LOSSLESS TRANSMISSION LINE AND ITS CONDITION

CHARACTERISTIC IMPEDANCE ,Z₀

DEFINITION I:- Z₀ is defined as the ratio of the square root of series impedance per unit length ,Z to the square root of shut admittance per unit length,Ỳ

Z₀=Z/Y =R+jωL/G+jωc

DEFINITION 2:- The characteristic impedance , Z₀ of a line is defined as the ratio of the forward voltage wave ,V_fto the forward current wave ,I_f at any point on the line.

Z₀=V_f/I_f

DEFINITION 3:- z₀ IS defined as the minus of the ratio of the reflected voltage wave . V_r to the reflected current wave , I_r at any point on the line,

Z₀= -V_r/I_r

Characteristic impedance ,Z₀ is also called Surge imperdance.

LOSSLESS TRANSMISSION LINES

A transmission line is said to be lossless if the conductors of line are perfect that is cnductivity σ_c=∞ and the dielectric medium between the lines is lossless that is conductivity σ_d=0

Condition for a line to be lossless

R=0=G

For loss less line,

(a)    Attenuation Constant       α=0

(b)   Propagation constant

Ỳ=α+jβ=jβ                                               (α=0)

Also as                      Ỳ=(R+jωL)(G+jωC)

As                              R=0, G=0

Thus propagation constant     Ỳ=j ω LC

(c) Phase shift constant

By comparing imaginary parts of Ỳ, we get

Phase shift constant β=ω LC

(d) Characteristic impedance,

Z₀=R+jωL/G+jωc

As                                          R=0=G

Z₀=L/C

Thus Z₀ is pure reactance (that is in the form of inductance or capacitance).

(e) Phase velocity or the velocity of propagation in lossless line,

V_p=ω/β

By putting value of β,we get

Thus                          v_p=ω/ω LC

Or                                      v_p=1/LC

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