Editing Q factor (section)

== 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>
|}