Quality factor (Q)

The quality factor Q characterises how sharply an RLC circuit (or any second-order resonator) responds at its natural frequency. For a series RLC at resonance frequency ω₀ = 1/√(LC), Q = ω₀L/R = 1/(ω₀RC) = (1/R)√(L/C). Physically, Q equals 2π times the ratio of energy stored to energy dissipated per cycle: high-Q systems ring for many cycles; low-Q systems damp out in one or two.

The frequency-response curve has a peak at ω₀ with a full width at half maximum of Δω = ω₀/Q. So a Q = 100 circuit has a fractional bandwidth of 1% around ω₀ — narrow and sharp. A Q = 10 circuit has 10% bandwidth — broad and gentle. Q = 1/2 is the critical-damping boundary between underdamped (oscillatory decay) and overdamped (monotonic decay) free response; Q < 1/2 is overdamped, Q > 1/2 is underdamped with the ringing period 2π/ω_d where ω_d = ω₀√(1 − 1/(4Q²)).

Typical values span many orders of magnitude. An RLC circuit on a breadboard: Q ~ 10. A good LC oscillator tank circuit: Q ~ 100–1000. A quartz crystal resonator: Q ~ 10⁴–10⁶. A superconducting microwave cavity for particle accelerators: Q ~ 10⁹–10¹⁰. An atomic clock's trapped-ion transition: Q ~ 10¹⁵ — which is why timekeeping at the 10⁻¹⁸ level is feasible at all. In radio receivers, Q sets selectivity (ability to pick out one station from neighbours); in mechanical engineering, Q sets how long a tuning fork rings; in laser physics, Q of the optical cavity sets the threshold pump power. Same dimensionless number, same physical meaning, same formula.