How Does Resonance Frequency Calculation Work in LC Circuits?

[Little Dump]

This isn't really a homework question. I'm just trying to figure things out for myself about resonance in a circuit that they haven't taught us yet.

Say you have a circuit

Rp || L || C and you have a sine source and some load connected to it call it Re and you want to mesaure the gain. Vi being across the sine source, Vo being across the load resistor.

So you get somehing like this:

$$H = \frac{V_o}{V_i} = \frac{R_e*R_p + R_e*j*w*L - R_e*w^2*L*C*R_p}{R_e*R_p + R_e*j*w*L - R_e*w^2*L*C*R_p + j*w*L*Rp}$$

So you take the derivative with respect to j*w and get something even uglier then that.

Set it equal 0 and solve for j*w and you get

$$j*w = \frac{1}{\sqrt{L*C}}$$

But don't you actually want to solve for w? Not j*w? In which case if you solve for w you get a complex resonant frequency. Which to me makes no sense. And is wrong because I know the formula for the circuit in question is:

$$w_o = \frac{1}{\sqrt{L*C}}$$

What did I over look?

 

[learningphysics]

This isn't really a homework question. I'm just trying to figure things out for myself about resonance in a circuit that they haven't taught us yet.

Say you have a circuit

Rp || L || C and you have a sine source and some load connected to it call it Re and you want to mesaure the gain. Vi being across the sine source, Vo being across the load resistor.

So you get somehing like this:

$$H = \frac{V_o}{V_i} = \frac{R_e*R_p + R_e*j*w*L - R_e*w^2*L*C*R_p}{R_e*R_p + R_e*j*w*L - R_e*w^2*L*C*R_p + j*w*L*Rp}$$

So you take the derivative with respect to j*w and get something even uglier then that.

Set it equal 0 and solve for j*w and you get

$$j*w = \frac{1}{\sqrt{L*C}}$$

But don't you actually want to solve for w? Not j*w? In which case if you solve for w you get a complex resonant frequency. Which to me makes no sense. And is wrong because I know the formula for the circuit in question is:

$$w_o = \frac{1}{\sqrt{L*C}}$$

What did I over look?

Did you take the derivative with respect to jw properly... wherever
you have w^2, rewrite it as -(jw)^2. Also, I'd then replace all the jw, with a variable x, and take the derivative with respect to x. This just makes things a little easier to look at...

 

[Little Dump]

That's exactly how I did it. I got maple to do it for me, and then I did it by hand. Same thing.

 

[clive]

Try again with
$$H=\frac{|V_0|}{|V_i|}$$