Decrease of amplitude in RLC circuit

[Antoha1]

Homework Statement

The oscillatory circuit consists of a coil with inductance L = 40 mH and a capacitor with capacitance C=0.25 µF. The active resistance of the circuit is R = 4.0 Ω. Determine how many times the amplitude of the oscillations decreases when a time equal to one oscillation period elapses.

Relevant Equations

$α=\frac{R}{2L},ω_{0}=\frac{1}{\sqrt{LC}} ,ω_{d}=\sqrt{ω_{0}^2−α^2} ,A(t)=A_{0}e^{−αt.}$

Hello. I do not have any literature on this topic because I am still in school and I have not solved any problems about damping vibrations. In school we do not dig into damping vibrations.
But I have dug deeper into it on the internet and found some relevant equations but I am still not sure about how exactly does it work.
So basically, I am thinking that I need damping time period $T=\frac{2\pi}{\omega_{d}}$ and put it this equation $A(t)=A_{0}e^{−αt.} \Rightarrow \frac{A(T)}{A_{0}}=e^{−αT}$

Also, what I am concerned about is damping because I have found that in series it is α=R/2L and in parallel α=1/2RC, so this is the point where I think searching other ways of solution would be necessary.

 

[DaveE]

I would respond with two answers, one for the series RLC and one for the parallel version. They have very different responses. You don't need a different method, you need a clearer problem description.

https://ocw.mit.edu/courses/6-071j-...96839a7bf1ca2631cd87e711_16_transint_rlc2.pdf

 

[Antoha1]

I would respond with two answers, one for the series RLC and one for the parallel version. They have very different responses. You don't need a different method, you need a clearer problem description.

https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/a929d33896839a7bf1ca2631cd87e711_16_transint_rlc

okay, but what if components are connected in mixed ways?

 

[DaveE]

okay, but what if components are connected in mixed ways?

IDK. Different circuits have different solutions. Could you be more specific? Maybe a schematic?

 

[Antoha1]

I am thinking that this problem has a few different answers depending on way of connecting the components. But in the statement it is not said how the components are connected together, so I am asking should I just write all possible or leave it just with series and parallel

 

[DaveE]

The question asked "Determine how many times the amplitude of the oscillations decreases when a time equal to one oscillation period elapses." Which is pretty nonsensical if you read carefully. The oscillations of a sinusoid will decrease exactly 1 time in each period. Think of sin(x) over x∈(π/2, 3π/2). It doesn't matter what the amplitude or period is. The concept of the amplitude of a sinusoid over only 1 period is likewise a little tricky, but I think it does exist as an envelope function which multiplies the sinusoid as a time varying amplitude. Something like the exponential part of $f(t)=e^{\alpha t} sin(\omega t)$.

They also didn't (apparently) actually describe the connections (schematic).

So, did you accurately reproduce the question, is it a trick that requires no calculation, or do you have a bad instructor?

I would just explain how the envelope function of the oscillations behaves for each case (series and parallel). This circuit's response will have three parameters, the oscillation frequency, the initial amplitude, and the time constant of the amplitude decay. The natural oscillation frequency will not depend on the circuit configuration, all voltages and currents will share that.

Of course all of this assumes he's asking about the natural response of the circuit to some non-zero initial condition, which must be the case.

Sorry I can't otherwise divine what this guy expects from you.