Q Factor⚠ unverified
Electrical / Filters · Compute the quality factor of an RLC circuit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| R | R | — | 1.0 | Resistance |
| L | L | H | 1.0 | Inductance |
| C | C | F | 1.0 | Capacitance |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Q | — | Quality factor (dimensionless) |
The science & history
Understanding the Parameters
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$R$ is the loss element — the only thing that lowers $Q$; ideal (loss‑free) resonators have $Q\to\infty$.
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$L/C$ ratio sets the characteristic impedance $\sqrt{L/C}$; a larger ratio (relative to $R$) gives a higher $Q$.
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$Q$ — roughly the number of radians of oscillation before the energy decays by $1/e$, and equal to $f_0/\text{BW}$ (see Rlc Bandwidth).
Derivation (Approaching a Proof)
Start from the energy definition, $Q = \omega_0 \dfrac{\text{energy stored}}{\text{power dissipated}}$. For a series RLC at resonance this evaluates to $Q = \dfrac{\omega_0 L}{R}$ (see Quality Factor). Substituting the resonant frequency $\omega_0 = 1/\sqrt{LC}$ (see Rlc Series Resonant):
$$Q = \frac{\omega_0 L}{R} = \frac{L}{R\sqrt{LC}} = \frac{1}{R}\sqrt{\frac{L}{C}}.$$
Equivalently $Q = \dfrac{1}{\omega_0 R C} = \dfrac{f_0}{\text{BW}}$, tying together resonance, sharpness, and bandwidth: a high‑$Q$ circuit has a narrow $-3\,$dB band $\text{BW} = f_0/Q$.
History
"$Q$" for a coil's quality comes from 1920s Bell Labs work on inductor loss; it was soon generalised to any resonator. In filter and RF design $Q$ is the single number that captures selectivity — how well a tuned circuit distinguishes its resonant frequency from neighbours.
Related Concepts: Quality Factor, Rlc Series Resonant, Rlc Bandwidth, Damping Ratio
Notes: Registry calculator q-factor (unverified; R mislabelled dimensionless — should be ohms).
Same physical quantity as Quality Factor (which uses $\omega L/R$); one of three "Q" calculators in
the catalog — see Known Issues.