qualityfactor

Quality factor, denoted Q, is a dimensionless parameter that describes how underdamped a resonant system is. It expresses the ratio of energy stored to energy dissipated per cycle and provides a measure of how sharply the system resonates. A common definition is Q = 2π × (energy stored) / (energy dissipated per cycle) = ω0 × (stored energy) / (power loss), where ω0 is the angular resonant frequency. For a resonant frequency f0, Q is also often written as Q ≈ f0 / Δf, where Δf is the bandwidth over which the power is half its peak value (the −3 dB points). Higher Q indicates lower damping and a narrower resonance.

In practical terms, Q relates to selectivity: a higher Q yields a narrower bandwidth and a more selective response, while a lower Q broadens the resonance. Q is used across disciplines to characterize electrical, mechanical, optical, and acoustic resonators, oscillators, and filters. It also serves as a convenient shorthand for damping: in lightly damped second-order systems, Q is approximately the reciprocal of twice the damping ratio, Q ≈ 1/(2ζ).

Formulas for common resonators include standard circuit models. For a series RLC circuit with resonance at ω0 = 1/√(LC), Q = ω0L/R = 1/(ω0RC). For a parallel RLC circuit, Q = R/(ω0L) = ω0RC = R√(C/L). In all cases, Q is proportional to the energy stored and inversely proportional to the energy lost as heat or radiation per cycle.

Limitations include frequency dependence, temperature, and material properties, which can cause Q to vary with operating conditions. Therefore Q is a useful but not complete descriptor of a system’s dynamic performance.