EMLAB

1

Chapter 5. Impedance matching and tuning

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2

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, VV

out,

V

LZ+V-

SZ[ns]3d TZs = 20

Z0= 50

ZL= 1k

Maximum power delivered when applied to matched load

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1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.91.0 2.0

-1

0

1

2

-2

3

time, usec

Vin

, VV

out,

V

Vs Vin Vout

RR1R=10 Ohm

VtPulseSRC1

Period=50 nsecWidth=25 nsecFall=1 nsecRise=1 nsecEdge=cosineDelay=0 nsecVhigh=1 VVlow=0 V

t

RR2R=1000 Ohm

MLINTL1

L=5 meterW=0.242 mmSubst="MSub1"

Signal source

Load

~ 10SZ

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.91.0 2.0

-0.0

0.2

0.4

0.6

0.8

-0.2

1.0

time, usec

Vin

, VV

out,

V

kZL 1500Z

Mismatched load

Ringing

EMLAB

4Impedance matching - Digital

~

10SZ

kZL 1500Z

R1

1k

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.950.00 1.00

0.0

0.5

-0.5

1.0

time, usec

Vin

, VV

out,

V

40

~ 10SZ

kZL 1

500Z 53

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.950.00 1.00

-0.0

0.2

0.4

0.6

0.8

-0.2

1.0

time, usec

Vin

, VV

out,

V

Source matching

Load matching

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~

10SZ kZL 1

500Z 53

R1

1k 40

Impedance matching – RF 회로

Maximum power transfer

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.950.00 1.00

-0.0

0.2

0.4

0.6

0.8

-0.2

1.0

time, usec

Vin

, VV

out,

V

Source matching

Load matching

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Matching with lumped elements

Figure 5.2 (p. 223)L-section matching networks. (a) Network for zL inside the 1 + jx circle. (b) Network for zL outside the 1 + jx circle.

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LL

L

LL

LLLLL

LLL

LLL

LL

LL

L

BR

Z

R

ZX

BX

XR

RZXRZRXB

XRBZBXX

ZRZXXRB

jXRjB

jXZZ

jxzZ

Zz

00

220

220

0

00

0in

0

1

/

)1(

)(

11

circle1,

LLL

LL

LL

LL

LL

LL

L

XRZRX

Z

RRZB

RBZXX

RZXXBZ

XXjRjB

ZY

jxzZ

Zz

)(

/)(

)(

)(

)(

11

circle1,

0

0

0

0

00

0in

0

Analytic solution

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Figure 5.3a (p. 226)Solution to Example 5.1. (a) Smith chart for the L-section matching networks.

ZL= 200-j 100

Z0= 100

f = 500MHz

1

2

3

4

5

Example 5.1

Smith chart – impedance chart

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Impedance-admittance chart

1

ZL= 200-j 100

Z0= 100

f = 500MHz

0.2

0.5

Add shunt C

Add series L

pF][95.03.0

3.03.0

0

00

ZC

CZ

YB

1.2

0.0

nH][2.382.1

2.12.1

0

00

ZL

ZLXZ

X

EMLAB

10Basic Smith chart operation

1. Translation

2. Add series element

)0( z)( lz

ljezlz 2)0()(

L

C

l 2

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11

L

C

3. Add shunt element

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12

x

R0 0.5 1 2

r=2

r=1

r=0.5

r=0

x

R

2

1

0.5

0.5

1

2

x=2x=0.5

x=1

x=-0.5

x=-1

x=-2

real

imag

real

imag

Constant resistance, reactance circles

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13

1

0.314

11.01 d

6.12.1 jzL

0.422

1

0.314

266.02 d 6.12.1 jzL

4.111 jy

5.2 Single stub tuning

Translate by ‘d’

ZL= 60-j 80

Z0= 50

f = 2GHz

D 를 변화시켜 1+jb 원의 원주 상에 yL 이 오도록 한다 .

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4.111 jy

096.01 l

Add shunt stub (shorted)

405.02 l

4.111 jy

1+jb 원의 원주 상의 지점을 shunt stub( 병렬 stub) 을 달아서 Γ 원의 원점으로 옮기면 impedance matching 이 완료됨 .

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15

1

0.314

11.01 d

0.422

266.02 d

6.12.1 jzL 4.111 jy

zL 이 1+jb 원의 원주 상에 올 수 있도록 d1 을 조절한다 .

( 점선 원 ) 상에 zL 이 옮겨 올 수 있도록 L1 을 조절한다 .

Impedance matching 순서

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Figure 5.5b (p. 231)(b) The two shunt-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

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5.3 Double stub tuning5.3 Double stub tuning

Figure 5.7 (p. 236)Double-stub tuning. (a) Original circuit with the load an arbitrary distance from the first stub. (b) Equivalent-circuit with load at the first stub.

Single stub 의 경우 d 와 L 을 둘 다 변화시켜야 하는 불편한 점이 있었으나 , double stub 인 경우 d는 고정되는 장점이 있다 .

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1

d 1l

2l

d 가 고정되어 있으므로 1+jb 원을 반시계 방향으로 2βd (rad) 만큼 돌린다 ( 점선 원 ).

( 점선 원 ) 상에 zL 이 옮겨 올 수 있도록 L1 을 조절한다 .

( 실선 원 ) 상에 있는 임피던스가 원점으로 옮겨 올 수 있도록 L2 를 조절한다 .

Impedance matching 순서

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19

1

d 1l

2l

EMLAB

20Single stub used in microwave circuits

Frequency doublermixer

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215.4 Quarter-wave transformer

Figure 5.10 (p. 241)A single-section quarter-wave matching transformer.

40

at the design frequency f0.

ljZZ

ljZZZZ

L

Lin

tan

tan

1

11

0

0

ZZ

ZZ

in

in

LZZZ 01 4/

21

lL

in Z

ZZ

Single section transformer

EMLAB

225.5 Theory of small reflections

21

1212

21

2121

2

23

1212

121

21,

21

,

ZZ

ZT

ZZ

ZT

ZZ

ZZ

ZZ

ZZ

L

L

231

231

232

232112

1

23

02

2321121

42

232112

2321121

1

1

j

j

j

j

njn

n

nj

jj

e

e

e

eTT

eeTT

eTTeTT

)1( 312

31 je

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Multi-section transformer

NL

NLN

nn

nnn ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

,,,,1

1

01

010

NjN

jj eee 242

210)(

(Zn 은 단조 증가 또는 감소 .)

(Γn 이 대칭적인 경우 )

})()({[)( )2()2(10 NjNjjNjNjN eeeee

,, 110 NN

]2

1

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

2/

10

N

njN nNNNe

EMLAB

245.6 Binomial multi-section matching transformer

NjeA )1()( 2 NNNj AeA cos2)1()( 2

0

0

0

0 22)0(ZZ

ZZA

ZZ

ZZA

L

LN

L

LN

N

n

jnnN

Nj eCAeA0

22 )1()(

!)!(

!

nnN

NCnN

NjN

jj eee 242

210)(

)1()1()( 42

21

2 jN

jN

Nj eCeCAeA

)331(2)(

3

642

0

03 jjj

L

L eeeZZ

ZZ

N

0

032

0

032

0

031

0

030

2,32

32,2

ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

L

L

L

L

L

L

L

L

Example

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)331(0417.0)(

0417.010050

1005022

642

3

0

0

jjj

L

LN

eee

ZZ

ZZA

0417.0,125.0

125.0,0417.0

22

10

n

n

n

n

nn

nnn Z

Z

ZZ

ZZ

1

11

1

1

Example 5.6

Design a three-section binomial transformer to match a 50Ω load to a 100 Ω line.

3,100,50 0 NZZL

NjeA )1()( 2

8.651

1

5.711

1

921

1

1

1

,100

2

223

1

112

0

001

1

0

ZZ

ZZ

ZZ

ZZ

Z

n

nnn

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Figure 5.15 (p. 250)Reflection coefficient magnitude versus frequency for multisection binomial matching transformers of Example 5.6 ZL = 50Ω and Z0 = 100Ω.

N

mm

mNN

m

A

A/1

1

2

1cos

cos2

N

mm

mmm

A

f

ff

f

f

/1

1

0

0

0

2

1cos

42

42

2/

)(2

Гm=5% bandwidth 70%

EMLAB

275.7 Chebyshev multi-section matching transformer

)coshcosh()(

)coscos()(1

1

xnxT

xnxT

n

n

1||for

1||for

x

x

Figure 5.16 (p. 251)The first four Chebyshev

polynomials Tn(x).

)()(2)(

34)(

12)(

)(

21

33

22

1

xTxxTxT

xxxT

xxT

xxT

nnn

nTn cos)(cos

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NjN

jj eee 242

210)(

mN

jN

NnjN

TAe

nNNNe

cos

cos

]2

1)2cos()2cos(cos[2)( 2/10

mmm

N

mmm

forT

for

1cos

cos

1cos

cos

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1)12(cossec4)32cos44(cossec)cos(sec

cossec3)cos33(cossec)cos(sec

1)cos21(sec)cos(sec

cossec)cos(sec

234

33

22

1

mmm

mmm

mm

mm

T

T

T

T

Design of Chebyshev transformers

mN

jN

NnjN

TAe

nNNNe

cos

cos

]2

1)2cos()2cos(cos[2)( 2/10

mNL

L

L

LmN TZZ

ZZA

ZZ

ZZAT

sec

1sec)0(

0

0

0

0

Determine A

Determine BW

0

0

0

0

1sec

sec

ZZ

ZZT

ZZ

ZZAAT

L

L

mmN

L

LmNm

EMLAB

305.8 Tapered lines

zdz

dZ

zZzZ

zdzdZ

zZzzZ

zZzzZz

)(2

1

)(2

)()(

)()()(

0

ln2

1

)(2

1)(

Z

Z

dz

d

dz

dZ

zZdz

zd

dzZ

Z

dz

dez

L zj

00

2 ln2

1)(

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31

L

az

ZLZ

eZzZ

)(

Lz0for )( 0

L

Le

L

ZZ

dzeL

ZZ

dzedz

de

LjL

L zjL

L azzj

sin

2

)/ln(2

)/ln(

)(ln2

1

0

0

20

0

2

Exponential taper