Damping in RLC Circuits

This Demonstration shows the variation with time of the current I in a series RLC circuit (resistor, inductor, capacitor) in which the capacitor C is initially charged to a voltage V. The resonant frequency of the circuit is f=(10^8) Hz and the plotted normalized current is (I/(2π f V C)). There are three types of behavior depending on the value of the quality factor Q: overdamping when Q<1/2 (no oscillation); critical damping when Q=1/2, (no oscillation and the most rapid damping); and underdamping when Q>1/2 (damped oscillations).

THINGS TO TRY

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DETAILS

Snapshot 1: overdamping

Snapshot 2: critical damping

Snapshot 3: underdamping

The capacitor charge q satisfies the differential equation (Overscript[q, . .])+((Subscript[ω, 0])/Q)(OverDot[q])+((Subscript[ω, 0])^2)q=0, where the resonance frequency f is given by (Subscript[ω, 0])=2π f=(1/(Sqrt[L C]))and the quality factor Q=(((Subscript[ω, 0])L)/R). It is convenient to work with a normalized current I=((-(OverDot[q]))/((Subscript[ω, 0])V C)). Initially the capacitor is charged to voltage V and the current is 0.

For critical damping, Q=1/2 and the current is I=(Subscript[ω, 0])t (e^(-(Subscript[ω, 0])t)). For overdamping and underdamping, Q ≠ 1/2 and the current is I=((Subscript[ω, 0])/((Subscript[r, 1])-(Subscript[r, 2])))((e^((Subscript[r, 1])t-(Subscript[r, 2])t))), where (Subscript[r, 1]),(Subscript[r, 2])=-((Subscript[ω, 0])/(2Q))±(Subscript[ω, 0])(Sqrt[-1+(1/(4(Q^2)))]). So when Q > 1/2, both (Subscript[r, 1]) and (Subscript[r, 2]) are complex, leading to a damped oscillating current. But when Q < 1/2, both (Subscript[r, 1]) and (Subscript[r, 2]) 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.

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Contributed by: James Burgess

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