微波工程講義

Chapter 7 Two-port network7.1 Impedance parameters

definition, examples7.2 Admittance parameters

definition, examples7.3 Hybrid parameters

definition, examples7.4 Transmission parameters

definition, examples7.5 Conversion of the impedance, admittance, chain, and hybrid parameters7.6 Scattering parameters

definition, characteristics, examples7.7 Conversion from impedance, admittance, chain, and hybrid parameters

to scattering parameters or vice versa7.8 Chain scattering parameters

7-1

微波工程講義

7.1 Impedance parametersBasics1.

7-2

[ ] [ ][ ] [ ] [ ]

2port at impedanceinput circuit -open:

dance transimpeforwardcircuit -open:

dance transimpereversecircuit -open:

1port at impedanceinput circuit -open:

,

response:,source:,

02

222

01

221

02

112

01

111

2221212

2121111

2

1

2221

1211

2

1

1

2

1

2

=

=

=

=

=

=

=

=

+=+=

=

=

I

I

I

I

IVZ

IVZ

IVZ

IVZ

IZIZVIZIZV

II

ZZZZ

VV

VIIZV

linearnetworkV1

I1

port 1 V2

I2

port 2

referenceplane 1

referenceplane 2

1/9/2003 2

How to Determine Z Parameters?

1 11 1 12 2

2 21 1 22 2

V Z I Z IV Z I Z I

= ⋅ + ⋅= ⋅ + ⋅

2 0I =

2 2

1 211 21

1 10 0I I

V VZ ZI I= =

= =

1 0I =

1 1

2 122 12

2 20 0I I

V VZ ZI I= =

= =

Two Port Network

I1=0

O.C.

Z22 I2

V2 V1 Reciprocity:

12 21Z Z=

Lecture 4

Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-3

Discussion1. Ex.7.1

V1

I1

V2

I2

6Ω[ ]

====

===Ω==

=

=

6666

,66

66

,6

11

1

01

221

2

2

02

1122211

2

1

ZII

IVZ

II

IVZZZ

I

I

2. Ex.7.2

V1

I1

V2

I2

12Ω3Ω

[ ]

0,0

30012

,3,12

01

221

02

112

22211

21

====

===

== II IVZ

IVZ

ZZZ

V1

I1

V2

I2

12Ω

6Ω

3Ω

3. Ex.7.3[ ] [ ] [ ]

66

,66

Z96618

,9,18

1

1

01

221

2

2

02

112

2132211

21

======

+=

===

==II

IVZ

II

IVZ

ZZZZ

II

4. Z12=Z21 → reciprocal circuitZ12=Z21, Z11=Z22→ symmetrical and reciprocal circuit

5. Useful for series circuits analysis.

1/9/2003 10

Series Connection of Two-Port Networks

V1

ZB

I2

ZA

I1

V2

V1A

V1B

V2A

V2B

I1 I2

[ ] [ ] [ ]A BZ Z Z= +

Lecture 4

ELG4105: Microwave Circuits © S. Loyka, Winter 2003

1/9/2003 5

Example: T-Network

2

2

111

1 0

221

1 0

A CI

CI

VZ Z ZI

VZ ZI

=

=

= = +

= =

1

12 21

222

2 0

C

B CI

Z Z Z

VZ Z ZI =

= = = = +

I1 I2

V2 V1

ZA ZB

ZC

1 0I =

2 0I =

Lecture 4

Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-4

6. Ex. 7.4

V1

I1

V2

I2

Zo,γ

l

[ ]

=====

==−

=−

==

==−+=

−

+==

=−=

=+=→

=Γ

−−

−

=

−

−

−

−

=

−

−−

ljZ

ljZ

ljZ

ljZ

Zlj

ZZZlj

ZZZ

Zl

Zee

ZeV

ZV

eVIVZ

Zl

ZeeeeZ

eVZV

eVVIVZ

IeV

ZV

I

eVVeVVV

V

oo

oo

oo

ollo

o

lin

o

in

lin

I

oll

ll

o

o

lin

o

in

linin

I

o

lin

o

in

lin

linin

openin

ββ

ββββ

γ

γ

γγγ

γ

γγ

γγ

γ

γ

γ

γγ

tansin

sintan,sin

,tan

line lossless

sinh2

Z

2

tanhZ

0,Z

2,

1open :2port , source:port1

21122211

12201

221

222

2

01

111

2

2

1

22

1

2

2

微波工程講義7-5

7.2 Admittance parametersBasics1.

[ ] [ ][ ] [ ] [ ]

2port at admittanceinput circuit -short:

ttance transadmiforwardcircuit -short:

ttance transadmireversecircuit -short:

1port at admittanceinput circuit -short:

,

response:,source:,

02

222

01

221

02

112

01

111

2221212

2121111

2

1

2221

1211

2

1

1

2

1

2

=

=

=

=

=

=

=

=

+=+=

=

=

V

V

V

V

VIY

VIY

VIY

VIY

VYVYIVYVYI

VV

YYYY

II

IVVYI

linearnetworkV1

I1

port 1 V2

I2

port 2

referenceplane 1

referenceplane 2

1/9/2003 3

Y Parameters

[ ] [ ] [ ]V Z I= ⋅ [ ] [ ] [ ]1I Z V−= ⋅ [ ] [ ] 1Y Z −=

1 11 1 12 2

2 21 1 22 2

I Y V Y VI Y V Y V

= ⋅ + ⋅= ⋅ + ⋅

2 0V =

2 2

1 211 21

1 10 0V V

I IY YV V= =

= =

1 0V =

1 1

2 122 12

2 20 0V V

I IY YV V= =

= =

Two Port Network

I1 S.C. Y22 I2

V2 V1=0

Lecture 4

Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-6

Discussion1. Ex.7.5

V1

I1

V2

I2

0.05S[ ]

212

2

02

112

12211

05.005.0

05.005.005.005.0

,05.0

1

YVV

VIY

YYY

V

=−=−==

−−

===

=

V1

I1

V2

I2

0.1S

0.025S

0.2S

2. Ex.7.6

[ ]

−−

=

=−=→−=−=

=+

===

=

=++

+=

=++

+=

=

0769.00615.00615.00692.0

0615.00615.01.0

615.0025.01.0

V:0.1S across voltage,0769.0I

short 1port ,V source:

0769.0025.02.01.0

)025.01.0(2.0

0692.0025.02.01.0

)025.02.0(1.0

2

211221

22

m22222

202

112

22

11

1

Y

YYVVI

VIVVY

VIY

Y

Y

m

V

微波工程講義7-7

V1

I1

V2

I2

0.1S

0.025S

0.2S

0.05S3. Ex.7.7

[ ] [ ] [ ]213

21122221

22

22

2222222

202

112

22

11

1269.01115.01115.01192.0

1115.01115.0)05.00615.0(

0615.00769.00.0250.1

0.1:0.1Srough current th

0769.0:0.2Srough current th

05.005.01269.0

05.0:0.05Srough current th,1269.0I

short 1port ,V source:

1269.0025.02.01.0

)025.01.0(2.005.0

1192.0025.02.01.0

)025.02.0(1.005.0

1

YYY

YYVVVI

VV

VII

IIIVVY

VIY

Y

Y

n

n

V

+=

−−

=

=−=→−=+−=

=+

=−

=+

===

=

=++

++=

=++

++=

=

4. Useful for parallel circuits analysis.

1/9/2003 11

Parallel Connection of Two-Port Networks

V1

YB

I2

YA

I1

V2

I1A I2A

I2A I2A [ ] [ ] [ ]A BY Y Y= +

Lecture 4

ELG4105: Microwave Circuits © S. Loyka, Winter 2003

1/9/2003 6

Example: Pi-Network

2

2

111

1 0

221

1 0

A CV

CV

IY Y YV

IY YV

=

=

= = +

= = −

1

12 21

222

2 0

C

B CV

Y Y Y

IY Y YV =

= = − = = +

YA YB

YC I1 I2

V2V1

1 0V =

2 0V =

Lecture 4

Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-8

5. Ex.7.8

V1

I1

V2

I2

Zo,γ

l

[ ]

=====

=−=−

−=−

−==

==−+=

−

+==

−=+=

=−=→

−=Γ

−−

−

=

−

−

−

−

=

−−

−

ljZlZj

lZj

ljZYlZ

jYYljZ

YY

YlZeeZeVV

eZV

VIY

YlZe

eZeVV

eVZV

VIY

eZV

IeV

ZV

I

VeVVVV

oo

oo

oo

oll

ol

inin

l

o

in

V

ol

l

ol

inin

o

lin

o

in

V

l

o

in

o

lin

o

in

linin

shortin

ββ

ββββ

γ

γ

γγγ

γ

γ

γ

γ

γ

γγ

γ

tan1

sin

sintan1

,sin

,tan1 line lossless

sinh121

2

tanh1

111Z

2,

Z

0,

1short :2port , source:port1

21122211

12201

221

222

2

2

2

01

111

2

2

1

22

1

2

2

微波工程講義7-9

7.3 Hybrid parametersBasics1.

2port at admittanceinput circuit -open:

gaincurrent forwardcircuit -short:

gain voltagereversecircuit -open:

1port at admittanceinput circuit -short:

responses:, sources,:,

,

02

222

01

221

02

112

01

111

2121

2221212

2121111

2

1

2221

1211

2

1

1

2

1

2

=

=

=

=

=

=

=

=

+=+=

=

I

V

V

V

VIh

IIh

VVh

VIh

IVVIVhIhIVhIhV

VI

hhhh

IV

linearnetworkV1

I1

port 1 V2

I2

port 2

referenceplane 1

referenceplane 2

微波工程講義7-10

Discussion1. Useful for transistor circuits analysis.2. Ex.7.9

V1

I1

V2

I2

12Ω

6Ω

3Ω

32

366,

33

96

91

631

14636312

01

221

2

2

02

112

02

222

01

111

21

1

1

−=+

−=====

=+

==

=+×+==

==

=

=

VI

I

V

IIh

II

VVh

VIh

IVh

微波工程講義7-11

7.4 Transmission (ABCD, chain) parametersBasics1.

gaincurrent reversecircuit -short:

ttance transadmireversecircuit -open:

dance transimpereversecircuit -short:

gain voltagereversecircuit -open:

,

02

1

02

1

02

1

02

1

221

221

2

2

1

1

2

2

2

2

=

=

=

=

−=

=

−=

=

−=−=

−

=

V

I

V

I

IID

VIC

IVB

VVA

DICVIBIAVV

IV

DCBA

IV

linearnetworkV1

I1

port 1 V2

I2

port 2

referenceplane 1

referenceplane 2

1/9/2003 7

ABCD Parameters

1 2 2

1 2 2

V A V B II C V D I

= ⋅ + ⋅= ⋅ + ⋅

1 2

1 2

V VA BI IC D

= ⋅

Two Port Network

I1

V1 V2

I2

!!!

Z

Z

1)

2)

3)

[ ] ?ABCD −

[ ] ?ABCD −

[ ] ?ABCD −

Lecture 4

ELG4105: Microwave Circuits © S. Loyka, Winter 2003

1/9/2003 12

Cascade Connection of Two-Port Networks

V1A V2A V1B V2B ABCDB ABCDA

I1A I2A I2BI1B

[ ] [ ] [ ]A BABCD ABCD ABCD= +

Lecture 4

ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-12

V1

I1

V2

I2

1Ω

Discussion1. Useful for cascade circuits analysis.2. Ex.7.10

===

−=

=−

===

==

==

1011

,0,1

1,1

02

1

02

1

02

1

02

1

22

22

IV

VI

VIC

IVB

IID

VVA

3. Ex.7.11

===

−=

=−

===

==

==

101

,,0

1,1

02

1

02

1

02

1

02

1

22

22

jwjw

VIC

IVB

IID

VVA

IV

VI

V1

I1

V2

I21Ω

jwS

1Ω

V1

I1

V2

I2

jwS

4. Ex.7.12

+++

=

+

+=

jwjwjwjw

jwjw

jwjw

jw

121

1011

111

111

101

1011

5. AD-BC=1→ reciprocal circuitA=D, AD-BC=1→ symmetrical and reciprocal circuit

微波工程講義7-13

6. Ex.7.13

V1

I1

V2

I2

Zo,γ

l

=−

===+

==

=−==+=

===

=+=+

==−=−

=

−=+==−=

=−

=−

=

−

−

=−

−

=

−−−

==

−

−

−

−

−

−

−

−

−−

−

==

llZj

ljZl

lZeV

eVZV

VICl

eVeVV

VVA

IeV

ZV

IeVVeVVV

ΓVIC

VVA

lee

eZV

eVZV

DlZeeZ

eZV

eVVB

eZVIeV

ZVIVeVVV

-ΓIID

IVB

o

o

ol

in

o

lin

o

in

Il

in

linin

I

o

lin

o

inlin

linin

II

l

l

l

o

in

o

lin

o

in

ol

l

ol

o

in

linin

l

o

in

o

lin

o

inlinin

VV

ββ

ββ

γγ

γγ

γ

γ

γ

γ

γγγ

γ

γ

γ

γ

γ

γ

γ

γ

γγ

γ

cossinh1sincos

line lossless

sinh12

Z,cosh

2

0,Z

,2,

1open :2port 1,port :source,,

cosh2

12

Z,sinh

21

2

2,

Z,0,

1short :2port 1,port :source,,

2

02

12

02

1

2

2

122

1

02

1

02

1

2

2

22

2

2

122

1

02

1

02

1

22

22

22

1/9/2003 8

ABCD Parameters of TL I1 I2

V2V1 ,l β

0 l

( )

( )

1 10

2

20

1,

1

j l j l

j l j l

V V V I V VZ

V V e V e

I V e V eZ

+ − + −

+ − β − β

+ − β − β

= + = −

= + = −

2 1 1 0

12 1

0

cos sin

sin cos

V V l jI Z lVI j l I lZ

= β + β = β + β

[ ] 0

0

cos sinsin cos

l jZ lABCD

jY l lβ β

= β β

Lecture 4

ELG4105: Microwave Circuits © S. Loyka, Winter 2003

1/9/2003 9

Transformation Between Different Sets of Parameters

• Any set of parameters can be transformed into any other set of parameters

Lecture 4

ELG4105: Microwave Circuits © S. Loyka, Winter 2003

R. L

udw

ig a

nd P

. Bre

tchk

o, R

F C

ircui

t Des

ign:

The

ory

and

Appl

icat

ions

, Pre

ntic

e H

all

1/16/2003 1

S-Parameters• Why S (scattering) parameters?• Z, Y and ABCD parameters: O.C. or S.C. terminations

– very difficult at microwave frequencies• O.C. & S.C. : standing waves make measurements

difficult and can destroy elements• S-parameters: defined in terms of incident/reflected

waves• Easy to measure at microwaves: matched terminations

Lecture 5

Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-14

7.5 Conversion of the impedance, admittance, chain, and hybrid parameterssee p.267, Table 7.1

7.6 Scattering parametersBasics1. Z, Y, H, and ABCD parameters require an open or short circuit at

port. It is difficult or impossible to determine the parameters of a network at radio and microwave frequencies.

2.

linearnetwork

b1

a1

port 1

a2

b2

port 2

referenceplane 1

referenceplane 2

ibia

aSaSbaSaSb

aa

SSSS

bb

i

i

port at wave(power) reflected:port at wave(power)incident :

,2221212

2121111

2

1

2221

1211

2

1

+=+=

=

1/16/2003 3

Definition of S-Parameters

2 2

1 1

1 111

1 0 1 0

2 222

2 0 2 0

a V

a V

b VSa V

b VSa V

+

+

−

+= =

−

+= =

= = =

= = =

reflected wave at port 1

incident wave at port 1

reflected power at port 2

incident power at port 2

2 2

1 1

2 221

1 0 1 0

1 112

2 0 2 0

a V

a V

b VSa V

b VSa V

+

+

−

+= =

−

+= =

= = =

= = =

transmitted power at port 2

incident power at port 1

transmitted power at port 1

incident power at port 2

Two-

port

netw

ork

Inpu

t po

rtO

utpu

t po

rt

Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-15

3. Measurable parameters for radio frequency and microwave frequency

match aby d terminate1port with 2port at t coefficien reflection:

match aby d terminate1port with 1port to2port fromt coefficienon transmissi:

match aby d terminate2port with 2port to1port fromt coefficienon transmissi:

match aby d terminate2port with 1port at t coefficien reflection:

02

222

02

112

01

221

01

111

1

1

2

2

=

=

=

=

=

=

=

=

a

a

a

a

abS

abS

abS

abS

linearnetwork

b1

a1

port 1

a2

b2

port 2

referenceplane 1

referenceplane 2

微波工程講義7-16

4. shifting the reference planes

linearnetwork

b1

a1

port 1

a2

b2

port 2

referenceplane 1

referenceplane 2

b'1

a'1

referenceplane 1'

a'2

b'2

referenceplane 2'

port 1' port 2'

l1 l2

[ ]

=

====

====

====

−+−

+−−

−

=

+−

=

+−

=

−

=

−−−−

221

211

2

1

21

1

21

2

1

2

2211

222

)(21

)(12

211

222

02

222

)(12

02

112

)(21

01

221

211

0'1

111

22221111

'

','

','''

',',','

ljllj

lljlj

lj

a

llj

a

llj

a

lj

a

ljljljlj

eSeSeSeS

S

eSabSeS

abS

eSabSeS

abS

ebbeaaebbeaa

ββ

ββ

ββ

ββ

ββββ

微波工程講義7-17

Discussion1.

22,

2

2

,,,,,,

22

,,,,,,

,

,

,

,

,,

,,

:iport the todeliveredpower

2)

*Re(

21)*Re(

21

2)

*Re(

21)*Re(

21

)(22

1222

)(22

1222

2

2)(1

iiid

ioi

iref

oi

irefirefirefirefiref

ioi

iin

oi

iiniiniiniiniavs

ioioi

i

oi

ioii

oi

irefi

ioioi

i

oi

ioii

oi

iini

ioiiiref

ioiiiin

irefiinoi

i

irefiini

baP

bZ

VZ

VVIVP

aZ

VZV

VIVP

IZZV

ZIZV

Z

Vb

IZZV

ZIZV

ZV

a

IZVV

IZVV

VVZ

I

VVV

−=

====

====

−=−

=≡

+=+

=≡

−=

+=

→

−=

+=

linearnetwork

Vref,1

Vin,1

Vin,2

Vref,2

linearnetworkV1

I1

port 1 V2

I2

port 2

referenceplane 1

referenceplane 2

02,

2,

02

222

02,

1,

02

112

01,

2,

01

221

01,

1,

01

11121

1,1

1,12,22,2

,, if

==

======

==

======→=

in

ininin

Vin

ref

a

Vin

ref

aVin

ref

aVin

ref

aoo

VV

abS

VV

abS

VV

abS

VV

abSZZ

微波工程講義7-18

2.

linearnetwork

b1

a1

a2

b2

Zs

Vs1

Z1, Γ1 Z2, Γ2

ZL

s

s

L

L

L

LL

s

ss

SSSS

aaSS

ab

SSSS

aaSS

ab

SS

aa

baS

aaS

abaSaSb

SS

aa

ba

aaSS

abaSaSb

Γ−Γ

+=+==Γ

Γ−Γ

+=+==Γ

ΓΓ−

=→=Γ+=→+=

ΓΓ−

=→=Γ+=→+=

11

211222

2

12122

2

22

22

211211

1

21211

1

11

21

22

2

1

2

222

2

121

2

22221212

12

11

1

2

1

1

1

21211

1

12121111

1

1

1,

1,

1/16/2003 4

Measurement of S-ParametersMeasurement Setup

DUT

0LZ Z=

1 1 2 211 1 21

1 11 1,b V b VS S

a aV V

− −

+ += = = Γ = =

Forward voltage gain: 221

1

2

G

VSV

=

Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

微波工程講義7-19

3. Ex.7.14 2,1,2,1,2211 ,50,10015.0,3012,401.0,010 refino VVZIVIV →=°∠=°∠=°∠=°∠=

2,1,2,1,

,,

,,,2

,2

refrefinin

ioiiiref

ioiiiin

VVVV

IZVVIZVV

→

−=

+=

4. Ex.7.15

12

1,

1,1,

1,

1

1,

2

1,

2,21

2211

22

222

2

02,

02,

02,02,

SZZ

ZZZ

ZZZZZ

ZZZ

VVV

VZZ

ZV

VV

VV

S

ZZZ

ZZZZZZSS

o

o

o

o

o

o

o

o

Vin

refin

V

in

o

o

VinVin

ref

ooo

oo

in

in

inin

=+

=++

+=

++

=+

===

+=

++−+

==

=

=

==

ZZO ZO

5. S12=S21 → reciprocal circuitS12=S21, S11=S22→ symmetrical and reciprocal circuit

微波工程講義7-20

6. Lossless circuitEx.7.15

≠=

=∑ kjkj

SS ikij for 0for 1*

jXZO ZO

0

22

2222

222

22

2

14

44

22

22

22

*2122

*1112

*2221

*1211

*

22

2

22

22

222

11*

1221

2211

=

+−++

+−−

+=+

+−−

++

+−+=+

=

=+

++

=+=

+=

+==

+=

+==

∑

∑

≠

=

o

o

ooo

o

oo

o

o

o

oik

kjij

o

o

oik

kjij

o

o

o

o

oo

ZjXZ

ZjXjX

ZjXjX

ZjXZ

SSSS

ZjXjX

ZjXZ

ZjXZ

ZjXjXSSSS

SS

ZXZ

ZXXSSSS

ZjXZ

ZZZSS

ZjXjX

ZZZSS

微波工程講義7-21

YZO ZO

7. Ex.7.16

12111,

1,1,

1,

1

1,

2

1,

2,21

1

1

1

122111

22

1

211

11

,

02,02,02,02,

SYY

YS

VVV

VV

VV

VV

S

YYY

YYYY

YY

YYSSYYY

o

o

Vin

refin

VinVinVin

ref

oo

o

o

oo

inininin

=+

=+=+

====

+−=

+−

=+

−==+=

====

8. Ex.7.17

12211

1,

1,1,

1,

1

1,

2

1,

2,21

2

2

2

2

222

2

2

2

11

1211/

11,

11

02,02,02,02,

Snn

nS

VVV

nVnV

VV

VV

S

nn

ZnZ

ZnZ

Snn

ZZnZZnS

inininin Vin

refin

VinVinVin

ref

oo

oo

oo

oo

=+

=+

=+

====

+−=

+

−=

+−=

+−

=

====

n:1

V2

1I 2I

1VZO ZO

微波工程講義7-22

9. Ex.7.18

Zo,γ

l

a2

b2

b1

a1

121,

1,

1,

2,21

2211

02,02,

0

SeVeV

VV

S

ZZZZSS

l

Vin

lin

Vin

ref

oo

oo

inin

====

=+−

==

−−

==

γγ

10. Ex.7.19

-j25

j50

ΖΟ=50 ΖΟ=50

121,

1

1,

2

1,

2,21

222

11

1

9067.0)306030401(

301020103010

2010

11774.0503010503010,3010

255050)25()5050(

11774.0503010503010

30102010502550

)25(5050

02,

02,02,

Sjj

jj

VjjV

VV

VV

S

jjSj

jjjjZ

jjS

jjjjjjZ

in

inin

V

inVinVin

ref =°−∠=++−+

+−=+

−

===

°−∠=+−−−=−=

−+−×+=

°∠=++−+=

+=−+=−−×+=

=

==

1/16/2003 5

Example: T-Network I1 I2

V2V1

R1 R2

R3

1 2 3

0

25 , 10075

R R RZ

= = Ω = Ω= Ω

S-Parameters ???

Zin

R1 R2

R3 Z0

S11 , S21

( )1 3 2 0 75inZ R R R Z= + + = Ω

11 220inS S= Γ = =

( )( )

3 2 0021

2 0 1 3 2 0

12

R R ZZSR Z R R R Z

+= =

+ + +

12 21S S=

Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003

The Scattering MatrixEXAMPLE

Find the S-parameters of the 3 dB attenuator circuit.

SolS11 can be found as the reflection coefficient seen at port 1 when port 2 is terminated in a matched loads (Note that Zo= 50Ω and Zin

(2) =50Ω)

But, , so . Because of the symmetry of the circuit, .

Ω=++++= 50)5056.88.141(/)]5056.8(8.141[56.8)1(inZ

Zin(2)Zin

(1)

011 =S

22 0S =

11 121 1

21 222 2

S SV VS SV V

− +

− +

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

The Scattering Matrix

S21 can be found by applying an incident wave at port 1, V1+ , and

measuring the outcoming wave at port 2, V2-.

This is equivalent to the transmission coefficient from port 1 to port 2 when port 2 is matched,

From the fact that S11=S22=0, we know that V1- =0 when port 2 is

matched to 50Ω. This also implies that V2+=0.

In this case we then have that and .

Zin(2)Zin

(1)

11 121 1

21 222 2

S SV VS SV V

− +

− +

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

22 VV =−11 VV =+

The Scattering Matrix

So by applying a voltage V1 at port 1 ( ) and the voltage cross the 50 load resistor at port 2 is ( )

where .Thus, .If the input power is , then the output power is

which is one-self (-3 dB) of the input power.

11 VV =+

)56.588.141(/)56.58(8.14144.41 +=707.02112 == SS

0

2

1 2/ ZV +

0

2

1

2

10

2

2102

1210

2

2 4/2/2/||2/ ZVVZSZVSZV +++− ===

Zin(2)Zin

(1)

11 121 1

21 222 2

S SV VS SV V

− +

− +

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

2 2V V− =

The Scattering MatrixEXAMPLE EXAMPLE

A two-port net work is measured and the following scattering matrix is obtained:

a) determine whether the network is reciprocal or lossless. b) If port two is terminated with a matched load, what will be the return

loss at port 1?c) If a short-circuit is placed on port 2, what will be the resulting return

loss at port 1?SolSola) Since [S] is not symmetry, the net work is reciprocal.

To be lossless, the [S] parameters must satisfy (4.53). Since

Thus, the network is not lossless.

0.15 0 0.85 45[ ]

0.85 45 0.2 0

o o

o oS

⎡ ⎤∠ ∠−= ⎢ ⎥∠ ∠⎣ ⎦

2 2 2 211 22| | | | 0.15 0.85 0.745 1S S+ = + = ≠

The Scattering Matrixb) When port 2 is terminated with matched load, the reflection coefficient at port 1 is Γ = S11 = 0.15, Thus,

When port 2 is terminated with a short circuit, we have . Thus,

The second equation gives

Substituting into the first equation we have

20log | | 20 log(0.15) 16.5RL dB= − Γ = − =

−+ −= 22 VV

(0.85 45 )(0.85 45 )0.15 0.4521 0.2

o o∠− ∠= − = −

+

The Scattering MatrixSo the return loss is

NOTENOTE

The reflection coefficient looking into port n is not equal to Snnunless all other ports are matched.

Similarly, the transmission coefficient from port m to port n is not equal to Snm, unless all other ports are matched.

The parameters of a network are properties only of the network itself (assuming the network is linear), and are defined under the condition that all ports are matched.

20log | | 20 log(0.452) 6.9RL dB= − Γ = − =

微波工程講義7-23

7.7 Conversion from impedance, admittance, chain, and hybrid parameters to scattering parameters o.r vice versa

Discussion1. See p.288, Table 7.42. Derivation ABCD→S

DCZZADVVCZZVAVV

abS

DCZZADCZZA

DVVCZZVAVDVVCZZVAV

abS

ZV

ZIZVb

ZDVVCZZVAV

ZIDZVCZIAV

ZIZV

b

ZDVVCZZVAV

ZIDZVCZIAV

ZIZVa

IZV

ZZZZIZV

bZIZV

aIV

DCBA

IV

ooooa

oo

oo

oo

oo

a

oo

o

o

oo

o

oo

o

o

o

oo

o

oo

o

o

o

ooooi

ioiii

oi

ioiii

+++=

+++==

+++−−+

=+++−−+

==

=−

=

−−+=

+−−=

−=

+++=

−+−=

+=

−=

==−

=+

=

−

=

=

=

/B2

/B2

/B/B

/B/B

222

22

22/B

22B

22

22/B

22B

22

2,port at load matched )1(

,22

,22

,

2222

2

01

221

2222

2222

01

111

2222

22222222111

22222222111

22

212

2

1

1

2

2

微波工程講義7-24

++++−+−

+++

+++∆

+++−−+

=

+++∆=

+++∆

==

++++−+−

=+++−−+

==

=−

=

−−+∆

=+−−

∆=

−=

+++∆

=−+−

∆=

+=

−=

==−

=+

=

−

∆=

=

=

DCZZADCZZA

DCZZA

DCZZADCZZADCZZA

SSSS

DCZZAAVVCZZVDVV

abS

DCZZADCZZA

AVVCZZVDVAVVCZZVDV

abS

ZV

ZIZV

b

ZAVVCZZVDV

ZIAZVCZIDV

ZIZV

b

ZAVVCZZVDV

ZIAZVCZIDV

ZIZV

a

IZV

ZZZZIZVb

ZIZVa

IV

ACBD

IV

oo

oo

oo

oooo

oo

ooooa

oo

oo

oo

oo

a

oo

o

o

oo

o

oo

o

o

o

oo

o

oo

o

o

o

ooooi

ioiii

oi

ioiii

/B/B

/B2

/B2

/B/B

/B2

/B2

/B/B

/B/B

222

22

22/B1

22B1

22

22/B1

22B1

22

1,port at load matched )2(

,22

,22

,1

2221

1211

1111

1

02

112

1111

1111

02

222

1111

11111111222

11111111222

11

211

1

2

2

1

1

微波工程講義7-25

7.8 Chain scattering (scattering transfer, T-) parametersBasics1.

2222211

2122111

2

2

2221

1211

1

1 ,aTbTbaTbTa

ab

TTTT

ba

+=+=

=

Discussion1. See p.289 Table 7.5 for T- and S-parameters conversion

21

21122211

21

2211211222211222

02

1

02

22122

21

22

02

21112212211

1

21

11

121

111

02

121

21121

1

02

111

12121111

2

2

1

2221

1211

2

1

,

,0

01,port at load matched )2(

,1

,02,port at load matched )1(

11

1

22

SSSSS

SSSSSSTST

ab

abTT

SS

abTTaTbT

a

SS

aSaS

bbT

SaSa

baT

aSbaSba

aa

SSSS

bb

aa

a

aa

−−=

−=−==+

−=−==+

=

======

===

=

==

=

==

Hw #6(due 2 weeks)2, 6, 11,21

微波工程講義

Chapter 9 Signal-flow graphs and applications9.1 Definitions and manipulation of signal-flow graphs

definition, reduction rules9.2 Signal-flow graph representation of a voltage source9.3 Signal-flow graph representation of a passive single-port device9.4 Power gain equations

transducer power gain, operating power gain, available power gain

9-1

微波工程講義

9.1 Definitions and manipulation of signal-flow graphsBasics1. Definitions: signal-flow graph: representation of a linear systemnode (input and output nodes) : representation of a variablebranch: representation of direction and relation between nodespath: a continuous succession of branches traversed in the same

directionloop: a path originates and ends at the same node without

encountering other nodes more than once along its traverse

9-2

微波工程講義9-3

2. Reduction rules(1) Rule 1

(2) Rule 2

Discussion1. Scattering parameters

linearnetwork

b1

a1

port 1

a2

b2

port 2

referenceplane 1

referenceplane 2

2221212

2121111

2

1

2221

1211

2

1 ,aSaSbaSaSb

aa

SSSS

bb

+=+=

=

a1

b1

b2

a2

S11 S22

S12

S21

node

branch

5s 2 10s

5s

3 5s+3

微波工程講義9-4

(3) Rule 3

sabasbbsbab21

44224−

=→=−→+=

42s

s214

−

(4) Rule 4

C1C2

C3

C4

C2

C1

C3

C4C4

C4

(5) Rule 5

C1

C2 C4

C3

C1

C2

C1

C1

C4

C3

微波工程講義9-5

3. Mason’s gain rule:

pointany at t touch don' that loopsorder first 3 ofproduct :gain looporder 3rdpointany at t touch don' that loopsorder first 2 ofproduct :gain looporder 2nd

nodeany at Ppath not touch do that gains looporder -first all of sum:1

nodeany at Ppath not touch do that gains looporder -second all of sum:2

nodeany at Ppath not touch do that gains looporder -first all of sum:1

gains looporder -second all of sum:2

gains looporder -first all of sum:1

....3211

.....3211

....3211path forwardith theofgain :

....)(function transfer

2)2(

1)1(

1)1(

)2()2()2(2

)1()1()1(1

332211

∑

∑

∑

∑

∑

∑∑ ∑

∑∑ ∑

∑∑ ∑

+−+=∆

+−+=∆

+−+=∆

∆+∆+∆+∆

=

)L(

)L(

)L(

)L(

)L(

)L()L()L(-

)L()L()L(-

)L()L()L(-P

PPPsT

i

微波工程講義9-6

4. Ex. 9.5 find the transfer function

2121

23212121

2

1

3

2

1

1)1()1(

)(

251

3

loops 2

141)4(

1111

6161)2)(1(

3132

11

111

paths forward 3

LLLLLPLLLLPPsH

ssL

sL

ssP

Psss

ss

sP

+−−−++−−+

=

+−=+

−=

+−=×−×

+××=

=××=++

=××+

××+

××=

1 1

-31 1

6-4

-53

11+s 2+s

s

P1

P2 P3

微波工程講義9-7

9.2 Signal-flow graph representation of a voltage source

Es

Zs

Γs

asbG 1 bs

Γs

asbs

bs

as

1

1Vs

Is

ssGs

o

ins

so

os

o

s

so

o

o

refs

insso

sos

so

orefs

refso

sins

o

srefsinss

o

refsins

refsinssrefsinsssss

abbZ

VZZZZ

ZE

ZZZ

Z

V

VZZZZ

EZZ

ZV

VZZV

ZZVVZ

ZVV

VVZIIVZIE

Γ+=→

+−

++

=→

+−

−+

=→

++−=++−

−=

+++−=+−=

222

)1()1(

)()(

,,

,,

,,,,,,

,,,,

微波工程講義9-8

9.3 Signal-flow graph representation of a passive single-port device

ZLΓL

aLaL

ΓL

bLbL

1

1VL

IL

LLL

o

inL

oL

oL

o

refL

inLoL

oLrefL

refLinL

Lo

refLinLLrefLinLLLL

abZ

VZZZZ

Z

V

VZZZZ

V

VV

ZZ

VVZIIZIV

Γ=→

+−

=→

+−

=→

+=

−=+==

22

)(

,,

,,

,,

,,,,

微波工程講義9-9

Discussion1. Ex.9.6 find Γin

Γin two-portnetwork ZL

a1

b1

b2

a2

S11 S22

S12

S21

ΓL

L

L

L

LLin

L

L

L

LLL

LLLLLLin

SSSS

SSSSS

LPLP

SLSSP

SP

SSSSSSSS

SSSSSSSSSS

Γ−Γ

+=Γ−

Γ+Γ−=

−+−

=Γ

Γ=

Γ==

Γ−Γ

+=+Γ+Γ+=

+ΓΓΓ+ΓΓ+Γ+=Γ

22

122111

22

12212211

1

211

221

12212

111

22

12211122122111

22221221221221122111

11)1(

1)1(

loop 1

paths forward 2

rule sMason')2(1

...)1(

.... )1(

Γin

微波工程講義9-10

2. Ex.9.7 find ΓoutΓout

two-portnetwork ZL

Zs

Vsa1

b1

b2

a2

S11 S22

S12

S21

Γs Γout

s

s

s

ssout

s

s

s

sss

ssLsssout

SSS

SS

SSSSL

PLPSL

SSPSP

SSSSSSSS

SSSSSSSSSS

Γ−Γ

+=Γ−

Γ+Γ−=

−+−

=Γ

Γ=

Γ==

Γ−Γ

+=+Γ+Γ+=

+ΓΓΓ+ΓΓ+Γ+=Γ

11

122122

11

12211122

1

211

111

12212

221

11

12211111122122

11111221111221122122

11)1(

1)1(

loop 1

paths forward 2

rule sMason')2(1

...)1(

.... )1(

微波工程講義9-11

3. Ex.9.8 find Pd:power delivered from source, PL:power delivered to the load, Pavs:maximum power available from source

bs

as

ΓL

bL

aL

Γs

bG 1 1

1

source load

2

22

2

2s

s

222222

2222

L

2

L

2

L

22

LL

L

1)1(

1

:conditionmatch conjugate

)1()1(

)1()1(111

1)

1(1

1

s

Gs

Gdavs

L

LsLLLLL

LLsLs

G

s

LG

s

Gssd

s

LGG

s

G

ss

Gss

s

Gs

sLsGsLLGsLGssGs

bbPP

babaP

PbbbbabP

bbbbba

bb

bbabbbabb

Ls Γ−=Γ−

Γ−==

Γ=Γ

Γ−=Γ−=−=

=Γ−=Γ−ΓΓ−

=ΓΓ−

Γ−

ΓΓ−=−=

ΓΓ−Γ

=−ΓΓ−Γ

=Γ−

=

ΓΓ−=

→

ΓΓ+=ΓΓ+=Γ+=Γ+=

∗Γ=Γ

∗

Pd PL

微波工程講義9-12

4. Ex.9.9 find b3/bs

3-portnetwork

b1

a1

a2

b2

Zs

Vs ZL

ZD a3b3

a1

b1

b2

a2

S11 S22

S12

S21Γs ΓL

bG

b3 a3

S31

ΓD

S33

S32 S31 S23

321

827143323121

81

2221

22113

13318

32237

3221136

1223315

12214

223

222

111

32212

311

)3(

)2(

...)1(1,1

)3()2()1(1

loop 8

paths forward 2

LLLL

LLLLLLLLLLLLL

LLLS

LLLPP

bb

SSLSSLSSSLSSSLSSL

SLSLSL

SSPSP

L

s

Ds

DL

DLs

DLs

Ls

D

L

s

L

=

+++++=

++==∆Γ−=∆

−+−∆+∆

=

ΓΓ=ΓΓ=

ΓΓΓ=ΓΓΓ=

ΓΓ=Γ=Γ=Γ=

Γ==

∑

∑

∑

∑ ∑ ∑bs

as

微波工程講義9-13

9.4 Power gain equationsBasics

[ S ]Vs

Zs

ZL

Γs Γin Γout ΓL

** ,

),,(gain power transducer

),(gain power available

),(gain power operating

outLSinLavninavs

LSavs

LT

Savs

avnA

Lin

LP

PPPP

SPPG

SPPG

SPPG

Γ=ΓΓ=Γ==

ΓΓ≡

Γ≡

Γ≡

Pavs PavnPin PL

微波工程講義

Discussion1.

a1

b1

b2

a2

S11 S22

S21

Γs

bs

as

bG

ΓL

Pavs Pin Pavn PL

source amplifier load

2

2

222

11

221

2

22

2221122

22

22

2221

222

22

22212222

22

21

22

1212

22

222

12

12

1

1

11

)1(1

1

)1(11,

11

)1(

11

)1()1(

)1)(1(1

1

)1()1(

1

Gs

inavs

Gouts

out

outsinsG

insout

outLavn

GinsL

LLLLLL

insL

G

LL

Gins

ininin

ins

GssinsGs

bPP

bS

S

S

Sb

S

SPP

bS

SabaP

SbS

SaSba

babaP

abbbbb

ins

outLoutL

Γ−==

Γ−Γ−=

Γ−

Γ−Γ−=ΓΓ−

ΓΓ−Γ−

Γ−==

ΓΓ−Γ−

Γ−=Γ−=−=

ΓΓ−Γ−=

Γ−==

ΓΓ−

Γ−=Γ−=−=

=ΓΓ−

=→ΓΓ+=

∗

∗∗

Γ=Γ

∗Γ=Γ∗Γ=Γr

r

r

aL

bL

9-14

微波工程講義

2.

2211

22221

2222

22221

2211

2221

2222

2221

22

222

11

221

222

22

22212

2

2

11

)1)(1(

11

)1)(1(

)1(1

)1(

)1(1

)1(

11,

)1(1

11

)1(,

1

)1(

outLs

sL

insL

sL

avs

LT

outs

s

avs

avnA

inL

L

in

LP

Gs

inavsGouts

Lavn

GinsL

LLG

ins

inin

S

S

S

SPPG

S

SPP

G

S

SPPG

bPPbS

SPP

bS

SPbP

insoutL

ΓΓ−Γ−

Γ−Γ−=

ΓΓ−Γ−

Γ−Γ−==

Γ−Γ−

Γ−==

Γ−Γ−

Γ−==

Γ−==

Γ−Γ−==

ΓΓ−Γ−

Γ−=

ΓΓ−

Γ−=

∗∗ Γ=ΓΓ=Γ

9-15

微波工程講義9-16

222

2212

11

max2211

222

22

21211

211in12

11

11,

1

1

1

1

0

SS

SGSS

GGGS

SS

G

SS

TULs

LoSL

L

s

sTU

−−=→Γ=Γ=

=Γ−

Γ−

Γ−

Γ−=

=Γ→=

∗∗

5. Unilateral transducer power gain GTU

[ S ]Go

Zo Zo

Γs Γin Γout ΓL

Outputmatchingcircuit GL

Inputmatchingcircuit Gs

微波工程講義9-17

6. A 800MHz amplifier (Zo=50Ω) with S11=0.45∠150°, S12=0.01∠-10°, S21=2.0∠10°, S22=0.4∠-150°, Zs=20Ω, ZL=30Ω→ GT, GP, GA

dBS

SPPG

dBS

SPP

G

dBS

SPPG

SSSS

SSSS

ZZZZ

ZZZZ

Lins

Ls

avs

LT

sout

s

avs

avnA

Lin

L

in

LP

s

sout

L

Lin

oL

oL

os

os

4.7487.511

)1)(1(

7.7855.51)1(

)1(

7.7937.51)1(

)1(

87.150408.01

,32.150455.01

25.0,429.0

222

2

22221

211

2

2221

222

2

2221

11

122122

22

122111

Ls

==Γ−ΓΓ−

Γ−Γ−=≡

==Γ−Γ−

Γ−=≡

==Γ−Γ−

Γ−=≡

°∠=Γ−Γ

+=Γ°∠=Γ−Γ

+=Γ

−=−−

=Γ−=−−

=Γ

微波工程講義9-18

dBS

SS

GGGG

S

SPPG

S

SPPG

S

SPPG

SSSSS

TUPAT

Lins

Ls

avs

LT

sout

s

avs

avnA

Lin

L

in

LP

outin

2542.3221

11

111

)1)(1(

1)1(

)1(,

1)1(

)1(

,

,0

222

2212

11max

222

2

22221

211

2

2221

222

2

2221

22L11s

221112

==−−

====

Γ−ΓΓ−

Γ−Γ−=≡

Γ−Γ−

Γ−=≡

Γ−Γ−

Γ−=≡

=Γ=Γ

=Γ=Γ→=∗∗

7. A 2GHz amplifier (Zo=50Ω) with S11=0.97∠-43°, S12=0, S21=3.39∠140°, S22=0.63∠-32°, Γs =0.97∠43°, ΓL =0.63∠32°, → GT, GP, GA

Hw #7 (due 2 weeks)5, 10, 14