Impedance vs frequency

In this video, we're going to continue talking about AC analysis and the concept of impedance as the ratio of voltage to current in an AC situation. Just as a reminder of the assumptions we've made for AC analysis, we've assumed that all of our signals are of the form of some kind of a sinusoid, like that cosine is the typical one we pick: cosine(Ωt) plus some offset angle.

We're also going to use Euler's equation, which says we can decompose our cosine into two complex exponentials. So one of the complex exponentials is e^(+jΩt + φ), and the j multiplies both of them, plus e^(-jΩt + φ). When we use this theorem, what we do is we love putting these things through differential equations, so we track each of these one at a time through our circuits. Typically, what I'll focus on is the plus term, but the minus term, all the math is the same except there's that one little minus sign.

When we made up the concept of impedance, we assumed that V or I was of this form, this complex exponential. This made it so when we put that through our components, we came up with the idea of impedance. As a review for a resistor, the impedance of a resistor turned out to be just R—that's the ratio of V to I in a resistor. For an inductor, we decided Z of an inductor was equal to jΩL, and finally, for a capacitor of value C, Z_C equals 1 / (jΩC).

So those are the three forms of impedance for our three favorite passive components. The units of these are in ohms. R is measured in ohms—that's the normal Ohm's law—and we use the same units for the impedance of inductors and capacitors. This is in ohms, and this is in ohms. It may seem a little funny because there's this frequency term here, but this is what it is.

In the rest of this video, I want to qualitatively look at the value of these impedances and specifically look at what happens when there's a range of values of the Ω term: what happens at zero frequency, low frequency, high frequency, and infinite frequency. So let's go ahead and do that. We're going to look at the impedance terms at different frequencies, and we'll measure frequency as radian frequency in this.

Let's talk about zero frequency, low frequency, high frequency, and infinite frequency. We'll build a little chart here. All right, so these are the values of Ω here, and now we'll do our components. First, we'll do our resistor, and we're going to fill in the table for Z_R, and we know that equals R. So at any frequency, R is just R; couldn't be simpler. At zero frequency, which is just called DC or a battery, R is R. At any low frequency, R is R; R is R at infinite frequency. So there's no dependence on frequency in R.

Now, let's do the inductor. We decided that Z of an inductor was jΩL. Let me do something very specific; I'm going to get rid of this j. I'm going to basically say I want to just look at the magnitude of the impedance. If we just look at the magnitudes, the magnitude of Z inductor is ΩL.

So now let's fill our table in for ΩL. When Ω is zero, the magnitude of the impedance is zero for an inductor. When the frequency is low, when Ω is low, the impedance is going to be relatively low. As the frequency gets high, then ΩL becomes a larger number, so it becomes high. If we let the frequency go to infinity, then Ω becomes infinity and this becomes infinity.

So we see in an inductor, from low frequency to high frequency, the impedance of the inductor goes from zero up to infinity. All right, let's fill in the last one here. Here's our capacitor, and Z of a capacitor equals 1 / (jΩC). The magnitude of the impedance is just 1 / (ΩC). So let's plot that out. What is 1 / (ΩC) when Ω is zero? Well, it's infinity.

And what if Ω is a small number? If Ω is a small number, then this is a large number; this is a high number. If the frequency gets very high, the higher Ω gets, the smaller the magnitude of Z gets, so we get low here. You see it's anti-symmetric with the inductor. Finally, if we let the frequency go to infinity, 1 / (∞ * C) is about zero.

With this chart, I can show you some of the words that we use—some of the slang words or jargon words that we use in electrical engineering—to describe the behavior of L and C at different frequencies. As we said, R is R at any frequency, so resistance is constant. An inductor changes over frequency, over this big range of frequency.

At zero frequency or low frequency, the impedance is very low, so this leads to the expression: we say an inductor looks like a short. An inductor looks like a short at low frequency or zero frequency. Now, let's go to high frequency: the impedance gets high, and it eventually goes to infinity—that's the impedance of an open circuit. So we say L equals an open at high frequency, and it's a short at low frequency.

Let's do the same sort of jargon investigation here for our capacitor. A capacitor at zero frequency has an infinite impedance, so it looks like an open at low frequency. Now, let's look at high frequencies: at high frequencies, the impedance becomes very low, and at infinite frequency, it becomes zero. That's the impedance of a perfect short, so the expression, the slang expression you'll hear is that a capacitor is a short circuit at high frequency.

The reason I'm telling you about this is that very often experienced engineers will talk to beginners and use these kinds of terms—that "an inductor looks like a short at low frequency," and I wanted to show you where these things come from, where these terms come from, and what they mean. This is not a simple idea. You saw how many assumptions we made and how much work we did to get to this idea of these sort of common or familiar expressions for how our components work.

I hope that helped for you to see how this comes about.