Transmission Lines - FEUP

Faculdade de Engenharia

Transmission Lines

ELECTROMAGNETIC ENGINEERINGMAP – TELE 2008/2009

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Faculdade de EngenhariaTransmission Lines

transmission lines à waveguides supporting TEM waves

parallel-plate waveguides

coaxial waveguides

two-wire waveguides

most common types

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Faculdade de EngenhariaTransmission Lines

general transmission line equations

time-harmonic solutions

finite transmission lines

voltage, current and impedance along the line

transmission lines in circuits

Smith chart

impedance matching

λ/4 transformer

reactive elements

single-stub

double-stub

transients

next week

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Faculdade de EngenhariaTEM waves in parallel-plate waveguides

η−=

βγ j= xeE

inside the guide:

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Faculdade de EngenhariaVoltage between the plates

voltage between the plates: zjebE β−−= 0

∫ ⋅−=−2

ldEVVrr

rinside the guide:

zjy eEE β−= 0

( ) ∫−=b

ydyEzV0

voltage à

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Faculdade de EngenhariaCurrent density on the plates

current density on the plates:

upper plate:

yan ˆˆ −=

inside the guide:

lower plate:

yan ˆˆ = 1

( )21ˆ HHaJ ns

rrr−×=

02 =Hr

xeEH zj ˆ01

η−−=

r( ) zeEbyJ zj

s ˆ0 β

η−−==

02 =Hr

xeEH zj ˆ01

η−−=

r ( ) zeEyJ zjs ˆ0 0 β

η−==

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Faculdade de EngenhariaCurrent on the plates

upper plate current:

∫ ⋅=A

sdJIrr

zjeEW β

η−−= 0

rinside the guide:

( ) zeEbyJ zjs ˆ0 β

η−−==

currentà

( ) ∫ ⋅=W

s zdxJzI ˆr

( ) zeE

byJ zjs ˆ0 β

η−−==

( ) zeE

yJ zjs ˆ0 0 β

η−==

current density:

lower plate current: zjeEW β

η−+= 0( ) ∫ ⋅=

Ws zdxJzI ˆ

EE 0809Lines 8

Faculdade de EngenhariaLossless transmission line equations

( ) zjebEzV β−−= 0

( ) zjeEWzI β

η−−= 0 zj

WjdzdI

ebEjdzdV

εµη

εµωβ

−= ( )H/mW

( )C/mbW

= VCjdzdI

ILjdzdV

eqs. for V e I in a losslesstransmission line

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Faculdade de EngenhariaEquivalent circuit of a lossless transmission line

differential length ∆z of a transmission line:

i(z+∆z,t)i(z,t)

--v(z,t) v(z+∆z,t)

tzizLvL ∂

∂∆=

tzzvzCiC ∂

∆+∂∆=

( ) ( ) ( )

( ) ( ) ( ) 0,,

=∆++∂

∆+∂∆+−

∆++∂

∂∆=

tzzvzCtzi

tzizLtzv

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Faculdade de EngenhariaEquivalent circuit of a lossless transmission line

i(z,t)

-v(z,t)

( ) ( ) ( )

( ) ( ) ( ) 0,,

=∆++∂

∆+∂∆+−

∆++∂

∂∆=

tzzvzCtzi

tzizLtzv

( ) ( )

( ) ( )t

∂∂=

∂∂−

∂∂=

∂∂−

,, ( ) ( )

( ) ( )zVCjdz

zILjdz

0lim →∆z

phasor notation0

same as before

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Faculdade de EngenhariaEquivalent circuit of a lossy transmission line

differential length ∆z of a transmission line:

zR∆ zL∆

zG∆zC∆

i(z+∆z,t)i(z,t)

--v(z,t) v(z+∆z,t)

( )( )

tzizRv

∂∂

( )( )

tzzvzGi

∂∆+∂

∆+∆=,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 0,,

=∆++∂

∆+∂∆+∆+∆+−

∆++∂

∂∆+∆=

tzzvzCtzzvzGtzi

tzizLtzizRtzv( ) ( ) ( )

( ) ( ) ( )t

tzvCtzvG

LtziRz

∂∂

∂−

∂∂

∂−

,0lim →∆z

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Faculdade de EngenhariaGeneral transmission line equations

( ) ( ) ( )

( ) ( ) ( )t

tzvCtzvG

LtziRz

∂∂

∂−

∂∂

∂−

( ) ( ) ( )

( ) ( ) ( )zVCjGdz

zILjRdz

general solution

i(z,t)

-v(z,t)

( )( )CjGLjR ωωγ ++=

( ) ( )

( ) ( )zIdz

βαγ j+= ( )( ) zz

eIeIzI

eVeVzVγγ

−−+

propagation constant

attenuation constant

phase constant

phasor notation

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( ) zz eVeVzV γγ −−+ += 00

Attenuation and phase constants

−( )zV

( ) ( ) tjzz eeVeVtzv ωγγ −−+ += 00Re,

( ) ( ) ztjzztjz eeVeeV βωαβωα +−−−+ += 00Re

−+00 and VVif are real

( ) ( )zteVzteV zz βωβω αα ++−= −−+ coscos 00

atenuation phase

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( ) zz eVeVzV γγ −−+ += 00

Voltage and current in transmission line

( ) ( ) ( )

( ) ( ) ( )zVCjGdz

zILjRdz

( ) zz eIeIzI γγ −−+ += 00

γωLjR

0 only 2 constantsare required

4 constants required to define voltage and current

+= 00 V

−−

+−= 00 V

−−00 , IV

++00 , IV

−( )zV

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Faculdade de EngenhariaCharacteristic impedance

Characteristic impedanceà

( )Ω++

=CjGLjR

infinite lineà no reflections

ratio between voltage and current for an infinite length transmission line

( ) zeIzI γ−+= 0

( ) zeVzV γ−+= 0

γωLjR +

characteristic impedance

( )( )CjGLjR ωωγ ++=

−−00 , IV

++00 , IV

−( )zV

( ) ( )( )zIzV

zZ = ( ) 0ZzZ =

note: in general 00

=−= −

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Faculdade de EngenhariaSummary

Propagation constantà ( )( ) ( )1m−++=+= CjGLjRj ωωβαγ

( )Ω++

=CjGLjR

Propagation velocityà

Characteristic impedanceà

( )1ms −=βω

Wavelengthà ( )m2βπ

General case

•frequency dependent attenuation

•frequency dependent velocity

SIGNAL DISTORTION

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Faculdade de EngenhariaTransmission lines – special cases

LCjωγ =

Z =0 LCv

Lossless lines

NO DISTORTION

0== GR ( )( )CjGLjRj ωωβαγ ++=+=

CjGLjR

=0 βω

LCωβ

Distortionless linesCG

LjR ωγ +=

Z =0LC

•zero or constant attenuation•constant velocity•constant and real characteristic impedance

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Faculdade de EngenhariaTransmission-line parameters

In turn, these parameters depend on the line geometry and on the materials thatconstitute the line

Letσ à dielectric conductivityσC à conductor conductiviityε à electric permitivitty of the dielectricµ à magnetic permeability of the dielectricµC à magnetic permeability of the conductor

The behaviour of a transmission line depends on the operating frequency andon parameters R, L, G and C

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Faculdade de EngenhariaTransmission-line parameters

coaxial two-wire conductor over ground parallelplate

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Faculdade de EngenhariaFinite transmission lines

LLL IZV =

( ) ( ) ( )[ ]( ) ( ) ( )[ ]z

eZZeZZIZ

eZZeZZIzV

−−+=

−++=

( ) zozo

eVeVzV

−−

−−+

−( )zV

zl−0

=−=−

( )( ) zz

eIeIzI

eVeVzVγγ

−−+

+= ( )

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Faculdade de EngenhariaImpedance along the transmission line

LZ±gV

−( )zV

( ) ( )( )

( ) ( )( ) ( ) z

eZZeZZeZZeZZ

zZ γγ

−−+−++

== −

( ) ( ) ( )[ ]( ) ( ) ( )[ ]z

eZZeZZIZ

eZZeZZIzV

−−+=

−++=

( ) ( ) ( )( ) ( ) L

ZeeZeeZeeZee

ZzZ γγγγ

γγγγ

−−+−−+

= −−

−−

( ) ( )( )zZZ

zZZZzZ

tanhtanh

00 −

( ) ( )( )'tanh

'tanh'

00 zZZ

zZZZzZ

=)tanh(

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Faculdade de EngenhariaInput impedance – lossless transmission line

LZ±gV

−( )zV

lossless line βγ j=( ) ( )xjjx tantanh =

( ) ( )( )'tan

00 zjZZ

zjZZZzZ

length l

( )( )ljZZ

ljZZZZ

Lin β

βtantan

( ) ( )( )'tanh

'tanh'

00 zZZ

zZZZzZ

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Faculdade de EngenhariaInput impedance of lossless transmission lines – special cases

lossless transmission line of length l ( )( )ljZZ

ljZZZZ

Lin β

βtantan

0ZZ L = 0ZZ in =

∞=LZ ( )ljZZ in βcotg0−=

nl = Lin ZZ =

0=LZ ( )lanjZZ in βt0=

−= nlL

always imaginary

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Faculdade de EngenhariaReflection coefficient at the load

Reflection coefficient (voltage)à ratio between reflected and incident voltages

( )( ) +

at the load:

−=Γ

Special cases:

0ZZ L = 0=ΓL

∞=LZ 1=ΓL

0=LZ 1−=ΓL

no reflections MATCHED LINE

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Faculdade de EngenhariaReflection coefficient at the load

−=Γ

Notes:

1. For current

2. Most often, is complex àLΓ ΓΓ=Γ θjLL e||

IΓ−=−===Γ +

−=Γ

( )( ) LL

jxrjxr

+++−

LL jxr +=

1||, ≤ΓL

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Faculdade de EngenhariaReflection coefficient along the line

( )( ) +

refLat the load: ΓΓ=

=Γ θjL

along the line:( )( )

ref eeVeV

zVz γ

)( Γ===Γ −+

zz −='

'2)'( zLez γ−Γ=Γ

lossless line: βγ j= ( )'2)'( zjL ez βθ −ΓΓ=Γ absolute value is constant

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Faculdade de EngenhariaVoltage along the line

( ) zjzj eVeVzV ββ −−+ += 00

( ) ( ) ( )zjzjzj eeVeVVzV βββ −−−−+ ++−= 000

( ) ( ) ( )zVeVVzV zj ββ cos2 000−−−+ +−=

cosjxjx ee

propagating wave

stationary wave

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Faculdade de EngenhariaNote – propagating and stationary waves

•let ( ) zjAezV β−=•let ( ) ( ) ( )ztAAeeAetzv ztjtjzj βωβωωβ −=== −− cosReRe,

zpropagating wave

( ) ( )zAzV βcos=•let ( ) ( ) ( ) ( )tzAezAtzv tj ωββ ω coscoscosRe, ==

stationary wavez

nodes( v=0 for every t )

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Faculdade de EngenhariaVoltage along the line

( ) zjzj eVeVzV ββ −−+ += 00

propagating + stationary waves

( ) ( )zjL

zj eeVzV ββ 20 1 Γ+= −+

( ) ( )( ) ( )( )( )'2cos21

'2sin'2cos1'

βθβθ

−Γ+Γ+=

−Γ+−Γ+=

periodic termperiod=λ/2

( ) ( )( )( )'2'

eeVzVβθβ

ΓΓ+=

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Faculdade de EngenhariaVoltage along the line - example

( ) ( )'2cos21'2

0 zVzV LL βθ −Γ+Γ+= Γ+

( )m2m1

πλβ

0123456789100

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Faculdade de EngenhariaVoltage maxima and minima

•voltage minima: ( ) 1'2cos −=−Γ zβθ

( ) ( )'2cos21'2

•location: πβθ nzM 22 / −=−Γ ( )Γ+= θπβ

nzM 221/

z 0'≥

integer

•location: ( )πβθ 122 / +−=−Γ nzm ( )[ ]Γ++= θπβ

1221/ nzm

z 0'≥

integer

•value: LLVV Γ−Γ+= + 212

0min( )LVV Γ−= + 10min

•value: LLMAXVV Γ+Γ+= + 21

20 ( )LMAX

VV Γ+= + 10

•voltage maxima: ( ) 1'2cos +=−Γ zβθ

EE 0809Lines 32

Faculdade de EngenhariaVoltage along the line - example

( ) ( )'2cos21'2

( )m2m1

πλβ

0123456789100

( ) 5.110 =Γ+= +LMAX

( ) 5.010min=Γ−= +

/ ππ += nzM

85/ π

π += nzm

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Faculdade de EngenhariaSWR

SWR (Voltage Standing Wave Ratio)à ratio between voltage maxima and minima

( )( )L

LSWRΓ−

=ΓSWRSWR

Note: 1≥SWR

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Faculdade de EngenhariaSWR – particular cases

−=Γ

LSWRΓ−

=ΓSWRSWR

Particular cases:

0ZZ L = 0=ΓL minVV MAX =

no reflections

no stationary wave

1=SWR 0=ΓL

matched line 1=SWR

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Faculdade de EngenhariaSWR – particular cases

−=Γ

LSWRΓ−

=ΓSWRSWR

Particular cases:

∞=LZ 1=ΓL

0=LZ 1−=ΓL ∞=SWR

( ) ++ =Γ+= 00 21 VVV LMAX

( ) 010min=Γ−= +

∞=SWR

+= 02VVMAX

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Faculdade de EngenhariaCurrent along the line

( ) zjzj eIeIzI ββ −−+ += 00

propagating + stationary waves

( ) ( )zjL

zj eeZV

zI ββ 2

0 1 Γ−= −+

( ) ( )( ) ( )( )

( )'2cos21

'2sin'2cos1'

βθβθ

−Γ−Γ+=

−Γ−+−Γ−=

periodic termperiod=λ/2

( ) ( )

( )( )'2'

βθβ

ΓΓ−=

Γ−=

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Faculdade de EngenhariaCurrent maxima and minima

•current minima: ( ) 1'2cos =−Γ zβθ

( ) ( )'2cos21' 2

VzI LL βθ −Γ−Γ+= Γ

•location: ( )πβθ 12'2 +−=−Γ nz ( )[ ]Γ++= θπβ

z 0'≥

integer

•location: πβθ nz 2'2 −=−Γ ( )Γ+= θπβ

nz 221

z 0'≥

integer

•value: LLZ

VI Γ−Γ+=

min( )LZ

VI Γ−=

•value: LLMAX Z

VI Γ+Γ+=

0 ( )LMAX Z

VI Γ+=

•current maxima: ( ) 1'2cos −=−Γ zβθ

EE 0809Lines 38

Faculdade de EngenhariaVoltage and current – maxima and minima location

( ) 1'2cos =−Γ zβθ

( ) ( )'2cos21' 2

( )[ ]Γ++= θπβ

1221/ nzm

z 0'≥

integer

( )Γ+= θπβ

nzM 221/

z 0'≥

integer

( ) 1'2cos −=−Γ zβθ

( ) ( )'2cos21' 20 zVzV LL βθ −Γ+Γ+= Γ+

máximos de tensãoe

mínimos de corrente

voltage maximaAND

current minima

voltage minimaAND

current maxima

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Faculdade de EngenhariaTransmission lines in circuits

−inV

( )( )lzII

inging

−==−==

( ) zozo

eVeVzV

−−

−−+

zl−+− Γ= 00 VV L

[ ][ ]l

Γ−=

( ) ( )[ ]lL

lg eZeZe

V γγγ 20

0 11 −−+

Γ++Γ−=

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−inV

( ) ( )[ ]lL

lg eZeZe

V γγγ 20

0 11 −−+

Γ++Γ−=

( )[ ]lLgg

00 −

Γ−++=

−=Γ (reflection coefficient at the source)

[ ]lLg

ΓΓ−+=

ΓΓ−Γ−

ΓΓ−Γ+

−−

voltage and current as functions of

LZload:

source:

lZ ,,0 γ

gg ZV ,

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−inV

ΓΓ−Γ−

ΓΓ−Γ+

−−

( ) ( )( ) 122

0 11−−−

ΓΓ−Γ++

lg eee

eVZzV γγγ

( ) ( ) ( ) ( )

+ΓΓΓ+ΓΓ+ΓΓΓ+ΓΓ+Γ+

+= −−−−−−−

lg eeeeeeeeee

eVZ γγγγγγγγγγγ

222222

( ) ( )

+ΓΓ+ΓΓ+Γ+

+= −−−

0 1 lLg

lg eeee

eVZ γγγγγ

L++++=−

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−inV

( ) ( ) ( ) ( ) ( )

+ΓΓΓ+ΓΓ+ΓΓΓ+ΓΓ+Γ+

+= −−−−−−−

lg eeeeeeeeee

eVZzV γγγγγγγγγγ

γ222222

( ) L++++++= −−+−−+−−+ zzzzzz eVeVeVeVeVeVzV γγγγγγ332211

−−+ Γ= 12

2 VeV lg

+− Γ= 33 VV L

−−+ Γ= 22

3 VeV lg

+− Γ= 22 VV L

+− Γ= 11 VV L

EE 0809Lines 43

−inV

( ) L++++++= −−+−−+−−+ zzzzzz eVeVeVeVeVeVzV γγγγγγ332211

( ) ( ) ( ) zzzz eVeVeVVVeVVVzV γγγγ −−+−−−−+++ +=+++++++= 00321321 LL

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Faculdade de EngenhariaPower in lossless transmission lines

( ) ( )( )( ) ( )( )'2'

βθβ

Γ−=

(lossless transmission line)

( ) ( ) ( ) ''Re21

' * zIzVzPav =

( ) ( )( ) ( ) ( )( )

Γ−Γ+= −−−+

−+ ΓΓ '2'

0 11Re21

zjav ee

eeVzP βθββθβ

( ) ( )( ) '2'22

LL eeZ

Vβθβθ −−−

ΓΓ −Γ+Γ−=

( ) '2sin21Re2

VLL βθ −Γ+Γ−= Γ

( ) ( ) constant12

0=Γ−=

incident reflected

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Faculdade de EngenhariaPower in transmission lines – general case

( ) ( )( )( ) ( )( )'2'2''

'2'2''0

eeeeZV

eeeeVzV

βθαβα

−−+

Γ−=

Γ+=( ) ( ) ( ) ''Re

' * zIzVzPav =

( ) ( )( ) ( ) ( )( )

Γ−Γ+= −−−−+

−−+ ΓΓ '2'2''

*0'2'2''

0 11Re21

' zjzL

zjzzjzL

zjzav eeee

eeeeVzP βθαβαβθαβα

( ) '2sin21Re2

'2'42'2

0zejee

Lz βθααα −Γ+Γ−= Γ

−−+

( ) ( )'22'2

av eeR

VzP αα −

Γ−= ( ) ( )2

20' LavLav R

VzPP Γ−===

( ) ( )lL

lavinav ee

VlzPP αα 222

' −+

Γ−===

00 RZ =if

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Faculdade de EngenhariaProblem

formulae

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formulae

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formulae

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formulae

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( ) 22

Γ+Γ−

Γ−Γ−=

Load impedance ó reflection coefficient

zwhere

z LL = (normalized load impedance)

00 RZjXRZ LLL

(lossless line)

LLL jxrz +=

LL je Γ+Γ=Γ=Γ Γθ

Γ−Γ+

( )( ) imre

imreLL j

Γ−Γ−Γ+Γ+

EE 0809Lines 51

( ) 22

Γ+Γ−

Γ−Γ−=

Load impedance ó refelction coefficient

−ΓL

Lre rr

( ) ( ) 220

20 Ryyxx =−+−

LLre rrcentered at

circle of radius ( )Lr+11

the reflection coefficients of all ZLwhose real part is rL are in this circle

EE 0809Lines 52

−ΓL

Lre rr

curve does not depend on xL

0=Γim 111

,, =Γ∨+−

=Γ rreL

Llre r

0=Lr 1, −=Γ lre

for any ZL

∞=Lr 1, =Γ lre

open circuit

EE 0809Lines 53

( ) 22

Γ+Γ−

Γ−Γ−=

−Γ+−Γ

LLimre xx

( ) ( ) 220

20 Ryyxx =−+−

circle of radius Lx1

=Γ=Γcentered at

1≤ΓL

the reflection coefficients of all ZLwhose imaginary part is xL are here

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curve does not depend on rL

−Γ+−Γ

LLimre xx

0=Lx infinite radius

symmetrical curves for xL < 0

EE 0809Lines 55

Faculdade de EngenhariaSmith chart

xL constant

rL constant

EE 0809Lines 56

Faculdade de EngenhariaSmith chart

EE 0809Lines 57

Smith chart

•from:

point in chart ( intersection of curves corresponding to rL and xL )

ΓθandLΓ

rL and xL

•from:

EE 0809Lines 58

Faculdade de EngenhariaReflection coefficient along the line

along the line:

zz −='

'2)'( zLez γ−Γ=Γ

lossless line: βγ j= ( )'2)'( zjL ez βθ −ΓΓ=Γ

constant magnitude

phase decreases with z’

toward generator

toward load

( )( ) 0

0)(ZzZZzZ

( )( )

ref eeVeV

zVz γ

)( Γ===Γ −+

Smith chart can be used to obtain from ( )zZ )(zΓ

EE 0809Lines 59

Faculdade de EngenhariaDistances in the Smith chart

in Smith chart the distances are measured as fractions of λ

toward generator

toward load

( )'2)'( zjL ez βθ −ΓΓ=Γ when πβ 2'2 =z

a complete turn (360º)

corresponds to a distance = λ/2

initial position

EE 0809Lines 60

Faculdade de EngenhariaInput impedance

1. draw the point corresponding to the normalized load impedance zL à point P1

2. draw the circle centered at the origin with radius OP1

3. draw the straight line from O to P1

4. draw the straight line from O that corresponds to a rotation of l toward the generator

5. intersection of this line with previous circle à point P2

6. obtain , where zin is read from P2

0ZzZ inin ⋅=

EE 0809Lines 61

Faculdade de EngenhariaAdmittance

( ) ( )( )'tan

00 zjZZ

zjZZZzZ

= ( )( ) LL

jZZjZZ

00 2tan

ππλ

λ( ) Lyz =4λ

º3602 ⇔λ

º1804 ⇔λ

1. draw zL

2. rotate 180º

EE 0809Lines 62

Faculdade de EngenhariaMaxima and minima location

( ) ( )'2cos21'2

( ) ( )'2cos21' 20 zVzV LL βθ −Γ+Γ+= Γ+

( ) 1'2cos =−Γ zβθ à voltage maxima and current minima

à voltage minima and current maxima( ) 1'2cos −=−Γ zβθ

( ) ( )'2' zjL ez βθ −ΓΓ=Γ

voltage maxima where ( ) πnz 2' =Γ∠

voltage minima where ( ) ( )π12' +=Γ∠ nz

EE 0809Lines 63

Faculdade de EngenhariaMaxima and minima location

voltage maxima where ( ) πnz 2' =Γ∠

voltage minima where ( ) ( )π12' +=Γ∠ nz

voltage maxima

voltage minima

1. maxima and minima where input

impedance is real

2. maxima (minima) points are separated

by nλ/2

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