ENEE482-Dr. Zaki 1

Impedance Matching with Lumped Elements

1G when usedCircuit L YL

jX1

jB2

LL

L

LL

LLLLL

LLL

LLL

LLL

in

RB

Z

R

ZX

BX

XR

RZXRZRXB

XRZBXBX

ZRZXRXB

Z

jXRjB

jX

ZjB

jXZ

2

00

21

220

220

2

0221

0012

0

2

1

2

1

1

/

)1(

)(

11

11

ENEE482-Dr. Zaki 2

jB2

jX1

ZL

1R when usedCircuit L

0

02

01

022

0102

01

2

/)(

)(

)(

)(

/1)(

1

Z

RRZB

XRZRX

RZBXX

RZXXZB

ZXXjR

jBY

LL

LLL

LL

LL

LLin

ENEE482-Dr. Zaki 3

Single-Stub Matching

Load impedance jBYin 1

1tan

2length stub The

1

1cos

4

1

epoint wher minimum- voltage thefrom distance thebe Let

real ist coefficien reflection then thereal, is If1

1

10

10

0

S

S

S

Sd

jBY

d

Y

SY

in

L

in

Input admittance=S

ENEE482-Dr. Zaki 4

Series Stub

Input impedance=1/S

Voltage minimum

S

S

Xjj

dS

X

S

Sd

djjS

djSXjZ

SZ

in

in

1tan

2

tan

tan)1

1(

1

1cos

4

tan1

tan1

/1

10

0

0

10

01

01

ENEE482-Dr. Zaki 5

Double Stub Matching Network

jB1jB2YL

ab

2

1

P tocircle econductancconstant along

point themoves which Bj esusceptanc a adds stubfirst The

aa plane at the

into ed transformis

LLL

LL

BjGY

YY

b a

ENEE482-Dr. Zaki 6

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit Popen circuit

r=0.5

Smith Chart

YL

. cancel willstub The circle. 1G on the liemust P The

.Y is admittanceinput theb-b plane At the

d2

anglean through circle radiusconstant a along P toP from Move

3

b

32

b

bb

Bj

BjG

ENEE482-Dr. Zaki 7

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit Popen circuit

r=0.5

Smith Chart

YL

Rotate the the G=1 circle through an angle -The intersection of G=1 and the GL circle determine

The point P2

P2

P3

G1=1

ENEE482-Dr. Zaki 8

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit Popen circuit

r=0.5

Smith Chart

YL

The shaded range is for the load impedance whichcannot be matched when d=1/8 wavelength

ENEE482-Dr. Zaki 9

Quarter-Wave Transformers

ZL

4

ZC=Z0 ZC=Z1

LL

L

in

in

L

L

L

LL

Lin

ZZjtZZ

ZZ

ZZ

ZZ

fttjZZ

tjZZZ

ZZZ

ZZ

Z

jZZ

jZZZZ

00

0

0

0

1

11in

01

0

21

1

11

2

)(tantan , Z

matchperfect

)4/tan(

)4/tan(

ENEE482-Dr. Zaki 10

small very is )(f2f width band The

1)(

2cos

is of valueingcorrespond The

toleratedbecan that

t coefficien reflection of valuemaximum theis If

cos2

1sec and /2 then fnear is f If

sec2

1

1

0

21

11

m

m

1

1

20

2

1

1

m

mL

Lmm

L

L

L

L

f

ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

ENEE482-Dr. Zaki 11

m

2

m

Bandwidth characteristic for a singleSection quarter wave transformer

m2

2Bandwidth

ENEE482-Dr. Zaki 12

Theory of Small Reflection

2

23

12

112

12

2121

1212

121

2

, 2

1

,

ZZ

ZZ

ZZ

ZT

ZZ

ZT

ZZ

ZZ

L

L

L1

2

1

ENEE482-Dr. Zaki 13

unity tocompared small are and If

1

1 , 11for Substitute

1

...

31

2312

31

231

1211212

232

232112

12

30

22

321121

42

232112

2321121

jj

j

j

jjnn

n

nj

jj

ee

e

TT

e

eTTeeTT

eTTeTT

1 3

21T je

ENEE482-Dr. Zaki 14

1 21

jeTT 232112

jeTT 4232112

jeTT 623

222112

jeT 2321

jeT 423212

jeT 623

2221

jeT 23221

jeT 423

2221

Multiple reflection of waves for a circuitwith two reflection junctions

ENEE482-Dr. Zaki 15

Approximate Theory for Multi-Section Quarter Wave Transformers

L

021 N

A multi-section quarter-wave transformer

NNL

NLN

nnn

nnn

L

ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

Z

ist coefficien reflectionlast The

,

real is Assume

1

10

01

010

ENEE482-Dr. Zaki 16

even Nfor ]cos

...)cos(...)2cos(cos[2

odd Nfor ]cos

...)2cos(...)2cos(cos[2

...])()([

...etc , , ,

i.e. lsymmetrica isr tranforme that theAssume

....

2/)(

10

2/)1(

10

)2()2(10

22110

242

210

N

njN

N

njN

NjNjjNjNjN

NNN

jNN

jj

nNNNe

nNNNe

eeeee

eee

ENEE482-Dr. Zaki 17

Binomial Transformer

0L

0L

0

n

2j-

0

Z

Z)0( , or 0when

/4for which ffrequency center the toscorrespond /2

1-N1,2,...,nfor /2at 00/)(d

and /2for 0)( that Note

)(cos2

)(

)eA(1)( Choose

/2. wherefrequency matching

at the vanish )or (frequency w.r.t sderivative 1-Nfirst theand

if obtained is sticscharacteri passbandflat maximally A

Z

Z

d

A

eeeA

f

n

NN

NjjNj

N

ENEE482-Dr. Zaki 18

nNNn

L

LNn

jNN

jjjn

NN

NNNnN

Nn

CZZ

ZZ

eeee

CNCCCC

0

0

242

210

2N

0n

Nn

110

2

....CA)(

:bygiven

response actual the toresponse passband desired theEquate

..... , 1, ,

!)!(

!

!

)1)...(2)(1(

Z

Z2)1(

Z

Z2)(

Z

Z2A , 2)0(

2

00L

0L2

0L

0L

0L

0L

nnN

N

n

nNNNNC

eCZ

Ze

Z

Z

Z

ZA

Nn

jN

n

Nn

NNjN

NN

ENEE482-Dr. Zaki 19

01

2

0

1

1

2

0

1

0

)1(

00

01

21

1

11

nn

ln2

1)...ln(

2

1

ln...lnln2

1)0(2

)ln(2

ln2]2[22ln

ln2

1

thefrom found becan Zimpedances sticcharacteri The

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

ZA

Z

ZA

Z

ZCC

ZZ

ZZ

Z

Z

eZZ

ZZ

ZZ

Z

Z

L

N

L

N

LN

LN

LNn

NNn

L

LNn

n

n

nn

nnn

nn

n

n

n

1

12l : Use

x

xnx

ENEE482-Dr. Zaki 20

N

mmm

N

m

m

mNN

m

Af

ff

A

A

/1

1

0

0

0

/1

1m

2

1cos

42

42

)(2

f

f

:isbandwidth fractional The

2

1cos

passband theof edgelower theis 2/

cos2

is passband over the

toleratedbecan that coeficient reflection of valuemaximum The

ENEE482-Dr. Zaki 21

Example

ZLZ2Z1Z0

Z3

Design a three section binomial transformer to match a 50 Ohms load to a 100 Ohms line. Calculate the bandwidthFor max reflection =.05 over the passband.

1!3)!33(

!3 ,3

!2)!23(

!3

, 3!1)!13(

!3 , 1

!0)!03(

!3

ln2

12

100 Z,50 Z3,N sections 3For

33

32

31

30

04

0

03

0L

CC

CC

Z

Z

ZZ

ZZA L

L

L

ENEE482-Dr. Zaki 22

5.5400.4100

50ln)3(27.70lnln

ln2lnlnZ :2

7.7026.4100

50ln)3(27.91lnln

ln2lnlnZ :1

7.91518.4100

50ln2100lnln

ln2lnlnZ :0

33

3

0

32

323

23

2

0

31

312

13

1

0

30

301

ZZ

Z

ZCZn

ZZ

Z

ZCZn

ZZ

Z

ZCZn

L

L

L

ENEE482-Dr. Zaki 23

Chebyshev Transformer

nT

xTxxTxT

xxxT

xxT

xxT

xT

n

nnn

n

cos)(cos

)()(2)(

34)(

12)(

)(

n degree of polynomial Chebyshev :)(

21

33

22

1

m

m

0

0

ZZ

ZZ

L

L

m

ENEE482-Dr. Zaki 24

)(sec T

)cos(sec T

Z

Zln

2

1

)(sec T2

)/ln(

ln2

1)(sec TA

Z

Z

: have we0When

)cos(sec T A

.......] )2cos(

....)2cos(cos[2

cos

coscoscos

cos

cos Consider

mN

mN

0

L

mN

0

0mN

0L

0L

mN

10

1

jN

L

L

jN

n

jN

mmn

e

ZZA

Z

Z

Z

Z

e

nN

NNe

nT

ENEE482-Dr. Zaki 25

)12(cossec4)34cos2cos4(sec)cos(secT

cossec3)3coscos3(sec)cos(sec T

1)2cos1(sec1)cossec(2)cos(sec T

cossec)cos(sec T

....])2cos(.......)2cos(cos[2

2)1(2)(cos

2

)/ln(cos

1cossec , )/ln(

2

1)(sec T

)(sec T2

)/ln(

unity is )cos(sec T of valuemaximum thepassband In the

2m

4m4

m3

m3

22mm2

mm1

101

2

0

2

01m0

1mN

mN

0

mN

mm

mm

m

nm

nnn

jmn

m

nm

jnnnjjnnn

m

LLm

Lm

mnCnCnC

eCeee

ZZ

NZZ

ZZ

ENEE482-Dr. Zaki 26

Example

Design a two section Chebyshev transformer (two sections) toMatch a line of load impedance =2. The maximum tolerance Value of is 0.05.

639.1 ,219.1

148.0)1sec(

099.0sec2

1

)1(seccos2sec

)cos(secT2cos2

04.1 and ,96.1sec

67.6)05.0(3

11sec2)(secT

12

202

1

m2

1

2m2

m0

m2

m2

m210

mm

2m2

10

ZeZZeZ

m

mm

m

m

ENEE482-Dr. Zaki 27

ENEE482-Dr. Zaki 28

Design of Complex Impedance Termination

Amplifier ZcZc

Input Matchingnetwork

Output Matchingnetwork

ZsZL

Microwave amplifier circuit

ENEE482-Dr. Zaki 29

MM

PMR

V

ZZ

RR

R

VR

jXRZ

R

V

L

inLT

c

LT

LT

T

cL

TTT

T

c

42

14

42

1 todeliveredpower The

42

1 :isnetwork thefrompower available The

2

0

2

2

0

2

0

ENEE482-Dr. Zaki 30

jB1

Stub

ZL ZcZL

l l

G=1

jB1ZL G=1jB2ZL G=1

jX1 jX2

=

Transmission Line Matching Network

Alternative Matching Networks

ENEE482-Dr. Zaki 31

Gen

erat

or

Load

l/ =

0.43

75l/ = 0.0626j1

-j1

YL

ZL= 0.4-j0.2

Y’in1

2

G=1

G=2

Y”in

Design Procedure for the Matching Network with Shunt Stub

ENEE482-Dr. Zaki 32

Impedance Mismatch Factor

factormismatch impedance thecalled is 4

4

42

1

42

1

obtained ispower available Maximum

X , then ZIf

2

1

2

2

2

2

in*

in

2

Sin

Sin

ava

Sin

Sin

S

Sin

S

Sava

SSinS

ininS

Sin

inS

S

ZZ

RRM

MPZZ

RR

R

VP

R

VP

XRRZ

RZZ

VP

ZZ

VI

ENEE482-Dr. Zaki 33

Z11-Z12Z22-Z12

Z12

ZL

ZS

VS

Zin

ZT

Voc

ZL

ML

A T matching network Thevenin equivalentnetwork

2122211

11

22T

11

12 Z,

ZZZ

ZZ

ZZ

ZZ

VZV

S

S

S

Soc

ENEE482-Dr. Zaki 34

Impedance Transformation and MatchingReview of Transmission Lines and Smith Chart

0

0

0

0

0

0

00

0

0

0

00

00

TCOEFFICEIN REFLECTION

I(z)

T.L. theof Impedance ticCharacters :

,)()()(

)()()(

V

V

eZ

Ve

Z

V

ZI

V

I

V

eIeIzIzIzI

eVeVzVzVzV

zz

zz

zz

Zg

ZL

Z0Vg

Finite Transmission Line terminated with load impedance

Z=LZ=0

L

ENEE482-Dr. Zaki 35

LZZ

LZZZLzZ

zZZ

zZZZ

zI

zVzZ

zZzZZ

IzI

zZzZIzV

zLz

eZZeZZZ

IzI

eZZeZZI

zV

eZIVVeZIVV

eZ

Ve

Z

VIeVeVV

L

Lin

L

L

LL

LL

zLL

zLL

L

zLL

zLL

L

LLL

LLL

LLL

LLL

tanh

tanh)(

tanh

tanh

)(

)()(

)coshsinh()(

)sinhcosh()(

Let

])()[(2

)(

])()[(2

)(

)(2

1 )(

2

1

, ,

0

00

0

00

00

0

)(0

)(0

0

)(0

)(0

0000

0

0

0

000

ENEE482-Dr. Zaki 36

Standing wave ration (SWR) S:

1S

1-S ;

1

1

min

max

V

VS

Smith Chart:

jjL

jjL

jL

jLin

in

j

j

LL

Lir

L

LL

L

L

L

L

L

L

L

e

e

e

e

Z

ZZ

e

ez

z

z

jxrZ

Z

eZZ

ZZ

2

2

2

2

0

0L

L

0

0

1

1

1

1

)(1

)(1

1

1 ;

1

1

1

1z

z impedance normalized The

ENEE482-Dr. Zaki 37

x

xxx

r

r

rrr

r

xr

j

jjxr

ir

ir

ir

ir

ir

i

ir

ir

ir

ir

1 and 1at centered and

1 radius of circle a ofEquation ;

11)1(

0 and 1

at centered and

1

1 radius a of circle a ofEquation ;

1

1)

1(

)1(

2 ;

)1(

1

)1(

)1(

222

222

2222

22

ENEE482-Dr. Zaki 38

Imaginary part ofRefl. Coeff.

0

r=1

x=1

x=-1

Real part ofRefl. Coeff.

Pshort circuit

Popen circuit

r=0.5

Smith Chart

ENEE482-Dr. Zaki 39

Review of Transmission Lines and Smith Chart

0

0

0

0

0

0

00

0

0

0

00

00

TCOEFFICEIN REFLECTION

I(z)

T.L. theof Impedance ticCharacters :

,)()()(

)()()(

V

V

eZ

Ve

Z

V

ZI

V

I

V

eIeIzIzIzI

eVeVzVzVzV

zz

zz

zz

Zg

ZL

Z0Vg

Finite Transmission Line terminated with load impedance

Z=LZ=0

L