1_Microwave Network Analysis

8/3/2019 1_Microwave Network Analysis

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nth terminal at z =0

I I

V V

nnn

nnn

I V

2



nnnn

n

n I

I

I

Z Z

Z

Z Z Z

V

V

V

..

..

.....

..........

..........

........

....

..

..

2

1

1

21

11211

2

1

1

Z Y

V Y I

I Z V

jk for I j

iij

k

I

V Z

,0

Driving port j with current Ij ,

open circuiting all the otherports ( I k = o for k j &

measuring open circuit voltage

at port i

jk for V j

iij

k

V

I Y

,0

Driving port j with voltage V j , shortcircuiting all the other ports ( V k = o

for k j & measuring short circuit

current at port i

3



Only passive devices

Lorentz reciprocity theorem for EM fields

nS

V

Ma

Ja

Jb

Mb

bbb

bbb

aaa

aaa

J E j H

M H j E

J E j H

M H j E

0

BH = M

4



baababbaabba H M H M E J E J H E H E

dv M H M H J E J E

dS H Eb H E dv H E H E

V abbabaab

abaSv

abba

baba M M J J If

)1( dS H E dS H E abS

baS

5



Two port with fields j jii H E H E ,,,

0,0 tantan

E E StructuresOpenBoundary Mettalic

)1( dS H E dS H E i j

S

ji

S

Non zero component will be the normal components from the

cross sectional areas of port 1 & 2 (terminal plane)

dl E V

C

dl H I

6

2)()( i j j jii I V I V

(Ii )j is the current arising at the port 1 due to

the applied voltage at 2



(3) in (2)

Symmetric

symmetric is Z

7

inVnV Y I I

jnVnV Y I I

i jii j j

jij jii

,0,)(

,0,)(

for

for

2)()( i j j jii I V I V

jiij

i jij ji

Y Y

YjiV V Y V V

(3)



Consider a reciprocal lossless N port junction

◦ the impedance & admittance are purely imaginary

◦ Or the net real power delivered to the network is zero

RePav = 0

N

n

N

mnmnm

t t

t t

av

I Z I

I Z I I Z I I Z I

I Z I I I Z

I V H E P

1 1

121221211111

*2

1

...........*2

1

*2

1

*2

1

*2

1*

2

1

*2

1

*2

1

(4)

8



50* nmnm I Z I In’s are independent

As we could set all port currents = 0

expect for nth current

0Re

0Re

mn

mn

Y

Z n& many for

9

60Re

0Re*Re2

mn

mnnnmnm

Z

Z I I Z I

Purely

imaginary



Provides complete description of the network

Relates the voltage waves incident on the ports &

reflected from the ports

Measurement using NWA

10



V SV

nnnn

n

n V

V

V

SS

S

SSS

V

V

V

..

..

.....

..........

..........

........

....

..

..

2

1

1

21

11211

2

1

+ incident

- reflected

jk for i

iij

V k V

V S

0

Driving port j with an incident voltage wave of

voltage V + j & measuring the reflected wave

amplitude V - j coming out of port I All ports expect jth port is given matched

termination to avoid reflection

iiS Reflection coefficient looking into port i when all the other

ports are terminated by matched load

ijS Transmission coefficient from port j to port i when all the

other ports are terminated by matched load

ijS IL log20

iiS RL log20

Insertion loss

Return loss



12

V V V

V Z V Z I Z

V V I I I

V V V

U Z V U Z V

V Z V V V Z

V V V Z V Z

1

U Z U Z V

V S

[U] unit or identitymatrix[U]= [U] t



S matrix Symmetric reciprocal

S matrix Unitary lossless

13

V V I I

V V I I

V V

nnnnnn

nn

n

nnn

nnn

I Z

Z I

V

,1

1

0

0

At any port the total current & voltage is:



14

1

)1(

)2( U Z U Z

V

V S

22

1

2

1

2

1

121

21

2

1

I U Z I I Z V

I V

I U Z I I Z V

I V

nnn

nnn

V

V

[U] unit or identitymatrix[U]= [U] t



15

[Z] symmetric [Z] = [Z] t

[U] = [U] t

1 U Z U Z S

For reciprocal networks [S] = [S] t

Lossless Network net real power delivered to the network is zero RePav = 0

0*Re2

1*Re

2

1 I V H E P

t t av



16

0)1(

0)1(

0

)1(

01

111

2

2

Z

Z S

Z

Z

V

V

in

in

V

V

For a 2 port network

0)2(

0)2(

0

)2(

02

222

1

1

Z

Z S

Z

Z

V

V

in

in

V

V

)1(

0

)1(

01

2

21 12

2

V

V T S V

V

0)1(

)1(2

Z Z

Z

in

in

)2(

0

)2(

02

112 1

2

1

V

V

T S

V

V

0

)2(

)2(2

Z Z

Z

in

in

Z0 = 50 & normalizing1

1

11

1111

z

zS



17

I SS

SSSS

SSSSSSSS

SSSS

t

*

0**

0**1**

1**

21221112

22211211

22221212

21211111

For a lossless 2 port network

1

1

0

11

2211

VSWR

VSWRS

SS

tcoefficienReflection

For a matched 2 port network

No reflection

For a reciprocal 2 port network

2112 SS

12

11

-20logS20log -IL

-20logS20log -RL

T



zn 0 [S]

zn ln [S’]

18

'' ' V SV

V SV

n

n

jnn

jnn

eV V

eV V

'

'nnn l

Electrical length of the outward shift

of the reference plane of port n



19

nmn j

nm SeS 2

'

Phase of Snm is shifted by twice theelectrical length of the shift in the

terminal plane n as the wave travels

twice over the length upon incidence &

refelection



20

Z050

Z0n characteristic

impedance of nth port

Wave amplitudes are given by

nportfrom wavereflected

nportat waveincident

n

nn

n

nn

Z

V b

Z V a

0

0



21

nnn

nnn

nnn

nnnnnn

ba Z Z I

ba Z V

V V I I

V V

00

0

11

22

22

21

21

**Re2

1*Re

2

1

nn

nnnnnnnnav

ba

ababba I V P

Purely imaginary, looking for

power delivered, so neglect

Generalized S parameters aSb

jk V i j

ji

jk a j

iij

k k

Z V

Z V

a

bS

forfor 00

0

0

1_Microwave Network Analysis

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