Passive Devices (3 port network)

S-matrix of 3 port networks

333231

232221

131211

SSS

SSS

SSS

S

(1) All ports are matched and network is reciprocal

0

0

0

2313

2312

1312

SS

SS

SS

S

(2) can be lossless?

? * USS T

10.0 ,1

1 0 ,0 ,1

1 0 ,0 ,1

2

23

2

1313*12

2

23

2

13

2

12*2312

*23

2

23

2

12

2

12*1323

*13

2

13

2

12

SSSSSS

SSletSSSS

SSletSSSS

3 Port Network can not be lossless, reciprocal, and match at all port

Matched, lossless, and nonreciprocal 3 port network

Circulator

0

0

0

3231

2321

1312

SS

SS

SS

S

00

00

00

1,0)2(

00

00

00

1,0)1(

.0 ,1

,0 ,1

,0 ,1

3

2

1

3

2

1

312312133221

313221312312

13*12

2

32

2

31

21*23

2

23

2

21

23*31

2

13

2

12

j

j

j

j

j

j

e

e

e

S

SSSSSS

e

e

e

S

SSSSSS

SSSS

SSSS

SSSS

Reciprocal, matched, and lossy 3 port network

Reciprocal, lossless, and only 2 ports matched

0

0

332313

2312

1312

SSS

SS

SS

S

1.0 ,1

1 0 ,0 ,1

1 0 ,0 ,1

2

3333*2313

*12

2

33

2

23

2

13

2

12*2313

*3312

*23

2

23

2

12

2

12*1323

*13

2

13

2

12

SSSSSSSS

SSletSSSSSS

SSletSSSS

00

00

00

j

j

j

e

e

e

S Port 3 is isolated

0

0

0

2313

2312

1312

SS

SS

SS

S Lossy power divider

Passive Devices (4 port network)S-matrix of reciprocal and matched

network

0

0

0

0

342414

342313

242312

141312

SSS

SSS

SSS

SSS

S

For lossless, S-matrix must be unitary matrix

0

0

0

0:)3,2(

,0:(2,4) ,1:)4,4(

,0:(1,3) ,1:)3,3(

,0:(3,4) ,1:)2,2(

,0:(1,2) ,1:)1,1(

2314

2

34

2

122323*3412

*143434

*1423

*1212

2

13

2

24*1423

*2413

*14

*1324

*1423

*13

*24

34*2413

*12

23*3412

*14

2

34

2

24

2

14

34*1423

*12

2

34

2

23

2

13

23*2413

*14

2

24

2

23

2

12

24*1423

*13

2

14

2

13

2

12

SS

SSSSSSSSSSSSS

SSSSSSSSSSSSS

SSSS

SSSSSSS

SSSSSSS

SSSSSSS

SSSSSSS

S

00

00

00

00

00

00

00

00

,,0 ,0 ,2/ ,0

)2( )1(

)12(

0

,,,

1

1,1

,1,1

2413341224133412

34241312

)()(34

*2413

*12

24133412

22

22

34

2

12

2

34

2

24

2

24

2

12

22

24

2

13

2

24

2

12

2

13

2

12

34241312

24133412

BA

BA

AB

AB

S

BjA

BjA

jAB

jAB

S

andletandlet

CouplerricAntisymmetCouplerSymmetric

n

ABeABeSSSS

AeSAeSBeSBeS

BA

BSSSSSS

ASSSSSS

jj

jjjj

dB

dB log20log10

dB log20log10

dB log20log10

144

1

144

3

3

1

CDI

SP

PIIsolation

S

A

P

PDyDirectivit

AP

PCCoupling

00

00

00

00

BjA

BjA

jAB

jAB

S

Hybrid Coupler(3 dB Coupler)

0110

1001

1001

0110

2

1

010

100

001

010

2

1

S

j

j

j

j

S

couplerhybridricAntisymmetcouplerhybridSymmetric

BA

Even-Odd Mode Circuit Analysis

•If circuit can be decomposed into the superposition of an even-mode excitation and odd-mode excitation

•Adding two sets( even & odd) of excitations produces the original excitation

,2

1

2

1,

2

1

2

1

,2

1

2

1,

2

1

2

1

43

21

oeoe

oeoe

BTTB

TTBB

3-Port Network: Power Divider

(1)Reciprocal and Lossless

-> some ports mismatched

Lossless Power Divider: 21 PPPin

inYInput Admittance at Input Port

21

11

ZZjBYin

for Input Port Matching, B=0 and

oin ZZZ

Y111

21

inP

1P

2P

Port #1Port

#2

Port #3

1221

2

2

21

2

1

2

::

2

1,

2

1,

2

1

ZZPP

Z

VP

Z

VP

Z

VP oo

o

oin

Ex) 1:2 divider (with B=0 & Port #1 matched: )

75,15031311

22

21121

2112

ZZZZZZZ

ZZPP

oo

666.01507550

15075501

Reflection at Port #2:

333.07515050

75150502

Reflection at Port #3:

22S

33S

011 S

(2)Resistive Divider – All ports are matched

- output ports are not isolated

inin

inooo

in

oo

o

oo

o

ooo

ooo

in

PPPP

PZ

V

Z

VPP

Z

VP

S

SS

VVZZ

ZVVV

VZZ

Z

VV

ZZZ

ZZZ

Z

2

1

4

1

8

1

2

1,

2

1

011

101

110

2

1

2

1

2

1

4

3

3

3

2

32

3

32

333

32

21

22

32

21

3121

132

11

023 S

Wilkinson Power Divider( equal power division)

Resistive divider -all ports are matched

-out-ports are isolated

Microstrip Wilkinson Power Divider

Equivalent Circuit

inP

2P

3P32 PPPin

Wilkinson Power Divider

even mode

odd mode

232 gg VV

232 gg VV

Equivalent Circuit in normalized and symmetric form

Wilkinson Power Divider: even mode

einZ

)0,0( 22 ,33,22

2

eeein SSmatchingforZ

ZZ

Voltage on transmission line)(xV

) ( 0

707.02

1

22

22

2

1

1

1)1()()4(

1

1,1)1()0(

)()(

,23,32

2

11312

1

1

2

openportS S

jjV

VS S

jV

jjVjjVVxV

VVVxV

eeVeVeVxV

oe

,e,e

xjxjxjxj

Wilkinson Power Divider: odd mode

) ( 0

0

1 0

)0,0( 22

,23,32

2

1,13,12

1

,33,22

shortportSS

V

VSS

shortedisportV

SSmatchingforrr

Z

oo

oo

oooin

oinZ

All power from Port #2 is delivered to r/2 resister, and all dissipated

No power is delivered to Port #1

Wilkinson Power Divider: input port

inZ

inZ

001

21

2

11

2

SZZZ

Z

inininin

in

Wilkinson Power Divider: S-matrix

00

00

0

j

j

jj

S

•All ports are matched•Reciprocal•Lossy•Output ports are isolated

•Symmetric•Not Unitary

Wilkinson Power Divider: unequal power division

inP

2P

3P

.1

,1

,1

,

23

22

3

2

3

2

32

KKZR

KKZZKZ

K

KZZ

P

PKLet

o

ooo

oo

Wilkinson Power Divider: N-way Divider

4-Port Network: Quadrature Hybrid

3 dB Directional Coupler with Quadrature Phase Difference between two output ports

Reciprocal, Lossless, and all ports are matched.

010

001

100

010

2

1

j

j

j

j

S

Even-mode

1

1

2

1

1

01

02

2/0

1

01

j

j

jj

j

jDC

BA

j

jjDCBAT

jj

jj

DCBA

DCBA

e

e

12

1

2/11

22

02/11

2/11

Odd-mode

1

1

2

1

1

01

02

2/0

1

01

j

j

jj

j

jDC

BA

j

jjDCBAT

jj

jj

DCBA

DCBA

o

e

12

1

2/11

22

02/11

2/11

02

,2

1

2

,2

1

2

,02

41

31

21

11

oe

oe

oe

oe

S

TTS

jTT

S

S

010

001

100

010

2

1

j

j

j

j

S

Coupled Line Theory

Propagating Modes on Symmetric Coupled Line

Even Mode Odd Mode

ground

Even Mode

Line Characteristics for even mode propagation

•Capacitance per unit length w/o dielectrics:

•Inductance per unit length:

•Capacitance per unit length with dielectrics:

•Characteristic Impedance:

•Propagation Constant:

aaea CCC 2211

eaoo CL /

eaeeffdded CCCC 2211

edo CLZ eeffooe

e LC

Odd Mode

Line Characteristics for even mode propagation

•Capacitance per unit length w/o dielectrics:

•Inductance per unit length:

•Capacitance per unit length with dielectrics:

•Characteristic Impedance:

•Propagation Constant:

aaaaoa CCCCC 12221211 22

oaoo CL /

oaoeffdd

ddod

CCC

CCC

1222

1211

2

2

edo CLZ oeffooo

o LC

(1)Voltage and Current Waves of Even Mode Excitation

eoeZ ,

eeee VVVV 2121 ,

zj

oe

ezj

oe

ezje

zje

zje

zje

eeee

ee

eZ

Ve

Z

VeIeIzi

eVeVzv

11111

111

)(

)(

zj

oe

ezj

oe

ezje

zje

zje

zje

eeee

ee

eZ

Ve

Z

VeIeIzi

eVeVzv

22222

222

)(

)(

(1)Voltage and Current Waves of Odd Mode Excitation

oooZ ,

zj

oo

ozj

oo

ozjo

zjo

zjo

zjo

oooo

oo

eZ

Ve

Z

VeIeIzi

eVeVzv

11111

111

)(

)(

zj

oo

ozj

oo

ozjo

zjo

zjo

zjo

oooo

oo

eZ

Ve

Z

VeIeIzi

eVeVzv

22222

222

)(

)(

oooo VVVV 2121 ,

Voltage and Current Waves on Coupled Lines

(1)Voltage and Current Waves on Line #1

zj

oo

ozj

oo

ozj

oe

ezj

oe

e

zjo

zjo

zje

zje

zjo

zjo

zje

zje

ooee

ooee

ooee

eZ

Ve

Z

Ve

Z

Ve

Z

V

eIeIeIeIzi

eVeVeVeVzv

)(

)(

1

1

(2)Voltage and Current Waves on Line #2

zj

oo

ozj

oo

ozj

oe

ezj

oe

e

zjo

zjo

zje

zje

zjo

zjo

zje

zje

ooee

ooee

ooee

eZ

Ve

Z

Ve

Z

Ve

Z

V

eIeIeIeIzi

eVeVeVeVzv

)(

)(

2

2

ooo

eoe ZZ ,,,

)(),( 11 zizv

)(),( 22 zizv

o

e

o

e

Ljoo

Ljoe

Ljoo

Ljoe

oooeoooe

Ljoe

Ljoe

Ljoo

Ljoe

oooeoooe

o

e

o

e

oo

oe

oo

oe

LjLjLjLj

LjLjLjLj

o

e

o

e

LjLjLjLj

LjLjLjLj

o

e

o

e

LjLjLjLj

LjLjLjLj

I

I

I

I

eZeZeZeZ

ZZZZ

eZeZeZeZ

ZZZZ

I

I

I

I

Z

Z

Z

Z

eeee

eeee

V

V

V

V

eeee

eeee

V

V

V

V

I

I

I

I

eeee

eeee

I

I

I

I

oeoe

oeoe

oeoe

oeoe

oeoe

oeoe

oeoe

oeoe

000

000

000

000

1111

1111

1111

1111

1111

1111

4

3

2

1

4

3

2

1

Impedance Matrix Z11 = Z22=Z33= Z44

0,0 4321

01

111

1

IIIII

VZ

kkI

&

1111

1111

0

0

01

jLjLjeo

o

e

o

e

LjLjLjLj

LjLjLjLj

eeeβlet β

I

I

I

I

eeee

eeee

I

oe

oeoe

oeoe

12

2

1

2

21

21

122

22

22

1

21

1

12,

12

111

011

,

.0

,0

,0

,

IZZ

e

eIZIZIZIZ

VVVVV

e

eIII

e

IIIIeIeI

IIeIeI

eIIeII

eIeIeIeI

eIeIeIeI

IIII

IIIII

oooej

j

ooooooeoeeoe

ooee

j

j

oejoej

oj

e

oej

oj

e

joo

jee

jo

je

jo

je

jo

je

jo

je

oeoe

oeoe

cot2

01

111

1

oooe

I

ZZj

I

VZ

kk

Impedance Matrix Z12=Z21=Z34=Z43

0,0 4312

02

112

2

IIIII

VZ

kkI

1111

1111

0

0

0

2

o

e

o

e

jjjj

jjjj

I

I

I

I

eeee

eeeeI

22

2

1

2

22

22

222

22

22

2

21

1

12,

12

111

011

,

.0

,0

,

,0

IZZ

e

eIZIZIZIZ

VVVVV

e

eIII

e

IIIIeIeI

IIeIeI

eIIeII

eIeIeIeI

eIeIeIeI

IIIII

IIII

oooej

j

ooooooeoeeoe

ooee

j

j

oejoej

oj

e

oej

oj

e

joo

jee

jo

je

jo

je

jo

je

jo

je

oeoe

oeoe

cot2

02

112

2

oooe

I

ZZj

I

VZ

kk

Impedance Matrix Z13 = Z31 = Z24 = Z42

0,0 4213

03

113

3

IIIII

VZ

kkI

1111

1111

0

0

0

3

o

e

o

e

jjjj

jjjj

I

I

I

I

eeee

eeee

I

3

3

1

333

3

sin2

2,

2

0

,

.0

,

,0

,0

IZZ

j

Iee

ZZIZIZIZIZ

VVVVV

ee

III

ee

IIIIeeIeeI

IIeeIeeI

IIII

eIeIeIeI

IeIeIeIeI

IIII

IIII

oooe

jjoeoe

oooeooeoeeoe

ooee

jjoejjoejj

ojj

e

oejj

ojj

e

ooee

jo

je

jo

je

jo

je

jo

je

oeoe

oeoe

sin203

113

3

oooe

I

ZZj

I

VZ

kk

Impedance Matrix Z14 = Z41 = Z32 = Z23

0,0 3214

04

114

4

IIIII

VZ

kkI

1111

1111

0

0

0

4

o

e

o

e

jjjj

jjjj

I

I

I

I

eeee

eeee

I

4

4

1

444

4

sin2

2,

2

0

,

.

,0

,0

,0

IZZ

j

Iee

ZZIZIZIZIZ

VVVVV

ee

III

ee

IIIIeeIeeI

IIeeIeeI

IIII

IeIeIeIeI

eIeIeIeI

IIII

IIII

oooe

jjoooe

ooooooeoeeoe

ooee

jjoejjoejj

ojj

e

oejj

ojj

e

ooee

jo

je

jo

je

jo

je

jo

je

oeoe

oeoe

sin204

114

3

oooe

I

ZZj

I

VZ

kk

tantansinsin

tantansinsin

sinsintantan

sinsintantan

2

4

3

2

1

4

3

2

1

I

I

I

I

ZZZZZZZZ

ZZZZZZZZ

ZZZZZZZZ

ZZZZZZZZ

j

V

V

V

V

oooeoooeoooeoooe

oooeoooeoooeoooe

oooeoooeoooeoooe

oooeoooeoooeoooe

LLLet oe

ooo

eoe ZZ ,,,

L

4-Port Network: Coupled Line Coupler

Single Section Coupled Line Coupler

o

o

o

oo

ZIV

ZIV

ZIV

ZIEVVZIE

44

33

22

111111

,

,

,

Boundary Condition at each Port

kokekkokek IIIVVV ,

oeine

oein

eine

ooe

oeooeein

ZZ

VI

ZZ

ZVV

jZZ

jZZZZ

,1

,

,1

,

,

tan

tan

einZ ,oinZ ,

ooine

ooin

oino

ooo

ooooooin

ZZ

VI

ZZ

ZVV

jZZ

jZZZZ

,1

,

,1

,

,

tan

tan

ooinein

ooineino

ooinein

oeinoinooinein

oe

oein

ZZZ

ZZZZ

ZZZ

ZZZZZZ

II

VV

I

VZ

22

2

,,

2,,

,,

,,,,

11

11

1

1

inZ

2,,

1

oeinein

oin

ZZZ

ZZ matching:# portfor

2,, oeinein ZZZfor

0

)1(

)1(

tan

11

222

222

2

22

22

,,

S

ZtZZjZtZ

tZZjZtZZ

ZZ Z

tZZjZtZZZ

tZZjZtZZZZZ

tjZZtjZZ

tjZZtjZZZZ

tjZZ

tjZZZ

tjZZ

tjZZZZZ

t

ooooeoo

oooeooo

oooeo

oooeoooooe

oooeooooeooooe

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ooooeooooe

ooo

ooooo

ooe

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VjtC

jCtV

ZZZCZZ

ZZClet

tjZZZ

tZZZZjV

tZZjZ

tZZjV

tZZjZ

tjZZ

tZZjZ

tjZZV

ZZ

Z

ZZ

ZV

VVVVV

oooeooooe

oooe

oooeo

oooeoooe

oooeo

oooe

oooeo

ooo

oooeo

oeo

ooin

oin

oein

ein

oeoe

23

2

,

,

,

,

11333

1

/21,

/2

/

2

2 2

tan1

tan2

3

jC

jC

V

V

, 222 oe VVVfor

z0zLz

cos2tan

tancos2

)(

tancos2

)(

,

1

1

e

oeo

oeoe

oeooeo

e

e

oeooeo

e

zj

oeo

oeozje

zje

zje

V

jZZ

ZZV

jZZZZ

V

VLzV

zjZZZZ

zV

eZZ

ZZeV

eVeVzV

on line

oeo

oeoe ZZ

ZZz

0

tan2costancos

1

costan

1

,

,

1

2

oooeo

o

oein

ein

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o

e

oeo

o

oeo

oeoeeee

ZZjZ

ZVV

ZZ

Z

jZZ

Z

V

jZZ

Z

ZZ

ZZVVVV

tZZjZ

tjZZ

tZZjZZ

tjZZZ

tZjZZtZjZZ

tZjZZ

tZjZZZtZjZZ

tZjZZ

ZtZZjZ

tjZZZZ

tZZjZ

tjZZZZ

ZZ

Z

oooeo

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oooeoooe

oeoooe

oooeoooeoooe

oeoooe

oooeoooeoeoooe

oeoooe

o

oooeoe

oeoooeoe

oooeoe

oeoooeoe

oein

ein

2

2

,

,

z0zLz

cos2tan

tancos2

)(

tancos2

)(

,

1

1

o

ooo

oooo

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e

zj

ooo

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zjo

zjo

V

jZZ

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jZZZZ

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zjZZZZ

zV

eZZ

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eVeVzV

on line

ooo

oooo ZZ

ZZz

0

tZZjZ

tjZZ

tZZjZZ

tjZZZ

tZjZZtZjZZ

tZjZZ

tZjZZZtZjZZ

tZjZZ

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tjZZZZ

tZZjZ

tjZZZZ

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Z

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2

2

,

,

tan2costancos

1

costan

1

,

,

1

2

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o

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o

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o

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ZVV

ZZ

Z

jZZ

Z

V

jZZ

Z

ZZ

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oe VV 22

0

sincos1

1

sincos/2

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sincos2

2

tan2cos22

224

2

2

2222

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oooeo

o

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VjC

C

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Z

ZZjZ

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0

1sincos1

1

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tan

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,

4

2

2

22

2

3

V

V

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V

V

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V

V

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1

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