# Microwave-Engineering : Field Analysis of Lines

By Rahul Ranjan|Updated : December 26th, 2019

### FIELD ANALYSIS OF LINES

The idea is to find relations between 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) Then 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.   Then, using this       Thanks GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com