This Demonstration shows the variation with time of the current I in a series RLC circuit (resistor, inductor, capacitor) in which the capacitor is initially charged to a voltage . The resonant frequency of the circuit is and the plotted normalized current is . There are three types of behavior depending on the value of the quality factor : overdamping when (no oscillation); critical damping when , (no oscillation and the most rapid damping); and underdamping when (damped oscillations).

The capacitor charge satisfies the differential equation , where the resonance frequency is given by and the quality factor . It is convenient to work with a normalized current . Initially the capacitor is charged to voltage and the current is 0.

For critical damping, and the current is For overdamping and underdamping, and the current is , where . So when , both and are complex, leading to a damped oscillating current. But when , both and are real and negative, so that the current is damped without any oscillations.

Reference:

J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, New York: Addison–Wesley, 1993.