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Transmission lines or T-lines are used to guide propagation of EM waves at high frequencies.

Examples:› Transmitter and antenna› Connections between computers in a network› Interconnects between components of a stereo system› Connection between a cable service provider and aTV set.› Connection between devices on circuit board

Distances between devices are separated by much larger order of wavelength than those in the normal

electrical circuits causing time delay.

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Properties to address:› time delay› reflections› attenuation› distortion

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Types of transmission lines

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The differential segment of the transmission line

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R’ = resistance per unit lengthL’= inductance per unit lengthC’= capacitor per unit lengthG’= conductance per unit length

General transmission lines equations:

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( , ) ( , )( , ) ' '

( , ) ( , )( , ) ' '

v z t i z ti z t R L

z ti z t v z t

v z t G Cz t

Time-harmonic waves on transmission lines

After arranging we have

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( )( ' ') ( )

( )( ' ') ( )

dV zR j L I z

dzdI z

G j C V zdz

22( )( ) 0

( ' ')( ' ') .

d V zV z

dz

R j L G j C j

where

Instantaneous form

Phasor form

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0 0

0 0

( , ) cos( ) cos( )

( , ) cos( ) cos( )

z z

z z

v z t V e t z V e t z

i z t I e t z I e t z

0 0

0 0

( )

( )

z z

z z

V z V e V e

I z I e I e

lossless when R’ = 0 and G’ = 0

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0

' 'j j L C

' 'L C

1

' 'pu

L C

and

low loss when R’ << L’ and G’ << C’

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1/ 2 1/ 2' ' ( ' ')j R j L G j C 1/ 2 1/ 2

' '' ' 1 1

' 'R G

j L Cj L j C

Expanding in binomial series gives1 x2

1 1 ......2 8x x

x for x << 1

Therefore, we get

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1 ' '( ' ' )2 ' '

C LR G

L C

1 ' '1 ( )8 ' 'G R

LCC L

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0 00

0 0

V VZ

I I

or

For lossless line,

0

' '.

' 'R j L

ZG j C

Characteristic impedance Z0 is defined as the the ratio of the traveling voltage wave amplitude to the traveling current wave amplitude.

0

'.'L

ZC

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Power transmitted over a specific distance is calculated.

The instantaneous power in the +z traveling wave at any point along the transmission line can be shown as

The time-averaged power can be shown as

22 20

0

( , ) ( , ) ( , ) cos ( ).zi

VP z t v z t i z t e t z

Z

22 20

0 00

1 1( ) ( , ) cos ( ) .

T Tz

avg i

VP z P z t dt e t z dt

T Z T

220

0

( ) zavg

VP z e

Z

W.

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A convenient way to measure power ratios

Power gain (dB)

Power loss (dB)

1 Np = 8.686 dB

( ) 10log( )out

in

PG dB

P

( ) 10log( )in

out

Pattenuation dB

P dB

dB

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Representation of absolute power levels is the dBm scale

( ) 10log( )1m

PG dB

mW dBm

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a) what fraction of the input power does it reach the output?

b) What fraction of the input power does it reach the midpoint of the line?

c) What is the attenuation constant?

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To satisfy boundary conditions between two dissimilar lines

If the line is lossy, Z0 will be complex.

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The phasor voltage along the line can be shown as

The phasor voltage and current at the load is the sum of incident and reflected values evaluated at z = 0.

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0

0

( )

( )

z j zi i

z j zr r

V z V e e

V z V e e

0 0

0 00 0

0

L i r

i rL i r

V V V

V VI I I

Z

Reflection coefficient

A reflected wave will experience a reduction in amplitude and a phase shift.

Transmission coefficient

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0 0

0 0

rjr LL

i L

V Z Ze

V Z Z

0 0

21 tjL L

Li L

V Ze

V Z Z

21

2

02 20 0,

0 0

20 0,

0

22 0 2

0

1 1 1Re Re2 2 2

( )( )1 1Re Re2 2

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L Lavg i i i

Lavg r r r

L

VV VP V I e e

Z Z

V VP V I e

Z

Ve

Z

2,

,

2,

,

1

avg r

avg i

avg t

avg i

P

P

P

P

W

W

W

The main objective in transmitting power to a load is to configure line/load combination such that there is no reflection, that means.

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0

0

.LZ Z

Incident and reflected waves create “Standing wave”.

Knowing standing waves or the voltage amplitude as a function of position helps determine load and input impedances

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max

min

VVSWR

V

Voltage standing wave ratio

If a load is matched then no reflected wave occurs, the voltage will be the same at every point.

If the load is terminated in short or open circuit, the total voltage form becomes a standing wave.

If the reflected voltage is neither 0 nor 100 percent of the incident voltage then the total voltage will compose of both traveling and standing waves.

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let a load be position at z = 0 and the input wave amplitude is V0,

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0 0

0

0

( )

.

j z j zT

jL

L

V z V e V e

Z Ze

Z Z

where

( )0( ) ( )j z j z

TV z V e e

/ 2 / 2 / 20 ( )j j z j j z jV e e e e e

we can show that

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/ 20 0( ) (1 ) 2 cos( ).

2j z j

TV z V e V e z

traveling wave standing wave

The maximum amplitude occurs when

The minimum amplitude occurs when standing waves become null,

0( ) (1 ).TV z V

0( ) (1 ).TV z V

The minimum voltage amplitude occurs when two phase terms have a phase difference of odd multiples of .

The maximum voltage amplitude occurs when two phase terms are the same or have a phase difference of even multiples of .

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( ) (2 1) ; 0,1,2,...z z m m

min ( (2 1) )4

z m

( ) 2 ; 0,1,2,...z z m m

max ( 2 )4

z m

If = 0, is real and positive

and

Each zmin are separated by multiples of one-half wavelength, the same applies to zmax. The distance between zmin and zmax is a quarter wavelength.

We can show that28

min (2 1)4

z m

,max

,min

1.

1T

T

VVSWR

V

max .2m

z

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