# Transmission Lines

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

Characteristic Impedance

Z_{0} =

Propagation constant Y =

Z_{0} =

- At low frequency, R >> ωL and G >> ωC

∴ Z_{0 }= - At high frequency, R << ωL and G << ωC

∴ Z_{0 }=

Characteristics impedance for parallel line,

Z_{0} = where, *d* = diameter of each wire

S = separation between two wires

Characteristics impedance for co-axial line wire,

Z_{0 }=

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

Z_{x}_{ }for open circuited and short circuited line

- If line is open circuited, Z
_{R }= ∞

Z= Z_{oc}_{0}cot*h*γ*x* - If line is short circuited, Z
_{R}= 0

Zh γ_{sc}= Z_{0}tan*x*

∴∇

*For a lossless line, *Z* _{x}* = Z

_{0}

If lossless line is open circuited, i.e. Z_{R} = ∞

∇

If lossless line is short circuited, i.e., Z_{R} = 0

Z* _{x}* =

*j*Z

_{0 }tan β

*x*= Z

_{sc}** **

# 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, Z_{0} 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. | E_{max} | = 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)