Microwave-Engineering : Field Analysis of Lines

By Rahul Ranjan|Updated : December 26th, 2019


The idea is to find relations between byjusexamprep and  with R, L, G, C.

where and We, Wm, Pd, PC are time-averaged values calculated for 1m length line and S is the line cross-sectional area.

Wave Propagation along the Line

Using the frequency domain Telegrapher equation

where  is propagation constant.

The solution of Telegrapher equation is:

Then, taking the z derivation of v(z), the calculation of i(z)


Converting to time domain by using  


where  is the phase.

Lossless Transmission Lines

When the lossless line is terminated by a load ZL.

 Reflected waves occur.  

Reflection coefficient at the load, z = 0

where v(z) and i(z) consists of a superposition of an incident and reflected waves called Standing Waves

  • Time Average Power Flow:


This shows Pav is constant at anywhere on the line.  When the line is matched  

 is constant.

When the line is mismatched (Return Loss

Matched load (No reflected power, maximum power is delivered).

 Total reflection, (All power reflected).


The maximum value  occurs when The minimum value  occurs when  When  increases, Vmax/Vmin increases as a measure of mismatch.

  • Standing Wave Ratio () is


When SWR = 1 means matched line. In that case at z = -l, the reflection coefficient and input impedance

Using the definition of , more useful form known as Transmission Line Impedance Equation as

  • Transmission Coefficient: Some part of EM wave is also transmitted to the second region as

  • Insertion Loss:

  • Short Circuit:

  • Open Circuit:

The proper length of open or short circuited transmission line can provide any desired reactance or susceptance.

The same impedance is observed at the input.

(Quarter Wave Transform)

Lossy Transmission Lines

In practice, finite conductivity (or lossy dielectrics) lines can be evaluated as a Lossy Line.  

In the lossy line; can be approximated to the lossless line.

  • Distortionless Line: For the lossy line, in fact the exact  is not a linear function of frequency means dispersive. But specifically, if the following condition holds

   then mean that the lossy line behaves as a lossless (distortionless) line.

  • Terminated Lossy Line: Loss is assumed small that  

  • Power lost in the line:

  • Perturbation Method for Calculating Attenuation

  • Attenuation constant:

  • Taylor series Inductance Rule:

Smith Chart

where R is Resistance, X is Reactance, G is Conductance and B is Susceptance. Whenever z = Z/Z0 is normalized impedance

The apsis and ordinate of Smith chart are  and  .

Rearranging them

These are two families of circles as rL and xL. Superposition of Smith Chart and its 180º  rotated version is known as Combined Impedance-Admittance Smith Chart.

Slotted Line

This device is used to find ZL as first Vmin.

Generator & Load Mismatches

Then, using this



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