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

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

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

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.

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

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

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

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

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

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

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.

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

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

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

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

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

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

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*

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

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

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

微波電路講義12-22

3. Two approaches for multi-stage amplifier design

(1)

Zo ZoZo Zo

Zout ZinZo

Zo

(2)

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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