Now, the output voltage going back to here is that Z2 impedance, so here it is

right here. Divided by the sum of R plus that Z2.

So, here is the output voltage. Now, we can rewrite that.

I want to take and clean things up a little bit by multiplying this through

the denominator. And this is a, a little exercise for you

to start with this formula and reduce it to this format.

Now, we're going to, to clean things up a little bit.

We're going to define two quantities here.

One of them, omega 0, is the resonant frequency of this LC tank circuit here.

And that's 1 over the square root of L times C.

And then, the other thing that we defined we talked about this a couple of weeks

ago, is the Q factor. The Q factor is omega 0.

The resonant frequency times L over R. And so, if you take, and you, you work on

this, work on the algebra of this a little bit, you should be able to reduce

it to this expression. now, what I did also is there is an extra

step here. You work out.

This is the full complex equation here. I want to compute the magnitude of this.

And so, after I have written this all out, I then take the the real part

squared plus the imaginary part squared, square root and that's where this comes

from. So this is a a little bit of a challenge

for you. But I want you to work this out.

this is the output as a function of the input.

And so, it has a Q up here, there's a Q down there and it has this interesting

resonance response. Now, if we take then and, and plot what

this looks like. So, this is the magnitude, remember.

And you can do this in Excel. I, I did this in Excel myself.

And so, I'll plot this as V out over V in as a function of the frequency.

But I'm going to normalize the frequency to the resonant frequency.

So, it's omega over omega nought is the variable that I'm going to use because it

always appears in that combination. Omega over omega nought.

So, my frequencies are always expressed just as a fraction of the resonant

frequency. So, when omega equals omega nought,

you're on resonance for that LC combination.

Now, this what I did here was just showed this curve for different values of omega

nought. one, a half, and two.

And I assumed that the cube was one over the square root of two in each case.

And so just by changing the value of omega nought, plotting this out versus

frequency, you see this kind of response. When omega equals omega nought.

So, let's look at the green curve. And, and the when omega is, when you're

on resonance here. What happens, omega over omega nought is

1. And then, this factor is going to go

away. I've got 1 minus 1, that's gone.

This is 1. There's a Q squared down here.