general the purpose of distributing the power delivered to a given port (for example port 1) between two
142
Directional couplers and power dividers
Figure 4.9 Behavior
of even and oddo mode effective permettivities as a function of the ratio W/h for coupled microstrips on a GaAs substrate, h = 300 mu m.
other ports (for example 2 and 3) maintaining port 4 isolated. The port where power enters is named the incident port, the two ports where power is divided are called coupled and transmitted port; the other port called the isolated port, as no (or very little) power comes out of it (Fig. 4.10). From this point of view, the directional coupler would appear similar to the so-called power divider : this is an n-port where the power entering port 1 is divided, equally or according to some repartition scheme, into the remining n 1 ports. While couplers can be exploited as power dividers, this component allows for a more specific function, i.e. imposing a specific phase relationship between the power wave at the coupled and transmitted ports, typically either 90 or 180 degrees. The directional coupler has several applications in the field of microwave circuits; it is used (in passive circuits) in the realization of delay lines, filters and matching networks; in active circuits, is a major building block in balanced amplifiers, mixers, attenuators, modulators and phase shifters. It should be remembered that directional couplers have a behavior that depends on frequency: it is ideal at centerband, and exhibits a gradual deterioration when moving away from the design frequency. Usually, directional couplers are rather narrowband (for example 20% of the center frequency). The coupler is chracterized by a number of parameters. Consider port 1 as the incident port, port 2 as the coupled port, port 3 as the transmission port and
−
4.2 The directional coupler
Figure 4.10 Scheme
143
of a directional coupler.
port 4 as the isolated port. We now define the power coupling coefficient:
−10log10 P P 21 , where the isolation of port 4 (ideally zero or −∞ in dB): P 4 I |dB = −10log10 . P 1 K dB =
|
The (power) transmision coefficient to port 3 is: T dB =
|
−10 log10
P 3 P 1
.
Finally the power reflection coefficient R at port 1 (ideally zero or is:
−∞ in dB)
R dB = 20 log10 (Γ1 )
|
where Γ1 is the voltage reflection cofficient. Usually, directional couplers are reactive elements, i.e. show low power dissipation; this implies P 1 P 2 + P 3 , i.e. the input power is partitioned between ports 2 and 3. A further figure of merit of the coupler is the directivity (ideally infinite):
≈
D
|dB = 10log10
P 2 P 4
= I dB
| − K |dB .
Fig. 4.11 shows some examples of directional couplers. The couplers can be divided into various categories:
Simple coupled-line couplers; can be uniform or non-uniform. In non-uniform couplers the bandwidth can be increased, at the expense of a larger footprint (in uniform couplers the centerband length is of the order of a quarter wavelength). The coupling achieved by these structures is low in microstrips (it is typically impossible to obtain a 1/2:1/2 power distribution on the coupled and transmission ports, which corresponds to the so-called 3 dB coupler ), high for stripline structures (in particular, broadside coupled striplines). The coupled
144
Directional couplers and power dividers
line couplers show a centerband phase shift of 90 degrees between the coupled port and the port in transmission; often they are referred to as 90 degrees hybrids . Interdigitated couplers, such as the Lange coupler; they are similar in behaviour and operation principle to coupled line couplers but can reach higher coupling (in particular 3 dB); Branch-line couplers, they are based on an interference principle and permit to obtain 3 dB couplers but they hve a large footprint (typically square with a centerband side of a quarter wavelength) and narrow band. They also are 90 degrees hybrids. The hybrid ring (also called rat race coupler) is also based on interference principles but also allows to impose a 180 degrees shift between the output ports at centerband. They are often used to generate signals balanced with respect to ground (or differential signals; this is particularly useful in mixers). The footprint is large (the periphery is of the order of a wavelength at centerband). Other types are the tandem couplers, the meander line couplers and the transformer couplers.
It should be emphasized that branch-line couplers and the hybrid ring can also be implemented in a concentrated form (when technologically possible), thus obtaining structures that are much more compact than the distributed ones.
4.3
The two-conductor coupled line coupler We start from considering a symmetric two-conductor coupled line carrying an even and an odd TEM or qusi-TEM propagation mode. In this case the coupler can be anaylzed through a superposition of even and odd mode excitation. Let us suppose to consider a coupled line section of length l (i.e. a four port) closed on the reference resistance R0 on all ports, see Fig. 4.12. Port 1 is connected to a signal generator with internal resistance R0 . The excitation can be decomposed into an even and odd excitation as shown in Fig. 4.13; each of them only excites the corresponding mode. Le us call V io and V ie the odd and even mode input voltages and V oo and V oe the output voltages; we have: V 1 = V ie + V io
(4.35)
V 2 = V ie
(4.36)
V 3 = V oe + V oo
(4.37)
V 4 = V oe
(4.38)
− V io
− V oo .
The even and odd mode voltages can be evaluated from the analysis of the loaded two-port in Fig. 4.14. For definiteness let us refer to the even mode. The
145
4.3 The two-conductor coupled line coupler
Figure 4.11 Examples
of directional couplers.
scattering matrix of the even mode line (characteristic impedance Z 0e and guided wavelength λ e ) vs. the reference impedance Z 0e can be trivially written as: Se =
0 exp( jθe ) exp( jθe ) 0
−
where θ e = 2πl/λe is the electrical line length.
−
146
Directional couplers and power dividers
Figure 4.12 Coupled
two-line coupler.
Figure 4.13 Decomposing
the excitation in even and odd modes.
147
4.3 The two-conductor coupled line coupler
Figure 4.14 Even-mode
line as a loaded two-port.
The reflection coefficients of the load and generator with impedance R 0 are: Γ0e =
R0 Z 0e R0 + Z 0e
−
while the even mode forward wave generator at the input has the value:
√
E g Z 0e b0e = . 2 Z 0e + R0 From the analysis of the loaded two-port we find that: e− jθe 1 Γ20e e−2jθe Γ0e e− jθe aoe = b 0e 1 Γ20e e−2jθe boe = b 0e
− −
Γ0e e−2jθe 1 Γ20e e−2jθe 1 aie = b 0e 2 1 Γ0e e−2jθe bie = b 0e
− −
and voltages can be recovered from: V ie = a ie V oe = b oe
Z 0e + bie
boo = b 0o aoo = b 0o
(4.41) (4.42)
Z 0e Z 0e
e− jθo Γ20o e−2jθo
(4.43)
Γ0e e− jθo 1 Γ20e e−2jθo
(4.44)
1
− −
Γ0e e−2jθo 1 Γ20e e−2jθo 1 aio = b 0o 2 1 Γ0o e−2jθo bio = b 0o
(4.40)
Z 0e + aoe
A similar results holds for the odd mode line:
(4.39)
− −
(4.45) (4.46)
148
Directional couplers and power dividers
where: R0 Z 0o R0 + Z 0o
−
Γ0o =
and the odd mode forward wave generator is:
√
E g Z 0o b0o = 2 R0 + Z 0o while θo = 2πl/λo . Substituting the foward wave generator value we finally obtain: V ie =
E g 1 + Γ 0e e−2jθe Z 0e 2 2j − θ e R + Z 2 1 Γ0e e 0 0e
(4.47)
V io =
E g 1 + Γ 0o e−2jθo Z 0o 2 − θ 2 1 Γ0o e 2j o R0 + Z 0o
(4.48)
V oe =
E g (1 + Γ0e )e− jθe Z 0e 2 − 2 1 Γ0e e 2jθe R0 + Z 0e
(4.49)
V oo =
E g (1 + Γ0o )e− jθo Z 0o . 2 2 1 Γ0o e−2jθo R0 + Z 0o
− −
−
−
(4.50)
As a first step, let us evaluate the resistance R 0 allowing for matching at port 1. To this purpose suppose that the phase velocities of the even and odd modes are equal: θe = θo = θ. We anticipate that the centerband frequency of the coupler corresponds to l = λg /4, i.e. e− jθ = j, e−2jθ = 1. In this condition vi have that the port 1 voltage V 1 = V ie + V io is:
−
−
E g Z 0e 1 Γ0e E g Z 0o 1 Γ0o E g V 1 = + = 2 2 2 R0 + Z 0e 1 + Γ 0e 2 R0 + Z 0o 1 + Γ 0o 2
−
−
Z 02o Z 02e + R20 + Z 02o R02 + Z 02e
Port 1 is matched if V 1 = E g /2, i.e.: Z 02e Z 02o + 2 = 1, R02 + Z 02e R0 + Z 02o from which the matching condition: R0 =
Z 0e Z 0o .
(4.51)
Taking into account this condition the even and odd mode reflection coefficients result as: R0 Z 0o Γ0o = = R0 + Z 0o R0 Z 0e Γ0e = = R0 + Z 0e
− −
√ Z − √ Z √ Z 0e + √ Z 0o = Γ √ Z 0e − √ Z 0o √ 0o √ 0e = −Γ. Z 0o +
Z 0e
If we move from the centerband frequency, taking into account that:
.
4.3 The two-conductor coupled line coupler
√ Z − √ Z √ 2 Z 0e 0e 0o √ = √ Z + √ Z 1 + Γ = 1 + √ Z 0e + Z 0o 0e 0o √ Z − √ Z √ 0e √ 0o = √ 2 Z 0√ o 1 − Γ = 1 − √ Z 0e +
Z 0o
Z 0e +
Z 0o
149
(4.52) (4.53)
we obtain that the even and odd mode voltages can be expressed as: E g 1 Γe−2jθ (1 + Γ) 4 1 Γ2 e−2jθ E g 1 + Γe−2jθ V io = (1 Γ) 4 1 Γ2 e−2jθ E g (1 Γ)e− jθ V oe = (1 + Γ) 4 1 Γ2 e−2jθ E g (1 + Γ)e− jθ V oo = (1 Γ) . 4 1 Γ2 e−2jθ
− −
V ie =
−
− − −
−
−
We can readily evaluate V 1 , V 2 , V 3 , V 4 summing or substracting the even and odd mode input and output even and odd mode voltages; we obtain: E g 2 E g Γ(1 e−2jθ ) V 2 = V ie V io = 2 1 Γ2 e−2jθ E g (1 Γ2 )e− jθ V 3 = V oe + V oo = 2 1 Γ2 e−2jθ V 4 = V oe V oo = 0 V 1 = V ie + V io =
−
−
−
− − −
We conclude that port 1 is always matched, independent on frequency, while port 4 is always isolated from port 1, i.e. S 11 = S 41 = 0 for all frequencies. Transmission from port 1 to 2 and 3 is possible and port 2 is called coupled, port 3 is called transmitted. Taking into account the structure twofold symmetry we have: S 11 = S 22 = S 33 = S 44 = 0 (all ports are matched): S 41 = S 32 = S 23 = S 14 = 0 (defining isolation of ports with respect to the input), and then: S 21 = S 12 = S 34 = S 43 (defining coupling), and finally: S 31 = S 13 = S 42 = S 24 (defining trasmission). Some of those relationships are anyway imposed by reciprocity. Taking into account that the reference impedance is the same for all ports and that port 1 is matched (the total voltage V 1 coincides with the progressive
150
Directional couplers and power dividers
value) we simply have S ji = V j /V i ; the scattering parameters of the synchronous coupler are then immediately evaluated.
4.3.1
Frequency behaviour of the synchronous coupler In the synchronous case the scattering matrix of the coupler as a function of the electrical length θ is:
S(θ) =
where:
0 Γ(1 e−2jθ ) 1 Γ2 e−2jθ (1 Γ2 )e− jθ 1 Γ2 e−2jθ
− − − −
0
Γ(1 e−2jθ ) (1 1 Γ2 e−2jθ 1
−
−
0
− Γ22)e−2j− jθθ −Γ e
(1 Γ2 )e− jθ 1 Γ2 e−2jθ Γ(1 e−2jθ ) 1 Γ2 e−2jθ
− − − −
0
0 (1 1
0
0
− Γ2)e−− jθθ Γ(1 − e−−2jθθ) − Γ2e 2j 1 − Γ2e 2j √ ω eff l 2πl θ =
λg
=
c0
0
(4.54)
.
At centerband (i.e. at the frequency where the line length is a quarter wavelength) θ = π/2 and the nonzero scattering matrix elements are: 2Γ Z 0e Z 0o = C 1 + Γ2 Z 0e + Z 0o 1 Γ2 2 Z 0e Z 0o S 31 = j = j = j 1 2 1+Γ Z 0e + Z 0o
−
S 21 =
− −
−
√
≡
−
−
(4.55)
C 2 .
(4.56)
The parameter C is the coupling of the coupler; the power coupling is K = C . For the 3 dB coupler (equal power division between the coupled and the transmission port) K = 1/2 and C = 1/ 2. The centerband scattering matrix can be therefore written as: 2
√
S(π/2) =
− √ −
0 C j 1 C 2 0 C 0 0 j 1 C 2 j 1 C 2 0 0 C 2 0 j 1 C C 0
√ − −
− √ −
− √ −
.
(4.57)
We summarize here for convenience the formulae for the design and analysis of a coupler relating the termination resistance R 0 and coupling with the even and odd mode impedances: Z 0e Z 0o C = , Z 0e + Z 0o
−
R0 =
Z 0o Z 0e ,
Z 0e = R0
1 + C , 1 C
−
Z 0o = R0
−
1 C . 1 + C
3 dB couplers and in general couplers with high coupling cannot be realized with coupled microstrips. In fact, to obtain a 3 dB coupler on 50 Ω we need Z 0e = 121 Ω, Z 0o = 21 Ω; such values would require an extremely close spacing (a few microns) between the strips, that is technologically inconvenient to obtain
151
4.3 The two-conductor coupled line coupler
Figure 4.15 Magnitude
and phase of S 21 and S 31 for a 3 dB coupler.
Figure 4.16 Magnitude
of S 11 , S 41 (left) and S 21 , S 31 (right) of a 3 dB coupler. The ratio between the effective permittivities of the two modes is 0.9.
in hybrid or integrated implementations (besides leading to large losses), cfr. Example 4.1. In such cases we can exploit multiconductor couplers such as the Lange coupler. In Fig. 4.15 we show the magnitude (in dB) of S 21 and S 31 for a 3 dB coupler. The coupler exploits line with even and odd effective permettivities equal to 5; the centerband frequency is 5 GHz. The coupler is comparatively narrowband as far as coupling and transmission are concerned.
Example 4.1:
• Try and design a 3 dB coupler on 50 Ω in a coupled microstrip using a GaAs substrate with thickness h = 300 µm. ◦ As already seen we have Z = 121 Ω, Z = 21 Ω. From Fig. 4.18, the needed 0e
0o
width and slot values are W/h = 0.2, S/h = 0.0018, i.e. W = 60 µm, s = 0.5 µm. The slot width is far too smal to be implemented from a technological standpoint.
152
Directional couplers and power dividers
Figure 4.17 Magnitude
of the S paramters of a 3 dB coupler in the presence of velocity
mismatch.
4.3.2
Effect of velocity mismatch and compensation techniques In quasi-TEM couplers the even and odd mode velocities are in principle different. Supposing R 0 = Z 0o Z 0e , if θe = θ o we have:
√
V ie = V io = V oe = V oo =
E g 1 Γe−2jθe (1 + Γ) 4 1 Γ2 e−2jθe E g 1 + Γe−2jθo (1 Γ) 4 1 Γ2 e−2jθo E g (1 Γ)e− jθe (1 + Γ) 4 1 Γ2 e−2jθe E g (1 + Γ)e− jθo (1 Γ) . 4 1 Γ2 e−2jθo
− −
−
− − −
−
−
from which V 1 , V 2 , V 3 , V 4 can be recovered as usual. We then obtain: S j 1 = 2
V j , E g
j = 2, 3, 4
taking into account that R0 is the same for all ports. The reflection coefficient at port 1 can be evaluated considering that: I 1 =
E g
− V 1 ; R0
153
4.3 The two-conductor coupled line coupler
Figure 4.18 Level
curves of Z 0e and Z 0o (Ω) as a function of S/h , W/h (log units) for
r = 13.
but the forward and backward voltages at port 1 can be written as: V 1 + R0 I 1 E g = 2 2 V R I 1 0 1 V 1− = = V 1 2
V 1+ =
−
− E 2g
(4.58)
(4.59)
from which: S 11 =
V 1− 2V 1 + = E g V 1
−1
We do not carry out the computations in detail but come to the main consequences. Velocity mismatch leads to an impedance mismatche at port 1 and to a decrease of the isolation at port 4, while coupling and transmission are affected but not dramatically. Fig. 4.16 shows the parameters of the already analyzed 3D dB coupler; the coupler length was assumed according to the empirical recipe: l =
λe λ o + 4 4
1 2
In the example, eff = 5 for the even and eff = 4.5 for the odd mode. Among the unfavourable consequences of mismatch perhaps the more serious one is the decreased isolation, that may cause trouble in systems where the coupler is the interface between a transmitter (strong signals) and a receiver (weak signals) and therefore a leakage from the strong signal environment and the weak one can impair the system operation.
154
Directional couplers and power dividers
Figure 4.19 Shielded
directional coupler (a) and with dielectric overlay (b).
An approximate estimate of the centerbnd directivity vs. velocity mismatch is obtained as follows: S 41
≈ δ (1 − C 2)
where: δ =
|v p − vd| v p + vd
An example of the behaviour of the scattering parameters magnitude as a function of velocity mismatch is shown in Fig. 4.17. The directivity of a non-synchronous microstrip directional coupler can be improved through various means. The phase velocity of the even and odd mode can be equalized through several expedients:
Use of grounded metal screens. For symmetry, if d = h the effective permittivities of the two modes become equal to ( r + 1)/2, see Fig. 4.19, (a). Using dielectric overlays. A dielectric layer of suitable thickness and permettivit can compensate for the phase velocity mismatch, as shown in Figure 4.19, because it independently changes odd and even capacitances.
There are other techniques to correct the phase velocity mismatch through distributed or concentrated techniques:
External loading capacitances, as shown in Fig. 4.20, (a). This modifies the odd mode capacitance only. Take into account that the odd efective permittivity is typically lower than the even one, so that the odd mode electrical length is lower than the even mode electical length for the same physical line length. A concentrated additional capacitance C = C 1 + C 2 is an increase of the oddmode line electrical length; in fact, is o is the p.u.l. odd mode capacitance, we have ∆l o = C , from which an odd mode phase shift results:
C
C
∆θo = ∆lω
C L o
o = ωCZ 0o .
This makes possibile to design, at least at centerband, the compensation capacitance. The use of serrated of wiggling lines, see Fig. 4.20, (b); the wiggling does not greatly affects the even mode capacitance but has a strong impact on the odd mode capacitance that depends on the edge coupling between lines.
155
4.4 Multiconductor line couplers
Figure 4.20 Compensation
4.4
through concentrated capacitances (a) e and wiggle (b).
Multiconductor line couplers Multiconductor (or interdigitated) line couplers are equivalent two-conductor couplers (by connecting a number of parallele conductors through wires or air bridges) allowing to obtain high coupling with technologically feasible geometries. We start from the analysis of a multiconductor line with an even number of k parallel strips, see Fig. 4.21. As a first approximation we neglect the mutual capacitance between non-neighbouring strips and we suppose that the mutual capacitance between neighbouring strips ( 12 ) and the strip capacitance to ground ( 20 ) is the same independent on the strip position, apart from the two external strips, whose capacitance towards ground will be denoted as 10 . The following approximate relationship can be shown to hold:
C
C
C
C20 ≈ C10 − C10C10 +C12C12 .
In fact, the capacitance towards ground of the external strip is similar to the capacitance towards ground of a strip at whose right there is an infinite number of floating strips (i.e. having zero total charge), see Fig. 4.22. Such capacitance can be estimated as the iterative capacitance of an infinite set of metal strips, as shown Fig. 4.23. We thus obtain:
C10 = C20 + C10C10 +C12C12 .
from which we obtain 20 . Notice once and for all that 12 , 10 are the capacitance between the two strips and between each strip and the ground of two coupled microstrips, that we suppose to be able to evaluate as a function of the line geometry and dielectric parameters. Suppose now that the lines of the multiconductor structure are alternatively connected so as to give rise to two equipotential conductors. The connection is typically not done in a continuous way but only at intervals, close enough (e.g. less than a quart wavelength), by means of airbridges or bonding wires, see Fig. 4.24.
C
C C
156
Directional couplers and power dividers
Figure 4.21 Multiconductor
Figure 4.22
line and associate capacitances.
Evaluating the capacitance of the two extreme lines - I.
Figure 4.23
Evaluating the capacitance of the two extreme lines - II.
It is then possibile to define an equivalent two-conductor line, able to carry an even and an odd mode. The even and odd mode capacitances are then the capacitance towards ground of a set of k/2 strips when the two set have the same
157
4.4 Multiconductor line couplers
Figure 4.24 Multiconductor
line with conductors connected though wire bondings.
or opposite potential, respectively. From Fig. 4.25 we obtain: Ce (k) = (k/2 − 1)C20 + C10 Co (k) = (k/2 − 1)C20 + C10 + 2(k − 1)C12
(4.60)
(4.61)
where all capacitances are per unit lenght. Taking into account that for two strips we have: (4.62) Ce (2) = C10 (4.63) Co (2) = C10 + 2C12 we can obtain C10 and C12 as a function of Ce (2) and Co (2) and therefore express the even and odd mode capacitances of the equivalent two-conductor line derived from the multiconductor one by connecting strips as a function of the even and odd mode capacitances of the two-conductor line. We obtain: 2
+ (k − 1)Ce (2) Ce(k) = Co (2)CCe(2) o (2) + Ce (2) + (k − 1)Co2 (2) . Co(k) = Co (2)CCe(2) o (2) + Ce (2)
(4.64)
(4.65)
We can now obtain the characteristics (termination impedance and centerband coupling) of the coupler. Taking into account that: C (k) =
Z 0e (k) Z 0o (k) = Z 0e (k) + Z 0o (k)
−
Co(k) − Ce(k) Ce(k) + Co(k)
158
Directional couplers and power dividers
Figure 4.25 Evaluating
the even and odd mode capacitances.
and that, for velocity matching, Z 0e /Z 0o =
where R is defined as:
Co /Ce, we obtain: (k − 1)(1 − R2 ) C (k) = (k − 1)(1 + R2 ) + 2R R =
Z 0o (2) . Z 0e (2)
(4.66)
(4.67)
We similarly show that the matching impedance is: R02 = Z 0e (2)Z 0o (2)
[(k
−
(1 + R)2 1)R + 1][(k 1) + R]
−
(4.68)
The behaviour of the coupling (in dB) as a function of R and for different values of the line number k is shown in Fig. 4.26. To obtain large coupling we need small values of R (and therefore very different even and odd mode impedances) for k = 2, while R increases with growing k. The improvement is however marginal for k > 8. Similar remarks can be made on the closing impedance normalized vs. the two-strip closing (or matching) impdance, see Fig. 4.27. Note that for a termination impedance of 50 Ω we have (for two conductors) Z 0e (2)Z 0o (2) = 50 Ω, while for four conductors Z 0e (2)Z 0o (2) 100, i.e. the strip width needed
≈
159
4.4 Multiconductor line couplers
Figure 4.26 Coupling
as a function of R for several values of k .
decreases with increasing k. The effect is negative for large k since it implies that very narrow strips are needed to implement the coupler with a reasonable impedance level. Therefore, the use of multiconductor couplers allows to design a technologically feasible 3 dB coupler; benefits are maximum with four or six wires, while increasing the number of conductors beyond this value increases the complexity of the structure and the losses due to the very narrow strips. The length of the Lange coupler can be determined from the effective permittivity of the even and odd modes. Taking into account that for a two-line structure we have:
Ce (2) = Cae (2)eff e(2) Co (2) = Cao (2)eff o(2)
(4.69) (4.70)
and that:
Cae(2) = Z 0ao (2) ≡ Ra Cao(2) Z 0ae(2)
(4.71)
160
Directional couplers and power dividers
Figure 4.27 Closing
impedance as a function of R for several values of k .
the ratio of in-air characteristic impedances, we can write: 2 2 a + Ra (k − 1)eff e (2) Ce (k) = Cao(2) eff e(2)eff eff o (2)R o (2) + Ra eff e (2) 2 + (k − 1)Cae (2) Ra + Ra2 (k − 1) = Cao (2) Cae (k) = Cao(2)CCaeao(2) (2) + Cae (2) 1 + Ra 2 o (2)Ra + (k − 1)eff e (2) Co (k) = Cao(2) eff e(2)eff eff o (2) + Ra eff e (2) 2 + (k − 1)Cao (2) Ra + (k − 1) = Cao (2) Cao (k) = Cao(2)CCaeao(2) (2) + Cae (2) 1 + Ra
(4.72) (4.73) (4.74) (4.75)
from which:
Ce (k) = eff e (k) = Cae(k) eff o (k) =
Co (k) = Cao(k)
eff e (2)eff o (2)Ra + Ra2 (k 1)2eff e (2) eff o (2) + Ra eff e (2)
−
eff e (2)eff o (2)Ra + (k 1)2eff o (2) eff o (2) + Ra eff e (2)
−
1 + Ra Ra + Ra2 (k
− 1)
(4.76)
1 + Ra Ra + (k 1)
−
.
(4.77)
In multiconductor lines the even and odd permittivities are rather different; for large coupling we have Ra 0 which implies that the even and odd mode permittivities are similar to the case of the two-conductor line. Also in this case we can approximate the centerband length through the arithmetic media of the even and odd mode quarter wavelengths.
→
4.4 Multiconductor line couplers
161
Figure 4.28 Lange
4.4.1
coupler, unfolded version.
The Lange coupler The Lange coupler (named after Julius Lange, who proposed it in 1969) is an interdigitated microstrip coupler consisting of four parallel lines alternately connected in pairs, as shown in Fig. 4.28 (a version called unfolded ) and in Fig. 4.29 (a version called folded , mre common in practice). Note that the unfolded version behaves like a directional coupler with two conductors with regard to the direct and coupled ports; it is however made by four lines connected two by two at the ends (and therefore virtually equipotential throughout the coupler). In the folded version the transmission and isolated ports are exchanged, so that the coupled and the transmission port are on the same side of the coupler. There is also a DC path between the upper and the lower side of the coupler. These characteristics make the folded version more common and convenient than the unfolded version. To design a Lange coupler, we can express, as a function of the centerband coupling C and of the closing impedance R 0 , the even and odd mode impedances of the two-conductor coupled lines having the needed design parameters w and s. We obtain:
−
1 C (k 1)(1 + q ) 1 + C (C + q ) + (k 1)(1 C + q Z 0e (2) = Z 0o (2) (k 1)(1 C )
Z 0o (2) = R0
−
−
where: q =
C 2 + (1
−
−
− C )
(4.78) (4.79)
− C 2)(k − 1)2.
In the most common case, the Lange coupler has k = 4. An approximate design technique is as follows:
162
Directional couplers and power dividers
Figure 4.29 Lange
coupler, folded version.
1. Starting from the centerband coupling and the closing impedance we derive the even and odd mode impedances of the two-conductor coupler having the same dimensions w and s, Z 0o (2) and Z 0e (2). 2. We derive the ratio W/h needed to obtain Z 0o (2)/2 and Z 0e (2)/2, respectively. To this purpose we can exploit the Wheeler appromations, yielding W/h as a function of the characteristic impedance: – For Z 44 2r Ω (where Z = Z 0e (2)/2 or Z 0o (2)/2):
≥ −
W = h
exp(B) 8
−
1 4 exp(B)
−1
where: Z B = 60
r + 1 1 + 2 2
− r 1 r + 1
0.2416 0.4516 + r
.
– For Z < 44
− 2r Ω (where Z = Z 0e(2)/2 or Z 0o(2)/2): W 2 2 r − 1 0.517 = (d − 1) − log(2d − 1) + log(d − 1) + 0.293 − h π π πr r
where: d =
60π 2 . Z r
√
3. The ratios W/h found are addressed as (W/h)e and (W/h)o . The real parameters S/h and W/h can be derived from the following equations, to be inverted
163
4.4 Multiconductor line couplers
numerically:
W h W h
= f e (W/h, S/h)
(4.80)
= f o (W/h,S/h)
(4.81)
e
o
where: 2 f e = cosh−1 π
2a
− g + 1
g + 1
(4.82)
and: 2 f o = cosh−1 π f o =
2 cosh−1 π
2a
2a
−
−g−1 g−1 −g g−1
1
4 W/h + cosh−1 1 + 2 π(1 + r /2) S/h +
1 W/h cosh−1 1 + 2 π S/h
r
≥ 6.
r
≤ 6 (4.83)
(4.84)
The parameters g and a are: g = cosh a = cosh
πS 2h πW π S + . h 2h
(4.85)
(4.86)
The previous formulae allow to evaluate from the even and odd W/h the rations W/h and S/h for the multiconductor line. For the coupler centerband length, a common approximation is to use the average of the quarter wavelength for the even and odd modes of the two-conductor line.
Example 4.2:
• Design a four-conductor Lange coupler on allumina at 10 GHz. The reference
impedance is 50 Ω, the substrate dielectric constant is 9, the substrate thickness is h = 25 mils (0.635 mm); we want 3 dB centerband coupling.
◦ We have C = 0.707, k = 4, R = 50 Ω; from the design formulae: 0
1 C 3(1 + q ) 1 + C (C + q ) + 3(1 C + q Z 0e (2) = Z 0o (2) 3(1 C )
Z 0o (2) = R 0
−
− C )
−
where: q =
C 2 + 9(1
− C 2) = √ 0.5 + 9 · 0.5 =
√
5 = 2.24
164
Directional couplers and power dividers
Figure 4.30 Graph
of f e (W/h, S/h) = (W/h)e and f o (W/h, S/h) = (W/h )o , see
Example 4.2.
we obtain Z 0o (2) = 52.6 Ω, Z 0e (2) = 176.2 Ω. The even mode W/h ratio therefore refers to Z 0e (2)/2 = 88.1 Ω, while the odd mode W/h ratio is derived from Z 0o (2)/2 = 26.3 Ω. Since 44 2r = 26 Ω, we need to use the first expression; we obtain (W/h)e = 0.25, (W/h)o = 3.07. Inverting or exploiting the chart in Fig. 4.30, we obtain S/h = 0.076, W/h = 0.09 from which s 46 µm, w 57 µm. For the even and odd mode permittivities of the four-conductor line we have eff o 5 and eff e 5.6 from which:
−
≈
≈
≈
√
λe /4 = 30/ 5.6/4 = 3.17 mm,
≈
√
λe /4 = 30/ 5/4 = 3.35 mm
i.e. a mean length l 3.26 mm. As a first approximation we note that eff (r + 1)/2 = 5 for both modes, from which l = 3.35 mm.
≈
4.5
≈
Interference couplers Coupled microstrip line couplers allow for high coupling only in multiconductor form, as in the Lange coupler. Other coupler obtain power division and isolation through an interference principle; typical examples are the branch line couplers and the hybrid ring or rat-race coupler.
165
4.5 Interference couplers
Figure 4.31 Branch-line
4.5.1
coupler.
Branch-line coupler The analysis of the branch-line coupler, see Fig. 4.31 for the microstrip layout, can be carried out exploiting the structure quadrantal symmetry, see Example 4.3. For symmetry the scattering matrix results:
S =
S 11 S 12 S 13 S 12 S 11 S 14 S 13 S 14 S 11 S 14 S 13 S 12
S 14 S 13 S 12 S 11
where, as shown in Example 4.3:
Γa + Γ b + Γ c + Γ d 4 Γa + Γ b Γc Γd S 21 = 4 Γa Γb + Γ c Γd S 31 = 4 Γa Γb Γc + Γ d S 41 = . 4 S 11 =
− − − − − −
(4.87)
(4.88a)
(4.88b)
(4.88c)
(4.88d)
The four indices refer to even and odd excitations with respect to the vertical and horizontal directions. Case (a) is even in bot directions, cased (d) is odd in both directions, (b) is even horizontally and odd vertically, (c) is odd horizontally and even vertically. At centerband, i.e. for θ1 = θ2 = π/4 (this means that the
166
Directional couplers and power dividers
lengths of the two lines are a quarte wavelength) we obtain: Y 0 j Y 01 j Y 02 Y 0 + j Y 01 + j Y 02 Y 0 j Y 01 + j Y 02 Γb = Y 0 + j Y 01 j Y 02 Y 0 + j Y 01 j Y 02 Γc = Y 0 j Y 01 + j Y 02 Y 0 + j Y 01 + j Y 02 Γd = Y 0 j Y 01 j Y 02
−
Γa =
−
−
(4.89) (4.90)
− −
− −
(4.91) (4.92)
−
from which, substituting into (4.88) we find for S 11 : 2 2 2 Y 04 (Y 01 Y 02 ) S 11 = 4 2 2 2 Y 0 + 2Y 0 (Y 01 + Y 02 ) + (Y 01
−
−
− Y 02)4
meaning that match is obtained with respect to the reference impedance if: 2 Y 02 = Y 01
| − Y 022 |
(4.93)
Taking into account of this last expression and imposing for instance Y 01 > Y 02 , 2 2 (4.93) becomes Y 02 = Y 01 Y 02 , i.e.:2
−
0 − Y Y 01
S 21 = j
(4.94)
S 31 = 0
(4.95)
− Y Y 0021 .
S 41 =
(4.96)
Expressions (4.93), (4.95) and (4.96) can be used as design equations. They also imply that a phase relationship between port 2 (coupled) and port 4 (transmission) exist a centerband as in the coupled line coupler, i.e. a phase shift of 90 degrees. The centerband scattering matrix therefore is:
− −
0
j
S =
2
Y 0 Y 01 0
Y 02 Y 01
Y 0 j Y 01 0
−
−
Y 02 Y 01 0
0
− Y Y 0021 0 0 − j Y Y 01
−
Y 02 Y 01 0
Y 0 j Y 01 0
−
2 If we impose instead Y 01 < Y 02 or Y 02 = Y 02 S 31 = jY 0 /Y 02 and S 41 = Y 01 /Y 02 .
−
−
− −
0
j
=
2 01
− Y
Z 01 Z 0 0
Z 01 Z 02
Z 01 j Z 0 0
−
−
Z 01 Z 02 0
0 01 − Z Z 02
0
− j Z Z 001
−
Z 01 Z 02 0
Z 01 j Z 0 0
−
.
(4.97)
we obtain by symmetry S 11 = 0, S 21 = 0,
4.5 Interference couplers
167
The coupling between port 2 and 1 is now C = Z 01 /Z 0 and, for power conservation in a lossless structure, we have 1 C 2 = Z 01 /Z 02 from which:
√ −
Z 01 = C Z 0 CZ 0 Z 02 = . 1 C 2
√
(4.98)
√ −
(4.99)
For a 3 dB coupler we have C = 1/ 2, i.e.: Z 02 = Z 0
√
Z 01 = Z 0 / 2.
(4.100)
(4.101)
Starting from an access line of 50 Ω, we find Z 01 = 35.35 Ωand Z 02 = 50 Ω, easily implemented in microstrip. 3 dB couplers with a 90 degrees shift beween the output ports are called 90 degrees hybrids .
Example 4.3:
• Evaluate the scattering matrix of a branch-line coupler. ◦ We start from the following remarks:
1. Due to symmetry and reciprocity, the scattering matrix is completely identified by the first row of the matrix. 2. Since the circuit is linear, we can apply the superposition principle, i.e. decompose the excitation required to evaluate the elements of the first row of the S-matrix into more convenient even and odd excitations. 3. Even and odd excitations can be conveniently assumed under the form of progressive voltages (or power waves) rather than of total voltages. Let us imagine to impress (through a forward wave generato) the forward voltage V n+ entering port n (n =1, 2, 3, 4) and let us denote with V n− the corresponding backward or reflected voltages. Consider now the following excitations at the four ports: 1. 2. 3. 4.
Case Case Case Case
(a): V 1+a = V 2+a = V 3+a = V 4+a = V + (b): V 1+b = V 2+b = V + , V 3+b = V 4+b = V + (c): V 1+c = V 3+c = V + , V 2+c = V 4+c = V + (d): V 1+d = V 4+d = V + , V 2+d = V 3+d = V + .
− − −
Notice that the four excitations have the already mentioned even and/or odd character with respect to the vertical and horizontal plane: Superinposing we have: V 1+ =4V + V 2+ =V 3+ = V 4+ = 0,
168
Directional couplers and power dividers
that are exactly the excitation conditions needed to evaluate the elements of the first row of the scattering matrix. Owing to the structure symmetry we also have for the reflected waves: 1. 2. 3. 4.
V 1−a = V 2−a = V 3−a = V 4−a V 1−b = V 2−b = V 3−b = V 4−b V 1−c = V 2−c = V 3−c = V 4−c V 1−d = V 2−d = V 3−d = V 4−d ;
−
− −
− −
−
it follows that: V 1−a + V 1−b + V 1−c + V 1−d V 1−a + V 1−b + V 1−c + V 1−d S 11 = + = 4V + V 1a + V 1+b + V 1+c + V 1+d V 2−a + V 2−b + V 2−c + V 2−d V 1−a + V 1−b V 1−c S 21 = + = 4V + V 1a + V 1+b + V 1+c + V 1+d
−
V 3−a + V 3−b + V 3−c + V 3−d V 1−a = V 1+a + V 1+b + V 1+c + V 1+d V − + V 4−b + V 4−c + V 4−d V 1−a S 41 = 4+a = V 1a + V 1+b + V 1+c + V 1+d S 31 =
− V 1−d
− V 1−b + V 1−c − V 1−d 4V +
− V 1−b − V 1−c + V 1−d 4V +
and taking into account that V 1+a = V 1+b = V 1+c = V 1+c = V + we can also write: 1 S 11 = 4 S 21 = S 31 = S 41 =
1 4 1 4
V 1−b V 1−d V 1−a V 1−c + ++ ++ + V 1+a V 1b V 1c V 1c V 1−b V 1−a + V 1+a V 1+b V 1−a V 1+a
1 V 1−a 4
V 1+a
−
−
1c
1c
−
−
−
1b
1c
1c
V −
V −
V −
1b
1c
1c
− V V 1+c − V V 1+d
− V V 1+b + V V 1+c − V V 1+d − V 1+b − V 1+c + V 1+d
= = = =
Γa + Γ b + Γ c + Γ d 4 Γa + Γ b
− Γc − Γd 4
Γa
− Γb + Γ c − Γd 4
Γa
− Γb − Γc + Γd 4
where Γa , Γb , Γc and Γd are the reflection coefficients at port 1 obtained for the four excitations. Application of the four excitations physically corresponds to introducing in the structure magnetic planes (planes of even symmetry implying zero current) or electric planes (planes of odd symmetry implying a short circuit) as shown in Fig. 4.32a, 4.32b, 4.32c, 4.32d. It follows that the reflection cofficients at port 1 corresponding to the four excitations can be simply derived from the circuits shown in Fig. 4.33a, 4.33b, 4.33c, 4.33d. To analyze the four configurations we only need to report the expression of a short circuit and open circuit line input admittances: sc Y in =j Y 0 tan (θ) oc Y in =
− j Y 0 cot (θ)
4.5 Interference couplers
Figure 4.32 Even
and/or odd excitation of a symmetric branch-line coupler.
Figure 4.33 Equivalent
circuit for the four excitations reported in Fig. 4.32.
169
170
Directional couplers and power dividers
where the electrical length is θ = 2πL/λ; in the four cases we have that the input admittance can be obtained as the parallel of two open or short circuit admittances as follows: sc sc Y a = Y in 1 + Y in2 = j Y 01 tan (θ1 ) + j Y 02 tan (θ2 ) sc oc Y b = Y in 1 + Y in2 = j Y 01 tan (θ1 )
− j Y 02 cot (θ2)
oc sc Y c = Y in 1 + Y in2 = j Y 01 cot (θ1 ) + j Y 02 tan (θ2 ) oc Y d = Y in 1 +
− − j Y 01 cot (θ1) − j Y 02 cot (θ2)
oc Y in 2 =
where Y 01 and θ1 = πl1 /λ (Y 02 and θ2 = πl2 /λ) are the characteristic admittance and electrical length of the horizontal (vertical) lines. (The line length considered is half of the total side, see Fig. 4.31.). Taking into account the definition of the reflection coefficient in terms of admittances: Γk =
Y 0 Y k , Y 0 + Y k
−
k = a,b,c,d
we immediately have: Y 0 j Y 01 tan (θ1 ) j Y 02 tan (θ2 ) Y 0 + j Y 01 tan (θ1 ) + j Y 02 tan (θ2 ) Y 0 j Y 01 tan (θ1 ) + j Y 02 cot (θ2 ) Γb = Y 0 + j Y 01 tan (θ1 ) j Y 02 cot (θ2 ) Y 0 + j Y 01 cot (θ1 ) j Y 02 tan (θ2 ) Γc = Y 0 j Y 01 cot (θ1 ) + j Y 02 tan (θ2 ) Y 0 + j Y 01 cot (θ1 ) + j Y 02 cot (θ2 ) Γd = . Y 0 j Y 01 cot (θ1 ) j Y 02 cot (θ2 )
Γa =
−
−
−
− −
− −
−
(4.102)
(4.103)
(4.104)
(4.105)
with the centerband values (θ1 = θ 2 = π/4): Y 0 j Y 01 j Y 02 Y 0 + j Y 01 + j Y 02 Y 0 j Y 01 + j Y 02 Γb (f 0 ) = Y 0 + j Y 01 j Y 02 Y 0 + j Y 01 j Y 02 Γc (f 0 ) = Y 0 j Y 01 + j Y 02 Y 0 + j Y 01 + j Y 02 Γd (f 0 ) = . Y 0 j Y 01 j Y 02
Γa (f 0 ) =
− − − −
4.5.2
−
(4.106) (4.107)
− − −
(4.108)
(4.109)
Lumped-parameter directional couplers The distributed coupling typical of coupled-line coupling, but also the interference effects on which branch-line couplers are based, can be implemented not only through distributed elements, but also though lumped-parameter elements.
4.5 Interference couplers
171
Figure 4.34 Lumped
parameter π section replacing a quarter-wave transformer.
We only show examples concerning branch-line couplers, that mimic in a more straightforward way their distributed counterpart. Lumped couplers have an important advantage in terms of size when compared to distributed ones, but they are limited by the operation frequency and by losses; moreover, in their implemetation designers try to avoid inductors that have a larger size, more strict frequency limitation and larger losses. We start from an example showing that a π network made of two parallel capacitors C and one series inductor L shows, around a resonant frequency ω 0 = 1/ LC , a behaviour similar to a quarter-wave transmission line (see Fig. 4.34). In fact, imposing the resonance at ω0 , the capacitor admittance and inductor impedance can be written as:
√
ω Y c = j Y, ω0
ω 1 Z l = j , ω0 Y
Y =
C . L
Suppose now that the π section is loaded by an impedance Z L (admittance Y L ); it is straightforward to evaluate the input admittance of the loaded section Y in as:
− −
ω j + ω0 Y in = Y
Y L ω +j Y ω0
1
ω ω0
1
2
+j
ω ω0
2
ω Y L ω0 Y
At the resonant frequency we have: Y in (ω0 ) =
Y 2 Y L
i.e. the same behaviour as a quarter-wave transformer. A comparison of the frequency behaviours of the distributed and lumped transformer is shown in Fig. 4.35; the center frequency is 5 GHz, the load is 50 Ω and the equivalent characteristic impedance is Z = 1/Y = 100 Ω. While it is clear that the frequency behaviour of the lumped design is slightly less favourable than in the distributed case, the centerband behaviour is exactly the same.
172
Directional couplers and power dividers
Figure 4.35 Comparison
between the frequency behaviour of a lumped and distributed quarter-wave transformer closed on 50 Ω and with a center frequency of 5 GHz; the equivalent line impedance is 100 Ω.
Figure 4.36 Lumped
parameter branch-line coupler: left, origin from π sections; right, practical implementation.
Taking into account that a branchline coupler has (see Fig. 4.31) two horizontal quarter-wave lines of impedance Z 01 and two vertical lines of impedance Z 02 , we could suppose to mimic the centerband behaviour of the coupler by subtituting a lumped equivalent with capacitances and inductances C 1 , L1 and C 2 , L2 respectively, see Fig. 4.36. Defining the susceptances at centerband (the resonant condition imposes that at centerband the inductor and capacitor susceptances
173
4.5 Interference couplers
Figure 4.37 Equivalent
circuits at port 1 resulting from odd and even excitations along the vertical and horizontal planes.
are equal and opposite: jω0 C 1 = jB1 , jω0 C 2 = jB2 1 1 1 1 = = jB1 , = = jB2 jω0 L1 jX 1 jω0 L2 jX 2
−
−
we can carry out the same analysis already discussed for the distributed coupler and based on the superposition of even and odd excitations with respect to the horizontal and vertical planes. The four cases (a)-(d) analyzed in Example 4.3 now lead, for the input admittance at port 1, to the configurations shown in Fig. 4.37; in correspondance of a magnetic wall (even excitation) the inductive element is splitted in a series of two and terminated by an open, while in the presence of an electric wall (odd excitation) the element is splitted in a series of two and terminated by a short. Finally, the input admittances at port 1 and the related reflection coefficients with respect to the termination admittance Y 0 corresponding to the four cases are: Y 2 − 2jY 0 (B1 + B2 ) − (B1 + B2 )2 1 + B2 ) = 0 → Γa = Y Y 00 +− j(B j(B1 + B2 ) Y 02 + (B1 + B2 )2 Y 0 − j(B1 − B2 ) Y 2 − 2jY 0 (B1 − B2 ) − (B1 − B2 )2 b Y in = j(B1 − B2 ) → Γb = = 0 Y 0 + j(B1 − B2 ) Y 02 + (B1 − B2 )2 Y 0 + j(B1 − B2 ) Y 2 + 2jY 0 (B1 − B2 ) − (B1 − B2 )2 c Y in = − j(B1 − B2 ) → Γc = = 0 Y 0 − j(B1 − B2 ) Y 02 + (B1 − B2 )2 Y 0 + j(B1 + B2 ) Y 2 + 2jY 0 (B1 + B2 ) − (B1 + B2 )2 d Y in = − j(B1 + B2 ) → Γd = = 0 Y 0 − j(B1 + B2 ) Y 02 + (B1 + B2 )2 a Y in = j(B1 + B2 )
174
Directional couplers and power dividers
Straightforward but lengthy computations lead to the centerband result: Γa + Γ b + Γ c + Γ d Y 04 (B12 B22 )2 S 11 = = 2 4 [Y 0 + (B1 + B2 )2 ] [Y 02 + (B1
−
−
Matching at port 1 requires the condition:
B12
− B22
− B2)2]
(4.110)
= Y 02
and, assuming for instance B 1 > B2 this leads to: B12
− B22 = Y 02
Applying this conditions the other elements of the first row of the scattering matrix turn out to assume the value: S 21 =
Γa + Γ b Γa
− Γc − Γd =
− j2Y 0B1
Y 02
+ B12 +
B22
4 Γb + Γ c Γd =0 4 Γb Γc + Γ d 2B1 B2 = 2 = 4 Y 0 + B12 + B22
− − Γa − − S 41 = S 31 =
−
− BY 01
= j
(4.111a) (4.111b)
− BB21
(4.111c)
Thus port 3 is isolated while the phase difference between port 2 (coupled) and port 4 (in transmission) is again 90 degrees as in the distributed branchline coupler. For 3 dB coupling we have again:
|S 21| = BY 01 = √ 12 → B1 =
√
2Y 0
(4.112a)
|S 41| = BB21 = √ 12 → B2 = √ 12 B1 = Y 0 (4.112b) Notice that this identically satisfies the condition B12 − B22 = Y 02 . The result
obtained is very similar to the distributed counterpart. In practice the branchline lumped coupler is replaced by solutions requiring a lower number of inductors, but coupled together, i.e. at least one transformer that plays the same role of coupled lines in the lumped design.
Example 4.4:
• Design a lumped coupler with center frequency 10 GHz, 3 dB coupling, closed on 50 Ω. ◦ We have: √ √ B1 =
2Y 0 =
2/50 = 2. 8284
B2 = Y 0 = 1/50 = 0.02 S.
× 10−2 S
4.5 Interference couplers
175
Figure 4.38 The
hybrid ring.
We then have: B1 2. 8284 10−2 = = 0.45 pF ω0 2π 1010 B2 2. 10−2 C 2 = = = 0.318 pF ω0 2π 1010 1 1 L1 = = = 56, 27 nH 10 ω0 B1 2π 10 2. 8284 10−2 1 1 L2 = = = 79, 5 nH 10 ω0 B2 2π 10 2. 10−2 C 1 =
× ×
×
×
× ×
4.5.3
· × · ×
The hybrid ring The hybrid ring is a four-port coupler shown in Fig. 4.38. At centerband the scattering matrix can be shown to be:
−
j
S =
j
0 Z 0 Z 02 0
Z 0 Z 01
Z 0 j Z 02
0
−
0 Z 0 j Z 01 0
j
Z 0 Z 01 0
0 − j Z Z 02
Z 0 j Z 01 0 Z 0 j Z 02
−
0
(4.113)
176
Directional couplers and power dividers
where Z 01 and Z 02 satisfy condition: Z 02 Z 02 2 + Z 2 = 1 Z 01 02 related to power conservation. The coupler is matched at all ports at centerband; port 2 is coupled to port 1, port 3 is isolated, port 4 is transmitted but with 180 ◦ phase; the 180 degrees hybrid follow. Given Z 0 , we only have one degree of freedom to establish the structure coupling. In particular, if we identify with port 2 the coupled port, we have C = Z 0 /Z 02 , 1 C 2 = Z 0 /Z 01 , i.e.:
√ −
Z 01 =
√ 1Z −0 C 2
(4.114)
Z 0 (4.115) C In a 3 dB coupler we only have to impose Z 01 = Z 02 from which we obtain: Z 02 =
Z 01 = Z 02 =
√
2Z 0 .
Thus, a 3 dB coupler 50 Ω requires 70.7 Ω microstrip impedances, that are easily realized. Notice that the 3 dB coupler has uniform impedances along the whole ring.
4.6
Power combiners and dividers In power amplifiers, there is often the problem of dividing the input power between several active devices and then to suitably combine the outputs of such devices. Limitations on the maximum power than can be achieved per unit gate width (of the order of 2 W/mm on GaAs; for the sake of definiteness we concentrate on FETs but the same remarks hold for bipolars) and on the maximum gate width per single device poses an upper limit to the maximum power that can be extracted from a single lumped or integrated devices. Series or parallel combinations (that combine the output voltages and currents, respectively; similar remarks hold for voltage or current input dividers) of active devices have a basic shortcoming, they do not preserve the impedance level on which the single device is matched. Suppose e.g. to parallel two devices with output current I and output voltage V matched on Z ; this roughly implies V = Z I ; but two devices in parallel have 2I so that the matching impedance should be changed to Z/2 to allow each device to operate with V , meaning that if the Z matching level has to be preserved at the output a matching section has to be added transforming Z/2 into Z . In conclusion, power dividing and combining has better be made by structures able to preserve the matching level, at least at centerband. The most common case of power division and combination is the division by 2 and combination of 2 elements; powers of 2 can be obtained by properly cascading
4.6 Power combiners and dividers
177
Figure 4.39 Wilkinson
divider.
power dividers and combiners. Although division and combination can be carried out by 3 dB directional couplers, there are other structures more specific to the purpose, which also have the advantage of greater compactness and simplicity. Several power divider structures have been proposed in the past; we concentrate here on the so-called Wilkinson divider; as in interference couplers, this structure is easily introduced as distributed, but can be also implemented in concentrated form.
4.6.1
Wilkinson distributed dividers The structure of a distributed Wilkinson divider is shown in Fig. 4.39. The analysis can be carried out by taking into account that the divider results from the connection of 3 two-ports, two transmission lines with characteristic impedance Z 01 (admittance Y 01 ) and electrical length θ, and a π structure with zero parallel conductance and series resistance R (conductance G), see Fig. 4.40. As shown in Example 4.6 the admittance matrix of the divider is:
Y =
l 2Y 11
l Y 12
l l R Y 12 Y 11 + Y 11 l Y 12
R Y 12
l Y 12 R Y 12 l R Y 11 + Y 11
178
Directional couplers and power dividers
where the elements of the admittance matrix of the two lines of electrical length θ are (see Example 4.5): l l Y 11 = Y 22 = jY 01 cot θ
(4.116)
l l Y 12 = Y 21 = jY 01 / sin θ
(4.117)
−
while the admittance matrix of the π structure is: R R Y 11 = Y 22 = G R Y 12
R = Y 21
=
−G.
(4.118)
(4.119)
We therefore obtain:
−
j2Y 01 cot θ
Y =
jY 01 / sin θ
jY 01 / sin θ G jY 01 / sin θ
jY 01 / sin θ
− j2Y 01 cot θ −G . G − j2Y 01 cot θ −G
At centerband θ = π/2, l = λ/4; thus, the normalized matrix on Z 0 is:
y = Y Z 0 =
0
jY 01 Z 0 jY 01 Z 0
jY 01 Z 0 GZ 0 jY 01 Z 0
−GZ 0
− GZ 0
GZ 0
from which the scattering matrix:
S =
2 Z 2 1 2Y 01 2jY 01 Z 0 2jY 01 Z 0 0 2 2 2 2 2 Z 2 1 + 2Y 01 Z 0 1 + 2Y 01 Z 0 1 + 2Y 01 0 2 2 3 2 G Y 01 Z 0 Z 0 2jY 01 Z 0 1 4Y 01 Z 0 G 2 2 2 2 2 3 2 Z 2 + 4Y 2 Z 3 G 1 + 2Y 01 Z 0 1 + 2GZ 0 + 2Y 01 Z 0 + 4Y 01 Z 0 G 1 + 2GZ 0 + 2Y 01 0 01 0 2 2 3 2 G Y 01 Z 0 Z 0 2jY 01 Z 0 1 4Y 01 Z 0 G 2 2 2 2 2 3 2 Z 2 + 4Y 2 Z 3 G 1 + 2Y 01 Z 0 1 + 2GZ 0 + 2Y 01 Z 0 + 4Y 01 Z 0 G 1 + 2GZ 0 + 2Y 01 0 01 0
− − −
−
−
−
−
−
−
−
−
Imposing matching at port 1 we obtain:
√
Y 01 = 1/( 2Z 0 ) i.e.: Z 01 =
√
2Z 0 ,
from which:
− √ − √ 0
S =
1 j 2 1 j 2 G 2
1 j 2 1 2GZ 0 2 + 4GZ 0 1 Z 0 2Z 0 2 + 4GZ 0
− √ −
−
1 j 2
− √
2 G
−
1 Z 0 2Z 0 2 + 4GZ 0 1 2GZ 0 2 + 4GZ 0
−
;
.
4.6 Power combiners and dividers
179
Figure 4.40 Scheme
for the analysis of the Wilkinson divider.
Matching at ports 2 and 3 implies: R = 1/G = 2Z 0 , from which, finally:
− √ 0
S =
1 j 2 1 j 2
− √
1 j 2
1 j 2
− √
− √ 0
0
0
0
The structure of the scattering matrix implies matching at all ports, isolation between ports 2 and 3, transmission from 1 to 2 and from 1 to 3 with 3 dB power division, phase shift of π/2 between input and output. Notice that, contrarily with the Lange ocupler, the two outputs are in phase. The divider is narrowband but the bandwidth can be increased by multisection structures.
−
Example 4.5:
• Evaluate the admittance matrix of a line with electrical length θ and characteristic admittance Y . ◦ A line is a symmetric and reciprocal two-port; from the definition of the admit0
tance matrix we have;
I 1 Y 11 = Y 22 = V 1 Y 21 = Y 12 =
I 2 V 1
V 2 =0
V 2 =0
Taking into account the expression of the voltages and currents in terms of forward and backward propagating waves we have (we assume the line length to
180
Directional couplers and power dividers
be L and the guided walevelength λ g ): V (z) = V + (z) + V − (z) I (z) = Y 0 V + (z)
− V −(z)
− − − − −
L z λg L z V − (z) = V − (L)exp j2π λg L z I + (z) = Y 0 V + (L)exp j2π λg L z I − (z) = Y 0 V − (L)exp j2π λg +
+
V (z) = V (L)exp( j2π
−
where section z = 0 is port 1 and z = L is port 2. If port 2 is shorted, then V 2 = V (L) = 0 or: V + (L) =
−V −(L)
I (L) = 2Y 0 V + (L) The port voltages and currents at port 1 and 2 can therefore expressed as: V 1 =V (0) = V + (0) + V − (0) = V + (L)[exp(jθ)
− exp(− jθ)] = 2jV +(L)sin θ
I 1 =I (0) = Y 0 V + (L)(exp(jθ) + exp( jθ)) = 2Y 0 V + (L)cos θ
−
V 2 =V (L) = 0 I 2 =
− I (L) = −2Y 0V +(L)
thus: I 1 Y 11 = Y 22 = V 1 Y 21 = Y 12 =
I 2 V 1
= V 2 =0
− jY 0 cot θ
=jY 0 / sin(θ) V 2 =0
Example 4.6:
• Consider three two-ports wih admittance matrices
Y a , Y b , Y c combined in
a triangle so that at port 1 the inputs of a and b are in parallel, at port 2 the output of a is in parallel with the input of c and at port 3 the output of c is in parallel with the output of b. Derive the admittance matrix of the three-port with ports 1, 2 and 3.
4.6 Power combiners and dividers
◦ We have the following equations:
a a a a I 1a = Y 11 V 1 + Y 12 V 2
a a a a I 2a = Y 21 V 1 + Y 22 V 2
b b I 1b = Y 11 V 1b + Y 12 V 2b
b b I 2b = Y 21 V 1b + Y 22 V 2b
c c I 1c = Y 11 V 1c + Y 12 V 2c
c c I 2c = Y 21 V 1c + Y 22 V 2c
181
but V 1a = V 1b = V 1 ; V 2a = V 1c = V 2 ; V 2c = V 2b = V 3 ; moreover I 1 = I 1a + I 1b ; I 2 = I 2a + I 1c ; I 3 = I 2b + I 2c . Substituting: a a I 1a = Y 11 V 1 + Y 12 V 2
a a I 2a = Y 21 V 1 + Y 22 V 2
b b I 1b = Y 11 V 1 + Y 12 V 3
b b I 2b = Y 21 V 1 + Y 22 V 3
c c I 1c = Y 11 V 2 + Y 12 V 3
c c I 2c = Y 21 V 2 + Y 22 V 3
and then summing we have: a b a b I 1 = I 1a + I 1b = Y 11 + Y 11 V 1 + Y 12 V 2 + Y 12 V 3
a a c c I 2 = I 2a + I 1c = Y 21 V 1 + (Y 22 + Y 11 ) V 2 + Y 12 V 3 b c b c I 3 = I 2b + I 2c = Y 21 V 1 + Y 21 V 2 + Y 22 + Y 22 V 3
which immediately yields the admittance matrix elements. In the distributed Wilkinson divider case we have: a a b b l Y 11 = Y 22 = Y 11 = Y 22 = Y 11 a a b b l Y 12 = Y 21 = Y 12 = Y 21 = Y 12 c c R Y 11 = Y 22 = Y 11 c c R = Y 21 = Y 12 Y 12
leading to the parallel representation: l l l I 1 = 2Y 11 V 1 + Y 12 V 2 + Y 12 V 3 l l R R I 2 = Y 12 V 1 + Y 11 + Y 11 V 2 + Y 11 V 3
l R l R I 3 = Y 12 V 1 + Y 12 V 2 + Y 11 + Y 11 V 3
4.6.2
Wilkinson lumped dividers The structure of a lumped Wilkinson divider is shown in Fig. 4.41. The two transmission lines are replaced by lumped element quarterwave equivalents. We shall limit the analysis to centerband where: jω 0 C = jB 1 1 = = jB jω 0 L jX
−
182
Directional couplers and power dividers
Figure 4.41 Lumped
Wilkinson divider.
leading to the admittance matrix elements of the lumped quarterwave equivalent: l l Y 11 = Y 22 =0 l l Y 12 = Y 21 =
(4.120)
jB
(4.121)
while the admittance matrix of the π resistive structure is always: R R Y 11 = Y 22 = G R R Y 12 = Y 21 =
−G.
(4.122)
(4.123)
Exploiting again the result of Example 4.6, the centerband admittance matrix of the divider is:
Y =
l 2Y 11
l Y 12
l Y 12
l l R Y 12 Y 11 + Y 11 l Y 12
R Y 12
R Y 12 l R Y 11 + Y 11
with normalized matrix:
y = Y Z 0 =
0
S =
=
0 jB jB
jB G
−GZ 0
−G
jB
jBZ 0 jBZ 0
jBZ 0 GZ 0 jBZ 0
and scattering matrix:
−GZ 0 GZ 0
−G
G
1 2B 2 Z 02 2jBZ 0 2jBZ 0 2 2 2 2 1 + 2B Z 0 1 + 2B Z 0 1 + 2B 2 Z 02 2 G B 2 Z 0 Z 0 2jBZ 0 1 4B 2 Z 03 G 1 + 2B 2 Z 02 1 + 2GZ 0 + 2B 2 Z 02 + 4B 2 Z 03 G 1 + 2GZ 0 + 2B 2 Z 02 + 4B 2 Z 03 G 2 G B 2 Z 0 Z 0 2jBZ 0 1 4B 2 Z 03 G 1 + 2B 2 Z 02 1 + 2GZ 0 + 2B 2 Z 02 + 4B 2 Z 03 G 1 + 2GZ 0 + 2B 2 Z 02 + 4B 2 Z 03 G
− − −
−
−
−
−
−
−
−
−
.
4.7 Conclusions
183
Imposing matching at port 1 we obtain:
√
B = 1/( 2Z 0 ) while matching t ports 2 and 3 implies: 1
− 4B2Z 03G = 1 − 2Z 0G = 0 → R = 2Z 0
leading to the centerband scattering matrix of the matched coupler:
− √ 0
S =
1 j 2 1 j 2
− √
1 j 2
− √ 0 0
exactly as in the distributed implementation.
4.7
1 j 2
− √ 0 0
Conclusions Fig. 4.42 summarizes the design rules and the centerband behaviour of the directional couplers and power dividers examined in this chapter. Some final remarks may be helpful:
4.8
The coupled line and branch line couplers have a phase shift of 90 degrees between the two coupled and transmission ports, and are therefore called 90 degrees hybrid; the hybrid loop or rat race coupler introduces a phase shift of 180 degrees between the two outputs (180 degrees hybrid), while in the Wilkinson divider the two outputs are in phase. The branch line and hybrdi ring couplers easily permit to have high coupling, while they are critical for low coupling; on the other hand, coupled line couplers do not allow for high coupling, apart from the multicunductor version (the Lange coupler). All distributed directional couplers have a rather large layout. Therefore distributed couplers are seldom used in integrated circuits, especially at relatively low frequencies, where lumped parameter couplers and dividers are preferred.
Questions and problems 1. Q In a coupled two-conductor microstrip the even mode permittivity is 8 while the odd mode permittivity is 6. The odd and even mode impedances are 75 and 40 Ω, respectively. Are the previous data physically correct? 2. P What is a directional coupler? Imagine that an ideal 3dB, 90 ◦ coupler is fed with a 100 mW signal. What is the power on the coupled and the transmission
184
Directional couplers and power dividers
Figure 4.42 Summary
3. 4. 5. 6. 7.
of the main distributed parameter combiners and dividers.
port, respectively? What is the power on the insulated port? What is the phase difference between the coupled and transmission ports? Q Sketch the layout of a Lange coupler and of a branch-line coupler and indicate the centerband dimensions. Q Is the hybrid ring a 90 ◦ or 180◦ coupler? P A Wilkinson divider on 50 Ω loads operates at 10 GHz. Assuming eff = 4 evaluate the lengths and characteristic impedance of the divider arms. Q Explain the difference between a 90 ◦ and 180◦ m, degrees hybrid. Q Sketch the layout of a branch-line directional coupler.
4.8 Questions and problems
185
8. Q Sketch the layout of a Wilkinson power divider. 9. Q Explain the difference between an unfolded and a folded Lange coupler. 10. Q Justify the fact that 3 dB couplers can be implemented through multiconductor coupled microstrips but not, in practice, through two-conductor coupled microstrips. 11. Q Exaplain why a low-coupling coupler (e.g. a 10 dB coupler) cannot easily be implemented through a branch-line coupler. 12. P Design a 10 dB coupler on two-conductor coupled microstrips; the substrate is teflon (effective permittivity 3) with thickness 0.5 mm; the centerband frequency is 10 GHz and the closing impedance 50 Ω. 13. P Design a four condcutor Lange 3 dB coupler; the substrate is teflon (effective permittivity 3) with thickness 0.5 mm; the centerband frequency is 10 GHz and the closing impedance 50 Ω. 14. Q In two coupled microstrips: a. The even and odd mode are always velocity matched b. The even and odd mode are typically velocity mismatched c. The even mode velocity is larger than the odd mode velocity. 15. Q Discuss the opportunity of implementing Lange couplers with a large number of conductors (say greater than 6). 16. Q Velocity mismatch in a directional coupler is particularly critical for: a. input matching b. coupling and transmission c. isolation. 17. Q Discuss available compensation techniques for the coupler velocity mismatch. 18. Q Sketch a hybrid ring coupler and explain its operation. 19. Q Design (dimensions and impedances) a hybrid ring with 3 dB coupling on 50 Ω at 5 GHz. Assume that the line effective permittivity is 5. 20. Q Design (dimensions and impedances) a branch-line coupler with 3 dB coupling on 100 Ω at 20 GHz. Assume that the line effective permittivity is 5. 21. Q Design (dimensions and impedances) a Wilkinson divider on 70 Ω at 30 GHz. Assume that the line effective permittivity is 2.
5
Active microwave devices and device models
186
6
Noise and noise models
187
7
The linear amplifiers
188
8
Power amplifiers
189
