# Transmission Lines

A transmission line conveys electric energy and signals from one point to another i.e. from source to load.

Characteristic Impedance

Z0 =

Propagation constant Y =

Z0 =

1. At low frequency, R >> ωL and G >> ωC
Z0
2. At high frequency, R << ωL and G << ωC
Z0

Characteristics impedance for parallel line,

Z0 =
where, d = diameter of each wire

S = separation between two wires

Characteristics impedance for co-axial line wire,

Z0

where, D = diameter of outer covering

d = diameter of inner covering

K = dielectric constant

Line impedance as a function of distance x from the load

Zx for open circuited and short circuited line

1. If line is open circuited, ZR = ∞
Zoc = Z0 cot h γx
2. If line is short circuited, ZR = 0
Zsc = Z0 tan h γx

∴∇

For a lossless line, Zx = Z0

If lossless line is open circuited, i.e. ZR = ∞

If lossless line is short circuited, i.e., ZR = 0

Zx = jZ0 tan βx = Zsc

# Lossless Transmission Line

It is said to be lossless if both its conductor and dielectric are lossless.

For lossless transmission line, R = G = 0.

Case I :
At radio frequency, it behaves as lossless line and

Case II : At very small frequency

In both the cases, Z0 is real.

Distortionless Line

When R and G are small, but not as small to be neglected, then characteristic impedance is expressed approximately in complex form as

For a transmission line to be distortionless,

⇒ LG = RC

In a distortionless line, attenuation constant α is frequency independent but phase constant β is nearly dependent.

Standing Wave Ratio (SWR)

It is the ratio the maximum to the minimum magnitude of voltage or current on the line having waves

ie. SWR =

As
=  and  =

The voltage at a point on the line will be maximum if two wave components E+ and E satisfy in phase condition

i.e. | Emax | =  so that

s =

where | ρ | is reflection coefficient.

If SWR (S) = 1 then | ρ | = 0 perfect matching condition.

If SWR (S) = ∞ then | ρ | = 1 perfect mismatch

(This is in case of short and open circuited lines)