More Than You Ever Wanted to Know About Electrical Engineering, Part 22: Resistance, Reactance, and Impedance

We’re starting to look at circuits with time-varying sources. For a lot of the stuff we want to do, this means a sinusoidal source whose voltage and current will look something like this:

What exactly does a time-varying source like this mean for circuit elements?

For resistors, it doesn’t really matter. A resistor just dams up the pipes a little, makes it harder for current to get through. It’ll present the same resistance to any source you throw at it.

Capacitors and inductors are another story.

Remember, these are elements where time is important - they require time to charge and time to discharge. So think about what happens when you expose a capacitor to the time-varying voltage above.

Let’s say the capacitor starts out fully charged to Vpk at t0. When it’s fully charged, remember that it acts like an open circuit - no current can flow through it. That’s going to change, though - the source voltage starts falling as soon as it passes t0. Now the capacitor is holding onto the highest voltage in the circuit and it starts to discharge. It’ll discharge increasingly rapidly as the source voltage drops, allowing more and more current to flow, and then once the source voltage starts climbing again, it’ll start recharging and letting less current flow as it gets closer to a full charge.

So in a circuit with a time-varying voltage like this, you are forcing capacitors and inductors to constantly charge and discharge. As a capacitor charges (or an inductor discharges), it lets less and less current through. When it starts discharging (or charging in the inductor’s case), more and more current can flow. Since the changing source forces the element to rubberband between these two states, there’s going to be some average impediment to the flow of current caused by a capacitor or inductor. This average quantity is called *reactance*, usually written as X. Its exact value will depend on the frequency of the source - that is, how long the element is allowed to charge or discharge - and the capacitance or inductance of the element - that is, how much of a charge it can hold. Here’s what the reactance looks like for a capacitor and an inductor:

Note that at higher frequencies, a capacitor has less reactance, but an inductor has more reactance.

The combined resistance and reactance of a circuit is called the impedance, Z. Z is written like this:

Note the imaginary number cropping up again and the similarity to the way we described sinusoids. You can convert this to polar form if you like, same as with sinusoids, and it will make dealing with circuits with time-varying sources and capacitance or inductance a thousand times easier to deal with mathematically. You’ll sometimes see resistance called the real component of impedance and reactance called the imaginary component. It’s a misleading way of talking about it - reactance is very much a real phenomenon and has a definite impact on circuit behavior. The “imaginary” indicates that what’s impeding the flow of current doesn’t really have to do with a “blockage” per se - it’s just a consequence of the way capacitors and inductors store energy. A resistor dissipates energy - a capacitor or inductor holds some of it back so that it’s not available for immediate use.