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 ,Vfto the forward current wave ,If at any point on the line.
Z0=Vf/If
DEFINITION 3:- z0 IS defined as the minus of the ratio of the reflected voltage wave . Vr to the reflected current wave , Ir at any point on the line,
Z0= -Vr/Ir
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 σc=∞ and the dielectric medium between the lines is lossless that is conductivity σd=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,
Vp=ω/β
By putting value of β,we get
Thus vp=ω/ω LC
Or vp=1/LC
If R=0 then how G =0
How R and G can be zero simultaneously in lossless transmission lines?
R is the series resistance where as G is the shunt conductance resulting in leakage current. This R is not related to this G, i.e., this G is not equal to this 1/R. Just to clarify with numerical values R is ,say about 0.3 ohm/km and G about, say, 5 micro S/km.
As high line current flows through R producing losses R is made zero for a lossless line.
As G is the shunt leakage conductance, there is a very small leakage current flowing to ground through G producing a very small loss. So G is also made zero ( or leakage resistance, Rg = 1/G , is made infinity) for a lossless .