Impedance

Now we're going to talk about the idea of impedance. This is a really important idea in electronics, and it's something that comes from the study of AC analysis. AC analysis is where we limit ourselves to inputs to our circuits that look like sinusoids, cosines, or sines. Of all the signals that we could possibly have in the entire universe, we're going to limit ourselves just for the moment to sine waves.

There's some great simplifications that emerge from this. So in this video, we're going to look at— we're going to develop basically the IV equations for our three favorite passive components: resistor, inductor, and capacitor. We're going to look at those when the input is a sinusoid. So that means that I or V, the voltage or the current, is in the shape of a sinusoid. We're going to see what that means for the IV equations for our favorite devices.

As we look at our IV equations with sinusoid inputs, we're actually going to break down sinusoids into complex exponentials. When we studied sinusoids, we found out that we could disassemble sinusoids into complex exponentials using Euler's equation. For example, if we have a cosine wave, if we have a cosine as a function of time, omega t, we could express that in terms of complex exponentials like this: one half e to the plus j omega t plus e to the minus j omega t, like that.

What I'm going to do now is I'm going to say, let's look at what happens when we use this as an input signal. This is not a real input signal; it's an imaginary rotating vector. But if I have two of these, I can reassemble them into a cosine wave. We like to use these exponentials because they go through the differential equations of a circuit really easily. These are the inputs that we know how to solve when we do differential equations.

So what I'm going to do is develop the IV equations for the resistor, inductor, and capacitor in terms of this kind of an input. When the voltage or the current looks like this, what do those equations look like? So we're going to start with— we'll start with the resistor. Here's a resistor, and we know from that that just Ohm's law is V equals I times R. Just for the moment, I'm going to assume that the current, let's assume that I equals e to the plus j omega t.

So if this is I, what is V for a resistor? Well, we just plug I in here, and we get V equals R times e to the plus j omega t. All right, now I'm going to do something that looks like it's a little too simple, but it's going to get interesting soon. I want to look at the ratio of voltage to current in this situation where we're driving with this complex exponential.

So voltage turned out to be R e to the plus j omega t, and that's the voltage I get if I put a current through the resistor of e to the plus j omega t. What does that equal to? Well, these two are the same, so they cancel, and I get the ratio of voltage to current is R for a resistor. So for a resistor, we just proved that V over I equals R. This isn't news; we didn't make a discovery here; this is just Ohm's law, and for a resistor, the voltage over the current is always equal to the resistance.

All right, this is going to get more interesting now as we go do inductors and capacitors. So let's do an inductor. It has a value of L henries, and for an inductor, we know that voltage equals L dI/dt. All right, and let's do the same thing again; let's let I equal e to the plus j omega t. So it's a complex exponential current that we're forcing through our inductor, and let's go ahead and work out what V is.

So V equals L times d/dt of this value here, e to the plus j omega t. Or V equals— now we take the derivative, and the j omega term comes down to multiply L, so we get j omega L times what? Times the same thing, e to the plus j omega t. This is the beautiful thing about exponentials; they give us back themselves.

All right, so now let's do this. Let's take once more— what's the ratio of voltage to current? And that equals— here's the voltage, j omega L times e to the plus j omega t. And let's divide that by I, which is I is right here; I is e to the plus j omega t. So those cancel, and we get V over I equals j omega L.

So now we have an equation for V over I for an inductor, and this is interesting. This time we have the inductance value, which we expected, and there's also this omega j omega term that comes in. So this tells us— this is frequency; omega is frequency. So this tells us that the ratio of V to I for an inductor is dependent on frequency.

Now we'll do the same thing for a capacitor. So here's a capacitor, and there's its capacitance value, inference. For the capacitor, we know that I equals C times dV/dt. And this time, let's let V of t— let’s let V equal e to the plus j omega t. So this time we're going to force a voltage across our capacitor that is this imaginary, this complex exponential.

That gives us— let's plug that in here; I equals C times d/dt of e to the plus j omega t. And let's take that derivative; I equals— same thing happens, j omega comes down to multiply out in front with C, and we get the same thing over here. So we get j omega C times e to the plus j omega t.

Now we'll ask the same question again that we did before, which is: what is V over I for a capacitor? We can fill this in; V is sitting right here. V is e to the j omega t, e to the plus j omega t, and the current is— we work that out down here; that's j omega C times e to the plus j omega t.

Once again, we get this nice cancellation here; this cancels with this, and for a capacitor, we get V over I equals 1 over j omega C. Let me put a box around that one too. This says that the ratio of voltage to current in a capacitor depends on the value of the capacitor, of course, and it also depends on frequency. So just like over with the inductor, we have a frequency term in here.

Now we're going to give this ratio of voltage to current in all three cases a special name, and that name is impedance. The symbol we often use for impedance is Z. So this word impedance is the general notion of the ratio of voltage to current, and we can do that for all three of our passive components. For a resistor, the impedance is its resistance R.

So the word impedance is like the word resistance, except it's a more general concept; it's the general concept of voltage divided by current. For a resistor, the impedance is the resistance. For an inductor, the impedance V over I is j omega L, and down here for a capacitor, the impedance is 1 over j omega C for a capacitance.

So this is where the idea of impedance— this word impedance— this is where it comes from. The idea includes both the values of the components and the effect that frequency has on the voltage-to-current ratio. So both of those things are combined into one idea.

Just a quick summary of impedance: if we say the impedance of a resistor, that equals R; the impedance of an inductor equals j omega L; and the impedance of a capacitor equals 1 over j omega C. As a reminder of the assumptions we made, we said that we're only going to consider sinusoidal inputs, and what we did is we broke up our sinusoid, our cosine wave, into these— we looked at how these rotating vectors pass through our components in the form of voltages and currents.

So there's no new physics here. What happened was we took these j omegas that came out of the exponentials when we took the derivatives; we associated those with the component itself. We did that here, and we did that here, and you can see it here. So we've sort of just done a— it's somewhat of a notational trick.

We associate this frequency dependence not with the inputs and the voltages and currents, but with the components themselves. This is something we call— this is referred to as transforming. We've transformed our components, and that's the phrase that's used there. But from this comes the idea of impedance in the general sense of the voltage-to-current ratio.