The above animation shows a rotating wheel. On the wheel there is a blue blob which goes round and round. When viewed 'flat on' we can see that the blob is moving around in a circle at a steady (if you computer is working OK!) rate. However, if we look at the wheel from the side we get a very different picture. From the side the blob seems to be oscillating up and down. If we plot a graph of the blob's position (viewed from the side) against time we find that it traces out a sinewave shape which oscillates through one cycle each time the wheel completes a rotation. Here, the sine-wave behaviour we see when looking from the side 'hides' the underlaying behaviour which is a continuous rotation.

We can now reverse the above argument when considering a.c. (sinewave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as 'side views' of something which is actually 'rotating' at a steady rate. We can only see the 'real' part of this, of course, so we have to 'imagine' the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the 'real' portion' of a 'complex' quantity that also has an 'imaginary' part. At any instant what we see is determined by a

The smooth rotation 'hidden' by our sideways view means that this phase angle varies at a steady rate which we can represent in terms of the signal frequency, '

We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can't. At first it seems pointless to create something we can't see or measure, but it turns out to be useful in a number of ways.

Firstly, it helps us understand the behaviour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals. Secondly, it gives us a new way to think about oscillations. This is useful when we want to apply concepts like the conservation of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.

using HTMLEdit3 on a StrongARM powered RISCOS machine.

University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.