Page 1
微波電路講義12-1
Chapter 12 Microwave Amplifier Design
12.1 Two-port power gains
power gains G, GT, GA
12.2 Stability
input and output stability circles, stability criterion
12.3 Single-stage transistor amplifier design
conjugate match, constant gain circle, noise parameters, constant
noise figure circle, LNA (low noise amplifier)
12.4 Broadband transistor amplifier design
balanced amplifier, distributed amplifier, differential amplifier
12.5 Power amplifier
nonlinear operation
Page 2
微波電路講義12-2
12.1 Two-port power gains
transistor
[ S ]
(Zo)Vs
Zs
ZL
+
V1
-
s in, Pin, Zin out L, PL, ZL
+
V2
-
* *
power gain ( , )
available power gain ( , )
transducer power gain ( , , )
( ), ( ) , ( ), ( )in S L out
LL
in
avnA S
avs
LT S L
avs
in in avs s in L L avn out L
PG S
P
PG S
P
PG S
P
P P P P P P
s L
in out
Page 3
微波電路講義12-3
Discussion
1. 1 11 1
1
22 2
12 2 2
2
(1 Γ ),
Γ ,Γ
1 Γ
1 Γ 2 1 Γ Γ
1 Γ1(1 Γ ) (1 Γ ) (1 Γ )
2 8 1 Γ Γ
ins in
s in
in o s oin s
in o s o
s in s s
in s in s in
s sin s in in in
o o s in
ZV V V V V
Z Z
Z Z Z Z
Z Z Z Z
V Z VV
Z Z
V VP P
Z Z
2. 2 21 1 22 2 2 2 1
212
22
2 2 2 2
2 2 22 21
2 2
22
1, ,
2 1
(1 )
2 (1 )(1 )
11(1 ) (1 ) (1 )
2 8 1 1
s sL
s in
s s
L s in
s s
L out L L L
o o L s in
VV S V S V V V V
V SV
S
V V SP P
Z Z S
Page 4
微波電路講義12-4
*
*
2 2
2
2 2 2 2
21 12 21112 2*
2222
2 2 2
21
2 2
11
1
8 1
1 (1 ),
8 11 1
1
8 1 (1 )
in S
L out
s s
avs in
o s
s s out Lavn L in
o Lout s in
s s
avn
o s out
VP P
Z
V S S SP P S
Z SS
V SP
Z S
3.
2 2
21
2 2
22
2 2
21
2 2
11
2 2 2
221
212 2
22
(1 ) ( , )
(1 ) 1
(1 )( , )
(1 ) 1
(1 )(1 )( , , ) ( , if 0)
1 1
LLL
in in L
savnA s
avs out s
s LLT s L s L
avs s in L
SPG S
P S
SPG S
P S
SPG S S
P S
4.
Page 5
微波電路講義12-5
5.
transistor
[ S ]
Go
ZoZo
s in out L
output
matching
circuit GL
input
matching
circuit Gs
2 2
2
212 2
22
2
2* *
max 212 2
22
2 2
2
12 212 2
11 22
2* *
11 22 max 212 2
11 22
1 1
1 1
11,
1 1
1 10, unilateral transducer gain
1 1
1 1
1 1
s L
T s o L
s in L
L
in s out L T
s L
s L
TU
s L
s L TU
G S G G GS
G SS
S G SS S
S S G S seleS S
, ction of transistor
Page 6
微波電路講義12-6
6. Ex.12.1 A Si BJT@1GHz
Zs=25, ZL=40, Zo=50
11 12 21 220.38 158 , 0.11 54 , 3.5 80 , 0.4 43S S S S
12 2111
22
12 2122
11
0.333, 0.111
0.365 1521
0.545 431
13.1, 19.8, 12.6
,
s o L os L
s o L o
Lin
L
sout
s
A T
avnL L LT A T
avs in avs avs
Z Z Z Z
Z Z Z Z
S SS
S
S SS
S
G G G
PP P PG G G G
P P P P
Page 7
微波電路講義12-7
* *
22
2
2 22
1,
1
2
1 1( )
2 2 ( ) : 6 / ,1 1 4 4 2
( ) /2 2
in S out L ds
gs
sgs
i gs
m gs dsm ds ds mL T
TU T
avs i gs i gss i
Z Z X Z Z wC BwC
VV
R jwC
g V Rg R R gP f
G dB octave fP w R C R f C
V R
7. conjugate match using FET equivalent circuit (S12=0, or Cgd=0)
Ri
Cgs gmVgsCds jB RdsRds
Ri jX
Vs+
Vgs
-
s in out L
Page 8
微波電路講義12-8
1,1,Z Z
1,1,Z Z
Ls
Ls
outin
outin
stable lconditiona
stable nalunconditio
12.2 Stability (S, f)
2. 12 2111 L
22
* *
22 11 12 21
2 2 2 2
22 22
12 2122 s
11
* *
11 22 12 21
2 2 2 2
11 11
1 output stability circle 1
( ),
1 input stability circle 1
( ),
(deriv
Lin L L
L
L L
sout s s
s
s s
S SS C R
S
S S S SC R
S S
S SS C R
S
S S S SC R
S S
ation in p.565)
Discussion
1. 12 in out 11 220, Γ 1, Γ 1 1, 1 S S S
Page 9
微波電路講義12-9
|S22|<1 |out|=1 |S22|>1
s-plane s-plane
Cs Rs
3. conditional stable
|S11|<1 |S11|>1
L-plane |in|=1 L-plane
CL RL
|in|<1
|out|<1
output
stability
circle
input
stability
circle
L =0
S 0
Page 10
微波電路講義12-10
4. unconditional stable, stability factor K
Cs
Rs
CL
RL
|S22|<1 |S11|<1
11 22
2 2 2
11 22
12 21
unconditional stable 1 1 1, 1
11, 1, det[ ] :Rollet's condition
2
(derivation in p.568 and 569)
,
s s L LS , S , C R C R
S SK S
S S
K selection of transistor
Cs
Rs
CL
RL
>1L LC R>1S SC R
Page 11
微波電路講義12-11
6. Ex.12.2 Triquint T1G6000528 @ 1.9GHz, Zo=50Ω
11 12 21
22
0.869 159 , 0.031 9 , 4.25 61 ,
0.507 117
0.336 1, 0.383 1
input stability circle 1.09 162 , 0.205
output stability circle 1.59 132 , 0.915
(p.570, Fig.12.6)
s s
L L
S S S
S
K
C R
C R
5. In practice, one should consider stability over a wide bandwidth for
the possible oscillations.
Page 12
微波電路講義12-12
12.3 Single-stage transistor amplifier design
• conjugate match (maximum transducer power gain)
* *
2
2 221max 212 2
1222
22
2 2 2* 12 2111
22 2
* 12 2122
11
if 1, 1
input and output simultaneously conjugate match ,
11( 1)
1 1
4
1 2
1
in s out L
L
T T
s L
Ls in L
L
sL out s
s
K
SG G S K K
SS
B B CS SS
S C
S SS
S
22
1 1 1
1
2 2 2 2 2 2
1 11 22 2 22 11
* *
1 11 22 2 22 11
4
2
1 , 1
,
(derivation in p. 571 and 572)
B B C
C
B S S B S S
C S S C S S
Page 13
微波電路講義12-13
Discussion
1. linear amplifier design procedure
if |<1, K>1 then uses input and output simultaneously conjugate
matches for GTmax
if K<1 then draws input and output stability circles to see if input
and output simultaneously conjugate matches possible, otherwise
selects proper s and L for gain or noise figure considerations.
2.
maxL
2
21maxs2
22
2
212
11
maxTU
*
22L
*
11s12
GSGS1
1S
S1
1G
S,S0S
3. Ex.12.3 FET @ 4GHz
11 12 21 22
s L
max
0.72 116 , 0.03 57 , 2.6 76 , 0.73 54
0.488 1, 1.195 1 0.872 123 , 0.876 61
6.2 8.3 2.22 16.7T
S S S S
K
G dB
Page 14
微波電路講義12-14
G
s* L*
1
2
1. y=1-j3.5
2. y=j3.5
0.12 0.206
0.206 0.206
frequency response (p.575, Fig.12.7)
s* L*
GT
f
-RL
61876.0
123872.0
*
*
L
s
Page 15
微波電路講義12-15
• constant gain circle (S12=0, unilateral assumption)2 2
2
21 12 112 2
22
2 2
max max2 2 2 2
11 11 22 22
S S
max
L
max
1 1, 0
1 1
1 11 1, , ,
1 1 1 1
constant gain circle in -plane
constant gain circle in -pl
s L
T in
s in L
s L
s s L L
s L
ss S S
s
LL
L
G S S SS
G G G GS S S S
Gg C R
G
Gg
G
L
2*1111
2 2
11 11
2*2222
2 2
22 22
ane
1 (1 ),
1 (1 ) 1 (1 )
1 (1 ),
1 (1 ) 1 (1 )
(derivation in p. 576 and 577)
L L
sss s
s s
LLL L
L L
C R
g Sg SC R
g S g S
g Sg SC R
g S g S
Page 16
微波電路講義12-16
Discussion
1. Gs=0dB and GL= 0dB circles pass through the Smith chart center.
maxL
*
22maxs
*
11 G,S ,GS
2
max2 2
11 11
2
max2 2
22 22
2 2
11 11
max max
2 2
22 22
max max
S L
*
11
1 1( 0 ) 1, ,
1 1
1 1( 0 ) 1,
1 1
11 1 ,
11 1
constant gain circles ,
1 (1
s
s s
s
L
L L
L
ss s
s s
LL L
L L
s s L L
ss
G dB GS S
G dB GS S
Gg S g S
G G
Gg S g S
G G
C R C R
g SC
g
2 2 2* *11 11 11 11 11 1111
2 4 2 2 4 2
11 11 11 11 11 11
2 2 2** *22 22 22 22 22 2222 22
2 4 2 2 4 2
22 22 22 22 22 22
(1 ) 1 (1 ) (1 ),
) 1 1 1 (1 ) 1 1
(1 ) 1 (1 ) (1 ),
1 (1 ) 1 1 1 (1 ) 1 1
s
s
s s
LLL L
L L
S S g S S S SSR
S S S g S S S
S S g S S S Sg S SC R
g S S S g S S S
, 0s s L L S LC R C R
GS=0dB GL= 0dB
Page 17
微波電路講義12-17
2.
merit of figure lunilateria )1)(1(
)1(
1
)1(
1
2
22
2
11
22122111
22
SS
SSSSU
UG
G
UTU
T
2. Centers of constant gain circles are distributed along the lines from
S11* and S22* to the Smith chart center, respectively.* *
11 max 22 max ,S LS ,G S G* *
11 22
2 2
11 22
* *
,1 11 ,1 22
* *
11 22,2 ,22 2
11 22
,1 ,2 ,1 ,21 1 1 1
,1 ,2 ,1 ,2
,1 (1 ) 1 (1 )
1, 1 ,
1, 1 ,1 1
Re( ) Re( ) Re( ) Re( )tan tan , tan tan
Im( ) Im( ) Im( ) Im(
s Ls L
s L
s L s L
s L s L
s s L L
s s L L
g S g SC C
g S g S
g g C S C S
S SG G C C
S S
C C C C
C C C C
)
GS=0dB GL= 0dB
Page 18
微波電路講義12-18
3. Ex.12.4 design an amplifier with GT=11dB @ 4GHz
11 12 21 22
max
0.75 120 , 0, 2.5 80 , 0.6 70
3.6 8 1.9 13.5
choose 2 8 1 11
TU
TU
S S S S
G dB
G dB
Gs=2dB GL=1dB
* * *
11 max 22 max s
*
L
, 1. 0.33 120 , 0.33 120
2. 0.22 70 , 0.22 70
S L s
L
S ,G S G
0.179 0.045
0.1 0.432
frequency response (p.579, Fig.12.8)
1 2
f
GT
-RL
* * s L
s* L*
Page 19
微波電路講義12-19
• constant noise figure circle
2
2
min min 22
min
2
2min
2
for a two-port amplifier
4
(1 ) 1
noise parameter: , , equivalent noise resistance of transistor
14 /1
constant noise fig
s optN Ns opt
s o s opt
opt N
s opt
opt
N os
R RF F Y Y F
G Z
F Y R
F FN
R Z
s
2
ure circle
( 1 ),
1 1
(derivation in p. 580 and 581)
F F
optopt
F F
C R
N NC R
N N
Page 20
微波電路講義12-20
dBSGdBS
GS
RCdBG
RCdBF
dBGGdBUG
G
U
U
TULL
ssss
FF
TUT
TU
T
53.825.17.125.11
1605.0
7553.015.0,6058.07.1
24.0,10056.02
53.05.0,13.1)1(
1
)1(
189.0
059.0
2
212
22
*
22
22
F=2dB Gs=1.7dB
0.226 0.25
0.144 0.136
Discussion
1. Ex.12.5 design a LNA with F=2dB and max. gain @ 4GHz
20,10062.0,6.1
605.0,819.1,2605.0,606.0
min
22211211
Nopt RdBF
SSSS
s*
L*
s
s* L*
L
Page 21
微波電路講義12-21
2. Approach for single-stage linear amplifier design
Calculate K and Δ
using eqs. (11.28) and (11.29)
Perform conjugate match design
using eq. (11.40a) to calculate ΓS
using eq. (11.40b) to calculate ΓL
Plot stability circles
plot input stability circle in ΓS-plane
plot output stability circle in ΓL-plane
Design input and output matching
circuits by properly selecting
ΓS and ΓL based on constant gain
circle consideration
Design input and output matching
circuits
Verify the stability over a wide
bandwidth
Design DC biasing circuits
and verify the stability again
K>1, úΔú1otherwise
YesNo No
Perform circuit layout
Verify realizability
Circuit implementation
NoNo
Yes
Given transistor S-parameters
Page 22
微波電路講義12-22
3. Two approaches for multi-stage amplifier design
(1)
Zo ZoZo Zo
Zout ZinZo
Zo
(2)
Page 23
微波電路講義12-23
12.4 Broadband transistor amplifier design
• Balanced amplifier
11 12 11 11 12 12
21 22 21 21 22 22
( )1
( ) ( )2
A B A B
A B A B
S S S S j S S
S S j S S S S
ú ú
AS
BS
1
1
b
a
2
2
a
b
A2
A2
a
b
B2
B2
a
b
A1
A1
b
a
B1
B1
b
a
Page 24
微波電路講義12-24
Discussion
1. Derivation of S-parameters
1 1
1 1o
1 1
1 , 11 , 12 ,
2 , 21 , 22
1 10 0 0 0
2 2 2 2
1 10 0 0 0
2 2 2 290 hybrid ,
1 10 0 0 0
2 2 2 2 0
1 10 0 0 0
2 2 2 2
A A
B B
A B A B A B
A B A B A
j j
b aj j
a b
j a j b
j j
b S S
b S S
ú ú ú ú
ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú
ú ú ú ú
ú
2 2
1 ,
, 2 , 2 2
2 2
10 0
2 2
10 0
02 2 ,
10 0
2 2
10 0
2 2
A A
A B
B A B
B B
j
a bj
a
a b aj
a b
j
ú ú
ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú
ú ú
1
1
b
a 1
1
A
A
b
a
1
1
B
B
b
a
2
2
A
A
b
a
2
2
B
B
b
a2
2
a
b
1 2
34
4
1
3
2
Page 25
微波電路講義12-25
)(2
1)(
2
)2
1
2(
2
1 )
22
1(
2
)(2
1 )(
2
2
1
2
)(2
)(2
1
)2
1
2(
2 )
22
1(
2
1
)(2
)(2
1
2
2
1
2222212121
222121222121
222121222121
222
2121211111
212111212111
212111212111
111
aSSaSSj
aSaj
Saj
SaSj
aSaSaSaSj
bbj
b
aSSj
aSS
aSaj
Sj
aj
SaS
aSaSj
aSaS
bj
bb
BABA
BBAA
BBBBAAAA
BA
BABA
BBAA
BBBBAAAA
BA
Page 26
微波電路講義12-26
2. amplifier A=amplifier B, good i/p and o/p match
good stability
ú
0
0
21
12
A
A
jS
jS
3. high reliablity and less tuning work
4. I/p and o/p matching are improved by two 90° hybrids, and
mismatch reflections are absorbed by two resistors.
5. If one transistors fails, gain drops 6dB.
graceful degradation
6. disadvantages: larger size and lower efficiency
7. Bandwidth is limited by two hybrids.
Page 27
微波電路講義12-27
9. Balanced amplifiers can be implemented in a tree structure with a
very high power in radar and communication applications.
10. Ex 12.7, two amplifiers of ex.12.4 are implemented as a balanced
amplifier to improve its i/p and o/p return loss at 4 GHz. Then, the
stub lengths are optimize to give better matching and gain flatness
from 3 to 5 GHz bandwidth.
frequency response (p.588, Fig.12.11)
8. Power amplifier application
1/4W 1W
6dB
1/2W 1/4W 1W
1/4W 1W 2W
6dB
Page 28
微波電路講義12-28
• Distributed (traveling wave) amplifier
TEM line “extreme wide operation bandwidth”
gate line
drain line
cut off frequency
FET equivalent
circuit
LCfc
1
Page 29
微波電路講義12-29
+
Vc1
-
Iogm Vc1
Discussion
1. unit cell of gate line
Lg
Cg
2 2small loss
1
/
/( )
1
( )2i gs
g
g
g gs g
gs g
g g g
i gs
i g gs gs
g gwR C
g g
g g
LZ
C C l
jwC ljwL jwC
jwR C
w R Z C Cjw L C
l l
j
G=1/Rilg
jB=jwCgs/lg
+
Vi
-
lg
ld
Page 30
微波電路講義12-30
(derivation of 1)
1 1 1
2 2 2
small loss
1
1
3 2
2
1( )
22
/,
1
/
/ (1 )( ) [ ]
1
( )( ) ( )
( )
i gs
i gs
gs g
g g
i gs
g
gwR C
g gs g
wR Cgs g gs i gs
g g g g g
i gs g
gs g i gs
g g
g g
a b a a b
jwC lZ jwL Y jwC
jwR C
LZZ
Y C C l
jwC l jwC jwR CZY jwL jwC jwL jwC
jwR C l
C jw L R Cjw L C
l l
jw L
3 2
2 2 22
( ) /1( )
2 ( / )
( ) ( )2 2
gs g i gs g
g g
g g g gs g
gs g i gs gs i gs g
g g g g g ggsg g g g
g
g
C jw L R C lC
l jw L C C l
C L R C C w R C Zwjw L C jw L C j
Cl l l lC
l
Lg
Cg
G=1/Rilg
jB=jwCgs/lg
Page 31
微波電路講義12-31
2. unit cell of drain line
Ld
Cd G = jB =
1/Rdsld jwCds/ld Id
small loss
/
1[ ( )]
( )2
dd
d ds d
dsd d d
ds d d
d dsd d
ds d d
d d
LZ
C C l
CjwL jw C
R l l
Z Cjw L C
R l l
j
3. o/p current
( 1)( )
1
( )
1
1 1, , ( )
2 1
2 2
g gd d
g g d d
g g g g d dd d
g g d d
Nn lN n l
o dn dn m cn cn i
n i gs
N l N lNl n l lN lm i m i
o l ln
I I e I g V V V ejwR C
g V g V e eI e e e
e e
Page 32
微波電路講義12-32
1 1 1
2 2 2
small loss
1
2
1( )
2
2
1, ( )
/
1[ ( )] ( ) ( )
1 1( )
2 ( ) ( / )
(
ds d
dsd d
ds d d
dd
R ld ds d
ds ds dd d d d d
ds d d d ds d
a b a a b
ds dd d
d ds dd d ds d
dsd d
CZ jwL Y jw C
R l l
LZZ
Y C C l
C C jwLZY jwL jw C jw L C
R l l l R l
C jwLjw L C
l R ljw L C C l
Cjw L C
l
1 1 1 1
) ( )2 2 /( / )
1( )
2
d ds dd d
d ds d d ds d d ds dd d ds d
ds dd d d d
d ds d
L C Ljw L C
R l l R l C C lL C C l
C Zjw L C j
l R l
(derivation of 2) Ld
Cd G = jB =
1/Rdsld jwCds/ld Id
Page 33
微波電路講義12-33
(derivation of 3)
( 1)( )
1
( )( )
1 1
( 1)( ) ( )
(
1 1, , ( )
2 1
(1 ),
2 2 1
2
g gd d
g g g g d dd d d d
g g d d g g d d
g gd d
Nn lN n l
o dn dn m cn cn i
n i gs
NN Nl n l lN n l N l nm m i
o cn
n n
N l l l llN lm i
I I e I g V V V ejwR C
g g V r rI V e e e e r
r
g V e ee e
e
)
( 1)( ) ( )
( 1)
1
2
2
d d
g g d d d d
g g d d g g d d
g gd d
g g d d
g g d d
g g d d
l
l l l
N l l l llN lm i
l l
N l N l
m i
l l
e
e
g V e ee e
e e
g V e e
e e
Page 34
微波電路講義12-34
2
22 2
2 2
22 ( ) ( )
( ) ( )
For matched i/p and o/p ports
24
2
4
under synchronization conditio
g g d d
g g d d
g g g g d d d d
g g g g d d d d
o dN l N l
o d g m d gout
l lin i i
g
N l j l N l j lm d g
l j l l j l
I ZI Z Z g Z ZP e e
GP e eV V
Z
g Z Z e e
e e
2 2
2
n ( )
( ), 0
4 ( )
ln( / )0
g g d d
g g d d
g g d d g d
N l N lm d g
l l
g g d d
opt
g g d d
l l
g Z Z e eG N G
e e
l ldGN
dN l l
4.
i ds
2 2
2 2
For a lossless amplifier (R =0, R ) and if
( ) ,4 2
d g o
m d g m o
Z Z Z
g Z Z N g Z NG G N
Page 35
微波電路講義12-35
5. Ex.12.8 Zd= Zg = Zo=50, Ri=5 , Rds=250 , Cgs=0.3pF,
gm=30mS
2 2
0.1@162
0.114@162
9.4, frequency response (p.593, Fig.12.16)
i gs o
g g
od d
ds
opt
w R C Zl GHz
Zl GHz
R
N
G
(dB)
f
N=16N=8
N=4
N=2
16 GHz
Page 36
微波電路講義12-36
• Differential amplifier
1
(1 )( )
( )=
( ) (1 )( )
igs
i gs
D ds m D dso m gs i
D ds i gs D ds
o o m D dsd
i i i gs D ds
VV
j R C
R R g R RV g V V
R R j R C R R
V V g R RA
V V j R C R R
0
0 1 1 0 0
1 0 0 1 0 2
1 0 0 1 02
20 1 1 0
0
ja
j
ja
a
ú ú ú ú
ú ú ú ú ú ú ú ú ú ú ú
balun
1
2
3
4
output swing and fT doubled
Page 37
微波電路講義12-37
11.5 Power amplifiers
• nonlinear operationS(input power, f, DC, T, ZL)
Discussion
1. power amplifier characteristics: efficiency, gain, intermodulation
product, thermal conduction
power added efficiency
FET nonlinear
equivalent
circuit (large-signal
S-parameter)
S G D
PAEout in
DC
P P
P
Page 38
微波電路講義12-38
2. DC bias consideration
Ids
Vds
Vgs = 0V
Vgs = - 1V
Vgs = Vp
LNA
high gain
high efficiency
high power
3. design consideration: large-signal source impedance Γs (source-pull
contour) and load impedance ΓL (load-pull contour)
[ S ]
GoZo
Zo
s in out L
Output
matching
circuit GL
Input
matching
circuit Gs
class A
Page 39
微波電路講義12-39
4. Ex.12.9 a transistor has small-signal S-parameters at 2.3GHz as
11 12 21 220.593 178 , 0.009 127 , 1.77 106 , 0.958 175
For class A operation at 28 and 0.6 , 10 , 16.4 ,
10 3 , 2.5 2.3 ,design the input and output matching
circuits.
DS D o
SP LP
S S S S
V V I A P W G dB
Z j Z j
max
From small-signal S-parameter, 0.579 1, 2.08 1 unconditional stable
From and 0.668 187 , 0.905 175
From small-signal S-parameter for 0.508 166 , 0.954 176
For 10 ,
SP LP SP LP
T S L
out in
K
Z Z
G
P W P
23.6 229
10 0.22925%
28 0.6
out
out inPAE
P G dBm mW
P Pη
VI
ADS examples: Ch12_prj