EMLAB

1

Network parameter transformation

EMLAB

S-matrix conversion from Z-matrix

)(1

0

nnnnn

nnn

VVZ

III

VVV

][][V

]][[]][[Z

1

]][[]][[]][[][

0

V

VZVZ

IZIZIZV

022021

012011

1

11

1111

//

//][

) ][][ () ][][ (][

1

1Zport -one

ZzZz

ZzZzZ

SUSUZ

ZS

][ ][][Z

1 ][ ][][

Z

1

00

VUZVUZ

][][Z

1 ][][

Z

1][

0

1

0

UZUZS

10

10

001

][

U

2

EMLAB

][][ ][][ Re

2

1

][][ Re2

1

**

0

*

VVVVZ

IVP

tt

tav

**

0

****

0

][][ ][][ Re2

1

][][ ][][ ][][ ][][ Re2

1

VVVVZ

VVVVVVVVZ

tt

tttt

incident power reflected power

Lossless network, incident power = reflected power

]][[][

][][ ][][ **

VSV

VVVV tt

][ ][

][][

][][][][ ][][

1*

*

***

t

t

ttt

SS

USS

VSSVVV

Unitary matrix

S-matrix property for Lossless network

N

kijkjkiSS

1

*

3

EMLAB

Example 4.5 S-parameter 계산

00.2450.85

45-0.8500.15[S]

1) Port 2 가 matched load 로 termination 된 경우 port 1 의 return loss2) Port 2 가 short 되었을 때 port 1 의 return loss.

111

1

2

2

1

2221

1211

2

1 0,)1

SV

V

VV

V

SS

SS

V

V

122

2112111

22

12122221212

2121111

22

2

1

2221

1211

2

1

1

1

,)2

VS

SSSV

S

VSVVSVSV

VSVSV

VVV

V

SS

SS

V

V

4

EMLAB

Reciprocal network

IZV

I

I

I

ZZ

ZZZ

V

V

V

NNNN

N

N

2

1

1

11211

2

1

xNMy

xMNy

IZV

IZV

xI

yV

port N

port M

yV

xI

jiij ZZ Port N 에 전류원 Ix 를 연결했을 때 port M 에 전압 Vy 가 측정되면 →

Port M 에 전류원 Ix 를 연결했을 때 port N 에 전압 Vy 가 측정된다 .

5

EMLAB

1

00

1

00

0

1

0

][][Z

1][][

Z

1

][][Z

1][][

Z

1

][][Z

1 ][][

Z

1][

UZUZ

UZUZ

UZUZS

tttt

t

t

S-matrix property for Reciprocal network

]])[[]([2

1][],])[[]([

2

1][

)(2

1),(

2

1

)(1

00

00

0

IUZZVIUZZV

IZVVIZVV

VVZ

III

VVV

nnnnnn

nnnnn

nnn

1

00

][][Z

1 ][][

Z

1][

UZUZS

tSS ][][

6

EMLAB

A shift in reference planes

jkzeVzV 0)(

nV

Ljn

jnn eVeVV n

nV

Ljn

jnn eVeVV n

]][[][ VSV

]][[][ VSV

][][2

1

n

j

j

j

n V

e

e

e

V

N

][][2

1

n

j

j

j

n V

e

e

e

V

N

NN j

j

j

j

j

j

e

e

e

S

e

e

e

S

2

1

2

1

][][

7

EMLAB

Generalized Scattering parameters

nnn

nnn

ZVb

ZVa

0

0

/

/

Port 별로 연결된 transmission line 의 임피던스가 다른 경우

)(1

)(1

)(

)(

00

0

0

nn

n

nnn

n

nnnnn

nnnnnn

baZ

VVZ

I

baZVV

baZVVV

jkVij

ji

jkaj

iij

kk

ZV

ZV

a

bS

for0

0

0

for0

8

EMLAB

Vector network analyzer9

EMLAB

4.4 Transmission (ABCD) parameter

2

2

1

1

221

221

I

V

DC

BA

I

V

DICVI

BIAVV

3

3

22

22

11

11

1

1

3

3

22

22

2

2

2

2

11

11

1

1 ,

I

V

DC

BA

DC

BA

I

V

I

V

DC

BA

I

V

I

V

DC

BA

I

V

10

EMLAB

Z-to-ABCD transform

2221212

2121111

IZIZV

IZIZV

2221212

2121111

VYVYI

VYVYI

Reciprocal network 인 경우 jiij ZZ

2221212

2121111

IZIZV

IZIZV

2

2

22

1221221111

21

2

2

22

121121

211

1

2

2

22

12

1

1

21

11

1

1

1

0

10

1

1

0

0

1

I

V

Z

ZZZZZ

Z

I

V

Z

ZZZ

ZI

V

I

V

Z

Z

I

V

Z

Z

11

EMLAB

12

EMLAB

Example 4.6

221

221

DICVI

BIAVV

1

0

/

1

02

1

02

1

1

1

02

1

02

1

2

2

2

2

I

I

V

I

I

ID

V

IC

ZZV

V

I

VB

V

VA

13

EMLAB

Figure 4.12 (p. 188) A coax-to-microstrip transition and equivalent circuit representations. (a) Geometry of the transition. (b) Representation of the transition by a “black box.” (c) A possible equivalent circuit for the transition [6].

14

EMLAB

Figure 4.23 (p. 199)Some common microstrip discontinuities. (a) Open-ended microstrip. (b) Gap in microstrip. (c) Change in width. (d) T-junction. (e) Coax-to-microstrip junction.

15

EMLAB

Properties of Z(ω) and Γ(ω)

deVt tj)(

2

1)( )()( * tt

deVdeVdeV tjtjtj )()()( **

)()( * VV

)()()()()()()()( **** IZVIZIZV

)()(),()(

)()()()(

)()(*

XXRR

jXRjXR

ZZ

)()()(

)()()(

)()(

)()(

)(

)()(

0

0*

0

0

0

0

jXZR

jXZR

jXZR

jXZR

ZZ

ZZ

)()(* ZZ

)()(*

2*2)()()()()()(

2

211111*1111

2

11 )(1)()()()()( SSSSSS

(A signal is a Real number function)

16

EMLAB

rE dV

CdI rH

Equivalent voltage and current

TEM 인 경우 unique

17

EMLAB

Figure 4.2 (p. 163)Electric field lines for the TE10 mode of a rectangular waveguide.

22

),(sin

),(sin

ak

eyxAhea

xA

ajH

eyxAeea

xA

ajE

zjx

zjx

zjy

zjy

k

H

EZ

x

yTE

TEZ

yxzyx

),(ˆˆ),(ˆ e

h

Rectangular waveguide Z018

EMLAB

k

ZTE GHz10f 2.54,ε

1.016cmb 2.286cm,a

r

10

12

20

12

20

4.2092

304

158

mc

fk

ma

k

ma

k

rd

a

316.0

6.259

500158

3774.209

00

00

0

000

ad

ad

dd

aa

ZZ

ZZ

Z

kZ

Example 4.2

Figure 4.3 (p. 167)Geometry of a partially filled waveguide and its transmission line equivalent for Example 4.2.

19

EMLAB

Electromagnetic energy and power flow

dW

dW

Vm

Ve

*

*

Re4

1

Re4

1

HB

ED • Time averaged electric field energy stored in volume V

• Time averaged magnetic field energy stored in volume V

daPS

)(Re2

1 *HE

Lem

VS

PWWj

djda

)(2

)(2

1)(

2

1 **

EEEDHBHE **

Poynting theorem

VLem

*2

***

)(2

2

1

2

1

2

1

1

PWWj

RIIC

IIILIjωIZIP

CjLjRjXRZ

Circuit analogy

20

EMLAB

Example

01Z 02Z

0102

0102

1

1

011

01111

/

/

ZZ

ZZ

V

V

ZV

ZVS

nnn

nnn

ZVb

ZVa

0

0

/

/

0102

0201

0102

01

01

02

2

1

01

02

022

01112

22

/

/

ZZ

ZZ

ZZ

Z

Z

Z

V

V

Z

Z

ZV

ZVS

0102

0201

0102

02

02

01

1

2

02

01

011

02221

22

/

/

ZZ

ZZ

ZZ

Z

Z

Z

V

V

Z

Z

ZV

ZVS

0201

0201

2

2

022

02222

/

/

ZZ

ZZ

V

V

ZV

ZVS

21