Editing Q factor (section)

== 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>