### 10.1 Wires as waveguides.

At low frequencies (up to about 1 GHz) we can carry electromagnetic signals over short distances using metal wires. These signals are usually characterised in terms of the voltage between & currents along the wires. The power being sent from place to place is usually written in terms of something like

where *V* is the voltage between the wires, *I* is the current flowing along them, and *P* is the power. This leads us into thinking about electrical signals using the “water in a hosepipe” model. Although useful, this picture is misleading since it leaves us with the impression that it is the movement of the electrons along the wires which carries the power.

In reality, the kinetic energy of the moving electrons is only a tiny fraction of the energy of an electronic signal. The overwhelming majority of the energy is carried by the __electromagnetic fields__ which link and surround the wires. Virtually all of this field is __outside__ the metal of the wires. Although we don't usually think in this way, it is correct to treat connecting wires (including 50Hz 230V mains cables and the copper patterns on printed circuit boards) as a form of * waveguide*. The cables, wires and pcb tracks ‘guide’ the electromagnetic fields which convey the power from place to place.

### 10.2 Modes and microwave guides.

Normal metal wiring only works well as a way of guiding EM waves when the dimensions of the wires & the distances between them are much smaller than the free space wavelength of the signals. As we progress to higher frequencies there is a tendency for wires to act as *antennas* and allow the signals to ‘radiate away’ instead of guiding them from place to place. To overcome this problem microwave engineers invented *metallic waveguide*.

A normal wire consists of a strip of conductor surrounded by free space (or a dielectric). Any fields which radiate from the metal can move away, never to return. A standard microwave guide turns this arrangement ‘inside out’ and consists of a strip of space (or dielectric) surrounded by metal — in effect, a metal pipe. The EM fields are now sent from place to place inside the pipe. Using this arrangement we find that any fields which try to radiate away from the metal simply bounce of the opposite wall of the pipe. Hence the EM power is ‘trapped’ and forced to flow along the pipe.

The field pattern which exists in the pipe must obey *Maxwell's Equations*. It must also obey the appropriate *boundary conditions*. For an EM field in contact with a metallic surface the main boundary condition is that the electric field component in the place of the metal surface must always be essentially zero. Using this condition for a rectangular pipe of the type illustrated in figure 10.1 we can solve Maxwell's equations for an EM field sinwave of frequency and discover a series of E-field vector solutions of the form

where

is a measure of the magnitude of the H-field in the guide

and a related set of equations specify the H-field pattern in the guide.

This set of solutions are called the *TE-modes* where ‘TE’ here stands for *Transverse Electric* since the electric field vector is always transverse to the long axis of the guide. In fact there are another, similar, set of solutions which are the *TM-modes* which have a transverse magnetic field. Most practical applications use the TE modes & we can make all the points we need just considering the TE electric field patterns, so we'll ignore the TM modes & the H-field equations.

The integers, & are called the *mode numbers*. The choice of these values determines the shape of the field distribution across the guide. Since each choice of *m* & *n* represents an independent valid solution of Maxwell's equations the field in the guide can, in general, be made from a linear superposition of these modes. i.e. we can imagine propagating a field pattern

where the values are a set of complex coefficients which determine the relative amplitudes and phases of each modal contribution to the total field.

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