Q_factor - Wikiparency

# Q factor *Not to be confused with [[Q_value|Q value]], the energy released by a nuclear or chemical reaction.* The **quality factor**, or **Q factor**, is a dimensionless parameter that describes how underdamped an oscillator or resonator is, and equivalently how many radians of oscillation it takes for the energy stored in it to fall off. It has two equivalent definitions: the ratio of a resonator's centre frequency to the width of the band of frequencies around it that it still responds to, Q = f_r/Δf, and the ratio, scaled by 2π, of the energy stored in the resonator to the energy it dissipates per cycle. The same number governs an [[RLC_circuit|RLC circuit]], a plucked string, a [[Resonator|resonator]] cavity and a laser cavity alike: a higher Q means a sharper resonance and a longer ring-down, a lower Q means a broader response that settles out fast. The microsim on this page is a driven, damped oscillator — the same equation that sits behind Q in every one of the systems below — with the damping ratio ζ = 1/(2Q) and the drive frequency both under the reader's hand, so the trade-off between sharpness and ringing is something to watch rather than take on faith. ## Explanation Q is used to describe how underdamped a resonator is, as well as characterizing a resonator's [[Bandwidth_(signal_processing)|bandwidth]] relative to its centre frequency. A high-Q resonator rings for a long time after being struck or driven, the way a tuning fork or a wine glass does; a low-Q one, like a car's door damper, absorbs the disturbance in a fraction of a cycle and does not ring at all. The same dimensionless number applies whether the resonator stores its energy as current and magnetic field in an inductor, as charge on a capacitor, as the kinetic and potential energy of a mass on a spring, or as sound pressure in an air cavity — which is why the electrical, mechanical, acoustical and optical sections below all reduce to the same handful of formulas. ## Definition Q is defined, for a resonator, as 2π times the ratio of the energy stored to the energy dissipated per cycle of oscillation. Equivalently, and more usefully for measurement, it is the resonant frequency divided by the width of the band of frequencies over which the resonator still responds appreciably. ### Bandwidth definition For a resonator with a well-defined resonant frequency f_r, the bandwidth definition is Q = f_r / Δf where Δf is the half-power bandwidth: the width of the range of frequencies for which the power is at least half its value at f_r. A driven circuit with a centre frequency of 100 kHz and a Q of 10, for example, has a bandwidth of 10 kHz, so it responds usefully to signals between 95 kHz and 105 kHz. ### Stored energy definition The general, energy-based definition is Q = 2π × (energy stored) / (energy dissipated per cycle) = ω × (energy stored) / (power loss) which is the form that generalizes cleanly to systems, such as an exponentially decaying resonator, that have no externally imposed drive frequency at all. ## Q-factor and damping Q is related to the more familiar [[Damping|damping ratio]] ζ by Q = 1/(2ζ). For a second-order linear system this splits the possible behaviour into three regimes: an **overdamped** system (ζ > 1, Q < 1/2) returns to equilibrium without oscillating at all; a **critically damped** system (ζ = 1, Q = 1/2) is the fastest-returning system that still does not overshoot; and an **underdamped** system (ζ < 1, Q > 1/2) oscillates at a frequency close to its natural frequency, with the oscillation's amplitude decaying exponentially. The higher the Q above one-half, the slower that decay, and the more cycles the system completes before it is negligible.  **Microsim — three.js (Wikitube framework, shared with the Acoustics portal):** *The Q factor: how sharp a resonance is* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Q_factor.html" data-title="Q factor"></div> The sim drives x″ + 2ζω₀x′ + ω₀²x = (F₀/m)cos(ωt) and reads out Q = 1/(2ζ) directly. Two things are worth trying. Leave the drive at ω/ω₀ = 1.00 and lower ζ from 0.02: Q climbs, the steady swing grows toward Q times the static deflection, and the ring-up time (about Q/π cycles to reach 1 − 1/e of that swing) stretches out — at ζ = 0.02 that is already Q = 25 and a ring-up of about 8 cycles. Then, at that same ζ, sweep the drive frequency ratio away from 1: the response falls off within the resonance's half-power width of ω₀/Q, which at Q = 25 is only 4% of ω₀ either side — the quantitative picture behind "a high-Q resonator has a sharp resonance." *Built from `MICROSIM_GUIDE/specs/acoustics/variants/Q_factor.json`, a variant of `acoustics/sims/Harmonic_oscillator`, sourced to Chapter 15.6 of *University Physics Volume 1*[^up1]. Registered to [[Electronics]] ([[PORTAL_Electronics]]); the same build is also embedded from the Acoustics portal, since the Q of a driven RLC circuit and the Q of a driven mechanical or acoustic resonator are the identical equation.*  ### Some examples Typical Q factors span an enormous range. A car's door damper, which should not oscillate at all when the door is released, has a Q of about one-half. A tuning fork has a Q of around 1,000. At the far end, atomic clocks, superconducting radio-frequency cavities and the best lasers reach Q factors of 10^11 or higher — resonators that, once excited, would ring for a very long time before their oscillation decayed away. ## Physical interpretation For a resonator with a high Q, the peak energy in each cycle is much larger than the energy lost in that cycle, so Q also has a direct time-domain reading: it is (2π times) the number of radians of oscillation required for the resonator's energy to fall to 1/e^(2π) of its initial value, and a high-Q oscillator loses only a small fraction of its energy per radian and continues oscillating for many cycles after any driving force is removed. The name follows the same idea informally — a high-Q resonator is a higher-"quality" one in the sense of dissipating less of the energy put into it. ## Electrical systems ### Relationship between Q and bandwidth In a driven [[RLC_circuit|RLC circuit]], Q measures how selective the circuit is: how narrow a band of frequencies it passes or rejects relative to its centre frequency. This selectivity is exactly what [[Selectivity_(radio)|makes a tuned circuit useful]] for picking one radio station out of many close in frequency, at the cost of a design that also rings for longer after a transient. ### RLC circuits For a series RLC circuit, Q = ωL/R = (1/R)√(L/C); for a parallel RLC circuit, Q = R√(C/L). In both cases raising the resistance (in series) or lowering it (in parallel) trades bandwidth for selectivity in the same way as raising the damping ratio does mechanically. ### Individual reactive components A real inductor or capacitor is not lossless, and each is assigned its own Q from the series resistance that models its loss: an inductor's Q is ωL/R_L, a capacitor's is 1/(ωR_C C). These per-component Q factors set an upper bound on the Q of any RLC circuit built from them. ## Mechanical systems For a damped mass-spring system with mass M, spring constant k and damping coefficient D, Q = √(Mk)/D. The same three-regime picture from "Q-factor and damping" applies directly: a heavily damped suspension is low-Q and does not oscillate after a bump, while a lightly damped one — a struck bell, a plucked string — is high-Q and rings. ## Acoustical systems An acoustic [[Resonator|resonator]], such as the air cavity of a wind instrument or a Helmholtz resonator, has a Q set by how much of its stored acoustic energy is lost to radiation and internal friction each cycle, exactly as in the mechanical case; a high-Q acoustic resonator sustains a pure tone for many cycles once excited, which is why instrument bodies are shaped to keep some resonances high-Q and others heavily damped. ## Optical systems An [[Optical_cavity|optical cavity]] or laser resonator is assigned a Q by the same energy-based definition, and because optical frequencies are so high, even a cavity that loses a significant fraction of its light per round trip can have an enormous Q by the standards of electrical or mechanical resonators — the basis of high-Q techniques such as Q-switching for producing short, intense laser pulses. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Q_factor) : [Wikitube](https://en.wikitube.io/wiki/Q_factor) Skeleton mirrored at revision 1374399236. Prose, emphasis and the microsim placement are Wikitube's own; the microsim itself is the Acoustics portal's `Q_factor` build, reused here rather than rebuilt, since the underlying physics is identical. ## See also - [[Resonance]] - [[Damping]] - [[RLC_circuit]] - [[Harmonic_oscillator]] - [[Bandwidth_(signal_processing)]] - [[Resonator]] - [[Optical_cavity]] - [[Electrical_reactance]] ## References The definitions, formulas and damping regimes above are standard textbook material and are not separately footnoted, per the Wikitube style guide §6.1, with the exception of the microsim's own worked figures (Q = 25, ring-up ≈ 8 cycles, half-power width ω₀/Q), which are pinned below. [^up1]: *University Physics Volume 1* (OpenStax, 2016), §15.6 "Forced Oscillations," pp. 737–741 (Fig. 15.32, the quality factor Q ≈ ω₀/(2ζ) for small damping). On-disk at `Portal Books/_text/077 University Physics Volume 1 (2016).txt`. *Citation needed:* the origin of the symbol Q, attributed to K. S. Johnson of Western Electric's engineering department around 1914, is reported in secondary sources but is not yet pinned to a primary or history-of-technology source in this project's own library; no source is asserted for it here. ## Further reading - Standard resonance and circuit-theory texts covering Q factor, to be pinned with the citation above. ## External links - To be pinned once a primary or history-of-technology source for the symbol's 1914 origin is added to the library.