What Mistakes Are Made in Solving Damped Oscillator LCR Circuit Problems?

[pondzo]

Homework Statement

Homework Equations

The Attempt at a Solution

For part (a) i did the following;
the time for it to decay to 40% is half the period of the square wave = 0.00002 seconds
So, 0.4q_m = q_m $e^(\frac{-0.00002R}{2L})cos(25000*2*\pi*0.00002)$
But the cosine term yields -1 which then makes the equation unsolvable, what am i doing wrong?

For part (b) I am a bit confused about the "17 ringing cylcles per half-cycle" but i tried ;

the time for one half oscillation of the square wave voltage is 0.5/(25E3) = 0.00002 seconds
during this time the LCR circuit rings 17 times so the period of oscillation of the LCR circuit is 0.00002/17 = 0.000001176
this corresponds to an angular freq of w = 5340707.511 rad.s^-1
Is this correct so far? and if so, does this mean there will be a different restance in part (b) than in part (a)?

 

[NascentOxygen]

Don't involve the cosine. For the decay all we are concerned with is the exponential envelope.

 

[pondzo]

Thank you, I should have realized that.
Do you know if what i did for part (b) is correct?

 

[NascentOxygen]

Your $\mathrm{\omega}$ looks right. You cannot estimate R from $\textrm{ω}$ because you don't know $\mathbf{ω}$ to the great precision necessary. The exponential decay is what allows you to determine R.

 

[HalfLostAndSearching]

I would like to clarify a few things about the content provided.

Firstly, a "damped oscillator LCR circuit" is a type of circuit that consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series. This type of circuit exhibits oscillatory behavior, where the energy stored in the capacitor and inductor is constantly being transferred between each other. However, due to the presence of a resistor, the oscillations gradually decrease in amplitude over time, leading to a phenomenon known as "damping".

Now, let's address the given homework statement. The first part (a) asks for the time it takes for the oscillations to decay to 40% of their initial amplitude. In order to solve this, you correctly identified that the time for one half oscillation of the square wave voltage is half the period of the square wave. However, the equation you used to calculate the decay time is incorrect. The correct equation should be:

decay time = 0.693 * (L/R)

This equation can be derived from the damped oscillations equation, by setting the amplitude to 40% of the initial amplitude. So, the correct equation for part (a) should be:

0.00002 seconds = 0.693 * (L/R)

Solving for L/R gives a value of 2.89 x 10^-6.

Moving on to part (b), the statement "17 ringing cycles per half-cycle" is a bit confusing and unclear. However, based on your attempt, it seems like the circuit is undergoing 17 oscillations during the time period of one half oscillation of the square wave. In this case, the period of oscillation of the LCR circuit would be 0.00002/17 = 1.176 x 10^-6 seconds. This would correspond to an angular frequency of w = 5340707.511 rad.s^-1. It is important to note that the resistance (R) in this part would be different from the one in part (a), as the circuit is undergoing different oscillations.

In conclusion, the equations and calculations used in both parts of the question are correct, but there are some minor errors in the equations used. It is important to carefully understand the concepts and equations involved in order to obtain accurate solutions.