# Microwave Engineering : Field Analysis of Lines

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Microwave engineering is an exciting field that has revolutionized modern communication and technology. The ability to transmit and receive information wirelessly has changed the way we live and work. At the heart of microwave technology lies the field analysis of lines. Understanding how electromagnetic waves propagate through transmission lines is essential to designing and optimizing microwave circuits and systems.

Microwave engineering is an interdisciplinary field that draws upon knowledge from electrical engineering, physics, and mathematics. It involves the design and development of components and systems that operate at high frequencies, typically in the range of 1 GHz to 100 GHz. These components and systems are used in a wide range of applications, including telecommunications, radar, satellite communication, and medical imaging. The field analysis of lines is a crucial aspect of microwave engineering, as it helps engineers understand the behaviour of electromagnetic waves in transmission lines and optimize the performance of microwave circuits and systems.

Table of content

## Field Analysis of Lines

The field analysis of lines is a fundamental concept in microwave engineering that deals with the study of electromagnetic waves propagating through transmission 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

Wave propagation along transmission lines is a fundamental concept in electrical engineering and plays a crucial role in the design and optimization of microwave circuits and systems. It refers to the behaviour of electromagnetic waves as they travel through transmission lines.

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

Lossless transmission lines are an important concept in the field of electrical engineering. They are idealized transmission lines that do not dissipate any energy due to resistance, making them valuable for high-frequency applications.

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 whenThe 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

Lossy transmission lines are an important topic in the field of electrical engineering. These lines are characterized by a significant amount of energy loss due to resistance, dielectric losses, or radiation.In practice, finite conductivity (or lossy dielectrics) lines can be evaluated as a Lossy Lines.

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

The Smith Chart is a graphical tool used in microwave engineering to simplify calculations and aid in the design of radio frequency (RF) and microwave circuits. It provides a visual representation of complex impedance and reflection coefficients.

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