# Transmission Lines

A transmission line is a two port network that connects a source to a load ## Modes

• Modes descibe the field pattern of propogating waves
• Can be found by solving Maxwell's equations in a transmission line
• In a transmission line, electric and magnetic fields are orthogonal to each other, and both orthogonal to the direction of propogation
• This is TEM (Transverse Electromagnetic) mode
• A TEM transmission line is represented by two parallel wires
• To reason about voltages and currents within it, we divide it into differential sections
• Each section is represented by an equivalent lumped element circuit • - the combined resistance of both conductors per unit length, in
• - the combined inductance of both conductors per unit length, in
• - the combined capacitance of both conductors per unit length, in
• - the conductance of the insulation medium between the two conductors per unit length, in

The table below gives parameters for some common transmission lines • Conductors have magnetic permeability and conductivity
• The insulating/spacing material has permittivity , permeability and conductivity
• All TEM transmission lines share the relations
• The constant propogation constant of a line
• is the attenuation constant (Np/m)
• is the phase constant (rad/m)
• The travelling wave solutions of a line are
• represents position along th eline
• represents the incident wave from source to load
• represents the reflected wave from load to source

We therefore have the characterisitic impedance of the TEM transmission line:

Both the voltage and current waves propagate with a phase velocity . The presence of the two waves propagating in opposite directions produces a standing wave.

## The Lossless Transmission Line

In most practical situations, we can assume a transmission line to be lossless:

• , and
• Assume , so
• Therefore, as :

This then gives velocity and wavelength:

As the insulating material is usually non-magnetic, we have

## Voltage Reflection Coefficient

Assume a transmission line in which the signals are produced by a generator with impedance and is terminated by a load impedance . At any position on the line, the total voltage and current is:

Using this we can find an expression for the ratio of backwards wave amplitude and forward wave amplitude. We obtain the equation below, the voltage reflection coefficient

• for a lossless line is a real number, but may be a complex quantity
• In general, the reflection coefficient is also complex,
• Note that , always
• A load is matched to a line when , as then
• No reflection by the load
• If then , then
• If then , then