chapter 4 microwave network analysis -...

微波電路講義4-1

Chapter 4 Microwave network analysis4.1 Impedance and equivalent voltages and currents

equivalent transmission line model (b, Zo)

4.2 Impedance and admittance matrices

not applicable in microwave circuits

4.3 The scattering matrix

properties, generalized scattering parameters, VNA measurement

4.4 The transmission (ABCD) matrix

cascade network

4.5 Signal flow graph

2-port circuit, TRL calibration

4.6 Discontinuities and modal analysis

microstrip discontinuities and compensation


4.1 Impedance and equivalent voltages and currents

• Equivalent voltages and currents

Microwave circuit approach

Interest: voltage and current at a set of terminals (ports), power flow

through a device, and how to find the response of a network

For a certain mode in the line, the line characteristics are represented by

it’s global quantities Zo, b, l.

Define: equivalent voltage (wave) transverse electric field

equivalent current (wave) transverse magnetic field

voltage (wave)/current (wave) = characteristic impedance or

wave impedance of the line

and voltage current = power flow of the mode

→ use transmission line theory to analyze microwave circuit

performance at the interested ports


• Impedance

characteristic impedance of the medium

wave impedance of the particular mode of wave

characteristic impedance of the line

input impedance at a port of circuit

t

tw

H

EZ

oV

ZI

( )( )

( )in

V zZ z

I z


Discussion

1. Transmission line model for the TE10 mode of a rectangular

waveguide

valueunique-non ,dependent x:

),(

dyEldEzxV y

transverse fields (Table 3.2, p.117) transmission line model

10

10

1

2

*

( )sin sin

1( )sin sin

1

2

j z j z

y

j z j z

x

TE

y

TE o

x

y x

x xE A e A e C V

a a

x xH A e A e C I

Z a a

E kZ Z

H

P E H dxdy

b b

b b

b

*

1

2

j z j zo o

j z j zo o

o oj z j z

o o

o oo

o o

o o

V V e V e

I I e I e

V Ve e

Z Z

V VZ

I I

P V I

b b

b b

b b

x

y


(derivation of C1 and C2)

10

10 10

10

1

1 1

2

2 2

2*

0 0

( )sin ( )sin

,

1( )sin ( )sin

,

1 1

2 4 2

j z j z j z j z

y o o

o o

j z j z j z j z

x o o

TE

o o

TE TE

a b

y x o

TE

x xE A e A e C V e V e

a a

A AV V

C C

x xH A e A e C I e I e

Z a a

A AI I

C Z C Z

abP E H dxdy A V

Z

b b b b

b b b b

10

10

10

2

* *

1 2*

1 2

2 2

1 1

1 2

2

2

=-1

2 2,

o

TE

TEoTE o

o

AI C C

C C Z ab

A C ZV CZ Z

I C A C

C Cab ab


2. Ex.4.2

2 2 2 2 2 2

1

, ,

1

, , ,

2 22, ,

/

band: 2.286 2 4.472 137 6.56 , 4.17

if 10 2 / 209 ,

r

od oa

od oa

o ooa od o o r o

a d

c ca c o d c c

c p r

c c c a c

o

Z Z

Z Z

k kZ Z k k k k

f fk k k k k

a v c

X a cm a cm k m f GHz f GHz

f GHz k f c m

b b

b b

b 1 1 1

1 1

1 1

10

158 , 2.54 333 , 304

500 , 259.6 , 0.316

if 6 201 in , 147 ,

126 in , 54 , "

a r d

oa od

r o c r d

o c a

m k m m

Z Z

f GHz k k m k m

k m k air j m TE non exis

b

b

b "t

roTE10 Zoa, ba Zod, bd

Q: What if the incident wave is

from the other direction? “N”

incident

wave


4.2 Impedance and admittance matrices

N-port

network

port 1

port N

t1

tN

V1¯, I1¯

V1+, I1+

VN¯, IN¯

VN+, IN+

reference plane

for port N

(plane for

)

reference plane

for port 1

(plane for

)

, ,

1( )

1 1Re{ *}, Re{ *}

2 2

i i i

i i i i ioi

i ioi

i i

inc i in i i ii i

V V V

I I I V VZ

V VZ

I I

P V I P V I

0NV

1 0V

V1, I1

IN ,VN

1oZ

onZ


• Admittance matrix

1 11 12 1 1

2 21 2 2

0, 0,

1 2

, ,

k k

N

N

i iij

j jV k j V k j

N N N NN N

I Y Y Y V

I Y Y VI response

I Y V YV source

I Y Y Y V

• Impedance matrix

1 11 12 1 1

2 21 2 2

0, 0,

1 2

, ,

k k

N

N

i iij

j jI k j I k j

N N N NN N

V Z Z Z I

V Z Z IV response

V Z I ZI source

V Z Z Z I


(derivation)

source port 1 port 2

a V1a, I1a V2a, I2a

b V1b, I1b V2b, I2b

reciprocity theorem: V1aI1b + V2aI2b =V1bI1a + V2bI2a

211212122112

2121121221211212

2222121121211122221211212111

2

1

2221

1211

2

1

0))((

)()()()(

ZZIIIIZZ

IIZIIZIIZIIZ

IIZIZIIZIZIIZIZIIZIZ

I

I

ZZ

ZZ

V

V

abba

ababbaba

abbabbbaabaa

Discussion

1. Reciprocal network

,

, and Y : symmetric matrix ,

t

ij ji

t

ij ji

Z Z Z ZZ

Y Y Y Y


2 2

1

1 11 12 1

2 21 22 2

1 211 1 3 21 3 12

1 10 0

222 2 3

2 0

3 12 1 11 12 2 22 12

( )

,

, ,

I I

I

derivation

V Z Z I

V Z Z I

V VZ Z Z Z Z Z

I I

VZ Z Z

I

Z Z Z Z Z Z Z Z

Z1 Z2

Z3

Z11-Z12 Z22-Z12

Z12

Z1 Z2

Z3I1 V1

I2=0

V2

2. T and Π networks

Z1 Z2

Z3 I2V1

I1=0

V2


2 2

1

1 11 12 1

2 21 22 2

1 211 1 3 21 3 12

1 10 0

222 2 3

2 0

1 11 12 3 12 2 22 12

( )

,

, ,

V V

V

derivation

I Y Y V

I Y Y V

I IY Y Y Y Y Y

V V

IY Y Y

V

Y Y Y Y Y Y Y Y

I1

V1

I2

V2=0Y2

Y3

Y1

I1

V1=0

I2

V2Y2

Y3

Y1

Y2

-Y12

Y11+Y12

Y3

Y22+Y12Y1


3. Z- and Y- matrices of a lossless transmission line section

1 1 2 2

1 1 1 1

2 1 1 1 1 1

( )

( ) , ( ) ( ),B.C. (0), (0), ( ), ( )

1, ( ) ( )

2

1 1( ) ( ) cos

2 2

j z j z j z j z

o o o o o

o o o o o o o

j l j l j l j l

o o o o

derivation

V z V e V e I z Y V e V e V V I I V V l I I l

V V V I Y V V V V Z I

V V e V e V Z I e V Z I e V l j

b b b b

b b b b

b 1

2 1 1 1 1 1 1

1 1 2

(3)

2 1 2 1

sin .....(1)

1 1( ) [ ( ) ( ) ] sin cos .....(2)

2 2

(2) cot csc .....(3)

(1) cos cot cot sin

o

j l j l j l j l

o o o o o o o

o o

o o o

Z I l

I Y V e V e Y V Z I e V Z I e jY V l I l

V jZ I l jZ I l

V jZ I l l jZ I l jZ I l

b b b b

b

b b

b b

b b b b

1 2

1 11 12 2 3 12

cot csc

csc cot = csc cot

(cot csc ) , csc

o

o o

o o

l lZ jZ

l ljZ I l jZ I l

Z Z Z jZ l l Z Z Z jZ l

b b

b b

b b

b b b

,oZ b

z0 l

1V 2V

1I 2I Z1 Z2

Z31 2

3

(cot csc )

csc

o

o

Z Z jZ l l

Z jZ l

b b

b


5. Problems to use Z- or Y-matrix in microwave circuits

1) difficult in defining voltage and current for non-TEM lines

2) no equipment available to measure voltage and current in

complex value (eg. sampling scope in microwave range,

impedance meter <3GHz)

3) difficult to make open and short circuits over broadband

4) active devices not stable as terminated with open or short circuit

4. Reciprocal lossless network

0}Re{ ijZ

1 2

3

(cot csc )

csc

o

o

Y Y jY l l

Y jY l

b b

b

2 1 1 1 1 2

(3)

2 1 1 1 1 2

1 2

1 11 12

(1) cos sin cot csc ...(3)

(2) sin cos sin ( cot csc )cos

csc cot

cot csc

csc cot

(co

o o o

o o o o

o o

o

o

V V l jZ I l I jY lV jY lV

I jY V l I l jY V l jY lV jY lV l

jY lV jY lV

l lY jY

l l

Y Y Y jY

b b b b

b b b b b b

b b

b b b b

2 3t csc ) , cscol l Y Y jY lb b b

Y2

Y3

Y1


4.3 The scattering matrix

N-port

network

11 12 11 1

21 22 2

0, 0,

1 2

response , ,

sourcek k

N

N

i iij

j jV k j V k j

N N NNN N

S S SV V

S SV VV

V S V SV

S S SV V

port 1

t1

V1¯V1+

port N

t1’

V1’+

V1’¯

tN

VN¯VN+

tN’

VN’¯VN’+

q1=bl1

qN=blN


Discussion

1. Ex 4.4 a 3dB attenuator (Zo=50Ω)

111 1 1 1 1 1

1

2 2 2 1 1 1 1 21 1

21 12

11 22

8.56 41.44 50

0( ) 0

41.44 50 1 10, 0.707

41.44 8.56 50 8.56 2 2

10

reciprocal 2: lossy

symmetric 10

2

inciden

in

in o

in o

Z

Z Z VS V V V V V

Z Z V

V V V V V V V S V

S SS

S S

2

1

,1

2

2 21

2 1

,2

1 2

1t power to port 1:

2

1

1 1 1 12transmitted power from port 2: : 3dB attenuation

2 2 2 2

input matchattenuator design ,

attenuation value

inc

o

trans

o o o

VP

Z

VV V

PZ Z Z

R R


2. T-type attenuator design

thE

oZ

oZ1V 2V2R

1R 1R

inZ

1 2 1//in o oZ R Z R R Z

2 12

21

1 1 2 1 1

//

//

o o

o o

R R Z ZVS

V R R R Z R Z

1

1

1oR Z

2 2

2

1oR Z

1

2 2

1 2 1

1 1 2 13 attenuator 2 2

2 21

o o

o o

R Z Z

dB

R Z Z


3. Relation of [Z], [Y], and [S]

11ZY,UZUZS

1

Let 1,

( )

( ) ( )

( ) ( )

( ) ( )

n n n

on

n n n

V V VZ

I V V

V Z I V V Z V V

Z V V Z V V

Z U V Z U V

V S V Z U S V Z U V

S Z U Z U

(derivation)


4. Reciprocal network

matrix symmetric: , SSS t

(derivation)

1 1

31 1

( ) / 2Let 1,

( ) / 2

1 1( ) ( )

2 2

1 1( ) ( )

2 2

( )( ) ( )( )

(( ) ) ( ) ( ) ( )

n n n n n n

on

n n n n n n

fromt t t

V V V V V IZ

I V V V V I

V V I Z U I

V V I Z U I

V Z U Z U V S Z U Z U

S Z U Z U Z U Z U S


5. Lossless network (unitary property)

0

1 , *

1

*

j i

j iSSUSS kj

N

k

ki

t

(derivation)

,

* **

* * *

Let 1

lossless (incident power=transmitted power) net averaged input power =0

1 1Re( ) Re[( ) ( )]

2 2

1Re[

2

on

in i

i

t t

in

t t t

Z

P

P V I V V V V

V V V V V V V

*

* * **

*

)] 0t

V S Vt t t t

t

V

V V V V V S S V

S S U

Im

6. Lossy network 1*

1

ki

N

k

kiSS


7. Ex.4.5

0 0

0.15 0 0.85 45=

0.85 45 0.2 0

o o

S

2 2

11 21

11

22

21

11

:not symmetric a non-reciprocal network

0.745 1 a lossy network

port 1 20log 16.5

port 2 20log 14

20log 1.4

port 2 terminated with a matched load 0

0.15

L

in

S

S S

RL S dB

RL S dB

IL S dB

S

12 2111

22

, 20log 0.15 16.5

port 2 terminated with a short circuit 1

0.452, 6.91

L

Lin

L

RL dB

S SS RL dB

S


9. S-matrix is not effected by the network arrangement.

1 1

2 2

'

0 0 0 0

0 0 0 0n-port network:

0 0 0 0N N

j j

j j

j j

e e

e eS S

e e

q q

q q

q q

8. Shift property

port 1

t1

V1¯V1+

port 2

t1’

V1’+

V1’¯

t2

V2¯V2+

t2’

V2’¯V2’+

q1=bl1 q2=bl2

1 1 2 2 1 22 2' ' ' '

11 11 21 21 12 12 22 22, , ,j j j j j jS e S S e S e S e S e S e S q q q q q q


10. Power waves on a lossless transmission line with Zoi

0 0

2 2*

, , ,

incident (power) wave =2

reflected (power) wave =2

1 1 1Re{ }

2 2 2

k k

i i oi ii

oi oi

i i oi ii

oi oi

i ojiij

j j oia ,k j V ,k j

in i i i i i inc i refl

V V Z I: a

Z Z

V V Z I: b

Z Z

V ZbV S V b S a , S

a V Z

P V I a b P P

2

, (1 )i inc i ii LP S P

a i =V i

+

Z oi,bi =

V i-

Z oi,V i =V i

+ +V i- = Z oi (a i +bi ),I i =

V i+ -V i

-

Z oi=a i - bi

Z oi

Pin,i =1

2Re{V iI i

*}=1

2Re{(a i +bi )(a i - bi )

*}=1

2Re{ a i

2- bi

2+ a i

*bi - a ibi*}

=1

2a i

2-

1

2bi

2=Pinc,i - Prefl ,i = Pinc ,i (1- S ii

2),S ii =

bi

a i a k=0 ,k¹i

=1

2Re{Pi

+ - Pi-}® Pi

+ =V i

+2

Z oi= a i

2,Pi

- =V i

-2

Z oi= bi

2

(derivation)

Im

ZLZoiiV

iI

Vgi

Pin,i


matched 1port with 2port at t coefficien reflection:

matched 1port t with coefficienion transmissreverse:

02

222

02

112

1

1

a

a

a

bS

a

bS

matched 2port t with coefficienion transmissforward:

matched 2port with 1port at t coefficien reflection:

01

221

01

111

2

2

a

a

a

bS

a

bS

11. Two-port device with its S-matrix

2222112

1221111

SaSab

SaSab

1 11 12 1

2 21 22 2

b S S a

b S S a


12. Reflection coefficient and S11, S22

Zg

g inZo, b

ZL

, , if then in g out L L o

in g out L g o L

in g out L L o

Z Z Z Z Z ZZ Z

Z Z Z Z Z Z

two-port

networkZo

Zo

two-port

networkZo

Zo

S11 S22

11in o

in o

Z ZS

Z Z

22out o

out o

Z ZS

Z Z

out L

Zin Zout

two-port

network

PL1 PL2

1

2

insertion loss 10log L

L

PIL(dB)

P

Zo

Zg

Zo

Zg


111

1

221

1

at port1: 20log 20log

from port 1 to port 2: 20log 20log

bRL - - S

a

bIL - - S

a

usually Zg=Zo

13. RL and IL

1a 1a2b

1b


15. Advantages to use S-matrix in microwave circuit

1) matched load available in broadband application

2) measurable quantity in terms of incident, reflected and

transmitted waves

3) termination with Zo causes no oscillation

4) convenient in the use of microwave network analysis

14. Two-port S-matrix measurement using VNA

DUT

a1 b1 b2 a2

→ S11 → S21

Zo

1V

1V

2V 2V


16. Generalized scattering parameters

incident power wave , : reference impedance2 Re

*reflected power wave

2 Re

* *power wave reflection coefficient

power reflection coeffici

i ri ii ri

ri

i ri ii

ri

i i ri i L riii

i i ri i L ri

V Z Ia Z

Z

V Z Ib

Z

b V Z I Z ZS

a V Z I Z Z

2 2

2 2

2 2

,

*ent ,conjugate match * 0

1 1Re{ *} ( )

2 2

Ref: K.Kurokawa,"Power waves and the scattering matrix", IEEE-MTT, pp.194-201, March 1965

traveling wave alo

i L riii L ri ii

i L ri

in i i i i i L

b Z ZS Z Z S

a Z Z

P V I a b P

0

0 0

0 0

ng a lossless line with real

, , ( ),

, , 2 2

,impedance mat

i i i i i ii i i i i oi i i i

oioi oi oi

i i i i ii i i i i oi i i

oi

i i oi i L oiii

i i oi i L oi

Z

V V V V a ba b V V V Z a b I

ZZ Z Z

V V Z I V Z Ia b a b I Z a b

Z Z Z

b V Z I Z ZS

a V Z I Z Z

ch 0L oi iiZ Z S

ZLZoiiV

iI

ZL

Zgi

iV

iI

Vgi

Pin,i


2

2

*

,

Re1Re{ *}

2 2

1

Re

2 Re 2 Reif *

1*

*0 0

2 Re 2 Re

L ri

giLi gi i

L gi L gi

gi LL i i

L gi

Lri

Z ZriL gi L gii ri i

i gi gi

L giri ri

L ri

Lri

L gi L gii ri ii gi ii

ri ri

VZV V I

Z Z Z Z

V ZP V I

Z Z

ZZ

ZZ Z Z ZV Z Ia V V

Z ZZ ZZ Z

ZZ

Z Z Z ZV Z Ib V S

Z Z

22

, , 2

2

Re1

2 2

1if * = :maximum power trasfer from source

8 Re

rigi

L in i i inc i

L gi

gi

L g L

L

ZVP P a P

Z Z

VZ Z P

Z

ZL

Zgi

iV

iI

Vgi

Pin,i


1

1

11

11

( )

( * )

0 . 0

0 . 0 0

0 0 . 0

0 0 0

Re 0 . 0

0 . 0 01,

0 0 . 02

0 0 0 Re

( *)( )

( *)( )

r

r

r

r

rN

r

rN

r r

r r

a F V Z I

b F V Z I

Z

Z

Z

Z

F

Z

V Z I b F Z Z Z Z F a

S F Z Z Z Z F


4.4 The transmission (ABCD) matrix

• Cascade network

1DC

BA

2DC

BA

3

3

212

2

11

1

I

V

DC

BA

DC

BA

I

V

DC

BA

I

V

Discussion

1. ABCD matrix of two-port circuits (p.190, Table 4.1)

2. Reciprocal network AD-BC=1

3. S-, Z-, Y-, ABCD-matrix relation of 2-port network (p.192,

Table 4.2)

4. Ex. 4.6 ABCD(Z)

V3V1 V2

I1I2 I3

+

_

+

_

+

_


(derivation) Z (ABCD)

221

02

222

022

2

01

221

2

2221

02

2

02

22

02

112

2

2

01

111

2

2

1

1

2

1

2221

1211

2

1

0,

1

)(

0,

,

1

22

111

2

DICVIC

D

I

VZ

CDICV

V

I

VZ

C

BCADB

C

DA

C

D

I

VDICVIB

I

VA

I

BIAV

I

VZ

C

A

CV

AV

I

VZ

I

V

DC

BA

I

V

I

I

ZZ

ZZ

V

V

I

II

III

I

reciprocal network

11

2112

BCADCC

BCAD

ZZ

symmetrical network

11 22Z Z A D


(derivation) S (ABCD)

2 2 2

1 11 12 1 1 2 1 1 1 1 1 1

2 21 22 2 1 2 1 1 1 1 1 1

1 1 2 2 2 21 111

1 1 1 1 2 20 0 0

( ) / 2, ,

( ) / ( ) / 2

o

o o

o o o

oa V V

b S S a V VA B V V V V V Z I

b S S a I IC D I V V Z V V Z I

V Z I AV BI Z CV Z DIb VS

a V V Z I AV BI

2

2 2 0

o o V

o o

o o

Z CV Z DI

A BY CZ D

A BY CZ D

oo YVIZIV

V

2222

2

,

0

2I

2V oZ

DCZBYA

DIZCVZBIAV

V

IZV

V

V

V

a

bS

oo

YVI

ooo

VV

Va

o

2

2

2/)(

22

22

222222

2

11

2

01

2

01

221


2 2 2 2 2 21 1

12

2 2 2 2 2 22 21 1

1 12 11

2 12 2

1 1

1 1 1 1 1 1

( ) / 2,

( ) / ( ) / 20 0

1 , ,

( ) / 2

2 2

( ) /

o

o o

o

o o o

V V V V V Z Ib VS

I V V Z V V Z Ia Va V

V V V VD BVAD BC

I IC AV Z I

V V

DV BI Z CV Z AI DV BY V

1 1

2Δ

o

o o

CZ V AV

A BY CZ D

oo YVIZIV 1111 ,

2I

2VoZ

DCZBYA

DCZBYA

VAYCVZVBYDV

VAYCVZVBYDV

AICVZBIDV

AICVZBIDV

IZV

IZV

V

V

a

bS

oo

oo

ooo

oooYVI

o

o

o

o

Va

o

)(

)(

)(

)(

1111

1111

1111

1111

22

22

02

2

02

222

11

11

1I

1V

1,

reciprocal

,

lsymmetrica

2112

2211

SS

DASS


5. Example

coaxial-microstrip transition

(a linear circuit)

[S] representation can be obtained from

measurement or calculation.

one possible equivalent circuit

t1

t2

Zoc

Zom

Zom

Zoc

[S]

C1 C2

L


4.5 Signal flow graphs

• 2-port representation

2222112

1221111

SaSab

SaSab

[S]

port 1 port 2

a1

b1

a2

b2

a1

b1

b2

a2

S11 S22

S12

S21

111

1

221

1

at port 1: 20log 20log

from port 1 to port 2: 20log 20log

bRL - - S

a

bIL - - S

a

1 11 12 1

2 21 22 2

b S S a

b S S a


Discussion

1. Source representation

Vs

Zs

s

as

b1 s

b1

} a1 as 1 a1

b1

2. Load representation

b2

a2

L ZL

b2

a2

L

3. Series, parallel, self-loop, splitting rules (p.196, Fig.4.16)

o

ss

so

oss

Z

ea

ZZ

ZVe

,

s


4. 2-port circuit representation

VsZs

ZLs in out L[S]

a1

b1

b2

a2

S11 S22

S12

S21

s L

as

a1

b1

s in

as

b2

a2

out L

12 211 1 11 1 21 12 22 1 11 1

22

1 12 2111

1 22

12 212 2 22 2 12 21 11 2 22 2

11

12 21222

2 11

(1 ...)1

1

(1 ...)1

1

LL L

L

Lin

L

SS S

S

Sout

S

S S Γb a S a S Γ S S Γ a S a

S Γ

b S S ΓΓ S

a S Γ

S S Γb a S a S Γ S S Γ a S a

S Γ

S S ΓbΓ S

a S Γ


5. TRL (Thru-Reflect-Line) calibration

Find [S]DUT from 2-port measurement using three calibrators

6 unknowns of [S]x and [S]y to be calibrated to acquire [S]DUT

T: Through 3 eqs., R: Reflection 2 eqs., L: Line 3 eqs.

R () and line length (l) can be unknown

DUT

ao

bo

b1

a1

e10

e11

e01

e00 S22

S12

S11

S21

a2

b2

e32

e33

e23

e22

b3

a3

x y

[S]


x y

x y

exp(l)

Calibrators

T: Through

3 eqs.

R: Reflection

2 eqs.

L: Line

3 eqs.

x y

Requirement: connectors and line have same characteristics for 3 calibrators

Limitation: operation bandwidth 20°≦βl≦160°


1 11 12 2

1 21 22 2

11 12 12 21 11 22 11

21 22 2221

11 12 11 12

21 22

( )

R matrix (wave cascade matrix)

b a

a b

1

1

error matrices , x y

brief derivation

R R

R R

R R S S S S SR

R R SS

x x y yR R

x x y

21 22

0 1 1 0Through:

1 0 0 1

0 0Line:

0 0 1 /

Reflection:

Thru measurement:

Line measurement:

T T

l l

L Ll l

mT x T y x y

mL x L y

y

S R

e eS R

e e

R R R R R R

R R R R


10 0100

11

23 3233

22

21

11 22 10 01 23 32 10 32 23 01

12

10 01

reflection measurement at port 1 1

reflection measurement at port 2 1

, , , ,

, ,

mx

my

mx

mT

my

mT

mT

e ee

e

e ee

e

ΓS

Γ e e e e e e e e e eS

Γ

e e e

23 32,

l

e

Γ, e

1 01 1000,

11

1 23 3233,

22

,

,

x x L mL mT

y L y mT mL

e eM R R R M R R e

e

e eR N R R N R R e

e


11 12 11 12 11 12

21 22 21 22 21 22

11 11 12 21 112

21 11 22 21 21

11 12 12 22 12

21 12 22 22 22

(det )

0

0 1 /

/

/

l

x x L l

l

l

l

l

ailed derivation

m m x x x x eM R R R

m m x x x x e

m x m x x em

m x m x x e

m x m x x e

m x m x x e

211 11 11 10 011 22 11 12 00

21 21 21 11

212 12 1221 22 11 12 00

22 22 22

10 01 00 11

11 12 11 12

21 22 21 22

( ) ( ) 0

( ) ( ) 0

: 0 , 1

0

0

l

y L y

x x x e em m m a e

x x x e

x x xm m m m b e

x x x

root choice e e a b e e a b

y y n n eR N R R

y y n n

11 12

21 22

211 11 12 21 11 11 11 1112 22 11 21

12 1221 11 22 21 12

221 2111 12 12 22 1212 22 11 21

22 2221 12 22 22 22

1 /

( ) ( ) 0

/( ) ( ) 0

/

l

l

l

l

l

y y

y ye

y n y n y e y y yn n n n c

y yy n y n y e

y yy n y n y en n n n

y yy n y n y e

23 3233

12 22

2133

22

23 32 22 33 : 0 , 1

e ee

y e

yd e

y

root choice e e c d e e c d


10 0100

11 11

23 3233

22 22

10 01 2200 11

11 22 22

2

11 22 10 01

11

1

1

1

1

1

1

1, , ( )

mxmx

mx

my

my

my

mTmT

mT

mymx mT mT

mx my mT mT

e e Γ b ΓΓ e Γ

e Γ e a Γ

d Γe e ΓΓ e Γ

e Γ e c Γ

e e e b ΓΓ e e

e e e a Γ

c Γb Γ b Γ b Γe e e e b a e

a Γ d Γ a Γ e a Γ

11 23 32 22

10 3221

11 22

10 32 21 11 22 23 01 12 11 22 10 01 23 32

23 0112

11 22

1211 11

11

, =( - )

1(1 ), = (1 ) , , ,

1

1 ...(also for selection),

mT

mT mT

mT

lmx

mx

e e c d e

e eS

e ee e S e e e e S e e e e e e

e eS

e e

b Γ mΓ e e m

e a Γ a


4.6 Discontinuities and modal analysis

• equivalent circuit components

E C, H L

constant E (V) parallel connection

constant H (I) serial connection

Discussion

1. Microstrip discontinuities

open-end gap

Cp Cp

Cg

Coc


step

T-junction

bend

C

C

C

L L

L1

L L

L2

L3


2. Microstrip discontinuity compensation

swept bend mitered bends

mitered step mitered T-junction

wr

r>3w

a

a=1.8w


Solved Problems

Prob. 4.11 Find [S] relative to Z0

Prob. 4.12 Find S21 of [SA] in cascade of [SB]

S21A

S22A

S12A

S11A

S21B

S22B

S12B

S11B

21 2121

22 111

A B

A B

S SS

S S

Zport

1

port

2Z

port

1

port

2

Yéëùû=

1/ Z -1/ Z

-1/ Z 1/ Z

é

ëê

ù

ûú,DY = (Y o +Y11)(Y o +Y 22 ) +Y12Y 21 =

Z + 2Z o

ZZ o2

Séëùû=

(Y o -Y11)(Y o +Y 22 ) +Y12Y 21

DY

-2Y12Y o

DY

-2Y 21Y o

DY

(Y o +Y11)(Y o -Y 22 ) +Y12Y 21

DY

é

ë

êêêê

ù

û

úúúú

=

1/ Z o2

DY

2 / ZZ o

DY

2 / ZZ o

DY

1/ Z o2

DY

é

ë

êêêê

ù

û

úúúú

=

Z

Z + 2Z o

2Z o

Z + 2Z o

2Z o

Z + 2Z o

Z

Z + 2Z o

é

ë

êêêêê

ù

û

úúúúú

,1- S 11 = S 21

211 22 12 21

11 22 12 21 12

21 11 22 12 21

2

2

, ( )( ) 2

( )( ) 2

2 ( )( )

22

2 2

22

2 2

o o o o

o o o

o o o

oo o

o o

oo o

o o

Z ZZ Z Z Z Z Z Z Z Z Z Z

Z Z

Z Z Z Z Z Z Z Z

Z ZS

Z Z Z Z Z Z Z Z

Z Z

Z ZZ ZZ

Z Z Z ZZ Z

Z ZZZ Z

Z Z Z ZZ Z

11 21,1

S S


Prob. 4.140 0 0

0 0

0 0

0 0

0.178 90 0.6 45 0.4 45 0

0.6 45 0 0 0.3 45

0.4 45 0 0 0.5 45

0 0.3 45 0.5 45 0

0

24

11

2

42

2

12

454port and 2port between delay phase

84.0log20log204port and 2port between lossinsertion )4(

201.0log20log20 1port at lossreturn )3(

lsymmetrica ])[2(

18.0)1(

dBS

dBS

reciprocalS

lossySS

0 0 01 1

0 02

0 03 3

0 04

03 1

0 0 0 0 01 1 3 1 1

0.178 90 0.6 45 0.4 45 0

00.6 45 0 0 0.3 45

0.4 45 0 0 0.5 45

00 0.3 45 0.5 45 0

0.4 45

0.178 90 0.4 45 0.178 90 0.16 90 0.018 90

b a

b

b b

b

b a

b a b a a a

1

111

1

0.018b

S ja

(5)reflection at port 1 as port 3 is connected to a short circuit


Prob. 4.19 given [Sij] of a two-port network normalized to Zo, find

its generalized [S’ij] in terms of Zo1 and Zo2

0 0

2 1' ' '

11 11 12 12 21 21 22

1 2

incident (power) wave =2

reflected (power) wave =2

, , ,

k k

i i oi ii

oi oi

i i oi ii

oi oi

i ojiij

j j oia ,k j V ,k j

o o

o o

V V Z I: a

Z Z

V V Z I: b

Z Z

V ZbV S V b S a , S

a V Z

Z ZS S S S S S S

Z Z

'

22S


Prob. 4.28 find P2/P1 and P3/P1

00

0

00

23

2312

12

S

SS

S

P1

P2

P3

Γ2

Γ3

a1

b1a2

b2

a3

b3

Γ2

Γ3

S12

S12 S23

S23

2

12 2 2312 2 121 1 1 2 1 3 12 2 2

2 3 23 2 3 23 2 3 23

2 2 2 2 2 2 2 2

2 2 2 2 12 2 12 22

2 2 2 2 2 2 22 2 2

1 21 1 1 12 2 2 3 23 12 22

2 3 23 22

2 3 23

2 2

3 33

1

, ,1 1 1

(1 ) (1 ) (1 )

(1 ) 11 (1 )

1

in

in

S SS Sb a a b a b a

S S S

b a b S SP

P a b a S S SS

S

b aP

P a

2 2 2 2 2 2 2 2 2 2

3 3 12 2 23 3 12 2 23 2

2 2 2 2 2 2 22 2 2

21 1 1 12 2 2 3 23 12 22

2 3 23 22

2 3 23

(1 ) (1 ) (1 )

(1 ) 11 (1 )

1

in

b S S S S

b a S S SS

S

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