Inphysics andengineering, thequality factor orQ factor is adimensionless parameter that describes howunderdamped anoscillator orresonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in oneradian of the cycle of oscillation.^[1]Q factor is alternatively defined as the ratio of a resonator's centre frequency to itsbandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results.^[2] HigherQ indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a highQ, while a pendulum immersed in oil has a low one. Resonators with high quality factors have lowdamping, so that they ring or vibrate longer.

TheQ factor is a parameter that describes theresonance behavior of an underdampedharmonic oscillator (resonator).Sinusoidally drivenresonators having higherQ factorsresonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the bandwidth. Thus, a high-Qtuned circuit in a radio receiver would be more difficult to tune, but would have moreselectivity; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High-Q oscillatorsoscillate with a smaller range of frequencies and are more stable.

The quality factor of oscillators varies substantially from system to system, depending on their construction. Systems for which damping is important (such as dampers keeping a door from slamming shut) haveQ near1⁄2. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability have high quality factors. Tuning forks have quality factors around 1000. The quality factor ofatomic clocks,superconducting RF cavities used in accelerators, and some high-Qlasers can reach as high as 10^11[3] and higher.^[4]

There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: thedamping ratio,relative bandwidth,linewidth and bandwidth measured inoctaves.

The concept ofQ originated with K. S. Johnson ofWestern Electric Company's Engineering Department while evaluating the quality of coils (inductors). His choice of the symbolQ was only because, at the time, all other letters of the alphabet were taken. The term was not intended as an abbreviation for "quality" or "quality factor", although these terms have grown to be associated with it.^[5][6][7]

The definition ofQ since its first use in 1914 has been generalized to apply to coils and condensers, resonant circuits, resonant devices, resonant transmission lines, cavity resonators,^[5] and has expanded beyond the electronics field to apply to dynamical systems in general: mechanical and acoustic resonators, materialQ and quantum systems such as spectral lines and particle resonances.

In the context of resonators, there are two common definitions forQ, which are not exactly equivalent. They become approximately equivalent asQ becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator:^[5]

$${\displaystyle Q\stackrel{\text{def}}{=}\frac{f_{\mathrm{r}}}{\mathrm{\Delta }f}=\frac{{\omega }_{\mathrm{r}}}{\mathrm{\Delta }\omega },}$$

wheref_r is the resonant frequency,Δf is theresonance width orfull width at half maximum (FWHM) i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency,ω_r = 2πf_r is theangular resonant frequency, andΔω is the angular half-power bandwidth.

Under this definition,Q is the reciprocal offractional bandwidth.

The other common nearly equivalent definition forQ is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes:^[8][9][5]

$${\displaystyle Q\stackrel{\text{def}}{=}2\pi \times \frac{\text{energy stored}}{\text{energy dissipated per cycle}}=2\pi f_{\mathrm{r}}\times \frac{\text{energy stored}}{\text{power loss}}.}$$

The factor2π makesQ expressible in simpler terms, involving only the coefficients of the second-order differential equation describing most resonant systems, electrical or mechanical. In electrical systems, the stored energy is the sum of energies stored in losslessinductors andcapacitors; the lost energy is the sum of the energies dissipated inresistors per cycle. In mechanical systems, the stored energy is the sum of thepotential andkinetic energies at some point in time; the lost energy is the work done by an externalforce, per cycle, to maintain amplitude.

More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition ofQ is used:^{[8][10][failed verification –see discussion][9]}

$${\displaystyle Q(\omega )=\omega \times \frac{\text{maximum energy stored}}{\text{power loss}},}$$

whereω is theangular frequency at which the stored energy and power loss are measured. This definition is consistent with its usage in describing circuits with a single reactive element (capacitor or inductor), where it can be shown to be equal to the ratio ofreactive power toreal power. (SeeIndividual reactive components.)

TheQ-factor determines thequalitative behavior of simple damped oscillators. (For mathematical details about these systems and their behavior seeharmonic oscillator andlinear time invariant (LTI) system.)

Starting from the stored energy definition for, it can be shown that${\displaystyle Q=\frac{1}{2\zeta }}$, where${\displaystyle \zeta }$ is thedamping ratio. There are three key distinct cases:

• A system withlow quality factor (Q <⁠1/2⁠) is said to beoverdamped. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it byexponential decay, approaching the steady state valueasymptotically. It has animpulse response that is the sum of twodecaying exponential functions with different rates of decay. As the quality factor decreases the slower decay mode becomes stronger relative to the faster mode and dominates the system's response resulting in a slower system. A second-orderlow-pass filter with a very low quality factor has a nearly first-order step response; the system's output responds to astep input by slowly rising toward an asymptote.
• A system withhigh quality factor (Q >⁠1/2⁠) is said to beunderdamped. Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude of the signal. Underdamped systems with a low quality factor (a little aboveQ =⁠1/2⁠) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. More generally, the output of a second-orderlow-pass filter with a very high quality factor responds to a step input by quickly rising above, oscillating around, and eventually converging to a steady-state value.
• A system with anintermediate quality factor (Q =⁠1/2⁠) is said to becritically damped. Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Like an underdamped response, the output of such a system responds quickly to a unit step input. Critical damping results in the fastest response (approach to the final value) possible without overshoot. Real system specifications usually allow some overshoot for a faster initial response or require a slower initial response to provide asafety margin against overshoot.

Innegative feedback systems, the dominant closed-loop response is often well-modeled by a second-order system. Thephase margin of the open-loop system sets the quality factorQ of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory (i.e., has a higher quality factor).

Physically speaking,Q is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation; or nearly equivalently, at high enoughQ values, 2π times the ratio of the total energy stored and the energy lost in a single cycle.^[14]

It is a dimensionless parameter that compares theexponential time constantτ for decay of anoscillating physical system'samplitude to its oscillationperiod. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy. More precisely, the frequency and period used should be based on the system's natural frequency, which at lowQ values is somewhat higher than the oscillation frequency as measured by zero crossings.

Equivalently (for large values ofQ), theQ factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off toe^−2π, or about1⁄535 or 0.2%, of its original energy.^[15] This means the amplitude falls off to approximatelye^−π or 4% of its original amplitude.^[16]

The width (bandwidth) of the resonance is given by (approximately):$${\displaystyle \mathrm{\Delta }f=\frac{f_{\mathrm{N}}}{Q},}$$wheref_N is thenatural frequency, andΔf, thebandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.

The resonant frequency is often expressed in natural units (radians per second), rather than using thef_N inhertz, as$${\displaystyle {\omega }_{\mathrm{N}}=2\pi f_{\mathrm{N}}.}$$

The factorsQ,damping ratioζ,natural frequencyω_N,attenuation rateα, andexponential time constantτ are related such that:^{[17][page needed]}

$${\displaystyle Q=\frac{1}{2\zeta }=\frac{{\omega }_{\mathrm{N}}}{2\alpha }=\frac{\tau {\omega }_{\mathrm{N}}}{2},}$$

and the damping ratio can be expressed as:

$${\displaystyle \zeta =\frac{1}{2Q}=\frac{\alpha }{{\omega }_{\mathrm{N}}}=\frac{1}{\tau {\omega }_{\mathrm{N}}}.}$$

The envelope of oscillation decays proportional toe^−αt ore^−t/τ, whereα andτ can be expressed as:

$${\displaystyle \alpha =\frac{{\omega }_{\mathrm{N}}}{2Q}=\zeta {\omega }_{\mathrm{N}}=\frac{1}{\tau }}$$and$${\displaystyle \tau =\frac{2Q}{{\omega }_{\mathrm{N}}}=\frac{1}{\zeta {\omega }_{\mathrm{N}}}=\frac{1}{\alpha }.}$$

The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, ase^−2αt ore^−2t/τ.

For a two-pole lowpass filter, thetransfer function of the filter is^[17]

$${\displaystyle H(s)=\frac{{\omega }_{\mathrm{N}}^2}{s^2+\underset{2\zeta {\omega }_{\mathrm{N}}=2\alpha }{\underbrace{\frac{{\omega }_{\mathrm{N}}}{Q}}}s+{\omega }_{\mathrm{N}}^2}}$$

For this system, whenQ >⁠1/2⁠ (i.e., when the system is underdamped), it has twocomplex conjugate poles that each have areal part of−α. That is, the attenuation parameterα represents the rate ofexponential decay of the oscillations (that is, of the output after animpulse) into the system. A higher quality factor implies a lower attenuation rate, and so high-Q systems oscillate for many cycles. For example, high-quality bells have an approximatelypure sinusoidal tone for a long time after being struck by a hammer.

Transfer functions for 2nd-order filters

Filter type (2nd order)	Transfer functionH(s)[18]
Lowpass	${\displaystyle \frac{{\omega }_{\mathrm{N}}^2}{s^2+\frac{{\omega }_{\mathrm{N}}}{Q}s+{\omega }_{\mathrm{N}}^2}}$
Bandpass	${\displaystyle \frac{\frac{{\omega }_{\mathrm{N}}}{Q}s}{s^2+\frac{{\omega }_{\mathrm{N}}}{Q}s+{\omega }_{\mathrm{N}}^2}}$
Notch (bandstop)	${\displaystyle \frac{s^2+{\omega }_{\mathrm{N}}^2}{s^2+\frac{{\omega }_{\mathrm{N}}}{Q}s+{\omega }_{\mathrm{N}}^2}}$
Highpass	${\displaystyle \frac{s^2}{s^2+\frac{{\omega }_{\mathrm{N}}}{Q}s+{\omega }_{\mathrm{N}}^2}}$

For an electrically resonant system, theQ factor represents the effect ofelectrical resistance and, for electromechanical resonators such asquartz crystals, mechanicalfriction.

The 2-sided bandwidth relative to a resonant frequency ofF₀ (Hz) is${\displaystyle \frac{F_0}{Q}}$.

For example, an antenna tuned to have aQ value of 10 and a centre frequency of 100 kHz would have a 3 dB bandwidth of 10 kHz.

In audio, bandwidth is often expressed in terms ofoctaves. Then the relationship betweenQ and bandwidth is

$${\displaystyle Q=\frac{2^{\frac{BW}{2}}}{2^{BW}-1}=\frac{1}{2\sinh \left(\frac{1}{2}\ln (2)BW\right)},}$$whereBW is the bandwidth in octaves.^[19]

In an ideal seriesRLC circuit, and in atuned radio frequency receiver (TRF) theQ factor is:^[20]

$${\displaystyle Q=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{{\omega }_{0}L}{R}=\frac{1}{{\omega }_{0}RC}}$$

whereR,L, andC are theresistance,inductance andcapacitance of the tuned circuit, respectively. Larger series resistances correspond to lower circuitQ values.

For a parallel RLC circuit, theQ factor is the inverse of the series case:^[21][20]

$${\displaystyle Q=R\sqrt{\frac{C}{L}}=\frac{R}{{\omega }_{0}L}={\omega }_{0}RC}$$^[22]

Consider a circuit whereR,L, andC are all in parallel. The lower the parallel resistance is, the more effect it will have in damping the circuit and thus result in lowerQ. This is useful in filter design to determine the bandwidth.

In a parallel LC circuit where the main loss is the resistance of the inductor,R, in series with the inductance,L,Q is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improveQ and narrow the bandwidth is the desired result.

TheQ of an individual reactive component depends on the frequency at which it is evaluated, which is typically the resonant frequency of the circuit that it is used in. TheQ of an inductor with a series loss resistance is theQ of a resonant circuit using that inductor (including its series loss) and a perfect capacitor.^[23]

$${\displaystyle Q_L=\frac{X_L}{R_L}=\frac{{\omega }_{0}L}{R_L}}$$

where:

• ω₀ is the resonance frequency in radians per second;
• L is the inductance;
• X_L is theinductive reactance; and
• R_L is the series resistance of the inductor.

TheQ of a capacitor with a series loss resistance is the same as theQ of a resonant circuit using that capacitor with a perfect inductor:^[23]

$${\displaystyle Q_C=\frac{-X_C}{R_C}=\frac{1}{{\omega }_{0}C{R}_C}}$$

where:

• ω₀ is the resonance frequency in radians per second;
• C is the capacitance;
• X_C is thecapacitive reactance; and
• R_C is the series resistance of the capacitor.

In general, theQ of a resonator involving a series combination of a capacitor and an inductor can be determined from theQ values of the components, whether their losses come from series resistance or otherwise:^[23]

${\displaystyle Q=\frac{1}{\frac{1}{Q_L}+\frac{1}{Q_C}}}$

For a single damped mass-spring system, theQ factor represents the effect of simplifiedviscous damping ordrag, where the damping force or drag force is proportional to velocity. The formula for theQ factor is:$${\displaystyle Q=\frac{\sqrt{Mk}}{D},}$$whereM is the mass,k is the spring constant, andD is the damping coefficient, defined by the equationF_damping = −Dv, wherev is the velocity.^[24]

TheQ of a musical instrument is critical; an excessively highQ in aresonator will not evenly amplify the multiple frequencies an instrument produces. For this reason, string instruments often have bodies with complex shapes, so that they produce a wide range of frequencies fairly evenly.

TheQ of abrass instrument orwind instrument needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed.By contrast, avuvuzela is made of flexible plastic, and therefore has a very lowQ for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higherQ values. An excessively highQ can make it harder to hit a note.Q in an instrument may vary across frequencies, but this may not be desirable.

Helmholtz resonators have a very highQ, as they are designed for picking out a very narrow range of frequencies.

Inoptics, theQ factor of aresonant cavity is given by$${\displaystyle Q=\frac{2\pi f_{o}E}{P},}$$wheref_o is the resonant frequency,E is the stored energy in the cavity, andP = −⁠dE/dt⁠ is the power dissipated. The opticalQ is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance. The average lifetime of a resonantphoton in the cavity is proportional to the cavity'sQ. If theQ factor of alaser's cavity is abruptly changed from a low value to a high one, the laser will emit apulse of light that is much more intense than the laser's normal continuous output. This technique is known asQ-switching.Q factor is of particular importance inplasmonics, where loss is linked to the damping of thesurface plasmon resonance.^[25] While loss is normally considered a hindrance in the development of plasmonic devices, it is possible to leverage this property to present new enhanced functionalities.^[26]

• Acoustic resonance
• Attenuation
• Chu–Harrington limit
• List of piezoelectric materials
• Phase margin
• Q meter
• Q multiplier
• Dissipation factor

Wikimedia Commons has media related toQuality factor.

• Calculating the cut-off frequencies when center frequency andQ factor is given
• Explanation ofQ factor in radio tuning circuits

• Electrical parameters
• Linear filters
• Mechanics
• Laser science
• Engineering ratios

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