Oscillations in an LRC Circuit

[Yosty22]

Homework Statement

(image attached)

For the circuit of Fig.30.17 in the textbook, let C = 13.0nF , L = 27.0mH , and R = 80.0Ω .

A). Calculate the oscillation frequency of the circuit once the capacitor has been charged and the switch has been connected to point a

B). How long will it take for the amplitude of the oscillation to decay to 10.0% of its original value?

C). What value of R would result in a critically damped circuit?

Homework Equations

The Attempt at a Solution

I was able to get part A and part C, but I am having a really hard time with B. For A, i got 8490 Hz and for C i got 2880 Ohms. I was able to understand those very well, but I cannot figure out B at all. I have no idea how to relate any of this information to the amplitude.

Any help would be greatly appreciated.

Thanks in advanced.

 

[gneill]

Investigate the Q of the circuit, and what Q represents.

 

[Yosty22]

I'm not sure, but I am still lost. My professor told me the equation

i=I_0e^-((R/L)t)

could be useful, but I cannot see how at all. I am really lost on this.

 

[gneill]

I'm not sure, but I am still lost. My professor told me the equation

i=I_0e^-((R/L)t)

could be useful, but I cannot see how at all. I am really lost on this.

Yes, that equation could help as it describes the decay of the maximum current in the oscillations due to energy dissipation in the resistance. It's the "envelope" of the sinusoidal current waveform.

I was thinking of using the Q of the circuit to find the number of cycles until the energy loss brought the amplitude down to the desired level. But if you can make use of the given equation, go for it!
But I think you'll find that the damping factor should be ##\frac{R}{2L}##.

If you look in your text, or notes, you should find the solution for the underdamped case to be a sinusoid multiplied by a decaying exponential...

 

[vanhees71]

The formula for [itex]i(t)[/itex] is valid for [itex]L=0[/itex] only. Here you need to derive the differential equation for the Circuit from Kirchhoff's Laws and then solve for it. As already said, this means to analyze how [itex]Q[/itex] on one capacitor plate behaves with time.

 

[gneill]

The formula for [itex]i(t)[/itex] is valid for [itex]L=0[/itex] only. Here you need to derive the differential equation for the Circuit from Kirchhoff's Laws and then solve for it. As already said, this means to analyze how [itex]Q[/itex] on one capacitor plate behaves with time.

If i(t) is interpreted as the envelope of the decaying sinusoidal current, then it should apply. Note that Io here is not the actual current at t=0 which, due to the presence of the inductor, will be zero. It's a current magnitude that you'd find if the initial energy was all in the inductor rather than in the capacitor.

If you solve the differential equation for this underdamped case it will have the form:
$$I(t) = I_o e^{-\alpha t} sin(ω_d t)$$
where ##I_o## and ##\alpha## depend upon the component values. It's that leading exponential term and constant that define the envelope. I suspect that this was the OP's professor's intention when suggesting that equation.