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Transmission Lines


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

Characteristic Impedance

Z0 = 613.png

Propagation constant Y = 


Z0 = 623.png

  1. At low frequency, R >> ωL and G >> ωC
    Z0 628.png
  2. At high frequency, R << ωL and G << ωC
    Z0 634.png

Characteristics impedance for parallel line,

Z0 = 
639.png where, d = diameter of each wire

S = separation between two wires

Characteristics impedance for co-axial line wire,

Z0 644.png

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 659.png

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 frequency679.png

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,

690.png ⇒ 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 = 695.png

700.png = 705.png and 710.png = 715.png

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 | = 720.png so that

s = 725.png

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)

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