For a resonator, the quality factor is times the stored energy divided by the energy dissipated per cycle, using a specified steady-state stored-energy convention. A series RLC circuit at resonance has , because its maximum magnetic energy is and one cycle of Joule heating dissipates .
For a linear oscillator with viscous drag, this parameter puts the underdamped/critical boundary at . The energy-per-cycle interpretation and narrow displacement linewidth are most useful at high quality factor; critical and overdamped free motion do not complete oscillation cycles. Amplitude and averaged-energy ringdown times differ by a factor of two. The damping classification is not the same as the condition for a nonzero-frequency displacement maximum.
Minimizing gives a displacement maximum at only when . An underdamped oscillator with consequently has no nonzero-frequency displacement peak. The absorbed-power response instead peaks at , with exact half-power bandwidth for this model. Comparing bandwidth definitions requires naming the measured response.
For a constant-amplitude force on , the cycle-averaged absorbed power is . It peaks at . Its half-power frequencies are , hence their separation is exactly and . A displacement-amplitude resonance has different half-power frequencies away from weak damping; the measured observable must be specified.
Converting the two half-power frequencies using gives . Since , this reduces exactly to the displayed expression. The wavelength endpoints are not symmetric around ; a symmetric narrow-line sketch is an approximation. This conversion concerns the same measured response versus a reparametrized axis, not a transformation between spectral density conventions.
For an isolated weakly damped resonator mode, . If , differentiating gives , so . Since energy is quadratic in wave amplitude, the wave amplitude exponential relaxation time is . A decay by an energy factor in a measured interval gives . For a conventional reverberation time, under this exponential single-rate approximation.
For two weakly damped normal modes with initial intensity ratio , independent exponential energy decay gives . An observed reversal to a ratio below one implies the displayed upper bound on , provided the denominator is positive. A strict intensity comparison does not determine a unique quality factor; equality fixes only the crossing threshold.
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The Q factor, or quality factor, is a dimensionless parameter that describes the damping of oscillatory systems, including mechanical, electrical, and optical systems. It is defined as the ratio of the resonant frequency of the system to the bandwidth over which the system can oscillate effectively. In simpler terms, it quantifies how underdamped an oscillator is, indicating the sharpness of its resonance peak.