# Waveguide theory

### - the basics of RF waveguide theory including TE waves, TM waves and TEM waves, etc. including waveguide propagation constant.

In order to be able to use waveguides to their best effect, it is necessary to have a basic understanding of waveguide theory, including propagation and the propagation constant. While waveguide theory can become particularly involved, it is not the aim here to delve too deeply into the waveguide theory mathematics.

Waveguide theory is based around electromagnetic wave theory because the waves propagating along waveguides are electromagnetic waves that have been constrained, typically within a hollow metal tube. The constraining boundaries of the metal tube prevent the electromagnetic wave from spreading out and thereby reducing in intensity according to the inverse square law. As a result, losses are very low.

## Waveguide theory of propagation

According to waveguide theory there are a number of different types of electromagnetic wave that can propagate within the waveguide. These different types of waves correspond to the different elements within an electromagnetic wave.

• TE waves:   Transverse electric waves, also sometimes called H waves, are characterised by the fact that the electric vector (E) is always perpendicular to the direction of propagation.
• TM waves:   Transverse magnetic waves, also called E waves are characterised by the fact that the magnetic vector (H vector) is always perpendicular to the direction of propagation.
• TEM waves:   The Transverse electromagnetic wave is cannot be propagated within a waveguide, but is included for completeness. It is the mode that is commonly used within coaxial and open wire feeders. The TEM wave is characterised by the fact that both the electric vector (E vector) and the magnetic vector (H vector) are perpendicular to the direction of propagation.

Text about waveguide theory often refers to the TE and TM waves with integers after them: TEm,n. The numerals M and N are always integers that can take on separate values from 0 or 1 to infinity. These indicate the wave modes within the waveguide.

Only a limited number of different m, n modes can be propagated along a waveguide dependent upon the waveguide dimensions and format.

For each mode there is a definite lower frequency limit. This is known as the cut-off frequency. Below this frequency no signals can propagate along the waveguide. As a result the waveguide can be seen as a high pass filter.

It is possible for many modes to propagate along a waveguide. The number of possible modes for a given size of waveguide increases with the frequency. It is also worth noting that there is only one possible mode, called the dominant mode for the lowest frequency that can be transmitted. It is the dominant mode in the waveguide that is normally used.

It should be remembered, that even though waveguide theory is expressed in terms of fields and waves, the wall of the waveguide conducts current. For many calculations it is assumed to be a perfect conductor. In reality this is not the case, and some losses are introduced as a result.

## Rules of thumb

There are a number of rules of thumb and common points that may be used when dealing with waveguide theory.

• For rectangular waveguides, the TE10 mode of propagation is the lowest mode that is supported.
• For rectangular waveguides, the width, i.e. the widest internal dimension of the cross section, determines the lower cut-off frequency and is equal to 1/2 wavelength of the lower cut-off frequency.
• For rectangular waveguides, the TE01 mode occurs when the height equals 1/2 wavelength of the cut-off frequency.
• For rectangular waveguides, the TE20, occurs when the width equals one wavelength of the lower cut-off frequency.

## Waveguide propagation constant

A quantity known as the propagation constant is denoted by the Greek letter gamma, γ. The waveguide propagation constant defines the phase and amplitude of each component of the wave as it propagates along the waveguide. The factor for each component of the wave can be expressed by:

exp[j ω t   -   γm,n z]

Where:
z = direction of propagation
ω = angular frequency, i.e. 2 π x frequency

It can be seen that if propagation constant, γm,n is real, the phase of each component is constant, and in this case the amplitude decreases exponentially as z increases. In this case no significant propagation takes place and the frequency used for the calculation is below the waveguide cut-off frequency.

It is actually found in this case that a small degree of propagation does occur, but as the levels of attenuation are very high, the signal only travels for a very small distance. As the results are very predictable, a short length of waveguide used below its cut-off frequency can be used as an attenuation with known attenuation.

The alternative case occurs when the propagation constant, γm,n is imaginary. Here it is found that the amplitude of each component remains constant, but the phase varies with the distance z. This means that propagation occurs within the waveguide.

The value of γm,n is contains purely imaginary when there is a totally lossless system. As in reality some loss always occurs, the propagation constant, γm,n will contain both real and imaginary parts, αm,n and βm,n respectively.

Accordingly it will be found that:

γm,n   =   αm,n   +   j βm,n

This waveguide theory and the waveguide equations are true for any waveguide regardless of whether they are rectangular or circular.

By Ian Poole

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