Chapter 4

Microwave Network Analysis

Equivalent Voltage and Current For non-TEM lines, the quantities of voltage, current, and impedance are nor unique, and are difficult to measured. Following considerations can provide useful result:

1) Voltage and current are defined only for a particular mode, and are defined so that the voltage is proportional to the transverse electric field, and the current is proportional to transverse magnetic field.2) The product of equivalent voltage and current equals to the power flow of the mode.3) The ratio of the voltage to the current for a single traveling wave should be equal to the characteristic impedance of the line. This impedance is usually selected as equal to the wave impedance of the line.

The Concept of Impedance Various types of impedance:

1) =(/)1/2 =intrinsic impedance of medium. This impedance is dependent only on the material parameters of medium, and is equal to the wave impedance of plane wave.2) Zw=Et /Ht=1/Yw =wave impedance, e.g. ZTEM, ZTM, ZTE. It may depend on the type of line or guide, the material, and the operating frequency.

3) Z0 =(L /C)1/2 =1/Y0 =characteristic impedance. It is the ratio of voltage to current. The characteristic impedance is unique definition for TEM mode but not for TM or TE modes.

The real and imaginary parts of impedance and reflection coefficient are even and odd in 0 respectively.

Impedance and Admittance Matrices The terminal plane (e.g. tN) is important in providing a phase reference for the voltage and current phasors.

At the nth terminal (reference) plane, the relations are given as:

1

for 0

for 0

|

matrix admittance is ;

|

matrix impedance is ;

YZ

V

IY

YVYI

I

VZ

ZIZV

jkVij

ijij

jkVij

ijij

k

k

Reciprocal Networks If the arbitrary network is reciprocal ( no active devices,

ferrites, or plasmas), [Y] and [Z] are symmetric matrices . Tjiij

Tjiij

YYYY

ZZ ZZ

Lossless Networks

If the arbitrary network is lossless, then the net real power delivered to the network must be zero. Besides

jiY

Z

ij

ij

,any for ,0}Re{

,0}Re{

Example4.1: Find the Z parameters of the two-port network?Solution

CCB

CI

CBI

CAI

ZZZ

Z

I

V

I

VZ

ZZI

VZ

ZZI

VZ

2

20

2

112

02

222

01

111

1

1

2

|

division By voltage

|

reciprocal isnetwork theBecause

|

The Scattering Matrices The scattering parameter Sij is the transmission coefficient from j port to port i when all other ports are terminated in matched loads.

jkVj

iij

KV

VS

for 0|

[Z] or [Y] [S]

10.0

0..

.10

0.01

matrixidentity or unit is ][

;][][

][][

U

UZ

UZS

[S] [Z]

][][

][][

SU

SUZ

Example4.2: Find the S parameters of the 3 dB attenuator circuit?

SolutionA matched 3B attenuator with a 50 Ω Characteristic impedance

02 2

2

111 0

11 on port 210 01 0

1

11 22

221 0

1

2 2 1

| | |

8.56 [141.8(8.56 50)]50

141.8 8.56 50So 0. From symmetry feature, 0.

|

41.44 50( )( ) 0.7041.44 8.56 50 8.56

inZV V

in

in

V

V Z ZS

V Z Z

Z

S S

VS

V

V V V

1

21 12

7

0.707

V

S S

Reciprocal Networks No real power delivers to

network.Besides tSS

Example4.3: Determine if the network is reciprocal, and lossless? If port 2 terminated with a matched load, what is the return loss at port 1? If port 2terminated with a short circuit, what is the return loss seen at port 1?

Solution

02.04585.0

4585.0015.0][S

Lossless Networks

jiSS

jiSS

N

kkjki

N

kkjki

for ;0

for ;1

1

*

1

*

[S] is symmetric matrix

Since [S] is not symmetric, the network is not reciprocal.

So the network is not lossless.

1745.0)85.0()15.0( 22221

211 SS

When port 2 terminated with a matched load, =S11=0.15.dB 5.16)15.0log(20log20 RL

When port 2 terminated with a short circuit,

dB 9.6)452.0log(20log20 So

452.02.01

)4585.0)(4585.0(15.0

1

result above theusing and by equationfirst theDividing

1

gives equation second The

22

211211

1

21211

1

1

1

22

2112

2221212221212

2121112121111

RL

S

SSS

V

VSS

V

V

V

S

SVV

VSVSVSVSV

VSVSVSVSV

Summary

Reciprocal Networks (symmetric)

No active elements, no anisotropic material

ttt SSYYZZ

Lossless Networks

No resistive material, no radiation

Unitary

matries , , of partsImaginary

0} Re{0}Re{

*

USS

SZY

YZ

T

Example4.4: Determine if the network is reciprocal, and lossless ?Solution

A matched 3B attenuator with a 50 Ω Characteristic impedance

36.1508.141

8.14136.150

8.14156.88.141

8.1418.14156.8Z

From the result of example 4.2

10

01

5.00

05.0][][

0707.0

707.00][

*SS

S

T

Since the network is reciprocal but not lossless, [S] should be symmetric but not unitary.

Since the network is reciprocal but not lossless, [Z] should be symmetric but not imaginary.

Example4.5: Determine if the network is reciprocal, and lossless ?Solution

2 2

1

2

1

111 0 0

1

112 0

2 2

2 121 0

1 1

2

222 0

2 2

| | 0 ( 0)

0| 0

0|

0 1|

0 0 [ ] 1

Since the network is neither reciprocal nor lossless,

gV V g

gs

Vds

mV m

mds

Vds

mds

IIY I

V V

IY

V I R

I g VY g

V V

VgI RY

V V R

Yg R

[ ] should be neither symmetric nor imaginary.Y

0Z0Z

D

S

GPort 1Port 2

dsRm gsg V

gsV

Port 1 Port 2

D

SS

G

unitary.nor symmetricneither be should ][

lossless,nor reciprocalneither isnetwork theSince

2

01][

/1

/1

0)/1)(0(

0)/1)(0(

))((

))((

2/1

2

0)/1)(0(

2

))((

2

00)/1)(0(

02

))((

2

1/1

/1

0)/1)(0(

0)/1)(0(

))((

))((

0

0

0

0

0

0

0

0

00

00

2112220110

211222011022

0

0

000

0

2112220110

02121

002112220110

01212

0

0

00

00

2112220110

211222011011

S

ZR

ZR

RZ

RZgS

ZR

ZR

RY

RY

gRYY

gRYY

YYYYYY

YYYYYYS

RZ

RZg

RY

g

gRYY

Yg

YYYYYY

YYS

gRYYYYYYYY

YYS

RY

RY

gRYY

gRYY

YYYYYY

YYYYYYS

ds

ds

ds

dsm

ds

ds

ds

ds

mds

mds

ds

dsm

ds

m

mds

m

mds

ds

ds

mds

mds

Problem 1Problem 1:Determine if the inductance networks are reciprocal, and lossless ?

Problem 2Problem 2:Determine if the capacitance networks are reciprocal, and lossless ?

1L 2L

3LPort 1 Port 2

1I 2I

1V 2V

Port 1 Port 2

1I 2I

1V 2V

3C

1C 2C

Three Port Network Reciprocal Network

333231

232221

131211

][

SSS

SSS

SSS

S322331132112 SSSSSS

Matching at all ports

0 0 0 332211 SSS Lossless Network

*

2 2 2 2 2 2

12 13 12 23 13 23

* * *13 23 23 12 12 13

[ ] is unitary [ ] [ ] [ ]

Condition A: 1 1 1

Condition B: 0 0 0

From condition B, two of the three

TS S S U

S S S S S S

S S S S S S

12 13 23 parameters, , , must be zero.

This can not satisfy condition A.

A three port network can not be lossless, reciprocal,

and match at all pors simultaneously.

S S S

[S] is unitary and matched at all ports, but not symmetric. Therefore, circulator is lossless and matched, but not reciprocal.

A counter-clockwise circulator

Power splitters

010

001

100

][S

Applications

002

1

002

12

12

10

][S

[S] is symmetric and matched at all ports, but not unitary. Therefore, circulator is reciprocal and matched, but not lossless.

Port 3

Port 2Port 1

Splitters

Twice the electric length represents that the wave travels twice over this length upon incidence and reflection.

A Shift in Reference Planes

mj

m

nnn

SeS

l

n

2'

Generalized Scattering

Parameters

jkVij

jiij

kZV

ZVS

,00

0|

If the characteristic impedances of a multi-port network are different,

Generalized Scattering Matrices The scattering parameter Sij defined earlier was based on the

assumption that all ports have the same characteristic impedances ( usually Z0=50). However, there are many cases where this may not apply and each port has a non-identical characteristic impedance.

A generalized scattering matrix can be applied for network with non-identical characteristic impedances, and is defined as following:

jkaj

iij

i

ii

j

ij

ka

bS

iZ

Vb

jZ

Va

for 0

0

0

|

,...2,1;

,..2,1;

i

jidentijjka

ij

jiidentnonij

Z

ZS

ZV

ZVS

k

0

0,for 0

0

0, |

Port 1 Port 2A Two-Port

Network

1a 2a

1b 2b

01Z 02Z

The Transmission (ABCD) Matrix

ABCD matrix has the advantage of cascade connection of multiple two-port networks.

2

2

1

1

I

V

DC

BA

I

V

2

2

22

22

11

11

1

1

I

V

DC

BA

DC

BA

I

V

Table 4-2 Conversions between two-port network parameters

1 BCAD

Example4.6: Find the S parameters of network?

Solution

For reciprocal network, [Z] is is symmetric. Hence, Z12=Z21

From Table 4-1

00022

00

0

0

0

0

1

01

0

0

1

01

1

01

2cos

2sin

2sin

2cos

1

01

CZjYZCj

jZCZ

CjjY

jZ

Cj

CjjY

jZ

Cj

DC

BA

Port 1 Port 2

4

C C0Z 0Z0,Z

4

C C0,Z

From Table 4-2

20

220

20

22

00

0022

20

220

001221

20

220

20

22

020

220

020

220

00

0011

22

22

2

2

22

)()(

)()(

ZCjjCZ

ZCj

DCZZBA

DCZZBAS

ZCjjCZ

DCZZBASS

ZCjjCZ

ZCj

CZZCjjjCZ

CZZCjjjCZ

DCZZBA

DCZZBAS

The Transmission [T] Matrix At low frequencies, ABCD matrix is defined in terms of net

voltages and currents. When at high frequencies, T matrix defined in terms of incident and reflected waves will become very useful to evaluate cascade networks.

2

2

2221

1211

1

1

V

V

TT

TT

V

V

2

2'

22'

21

'12

'11

2221

1211

1

1

V

V

TT

TT

TT

TT

V

V

21

221112

21

11

21

22

21

2221

1211

1

S

SSS

S

SS

S

STT

TT

Port 1 Port 2 T Two-Port

Network

1V 2V

1V 2V

Port 1 Port 2 T Two-Port

Network

T Two-Port

Network

1V

1V

'1V

'1V

2V

2V

'2V

'2V

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.

Equivalent Circuit for Two-Port Networks

Equivalent circuits for some common microstrip discontinuities. (a) Open-ended. (b) Gap. (c) Change in width. (d) T-junction.

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V

2

1

2221

1211

2

1

V

V

YY

YY

I

I

Equivalent circuits for a reciprocal two-port network. (a) T equivalent (b) equivalent

Example4.7: Find the network as equivalent T and model at 1GHz?Solution

From Table 4-1

0100

1000

2cos

1002sin

2sin100

2cos

24

2

jj

j

j

DC

BA

l

From Table 4-2

0

1001

0

22

2112

11

C

DZ

jC

ZZ

C

AZ

0100

1000

j

jZ

100

100

100

12

1222

1211

jZ

jZZ

jZZ

Port 1 Port 2

4

1 100Z

0 50Z 0 50Z

From Table 4-2

pF5915.110012

1

nL915.1510012

CCG

LLG

0100

1000

j

jY

100

100

100

12

1222

1211

jY

jYY

jYY

0

1001

0

22

2112

11

B

AY

jB

YY

B

DY

nL915.151001

12

1

pF5915.1100112

LLG

CCG

Equivalent T model

Equivalent model

15.915nH

Port 1 Port 21.592pF

15.915nH

100j

Port 1 Port 2

100j

100j

15.915nH

Port 1 Port 21.592pF

1

100

j

Port 1 Port 2

1.592pF

1

100

j 1

100

j

Example4.8: Find the equivalent model of microstrip-line inductor?Solution

From Table 4-1

lljY

ljZl

DC

BA

cossin

sincos

0

0

From Table 4-2

w

h

r

l

ljYljY

ljYljYY

cotcsc

csccot

00

00

ljYljZ

l

B

AY

ljYljZB

YY

ljYljZ

l

B

DY

cotsin

cos

cscsin

11

cotsin

cos

00

22

00

2112

00

11

2tan

2cos

2sin2

)2

sin2()cos1(

sin

csccot

0

20

0

001211

ljY

ll

ljY

ll

jY

ljYljYYY

Equivalent model

Port 1 Port 2

L

C

10 cscjY l

10 tan

2

ljY

Port 1 Port 2

C

10 tan

2

ljY

ljYY

ljY

ljYljYYY

csc2

tan

csccot

012

0

001222

0

0 00

0 0 0 00

0

0 0

0 0 0 0 0

Inductance of microstrip-line

1sin

csc

;(H)

Parasitic capacitance

tan2 2

2 2 ;(F)2

l

eff eff

l

eff eff

j L jZ l jZ ljY l

Z l Z lZ lL

c

l lj C jY jY

l lY Y Y l

Cc

Example4.9: Find the equivalent T model of microstrip-line capacitor?Solution

From Table 4-1

lljY

ljZl

DC

BA

cossin

sincos

0

0

From Table 4-2

w

h

r

l

ljZljY

l

C

DZ

ljZljYC

ZZ

ljZljY

l

C

AZ

cotsin

cos

cscsin

11

cotsin

cos

00

22

00

2112

00

11

ljZljZ

ljZljZZ

cotcsc

csccot

00

00

2tan

2cos

2sin2

)2

sin2()cos1(

sin

csccot

0

20

0

001211

ljZ

ll

ljZ

ll

jZ

ljZljZZZ

Equivalent T model

Port 1 Port 2

0 tan2

ljZ

Port 1 Port 2

L L

C

0 tan2

ljZ

0 cscjZ l

ljZZ

ljZ

ljZljZZZ

csc2

tan

csccot

012

0

001222

(H);222

2tan

2tan

inductance Parasitic

(F);sin

csc1

line-microstrip of eCapacitanc

00000

00

0

00

00

0

0

0

0

c

lZlZlZ

lZ

L

ljZLj

cZ

l

Z

l

Z

l

Z

lC

ljZCj

effeffl

effeffl

Problem3Problem3: Design a 6GHz attenuator ?

(Hint: -20logS21=6 S21=0.501 )

0501.0

501.00S

Problem4Problem4: Design a 6nH microstrip-line inductor on a 1.6mm thick FR4 substrate. The width of line is 0.25mm. Find the length (l ) and parasitic capacitance?

Problem5Problem5: Design a 2pF microstrip-line capacitor on a 1.6mm thick FR4 substrate. The width of line is 5mm. Find the length (l ) and parasitic inductance?

6nH0Z

0Z

l

w=0.25mm

h=1.6mm

r=4.5

2pF0Z 0Z

w=5mm

h=1.6mm

r=4.5

l