Editing Q factor

{{Short description|Parameter describing the longevity of energy in a resonator relative to its resonant frequency}}
{{For|other uses of the terms '''Q''', '''Q factor''', and '''Quality factor'''|Q value (disambiguation)}}
{{DISPLAYTITLE:''Q'' factor}}
[[File:Damped oscillation function plot.svg|thumb|upright=1.5| A damped oscillation.  A low {{mvar|Q}} factor – about 5 here – means the oscillation dies out rapidly.]]
In [[physics]] and [[engineering]], the '''quality factor''' or '''{{mvar|Q}} factor''' is a [[Dimensionless quantity|dimensionless]] parameter that describes how [[underdamped]] an [[oscillator]] or [[resonator]] is.  It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one [[radian]] of the cycle of oscillation.<ref>{{cite book |last1=Hickman |first1=Ian |title=Analog Electronics: Analog Circuitry Explained |date=2013 |publisher=Newnes |page=42 |url=https://books.google.com/books?id=l-AgBQAAQBAJ&pg=PA42 | isbn= 9781483162287}}</ref> {{mvar|Q}} factor is alternatively defined as the ratio of a resonator's centre frequency to its [[bandwidth (signal processing)|bandwidth]] when subject to an oscillating driving force.  These two definitions give numerically similar, but not identical, results.<ref>{{cite book
 |title       = Electronic circuits: fundamentals and applications
 |first       = Michael H.
 |last        = Tooley
 |publisher   = Newnes
 |year        = 2006
 |isbn        = 978-0-7506-6923-8
 |pages       = 77–78
 |url         = https://books.google.com/books?id=rwWWvtQYEO0C&pg=PA78
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20161201212653/https://books.google.com/books?id=8fuppV9O7xwC&pg=PA77&dq=q-factor+bandwidth#v=onepage&q=q-factor%20bandwidth&f=false
 |archive-date = 2016-12-01
}}</ref> Higher {{mvar|Q}} 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 high {{mvar|Q}}, while a pendulum immersed in oil has a low one. Resonators with high quality factors have low [[Damping ratio|damping]], so that they ring or vibrate longer.

==Explanation==
The {{mvar|Q}} factor is a parameter that describes the [[resonance]] behavior of an underdamped [[harmonic oscillator]] (resonator). [[sine wave|Sinusoidally]] driven [[resonator]]s having higher {{mvar|Q}} factors [[resonance|resonate]] 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-{{mvar|Q}} [[RLC circuit|tuned circuit]] in a radio receiver would be more difficult to tune, but would have more [[selectivity (radio)|selectivity]]; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High-{{mvar|Q}} oscillators [[oscillator phase noise|oscillate 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) have {{mvar|Q}} near {{1/2}}. 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 of [[atomic clock]]s, [[Superconducting Radio Frequency|superconducting RF]] cavities used in accelerators, and some high-{{mvar|Q}} [[optical cavity|lasers]] can reach as high as 10<sup>11</sup><ref>

[http://www.rp-photonics.com/q_factor.html Encyclopedia of Laser Physics and Technology: ''Q'' factor] {{webarchive
 | url=https://web.archive.org/web/20090224211703/http://www.rp-photonics.com/q_factor.html
 | date=2009-02-24
}}</ref> and higher.<ref>

[http://tf.nist.gov/general/enc-q.htm Time and Frequency from A to Z: Q to Ra] {{webarchive
 | url=https://web.archive.org/web/20080504160852/http://tf.nist.gov/general/enc-q.htm
 | date=2008-05-04
}}</ref>

There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the [[damping ratio]], [[bandwidth (signal processing)|relative bandwidth]], [[oscillator linewidth|linewidth]] and bandwidth measured in [[Octave (electronics)|octave]]s.

The concept of {{mvar|Q}} originated with K. S. Johnson of [[Western Electric Company]]'s Engineering Department while evaluating the quality of coils (inductors). His choice of the symbol {{mvar|Q}} 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.<ref name="Green">
{{Cite journal
 |last1=Green|first1=Estill I.
 |date=October 1955
 |title=The Story of Q
 |url=http://www.collinsaudio.com/Prosound_Workshop/The_story_of_Q.pdf
 |journal=[[American Scientist]]
 |volume=43|pages=584–594
 |archive-url=https://web.archive.org/web/20121203044200/http://www.collinsaudio.com/Prosound_Workshop/The_story_of_Q.pdf
 |archive-date=2012-12-03
 |access-date=2012-11-21
 |url-status=live
}}</ref><ref>B. Jeffreys, Q.Jl R. astr. Soc. (1985) 26, 51–52</ref><ref name="Paschotta">{{cite book
 |last        = Paschotta
 |first       = Rüdiger
 |title       = Encyclopedia of Laser Physics and Technology, Vol. 1: A-M
 |publisher   = Wiley-VCH
 |year        = 2008
 |pages       = 580
 |url         = https://books.google.com/books?id=hdkJ5ASTFjcC&q=%22Q+factor%22+definition+history&pg=PA580
 |isbn        = 978-3527408283
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20180511181437/https://books.google.com/books?id=hdkJ5ASTFjcC&pg=PA580&dq=%22Q+factor%22+definition+history
 |archive-date = 2018-05-11
}}</ref>

== Definition ==

The definition of {{mvar|Q}} since its first use in 1914 has been generalized to apply to coils and condensers, resonant circuits, resonant devices, resonant transmission lines, cavity resonators,<ref name="Green"/> and has expanded beyond the electronics field to apply to dynamical systems in general: mechanical and acoustic resonators, material {{mvar|Q}} and quantum systems such as spectral lines and particle resonances. 

=== Bandwidth definition ===
In the context of resonators, there are two common definitions for {{mvar|Q}}, which are not exactly equivalent. They become approximately equivalent as {{mvar|Q}} becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator:<ref name="Green"/>

<math display="block">Q \mathrel\stackrel{\text{def}}{=} \frac{f_\mathrm{r}}{\Delta f} = \frac{\omega_\mathrm{r}}{\Delta \omega},</math>

where {{math|''f''<sub>r</sub>}} is the resonant frequency, {{math|Δ''f''}} is the '''resonance width''' or [[full 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, {{math|1= ''ω''<sub>r</sub> = 2''πf''<sub>r</sub>}} is the [[angular frequency|angular]] resonant frequency, and {{math|Δ''ω''}} is the angular half-power bandwidth.

Under this definition, {{mvar|Q}} is the reciprocal of [[fractional bandwidth]].

=== Stored energy definition ===
The other common nearly equivalent definition for {{mvar|Q}} is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes:<ref name="IEEE">Slyusar V. I. 60 Years of Electrically Small Antennas Theory.//Proceedings of the 6-th International Conference on Antenna Theory and Techniques, 17–21 September 2007, Sevastopol, Ukraine. - Pp. 116 – 118. {{cite web|url=http://slyusar.kiev.ua/ICATT_2007_1.pdf|title=ANTENNA THEORY AND TECHNIQUES|url-status=live|archive-url=https://web.archive.org/web/20170828212548/http://www.slyusar.kiev.ua/ICATT_2007_1.pdf|archive-date=2017-08-28|access-date=2017-09-02}}</ref><ref name=":0">{{Cite book|url=https://books.google.com/books?id=kqjdcIV_CCUC&q=quality+factor|title=Network Analysis|last=U.A.Bakshi|first=A. V. Bakshi|date=2006|publisher=Technical Publications|isbn=9788189411237|pages=228|language=en}}</ref><ref name="Green"/>

<math display="block">Q \mathrel\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}}.</math>

The factor {{math|2''π''}} makes {{mvar|Q}} 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 lossless [[inductors]] and [[capacitors]]; the lost energy is the sum of the energies dissipated in [[resistors]] per cycle. In mechanical systems, the stored energy is the sum of the [[potential energy|potential]] and [[kinetic energy|kinetic]] energies at some point in time; the lost energy is the work done by an external [[force]], per cycle, to maintain amplitude.

More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition of {{mvar|Q}} is used:<ref name=IEEE/><ref>{{cite book|title=Electric Circuits|isbn=0-201-17288-7|author=James W. Nilsson|year=1989}}</ref>{{Failed verification|date=February 2015|talk=Q factor definition in the context of individual reactive components}}<ref name=":0" />

<math display="block">Q(\omega) = \omega \times \frac{\text{maximum energy stored}}{\text{power loss}},</math>

where {{mvar|ω}} is the [[angular 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 of [[reactive power]] to [[real power]]. (''See'' [[#Individual reactive components|Individual reactive components]].)

== {{mvar|Q}}-factor and damping ==
{{main|Damping|Linear time-invariant system|l2=linear time invariant (LTI) system}}

The {{mvar|Q}}-factor determines the [[qualitative data|qualitative]] behavior of simple damped oscillators. (For mathematical details about these systems and their behavior see [[harmonic oscillator]] and [[LTI system|linear time invariant (LTI) system]].)

Starting from the stored energy definition for, it can be shown that <math> Q = \frac{1}{2\zeta}</math>, where <math>\zeta</math> is the [[Damping#Damping ratio|damping ratio]]. There are three key distinct cases:

* A system with '''low quality factor''' ({{math|''Q'' < {{small|{{sfrac|1|2}}}}}}) is said to be '''overdamped'''. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by [[exponential decay]], approaching the steady state value [[asymptotic]]ally. It has an [[impulse response]] that is the sum of two [[exponential decay|decaying 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-order [[low-pass filter]] with a very low quality factor has a nearly first-order step response; the system's output responds to a [[Heaviside step function|step input]] by slowly rising toward an asymptote.
* A system with '''high quality factor''' ({{math|''Q'' > {{small|{{sfrac|1|2}}}}}}) is said to be '''underdamped'''. 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 above {{math|1=''Q'' = {{small|{{sfrac|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-order [[low-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 an '''intermediate quality factor''' ({{math|1=''Q'' = {{small|{{sfrac|1|2}}}}}}) is said to be '''critically 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 a [[factor of safety|safety margin]] against overshoot.

In [[negative feedback]] systems, the dominant closed-loop response is often well-modeled by a second-order system. The [[phase margin]] of the open-loop system sets the quality factor {{mvar|Q}} 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).

=== Some examples ===
{{Anchor|Quality factors of common systems|Some examples}}

{{bulleted list
| A unity-gain [[Sallen–Key topology#Application: low-pass filter|Sallen–Key lowpass filter topology]] with equal capacitors and equal resistors is critically damped (i.e., {{math|''Q'' {{=}} {{small|{{sfrac|1|2}}}}}}).
| A second-order [[Bessel filter]] (i.e., continuous-time filter with flattest [[group delay]]) has an underdamped {{math|''Q'' {{=}} {{small|{{sfrac|{{sqrt|3}}}}}}}}.
| A second-order [[Butterworth filter]] (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped {{math|''Q'' {{=}} {{small|{{sfrac|{{sqrt|2}}}}}}}}.<ref>{{cite book |last1=Sabah |first1=Nassir H. |title=Circuit Analysis with PSpice: A Simplified Approach |date=2017 |publisher=CRC Press |isbn=9781315402215 |page=446 |url=https://books.google.com/books?id=B7W8DgAAQBAJ&pg=PA446}}</ref>
| A pendulum's {{mvar|Q}}-factor is: {{math|''Q'' {{=}} ''Mω''/''Γ''}}, where {{mvar|M}} is the mass of the bob, {{math|''ω'' {{=}} 2''π''/''T''}} is the pendulum's radian frequency of oscillation, and {{mvar|Γ}} is the frictional damping force on the pendulum per unit velocity.
| The design of a high-energy (near [[terahertz (unit)|terahertz]]) [[gyrotron]] considers both diffractive Q-factor, <math display="inline">Q_D \approx 30 \left(\frac{L}{\lambda}\right)^2</math> as a function of resonator length {{mvar|L}}, wavelength {{mvar|λ}}, and ohmic {{mvar|Q}}-factor ({{math|TE{{sub|''m,p''}}}}–modes)
<math display="block">Q_\Omega = \frac{R_\mathrm{w}}{\delta} \frac{1 - m^2}{v^2_{m,p}},</math>
where {{math|''R''{{sub|w}}}} is the cavity wall radius, {{mvar|δ}} is the [[skin depth]] of the cavity wall, {{mvar|v{{sub|m,p}}}} is the [[eigenvalue]] scalar ({{mvar|m}} is the azimuth index, {{mvar|p}} is the radial index; in this application, skin depth is {{nowrap|<math display="inline">\delta = {1}/{\sqrt{ \pi f \sigma u_o}}</math>)}}<ref>{{cite web |title=Near THz Gyrotron: Theory, Design, and Applications |url=https://ireap.umd.edu/sites/ireap.umd.edu/files/documents/ruifengpu-ms.pdf |website=The Institute for Research in Electronics and Applied Physics |publisher=University of Maryland |access-date=5 January 2021}}</ref>
| In [[medical ultrasonography]], a transducer with a high {{mvar|Q}}-factor is suitable for [[doppler ultrasonography]] because of its long ring-down time, where it can measure the velocities of blood flow. Meanwhile, a transducer with a low {{mvar|Q}}-factor has a short ring-down time and is suitable for organ imaging because it can receive a broad range of reflected echoes from bodily organs.<ref>{{cite book |last1=Curry |first1=TS |last2=Dowdey |first2=JE |last3=Murry |first3=RC |title=Christensen's Physics of Diagnostic Radiology |date=1990 |publisher=Lippincott Williams & Wilkins |isbn=9780812113105 |page=331 |url=https://books.google.com/books?id=W2PrMwHqXl0C |access-date=22 January 2023}}</ref>
}}

== Physical interpretation ==

Physically speaking, {{mvar|Q}} is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation; or nearly equivalently, at high enough {{mvar|Q}} values, 2{{pi}} times the ratio of the total energy stored and the energy lost in a single cycle.<ref>
{{cite book | last = Jackson | first = R. | url = https://books.google.com/books?id=6CZZE9I0HbQC&dq=%22q+factor%22+energy&pg=PA28 | title = Novel Sensors and Sensing | publisher = Institute of Physics Pub | location = Bristol | year = 2004 | isbn = 0-7503-0989-X | pages = 28 }}
</ref>

It is a dimensionless parameter that compares the [[exponential decay#Mean lifetime|exponential time constant]] {{mvar|τ}} for decay of an [[oscillating]] physical system's [[amplitude]] to its oscillation [[frequency|period]]. 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 low {{mvar|Q}} values is somewhat higher than the oscillation frequency as measured by zero crossings.

Equivalently (for large values of {{mvar|Q}}), the {{mvar|Q}} factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to {{math|''e''<sup>−2''π''</sup>}}, or about {{frac|535}} or 0.2%, of its original energy.<ref>{{cite web | title = Light and Matter | author = Benjamin Crowell | year = 2006 | url = http://www.lightandmatter.com/lm | url-status = live | archive-url = https://web.archive.org/web/20110519093054/http://lightandmatter.com/lm/ | archive-date = 2011-05-19 }}, Ch. 18</ref> This means the amplitude falls off to approximately {{math|''e''<sup>−''π''</sup>}} or 4% of its original amplitude.<ref>{{Cite book|title=Foundations of analog & digital electronic circuits|last=Anant.|first=Agarwal|date=2005|publisher=Elsevier|others=Lang, Jeffrey (Jeffrey H.)|isbn=9781558607354|location=Amsterdam|pages=647|oclc=60245509}}</ref>

The width (bandwidth) of the resonance is given by (approximately):
<math display="block">\Delta f = \frac{f_\mathrm{N}}{Q}, \,</math>
where {{math|''f''<sub>N</sub>}} is the [[natural frequency]], and {{math|Δ''f''}}, the [[bandwidth (signal processing)|bandwidth]], 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 the {{math|''f''<sub>N</sub>}} in [[hertz]], as
<math display="block">\omega_\mathrm{N} = 2\pi f_\mathrm{N}.</math>

The factors {{mvar|Q}}, [[damping ratio]] {{mvar|ζ}}, [[natural frequency]] {{math|''ω''<sub>N</sub>}}, [[Exponential decay|attenuation rate]] {{mvar|α}}, and [[exponential decay#Mean lifetime|exponential time constant]] {{mvar|τ}} are related such that:<ref name=Siebert>{{cite book | title = Circuits, Signals, and Systems | first = William McC. | last = Siebert | publisher = MIT Press }}</ref>{{page needed|date=August 2022}}

<math display="block">Q = \frac{1}{2 \zeta} = \frac{ \omega_\mathrm{N} }{2 \alpha } = \frac{ \tau \omega_\mathrm{N} }{ 2 },</math>

and the damping ratio can be expressed as:

<math display="block">\zeta = \frac{1}{2 Q} = { \alpha \over \omega_\mathrm{N} } = { 1 \over \tau \omega_\mathrm{N} }.</math>

The envelope of oscillation decays proportional to {{math|''e''<sup>−''αt''</sup>}} or {{math|''e''<sup>−''t/τ''</sup>}}, where {{mvar|α}} and {{mvar|τ}} can be expressed as:

<math display="block">\alpha = { \omega_\mathrm{N} \over 2 Q } = \zeta \omega_\mathrm{N} = {1 \over \tau}</math>
and
<math display="block">\tau = { 2 Q \over \omega_\mathrm{N} } = {1 \over \zeta \omega_\mathrm{N}} = \frac{1}{\alpha}. </math>

The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as {{math|''e''<sup>−2''αt''</sup>}} or {{math|''e''<sup>−2''t/τ''</sup>}}.

For a two-pole lowpass filter, the [[transfer function]] of the filter is<ref name=Siebert/>

<math display="block">H(s) = \frac{ \omega_{\mathrm N}^2 }{ s^2 + \underbrace{ \frac{ \omega_{\mathrm N} }{Q} }_{2 \zeta \omega_{\mathrm N} = 2 \alpha }s + \omega_{\mathrm N}^2 } \,</math>

For this system, when {{math|''Q'' > {{small|{{sfrac|1|2}}}}}} (i.e., when the system is underdamped), it has two [[complex conjugate]] poles that each have a [[real part]] of {{mvar|−α}}. That is, the attenuation parameter {{mvar|α}} represents the rate of [[exponential decay]] of the oscillations (that is, of the output after an [[impulse response|impulse]]) into the system. A higher quality factor implies a lower attenuation rate, and so high-{{mvar|Q}} systems oscillate for many cycles. For example, high-quality bells have an approximately [[pure tone|pure sinusoidal tone]] for a long time after being struck by a hammer.

{| class="wikitable" style="text-align:center;"
|+ Transfer functions for 2nd-order filters
|-
! scope="col" | Filter type (2nd order)
! scope="col" | Transfer function {{math|''H''(''s'')}}<ref>{{cite web|url=http://www.analog.com/library/analogdialogue/archives/43-09/edch+8+filter.pdf|title=Analog Dialogue Technical Journal - Analog Devices|website=www.analog.com|url-status=live|archive-url=https://web.archive.org/web/20160804012051/http://www.analog.com/library/analogdialogue/archives/43-09/edch%208%20filter.pdf|archive-date=2016-08-04}}</ref>
|-
! scope="row" | Lowpass
| <math>\frac{ \omega_\mathrm{N}^2 }{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>
|-
! scope="row" | Bandpass
| <math>\frac{ \frac{\omega_\mathrm{N}}{Q}s}{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>
|-
! scope="row" | Notch (bandstop)
| <math>\frac{ s^2 + \omega_\mathrm{N}^2}{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>
|-
! scope="row" | Highpass
| <math>\frac{ s^2 }{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>
|}

== Electrical systems ==

[[Image:bandwidth.svg|upright=1.2|thumb|A graph of a filter's gain magnitude, illustrating the concept of −3&nbsp;dB at a voltage gain of 0.707 or half-power bandwidth. The frequency axis of this symbolic diagram can be linear or [[logarithm]]ically scaled.]]

For an electrically resonant system, the ''Q'' factor represents the effect of [[electrical resistance]] and, for electromechanical resonators such as [[Crystal oscillator|quartz crystals]], mechanical [[friction]].

=== Relationship between {{mvar|Q}} and bandwidth ===

The 2-sided bandwidth relative to a resonant frequency of {{math|''F''<sub>0</sub>}} (Hz) is <math>\frac{F_0}{Q}</math>.

For example, an antenna tuned to have a {{mvar|Q}} value of 10 and a centre frequency of 100&nbsp;kHz would have a 3&nbsp;dB bandwidth of 10&nbsp;kHz.

In audio, bandwidth is often expressed in terms of [[octave]]s.  Then the relationship between {{mvar|Q}} and bandwidth is

<math display="block">Q = \frac{2^\frac{BW}{2}}{2^{BW} - 1} = \frac{1}{2 \sinh\left(\frac{1}{2}\ln(2) BW \right)},</math>
where {{mvar|BW}} is the bandwidth in octaves.<ref>{{Cite web|url=http://www.rane.com/note170.html|title=Bandwidth in Octaves Versus Q in Bandpass Filters|last=Dennis Bohn, Rane|date=January 2008|website=www.rane.com|access-date=2019-11-20}}</ref>

=== RLC circuits ===

In an ideal series [[RLC circuit]], and in a [[tuned radio frequency receiver]] (TRF) the {{mvar|Q}} factor is:<ref name=":1">{{Cite book|url=https://books.google.com/books?id=iRQa6dfeaKIC&q=quality+factor|title=Electric Circuits|last1=U.A.Bakshi|last2=A.V.Bakshi|date=2008|publisher=Technical Publications|isbn=9788184314526|pages=2–79|language=en}}{{Dead link|date=November 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

<math display="block">Q = \frac{1}{R} \sqrt{\frac{L}{C}} = \frac{\omega_0 L}{R} = \frac {1} {\omega_0 R C}</math>

where {{mvar|R}}, {{mvar|L}}, and {{mvar|C}} are the [[electrical resistance|resistance]], [[inductance]] and [[capacitance]] of the tuned circuit, respectively. Larger series resistances correspond to lower circuit {{mvar|Q}} values.

For a parallel RLC circuit, the {{mvar|Q}} factor is the inverse of the series case:<ref>{{cite web|url=http://fourier.eng.hmc.edu/e84/lectures/ch3/node8.html|title=Complete Response I - Constant Input|website=fourier.eng.hmc.edu|url-status=live|archive-url=https://web.archive.org/web/20120110062257/http://fourier.eng.hmc.edu/e84/lectures/ch3/node8.html|archive-date=2012-01-10}}</ref><ref name=":1" />

<math display="block">Q = R \sqrt{\frac{C}{L}} = \frac{R}{\omega_0 L} = \omega_0 R C</math><ref>[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf Frequency Response: Resonance, Bandwidth, ''Q'' Factor] {{webarchive|url=https://web.archive.org/web/20141206172316/http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf|date=2014-12-06|title=}} ([[PDF]])</ref>

Consider a circuit where {{mvar|R}}, {{mvar|L}}, and {{mvar|C}} are all in parallel. The lower the parallel resistance is, the more effect it will have in damping the circuit and thus result in lower {{mvar|Q}}. 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, {{mvar|R}}, in series with the inductance, {{mvar|L}}, {{mvar|Q}} is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improve {{mvar|Q}} and narrow the bandwidth is the desired result.

=== Individual reactive components ===

The {{mvar|Q}} 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. The {{mvar|Q}} of an inductor with a series loss resistance is the {{mvar|Q}} of a resonant circuit using that inductor (including its series loss) and a perfect capacitor.<ref name=dipaolo>
{{cite book
 |title       = Networks and Devices Using Planar Transmission Lines
 |first       = Franco
 |last        = Di Paolo
 |publisher   = CRC Press
 |year        = 2000
 |isbn        = 9780849318351
 |pages       = 490–491
 |url         = https://books.google.com/books?id=z9CsA1ZvwW0C&pg=PA489
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20180511181437/https://books.google.com/books?id=z9CsA1ZvwW0C&pg=PA489
 |archive-date = 2018-05-11
}}</ref>

<math display="block">Q_L = \frac{X_L}{R_L}=\frac{\omega_0 L}{R_L}</math>

where:
* {{math|''ω''<sub>0</sub>}} is the resonance frequency in radians per second;
* {{mvar|L}} is the inductance;
* {{mvar|X<sub>L</sub>}} is the [[inductive reactance]]; and
* {{mvar|R<sub>L</sub>}} is the series resistance of the inductor.

The {{mvar|Q}} of a capacitor with a series loss resistance is the same as the {{mvar|Q}} of a resonant circuit using that capacitor with a perfect inductor:<ref name=dipaolo/>

<math display="block">Q_C = \frac{-X_C}{R_C}=\frac{1}{\omega_0 C R_C}</math>

where:
* {{math|''ω''<sub>0</sub>}} is the resonance frequency in radians per second;
* {{mvar|C}} is the capacitance;
* {{mvar|X<sub>C</sub>}} is the [[capacitive reactance]]; and
* {{mvar|R<sub>C</sub>}} is the series resistance of the capacitor.

In general, the {{mvar|Q}} of a resonator involving a series combination of a capacitor and an inductor can be determined from the {{mvar|Q}} values of the components, whether their losses come from series resistance or otherwise:<ref name=dipaolo/>

<math dislpay="block"> Q = \frac{1}{\frac{1}{Q_L} + \frac{1}{Q_C}} </math>

== Mechanical systems ==
For a single damped mass-spring system, the {{mvar|Q}} factor represents the effect of simplified [[viscosity|viscous]] damping or [[Drag (physics)|drag]], where the damping force or drag force is proportional to velocity. The formula for the {{mvar|Q}} factor is:
<math display="block">Q = \frac{\sqrt{M k}}{D}, \,</math><!-- To derive this equation, go to the reference link and substitute eqn (2) into eqn (3), then simplify -->
where {{mvar|M}} is the mass, {{mvar|k}} is the spring constant, and {{mvar|D}} is the damping coefficient, defined by the equation {{math|1= ''F''<sub>damping</sub> = −''Dv''}}, where {{mvar|v}} is the velocity.<ref>[http://units.physics.uwa.edu.au/__data/page/115450/lecture5_(amplifier_noise_etc).pdf Methods of Experimental Physics – Lecture 5: Fourier Transforms and Differential Equations] {{webarchive|url=https://web.archive.org/web/20120319163127/http://units.physics.uwa.edu.au/__data/page/115450/lecture5_(amplifier_noise_etc).pdf|date=2012-03-19|title=}} ([[PDF]])</ref>

== Acoustical systems ==
The {{mvar|Q}} of a musical instrument is critical; an excessively high {{mvar|Q}} in a [[resonator]] 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.

The {{mvar|Q}} of a [[brass instrument]] or [[wind instrument]] needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed.
By contrast, a [[vuvuzela]] is made of flexible plastic, and therefore has a very low {{mvar|Q}} for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higher {{mvar|Q}} values. An excessively high {{mvar|Q}} can make it harder to hit a note. {{mvar|Q}} in an instrument may vary across frequencies, but this may not be desirable.

[[Helmholtz resonator]]s have a very high {{mvar|Q}}, as they are designed for picking out a very narrow range of frequencies.

== Optical systems ==
In [[optics]], the {{mvar|Q}} factor of a [[resonant cavity]] is given by
<math display="block">Q = \frac{2\pi f_o\,E}{P}, \,</math>
where {{mvar|f<sub>o</sub>}} is the resonant frequency, {{mvar|E}} is the stored energy in the cavity, and {{math|1= ''P'' = −{{sfrac|''dE''|''dt''}}}} is the power dissipated. The optical {{mvar|Q}} is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance. The average lifetime of a resonant [[photon]] in the cavity is proportional to the cavity's {{mvar|Q}}. If the {{mvar|Q}} factor of a [[laser]]'s cavity is abruptly changed from a low value to a high one, the laser will emit a [[Pulse (physics)|pulse]] of light that is much more intense than the laser's normal continuous output. This technique is known as [[Q-switching|{{mvar|Q}}-switching]]. {{mvar|Q}} factor is of particular importance in [[plasmonics]], where loss is linked to the damping of the [[surface plasmon resonance]].<ref>{{Cite journal|last1=Tavakoli|first1=Mehdi|last2=Jalili|first2=Yousef Seyed|last3=Elahi|first3=Seyed Mohammad|date=2019-04-28|title=Rayleigh-Wood anomaly approximation with FDTD simulation of plasmonic gold nanohole array for determination of optimum extraordinary optical transmission characteristics|journal=Superlattices and Microstructures|language=en|volume=130|pages=454–471|doi=10.1016/j.spmi.2019.04.035|bibcode=2019SuMi..130..454T|s2cid=150365680}}</ref> 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.<ref>{{Cite journal|last1=Chen|first1=Gang|last2=Mahan|first2=Gerald|last3=Meroueh|first3=Laureen|last4=Huang|first4=Yi|last5=Tsurimaki|first5=Yoichiro|last6=Tong|first6=Jonathan K.|last7=Ni|first7=George|last8=Zeng|first8=Lingping|last9=Cooper|first9=Thomas Alan|date=2017-12-31|title=Losses in plasmonics: from mitigating energy dissipation to embracing loss-enabled functionalities|journal=[[Advances in Optics and Photonics]]|language=EN|volume=9|issue=4|pages=775–827|doi=10.1364/AOP.9.000775|bibcode=2017AdOP....9..775B|issn=1943-8206|doi-access=free|arxiv=1802.01469}}</ref>

== See also ==
* [[Acoustic resonance]]
* [[Attenuation]]
* [[Chu–Harrington limit]]
* [[List of piezoelectric materials]]
* [[Phase margin]]
* [[Q meter]]
* [[Q multiplier]]
* [[Dissipation factor]]

== References ==
{{Reflist}}

== Further reading ==
{{refbegin}}
* {{Cite book|last1=Agarwal|first1=Anant|author-link1=Anant Agarwal|last2=Lang|first2=Jeffrey|title=Foundations of Analog and Digital Electronic Circuits|year=2005|publisher=Morgan Kaufmann|isbn=1-55860-735-8|url = https://books.google.com/books?id=83onAAAACAAJ&q=intitle:%22Foundations+of+Analog+and+Digital+Electronic+Circuits%22 }}
{{refend}}

== External links ==
{{Commons category|Quality factor}}
* [http://www.sengpielaudio.com/calculator-cutoffFrequencies.htm Calculating the cut-off frequencies when center frequency and ''Q'' factor is given]
* [http://www.techlib.com/reference/q.htm Explanation of ''Q'' factor in radio tuning circuits]

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[[Category:Electrical parameters]]
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[[Category:Mechanics]]
[[Category:Laser science]]
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