# LOSSLESS TRANSMISSION LINE AND ITS CONDITION

**CHARACTERISTIC IMPEDANCE ,Z _{0}**

**DEFINITION I:- ** Z_{0} 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

Z_{0}=Z/Y =R+jωL/G+jωc

**DEFINITION 2:-** The characteristic impedance , Z_{0} of a line is defined as the ratio of the forward voltage wave ,V_{f}to the forward current wave ,I_{f }at any point on the line.

Z_{0}=V_{f}/I_{f}

**DEFINITION 3:-** z_{0} 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,

Z_{0}= -V_{r}/I_{r}

Characteristic impedance ,Z_{0} is also called **Surge imperdance.**

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**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,**

Z_{0}=R+jωL/G+jωc

As R=0=G

Z_{0}=L/C

Thus Z_{0} 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

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 .