TRANSMISSION LINE THEORY

(TEM Line)

A uniform transmission line is defined as the one

whose dimensions and electrical properties are

identical at all planes transverse to the direction of

propagation.

Circuit Representation of TL’s

A uniform TL may be modeled by the following circuit

representation:

R: Series resistance per unit length of line (for both

conductors (ohm/m)).

L: Series inductance per unit length of line (Henry/m).

G: Shunt conductance per unit length of line (mho/m).

C: Shunt capacitance per unit length of line (Farad/m).

The line is pictured as a cascade of identical sections,

each of z long.

Since z can always be chosen small compared to the

operating wavelength, an individual section of line may

be analyzed using ordinary ac circuit theory. In the

following analysis, we let 0z , so the results are

valid at all frequencies (hence for any physical time

variation).

Applying the Kirchhoff’s voltage law to the line section

gives:

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

i z tv z t R z i z t L z v z z t

t

Rearranging yields:

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

v z z t v z t i z tR i z t L

z t

Letting 0z , we get,

( , ) ( , )( , )

v z t i z tR i z t L

z t

Now applying Kirchhoff’s current law to the line

section gives:

( , )

( , ) , ( , )v z z t

i z t G z v z z t C z i z z tt

Rearranging yields:

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

i z z t i z t v z z tG v z z t C

z t

Letting 0z

( , ) ( , )( , )

i z t v z tG v z t C

z t

Then the time domain TL or telegrapher equations are:

( , ) ( , )( , )

v z t i z tR i z t L

z t

( , ) ( , )( , )

i z t v z tG v z t C

z t

The solution of these equations, together with the

electrical properties of the generator and load, allow

us to determine the instantaneous voltage and current

at any time t and any place z along the uniform TL.

Lossless Line: For the case of perfect conductors (R=0)

and insulators (G=0), the telegrapher equations reduce

to the following form:

( , ) ( , )v z t i z tL

z t

( , ) ( , )i z t v z tC

z t

( , ) ( , )v z t i z tL

z z z t

( , )i z t vL L C

t z t t

Or,

2 2

2 2

2 2

2

( , ) ( , )0

( , ) ( , )0

v z t i z tLC

z t

i z t i z tLC

z t

Wave equation’s for voltage and current on a lossless

TL.

Although real lines are never lossless, lolessness

approximation for practical TL’s is very usefull.

TRANSMISSION LINES WITH SINUSOIDAL EXCITATION

We will only consider the sinusoidal steady-state

solutions.

Transmission-Line Equations:

Under sinusoidal steady state conditions, the TL

equations take the form:

( )( ) ( )

( )( ) ( )

dV zR j L I z

dz

dI zG j C V z

dz

Where ( )V z and ( )I z are voltage and current phasors.

The real sinusoidal voltage and current waveforms are

obtained from:

( , ) Re ( )

( , ) Re ( )

j t

j t

v z t V z e

i z t I z e

Wave Propagation on a TL

The second order differential equations for ( )V z and

( )I z are:

22

2

22

2

( )( ) 0

( )( ) 0

d V zV z

dz

d I zI z

dz

where 1/2

j R j L G j C

complex propagation constant.

attenuation constant (Np/m).

phase constant (rad/m).

The solution for ( )V z is:

0 0( )

( ) ( ) ( )

z zV z V e V e

V z V z V z

Where 0V

and 0V

are constants independent of z, and

0

0

( )

( )

z

z

V z V e

V z V e

Represent voltage waves traveling on the line in the

positive and negative z directions respectively.

WAVE PROPAGATION ON A TL

The second order equations for V(z) and I(z) are:

22

2

22

2

( )( ) 0

( )( ) 0

d V zV z

dz

d I zI z

dz

The solution for V(z) is:

0 0( ) z zV z V e V e

Where 0V

is a constant of voltage waves traveling in the

forward direction, 0V

is a constant of voltage waves

traveling in the reverse direction, independent of z.

Now consider the equation:

( )

( ) ( )dV z

R j L I zdz

If we substitude the solution for V(z) into the above

equation we get,

0 0 ( ) ( )z zV e V e R j L I z or,

0 0( )( ) ( )

z zI z V e V eR j L R j L

1/2

( )

( )

R j L G j C

R j L R j L

1/2

( )

( )

G j C

R j L R j L

Define

1/2

0( )

R j LZ

G j C

the characteristic impedance of the TL. Then,

0

1

( )R j L Z

So,

0 00 0

0 0

( )

( ) ( )

z z z zV VI z e e I e I e

Z Z

I z I z

Where,

0 00 0

0 0

V VI I

Z Z

Constants independent of z.

0 0

0 0

z zV VI z e I z e

Z Z

Lossless Line:

For the lossless line R=0, G=0.

i) 0

LZ

C

ii)

R j L G j C

j L j C j LC

j

LC Propagation constant

0 (no loss)

iii)

0 0( ) j z j zV z V e V e

0 0

0

1( ) ( )j z j zI z V e V e

Z

iv) If 0 0 0 0

j jV V e V V e then

0 0( , ) cos cosv z t V t z V t z

0 0

0

1( , ) cos cosi z t V t z V t z

Z

v) 2 2 1

2 f LC f LC

vi) 1

puLC LC

So,

1 p

p

p

vu

fLC

u f

TERMINATED TRANSMISSION LINE

Infinitely Long TL

For the infinite line, only forward traveling waves exist and

therefore at z=0,

00 0

0

0in

VV V I I

Z

VZ Z

I

Suppose now that the infinite line is broken at z . Since the

line to the right of z is still infinite, its input impedance is 0Z

and therefore replacing it by a load impedance of the same value

does not change any of the conditions to the left of z . This

means that a finite line terminated in its characteristic impedance

is equivalent to an infinitely long line. Like the infinite line, a finite

length line, terminated in 0Z has no reflections. Also, its input

impedance is equal to 0Z and independent of the line length.

00

0

in g

g

ZV V V

Z Z

0

0

g

in

g

VI I

Z Z

0inV V

00

0

in

VI I

Z

So,

0

0

0

g

g

ZV V

Z Z

And

0

0

( ) z j z

g

g

ZV z V e e

Z Z

0

( )g z j z

g

VI z e e

Z Z

The time-averaged power absorbed by the load is:

*

* 0

0 0

1 1Re Re

2 2

gl j l l j l

L L L g

g g

VZP V I V e e e e

Z Z Z Z

2 2

2 2002

00

1

2 2

g gl l

L

gg

V VZP e Z e

Z ZZ Z

If 0 (lossless line)

2

0

0

1

2

g

L in

g

VP Z P

Z Z