The three regimes of a free RLC circuit

Discriminant and regimes

The sign of Delta = (omega0/Q)^2 - 4*omega0^2, or equivalently the comparison of Q to 1/2, determines the circuit's evolution regime:

Condition	Regime	Behavior
Q < 1/2	overdamped	slow return to zero, no oscillation
Q = 1/2	critically damped	fastest possible return, no oscillation
Q > 1/2	underdamped (pseudo-periodic)	damped oscillations

Underdamped regime (the most common in practice)

For Q > 1/2, the solution is written:

q(t) = A*exp(-t/tau)cos(omegat + phi)

with tau = 2Q/omega0 (characteristic damping time) and omega = omega0sqrt(1 - 1/(4*Q^2)), the pseudo-angular frequency, always slightly lower than omega0.

Concrete example

For L = 1 H, C = 1 uF, R = 100 ohms: omega0 = 1/sqrt(L*C) = 1000 rad/s, and Q = (1/R)sqrt(L/C) = 10. Since Q > 1/2, the circuit oscillates while damping slowly: the envelope decreases as exp(-t/tau) with tau = 2Q/omega0 = 0.02 s.

Limiting case Q >> 1

When Q is very large (R very small), omega tends toward omega0 and the damping becomes very slow: the circuit behaves almost like an ideal LC circuit, whose oscillations never die out.

Common pitfall

A frequent mistake is to believe that the critically damped regime is the fastest 'with' oscillation: it is in fact the exact boundary between the two non-oscillating regimes, and it is the fastest only among those.