13 transmission line theory

7/31/2019 13 Transmission Line Theory

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Transmission Line Theory


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Transmission Line Equations (1)

At low frequencies:

connect two components

At high frequencies:

Cannot simply use a wire to

connect two components

m105Hz60 6

0.5m103

MHz6006

8

f


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RzLz

Gz

Differential Length

CzSource Load

z z + zz

R: Series resistance per unit length ( /m)

G: Shunt conductance per unit length (S/m)

L: Series inductance per unit length (H/m)

C: Shunt capacitance per unit length (F/m)

Lossless Line: R = G = 0


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),( tzi ),( tzzi

Gz

z

Lz

Cz

+ +),( tzv ),( tzzv Source Load

z z + zz

Differential Length

Cannot apply Kirchhoff's Laws to the whole line

Can apply Kirchhoff's Laws to the differential length

KVL: tzzi ),(

tzziLtzzRi

tzzvtzv

t

),(

),(),(),(

,,,

t

tzi

LtzRiz

tzv

z

),(

),(

),(

0


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),( tzi ),( tzzi

Gz

z

Lz

Cz

+ +),( tzv ),( tzzv Source Load

z z + zz

Differential Length

tzzit

tzvzCzGtzvtzi

),(

),(),(),(:KCL

t

tzvCtzGv

z

tzzitzi

),(),(

),(),(

t

tzv

CtzGvz

tzi

z

),(

),(

),(

0


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t

tziLtzRi

tzv)1(

),(),(

),(

ttzvCtzGv

ztzi )2(),(),(),(

t

tzi

L

tzv

)3(

),(),(

neoss ess

t

tzvC

z

tzi)4(

),(),(

tzvtzv

ttzvLC

ztz

tL

ztzv

1,1,

,),(,(3)From

22

22

LCv

tvzp

p

222

Wave Equation


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Phasor Form Representation for

)cos( tu

Re eAe

2

j

AeU

)cos(

2t

kztu

jt

Re

jkzAeU

tje

jkzAe

U


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Sinusoidal Excitation of

),(),(

),(

tziLtzRi

tzv

),(),(

),(

t

tzvCtzGv

z

tzi

)( IZLIRIzdV

)()()()(

zVYzCVjzGVzdI

dz

Len ther UnitIm edanceSeries LjRZ

Lengthper UnitAdmittnaceShunt CjGY


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)(

zdV

)()()()()(

zVCjGzCVjzGvzdI

dz

From

2

z

)(

0)())((

2222

2

zVd

zVCjGLjRdz

)(

22

2

zVd

zdz

2

dz


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kzkz

zz eVeVzV

00 )(

ee

00

jkzjkzeVVeVV

00

Define

WaveBackwardWaveForward


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:osolution tabovetheSubstitute

)()()( zILjRd

zdV

)(00 zz

e

V

e

V

zI

II

where0

0

0

00

zze

ZVIe

ZVI

CjGLjRZ


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Characteristic Impedance

0 1LjRZ

If R=G=0 lossless case

CZ 0


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Propagation Constant

))((22 CjGLjRk

rwavenumbe:constant,npropagatio: k-

rad/mconstant,phase:nep/mconstant,nattenuatio:

zzzkz .


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Reflection Coefficient (1)

)(z

Source Load

Reflection Coefficient (or: Voltage Reflection Coefficient) at

zz0

any po nt a ong t e transm ss on ne s e ne as:

)()(

zVz

)()()( zVzVzV z jkz

eVzV 0)(

))(1)(( zzV jkz

eVzV

0)(


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Reflection Coefficient (2)

)( 1z )( 2z

Source Loadd

z0 1z 2z

11 201)( jkzjkzeVzV

z

1

01

1)(

jkzeVzV

22 2

2

2

2 )()0()0()(jkzjkz

ezez

jkdzzjkezezz

2

2

)(2

21 )()()(12

djdeez

22

2 )(


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Input Impedance (1)

)(zZin

SourceLoad

)(z

zz0

Defined as: )(zV )(zI

in

)](1[)()()( zVzVzVzV

)](1[)()()(0

zZVzIzIzI

0

0

0 )(

)(

)(or)(1

)(1

)( ZzZ

ZzZ

zz

z

ZzZin

in

in


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Input Impedance (2)

)( 1zZin )( 2zZin

SourceLoad

1 2

d

z0 1z 2z

From 01 )( ZzZ in 01

1)( ZzZ in

02

022

)(

)()(

ZzZ

ZzZz

in

in

dezz

2

21 )()(

)tanh()()( 0201

dZzZZzZ inin

20 in


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Input Impedance (3)

inZ

Sourced LZ

z0

Im edance Transformation Formula:

d 0

)tanh(00

dZZZZ Lin

0 L

)tanh(0 dZZin Note: as d

)tanh(00

dZZ inL


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Input Impedance (4)

inin ZY /1

Sourced LZ

z0

z

d 0

)(10

zY

ZY

in

in

)tanh()tanh(

0

00

dYYdYYYY

L

Lin


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Lossless Terminated Transmission Line (1)

inZ

Sourced LZ

z0

zd 0

real,,,0 0C

Zjk

02 ZZindj

0ZZinL constantL

tan dZZ

)tan(00

djZZ Lin


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inZ

Sourced LZ

z0

)tan( dZZ

d 0

)tan(00

djZZ Lin

Discussions:

2)12(or,

4When(1)

2

Z

ndnd

.0,if

;,

LL

inL

L

in

ZZ

ZZZ

Z


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inZ

Sourced LZ

z0

tan dZZ

d 0

)tan(00

djZZZZ

L

in

Lin ZZndn

d ,or,When(2)

coto.c.When4

)tan((s.c.)0When(3) 0

dZZZ

djZZZ inL

0,When(5)00

ZZZZinL


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Average Power on Lossless

Transmission Line)(zI

)()()( zVzVzV

SourceLoad

)(zV

00

)()()( Z

zV

Z

zVzI

zz0

)])(Re[(2

1

)]()(Re[2

)(

**

*

Z

V

Z

VVV

zIzVzP

]||||

Re[2

1

22

0

**

0

2

0

2

Z

VVVV

Z

V

Z

V

(constant)22

0

0

0

0 PPZZ


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Chain Matrix of Transmission Line (1)

)0(I )(dI

SourceLoad

)0(V )(dV

zd0

00)0( VVV

jkdjkd

eVeVdV

00)(

0

0

0

0)0(Z

V

Z

VI

0

0

0

0)(Z

eV

Z

eVdI

jkdjkd

)0(

)0(

cossin

sincos

)(

)( 0

I

V

kdkd

j

kdjZkd

dI

dV

0

Chain Matrix A


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Chain Matrix of Transmission Line (2)

Z01k1

Z03k3

Z02k2

Z0,N-2kN-2

Z0,N-1kN-1

Z0NkN

d3d

2 dN-2d

N-1

2321tot AAAAA NN

13 transmission line theory

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