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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.

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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.

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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.

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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.