Transmission Line Theory

Dr. M.A.Motawea

Introduction:

In an electronic system, the delivery of power requires the connection of

two wires between the source and the load. At low frequencies, power is

considered to be delivered to the load through the wire.

In the microwave frequency region, power is considered to be in electric

and magnetic fields that are guided from place to place by some physical structure.

Any physical structure that will guide an electromagnetic wave place is called a

Transmission Line.

Transmission lines are used in power distribution (at low frequencies), and

in communications (at high frequencies).

A transmission line consists of two or more parallel conductors used to

connect a source to a load., the source may be a generator, a transmitter, or

an oscillator and the load may be a factory, an antenna, or an oscilloscope,

respectively.

Transmission lines include coaxial cable, a two wire line, a parallel plate or

planar line, a wire above the conducting plane, and a micro-strip line

Cross sectional views of these lines consists of two conductors in figure

,each of these lines consists of two conductors in parallel

Coaxial cables are used in electrical laboratories and in connecting T.V sets

to T.V antennas

Micro-strip lines are important in integrated circuits where metallic strips

connecting electronic elements are deposited on dielectric substrates.

There are different types of modes propagatr between the two conductorsof

transmission line as:

- TE, transverse electric, i.e. 0,0 ZZ HE

- TM, transverse magnetic, i.e. 0,0 ZZ EH

- TEM, transverse electro-magnetic i.e. 0 ZZ EH

- Propagate in Z-direction, ZZ EH ,

Transmission line problems are usually solved using EM field theory and electric

theory, the two major theories on which electrical engineering is based, we use

circuit theory because it is easier to deal with mathematics.

Our analysis of transmission lines will include the derivation of transmission line

equations and characteristic quantities, the use of Smith chart , various practical

applications of transmission lines, and transients on transmission lines.

a b

c d e

Cross sectional view of transmission lines:

a-coaxial line b-two wire line c- planar line d- wire above conducting plane

e- microstrip line

RL

S IRS

Vg

coaxial line

loadgenerator

E

k

E & H fields in the coaxial line

For conductor ( 0,, CCC ), and for dielectric ( ),, .

Transmission Lines Parameters:

We must describe a transmission line in terms of its line parameters.

R[Ω/m] ….conductivity of conductor

L[H/m]….self inductance of wire

G[Ω-1

/m]….dielectric between two conductors

C[F/m]….proximity between two conductors.

We have : LC , ,CG

to generator

i(z,t) i(z+ z, t )

+

v(z,t)v(z+ z,t)

z

R z L z

C zG z

to load

+

- -

z z+ z

I

1- The line parameters R,L,G, and C are not discrete or lumped but distributed as

shown .by this we mean that the parameters are uniformly distributed along the

entire length.

2- For each line, the conductors are characterized by C , C , 0 C , and the

homogeneous dielectric separating the conductors is characterized by , ,

3- R

G1

; R is the ac resistance per unit length of the conductors comprising the line

and G is the conductance per length due to the dielectric medium separating the

conductors.

4- The external inductance per unit length; that is, extLL . The effects of internal

inductance )( RlLin are negligible at high frequencies at which most

communication systems operate.

5- For each line, LC and

C

G

Transmission Line Equations:

As mentioned above , two conductor transmission line supports TEM wave; the

electric and magnetic fields on the line are transverse to the direction of wave

propagation . an important property of TEM waves is that the fields E and H are

uniquely related to voltage V and current I respectively:

dlEV . dlHI .

In view of this , we will use circuit equations V and I in solving the transmission

line problems instead of solving field quantities E and H ( i.e solving Maxwell’s

equations and B.C.), the circuit model is simpler and more convenient.

Let us examine an incremental portion of length z of a two conductor transmission

line . we intend to find an equivalent circuit for this line and drive the line

equations. From the figure of distributed element model of transmission line, we

assume that the wave propagates along +z direction , from the generator to the load.

By applying Kirchoff’s voltage law to the outer loop of figure above , we obtain:

KVL: 0V

),(),(),(*),( tZZVtZIt

ZLtZIZRtZV

),(),(),(),(

tZIt

LtZRIZ

tZVtZZV

By taking the limit of this equation, as 0Z , leads to

),(),(),( tZIt

LtZRItZVZ

(1)

KCL: 0I at the main node of the circuit

ItZZItZI ),(),(

t

tZZVZCtZZZVGtZZItZI

),(),(),(),(

t

tZZVCtZZGV

Z

tZItZZI

),(),(

),(),(

By taking the limit of this equation, as 0Z , leads to

t

tZVCtZGV

Z

tZI

),(),(

),( (2)

If we assume harmonic time dependence so that :

])(Re[),( tj

s eZVtZV ])(Re[),( tj

s eZItZI

Where Vs(Z) and Is(Z) are the phasor forms of V(Z,t) and I(Z,t) respectively

Then eqn (1) becomes

),(),(),( tZIt

LtZRItZVZ

tj

s

tj

s

tj

s eZIt

LeZIReZVZ

)(Re)(Re)(Re

tj

s

tj

s

tj

s eZIt

LeZIReZVZ

)(Re)(Re)(Re

tj

s

tj

s

tj

s eZLIjeZRIeZVZ

)()()(

)()()()()( ZILjRZLIjZRIZVZ

ssss

)()()( ZILjRZVZ

ss

(3)

Also,

)()()( ZVCjGZIdZ

dss (4)

By taking d/dt of 3: )()()(2

2

ZIdZ

dLjRZV

dZ

dss

Substitute in 4: )())(()(2

2

ZVCjGLjRZVdZ

dss

0)())(()(2

2

ZVCjGLjRZVdZ

dss

0)()( 2

2

2

ZVZVdZ

dss Wave eq

n for voltage (5)

Also,

0)()( 2

2

2

ZIZIdZ

dss Wave eq

n for current (6)

These are wave equations for voltage and current similar in form to the wave

equations obtained for plane waves in previous chapter.

jCjGLjR ))(( (7)

is the propagation constant.

is attenuation factor, [Np/m, dB/m]. is phase constant [rad/sec].

2 is wavelength, [m].

fu is wave velocity, [m/s].

Solution of wave equation:

The solutions of the linear homogeneous differential equations, 5 and 6 are:

ZZ

s eVeVZV 00)( (8) and ZZ

s eIeIZI 00)( (9)

+z -z +z -z

Where :

,, 00

VV

00 , II are wave amplitudes; + and – sign respectively denote wave

traveling along +Z and –Z directions, as is also indicated by the arrows.

Thus we obtain the instantaneous expression for voltage as :

)cos()cos(])(Re[),( 00 zteVzteVeZVtzV ZZtj

s (10)

We define the C/C impedance Z0 of the line as the ratio of positively traveling

voltage wave to current wave at any point on the line. Z0 is analogous to η

(intrinsic impedance of the medium of wave propagation).

By substituting eqs, (8) and (9) into eqs. (5) , (6) and equating coefficients of terms

Zeand

Ze as:

We have:

ZZ

s eVeVZV 00)( , )()()( ZILjRZVZ

ss

ZZZZ eILjReILjReVeV 0000 )()(

By equating coefficient of the exponential , then:

00 )( ILjRV ,

)(

0

0 LjR

I

V

(11)

00 )( ILjRV ,

)(

0

0 LjR

I

V

(12)

CjG

LjR

CjGLjR

LjR

I

V

I

VZ

))((

)(

0

0

0

00 (13)

CjG

LjRZ

0 C/C impedance of the line

We have also:

ZZ

s eIeIZI 00)( )()()( ZVCjGZIdZ

dss

))(( 0000

ZZZZ eVeVCjGeIeI

By equating coft of the exponential , then:

00 )( VCjGI CjGI

V

0

0

(13)

00 )( VCjGI CjGI

V

0

0

(14)

CjG

LjR

CjG

CjGLjR

CjGI

V

I

VZ

))((

0

0

0

0

0 (15)

000 jXRCjG

LjRZ

Where R0 [Ω] is real part of Z0 , X0 [Ω] is imaginary part of Z0 . The propagation

constant and the C/C impedance Z0 are important properties of the line because

they both depend on the line parameters R,l,G, and C and the frequency of

operation. the reciprocal of z0 is the C/c admittance Y0, that is Y0 = 1/Z0.

We may now consider two special cases of lossless transmission line and

distortion-less line.

A- Lossless line:

A transmission line is said to be lossless if the conductors of the line are

perfect ( C ) and the dielectric medium separating them is lossless ( )0 . For

such line : GR 0 necessary condition for a line to be lossless.

Then: jCjGLjR ))((

jCjLj ))((

0 , LC ,

f

LCu

1

Also, C

LjXR

CjG

LjRZ

000

00 X , C

LRZ 00

It is desirable for power transmission,

B- Distortionless Line :

A signal normally consists of a band of frequencies; wave amplitudes of

different frequency components will be attenuated differently in a lossy line as

α is frequency dependent. This results in distortion. A distortion-less line is one

in which the attenuation constant α is frequency independent while the phase

constant β is linearly dependent on frequency. From the general expression for

α and β . it is evident that a distortion-less line results if the line parameters are

such that :

C

G

L

R

jG

CjRG

G

Cj

R

LjRGCjGLjR )1()1)(1())((

RG , LCC

LC

G

RC

G

CRG

f

LCu

1

Showing that α deos not depend on frequency, whereas β is a linear function of

frequency.

C

L

G

R

GCjG

RLjRjXR

CjG

LjRZ

)/1(

)/1(000

00 X , C

LRZ 00

Note that:

1. The phase velocity is independent of frequency because the phase

constant linearly depends on frequency. We have shape distortion of

signals unless u and are independent on frequency.

2. u and 0Z remain the same as for lossless line.

3. A lossless line is also distortion-less line,but a distortion-less line is not

necessarily lossless. Although lossless lines are desirable in power

transmission, Telephone lines are required to be distortion-less.

Microwave Engineering

Sheet #3-a

Transmission line

Q1: Define T.L.?

Q2: State types of mode which propagate in T.L.?

Q3: Deduce the formula of propagation constant , ?

Q4: Deduce the formula of C/C impedance, Z0 ?

Q5: Sketch E/H field in coaxial cable?

Q6: An air line has Z0 =70 Ω, and phase constant =3 rad/m at f= 100MHz

Calculate: - the inductance/m - the capacitance/m

Q7: A transmission line operating at f=500 MHz has Z0 =80 Ω , mN p /04.0

mrad /5.1 , Find the line parameters R,L,G, and C.

Q8: A distortion-less line has Z0= 60 Ω , mmN p /20 , u=0.6 c(velocity of light in vacuum).

Find: R,l,G,C and at f=100 MHz?

Q9: A telephone line has R= 30 Ω/km, L= 100 mH/km, G=0, and C= 20 µF/km at f= 1 kHz,

obtain: a- C/C impedance of the line

b- propagation constant

c- the phase velocity

Dr. M.A.Motawea

2- Input Impedance, Standing Wave Ratio, and Power:

Consider a transmission line of length l, characterized by and 0Z , connected to a

load LZ as shown in figure:

Zg I0

Vg V0

+

_

+

_

VLZL

Z=0 Z=l

ZinZin

Z l' = l-Z

IL

a- Input impedance due to a line terminated by a load

I0

Zin

Zg

Vg V0

+

_

b- Equivalent circuit for finding V0 and I0 in terms of Zin at the input

Looking into the line, the generator sees the line with the load as an input

impedance inpZ . It is our intention in this section to determine the input impedance,

the standing wave ratio (SWR), and power flow on the line.

Let the transmission line extend from z = 0 at the generator to z = l at the

load. First of all, we need 0V and 0I , as mention above,

ZZ

s eVeVzV 00)( (1)

ZZ

s eIeIzI 00)( ,

ZZ

s eZ

Ve

Z

VzI

0

0

0

0)( (2)

Where eqn of Z0 has been incorporated,

CjG

LjR

CjG

LjR

I

V

I

VZ

)(

0

0

0

00 (3)

To find

0V and

0V the terminal conditions must be given.

e.g. at the input , (at z = 0) :

),0(0 zVV (4)

),0(0 zII (5)

Substitute (4), (5) into (1) ,(2) results in :

)(2

10000 IZVV

(6)

)(2

10000 IZVV

(7)

If the input impedance at the input terminals is inpZ , the input voltage V0 and the

input current I0 are easily obtained from figure as:

g

gin

in VZZ

ZV

0 ,

gin

g

ZZ

VI

0 (8)

(at z = l ) :

On the other hand, if we are given the conditions at the load, say:

)( lzVVL , )( lzII L (9)

Substitute this into eqn (1), (2) gives:

l

LL eIZVV )(2

100

(10)

l

LL eIZVV )(2

100 (11)

at any point on the line:

Next we determine input impedance )(/)( zIzVZ ssin at any point on the line.

at the generator, for example,

00

000 )(

)(

)(

VV

VVZ

zI

zVZ

s

sin (12)

Substituting eqn (1), (11) into (12) yields:

)2

()2

(

)2

()2

(

2

)(

2

)(

2

)(

2

)(

)(

)(

)(

0

0

0

00

00

0

00

000

LL

L

LL

L

LL

L

LL

L

l

LL

l

LL

l

LL

l

LL

s

s

inee

IZee

V

eeIZ

eeV

ZeIZVeIZV

eIZVeIZV

ZVV

VVZ

zI

zVZ

lZZ

lZZZ

ZlZ

lZZZ

lZlZ

lZlZZ

lZlI

V

lZlI

V

Z

lZlI

V

lZlI

V

ZlIZlV

lIZlVZ

L

L

L

L

L

L

L

L

L

L

L

L

L

L

LL

LL

tanh

)tanh

tanh

)tanh

coshsinh

sinhcosh

coshsinh

sinhcosh

]

coshsinh

sinhcosh

[coshsinh

sinhcosh

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(13)

lZZ

lZZZZ

L

L

in

tanh

)tanh

0

0

0 * for lossy line (14)

ljZZ

ljZZZZ

L

Lin

tanh

)tanh

0

00 * for lossless line (15)

This indicates that the input impedance varies periodically with distance l from

the load. The quantity βl is usually referred to as the electrical length of the line

and can be expressed in degree or radians.

Voltage reflection coefficient ( L ):

We now defined L as the voltage reflection coefficient (at the load) .

L is the ratio of the voltage reflection wave to the incident wave at the load:

l

l

LeV

eV

0

0

,

by eqn 10, 11 gives:

0

0

ZZ

ZZ

L

LL

(16)

In general , the voltage reflection coefficient at any point on the line can be

defined as the ratio of the magnitude of the reflected voltage wave to that of the

incident wave , that is as.

z

l

l

L eV

V

eV

eVz

2

0

0

0

0)(

But 'llz , substituting and combining with eqn l

l

LeV

eV

0

0

, we get:

''

222

0

0)( l

L

ll

L eeeV

Vz

(17)

current reflection coefficient

The current reflection coefficient at any point on the line is negative of the voltage

reflection coefficient at that point. Thus , the current reflection coefficient at the

load is : L

ll eIeI 00 /

as we did in plane waves, we define the standing wave ratio (s) :

L

L

I

I

V

Vs

1

1

min

max

min

max

(18)

It is easy to show that 0maxmax / ZVI and 0minmin / ZVI .

The input impedance inpZ in eqn (14) has maxima and minima that occur

respectively at the maxima and minima of the voltage and current standing wave.

It is easily shown that:

0

min

max

maxsZ

I

VZ in (19)

And s

Z

I

VZ in

0

max

min

min

(20)

As a way of demonstrating these concepts, consider a lossless line with

characteristic of 500Z . For the sake of simplicity, we assume that the line is

terminated in a pure resistive load 100LZ and the voltage at the load is 100 V

(rms). The conditions on the line repeat themselves every half wavelength.

The average input power at a distance z from the load is given by an

equation similar to eqn of Pav,

)]()(Re[2

1 * zIzVP ssav (21)

voltage &current wave patterns on a lossless line terminated by a resistive load.

We now consider special cases when the line is connected to load 0LZ ,

LZ , and 0ZZL . These special cases can easily be derived from the general

case.

A. Shorted Line (ZL =0)

For this case , eq. (15) becomes:

ljZZZ

LZinsc tan00

(22)

Also, ,1L s

This impedance is pure reactance, which could be capacitive or inductive

depending on the value of l . The variation of Zin with l is shown in

figure (a).

Input impedance of a lossless line: (a) when shorted, (b) when open

B. Open Circuited Line ( LZ )

In this case , eq (15) becomes

ljZ

lj

ZZZ inzoc

L

cottan

lim 00 (23)

,1L s

The variation of Zin with l is shown in figure (b) . notice from eqs (22),(23) that:

2

0ZZZ scoc (24)

C. Matched Line ( )0ZZ in

This is the most desired case from the practical point of view. For this case, eq(15)

reduces to:

0ZZ in (25)

And

,0L 1s

That is , 00 V , the whole wave is transmitted and there is no reflection . the

incident power is fully absorbed by the load. Thus maximum power transfer is

possible when the transmission line is matched to the load.

Example:

A 30-m long lossless transmission line with Z0 =50 Ω operating at 2 MHz is

terminated with a load 4060 jZL . If cu 6.0 on the line. Find:

a- The reflection coefficient

b- The standing wave ratio s.

c- The input impedance.

0

0

0

5.903.003.02.0137

427

13700

4002700

)40110)(40110(

)40110)(4010(

40110

4010

504060

504060

anglejj

j

jj

jj

j

j

j

j

ZZ

ZZ

L

LL

06.197.

03.1

03.1

03.1

1

1

min

max

min

max

L

L

I

I

V

Vs

ljZZ

ljZZZZ

L

Lin

tanh

)tanh

0

00

3-The Smith Chart

It is a tool to solve problems of transmission lines , it is commonly used of the

graphical techniques. It is basically a graphical indication of a transmission line as

one moves along the line. We calculate L ,s, and Zin and so on.

Example :

a 100+j150 Ω load is connected to a 75 Ω lossless line . find :

a-

b- S

c- The load admittance LY

d- 4.0atZ in from the load

e- The location of Vmax and Vmin w.r.t. the load if the line is 0.6 λ long

f- Zin at the generator.

Smith Chart

Sheet #3-b

Transmission line

Input impedance, reflection coefficient & standing wave ratio

Q1: a transmission line has c/c impedance Z0=50 Ω is connected with a generator has frequency

f=25 MHz and terminated by a load impedance ZL, at l=25 m from the load Zin=50-j100 .

Required: ZL, L