Berkeley
Power Gain and Stability
Prof. Ali M. Niknejad
U.C. Berkeley
c 2014 by Ali M. Niknejad
Copyright September 17, 2014
Niknejad
Power Gain
Power Gain
Niknejad
Power Gain
Power Gain
PL
Pin
YS
!
+
vs
−
y11
y21
y12
y22
"
YL
Pav,l
Pav,s
We can define power gain in many different ways. The power
gain Gp is defined as follows
Gp =
PL
= f (YL , Yij ) 6= f (YS )
Pin
We note that this power gain is a function of the load
admittance YL and the two-port parameters Yij .
Niknejad
Power Gain
Power Gain (cont)
The available power gain is defined as follows
Ga =
Pav ,L
= f (YS , Yij ) 6= f (YL )
Pav ,S
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
= f (YL , YS , Yij )
Pav ,S
This is a measure of the efficacy of the two-port as it
compares the power at the load to a simple conjugate match.
Niknejad
Power Gain
Bi-Conjugate Match
When the input and output are simultaneously conjugately
matched, or a bi-conjugate match has been established, we
find that the transducer gain is maximized with respect to the
source and load impedance
GT ,max = Gp,max = Ga,max
This is thus the recipe for calculating the optimal source and
load impedance in to maximize gain
Yin = Y11 −
Y12 Y21
= YS∗
YL + Y22
Yout = Y22 −
Y12 Y21
= YL∗
YS + Y11
Solution of the above four equations (real/imag) results in the
optimal YS,opt and YL,opt .
Niknejad
Power Gain
Calculation of Optimal Source/Load
Another approach is to simply equate the partial derivatives of
GT with respect to the source/load admittance to find the
maximum point
∂GT
∂GT
= 0;
=0
∂GS
∂BS
∂GT
∂GT
= 0;
=0
∂GL
∂BL
Niknejad
Power Gain
Optimal Power Gain Derivation (cont)
Again we have four equations. But we should be smarter
about this and recall that the maximum gains are all equal.
Since Ga and Gp are only a function of the source or load, we
can get away with only solving two equations. For instance
∂Ga
∂Ga
= 0;
=0
∂GS
∂BS
∗ we can find the
This yields YS,opt and by setting YL = Yout
YL,opt .
Likewise we can also solve
∂Gp
∂Gp
= 0;
=0
∂GL
∂BL
And now use YS,opt = Yin∗ .
Niknejad
Power Gain
Optimal Power Gain Derivation
Let’s outline the procedure for the optimal power gain. We’ll
use the power gain Gp and take partials with respect to the
load. Let
Yjk = mjk + jnjk
YL = GL + jXL
Y12 Y21 = P + jQ = Le jφ
<(YL )
|Y21 |2
|Y21 |2
GL
=
|YL + Y22 |2 <(Yin )
D
Y12 Y21
<(Y12 Y21 (YL + Y22 )∗ )
< Y11 −
= m11 −
YL + Y22
|YL + Y22 |2
Gp =
D = m11 |YL + Y22 |2 − P(GL + m22 ) − Q(BL + n22 )
∂Gp
|Y21 |2 GL ∂D
=0=−
∂BL
D 2 ∂BL
Niknejad
Power Gain
Optimal Load (cont)
Solving the above equation we arrive at the following solution
BL,opt =
Q
− n22
2m11
In a similar fashion, solving for the optimal load conductance
q
1
GL,opt =
(2m11 m22 − P)2 − L2
2m11
If we substitute these values into the equation for Gp (lot’s of
algebra ...), we arrive at
Gp,max =
|Y |2
p 21
2m11 m22 − P + (2m11 m22 − P)2 − L2
Niknejad
Power Gain
Final Solution
Notice that for the solution to exists, GL must be a real
number. In other words
(2m11 m22 − P)2 > L2
(2m11 m22 − P) > L
2m11 m22 − P
>1
K=
L
This factor K plays an important role as we shall show that it
also corresponds to an unconditionally stable two-port. We
can recast all of the work up to here in terms of K
√
Y12 Y21 + |Y12 Y21 |(K + K 2 − 1)
YS,opt =
2<(Y22 )
√
Y12 Y21 + |Y12 Y21 |(K + K 2 − 1)
YL,opt =
2<(Y11 )
Y21
1
√
Gp,max = GT ,max = Ga,max =
Y12 K + K 2 − 1
Niknejad
Power Gain
Maximum Gain
The maximum gain is usually written in the following
insightful form
Gmax =
p
Y21
(K − K 2 − 1)
Y12
For a reciprocal network, such as a passive element, Y12 = Y21
and thus the maximum gain is given by the second factor
p
Gr ,max = K − K 2 − 1
Since K > 1, |Gr ,max | < 1. The reciprocal gain factor is
known as the efficiency of the reciprocal network.
The first factor, on the other hand, is a measure of the
non-reciprocity.
Niknejad
Power Gain
Unilateral Maximum Gain
For a unilateral network, the design for maximum gain is
trivial. For a bi-conjugate match
∗
YS = Y11
∗
YL = Y22
GT ,max =
Niknejad
|Y21 |2
4m11 m22
Power Gain
Ideal MOSFET
Cgs
+
vin
−
gm vin
ro
Cds
The AC equivalent circuit for a MOSFET at low to moderate
frequencies is shown above. Since |S11 | = 1, this circuit has
infinite power gain. This is a trivial fact since the gate
capacitance cannot dissipate power whereas the output can
deliver real power to the load.
Niknejad
Power Gain
Real MOSFET
Ri
+
vs
−
Ri
Cgs
+
vin
−
gm vin
R ds
Cds
R ds
A more realistic equivalent circuit is shown above. If we make
the unilateral assumption, then the input and output power
can be easily calculated. Assume we conjugate match the
input/output
|VS |2
Pavs =
8Ri
2
∗
1
1 gm V1 PL = <( 2 IL VL ) = 2 Rds
2 2
V1 2
GTU,max = gm Rds Ri VS
Niknejad
Power Gain
Real MOSFET (cont)
At the center resonant frequency, the voltage at the input of
the FET is given by
V1 =
GTU,max =
1 VS
jωCgs 2Ri
Rds (gm /Cgs )2
Ri
4ω 2
This can be written in terms of the device unity gain
frequency fT
1 Rds fT 2
GTU,max =
4 Ri
f
The above expression is very insightful. To maximum power
gain we should maximize the device fT and minimize the
input 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 a
negative conductance for a passive termination on the second
port. Under such a condtion, the two-port can oscillate.
Consider the input admittance
Y12 Y21
Yin = Gin + jBin = Y11 −
Y22 + YL
Using the following definitions
Y11 = g11 + jb11
Y12 Y21 = P + jQ = L∠φ
Y22 = g22 + jb22
YL = GL + jBL
Now substitute real/imag parts of the above quantities into
Yin
P + jQ
Yin = g11 + jb11 −
g22 + jb22 + GL + jBL
(P + jQ)(g22 + GL − j(b22 + BL ))
= g11 + jb11 −
(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. Note
that g11 > 0 is a requirement since otherwise oscillations
would occur for a short circuit at port 2.
The numerator can be factored into several positive terms
P
Q
(g22 + GL ) −
(b22 + BL )
g11
g11
2 2
P
Q
P2 + Q2
= GL + g22 −
+ BL + b22 −
−
2
2g11
2g11
4g11
N = (g22 + GL )2 + (b22 + BL )2 −
Niknejad
Power Gain
Input Conductance (cont)
Now note that the numerator can go negative only if the first
two terms are smaller than the last term. To minimize the
first two terms, choose GL = 0 and BL = − b22 − 2gQ11
(reactive load)
Nmin
P 2 P2 + Q2
= g22 −
−
2
2g11
4g11
And thus the above must remain positive, Nmin > 0, so
P 2 P2 + Q2
−
g22 +
>0
2
2g11
4g11
g11 g22 >
P +L
L
= (1 + cos φ)
2
2
Niknejad
Power Gain
Linvill/Llewellyn Stability Factors
Using the above equation, we define the Linvill stability factor
L < 2g11 g22 − P
C=
L
<1
2g11 g22 − P
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 stability
measure
2g11 g22 − P
>1
L
The above definition of stability is perhaps the most common
K=
2<(Y11 )<(Y22 ) − <(Y12 Y21 )
>1
|Y12 Y21 |
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 maximum
stable gain derived earlier. The connection is now more
obvious. 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 reflection
coefficient. For a general two-port with load ΓL we have
+
+
+
v2− = Γ−1
L v2 = S21 v1 + S22 v2
v2+ =
S21
v−
− S22 1
Γ−1
L
S12 S21 ΓL
v1+
= S11 +
1 − ΓL S22
S12 S21 ΓL
Γ = S11 +
1 − ΓL S22
If |Γ| < 1 for all ΓL , then the two-port is stable
v1−
Γ=
S11 (1 − S22 ΓL ) + S12 S21 ΓL
S11 + ΓL (S21 S12 − S11 S22 )
=
1 − S22 ΓL
1 − S22 ΓL
=
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S11 − ∆ΓL
1 − S22 ΓL
Power Gain
Stability Circle
To find the boundary between stability/instability, let’s set
|Γ| = 1
S11 − ∆ΓL 1 − S22 ΓL = 1
|S11 − ∆ΓL | = |1 − S22 ΓL |
After some algebraic manipulations, we arrive at the following
equation
∗
∗
ΓL − S22 − ∆ S11 = |S12 S21 |
2
2
|S22 | − |∆| |S22 |2 − |∆|2
This is of course an equation of a circle, |ΓL − C | = R, in the
complex plane with center at C and radius R
Thus a circle on the Smith Chart divides the region of
instability from stability.
Niknejad
Power Gain
Example: Stability Circle
In this example, the origin
of the circle lies outside
the stability circle but a
portion of the circle falls
inside the unit circle. Is
the region of stability
inside the circle or
outside?
RS
re
re gion
gio
n
CS
le
tab
uns table
s
|S11 | < 1
This is easily determined if
we note that if ΓL = 0,
then Γ = S11 . So if
S11 < 1, the origin should
be in the stable region.
Otherwise, if S11 > 1, the
origin should be in the
unstable region.
Niknejad
Power Gain
Stability: Unilateral Case
Consider the stability circle for a unilateral two-port
CS =
∗
∗ − (S ∗ S ∗ )S
S11
S11
11 22 22
=
|S11 |2 − |S11 S22 |2
|S11 |2
1
|CS | =
RS = 0
|S11 |
The cetner of the circle lies outside of the unit circle if
|S11 | < 1. The same is true of the load stability circle. Since
the radius is zero, stability is only determined by the location
of the center.
If S12 = 0, then the two-port is unconditionally stable if
S11 < 1 and S22 < 1.
This result is trivial since
ΓS |S12 =0 = S11
The stability of the source depends only on the device and not
on 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
∗ | + |S S | > 1
|S22 − ∆S11
12 21
The µ test not only is a test for unconditional stability, but
the magnitude of µ is a measure of the stability. In other
words, if one two port has a larger µ, it is more stable.
The advantage of the µ test is that only a single parameter
needs to be evaluated. There are no auxiliary conditions like
the K test derivation earlier.
The derivation of the µ test proceeds as follows. First let
ΓS = |ρs |e jφ and evaluate Γout
Γout =
Niknejad
S22 − ∆|ρs |e jφ
1 − S11 |ρs |e jφ
Power Gain
Mu Test (cont)
Next we can manipulate this equation into the following circle
|Γout − C | = R
p
∗ ∆−S |ρs ||S12 S21 |
|ρ
|S
s
22
11
=
Γout +
2
1 − |ρs ||S11 |
(1 − |ρs ||S11 |2 )
For a two-port to be unconditionally stable, we’d like Γout to
fall within the unit circle
||C | + R| < 1
p
∗
||ρs |S11
∆ − S22 | + |ρs ||S21 S12 | < 1 − |ρs ||S11 |2
p
∗
∆ − S22 | + |ρs ||S21 S12 | + |ρs ||S11 |2 < 1
||ρs |S11
The worse case stability occurs when |ρs | = 1 since it
maximizes the left-hand side of the equation. Therefore we
have
1 − |S11 |2
µ= ∗
>1
|S11 ∆ − S22 | + |S12 S21 |
Niknejad
Power Gain
K-∆ Test
The K stability test has already been derived using Y
parameters. We can also do a derivation based on S
parameters. This form of the equation has been attributed to
Rollett and Kurokawa.
The idea is very simple and similar to the µ test. We simply
require that all points in the instability region fall outside of
the unit circle.
The stability circle will intersect with the unit circle if
|CL | − RL > 1
or
∗ − ∆∗ S | − |S S |
|S22
11
12 21
>1
2
2
|S22 | − |∆|
This can be recast into the following form (assuming |∆| < 1)
K=
1 − |S11 |2 − |S22 |2 + |∆|2
>1
2|S12 ||S21 |
Niknejad
Power Gain
Two-Port Power and Scattering Parameters
The power flowing into a two-port can be represented by
Pin =
|V1+ |2
(1 − |Γin |2 )
2Z0
The power flowing to the load is likewise given by
PL =
|V2− |2
(1 − |ΓL |2 )
2Z0
We can solve for V1+ using circuit theory
V1+ + V1− = V1+ (1 + Γin ) =
Zin
VS
Zin + ZS
In terms of the input and source reflection coefficient
Zin =
1 + Γin
Z0
1 − Γin
ZS =
Niknejad
1 + ΓS
Z0
1 − ΓS
Power Gain
Two-Port Incident Wave
Solve for V1+
V1+ (1 + Γin ) =
VS (1 + Γin )(1 − ΓS )
(1 + Γin )(1 − ΓS ) + (1 + ΓS )(1 − Γin )
V1+ =
VS 1 − ΓS
2 1 − Γin ΓS
The voltage incident on the load is given by
V2− = S21 V1+ + S22 V2+ = S21 V1+ + S22 ΓL V2−
S21 V1+
1 − S22 ΓL
2
|S21 |2 V1+ 1 − |ΓL |2
PL =
|1 − S22 ΓL |2 2Z0
V2− =
Niknejad
Power Gain
Operating Gain and Available Power
The operating power gain can be written in terms of the
two-port s-parameters and the load reflection coefficient
Gp =
PL
|S21 |2 (1 − |ΓL |2 )
=
Pin
|1 − S22 ΓL |2 (1 − |Γin |2 )
The available power can be similarly derived from V1+
Pavs = Pin |Γin =Γ∗ =
S
+
V1a
= V1+ Γ
∗
in =ΓS
Pavs =
Niknejad
+ 2
V 1a
2Z0
=
(1 − |Γ∗S |2 )
VS 1 − Γ∗S
2 1 − |ΓS |2
|VS |2 |1 − ΓS |2
8Z0 1 − |ΓS |2
Power Gain
Transducer Gain
The transducer gain can be easily derived
GT =
|S21 |2 (1 − |ΓL |2 )(1 − |ΓS |2 )
PL
=
Pavs
|1 − Γin ΓS |2 |1 − S22 ΓL |2
Note that as expected, GT is a function of the two-port
s-parameters and the load and source impedance.
If the two port is connected to a source and load with
impedance Z0 , then we have ΓL = ΓS = 0 and
GT = |S21 |2
Niknejad
Power Gain
Unilateral Gain
Z0
+
vs
−
M1
!
S11
S21
0
S22
|S21 |
GS
"
2
M2
Z0
GL
If S12 ≈ 0, we can simplify the expression by just assuming
S12 = 0. This is the unilateral assumption
GTU =
1 − |ΓS |2
|1 − S11 ΓS |2
|S21 |2
1 − |ΓL |2
|1 − S22 ΓL |2
= GS |S21 |2 GL
The gain partitions into three terms, which can be interpreted
as the gain from the source matching network, the gain of the
two port, and the gain of the load.
Niknejad
Power Gain
Maximum Unilateral Gain
We know that the maximum gain occurs for the biconjugate
match
∗
ΓS = S11
∗
ΓL = S22
GS,max =
GL,max =
GTU,max =
1
1 − |S11 |2
1
1 − |S22 |2
|S21 |2
(1 − |S11 |2 )(1 − |S22 |2 )
Note that if |S11 | = 1 of |S22 | = 1, the maximum gain is
infinity. This is the unstable case since |Sii | > 1 is potentially
unstable.
Niknejad
Power Gain
Design for Gain
So far we have only discussed power gain using bi-conjugate
matching. This is possible when the device is unconditionally
stable. In many case, though, we’d like to design with a
potentially unstable device.
Moreover, we would like to introduce more flexibility in the
design. We can trade off gain for
bandwidth
noise
gain flatness
linearity
etc.
We can make this tradeoff by identifying a range of
source/load impedances that can realize a given value of
power gain. While maximum gain is acheived for a single
point on the Smith Chart, we will find that a lot more
flexibility if we back-off from the peak gain.
Niknejad
Power Gain
Unilateral Design
No real transistor is unilateral. But most are predominantly
unilateral, or else we use cascades of devices (such as the
cascode) to realize such a device.
The unilateral figure of merit can be used to test the validity
of 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 the
following inequality
1
GT
1
<
<
(1 + U)2
GTU
(1 − U)2
Where the actual power gain GT is compared to the power
gain under the unilateral assumption GTU . If the inequality is
tight, say on the order of 0.1 dB, then the amplifier can be
assumed to be unilateral with negligible error.
Niknejad
Power Gain
Gain Circles
We now can plot gain circles for the source and load. Let
gS =
gL =
GS
GS,max
GL
GL,max
By definition, 0 ≤ gS ≤ 1 and 0 ≤ gL ≤ 1. One can show that
a fixed value of gS represents a circle on the ΓS plane
√
2 ∗ g
1
−
g
(1
−
|S
|
)
S
11
S
11 S
ΓS −
=
2
2
|S11 | (gS − 1) + 1 |S11 | (gS − 1) + 1
More simply, |ΓS − CS | = RS . A similar equation can be
derived for the load. Note that for gS = 1, RS = 0, and
∗ corresponding to the maximum gain.
CS = S11
Niknejad
Power Gain
Gain Circles (cont)
0.11
0.1
45
50
1.4
1.2
1.0
0.9
0.8
1.6
1.8
0.6 60
8
2.0
0.0
6
0.4
4
70
5
0.4
20
3.0
0.6
0.3
CT
AN
C
0.8
IND
UCT
IV E
4.0
15
1.0
0.22
0.28
REA
gS = 0 dB
5.0
1.0
0.0
0.1
0.3
2
25
EC
75
0.4
5
OM
PO
NE
NT
(+
jX
/Z
o
0.4
10
0.2
0.8
0.27
0.1
50
20
10
5.0
4.0
3.0
1.8
2.0
1.6
1.4
1.2
1.0
50
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.0
0.2
0.1
0.4
20
LOAD <—
0.2
20
TA
EP
SC
SU
IVE
CT
DU
0.14
0.36
0.12
0.13
0.38
0.37
0.11
6
0.0
-70
-
0.5
-65
2.0
0.1
CE
CO
M
PO
NE
NT
(
7
0.0
0.6
1.8
1.6
1.4
0
1.0
0.9
0.8
1.2
0.35
5
-4
0.15
RE
AC
TA
N
0.09
-5
-4
0
-55
-35
0.16
0.34
0.7
0.17
0.33
IVE
-60
-30
2
CAP
AC
IT
-75
N
RI
), O
o
/Z
jX
0.3
0.2
4
0.4
-25
0.3
1
8
-80
1.0
)
/Yo
0.4
0.1
9
0.1
-85
jB
E (NC
0.6
-20
5
0.0
0.8
3.0
0.3
5
0.4
4.0
TO
GTHS
ELEN
WAV
<—
-90
1.0
-15
0.47
5.0
0.04
0.28
0.29
0.2
0.4
0.22
0.3
0.21
0.46
0.8
gS = −3 dB
0.27
0.6
gS = −2 dB
0.23
0.2
0.4
10
gS = −1 dB
0.1
-10
WARD
50
0.49
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.48
0.25
0.26
0.24
0.25
0.24
0.26
C
OF REFLECTION OEFFICIENT IN DEGREES
ANGLE
TRANSMISSION COEFFICIENT IN DEGRE
ES
∗
S11
OF
ANGLE
10
̸
0.23
0.6
0.04
0.17
0.33
30
0.2
0.21
80
0.16
0.34
35
0.29
0.46
C
AN
PT
o)
jB/Y
E (+
0.3
0.0
0.15
40
0.2
—> WAVEL
ENGTH
S T OW
AR D
0.49
GEN
ERA
0.48
TOR
—>
0.47
90
85
E
SC
SU
VE
ITI
0.14
0.36
1
0.3
RC
), O
AC
AP
0.37
9
0.1
0.5
65
3
0.4
0.13
0.38
0.7
0.08
7
0.0
0.12
0.35
55
0.41
0.42
0.39
0.4
0.09
0.4
3
0.08
0.42
0.41
0.4
0.39
All gain circles lie on the line given by the angle of Sii∗ . We
can select any desired value of source/load reflection
Power
Gain
coefficient to acheive Niknejad
the desired
gain.
To minimize the
Extended Smith Chart
For |Γ| > 1, we can still employ the Smith Chart if we make
the following mapping. The reflection coefficient for a
negative resistance is given by
Γ(−R + jX ) =
(R + Z0 ) − jX
−R + jX − Z0
=
−R + jX + Z0
(R − Z0 ) − jX
1
(R − Z0 ) + jX
=
∗
Γ
(R + Z0 ) + jX
We see that Γ can be mapped to the unit circle by taking
1/Γ∗ and reading the resistance value (and noting that it’s
actually negative).
Niknejad
Power Gain
Potentially Unstable Unilateral Amplifier
For a unilateral two-port with |S11 | > 1, we note that the
input impedance has a negative real part. Thus we can still
design a stable amplifier as long as the source resistance is
larger than <(Zin )
<(ZS ) > |<(Zin )|
The same is true of the load impedance if |S22 | > 1. Thus the
design procedure is identical to before as long as we avoid
source or load reflection coefficients with real part less than
the critical value.
Niknejad
Power Gain
Pot. Unstable Unilateral Amp Example
Consider a transistor with the following S-Parameters
S12 = 0
S11 = 2.02∠ − 130.4◦
S21 = 5.00∠60◦
S22 = 0.50∠ − 70◦
0.11
0.1
45
50
1.4
1.2
1.0
0.9
55
1.6
1.8
2.0
0.0
6
0.4
4
70
0.05
0.45
region
ble
sta
20
3.0
0.6
0.3
0.46
0.18
0.32
25
0.4
RS
0.8
GS = 5 dB
4.0
15
5.0
1.0
IND
UCT
IVE
1.0
0.27
0.6
10
0.25
0.26
0.24
0.25
0.24
0.26
CO
OF REFLECTION EFFICIENT IN DEGREES
ANGLE
ISSION COEFFICIENT IN D
EGREES
OF TRANSM
ANGLE
0.2
ΓS
0.23
CS
90
10
0.8
0.4
0.04
0.4
80
0.17
0.33
30
0.2
0.22
0.28
REA
75
CT
AN
CE
CO
MP
ON
EN
T(
+j
X/
Zo
6
4
0.21
0.47
0.1
0.3
35
0.29
—>
o)
jB/Y
E (+
NC
TA
EP
SC
SU
0.3
85
0.15
40
0.2
0.1
20
0.2
—> WAVEL
ENGTH
S TOW
ARD
0.49
GEN
ERA
0.48
TOR
E
TIV
CI
PA
CA
0.14
0.36
1
0.3
R
), O
0.37
9
0.1
65
0.5
0.43
0.13
0.38
0.7
0.6 60
0.07
0.12
0.35
1
0.08
0.42
0.39
0.4
9
0.4
0.8
0.0
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
50
20
0.4
0.1
1.0
o)
jB/Y
E (NC
TA
EP
SC
SU
-80
E
TIV
UC
ND
RI
,O
o)
/Z
jX
-75
0.05
0.6
0.43
CAP
AC
0.07
ITIV
ER
EA
CT
AN
CE
CO
M
PO
NE
NT
(-
2.0
0.5
1.8
-65
0.2
1.4
1.2
1.0
0
-4
0.15
0.35
0.9
6
5
-4
4
0.0
0
-5
-35
0.1
0.3
0.8
0.17
-55
Now any source inside this
circle is stable, since
<(ZS ) > <(Zin ).
We also draw the source
gain circle for GS = 5 dB.
0.33
0.7
1.6
-30
0.6
0.18
0.1
0.14
0.36
0.12
0.13
0.38
0.37
-60
-25
0.3
1
0.11
0.39
0.4
0.08
0.42
9
0.4
1
4
0.4
9
6
0.0
-70
0.4
0.1
0.32
0.04
0.8
3.0
-20
0.46
4.0
0.47
1.0
-15
0.3
0.45
0.28
0.2
0.4
0.29
0.21
0.3
5.0
-85
10
0.8
0.27
0.6
0.23
0.2
1
∗
S11
-10
0.49
50
0.2
0.22
0.48
<—
RD LOAD
TOWA
GTHS
ELEN
WAV
<—
-90
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
Since |S11 | > 1, the
amplifier is potentially
unstable. We begin by
∗ to find the
plotting 1/S11
negative real input
resistance.
Niknejad
Power Gain
Amp Example (cont)
∗ .
The input impedance is read off the Smith Chart from 1/S11
Note the real part is interpreted as negative
Zin = 50(−0.4 − 0.4j)
The GS = 5 dB gain circle is calculated as follows
gS = 3.15(1 − |S11 |2 )
√
1 − gS (1 − |S11 |2 )
= 0.236
RS =
1 − |S11 |2 (1 − gS )
∗
gS S11
CS =
= −.3 + 0.35j
1 − |S11 |2 (1 − gS )
We can select any point on this circle and obtain a stable gain
of 5 dB. In particular, we can pick a point near the origin (to
maximize the BW) but with as large of a real impedance as
possible:
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 is
a function of the load impedaance. Gp , on the other hand, is
only a function of the load.
|S21 |2 (1 − |ΓL |2 )
Gp = = |S21 |2 gp
S11 −∆ΓL 2
2
1 − 1−S22 ΓL |1 − S22 ΓL |
It can be shown that gp is a circle on the ΓL plane. The radius
and center are given by
q
1 − 2K |S12 S21 |gp + |S12 S21 |2 gp2
RL =
2
2
2
−1
−
|S
|
g
+
|∆|
g
22
p
p
CL =
∗ − ∆∗ S )
gp (S22
11
1 + gp (|S22 |2 − |∆|2 )
Niknejad
Power Gain
Bilateral Amp (cont)
We can also use this formula to find the maximum gain. We
know that this occurs when RL = 0, or
2
1 − 2K |S12 S21 |gp,max + |S12 S21 |2 gp,max
=0
p
1
K − K2 − 1
|S12 S21 |
p
S21 K − K2 − 1
=
S12 gp,max =
Gp,max
The design procedure is as follows
1
2
3
4
5
Specify gp
Draw operating gain circle.
Draw load stability circle. Select ΓL that is in the stable region
and not too close to the stability circle.
Draw source stability circle.
To maximize gain, calculate Γin and check to see if ΓS = Γ∗in is
in the stable region. If not, iterate on ΓL or compromise.
Niknejad
Power Gain