ELI the ICE man

- [Voiceover] Okay, it's time to introduce you to a new friend, ELI the ICE man. ELI the ICE man is a friend of every electrical engineer and what we've been taking about is A/C analysis, and in A/C analysis we limit ourselves to one type of signal and that's a sinusoid, and the sinusoid we like is called cosine. We say cosine of omega t plus phi. Omega represents the radiant frequency of the cosine, here it's shown in blue. That radiant frequency is omega, and phi is the phased layer, the phased shift. And 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 leed, and that's phi. So when phi is a positive number, this whole cosine wave is shifted a little bit to the left. That's what we mean by phase shift. So in these kind of signals, our input into our favorite components, we're gonna get a relationship between the voltage and the current in those components, and it's related by the impedance. We define 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 gonna use a different letter, I'm gonna use e. E is short for EMF or electromotive force, and it's really commonly used, almost as often as v, for representing voltage, and I'll show you why I wanna use e in a little bit. And another way I can write this just as easily, e equals z times i. And this looks a lot like Ohm's law, and what we're gonna find out here is we can apply this in addition to applying it to resisters, we can apply it to capacitors and inductors. So first off we're gonna look at our friend, the inductor. And we're gonna look at the equation e equals z i for an inductor. I'm going to assign i to be a sinusoid, so i is gonna be equal to some magnitude, we'll call it i naught, cosine omega t plus phi. So I'm gonna 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. And now let's write e in terms of this i here, so I can write e equals z, now what is z for an inductor? The impedance of an inductor is j omega l, and what is i? i is sitting right here, and I'm gonna represent i like this, I'm gonna represent i as a phasor, or a phasor representation. And we said that that could be represented as i, the magnitude of the current indicated at the angle of phi, so these are equivalent representations of i, this is the time domain representation, and this is the phasor 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 omega l. 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 gonna assume that this scaling factor omega l 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, multiplying something by j represents a rotation of plus 90 degrees. And so I can write this as e equals, let's put the scaling factor out there, and we'll have i naught, which is the original magnitude of the current, and phi, it's changed here, phi changes, phi becomes phi and this multiplication by j here corresponds to adding 90 degrees to phi. So multiplying by j corresponds to a 90 degree phase shift, and if I draw here, this is now e, and the phase shift, we decided this distance right here, this distance right here is phi, and this distance right here is a phase leed of 90 degrees. And you'll notice I key off the peaks of these wave forms, because that's the easiest place to see the leed. So when I move to the left, that corresponds to a lead of plus 90 degrees. So in an inductor, in an inductor, we say that e leeds i by 90 degrees. Alright, now let's do it for our capacitor. And we'll do the same kind of thing here for capacitor we'll assign the same current, we'll say i equals some current, i naught, times cosine of omega t plus phi, and 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 equals zi or I can write e in the capacitor equals z. Now what is impedance of a capacitor? It's one over j omega c. That's z, and i we represent the same way as we did before, i naught at an angle of phi. So now let's carefully do this multiplication. e equals one over j times one over omega c times i naught at an angle of phi. So here's this one over j term, now I can rewrite one over j as minus j. Now we're multiplying something by minus j, and multiplying by minus j corresponds to a rotation of minus 90 degrees, so I can write e one more time like this, e equals one over omega c, here's the scale factor, here's the original current magnitude, and I get the angle of phi, this time minus 90 degrees. So this minus sign 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, here we have a phase lag, where it's pointing to the right of 90 degrees. And that we call a lag. We can summarize that, we can say in a capacitor, we say e lags i. And an equivalent way to say this is we could say that i leeds e. I leeds 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, and I wanna introduce you to someone who can help you remember this, and his name is ELI the ICE man. So what can Eli tell us? ELI tells us that in an inductor, an l, voltage, leeds current, and over here in a capacitor, c, current leeds voltage. That's the message from ELI the ICE man, he helps us remember the order that voltage and current change in inductors and capacitors. He's gonna be your friend for a long time.