Now to do that we're going to define a couple of, of, parameters that

characterize this RLC circuit. The first one is its resonant frequency.

The resonant frequency squared is 1 over LC.

The other quantity is the quality factor, or the Q factor, and the Q is 1 over the

resonant frequency, times RC, all in the denominator.

Now, the Q, also to give it a little intuition.

I can write this as one over 2 pi times T over RC where T is the period of the

oscillation of this circuit. Now what happens in a circuit like this.

It's just like. It's the electrical analog of the mass on

a spring. The inductor is playing the role of the

mass. The capacitor plays the role of the

spring. And what happens is the, the electrical

energy here goes in one instant there's a, a large current flowing, and I have

magnetic flux stored in the inductor. And then, that current flows and the

capacitor gets charged up. So it's kind of, when the capacitor's

charged it's like compressing the spring. And so the capacitor's totally charged.

At some instant, the current stops flowing and it's going to turn around and

start flowing in the, in the other direction which is exactly like the

motion of the mass on the spring. This, the mass moves on, on to the end of

the compression of the spring, and then the spring is pushing back, like the, the

capacitor is pushing back. And then the, the mass starts to move in

the opposite direction. So, the current and this circuit is going

to oscillate back and forth, it's going to flow.

this way, first the capacitor's going to get charged up, after the capacitor is

fully charged it's going to push the current back through the inductor.

But the inductor is going to produce a voltage when I try to push a current

through it. It's going, and this is where Lenz's Law

is so important here. The voltage that the inductor tries to

induce back in the circuit is opposite the change, and so the inductor pushes

back. And so this current sort of sloshes back

and forth between the inductor and the capacitor in this circuit.

Now, the frequency at which that sloshing of charge takes place is this resonant

frequency. Now, this is the angular frequency.

To put this in hertz, I would take omega r and divide by 2 pi.

Now this factor, this time, that's 1 over the frequency, and the frequency is omega

over 2 pi, so that's where this 2 pi comes from.

But what the Q is really is the ratio of the characteristic time of the

oscillation to the RC time constant of the circuit.

So the RC time constant is a measure of the amount of loss in the circuit, and

capital T here is the time scale of the oscillation.

And so it's really the ratio of those two quantities that determine the quality

factor. So if the RC time constant is extremely

short then the Q factor becomes very large.

And vice versa if it's very long. Then the Q factor becomes small.

So this is just a way to really clean up this this formula here.

And it's, it's a way to compare oscillators with different R, L, and C

values to because you change L and C in complementary ways and, and retain the

resonant frequency, but then that's going to change the Q in, in predictable

but sometimes rather complex ways. So rewriting the, this expression using

these factors, omega r and Q. You get something like this.

So I'm not going to work that out for you right now but take and, with your paper

and pencil, work that out for yourself and I hope you get this formula.

I don't think they made any mistakes there.

So there is the output voltage. It's the signal voltage times this

factor. Now, what we're concerned with is going

to be the magnitude of the output voltage.

Now, the signal voltage, that's just real, so when I take the magnitude, it's

just that number still. But here's the factor down in the

denominator. It's going to be the, computing the

magnitude of this complex expression here.

And so, it's the real part squared plus the imaginary part squared.

And then it's the square root of that entire expression.

And so this term, if you look at it for a minute, when omega equals omega r, this

factor becomes 1, and that's going to cancel that.

And so this is going to go to 0 somewhere.

And so this some frequency omega equals omega r, where this part vanishes and

that I'm only left with this part. Now, at that point, that's going to be

the peak, in the response. Because, when I make the denominator as

small as possible, the, 1 over that, will become, as large, as it's ever going to

be. Now, on the peak, when omega equals omega

r we just have this expression and so, if I if I were to take and let this be zero

and then take this square root, it would just be Q omega r, over omega.

And so, the, height of the peak is proportional to Q.

So the higher the Q the higher the peak, in the response.

Now, this is probably, hard to visualize now but lets take a look at some actual

numbers and plot some of these transfer functions.