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ENE 429Antenna and Transmission lines Theory

Lecture 4 Transmission lines

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Transmission lines (1) 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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Transmission lines (2) Properties to address:

time delay reflections attenuation distortion

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Distributed-parameter model Types of transmission lines

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Distributed-parameter model The differential segment of the

transmission line

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

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Telegraphist’s equations General transmission lines equations:

( , ) ( , )( , ) ' '

( , ) ( , )( , ) ' '

v z t i z ti z t R L

z ti z t v z t

v z t G Cz t

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Telegraphist’s time-harmonic wave equations Time-harmonic waves on transmission

lines

After arranging we have

( )( ' ') ( )

( )( ' ') ( )

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

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Traveling wave equations for the transmission line Instantaneous form

Phasor form

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

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Lossless transmission line

lossless when R’ = 0 and G’ = 0

0

' 'j j L C

' 'L C

1

' 'pu

L C

and

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Low loss transmission line (1) low loss when R’ << L’ and G’ << C’

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

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Low loss transmission line (2)Therefore, we get

1 ' '( ' ' )2 ' '

C LR G

L C

1 ' '1 ( )8 ' 'G R

LCC L

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Characteristic impedance

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

Power transmission

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

Power ratios on the decibel scale (1)

( ) 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

Power ratios on the decibel scale (2)

( ) 10log( )1m

PG dB

mW dBm

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Ex1 A 12-dB amplifier is in series with a 4-dB attenuator. What is the overall gain of the circuit?

Ex2 If 1 W of power is inserted into a coaxial cable, and 1 W of power is measured 100m down the line, what is the line’s attenuation in dB/m?

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Ex3 A 20 m length of the transmission line is known to produce a 2 dB drop in the power from end to end,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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Wave reflection at discontinuities To satisfy boundary conditions between

two dissimilar lines

If the line is lossy, Z0 will be complex.

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Reflection coefficient at the load (1) 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.

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

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Reflection coefficient at the load (2) Reflection coefficient

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

Transmission coefficient

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

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Power transmission in terms of reflection coefficient

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

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Total power transmission (matched condition) The main objective in transmitting power to a

load is to configure line/load combination such that there is no reflection, that means.

0

0

.LZ Z

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Voltage standing wave ratio 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

max

min

VVSWR

V

Voltage standing wave ratio

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Forms of voltage (1)

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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Forms of voltage (2)

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

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

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Forms of voltage (3)

we can show that

/ 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

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The locations where minimum and maximum voltage amplitudes occur (1) 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 .

( ) (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

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The locations where minimum and maximum voltage amplitudes occur (2) 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 that

min (2 1)4

z m

,max

,min

1.

1T

T

VVSWR

V

max .2m

z

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Ex4 Slotted line measurements yield a VSWR of 5, a 15 cm between successive voltage maximum, and the first maximum is at a distance of 7.5 cm in front of the load. Determine load impedance, assuming Z0 = 50 .