Now that inductors and capacitors have been introduced, and the idea of a phasor has been shown to you. We're going to start trying to use phasors in AC, alternating current, circuit analysis. So now the first thing we have to do is let's assume that the current and voltage as functions of time are just sinusoidally varying functions. And so we're going to represent a general sinusoidally varying current as some number. I of omega, i with a tilde over it, of omega times e to the j omega t. So e to the j omega t is this unit phasor at frequency omega and I at that frequency. The same thing for the voltage. Now this I and V, those are the phasor amplitudes that we're going to talk about. These are both time independent quantities, they do not depend on time explicitly. All of the time dependencies here in this sinusoidally varying function, but these may be complex numbers. Now, this is worth a little bit of discussion. It looks like we, specialized our current voltage to be something very specific, just a sinusoidally varying function. Now the point here, and we'll talk more about this closer to the end of the course. The point here is we really have not lost any generality at all. We haven't specified the frequency. This can be any frequency that you want, from zero to infinity. And there's something that we're going to look at later on called Fourier analysis, which says that you can construct any arbitrary function by adding together appropriate combinations of sine and cosine functions. And so, if we're able to analyze a circuit for one frequency or in more to the point for any frequency. Omega is unspecified, then we can figure out what that circuit response is going to be in response to any arbitrary function. Because the this is something very important. All of the circuits that we're looking at here are linear, and I can add together the response of the circuit at every frequency and I can add those together these various frequencies together to create any arbitrary function of time. So, solving for the response of a circuit, at any old frequency, omega, an unspecified frequency, enables us to ultimately figure out the response of that circuit to any arbitrarily complicated function of time. So, for now, all we have to do is worry about what's the response of a circuit for a sinusoidal input at some frequency omega, where omega is general. It can be any frequency that you want. So we're really solving a whole bunch of problems at once by leaving omega explicitly as a variable. Okay, now, hopefully that didn't completely lose you in in the weeds, but let's just proceed, and I think sometimes it's better to just kind of plow through and start using things and some of the more subtle points will become more clear as you become more familiar with what's going on. So the first thing I'm going to do this I'm going to drop the omega dependence explicitly and just use I with a tilde and V with a tilde to represent the phasor current and voltage. Now let's go back and look at a capacitor again And we had the expression that related the current and voltage for a capacitor. It's I is C times the time rate of change of the voltage. Now, the nice thing here is if I assume this form, with an E to the J omega T and assume that the I and V in front, this I and this V. Are not time dependent. They're just functions of frequency. I can take that derivative. So Ie to the j omega t. That's the I. Is C times ddt. The time derivative of V. And v is written as v phaser e to j omega t. So now if I go and I take this derivative. I know this is a constant so it just comes out front. And then when I take the time derivative of e to the j omega t I get a j omega factor and the e to the j omega t back again. So I have this equation, i with the exponential factor equals CV times j omega multiplied by the same exponential factor. So I can cancel out the either the j omega t's on both sides, and I just have I is j omega C, times V. Or I can take and put this in the form of V equals something times I. So this is to make it look more like Ohm's law, like V equal IR. And this thing out in front, this factor out in front, 1 over j omega C. Is like a generalization of the idea of resistance and it's called impedance. And so, the impedance of a capacitor is 1 over J Omega C. So, this formula, here resembles Ohm's law. V = IR The generalization of that is v equals i times an impedance. And the impedance of the capacitor is 1 over j omega C. So the impedance, or the thing that the generalized resistance if you will of this capacitor. Now it depends upon frequency. another thing worth saying is that at omega equals 0, the impedance goes to infinity, and as omega becomes very large, the impedance becomes very small. So, let's take a look at inductors and do the same kind of analysis. So the inductor equation is v is l di dt. And I'll plug in v and i in that equation in the phaser form from up here and take the derivative and I'll bring down the j omega term. And so I had V where the exponential is LIj omega times the exponential. I can cancel these factors again, and I just get V is J omega L times I. So this is written in the same format as this equation, V is something times I. And that something is the impedance, and the impedance of an inductor is j omega L. So inductors behave oppositely from capacitors. At omega equals zero, the impedance of an inductor goes to zero. And. Add, as omega becomes very large, the impedance of the inductor becomes very large. Now the other thing is this J factor that's, here. That is going to tell you about the phase relationship between the current And the voltage in a capacitor and an inductor. And more about that as we go along.