Quality Factor⚠ unverified
Physics / Vibration · Compute the quality (Q) factor from the damping ratio
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| zeta | ζ | — | 1.0 | Damping ratio (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Q | — | Quality factor (dimensionless). Returns positive infinity if ``zeta`` is non-positive |
The science & history
Understanding the Parameters
-
$R$, $\omega$, $L$ (electrical) — reactance $\omega L$ over resistance $R$; more reactance relative to loss means a higher‑$Q$, sharper resonance.
-
$\zeta$ (mechanical) — the damping ratio; light damping ($\zeta \ll 1$) gives high $Q$.
- $Q$ — dimensionless. Roughly, a resonator "rings" for about $Q$ radians (≈ $Q/\pi$ cycles) before its amplitude decays appreciably.
Derivation (Approaching a Proof)
The energy‑based definition is universal:
$$Q = \omega_0 \,\frac{\text{energy stored}}{\text{average power dissipated}} = 2\pi\,\frac{\text{energy stored}}{\text{energy lost per cycle}}.$$
Electrical (series RLC). At resonance the peak stored energy is $\tfrac12 L I_{peak}^2$ and the power dissipated is $\tfrac12 I_{peak}^2 R$. Substituting and using $\omega_0 = 1/\sqrt{LC}$:
$$Q = \frac{\omega_0 (\tfrac12 L I_{peak}^2)}{\tfrac12 I_{peak}^2 R} = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 R C} = \frac{1}{R}\sqrt{\frac{L}{C}}.$$
The calculator uses $Q = \omega L/R$ (evaluated at the operating $\omega$).
Mechanical (damped oscillator). A system $\ddot{x} + 2\zeta\omega_0 \dot{x} + \omega_0^2 x = 0$ loses a fraction of its energy each cycle set by $\zeta$; carrying the energy ratio through gives
$$Q = \frac{1}{2\zeta}.$$
Bandwidth link. For a resonator, $Q = \dfrac{f_0}{\Delta f}$, where $\Delta f$ is the $-3\,$dB bandwidth — high $Q$ means a narrow, selective peak (the basis of radio tuning and sharp filters).
History
The term "quality factor" and the symbol $Q$ trace to 1920s Bell Labs work on the loss of inductor coils (often attributed to K. S. Johnson); "$Q$" originally flagged a coil's quality (low loss). The concept was quickly recognised as the same dimensionless damping measure used for any resonator, from mechanical vibrations to microwave cavities and atomic clocks.
Related Concepts: Resonance Frequency LC Circuit, Inductive Reactance, Damping Ratio, Capacitive Reactance
Notes: Serves two registry calculators — quality-factor (Electrical / AC, $\omega L/R$) and
quality-factor-2 (Physics / Vibration, $1/2\zeta$); both are the same dimensionless $Q$. Name collision
noted in Known Issues; the electrical card also mislabels the output units (should be dimensionless).