How are imaginary numbers used in electricity?

Answer 1

The very simplified answer is that imaginary numbers put together with real numbers (to make a complex number) can describe the timing of voltage relative to current, or current relative to voltage, in an AC circuit. Let's say that we're driving an AC electrical circuit with an oscillating current source, and measuring a resulting oscillating voltage. Here's the rub:

Purely Real: If you put a resistor in the circuit and measure the voltage oscillations across it, the voltage will be a purely real number. This means that the timing of the voltage peaks will match the timing of the current peaks exactly.

Purely Positive Imaginary: Now, put an inductor in the circuit instead of a resistor and measure the voltage oscillations. It will be a purely positive imaginary voltage. This does not mean that the voltage is non-existent (as many people think)! It simply means that the voltage peaks will be one quarter cycle ahead of the current peaks, or 90 degrees ahead. The voltage has physical value. If you were to touch the ends of the inductor, you would still get shocked! The imaginary property just tells you that the timing is ahead by a quarter cycle, that's all--nothing esoteric or "complicated." A good analogy to this would be if you were riding your bicycle side by side with your friend, and you were pedaling at the same rate, BUT your pedal was consistently a quarter turn ahead of his.. Your timing could be considered purely imaginary relative to him (or her).

Purely Negative Imaginary: Now, put a capacitor in the circuit and measure the voltage oscillations. It will be a purely negative imaginary voltage, which simply means that the voltage peaks will be one quarter cycle behind of the current peaks, or 90 degrees lagging.

Complex: By putting a combination of resistors, inductors, and capacitors in the circuit together, you get a complex voltage, allowing you to get "in between" values. For example, you could carefully size a resistor and inductor, put them in series, and force the voltage peaks to be 45 degrees ahead.

Hope this is clear. If it's still cloudy, I'll paste a link in the web link area that has a site out there with an interactive explanation showing how imaginary numbers can be used with complex numbers to represent both size and timing (it's actually my site, but for educational purposes only).

While these answers mainly deal with electric power [alternating current], the same concepts apply to waves in general which have a phase difference [difference in timing of peaks and valleys of the waves].

Please see the below link for a graph of the fields around current carrying

conductors by the formula: w=(z-1)/(z+1), z=x + iy.