# Natural Response of Parallel RLC Circuit

In the circuit shown below, I need to calculate the current through the capacitor, which is initially charged. I'm thinking I have to calculate the natural response of the circuit, but am unsure of how to do so. I can calculate the impedance and find the pole, but then what?

simulate this circuit – Schematic created using CircuitLab

Let's write down equations in laplace form :

Z total = R || LS || 1/CS

Z total = R / [1+R(CS+1/LS)]

It is total impedance of circuit. If we assume initial conditions Vc0- = V0, and IL0- = I0, then for Ic we can write this equation:

Ic = Vc / Zc

Ic = [I0 * Ztotal + V0] / (1/CS)

Ic = I0 * R / [ 1/CS + R(1+1/LCS^2)] + V0 * CS

This is current equation of Capacitor in laplace form.

Write down all the equations, then convert them all to common a unit. (I think I always end up working in Coulombs.) Solve that and then find the current. (Hint, it's a damped oscillator)

• I don't think it's correct to say that it's an oscillator. Oct 21, 2014 at 0:56
• @AlfredCentauri, Ah OK, for a large range of R values it will oscillate. (I call that an oscillator... for me it doesn't have to go on forever.) I'll agree to disagree. Oct 21, 2014 at 1:06
• @GeorgeHerold In the interests of harmony, perhaps we could agree to call it a (damped) "Harmonic Oscillator"? ;-) Jun 24, 2015 at 19:37
• @SpehroPefhany, fixed. Jun 25, 2015 at 3:03