How to Derive Impedance in an LCR Circuit at Various Frequencies?

[Blue Kangaroo]

Homework Statement

a. express the equation of motion
b derive impedance at the resonant frequency (ω=ω_o)
c derive impedance at very high frequencies (ω>>ω_o and ω>>R/L)

Homework Equations

V_o=ψ+(R/L)ψ+(1/LC)ψ
Z=(1/iω)K(ω)
K=1/((s-mω²)+ibω)

The Attempt at a Solution

Part a was simple, that was the first equation. I don't know how to put dots to signify derivatives though.
For b, since ω=ω_o, it will equal (1/LC)^0.5, so I think I got it right in the scratch work I uploaded.
For c, I'm not quite sure where to start. Since ω>>ω_o, can I just take ω_o to be zero and disregard it so that there will be no (1/LC)^0.5 term?

Sorry, my handwriting is not the best. I'm quadriplegic, so it's not the easiest thing for me.

 

[BvU]

Hi,

how to put dots to signify derivatives though

I wouldn't know how to do it with the symbol buttons. You could use single quotes and double quotes. For the dots, all I can do is show how to do it using $\LaTeX$ : You type $$ L \ \ddot q + R\ \dot q + \frac {Q}{C}\ q = V_0 $$ and you get
$$ L\ \ddot q + R\ \dot q + \frac {1}{C} \ q = V_0 $$ the equation for a series LCR circuit (which I have to assume your problem statement mentions ? )
( here is where I stole the notation -- I added the " \ " to get more spacing)

So it looks like your $\psi$ is actually $q L$ ? (did you miss the 'all variables' in part 1 of the template ? -- gives you the chance to also tell us what m and s and K stand for ...)Next question: are you familiar with complex impedances ? It makes the math a lot easier:
$$Z \equiv {V\over I} = R + j\omega L + {1\over j\omega C}$$and $|Z|$ should give a nice expression for $\omega_0^2 = {1\over LC}$

 

[Blue Kangaroo]

The problem never explicitly stated that it was a series circuit. I've uploaded the homework. It's the last problem on the page.

Yes, Ψ should be qL, so the equation is (1/L)V_o=q"+R/Lq'+(1/LC)q

Sorry, Z is impedance, K is compliance and I'm honestly not quite sure what s and b are.

I am not too familiar with complex impedances, but it would probably behoove me to learn.

 

[BvU]

never explicitly stated that it was a series circuit

But the picture surely does !

K is compliance

I feel somewhat stupid never having heard of that in circuit analysis. You write $Z=\displaystyle {1\over i\omega}K(\omega)$. Is that its definition ? $\ \ i\$ is the currrent, or $\ i\$ is something that has $i^2 = -1$ ? $\ \ i\$ also appears in the denominator of K ?

not quite sure what s and b are

you also use the symbol $\ m\$ ?. I see a lot of similarity with expressions in the link. But not for $s$...

PS make sure you distinguish between $\omega$ and $\omega_0$ when you write your expressions.

 

[Blue Kangaroo]

I see, I stand corrected.

That is the equation relating Z and K that we were given in class. I think I found what I was doing wrong though.

 

[BvU]

So what's the situation now ?