RLC Circuits Problem: Find Frequency, Decay Time & Current

[ayoubster]

Homework Statement

A certain circuit has a resistance of 30 Ω, inductance of 5.0 mH, and capacitance of 0.375 µF. At time t=0, the capacitor is charged with 4.0 µC on the top plate (and -4.0 µC on the bottom), and the switch is then thrown so that the capacitor can discharge through the inductor and resistor.
(a) What is the frequency at which this circuit will oscillate?
(b) How long will it take the circuit to decay to an amplitude of 1% the initial amplitude?
(c) Suppose we took out the capacitor, and instead allowed a current to decay through the resistor and inductor. How long would it take to reach 1% of the initial current?

Homework Equations

ω = √((1/LC) - (R/2L)^2)
ω = 1/√LC
q = q_oe^-t/τ
V = V_oe^-t/τ
where τ = RC

The Attempt at a Solution

Plugging in my values for a) gives me 22.9kHz, which seems reasonable, however I'm not sure if I should be using the first frequency equation or the second.
Moreover, I have no idea what to do for (b) and (c). I tried using the 3rd equation, where 1% of the charge would be .04μC, taking the ln of both sides would let me solve for t, however I'm really not sure about this..
I'm completely lost on (c), as I thought if they asked for current, I can solve for my voltage and my current using my values, however doesn't Q = CV only work for RC circuits?

 

[Simon Bridge]

Plugging in my values for a) gives me 22.9kHz, which seems reasonable, however I'm not sure if I should be using the first frequency equation or the second.

This is the problem with the plug-n-chug approach ... you have to guess which equation is the correct one.
To figure it out - go back through your course materials and find where the equations were derived - then see what situations they were derived for. Use the equation that fits the situation you have.

Moreover, I have no idea what to do for (b) and (c). I tried using the 3rd equation, where 1% of the charge would be .04μC, taking the ln of both sides would let me solve for t, however I'm really not sure about this..

... but "amplitude" of what? Does it matter?
$(q/q_0)=e^{-t/\tau}$ looks good doesn't it?
This is the plug-n-chug problem again ... did you try $q=q_0/100$ and see?
Do you know what the equation is telling you about the physics?

I'm completely lost on (c), as I thought if they asked for current, I can solve for my voltage and my current using my values, however doesn't Q = CV only work for RC circuits?

It is unclear what "took out the capacitor" means. You will have to interpret the question.
Maybe solve for an RL circuit with the same initial current?

 

[ayoubster]

I know where the derivations are coming from, and I know how it also relates to damping force and all the good stuff involving second order differential equations, however this is where I'm confused. I do see what the equation is telling me, and you're right, the amplitude of whatever equation I'm using doesn't matter. So for (b), would I use that equation, take the derivative q to solve for I? I = dq/dt, solve for I? but I don't have my initial I, would it just equal to Q/RC? And I should be good from there? I'm unfortunately stuck on whether or not Q = CV can only be used for RC circuits, I've been looking at derivations and other similar problems all day and still can't really wrap my head around this.

 

[ayoubster]

Also, I'm trying to avoid the plug and chug approach but I can't seem to find any similar problems to this to get a full understanding of both equations. I know that the first involves the resonance frequency, and the second involves the damping frequency, logically, I'd assume they're asking for the damping frequency since it's an RLC circuit and their asking for the 'oscillating frequency'

 

[Simon Bridge]

OK - for (a): you have two candidate equations for the frequency. And you tell me

I know where the derivations are coming from...

... great: you know the circumstances in which they are derived ... so what are those circumstances?
Which circumstance fits the situation you have?

 

[ayoubster]

To derive the second equation you need an RLC circuit, and for the first, well, it can be just an LC circuit or the resonance frequency of an RLC circuit, so I would use the second equation.

 

[Simon Bridge]

That seems fair reasoning - doesn't use much physics yet though: do you know what the frequencies are of?
Physically - what is that the frequency of?
ie is it the frequency of steady state oscillations when there is a sinusoidal driving force?
Is that the frequency you have to find?

What you have is $v_L(t)+v_C(t)+v_R(t) = 0 : v_C(0)=q_0/C$ ... this oscillator is damped, but not driven.
It is the electronic equivalent of a pendulum pulled to one side and let go - if the pivot is a bit stiff.
You know how that sort of thing behaves right?

 

[ayoubster]

The current is oscillating back and forth between the capacitor and inductor, the resistor damps the the voltage. When the impedance for the inductor and capacitor are the same, maximum output of voltage is attained. If max voltage is outputted, this is also the point where maximum current is flowing. So when you ask if I'm trying to find the frequency of steady state oscillations, yes, I'm trying to find the frequency of the current going back and forth between these two components until it damps to 0 due to power loss, etc. If the frequency was driven, I would be finding the frequency at which the voltage is being outputted. Am I understanding this correctly?

 

[Simon Bridge]

The current is oscillating back and forth between the capacitor and inductor, the resistor damps the the voltage. When the impedance for the inductor and capacitor are the same, maximum output of voltage is attained. If max voltage is outputted,

Does this circuit have an output?

...this is also the point where maximum current is flowing. So when you ask if I'm trying to find the frequency of steady state oscillations, yes, I'm trying to find the frequency of the current going back and forth between these two components until it damps to 0 due to power loss, etc. If the frequency was driven, I would be finding the frequency at which the voltage is being outputted. Am I understanding this correctly?

I don't think you understand what I am asking you.
It is unclear that you understand how the equations you have listed were arrived at or what they are actually for. You need to make the link between the maths and the physics.

Can you sketch/describe what behaviour you expect from the circuit given the initial conditions?
ie. let's say you keep track of the voltage across the capacitor.

 

[ayoubster]

The second equation I listed is for driven circuits, this one isn't driven. Thanks, I see where that's coming from now.

Keeping track of the voltage across the capicitor, the current leads the voltage by a certain phase depending on the impedance. Calculating the frequency of this circuit I should use ω = 1/√LC. When the capacitor charges, current is flowing out of it into the inductor, and current is flowing outside of the inductor in the opposite direction as it induces an emf, so the current lags behind the voltage in the inductor. Keeping the circuit this way as t→∞ the current will eventually stop flowing. When we take out the capacitor, we have the current flowing from it into the inductor and resistor, so I would use that current for my 3rd question as I = Q/RC. Since it's discharging, I would use the equations I listed rather than (1 + e^t/τ). Have I finally made a connection with the physics and math?

 

[Simon Bridge]

OK close enough.
The voltage across the capacitor will start high and then oscillate with an exponentially decreasing amplitude until it's gone.
... for the voltage across the capacitor, that is written like this:
$v_C(t) = V_{C}e^{-t/\tau}\cos\omega t$ (... assuming the circuit is not critically damped).

Your task amounts to finding $\tau$ and $\omega$ ... then comparing for when there is no capacitor.
What is the relationship between the initial current $I_0$ and the initial voltage across the capacitor $V_{C}$ ?

That stuff about the current leading or trailing the voltage ... that applies mainly to driven systems. The voltage that the current leads is the supply voltage.
However, the current through a component may lead or trail the voltage across it...