A square wave pulse (generated using an oscilloscope) is used to induce damped oscillations in a circuit that consits of an inductance L and a capacitance C connected in series. A resistance is present even though no resistor is present in the circuit.(adsbygoogle = window.adsbygoogle || []).push({});

a) Find the differential equation for the capacitor charge.

b) Find the underdamped solution. Hint: Understand why application of a square-wave corresponds to kicking a damped harmonic oscillator:

q(0)=0,q'(0)=0

Here are my attempts:

a)

LI(dI/dt) + (q/C) dq/dt = -I^2 / R and then using I = dq/dt,

[tex]L \frac{d^{2}q}{dt^{2}} + R \frac{dq}{dt} + \frac{q}{C}[/tex] = 0

I am quite sure about this part.

b)

Here is where I am getting really confused. How do they know that q(0) = 0, q'(0) = 0 are the necessary initial conditions?

These initial conditions imply that the capacitor is not charged initially. Taking these as given (even though I don't understand why) the solution to the differential equation HAS to be

q(t) = [tex]e^{-Rt/2L}sin(\omega t)[/tex]

This is the only way that I can ensure that q(0) = 0 because for a cosine solution, q(0) will not be zero.

Finally regarding the undercritical damping, the solution above is infact the undercritical case. By definition of underctirical damping, the frequency [tex]\omega[/tex] is essentially equal to the undamped frequency.

i.e. [tex]\omega = \sqrt \frac{1}{LC}[/tex].

Is my part b solution correct?

James

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# Underdamped oscillations in an LC circuit

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