ELI the ICE man

Okay, it's time to introduce you to a new friend: Eli the Iceman. Eli the Iceman is a friend of every electrical engineer, and what we've been talking about is AC analysis. In AC analysis, we limit ourselves to one type of signal, and that's a sinusoid. The sinusoid we like is called cosine. We say cosine of Omega t plus φ.

Omega represents the radian frequency of the cosine. Here, it's shown in blue that radian frequency is Omega, and φ is the phase delay or the phase shift. If we look here, we see this isn't really a cosine wave because the peak is a little before zero time equals zero. So this distance right here is the lead, the phase lead, and that's φ.

When φ is a positive number, this whole cosine wave is shifted a little bit to the left. That's what we mean by phase shift. When these kinds of signals are input into our favorite components, we're going to get a relationship between the voltage and the current in those components, and that's related by the impedance.

We defined the idea of impedance as the ratio of voltage to current. We gave that the symbol Z. Now, in this video, instead of using v as my variable for voltage, I'm going to use a different letter. I'm going to use e. e is short for EMF or electromotive force, and it's really commonly used almost as often as V for representing voltage. I'll show you why I want to use e in a little bit.

Another way I can write this just as easily is e = ZI, and this looks a lot like Ohm's law. What we're going to find out here is we can apply this; in addition to applying it to resistors, we can apply it to capacitors and inductors. So first off, we're going to look at our friend the inductor, and we're going to look at the equation E = ZI for an inductor.

I'm going to assign I to be a sinusoid, so I is going to be equal to some magnitude; we'll call it I_KN cosine(Ωt + φ). So I'm going to say my current is a cosine wave of this magnitude with this phase delay, and that's shown in blue here. So this here is I. Now let's write e in terms of this I here.

So I can write E = Z. Now, what is Z for an inductor? The impedance of an inductor is JΩL. What is I? I is sitting right here, and I'm going to represent I like this. I'm going to represent I as a phaser or a phaser representation. We said that that could be represented as I, the magnitude of the current, indicated at the angle of φ.

These are equivalent representations of I. This is the time domain representation, and this is the phaser representation. Now what we have out here in front of I is a scaling factor. There's this complex J that we'll take care of in a second, and there's ΩL. So Omega is the frequency, and L is the size of the inductor.

Now, for the purposes of this video, when I plot out the voltage over here in orange, we're going to assume that this scaling factor ΩL is one, just so that we can focus on the timing relationships between the current and the voltage. When we talked about complex numbers, multiplying by J represents a rotation of +90°.

So I can write this as E = (scaling factor) * I_KN, which is the original magnitude of the current, and φ gets changed here. φ changes; φ becomes φ + 90°. This multiplication by J corresponds to adding 90° to φ. So multiplying by J corresponds to a 90° phase shift, and if I draw here, this is now e.

The phase shift, we say this distance right here is φ, and this distance right here is a phase lead of 90°. You'll notice I key off the peaks of these waveforms because that's the easiest place to see the lead. So when I move to the left, that corresponds to a lead of +90°.

So in an inductor, we say that E leads I by 90°. Alright, now let's do it for a capacitor. We'll do the same kind of thing here for a capacitor. We'll assign the same current, we'll say I = I_KN cosine(Ωt + φ). Now let's work out the voltage across the capacitor.

So the voltage across the capacitor e is the same thing we have here, E = Z. Or, I can write e in the capacitor equals Z. Now, what is the impedance of a capacitor? It's 1/(JΩC). That's Z. And I we represent the same way as we did before: I_KN at an angle of φ.

So now, let's carefully do this. This multiplication e = 1/J * (1/(Ω I_KN)) at an angle of φ. So here's this 1/J term. Now, I can rewrite 1/J as -J. Now we're multiplying something by -J, and multiplying by -J corresponds to a rotation of -90°.

So I can write e one more time: e = (1/(ΩC)) * I_KN at an angle of (φ - 90°). This -90° here corresponds to a lag, a phase lag. So here's our original current here; let me label that. Here's I, and now we have our voltage e, looks like this. Here’s e, and what we see...

Let me go out here and measure it. Here we have a phase lag; we're pointing to the right of -90°, and that we call a lag. We can summarize that. We can say in a capacitor that e lags I. An equivalent way to say this is we can say that I leads e; I leads voltage.

So I can actually put boxes around these two results here and here. Now, there's a lot of sign flipping going on here, and there's actually an easy way to remember this. I want to introduce you to someone who can help you remember this, and his name is Eli the Iceman.

So what can Eli tell us? Eli tells us that in an inductor (L), voltage leads current, and over here in a capacitor (C), current leads voltage. That's the message from Eli the Iceman. He helps us remember the order that voltage and current change in inductors and capacitors. He's going to be your friend for a long time.