transmission line theorytransmission line theory...

Transmission Line TheoryTransmission Line TheorySmith Chart

1

Transmission Line Theorya s ss o e eo y

• Wire connections in Analog / Digital circuits:V, INo voltage drop on wire connectionsNo voltage drop on wire connections

• Transmission Lines in Microwave circuits:• Transmission Lines in Microwave circuits:V, IV, I waves on transmission lines

• Field analysis:E, HEM wavesEM waves

2


/m.in ,conductorsboth for length,unit per resistance series

R

H/m.in ,conductorsboth for length,unit per inductance seriesL

S/min length,unit per econductancshunt G

F/min length,unit per ecapacitancshunt C

Figure 2.1 (p. 50) Voltage and current definitions and equivalent circuit for an incremental length of transmission line. (a) Voltage and current definitions.

3

(b) Lumped-element equivalent circuit.


(2 1a)0)() ,( )()(

:law voltagesKirchhoff' From

tzzvtzizLtzizRtzv

:lawcurrent sKirchhoff' From

(2.1a) ,0) ,(

),( ),( tzzvt

zLtzizRtzv

:0

(2.1b) .0) ,(

),( ) ,( ) ,(

z

tzzit

tzzvzCtzzvzGtzi

(2.2a) , , ,

:0

ttziLtziR

ztzv

z

(2.2b)

, ,

, ttzvCtzvG

ztzi

tz

equations)pher / telegraequations linensmission domain tra-(time

4


:formphasor thecondition,statesteady sinusoidalFor the

0

0 (2.6a) ,)(

:solutions waveTravelingeVeVzV zz

(2.3a) ,

zILjRzdzVd

0

0

00

:(2.6a)to(2.3a)Applying(2.6b) .)(

( ),)(

eIeIzI zz

(2.3b) .

zVCjGzdzId

:)(and)(forequationsWave zIzV

0

0)(

( )( )pp y g

eVeVLjR

zI zz

(2.4a) ,0)(

:)(and)(for equations Wave

22

2

zVzdzVd

zIzV

(2 7)

impedance sticcharacteri

LjRLjRZ

(2.4b) .0)( 22

2

zIzdzIdzd

00

0 (2.7)

VZV

CjGZ

constantn propagatiocomplex theis

(2.5) where CjGLjRj 0

00

I

ZI

5


cos,

:domaintimein thewaveform voltageThe

0 eztVtzv z

)0(:LineLosslessThe

LCjj

GR

:line on thewavelength

(2.9) cos 0 eztV z

(2.12b) 0 (2.12a) LC

(2.10) 2

g

(2 13)

:impedance sticcharacteri The

0LZ

(2 11)

: velocityphase the

fv

:solutions general The

(2.13) 0 CZ

jj (2.11) p fv

(2.14b).)(

(2.14a) ,)(

0 0

00

eVeVzI

eVeVzV

zjzj

zjzj

.1 ,22

( ))(00

LCv

LC

ZZ

p

6

LCLC p

Field Analysis of Transmission Linese d a ys s o a s ss o es

(2 17)H/m

dsHHL

:energy electric storedfor Similarly

(2.17) H/m 20S dsHH

IL

Figure 2.2 (p. 53) Field lines on an ii it thd

, 4

Se dsEEW

arbitrary TEM transmission line.

:energymagneticstored average- timeThe 4

:giveseory circuit thand2

0e VCW

, 4

gygg

dsHHWSm

(2.18) F/m 2

SdsEEC

4

:giveseory circuit th and2

0ILWm

( )20SV

7


s ldHHRP

:conductor theoflossPower

:dielectriclossy theof lossPower

dsEEP

CCc

SH

ldHHP

) tol tangentiais assuming(

2

21

:giveseory circuit th and

,2

Sd dsEEP

c IRP 2

:giveseory circuit th and2

02 2

0

d VGP

s ldHHRR (2.19)/m

(2.20) S/m 20

S

dsEEV

G

ss

CC

R

ldHHI

R

1 where

(2.19) /m 21

20

8


• Table 2.1 Transmission Line Parameters for Some Common LinesCOAX TWO-WIRE PARALLEL PLATE

aa

D

w

db a

D d

dDb

wC

wd

aD

abL

2

2

cosh ln2

1

RRRR

daDabC

sss

2 11

2cosh

ln

1

dw

aDabG

waba

2cosh

ln

2

2

1

9

daDab 2coshln

Terminated Lossless Transmission Linee ated oss ess a s ss o e

Figure 2.4 (p. 58) A transmission line terminated in a load impedance ZL. (modified)

j eV zj ZV

00

0

ZV

eV zj

00

0

ZIV

eV j

LZI

0If ZZL (2.34a) ,)(

:line on theurrent voltage/ctotalThe

0

0 eVeVzV zjzj

00I

0I

0

0

0

V

eV zjL

(2.34b) .)(

0

0

0

0 eZVe

ZVzI zjzj

00

0 ZIV

.)0(

havemust we,0At

000 ZVVVZ

z

L

10

)0( 000 VVIL


VZZZZV

L

L

0

0

00

00

00 V

ZZZZV

L

L

ZZV L

00 (2.35)

: t,coefficien reflection Voltage

(2.35)

: t,coefficien reflection Voltage

00 ZZV L

ZZV L

00

:line on theurrent voltage/c totalThe

(2.35)

:line thealong flowpower average- timeThe

(2.35) 00 ZZV L

eeVzI

eeVzV

zjzj

zjzj

0

0

(2.36b) .)(

(2.36a) ,)(

)Im(2 )()(Re21

2

av

V

AjAAzIzVP

Z

0

0fl iN wavesstanding

( ))(

1 Re21

2

2 2 2

0

0 eeZV

zjzj

matchedZZL

0 0 :reflection No

10

.1 21 2

0

2

0

ZV

11


2 :PowerIncident

2

0in Z

VP

:line on the voltage theof magnitude,)(

0 eeVzV zjzj

:PowerReflected

2

in2

22

0

2

0r

0

PVV

P

Z

1

1)( 2

20

eV

eVzVlj

zj

1 :Power dTransmitte

22

in2

rint

in00

r

PPPP

ZZ

1

1 2

0

0

eV

eVlj

thedefined load, the todelivered ispower available of allnot ,mismatched is load When the

1

1

0min

0max

VV

VV

dB 0

dBlog20RL

:as dBin (RL) lossreturn

as defindebecan VSWR)/ (SWR

0

ratiowaveandingstvoltage

dB01

dB. log20RL

11

SWR

)(

min

max

VV

12

min


Zzl :at load the

dseen towar impedanceinput The

Zin eZZeZZ

eZZeZZZZ ljL

ljL

ljL

ljL

0

0

0

0

0in

ljZlZljZlZZ

L

L

sin cos sin cos

0

00

0)(

:tcoefficien reflection dgeneralize

2

0 eeVl ljlj

ljZZljZZZ

L

L (2.44) tan tan

0

00

dseen towar impedanceinput the

,0)( 0

eeV

l lj

equationimpedancelineontransmissi

:at load the

0

0

in ZeeVlVZ

zl

ljlj

ljlj

(2.43)

11 02

2

00

in

Ze

eeVlI

lj

lj

ljlj

13

( )1 0 2e lj


• Special case: short terminated

Figure 2.5 (p. 60) A transmission line terminated in a short circuit.

(2 45a)sin2)(:line on theurrent voltage/c totalThe

zjVeeVzV zjzj (2.45b) . cos2)(

(2.45a) ,sin2)(

0 0

00

zZVee

ZVzI

zjVeeVzV

zjzj

.tan:impedanceinput The

0i

00

ljZZ

ZZ

Figure 2.6 (p. 61) (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation

14

. tan 0in ljZZ along a short-circuited transmission line.


• Special case: open terminated

Figure 2.7 (p. 61) A transmission line terminated in an open circuitterminated in an open circuit.

(2 46a)cos2)(:line on theurrent voltage/c totalThe

zVeeVzV zjzj (2.45b) . sin2)(

(2.46a) ,cos2)(

0 0

00

zZjVee

ZVzI

zVeeVzV

zjzj

.cot:impedanceinput The

0i

00

ljZZ

ZZ

Figure 2.8 (p. 62) (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ) variation

15

. cot 0in ljZZ along an open-circuited transmission line.


ZZnnl

L (2.47) . :) 3, 2, ,1( 2/ If

in

(2.49).

:long infinitely line loading The

01 ZZ

ZZ

nnl

(2.48).

:) 3, 2, ,1( 2/4/ If20

i

(2.50a) ,0 ,)(

(2.49) .

0

01

zeeVzV

ZZzjzj

rtransformewavequarterZ

ZL

(2.48) . in

(2.51).21

(2.50b) .0 ,)(

1

0

1

ZT

zTeVzV zj

:

(2.51) .101

lossisertionZZ

T

Zin

dB log20 TIL

Figure 2.9 (p. 63) Reflection and transmission at the junction of two transmission lines with different characteristic impedances

16

different characteristic impedances.

Smith ChartS t C a t

Figure 2.10 (p. 65)The Smith chart

17

The Smith chart.


ofplot polar a:chartSmith The 1

: load a with )( line lossless The 0

jL

L

ezZZ

1)(radius:magnitude

e j

. where1

0LL

j

L

L

ZZz

ez

oo )180180( :angle

)(g

11

j

j

L ee

z

tcoefficien reflection :partsimaginary

and real of in term and Express Lz

e)(admittanc impedance normalized

LLL

ir

xjrzj

0 :

ZZzimpedancenormalized

.11

ir

irLL j

jxjr

18


ir

irLr 22

22

(2.55a) 11

Constant resistance (rL) circles

ixL

ir

iL

ir

x22 (2.55b)

12

xL

310r

+xL

0 1 3 rL

Lr2

22

1

:(2.55) grearrangin

CONSTANT RESISTANCE LINES IN THE zL=rL+jxL

PLANE

Li

L

Lr rr

r

22

2

11

11

1

Constant reactance (xL) circles

0 51i

L L j LPLANE

xL

lj

LLir

e

xx 2

2

1

.111

xL0

+xL

0.5 3

r

1

3

-1 rL

lj

j

eeZZ 20in 1

1

xL

-0.5-1

-3

r

-3

1 L

CONSTANT REACTANCE LINES IN THE zL=rL+jxL

19

LINES IN THE zL rL+jxLPLANE

PLANE


The constant r and the constant x loci form two families of orthogonal circles in the chartin the chart.

The constant r and constant x circles all pass through the point (r = 1, i = 0).

The upper half of the diagram represents +jx.

The lower half of the diagram represents jx.

For admittance the constant r circles become constant g circles and thebecome constant g circles, and the constant x circles become constant susceptance b circles.

The distance once around the Smith The distance once around the Smith chart is one-half wavelength ( / 2)

20

Smith Chart - ExampleS t C a t a p e

Locate in Smith Chart with following normalized impedancesimpedances

1. z1=1+j1

2. z2=0.4+j0.5

3 3 j33. z3=3-j3

4. z4=0.2-j0.6

6 z1 7 z

5. z5=0

6. z6=

7. z7=1

21


0 in

load impedance: 40 70 100 , 0.3 , find ?

jZ l Z

0

: 0.4 0.7L L

solutionz Z Z j

0.59 SWR 3.87

RL 4.6 dB WTG: 0.106

in 0 in

0.3 : 0.406 0.365 0.611 36.5 61.1

jZ Z z j

Figure 2.11 (p. 67)Smith chart for Example 2 2

22

Smith chart for Example 2.2.

Smith Chart – Z vs Y S t C a t vs

d ittli d/1liit il/4 admittancenormalized/1 :lineion transmisslong /4 in LL zzz

Z Smith chart Y Smith chart

23

ZY Smith chart

ZY Smith chart

24


/4 long transmission line:

in

/4 long transmission line: 1/

normalized admittanceLz z

0 in

normalized admittanceload impedance: 100 50

50 , 0.15 , find ?j

Z l Y

0 in

0

, ,:

2 1L L

solutionz Z Z j 0

0.4 0.2

0 008 0 004 S

L L

L

L

jy j

yY y Y j

00

0.008 0.004 S

WTG: 0

L LY y Y jZ

.214

0

0.15 : 0.364 0.61 0.66

0.0122 0.0132 S

y jyY yY j

25

00

0.0122 0.0132 SY yY jZ

Slotted LineS otted e

4 2cm2 2cmcm20z 4.2cm2.2cm,cm,2.0z

4 72cm2 72cmcm720z 4.72cm2.72cm,cm,72.0z

Figure 2.13 An X-band waveguide slotted line.

2/i i 4

3701 4872224cm 4

2/every repeat minima

l

1996.00126.02.0

4.8648.14

4

4.86

o

o

je j

2.015115.1

1SWR1-SWR

37.0cm1.4872.22.4min

l

7.193.4711

0 jZZ

j

L

26

15.11SWR

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

Figure 2.16 (p. 73)The quarter-wave matching transformer.

L ljZRZZ 1 tan

L

L

lZfZ

ljRZjZZ

21

1

11in

)2/(at

tan

L

L

RZZ

lR

fZ

01

10in )2/(at

Figure 2.18 (p. 75) Multiple reflection analysis of the quarter wave transformer

27

L01 analysis of the quarter-wave transformer.

The Quarter-Wave Transformere Qua te Wave a s o eExample 2.5

• Consider a load resistance RL = 100, to be matched to a 50 line with a quarter-wave transformer. Find the characteristic impedance of the matching section and plot the magnitude of the reflection coefficientmatching section and plot the magnitude of the reflection coefficient versus normalized frequency, f / f0, where f0 is the frequency at which the line is / 4 long.

• Solution: 71.701005001 LRZZ

0in

0in

ZZZZ

ljZZljZZZZ

L

L

tan tan

1

11in

0in

0

242

42

ff

fv

vf

l p

00 244 ffvp

Figure 2.17 (p. 74)Reflection coefficient versus normalized frequency f th t t f f E l 2 5

28

for the quarter-wave transformer of Example 2.5.

Generator and Load MismatchesGe e ato a d oad s atc es

Figure 2.19 (p. 77) Transmission line circuit for mismatched load and generator.

021:LinetoMatchedLoad

ZVP

2

:load the todeliveredpower The 22

0

:Line Loaded toMatchedGenerator

2 ggg XRZVP

in

2

in

in2

inin1Re

21Re

21

gg ZZZ

ZVIVP

in

in 0g

g

ZZZZ

2in2

in

in2

21

ggg XXRR

RV

22

2

421

gg

gg XR

RVP

29

gg

Generator and Load MismatchesGe e ato a d oad s atc es

maximize tofixed, :Matching ConjugatePZg , inin

gg XXRR

00 2

in2in

2

in

XXRRRP

gg

g

11

or 2

in g

VP

ZZ

00

ininin

XXXXP

gfrompower availablemaxmum

42max in,

g

g RVP

generator thep

30

Lossy Transmission Linesossy a s ss o es

CjGLjR

j

GR

:Line lessDistortion The

GRjLCj

1

:Line Loss-Low The

LCR

CL

constant

GRjLCj

CLjLCj

1

1

LCvLCL

p 1/

GZRLGCR

CLjLCj

11

21

LZ

0

constant

LC

GZZR

CLG

LCR

002

121

C0

CLZ

LC

0

31

C


22

0

2

0in 1

2 Re

21 el

ZV

lIlVP l

Figure 2 20 (p 82) A lossy transmission 2

2

0

0

1 2

00 Re21

22

ZV

IVP

Z

L

Figure 2.20 (p. 82) A lossy transmission line terminated in the impedance ZL.

:LineLossy Terminated The

22 2

2

0

0

11

22

eeV

PPP

Z

ll

VI

eeVzV

zz

zz

)(

)(

0

0

0inloss

for Methodon Perturbati The

112

eeZ

PPP L

el

eeZ

zI

l

zz

)(

2

0

0

20

20

)(22 )(

zPePPP

ePzP

zl

z

lZZlZZZZ

L

L

tanh tanh

0

00in

0

20

)(2

)(22

PzP

zPzP

zPePz

P

ll

l

32

02)(2 PzP


2W/m

Rule inductance lIncrementa Wheeler The

ldHRP ts

l

0

00

22 ss

ldZdZZ

2

20

2

W/m 2

ldHI

L

ldHP

C ts

C tl

00

2

s

ZdZldZdZ

2

2

22

2

LILILIRP

I

sl

C

0

00

0

4 s

c

ZdRldZd

ZZZα

200

22

2

ZL

PPα

P

lc

ssl

0

0

:surfaceconductorofroughness 2

s

ldZd

ZR

0

00

22

LvLLZ

ZP

p

c

21 4.1tan21

:surfaceconductor ofroughness

cc αα

0

0

ZZα

LCC

c

p

s

cc

33

0Z

transmission line theorytransmission line theory...

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