Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and 'imaginary' numbers to explain things in the real world seem crazy! Once you get used to them, however, they do make a lot of things clearer. The problem is understanding what they 'mean' and how to use them in the first place. To help you get a clear picture of how they're used and what they mean we can look at a mechanical example...
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 phase angle which varies smoothly with time
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, 'f'. The complete complex version of the signal has two parts which we can add together provided we remember to label the imaginary part with an 'i' or 'j' to remind us that it is imaginary. Note that, as so often in science and engineering, there are various ways to represent the quantities we're talking about here. For example: Engineers use a 'j' to indicate the square root of minus one since they tend to use 'i' as a current. Mathematicians use 'i' for this since they don't know a current from a hole in the ground! Similarly, you'll sometimes see the signal written as an exponential of an imaginary number, sometimes as a sum of a cosine and a sine. Sometimes the sign on the imaginary part may be negative. These are all slightly different conventions for representing the same things. (A bit like the way 'conventional' current and the actual electron flow go in opposite directions) The choice doesn't matter so long as you're consistent during a specific argument.
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.
Content and pages maintained by: Jim Lesurf (firstname.lastname@example.org)
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.