Underdamped oscillations in an LC circuit

[JamesJames]

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.

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,

L \frac{d^{2}q}{dt^{2}} + R \frac{dq}{dt} + \frac{q}{C} = 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) = e^{-Rt/2L}sin(\omega t)

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 \omega is essentially equal to the undamped frequency.
i.e. \omega = \sqrt \frac{1}{LC}.

Is my part b solution correct?

James

 

[JamesJames]

Anyone? I have made an attempt on my own.

 

[Andrew Mason]

a)
LI(dI/dt) + (q/C) dq/dt = -I^2 / R and then using I = dq/dt,
L \frac{d^{2}q}{dt^{2}} + R \frac{dq}{dt} + \frac{q}{C} = 0
I am quite sure about this part.

This is the equation for the damped unforced oscillator. When you apply a square wave pulse of voltage, would you not have:
L \frac{d^{2}q}{dt^{2}} + R \frac{dq}{dt} + \frac{q}{C} = V

where 0 \le t \le T (T being the duration of the pulse)?

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?

Consider the conditions at the very beginning of the pulse. There is no charge in the capacitor because the voltage has just been applied. The current is just beginning to start to flow. There is just a rapid rate of increase of current. So:

L\frac{d^2q}{dt^2} = V

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

The general solution of the underdamped oscillator should have a steady state term.

q(t) = Ae^{-\gamma}sin(\omega t + \phi) + ?

where \gamma = R/2L; \omega_0 = 1/LC and \omega^2 = \omega_0^2 - \gamma^2

AM

 

[JamesJames]

So what you are saying is that it is impossible to use square wave pulses to induce damped oscillations in a circuit consisting of L and C. Instead, application of such a wave pulse would induce FORCED or DRIVEN HARMONIC OSCILLATIONS.

Here' s the thing and I apologize for not posting this although I did not feel that it would be relevant: The section in the book is titled "Damped oscillations in an LC circuit" and states specifically that square waves are used to induced damped oscillations...there is no mention of driven.

The solution to the problem is supposed to be q ~ exp(-RT/2L)sin(omega*t).

Does that help?

 

[Andrew Mason]

So what you are saying is that it is impossible to use square wave pulses to induce damped oscillations in a circuit consisting of L and C. Instead, application of such a wave pulse would induce FORCED or DRIVEN HARMONIC OSCILLATIONS.

It is forced or driven while the wave pulse applies. It is unforced afterward.

Here' s the thing and I apologize for not posting this although I did not feel that it would be relevant: The section in the book is titled "Damped oscillations in an LC circuit" and states specifically that square waves are used to induced damped oscillations...there is no mention of driven.
The solution to the problem is supposed to be q ~ exp(-RT/2L)sin(omega*t).
Does that help?

For t>T (ie after the pulse ends) this would be the solution. What is the condition for underdamping (in terms of R, L and C)?

AM