Impedance Matching and Tuning -...

Impedance Matching and Tuning

1

Impedance Matching and Tuningpeda ce atc g a d u g

• Impedance matching or tuning is important for the following reasons:reasons: Maximum power is delivered

Improve the SNR of the system Improve the SNR of the system

Reduce amplitude and phase errors

Lin

Figure 5.1 (p. 223)A lossless network matching an arbitrary load impedance to a transmission line.

2

Impedance Matching and Tuningpeda ce atc g a d u g

Other Discussion Matching network usually use lossless components: L C transmission Matching network usually use lossless components: L, C, transmission

line, transformer, …

There are many possible solutions available There are many possible solutions available

Use Smith chart to find the optimal design

F i h l i f i l hi k Factors in the selection of a particular matching network: Complexity

Bandwidth

Implementation

Adjustability

3

Matching with Lumped Elementsatc g w t u ped e e ts

Figure 5.2 (p. 223)i hi k ( ) k fL-section matching networks. (a) Network for zL

inside the 1 + jx circle. (b) Network for zL outside the 1 + jx circle.

4


• Analytic Solutions (ZL = RL + j XL)LL ZRjxz 0 )( 1 theinside is :1 Case 0 )( 1 theoutside is :2 Case ZRjxz LL

jXZ01

:condition matching aFor

11

:condition matching aFor

jB

LL jXRjBjXZ0

:parts Re/Im into Separating1

0

:parts Re/Im into SeparatingXXjR

jBZ LL

LLL

LLL

XRBZBXXZRZXXRB

0

00

1

0

00

RBZXXRZXXBZ

LL

LL

LLLLL RZXRZRXB 0

220

:Solution

0

:Solution

XRZRX LLL

L

LL

ZZXX

XRB

00

22

1

0

0

ZRRZ

B LL

5LL BRRB

X


jB- jB-jX jXLL L

jB jBjX- jX-C C

L CZ Y

L

LL

CC

C

6

L C


• Smith chart solutionscircle1 theinsideis:1 Case jxzL jL

circle 1 jx

7


circle 1 theoutside is :2 Case jxzL

i l1 jb circle 1 jb

8


• Example 5.1 L-Section Impedance MatchingImpedance Matching

MHz 500, 100 , 100200 0

fZjZL

12:1Solution

jzL

jyjjbjyL

5.04.03.0 2.04.0

pF0 92bC

2.1 2.11 jxjz

nH838

pF0.92 2

0

0

ZxL

ZfC

9

nH 8.382 f

L


• Example 5.1 L-Section Impedance MatchingImpedance Matching

MHz 500, 100 , 100200 0

fZjZL

12:1Solution

jzL

jyjjbjyL

5.04.07.0 2.04.0

pF61.21

C

2.1 2.11 jxjz

nH1.462

p 2

0

0

bf

ZL

Zxf

10

2 bf


• Example 5.1 L-Section Impedance Matching

Figure 5.3b (p. 227) (b) The two possible L-section matching circuits. ( ) R fl i ffi i i d f f h hi i i f (b)

11

(c) Reflection coefficient magnitudes versus frequency for the matching circuits of (b).


• Lumped elements (l < /10): parasitic C/L, spurious resonances, fringing fields, loss and perturbations caused by a ground plane.

nH10 nH10

pF 5.0pF 25

12


frequency single:Bandwidth Estimating

may tuningeApproximatbandwidth

q yg

:ContoursFrequncy !! be Better

impedance of , as theoremreactance '

jXfsFoster

sadmittance and Impedances sadmittance of and

jB

increased. isfrequency as arcsclockwise echart tracSmith on the

13


• Constant Q circles:BX

RFQGB

RXQ

21

LR

R

RRF

Q

0

L

14


Q

matching LowBroadband

nRFQ 21

Q=1 074Q=1.074

15


One-section High Q Matching v. s. 3-sections Low Q Matching

16

Single-Stub TuningS g e Stub u g

jBY i jBYin

(a) No Lumped Elements(b) Easy to fabricate in microstrip or stripline.

jBY 0

jXZ jXZ in

Figure 5.4 (p. 229)

jXZ 0

g (p )Single-stub tuning circuits.

(a) Shunt stub. (b) Series stub.

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Single-Stub Tuning (Shunt)S g e Stub u g (S u t)

• Example 5.2

GHz 2 , 50 8060

0

fZjZL

S.C. yS.C.

i l1i t ti lSWR8.06.06.12.1

jbjyjz LL

4711for26047.11for 11.0

:circle 1intersectscircle SWR

11

jydjyd

jb

4050S C471095.0S.C.47.1

47.11for 26.0

1

22

ljlj

jyd

18

405.0S.C.47.1 2 lj


• Example 5.2

19


1 jXRYZ

0

0:

1

tjXRjZtjZjXRZZd

jXRYZ

LL

LLLL

1

0for ,tan21

1

1

ttd 0

1tan where

ZjBGYdt

tjXRjZ LL

0for ,tan21 1

BBBt

tt

ss

22

21 where

1

tZXRtRG

ZjBGY

L

11

stub, circuited-openan For

11

BBl so

ss

2200

20

tZXRZ

tZXtXZtRB

tZXR

LLL

LL

stub,circuited-shortaFor

tan2

1tan21

0

1

0

1

YB

Yl so

22

00

00

1 that sochosen is

ZXRZRX

ZYGdtZXRZ LL

tan21tan

21

,

0101

BY

BYl

s

s

0

0

022

0

f2

for

ZRZX

ZRZR

ZXRZRXt L

L

LLLL

2negative is resultant theIf

l

s

20

00 for 2 ZRZXt LL

Single-Stub Tuning (Series)S g e Stub u g (Se es)

• Example 5.3

50 80100

ZjZL

O.C. GHz 2,500

fZ

612 jzz

O.C.

3311for1200 :circle 1 intersects circle SWR

6.12

jzdjx

jzL

3970O C33133.11for 463.033.11for 120.0

22

11

ljjzdjzd

103.0O.C.33.1397.0O.C.33.1

2

1

ljlj

21


• Example 5.3

22


1 jBGZY

0

0 :

1

tjBGjYtjYjBGYYd

jBGZY

LL

LLLL

1

0for ,tan21

1

1

ttd 0

1tan where

YjXRZdt

tjBGjY LL

0for ,tan21 1

XXXt

tt

ss

22

21 where

1

tYBGtGR

YjXRZ

L

11

stub, circuited-openan For

0101

ZZlo

ss

2200

20

tYBGY

tYBtBYtGX

tYBG

LLL

LL

stub,circuited-shortaFor

tan21tan

21 0101

XXl

s

o

22

00

00

1 that sochosen is

YBGYGB

YZRdtYBGY LL

tan2

1tan21

,

0

1

0

1

ZX

ZXl ss

0

0

022

0

f2

for

YGYB

YGYG

YBGYGBt L

L

LLLL

2negative is resultant theIf

00

l

23

00 for 2 YGYBt LL

Double-Stub Tuningoub e Stub u g

24


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

25

first stub.


pointon intersecti No:region Forbidden

circle 1 Rotatedwith jb

region forbidden reducingfor reduce d

frequency :2/or 0 d sensitive

8/3or 8/ aschosen generally are

d

26


• Example 5.4060 80 , 50 LZ j Z

Stubs: open-circuited stubs,/ 8, 2 GHzd f

: a series resistor and capacitorSolution:

LZ

1 1

1.2 1.6 0.3 0.41.314 0.146

L Lz j y jb l

1 1

2

0.114 0.4821 3.381 1 38

b ly j

22

1 1.38y jb 23.38 0.204

1 38 0 350l

b l

27

2 21.38 0.350b l

Double-Stub Tuningoub e Stub u g0.995 pF

0.1460.204

0.995 pF

(c)0.4820.350

Figure 5.9b (p. 239)(b) The two double-stub tuning solutions. (c) Reflection coefficient magnitudes versus (b)

0.4820.350

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frequency for the tuning circuits of (b).(b)


• Analytic Solution

st :stub 1 theofleft theJust to

tBtBYt L

2

210

2

1410

:regionForbidden

11

linesion transmislength dBBjGY LL

YtYG

tY

02

2220

10

10

0102

nd :stub 2 theofleft thejust totYBBjGYY LL tGYGtY

BB

dtYG

LL

L

220

20

220

1

sin0

00

1002

fl/1 and tan where

YYZYdtBjBjGtjY LL

YGtGYGtYB

tBB

LLL

L

022

02

0

1

1

210

2

2

02

02

01

ofpart real

tBtBYtYGG

YY

LLL

tGB

L2

Blo 1tan1 :stub O.C.For

2

210

2

2

2

0

220

4111 tBtBYttYG

tt

LL

LL

Yl

Y

s 01

0

tan1 :stub S.C.For

2

29

2220

2012 tYtL

B2

The Quarter-Wave Transformere Qua te Wave a s o e

sec41

1

20 LZZ

Figure 5.10 (p. 241) A single-section quarter-t hi t f t th d i

2f

sec1

0

20

L

L

ZZ

ZZ

wave matching transformer. at the design frequency f0.

40

tZjZ

2near for cos2

0

0 L

L

ZZ

L

L

flttZjZtZjZZZ

1

11in

at2tantanwhere

LL

L

ZZtjZZZZ

ZZZZ

flt

00

0

0in

0in

0

2

at ,2tan tan where

L

LL

ZZZ

ZZtjZZZZ

02

1

000in 2

Figure 5.11 (p. 242) Approximate behavior of the reflection coefficient magnitude for a single-section quarter-wave transformer operating near

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its design frequency.

The Quarter-Wave Transformere Qua te Wave a s o e

22 :Bandwidth m

2

02 sec

211

ZZZZ L

0

2

0

2cosor

ZZZZ

ZZ

Lmm

Lm

0

2

thenlines, TEM assume weIf1 ZZLm

m

00 24

2 ff

fv

vfl p

p

Fi 5 12 ( 243) R fl ti ffi i t

0

0

2

1

0

2

1cos4242

ZZZZ

ff

L

L

m

mm

Figure 5.12 (p. 243) Reflection coefficient magnitude versus frequency for a single-section quarter-wave matching transformer with various load mismatches

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m with various load mismatches.

The Theory of Small Reflectionse eo y o S a e ect o s

• Single-Section Transformer jj eTTeTT 42

232112

2321121

n

jnnnj eeTT0

232

2321121

1

jn

n

eTT

xx

x

232112

01for ,

11

j

j

e

eeTT

231

232

321121 1

j

j

e

ee

231

231

31

1

Figure 5.13 (p. 244) Partial reflections and transmissions on a single section matching transformer

31

32

on a single-section matching transformer.

The Theory of Small Reflectionse eo y o S a e ect o s

• Multisection Transformer

Figure 5 14 (p 245) PartialFigure 5.14 (p. 245) Partial reflection coefficients for a multisection matching transformer.

010 ZZ

ZZ

jN

Njj eee

:lsymmetricamadebecanrTransforme

242

210

1

1

01

nnn ZZ

ZZZZ

jN

NjNjjNjNjN

nNNNe

eeeee 2cos2coscos2

:lsymmetrica made becan r Transforme22

10

1

NL

NLN

nn

ZZZZZZ

jNN

nj

NNN

NnNNNe

2cos2coscos2

even for ,21 2cos2coscos2

2/

10

llymonotonica vary :n

NL

Z

N

njN

ZZZ

NnNNNe

d iGi

odd for ,cos 2cos2coscos2

2/)1(

10

33

NZZZ , ,design ,Given 21

The Bode-Fano Criterione ode a o C te oCircuit limit Fano-Bode

Figure 5.22 (p. 262) The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (ω0 is the center frequency of the matching bandwidth). (a) Parallel RC. (b) S i RC

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(b) Series RC.

The Bode-Fano Criterione ode a o C te oCircuit limit Fano-Bode

Figure 5.22 (p. 262) The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (ω0 is the center frequency of the matching bandwidth). (c) Parallel RL. (d) S i RL

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(d) Series RL.

The Bode-Fano Criterione ode a o C te o

Figure 5.23 (p. 263)Illustrating the Bode-Fano criterion. (a) A possible reflection coefficient(a) A possible reflection coefficient response. (b) Nonrealizable and realizable reflection coefficient responsesreflection coefficient responses.

:Given 1. RC

fi il00 unless ,0 .2 m

m

or3sfrequencie ofnumber

finite aat only 0

CR

m

matchharder toisloadhighor

or .3

Q

CR

m

36

match harder to is load high Q

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