Berkeley

Power Gain and Stability

Prof. Ali M. Niknejad

U.C. BerkeleyCopyright c© 2014 by Ali M. Niknejad

September 17, 2014

Niknejad Power Gain

Power Gain

Niknejad Power Gain

Power Gain

+vs

!

YS

YL

!y11 y12

y21 y22

"

Pin PL

Pav,s Pav,l

We can define power gain in many different ways. The powergain Gp is defined as follows

Gp =PL

Pin= f (YL,Yij) 6= f (YS)

We note that this power gain is a function of the loadadmittance YL and the two-port parameters Yij .

Niknejad Power Gain

Power Gain (cont)

The available power gain is defined as follows

Ga =Pav ,L

Pav ,S= f (YS ,Yij) 6= f (YL)

The available power from the two-port is denoted Pav ,L

whereas the power available from the source is Pav ,S .

Finally, the transducer gain is defined by

GT =PL

Pav ,S= f (YL,YS ,Yij)

This is a measure of the efficacy of the two-port as itcompares the power at the load to a simple conjugate match.

Niknejad Power Gain

Bi-Conjugate Match

When the input and output are simultaneously conjugatelymatched, or a bi-conjugate match has been established, wefind that the transducer gain is maximized with respect to thesource and load impedance

GT ,max = Gp,max = Ga,max

This is thus the recipe for calculating the optimal source andload impedance in to maximize gain

Yin = Y11 −Y12Y21

YL + Y22= Y ∗

S

Yout = Y22 −Y12Y21

YS + Y11= Y ∗

L

Solution of the above four equations (real/imag) results in theoptimal YS ,opt and YL,opt .

Niknejad Power Gain

Calculation of Optimal Source/Load

Another approach is to simply equate the partial derivatives ofGT with respect to the source/load admittance to find themaximum point

∂GT

∂GS= 0;

∂GT

∂BS= 0

∂GT

∂GL= 0;

∂GT

∂BL= 0

Niknejad Power Gain

Optimal Power Gain Derivation (cont)

Again we have four equations. But we should be smarterabout this and recall that the maximum gains are all equal.Since Ga and Gp are only a function of the source or load, wecan get away with only solving two equations. For instance

∂Ga

∂GS= 0;

∂Ga

∂BS= 0

This yields YS,opt and by setting YL = Y ∗out we can find the

YL,opt .

Likewise we can also solve

∂Gp

∂GL= 0;

∂Gp

∂BL= 0

And now use YS ,opt = Y ∗in.

Niknejad Power Gain

Optimal Power Gain Derivation

Let’s outline the procedure for the optimal power gain. We’lluse the power gain Gp and take partials with respect to theload. Let

Yjk = mjk + jnjk

YL = GL + jXL

Y12Y21 = P + jQ = Le jφ

Gp =|Y21|2

|YL + Y22|2<(YL)

<(Yin)=|Y21|2

DGL

<(

Y11 −Y12Y21

YL + Y22

)= m11 −

<(Y12Y21(YL + Y22)∗)

|YL + Y22|2

D = m11|YL + Y22|2 − P(GL + m22)− Q(BL + n22)

∂Gp

∂BL= 0 = −|Y21|2GL

D2

∂D

∂BL

Niknejad Power Gain

Optimal Load (cont)

Solving the above equation we arrive at the following solution

BL,opt =Q

2m11− n22

In a similar fashion, solving for the optimal load conductance

GL,opt =1

2m11

√(2m11m22 − P)2 − L2

If we substitute these values into the equation for Gp (lot’s ofalgebra ...), we arrive at

Gp,max =|Y21|2

2m11m22 − P +√

(2m11m22 − P)2 − L2

Niknejad Power Gain

Final Solution

Notice that for the solution to exists, GL must be a realnumber. In other words

(2m11m22 − P)2 > L2

(2m11m22 − P) > L

K =2m11m22 − P

L> 1

This factor K plays an important role as we shall show that italso corresponds to an unconditionally stable two-port. Wecan recast all of the work up to here in terms of K

YS ,opt =Y12Y21 + |Y12Y21|(K +

√K 2 − 1)

2<(Y22)

YL,opt =Y12Y21 + |Y12Y21|(K +

√K 2 − 1)

2<(Y11)

Gp,max = GT ,max = Ga,max =Y21

Y12

1

K +√

K 2 − 1

Niknejad Power Gain

Maximum Gain

The maximum gain is usually written in the followinginsightful form

Gmax =Y21

Y12(K −

√K 2 − 1)

For a reciprocal network, such as a passive element, Y12 = Y21

and thus the maximum gain is given by the second factor

Gr ,max = K −√

K 2 − 1

Since K > 1, |Gr ,max | < 1. The reciprocal gain factor isknown as the efficiency of the reciprocal network.

The first factor, on the other hand, is a measure of thenon-reciprocity.

Niknejad Power Gain

Unilateral Maximum Gain

For a unilateral network, the design for maximum gain istrivial. For a bi-conjugate match

YS = Y ∗11

YL = Y ∗22

GT ,max =|Y21|2

4m11m22

Niknejad Power Gain

Ideal MOSFET

roCgs gmvin

+vin

−Cds

The AC equivalent circuit for a MOSFET at low to moderatefrequencies is shown above. Since |S11| = 1, this circuit hasinfinite power gain. This is a trivial fact since the gatecapacitance cannot dissipate power whereas the output candeliver real power to the load.

Niknejad Power Gain

Real MOSFET

Cgs gmvin

+vin

−

RiRi

CdsRds Rds

+vs

−

A more realistic equivalent circuit is shown above. If we makethe unilateral assumption, then the input and output powercan be easily calculated. Assume we conjugate match theinput/output

Pavs =|VS |28Ri

PL = <( 12 ILV ∗

L ) = 12

∣∣∣∣gmV1

2

∣∣∣∣2

Rds

GTU,max = g 2mRdsRi

∣∣∣∣V1

VS

∣∣∣∣2

Niknejad Power Gain

Real MOSFET (cont)

At the center resonant frequency, the voltage at the input ofthe FET is given by

V1 =1

jωCgs

VS

2Ri

GTU,max =Rds

Ri

(gm/Cgs)2

4ω2

This can be written in terms of the device unity gainfrequency fT

GTU,max =1

4

Rds

Ri

(fTf

)2

The above expression is very insightful. To maximum powergain we should maximize the device fT and minimize theinput resistance while maximizing the output resistance.

Niknejad Power Gain

Stability of a Two-Port

Niknejad Power Gain

Stability of a Two-Port

A two-port is unstable if the admittance of either port has anegative conductance for a passive termination on the secondport. Under such a condtion, the two-port can oscillate.Consider the input admittance

Yin = Gin + jBin = Y11 −Y12Y21

Y22 + YL

Using the following definitions

Y11 = g11 + jb11

Y22 = g22 + jb22

Y12Y21 = P + jQ = L∠φ

YL = GL + jBLNow substitute real/imag parts of the above quantities intoYin

Yin = g11 + jb11 −P + jQ

g22 + jb22 + GL + jBL

= g11 + jb11 −(P + jQ)(g22 + GL − j(b22 + BL))

(g22 + GL)2 + (b22 + BL)2

Niknejad Power Gain

Input Conductance

Taking the real part, we have the input conductance

<(Yin) = Gin = g11 −P(g22 + GL) + Q(b22 + BL)

(g22 + GL)2 + (b22 + BL)2

=(g22 + GL)2 + (b22 + BL)2 − P

g11(g22 + GL)− Q

g11(b22 + BL)

D

Since D > 0 if g11 > 0, we can focus on the numerator. Notethat g11 > 0 is a requirement since otherwise oscillationswould occur for a short circuit at port 2.

The numerator can be factored into several positive terms

N = (g22 +GL)2 +(b22 +BL)2− P

g11(g22 +GL)− Q

g11(b22 +BL)

=

(GL +

(g22 −

P

2g11

))2

+

(BL +

(b22 −

Q

2g11

))2

−P2 + Q2

4g 211

Niknejad Power Gain

Input Conductance (cont)

Now note that the numerator can go negative only if the firsttwo terms are smaller than the last term. To minimize thefirst two terms, choose GL = 0 and BL = −

(b22 − Q

2g11

)

(reactive load)

Nmin =

(g22 −

P

2g11

)2

− P2 + Q2

4g 211

And thus the above must remain positive, Nmin > 0, so

(g22 +

P

2g11

)2

− P2 + Q2

4g 211

> 0

g11g22 >P + L

2=

L

2(1 + cosφ)

Niknejad Power Gain

Linvill/Llewellyn Stability Factors

Using the above equation, we define the Linvill stability factor

L < 2g11g22 − P

C =L

2g11g22 − P< 1

The two-port is stable if 0 < C < 1.

Niknejad Power Gain

Stability (cont)

It’s more common to use the inverse of C as the stabilitymeasure

2g11g22 − P

L> 1

The above definition of stability is perhaps the most common

K =2<(Y11)<(Y22)−<(Y12Y21)

|Y12Y21|> 1

The above expression is identical if we interchange ports 1/2.Thus it’s the general condition for stability.

Note that K > 1 is the same condition for the maximumstable gain derived earlier. The connection is now moreobvious. If K < 1, then the maximum gain is infinity!

Niknejad Power Gain

Stability From Another Perspective

We can also derive stability in terms of the input reflectioncoefficient. For a general two-port with load ΓL we have

v−2 = Γ−1

L v +2 = S21v +

1 + S22v +2

v +2 =

S21

Γ−1L − S22

v−1

v−1 =

(S11 +

S12S21ΓL

1− ΓLS22

)v +

1

Γ = S11 +S12S21ΓL

1− ΓLS22

If |Γ| < 1 for all ΓL, then the two-port is stable

Γ =S11(1− S22ΓL) + S12S21ΓL

1− S22ΓL=

S11 + ΓL(S21S12 − S11S22)

1− S22ΓL

=S11 −∆ΓL

1− S22ΓL

Niknejad Power Gain

Stability Circle

To find the boundary between stability/instability, let’s set|Γ| = 1 ∣∣∣∣

S11 −∆ΓL

1− S22ΓL

∣∣∣∣ = 1

|S11 −∆ΓL| = |1− S22ΓL|After some algebraic manipulations, we arrive at the followingequation

∣∣∣∣ΓL −S∗

22 −∆∗S11

|S22|2 − |∆|2∣∣∣∣ =

|S12S21||S22|2 − |∆|2

This is of course an equation of a circle, |ΓL − C | = R, in thecomplex plane with center at C and radius R

Thus a circle on the Smith Chart divides the region ofinstability from stability.

Niknejad Power Gain

Example: Stability Circle

CS

RS

|S11| < 1

stable reg

ion

unstable reg

ion

In this example, the originof the circle lies outsidethe stability circle but aportion of the circle fallsinside the unit circle. Isthe region of stabilityinside the circle oroutside?

This is easily determined ifwe note that if ΓL = 0,then Γ = S11. So ifS11 < 1, the origin shouldbe in the stable region.Otherwise, if S11 > 1, theorigin should be in theunstable region.

Niknejad Power Gain

Stability: Unilateral Case

Consider the stability circle for a unilateral two-port

CS =S∗

11 − (S∗11S∗

22)S22

|S11|2 − |S11S22|2=

S∗11

|S11|2

RS = 0 |CS | =1

|S11|The cetner of the circle lies outside of the unit circle if|S11| < 1. The same is true of the load stability circle. Sincethe radius is zero, stability is only determined by the locationof the center.

If S12 = 0, then the two-port is unconditionally stable ifS11 < 1 and S22 < 1.

This result is trivial since

ΓS |S12=0 = S11

The stability of the source depends only on the device and noton the load.

Niknejad Power Gain

Mu Stability Test

If we want to determine if a two-port is unconditionally stable,then we should use the µ test

µ =1− |S11|2

|S22 −∆S∗11|+ |S12S21|

> 1

The µ test not only is a test for unconditional stability, butthe magnitude of µ is a measure of the stability. In otherwords, if one two port has a larger µ, it is more stable.

The advantage of the µ test is that only a single parameterneeds to be evaluated. There are no auxiliary conditions likethe K test derivation earlier.

The derivation of the µ test proceeds as follows. First letΓS = |ρs |e jφ and evaluate Γout

Γout =S22 −∆|ρs |e jφ1− S11|ρs |e jφ

Niknejad Power Gain

Mu Test (cont)

Next we can manipulate this equation into the following circle|Γout − C | = R

∣∣∣∣Γout +|ρs |S∗

11∆− S22

1− |ρs ||S11|2∣∣∣∣ =

√|ρs ||S12S21|

(1− |ρs ||S11|2)

For a two-port to be unconditionally stable, we’d like Γout tofall within the unit circle

||C |+ R| < 1

||ρs |S∗11∆− S22|+

√|ρs ||S21S12| < 1− |ρs ||S11|2

||ρs |S∗11∆− S22|+

√|ρs ||S21S12|+ |ρs ||S11|2 < 1

The worse case stability occurs when |ρs | = 1 since itmaximizes the left-hand side of the equation. Therefore wehave

µ =1− |S11|2

|S∗11∆− S22|+ |S12S21|

> 1

Niknejad Power Gain

K-∆ Test

The K stability test has already been derived using Yparameters. We can also do a derivation based on Sparameters. This form of the equation has been attributed toRollett and Kurokawa.

The idea is very simple and similar to the µ test. We simplyrequire that all points in the instability region fall outside ofthe unit circle.

The stability circle will intersect with the unit circle if

|CL| − RL > 1

or|S∗

22 −∆∗S11| − |S12S21||S22|2 − |∆|2

> 1

This can be recast into the following form (assuming |∆| < 1)

K =1− |S11|2 − |S22|2 + |∆|2

2|S12||S21|> 1

Niknejad Power Gain

Two-Port Power and Scattering Parameters

The power flowing into a two-port can be represented by

Pin =|V +

1 |22Z0

(1− |Γin|2)

The power flowing to the load is likewise given by

PL =|V −

2 |22Z0

(1− |ΓL|2)

We can solve for V +1 using circuit theory

V +1 + V −

1 = V +1 (1 + Γin) =

Zin

Zin + ZSVS

In terms of the input and source reflection coefficient

Zin =1 + Γin

1− ΓinZ0 ZS =

1 + ΓS

1− ΓSZ0

Niknejad Power Gain

Two-Port Incident Wave

Solve for V +1

V +1 (1 + Γin) =

VS(1 + Γin)(1− ΓS)

(1 + Γin)(1− ΓS) + (1 + ΓS)(1− Γin)

V +1 =

VS

2

1− ΓS

1− ΓinΓS

The voltage incident on the load is given by

V −2 = S21V +

1 + S22V +2 = S21V +

1 + S22ΓLV −2

V −2 =

S21V +1

1− S22ΓL

PL =|S21|2

∣∣V +1

∣∣2

|1− S22ΓL|21− |ΓL|2

2Z0

Niknejad Power Gain

Operating Gain and Available Power

The operating power gain can be written in terms of thetwo-port s-parameters and the load reflection coefficient

Gp =PL

Pin=

|S21|2 (1− |ΓL|2)

|1− S22ΓL|2 (1− |Γin|2)

The available power can be similarly derived from V +1

Pavs = Pin|Γin=Γ∗S

=

∣∣V +1a

∣∣2

2Z0(1− |Γ∗

S |2)

V +1a = V +

1

∣∣Γin=Γ∗

S=

VS

2

1− Γ∗S

1− |ΓS |2

Pavs =|VS |28Z0

|1− ΓS |2

1− |ΓS |2

Niknejad Power Gain

Transducer Gain

The transducer gain can be easily derived

GT =PL

Pavs=|S21|2 (1− |ΓL|2)(1− |ΓS |2)

|1− ΓinΓS |2 |1− S22ΓL|2

Note that as expected, GT is a function of the two-ports-parameters and the load and source impedance.

If the two port is connected to a source and load withimpedance Z0, then we have ΓL = ΓS = 0 and

GT = |S21|2

Niknejad Power Gain

Unilateral Gain

+vs

−

Z0

M1 M2

GS GL|S21|2

!S11 0S21 S22

"Z0

If S12 ≈ 0, we can simplify the expression by just assumingS12 = 0. This is the unilateral assumption

GTU =1− |ΓS |2

|1− S11ΓS |2|S21|2

1− |ΓL|2

|1− S22ΓL|2= GS |S21|2 GL

The gain partitions into three terms, which can be interpretedas the gain from the source matching network, the gain of thetwo port, and the gain of the load.

Niknejad Power Gain

Maximum Unilateral Gain

We know that the maximum gain occurs for the biconjugatematch

ΓS = S∗11

ΓL = S∗22

GS ,max =1

1− |S11|2

GL,max =1

1− |S22|2

GTU,max =|S21|2

(1− |S11|2)(1− |S22|2)

Note that if |S11| = 1 of |S22| = 1, the maximum gain isinfinity. This is the unstable case since |Sii | > 1 is potentiallyunstable.

Niknejad Power Gain

Design for Gain

So far we have only discussed power gain using bi-conjugatematching. This is possible when the device is unconditionallystable. In many case, though, we’d like to design with apotentially unstable device.

Moreover, we would like to introduce more flexibility in thedesign. We can trade off gain for

bandwidthnoisegain flatnesslinearityetc.

We can make this tradeoff by identifying a range ofsource/load impedances that can realize a given value ofpower gain. While maximum gain is acheived for a singlepoint on the Smith Chart, we will find that a lot moreflexibility if we back-off from the peak gain.

Niknejad Power Gain

Unilateral Design

No real transistor is unilateral. But most are predominantlyunilateral, or else we use cascades of devices (such as thecascode) to realize such a device.

The unilateral figure of merit can be used to test the validityof the unilateral assumption

Um =|S12|2 |S21|2 |S11|2 |S22|2

(1− |S11|2)(1− |S22|2)

It can be shown that the transducer gain satisfies thefollowing inequality

1

(1 + U)2<

GT

GTU<

1

(1− U)2

Where the actual power gain GT is compared to the powergain under the unilateral assumption GTU . If the inequality istight, say on the order of 0.1 dB, then the amplifier can beassumed to be unilateral with negligible error.

Niknejad Power Gain

Gain Circles

We now can plot gain circles for the source and load. Let

gS =GS

GS ,max

gL =GL

GL,max

By definition, 0 ≤ gS ≤ 1 and 0 ≤ gL ≤ 1. One can show thata fixed value of gS represents a circle on the ΓS plane

∣∣∣∣ΓS −S∗

11gS

|S11|2 (gS − 1) + 1

∣∣∣∣ =

∣∣∣∣∣

√1− gS(1− |S11|2)

|S11|2 (gS − 1) + 1

∣∣∣∣∣

More simply, |ΓS − CS | = RS . A similar equation can bederived for the load. Note that for gS = 1, RS = 0, andCS = S∗

11 corresponding to the maximum gain.

Niknejad Power Gain

Gain Circles (cont)

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AN

GLE O

F TRAN

SMISSIO

N C

OEFFIC

IENT IN

DEG

REES

AN

GLE O

F REF LECTIO

N C

OEFFIC

IENT IN

DEG

REES

—>

WA

VELE

NG

THS

TOW

ARD

GEN

ERA

TOR

—>

<— W

AVE

LEN

GTH

S TO

WA

RD L

OA

D <

—

IND

UCT

IVE

REAC

TANCE

COM

PONEN

T (+jX

/Zo), O

R CAPACITIVE SUSCEPTANCE (+jB/Yo)

CAPACITIVE REACTANCE COMPONENT (-jX/Z

o), OR IN

DUCTIVE

SUSC

EPTA

NCE

(-jB

/Yo)

RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)

S∗11

gS = −1 dB

gS = 0dB

gS = −2 dB

gS = −3 dB

All gain circles lie on the line given by the angle of S∗ii . We

can select any desired value of source/load reflectioncoefficient to acheive the desired gain. To minimize theimpedance mismatch, and thus maximize the bandwidth, weshould select a point closest to the origin.

Niknejad Power Gain

Extended Smith Chart

For |Γ| > 1, we can still employ the Smith Chart if we makethe following mapping. The reflection coefficient for anegative resistance is given by

Γ(−R + jX ) =−R + jX − Z0

−R + jX + Z0=

(R + Z0)− jX

(R − Z0)− jX

1

Γ∗ =(R − Z0) + jX

(R + Z0) + jX

We see that Γ can be mapped to the unit circle by taking1/Γ∗ and reading the resistance value (and noting that it’sactually negative).

Niknejad Power Gain

Potentially Unstable Unilateral Amplifier

For a unilateral two-port with |S11| > 1, we note that theinput impedance has a negative real part. Thus we can stilldesign a stable amplifier as long as the source resistance islarger than <(Zin)

<(ZS) > |<(Zin)|

The same is true of the load impedance if |S22| > 1. Thus thedesign procedure is identical to before as long as we avoidsource or load reflection coefficients with real part less thanthe critical value.

Niknejad Power Gain

Pot. Unstable Unilateral Amp Example

Consider a transistor with the following S-Parameters

S11 = 2.02∠− 130.4◦

S22 = 0.50∠− 70◦

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1.4

1.6

1.6

1.6

1.8

1.8

1.8

2.0

2.0

2.0

3.0

3.0

3.0

4.0

4.0

4.0

5.0

5.0

5.0

10

10

10

20

20

20

50

50

50

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

90-9

085

-85

80-8

0

75-7

5

70-7

0

65-6

5

60-6

0

55-5

5

50-5

0

45

-45

40

-40

35

-35

30

-30

25

-25

20

-20

15

-15

10

-10

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0.07

0.08

0.08

0.09

0.09

0.1

0.1

0.11

0.11

0.12

0.12

0.13

0.13

0.14

0.14

0.15

0.15

0.16

0.16

0.17

0.17

0.18

0.18

0.190.19

0.20.2

0.21

0.210.22

0.220.23

0.230.24

0.24

0.25

0.25

0.26

0.26

0.27

0.27

0.28

0.28

0.29

0.29

0.3

0.3

0.31

0.31

0.32

0.32

0.33

0.33

0.34

0.34

0.35

0.35

0.36

0.36

0.37

0.37

0.38

0.38

0.39

0.39

0.4

0.4

0.41

0.41

0.42

0.42

0.43

0.43

0.440.44

0.45

0.45

0.46

0.46

0.47

0.47

0.48

0.48

0.49

0.49

0.0

0.0

AN

GLE O

F TRAN

SMISSIO

N C

OEFFIC

IENT IN

DEG

REES

AN

GLE O

F REFLECTIO

N C

OEFFIC

IENT IN

DEG

REES

—>

WA

VELE

NG

THS

TOW

ARD

GEN

ERAT

OR

—>

<— W

AVE

LEN

GTH

S TO

WA

RD L

OA

D <

—

IND

UCT

IVE

REAC

TANCE

COM

PONEN

T (+jX

/Zo), O

R CAPACITIVE SUSCEPTANCE (+jB/Yo)

CAPACITIVE REACTANCE COMPONENT (-jX/Z

o), OR IN

DUCTIVE

SUSC

EPTA

NCE

(-jB

/Yo)

RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)

CS

RS

stable region

1

S∗11

GS = 5dB

ΓS

S12 = 0

S21 = 5.00∠60◦

Since |S11| > 1, theamplifier is potentiallyunstable. We begin byplotting 1/S∗

11 to find thenegative real inputresistance.

Now any source inside thiscircle is stable, since<(ZS) > <(Zin).

We also draw the sourcegain circle for GS = 5dB.

Niknejad Power Gain

Amp Example (cont)

The input impedance is read off the Smith Chart from 1/S∗11.

Note the real part is interpreted as negative

Zin = 50(−0.4− 0.4j)

The GS = 5dB gain circle is calculated as follows

gS = 3.15(1− |S11|2)

RS =

√1− gS(1− |S11|2)

1− |S11|2 (1− gS)= 0.236

CS =gSS∗

11

1− |S11|2 (1− gS)= −.3 + 0.35j

We can select any point on this circle and obtain a stable gainof 5 dB. In particular, we can pick a point near the origin (tomaximize the BW) but with as large of a real impedance aspossible:

ZS = 50(0.75 + 0.4j)

Niknejad Power Gain

Bilateral Amp Design

In the bilateral case, we will work with the power gain Gp.The transducer gain is not used since the source impedance isa function of the load impedaance. Gp, on the other hand, isonly a function of the load.

Gp =|S21|2 (1− |ΓL|2)(

1−∣∣∣S11−∆ΓL

1−S22ΓL

∣∣∣2)|1− S22ΓL|2

= |S21|2 gp

It can be shown that gp is a circle on the ΓL plane. The radiusand center are given by

RL =

√1− 2K |S12S21|gp + |S12S21|2g 2

p∣∣∣−1− |S22|2 gp + |∆|2 gp

∣∣∣2

CL =gp(S∗

22 −∆∗S11)

1 + gp(|S22|2 − |∆|2)

Niknejad Power Gain

Bilateral Amp (cont)

We can also use this formula to find the maximum gain. Weknow that this occurs when RL = 0, or

1− 2K |S12S21|gp,max + |S12S21|2g 2p,max = 0

gp,max =1

|S12S21|(

K −√

K 2 − 1)

Gp,max =

∣∣∣∣S21

S12

∣∣∣∣(

K −√

K 2 − 1)

The design procedure is as follows1 Specify gp

2 Draw operating gain circle.3 Draw load stability circle. Select ΓL that is in the stable region

and not too close to the stability circle.4 Draw source stability circle.5 To maximize gain, calculate Γin and check to see if ΓS = Γ∗

in isin the stable region. If not, iterate on ΓL or compromise.

Niknejad Power Gain