Definition DISTORTIONLESS LINE

A transmission line is said to be distortionless when attenuation constant ‘α’ is frequency independent and the phase shift constant ‘β’ is linearly dependent on the frequency.

Condition for line to be distortionless

R/L=G/C

(a)    Propagation constant

Ỳ= (R+jωL)(G+jωC)

Or                     Ỳ= RG(1+jωL/R) (1+jωC/G)

If                       R/L=G/C

Put value of R/L in equation of Ỳ

Thus                 Ỳ= RG(1+jωC/G)(1+jωC/G)

Or                     Ỳ= RG(1+jωC/G)

Also                   Ỳ=α+jβ

Comparing Real and Imaginary parts, we get

α=RG

and                       jβ= RGjωC/G

=jω RC/G

Or                                 β=ω L/C C                                      (R/G=L/C)

Thus                         β=ω LC

The above results show that α is frequency independent and β is frequency dependent

(b)   Characteristic impedance

Z₀= √R+jωL/G+jωC

Z₀ =R/G=L/C

(c)    Phase velocity:-

V_p=ω/β

Substituting value of β in above expression,we get

V_p=ω/ω LC

Thus                              v_p=1/ LC

Note: If you do not know about the basics of transmission line then please read the article transmission line and its types.

Last time I have also discussed the lossless transmission line and its condition.