Second order ODE for RLC circuit

1. Nov 24, 2014

icesalmon

if I consider a circuit consisting of a capacitor, an inductor and a resistor and using kirchoffs voltage rule for the circuit i come up with the following

L(Q''(t)) + R(Q'(t)) + (Q(t))/C = 0 I solve for the roots using a characteristic equation of the form
LM2 +MR +(1/C) = 0
solving this for m I obtain
m = [-R/L +/- sqrt((r/l2) - 4(1)(1/LC))]/2
i'm expecting an equation using both decaying exponentials and sinusoids so that the energy will tend towards zero after a long time. But this depends on the values of R, C and L. I'm having trouble moving forward from this point in deriving the equation for the charge on the capacitor as a function of time.
Q(t) = Ae-Btcos(w't+θ)
B = R/2L

Last edited: Nov 24, 2014
2. Nov 24, 2014

zoki85

3. Nov 24, 2014

icesalmon

so obviously the characteristic equation doesn't directly lead to that damped charge motion then, it depends on the discriminant. I thought I was SUPPOSED to get the decaying exponential out of it. Anyways, thanks.