# Waveguides and Cavity Resonators

## Search

Transmission line notes. Also, more links to relevant topics.
Microstrip.

Microwaves for satellite communications.
Scattering parameters, scattering matrix, s-parameters.
Four-port S-matrix relations and directional couplers.
Problems 3, on waveguides.

### Java applet of wave motion

This applet shows a visualisation of wave motion and you can add your waveguide walls with the mouse.

## Waves in one, two, and three dimensions.

The number of dimensions in the section heading refers to the number of dimensions in which the wave can move.

• One dimension.

On the transmission lines discussed elsewhere the wave is restricted to one dimension only. It can travel along the line from generator to load, or indeed back again, but it can't escape from the length of the line.

• Two dimensions.

Waves on the surface of water, for example your bath, or a pond, or the sea, have two dimensions they can travel. We might refer to these as "perpendicular to the shore" and "parallel to the shore", or x and y. Any two orthogonal directions may be chosen as the fundamental reference directions. "Orthogonal" means taking two directions at 90 degrees to each other.

In the case of waves on a pond, we may see various patterns of wave crests. Two important patterns are

1. when the crests form concentric circles, moving radially out from a point where maybe a pebble was cast, and
2. when the crests form ranks of straight lines, moving uniformly in a direction at right angles to the crests.

• Three dimensions.

The case which concerns us in general is when waves, perhaps radiation from an antenna, can move in a direction unrestricted in the three space dimensions. Various co-ordinate systems may be used to describe these waves. Three of the important patterns of wave crests are

1. when the crests form concentric spheres, moving radially outwards from a point. These are best described by spherical polar co-ordinates, having radial, azimuth, and elevation directions.
2. when the crests form nested cylinders, moving radially away from a straight line of source. These are usually described by cylindrical polar co-ordinates, having radial, azimuth, and linear-z directions.
3. when the crests form parallel planes, moving in a direction normal to the plane surfaces. These are described by rectangular Cartesian coordinates, in x y and z directions. Such waves are called "plane waves".

Electromagnetic waves are "transverse waves", that is the directions of local magnetic and electric field lie at right angles to each other and to the local direction of travel of the wave. There is a picture here .

Generally speaking, the direction of electric and magnetic field does not change as we move along with the wave motion, but stays fixed in space. An exception to this general rule occurs in "optically active" regions, where the plane of "polarisation" (electric field direction) can rotate as we advance with the wave. Also, it is possible to superpose two plane waves having orthogonal electric field directions which are out of time phase with each other by +/-90 degrees. In this case, the resultant wave appears to have its resultant electric field vector rotate as the wave advances; this has nothing to do with optical activity. It is referred to as "circular polarisation". If the electric field vector rotates clockwise as we advance with the wave, it is "right handed circular polarisation". If the electric field vector rotates anti-clockwise as we advance with the wave it is "left handed circular polarisation".

## What is a waveguide?

A waveguide restricts the three dimensional "free space" propagation of the electromagnetic wave to a single dimension. Usually waveguides are

• Low loss. That is, the wave travels along the guide without greatly attenuating as it goes.
• Routeable. This means that we can gently bend the guiding structure without losing contact with the wave, without generating reflections, and without incurring much additional loss.

Thus we see that waveguides are transmission lines, in terms of the properties described in the notes referred to above.

There are a great many different wave guiding structures. Usually they are uniform in the direction of travel of the guided wave; that is, one cannot tell where one is along the waveguide by physically looking around one. There is an exception in the case of corrugated waveguide and other slow wave structures which we shall return to later. These guides may have spatially periodic structure in the direction of wave travel.

The different co-ordinate systems described above are appropriate to different waveguide cross sectional shapes. Cylindrical polar co-ordinates are used to describe circular cross section waveguide, and coaxial cables. Rectangular Cartesian co-ordinates are preferred for rectangular waveguide. In the case of more exotic structures such as microstrip, fin-line, or coplanar waveguide, it is usual to use rectangular co-ordinates and solve approximately for the cross sectional field distributions by using "conformal mapping", a technique borrowed from complex variable theory.

## Rectangular metal pipe waveguide

### Boundary conditions.

If an electric field is to exist close to a conductor, we have to make sure that the conductor does not short out the electric field. In practice, this means that all electric fields meet conductors at right angles to the surface. The sources of electric field are the charges which reside on the metal surface. If the electric field represents a wave crest, the line along the crest of the wave meets the guide metal wall at right angles and moves laterally, parallel to the surface of the metal. The associated charges move along the guide and set up currents which are linked to magnetic fields, parallel to the guide metal surface and perpendicular to the electric fields.

There can be no time-dependent magnetic field meeting a metal wall at right angles, since the metal would act as a shorted turn and suppress the magnetic field.

### Constructing a waveguide.

Let us suppose we think about plane waves having electric field along the z direction, but moving in the x-y plane. We can clearly add a metal sheet in the x-y plane without disturbing the wave motion. The Ez field meets the sheet at right angles, and the wave travels at an arbitrary angle to the x axis (all 360 degrees of bearing are possible) with magnetic field at right angles to the direction of propagation, but parallel to the x-y plane. The currents in the x-y metal sheet are in the direction of propagation of the wave. If we have more than one wave, with more than one direction of travel, we merely add up the fields due to each to get the resultant field patterns. Of course, the addition has to be done vectorially.

Now let us add another sheet aligned in the x-y plane but at a different height z. The electric field will then start on one sheet and finish on the other. It is still lined up along the z direction. It starts on a positive charge and ends on a negative charge. Both charges are notionally moving in the direction of propagation, but they represent oppositely directed currents. The adjacent magnetic fields are in the same sense however, as they lie on opposite sides of the oppositely directed currents. One is above the current sheet and the other below.

We have made a Transverse Electromagnetic (TEM) wave guiding structure. There is no way the waves can escape from the space between the x-y metal sheets, which act as perfect screens. The electric field, magnetic field, and direction of wave travel are all mutually orthogonal; that is, they are each at right angles to the other two. The electric and magnetic fields are uniform (they don't depend on z) within the gap.

We have not yet made a waveguide, however. The wave still has a choice of direction in the x-y plane. Nevertheless, our structure is strongly reminiscent of a 2 wire transmission line, if we assume the line length to lie along the arbitrary direction (in the x-y plane) of propagation.

Exercise for the student. Draw a plan view of a few wave crests looking down along the z axis. You should see straight lines on your paper, orientated at an arbitrary angle to the x axis, but uniformly spaced apart (by a wavelength). Measure the angle alpha between the crest lines and the x axis. Measure the spacing lambda between the crest lines along the direction of wave travel. Measure the crest intersection distance lambda(x) along the x axis. Show by geometry, or with a calculator, that [lambda]/[lambda(x)] = cos (alpha). Similarly, measure the crest intersection distance lambda(y) along the y axis and show that [lambda]/[lambda(y)] = sin (alpha).

Then using Pythagoras' theorem (sin^2 + cos^2 = 1) show that

1/[lambda]^2 = 1/[lambda(x)]^2 + 1/[lambda(y)]^2

We are nearly there now. If we put walls in the x-z plane spaced by

a = (1/2)lambda(y)

we have made a tube whose axis lies along x. We have made a waveguide. Two waves are needed to make a standing wave pattern across this tube. Nulls are needed at a spacing of (1/2)lambda(y) to satisfy the boundary condition that the Ez field vanishes on these walls. We are left with the waveguide formula

1/[lambda]^2 = 1/(2a)^2 + 1/[lambda(x)]^2

and the wavelength along the guide is

lambda(x) = lambda/cos(alpha).

Lambda(x) is usually referred to as the guide wavelength lambda(g). It is longer than the free space wavelength lambda because cos(alpha) is always less than unity. The plane waves are travelling at an angle alpha to the direction along the waveguide. The energy travels at a velocity c*cos(alpha), but the wave pattern travels at the velocity given by f*lambda(g) = c/cos(alpha).

### Group and phase velocity.

The energy and the modulations on the microwave signal going down the waveguide both travel at the "group velocity" c*cos(alpha) which is necessarily less than the velocity of light c. The pattern however travels at the "phase velocity" c/cos(alpha) which is necessarily greater than the velocity of light. The product of (group velocity)*(phase velocity) = c^2.

### Modes.

The field pattern is formed from the superposition of two plane waves travelling in different directions, (the two directions are plus and minus alpha with respect to the direction along the waveguide). These two waves have the same free space wavelength, and give the standing wave pattern along y required to make the fields vanish at the side guide walls. The whole field pattern is called a "Mode". For the TE10 mode (transverse electric), if you plot out a plan view of the field patterns carefully, you will see that the electric field is always out of the paper, but that the magnetic field forms stadia loops with repetition distance equal to a guide wavelength; there are two stadia loops per guide wavelength, in opposite senses of magnetic field circulation.

The magnetic field is always parallel to the top and bottom guide walls, and turns to become parallel to the side guide walls where it approaches them. The guide wall currents are at right angles to the adjacent magnetic field lines.

### Other modes

The rectangular pipe has cross section a by b metres, with wall planes x-y and x-z. We chose the electric field to lie along the z direction in our first example. However, one can equally satisfy the boundary conditions with the electric field along the y direction, and the standing wave along z. Since the electric field in these modes is entirely transverse to the direction x of propagation, they are called "transverse electric" or TE modes. The two modes here are TE10 and TE01. The 1 refers to the number of half-wave loops across the guide. By convention, the TE10 mode has its single loop across the largest guide lateral dimension, and the TE01 mode has its loop across the smaller guide lateral dimension. Thus the cutoff frequency for the TE10 mode is the lowest frequency at which the waveguide will transmit without attenuation.

There are also transverse magnetic modes, TM modes. These require loops across both dimensions. The lowest TM mode is the TM11 mode. The magnetic field is entirely transverse; there is a component of longitudinal electric field in the x direction in the centre of the guide. Both parallel electric field and normal magnetic field fall to zero at the guide walls because of the standing wave pattern.

One can have higher order modes, TE20, TE30, ... TEn0

and also TE21, TE31,....TEmn

and the corresponding transverse magnetic modes TMmn, providing there is at least one loop in each transverse direction.

There are two pictures of the field patterns for some of the higher order modes in rectangular and circular waveguide.

Looking at our derivation of the waveguide formula, it is not difficult to see that for a general TEmn or TMmn mode the waveguide formula is

1/[lambda]^2 = {m/2a}^2 + {n/2b)^2 + 1/[lambda(g)]^2

Waveguides are generally used in a frequency range where only the lowest mode, or exceptionally the lowest few modes, will propagate. If one wants to launch a mode in a guide which supports more than one propagating mode, one drives it with probes or loops which have a symmetry that does not excite any of the other (unwanted) modes. Thus, for example, to excite a TE10 mode we might use a probe in the middle of the wide face of the guide; this would not excite the TE20 mode which has a null in the middle of the guide. To excite the TE20 mode we could drive two probes, spaced 1/4 and 3/4 of the way across the broad face, in anti-phase. This would be the wrong symmetry to excite the TE10 mode, for which the two probes thus placed would have to be driven in phase with each other. It is easy to see how to extend this method to other scenarios.

### Mode field patterns

Here we have shown a picture of the field and current patterns in the TE10 (H10) and TM11 (E11) modes in square cross section waveguide. The magnetic field lines are GREEN, the electric field lines are RED, and the currents in the guide walls are BLUE.

Nomenclature; originally the British called a TE mode an "H mode". In a TE mode there is no longitudinal E, but there is a longitudinal H field. However you can see that in the TE mode there is also a transverse component of magnetic field. The British thought that the unique longitudinal H field was a better descriptor than the transverse electric field with no longitudinal E component.

Thus in the diagram, H10 = TE10 and E11 = TM11.

The picture is a heavily processed image taken from an original in "The Services Textbook of Radio", vol 5, 1958 HMSO, wherein there are other pictures and a very clear exposition and discussion of the principles behind wave transmission and propagation. This book is long out of print, but if you can find a copy in the library it is well worth a read.

### A problem.

An exercise for the reader. The cutoff wavelength for these modes is lambda(c) where

1/[lambda(c)]^2 = {m/2a}^2 + {n/2b}^2, at which point the guide wavelength lambda(g) has gone to infinity, and the waves are bouncing back and forth across the guide and making no progress along it.

Calculate the cutoff frequency of a waveguide formed by the BB21 corridor, assumed to be 3 metres high and 2 metres wide. Hint. Find the lowest cutoff frequency of the mn modes; nothing can propagate below this frequency. Of course, we assume here that some kind soul has papered the walls with kitchen foil....

### Slots in waveguide walls.

If we look at the blue current lines in the waveguide walls, we find there are directions of flow where a slot, cut parallel to the flow lines, will not interrupt the currents. These slots are "non-radiating" slots.

However, if we deliberately cut a slot such as to interrupt the current flow lines, the currents have to go round the slot and this gives rise to a distortion of the electric field and magnetic field patterns inside the guide. One finds that there is a voltage difference between opposite edges of the slot, in the middle of the slot. The slot acts as a dipole antenna and will radiate, and the energy leaks out from the waveguide.

This principle is used to construct slot antennas in waveguide.

An important application of slotted waveguide is to construct a "slotted line" for measurements. This is usually done for the TE10 mode pattern, which is the only mode which will propagate for sufficiently low frequencies. The slot is cut parallel to the guide axis, in the middle of the wide face. It is a non-radiating slot, and the field pattern inside the guide is but little disturbed by it. A probe may be put into the slot to sample the local electric field strength, and moved up and down the slot along the waveguide to see the standing wave patterns and to measure the VSWR directly.

On the X band waveguide in the labs there is a ferrite collar around the slot on the slotted lines. This reduces residual radiation from the slot. There is some residual radiation because of the finite width of the slot, and because the probe and carriage distort the field patterns somewhat.

## Rectangular cavity resonators

Starting from a rectangular waveguide of cross section a by b metres, we can add short circuit walls in the y-z planes, along the direction of propagation.

This gives a rectangular box whose resonant frequency is given by f where (f*lambda) = c = 3*10^8, and

1/[lambda]^2 = {m/2a}^2 + {n/2b}^2 + {p/2d}^2

Here, there are m half wavelength loops along y, n half wavelength loops along z, and p half wavelength loops along x. It is possible for just one only of the loop numbers m, n, and p to take the value zero. The spacings of the walls are

d along x, a along y, and b along z. We see there are many modes of a rectangular cavity.

Exercise for the student. List the 30 lowest resonant frequencies of my microwave oven cavity, which has dimensions 0.36 metres by 0.33 metres by 0.23 metres. Assume it is empty, (no chickens, coffee cups, or turntables. ) Now justify the choice of operating frequency for such a microwave oven.

### Other shapes of cavities

Clearly, a cavity can be many other shapes than rectangular. The field theory for calculating the modes of arbitrary shaped cavities is straightforward, but often numerical methods are needed as there are no analytic solutions. Often, cylindrical cavities are used. It is possible to use more than one mode in a cavity filter, with tuning screws and stubs to convert energy from one mode to another.

Any equipment box will have microwave resonant frequencies, often in the range around 1GHz. This poses problems for the engineer faced with EMC (Electromagnetic compatibility) and interference problems, if there is any source of signal within the box having spectral frequency components somewhere near the box modes.

Another (salutary) exercise for the student. Do you have a cellnet phone? On what frequency does it transmit? Calculate the dimensions of an enclosure which is resonant at this frequency, and compare it with your car, the lift, and other metallic enclosed spaces in which you sometimes find yourself.

If you transmit in a screened enclosure, all the energy from the transmitter is concentrated in your body, which is very lossy to microwave energy.

## Launching waves in rectangular waveguide.

Slots and apertures may be cut in waveguide walls. If these slots or apertures do not interrupt the currents flowing on the interior surface of the guide walls, very little radiation escapes from the guide through them.

A probe or loop may be introduced through the opening, and used to excite electric fields (probes) or magnetic fields (loops).

For a waveguide supporting a TE10 mode, a non-radiating slot may be cut in the centreline of the wide face of the waveguide. This allows a probe to be inserted to sample the electric field in the waveguide. Alternatively, a small hole may be drilled on this centreline and a longer probe introduced, formed by the extension of the centre conductor of a coaxial cable.

If the waveguide is closed by a short circuit sheet, lambda(g)/4 away from the probe, the probe will be placed at a standing wave maximum, and will form a radiating antenna transition into the fields inside the waveguide.

As with all transitions in transmission line, if the impedance of the waveguide mode is matched to the characteristic impedance of the coaxial cable, then there is no reflection set up at the junction.

Various parameters may be adjusted to help in the matching process. The hole diameter, the penetration depth of the probe, the probe wire diameter, the spacing from the waveguide short, the dielectric properties and dimensions of any probe sleeving, are all adjustable. Usually it is desired to have a reasonably broadband match, over maybe 10% fractional bandwidth.

The design procedure is a job for the specialist, equipped with electromagnetic field modelling software and CAD packages.

#### Cavities and filters for satcoms applications

Waveguide filters and cavity resonators are larger and heavier than their microstrip, and other transmission line, equivalents. But they have the following advantages for Satcoms applications.

• Power handling capacity
• Lower losses (resistive and dielectric)
• Dimensional stability against vibration, changes in temperature, and pressure.
• Frequency and bandpass response stability.
• Ease of tuning after manufacture
• Robustness