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Q factor

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses of the terms Q, Q factor, and Quality factor, see <a href="Q_value_(disambiguation)" class="mw-redirect mw-disambig" title="Q value (disambiguation)">Q value (disambiguation)</a>.</div>

In <a href="Physics" title="Physics">physics</a> and <a href="Engineering" title="Engineering">engineering</a>, the quality factor or Q factor is a <a href="Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> parameter that describes how <a href="Underdamped" class="mw-redirect" title="Underdamped">underdamped</a> an <a href="Oscillator" class="mw-redirect" title="Oscillator">oscillator</a> or <a href="Resonator" title="Resonator">resonator</a> is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one <a href="Radian" title="Radian">radian</a> of the cycle of oscillation.<a href="#cite_note-1">[1]</a> Q factor is alternatively defined as the ratio of a resonator's centre frequency to its <a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a> when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results.<a href="#cite_note-2">[2]</a> Higher 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 Q, while a pendulum immersed in oil has a low one. Resonators with high quality factors have low <a href="Damping_ratio" class="mw-redirect" title="Damping ratio">damping</a>, so that they ring or vibrate longer.

<div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2></div>

The Q factor is a parameter that describes the <a href="Resonance" title="Resonance">resonance</a> behavior of an underdamped <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> (resonator). <a href="Sine_wave" title="Sine wave">Sinusoidally</a> driven <a href="Resonator" title="Resonator">resonators</a> having higher Q factors <a href="Resonance" title="Resonance">resonate</a> 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-Q <a href="RLC_circuit" title="RLC circuit">tuned circuit</a> in a radio receiver would be more difficult to tune, but would have more <a href="Selectivity_(radio)" title="Selectivity (radio)">selectivity</a>; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High-Q oscillators <a href="Oscillator_phase_noise" title="Oscillator phase noise">oscillate with a smaller range of frequencies</a> 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 Q near <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>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 <a href="Atomic_clock" title="Atomic clock">atomic clocks</a>, <a href="Superconducting_Radio_Frequency" class="mw-redirect" title="Superconducting Radio Frequency">superconducting RF</a> cavities used in accelerators, and some high-Q <a href="Optical_cavity" title="Optical cavity">lasers</a> can reach as high as 1011<a href="#cite_note-3">[3]</a> and higher.<a href="#cite_note-4">[4]</a>

There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the <a href="Damping_ratio" class="mw-redirect" title="Damping ratio">damping ratio</a>, <a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">relative bandwidth</a>, <a href="Oscillator_linewidth" title="Oscillator linewidth">linewidth</a> and bandwidth measured in <a href="Octave_(electronics)" title="Octave (electronics)">octaves</a>.

The concept of Q originated with K. S. Johnson of <a href="Western_Electric_Company" class="mw-redirect" title="Western Electric Company">Western Electric Company</a>'s Engineering Department while evaluating the quality of coils (inductors). His choice of the symbol 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.<a href="#cite_note-Green-5">[5]</a><a href="#cite_note-6">[6]</a><a href="#cite_note-Paschotta-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The definition of 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,<a href="#cite_note-Green-5">[5]</a> and has expanded beyond the electronics field to apply to dynamical systems in general: mechanical and acoustic resonators, material Q and quantum systems such as spectral lines and particle resonances.

<div class="mw-heading mw-heading3"><h3 id="Bandwidth_definition">Bandwidth definition</h3></div>

In the context of resonators, there are two common definitions for Q, which are not exactly equivalent. They become approximately equivalent as Q becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator:<a href="#cite_note-Green-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle Q\mathrel {\stackrel {\text{def}}{=}} {\frac {f_{\mathrm {r} }}{\Delta f}}={\frac {\omega _{\mathrm {r} }}{\Delta \omega }},}</annotation>

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where fr is the resonant frequency, Δf is the resonance width or <a href="Full_width_at_half_maximum" title="Full width at half maximum">full width at half maximum</a> (FWHM) i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ωr = 2πfr is the <a href="Angular_frequency" title="Angular frequency">angular</a> resonant frequency, and Δω is the angular half-power bandwidth.

Under this definition, Q is the reciprocal of <a href="Fractional_bandwidth" class="mw-redirect" title="Fractional bandwidth">fractional bandwidth</a>.

<div class="mw-heading mw-heading3"><h3 id="Stored_energy_definition">Stored energy definition</h3></div>

The other common nearly equivalent definition for Q is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes:<a href="#cite_note-IEEE-8">[8]</a><a href="#cite_note-:0-9">[9]</a><a href="#cite_note-Green-5">[5]</a>

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<mtext>energy stored</mtext>

<mtext>energy dissipated per cycle</mtext>

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<mtext>energy stored</mtext>

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<annotation encoding="application/x-tex">{\displaystyle 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}}}.}</annotation>

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The factor 2π makes 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 <a href="Inductors" class="mw-redirect" title="Inductors">inductors</a> and <a href="Capacitors" class="mw-redirect" title="Capacitors">capacitors</a>; the lost energy is the sum of the energies dissipated in <a href="Resistors" class="mw-redirect" title="Resistors">resistors</a> per cycle. In mechanical systems, the stored energy is the sum of the <a href="Potential_energy" title="Potential energy">potential</a> and <a href="Kinetic_energy" title="Kinetic energy">kinetic</a> energies at some point in time; the lost energy is the work done by an external <a href="Force" title="Force">force</a>, per cycle, to maintain amplitude.

More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition of Q is used:<a href="#cite_note-IEEE-8">[8]</a><a href="#cite_note-10">[10]</a><a href="#cite_note-:0-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle Q(\omega )=\omega \times {\frac {\text{maximum energy stored}}{\text{power loss}}},}</annotation>

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where ω is the <a href="Angular_frequency" title="Angular frequency">angular frequency</a> 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 <a href="Reactive_power" class="mw-redirect" title="Reactive power">reactive power</a> to <a href="Real_power" class="mw-redirect" title="Real power">real power</a>. (See <a href="#Individual_reactive_components">Individual reactive components</a>.)

<div class="mw-heading mw-heading2"><h2 id="Q-factor_and_damping">Q-factor and damping</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Damping" title="Damping">Damping</a> and <a href="Linear_time-invariant_system" title="Linear time-invariant system">linear time invariant (LTI) system</a></div>

The Q-factor determines the <a href="Qualitative_data" class="mw-redirect" title="Qualitative data">qualitative</a> behavior of simple damped oscillators. (For mathematical details about these systems and their behavior see <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> and <a href="LTI_system" class="mw-redirect" title="LTI system">linear time invariant (LTI) system</a>.)

Starting from the stored energy definition for, it can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\frac {1}{2\zeta }}}">

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<annotation encoding="application/x-tex">{\displaystyle Q={\frac {1}{2\zeta }}}</annotation>

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</math><img src="./ffff729cd776e404e2174542c8268005a710d2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.03ex; height:5.676ex;" alt="{\displaystyle Q={\frac {1}{2\zeta }}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> is the <a href="Damping#Damping_ratio" title="Damping">damping ratio</a>. There are three key distinct cases:

<ul><li>A system with low quality factor (Q < <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠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 <a href="Exponential_decay" title="Exponential decay">exponential decay</a>, approaching the steady state value <a href="Asymptotic" class="mw-redirect" title="Asymptotic">asymptotically</a>. It has an <a href="Impulse_response" title="Impulse response">impulse response</a> that is the sum of two <a href="Exponential_decay" title="Exponential decay">decaying exponential functions</a> 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 <a href="Low-pass_filter" title="Low-pass filter">low-pass filter</a> with a very low quality factor has a nearly first-order step response; the system's output responds to a <a href="Heaviside_step_function" title="Heaviside step function">step input</a> by slowly rising toward an asymptote.</li>

<li>A system with high quality factor (Q > ⁠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 Q = ⁠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 <a href="Low-pass_filter" title="Low-pass filter">low-pass filter</a> 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.</li>

<li>A system with an intermediate quality factor (Q = ⁠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 <a href="Factor_of_safety" title="Factor of safety">safety margin</a> against overshoot.</li></ul>

In <a href="Negative_feedback" title="Negative feedback">negative feedback</a> systems, the dominant closed-loop response is often well-modeled by a second-order system. The <a href="Phase_margin" title="Phase margin">phase margin</a> of the open-loop system sets the quality factor 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).

<div class="mw-heading mw-heading3"><h3 id="Some_examples">Some examples</h3></div>

<div><ul><li>A unity-gain <a href="Sallen%E2%80%93Key_topology#Application:_low-pass_filter" title="Sallen–Key topology">Sallen–Key lowpass filter topology</a> with equal capacitors and equal resistors is critically damped (i.e., Q = ⁠1/2⁠).</li><li>A second-order <a href="Bessel_filter" title="Bessel filter">Bessel filter</a> (i.e., continuous-time filter with flattest <a href="Group_delay" class="mw-redirect" title="Group delay">group delay</a>) has an underdamped Q = ⁠1/√3⁠.</li><li>A second-order <a href="Butterworth_filter" title="Butterworth filter">Butterworth filter</a> (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped Q = ⁠1/√2⁠.<a href="#cite_note-11">[11]</a></li><li>A pendulum's Q-factor is: Q = Mω/Γ, where M is the mass of the bob, ω = 2π/T is the pendulum's radian frequency of oscillation, and Γ is the frictional damping force on the pendulum per unit velocity.</li><li>The design of a high-energy (near <a href="Terahertz_(unit)" class="mw-redirect" title="Terahertz (unit)">terahertz</a>) <a href="Gyrotron" title="Gyrotron">gyrotron</a> considers both diffractive Q-factor, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Q_{D}\approx 30\left({\frac {L}{\lambda }}\right)^{2}}">

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<annotation encoding="application/x-tex">{\textstyle Q_{D}\approx 30\left({\frac {L}{\lambda }}\right)^{2}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle Q_{\Omega }={\frac {R_{\mathrm {w} }}{\delta }}{\frac {1-m^{2}}{v_{m,p}^{2}}},}</annotation>

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where Rw is the cavity wall radius, δ is the <a href="Skin_depth" class="mw-redirect" title="Skin depth">skin depth</a> of the cavity wall, vm,p is the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> scalar (m is the azimuth index, p is the radial index; in this application, skin depth is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta ={1}/{\sqrt {\pi f\sigma u_{o}}}}">

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<annotation encoding="application/x-tex">{\textstyle \delta ={1}/{\sqrt {\pi f\sigma u_{o}}}}</annotation>

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</math><img src="./125f0e4160deb36a4a3615377aca577ab35eb9f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.095ex; height:3.343ex;" alt="{\textstyle \delta ={1}/{\sqrt {\pi f\sigma u_{o}}}}" loading="lazy">)<a href="#cite_note-12">[12]</a></li><li>In <a href="Medical_ultrasonography" class="mw-redirect" title="Medical ultrasonography">medical ultrasonography</a>, a transducer with a high Q-factor is suitable for <a href="Doppler_ultrasonography" title="Doppler ultrasonography">doppler ultrasonography</a> because of its long ring-down time, where it can measure the velocities of blood flow. Meanwhile, a transducer with a low 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.<a href="#cite_note-13">[13]</a></li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="Physical_interpretation">Physical interpretation</h2></div>

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 enough Q values, 2π times the ratio of the total energy stored and the energy lost in a single cycle.<a href="#cite_note-14">[14]</a>

It is a dimensionless parameter that compares the <a href="Exponential_decay#Mean_lifetime" title="Exponential decay">exponential time constant</a> τ for decay of an <a href="Oscillating" class="mw-redirect" title="Oscillating">oscillating</a> physical system's <a href="Amplitude" title="Amplitude">amplitude</a> to its oscillation <a href="Frequency" title="Frequency">period</a>. 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 Q values is somewhat higher than the oscillation frequency as measured by zero crossings.

Equivalently (for large values of Q), the Q factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to e−2π, or about 1⁄535 or 0.2%, of its original energy.<a href="#cite_note-15">[15]</a> This means the amplitude falls off to approximately e−π or 4% of its original amplitude.<a href="#cite_note-16">[16]</a>

The width (bandwidth) of the resonance is given by (approximately):

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<annotation encoding="application/x-tex">{\displaystyle \Delta f={\frac {f_{\mathrm {N} }}{Q}},\,}</annotation>

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where fN is the <a href="Natural_frequency" title="Natural frequency">natural frequency</a>, and Δf, the <a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a>, 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 fN in <a href="Hertz" title="Hertz">hertz</a>, as

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<annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {N} }=2\pi f_{\mathrm {N} }.}</annotation>

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The factors Q, <a href="Damping_ratio" class="mw-redirect" title="Damping ratio">damping ratio</a> ζ, <a href="Natural_frequency" title="Natural frequency">natural frequency</a> ωN, <a href="Exponential_decay" title="Exponential decay">attenuation rate</a> α, and <a href="Exponential_decay#Mean_lifetime" title="Exponential decay">exponential time constant</a> τ are related such that:<a href="#cite_note-Siebert-17">[17]</a>

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<annotation encoding="application/x-tex">{\displaystyle Q={\frac {1}{2\zeta }}={\frac {\omega _{\mathrm {N} }}{2\alpha }}={\frac {\tau \omega _{\mathrm {N} }}{2}},}</annotation>

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and the damping ratio can be expressed as:

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<annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{\mathrm {N} }}={1 \over \tau \omega _{\mathrm {N} }}.}</annotation>

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The envelope of oscillation decays proportional to e−αt or e−t/τ, where α and τ can be expressed as:

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<annotation encoding="application/x-tex">{\displaystyle \alpha ={\omega _{\mathrm {N} } \over 2Q}=\zeta \omega _{\mathrm {N} }={1 \over \tau }}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle \tau ={2Q \over \omega _{\mathrm {N} }}={1 \over \zeta \omega _{\mathrm {N} }}={\frac {1}{\alpha }}.}</annotation>

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

For a two-pole lowpass filter, the <a href="Transfer_function" title="Transfer function">transfer function</a> of the filter is<a href="#cite_note-Siebert-17">[17]</a>

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<annotation encoding="application/x-tex">{\displaystyle 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}}}\,}</annotation>

</semantics>

</math>

For this system, when Q > ⁠1/2⁠ (i.e., when the system is underdamped), it has two <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> poles that each have a <a href="Real_part" class="mw-redirect" title="Real part">real part</a> of −α. That is, the attenuation parameter α represents the rate of <a href="Exponential_decay" title="Exponential decay">exponential decay</a> of the oscillations (that is, of the output after an <a href="Impulse_response" title="Impulse response">impulse</a>) 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 approximately <a href="Pure_tone" title="Pure tone">pure sinusoidal tone</a> for a long time after being struck by a hammer.

<caption>Transfer functions for 2nd-order filters

</caption>

<th scope="col">Filter type (2nd order)

</th>

<th scope="col">Transfer function H(s)<a href="#cite_note-18">[18]</a>

</th></tr>

<tr>

<th scope="row">Lowpass

</th>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\omega _{\mathrm {N} }^{2}}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}</annotation>

</semantics>

</math><img src="./6d5dc45895df34292a4b1f4b648ed1dc754ba07f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:15.686ex; height:8.009ex;" alt="{\displaystyle {\frac {\omega _{\mathrm {N} }^{2}}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}" loading="lazy">

</td></tr>

<tr>

<th scope="row">Bandpass

</th>

<mfrac>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\frac {\omega _{\mathrm {N} }}{Q}}s}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}</annotation>

</semantics>

</math><img src="./db5f30d4d4e4eb022418bc7789c985748f2ee656.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:15.686ex; height:8.676ex;" alt="{\displaystyle {\frac {{\frac {\omega _{\mathrm {N} }}{Q}}s}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}" loading="lazy">

</td></tr>

<tr>

<th scope="row">Notch (bandstop)

</th>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {s^{2}+\omega _{\mathrm {N} }^{2}}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}</annotation>

</semantics>

</math><img src="./733db7a43f3c43d89b70551629bc1af9e1b17b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:15.686ex; height:8.009ex;" alt="{\displaystyle {\frac {s^{2}+\omega _{\mathrm {N} }^{2}}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}" loading="lazy">

</td></tr>

<tr>

<th scope="row">Highpass

</th>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {s^{2}}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}</annotation>

</semantics>

</math><img src="./ec6116eb8bb90b9cceeed1da22f44b5c969f3bdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:15.686ex; height:7.676ex;" alt="{\displaystyle {\frac {s^{2}}{s^{2}+{\frac {\omega _{\mathrm {N} }}{Q}}s+\omega _{\mathrm {N} }^{2}}}}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Electrical_systems">Electrical systems</h2></div>

For an electrically resonant system, the Q factor represents the effect of <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> and, for electromechanical resonators such as <a href="Crystal_oscillator" title="Crystal oscillator">quartz crystals</a>, mechanical <a href="Friction" title="Friction">friction</a>.

<div class="mw-heading mw-heading3"><h3 id="Relationship_between_Q_and_bandwidth">Relationship between Q and bandwidth</h3></div>

The 2-sided bandwidth relative to a resonant frequency of F0 (Hz) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{0}}{Q}}}">

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {F_{0}}{Q}}}</annotation>

</semantics>

</math><img src="./0440692111d0de11719e59447a11863bc92632c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.385ex; height:5.843ex;" alt="{\displaystyle {\frac {F_{0}}{Q}}}" loading="lazy">.

For example, an antenna tuned to have a Q 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 of <a href="Octave" title="Octave">octaves</a>. Then the relationship between Q and bandwidth is

<mfrac>

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q={\frac {2^{\frac {BW}{2}}}{2^{BW}-1}}={\frac {1}{2\sinh \left({\frac {1}{2}}\ln(2)BW\right)}},}</annotation>

</semantics>

</math>

where BW is the bandwidth in octaves.<a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="RLC_circuits">RLC circuits</h3></div>

In an ideal series <a href="RLC_circuit" title="RLC circuit">RLC circuit</a>, and in a <a href="Tuned_radio_frequency_receiver" title="Tuned radio frequency receiver">tuned radio frequency receiver</a> (TRF) the Q factor is:<a href="#cite_note-:1-20">[20]</a>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q={\frac {1}{R}}{\sqrt {\frac {L}{C}}}={\frac {\omega _{0}L}{R}}={\frac {1}{\omega _{0}RC}}}</annotation>

</semantics>

</math>

where R, L, and C are the <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>, <a href="Inductance" title="Inductance">inductance</a> and <a href="Capacitance" title="Capacitance">capacitance</a> of the tuned circuit, respectively. Larger series resistances correspond to lower circuit Q values.

For a parallel RLC circuit, the Q factor is the inverse of the series case:<a href="#cite_note-21">[21]</a><a href="#cite_note-:1-20">[20]</a>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q=R{\sqrt {\frac {C}{L}}}={\frac {R}{\omega _{0}L}}=\omega _{0}RC}</annotation>

</semantics>

</math><a href="#cite_note-22">[22]</a>

Consider a circuit where R, L, and 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 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, 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 improve Q and narrow the bandwidth is the desired result.

<div class="mw-heading mw-heading3"><h3 id="Individual_reactive_components">Individual reactive components</h3></div>

The 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 Q of an inductor with a series loss resistance is the Q of a resonant circuit using that inductor (including its series loss) and a perfect capacitor.<a href="#cite_note-dipaolo-23">[23]</a>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

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</msub>

<msub>

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</mrow>

<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{L}={\frac {X_{L}}{R_{L}}}={\frac {\omega _{0}L}{R_{L}}}}</annotation>

</semantics>

</math>

where:

<ul><li>ω0 is the resonance frequency in radians per second;</li>

<li>L is the inductance;</li>

<li>XL is the <a href="Inductive_reactance" class="mw-redirect" title="Inductive reactance">inductive reactance</a>; and</li>

<li>RL is the series resistance of the inductor.</li></ul>

The Q of a capacitor with a series loss resistance is the same as the Q of a resonant circuit using that capacitor with a perfect inductor:<a href="#cite_note-dipaolo-23">[23]</a>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{C}={\frac {-X_{C}}{R_{C}}}={\frac {1}{\omega _{0}CR_{C}}}}</annotation>

</semantics>

</math>

where:

<ul><li>ω0 is the resonance frequency in radians per second;</li>

<li>C is the capacitance;</li>

<li>XC is the <a href="Capacitive_reactance" class="mw-redirect" title="Capacitive reactance">capacitive reactance</a>; and</li>

<li>RC is the series resistance of the capacitor.</li></ul>

In general, the Q of a resonator involving a series combination of a capacitor and an inductor can be determined from the Q values of the components, whether their losses come from series resistance or otherwise:<a href="#cite_note-dipaolo-23">[23]</a>

<mfrac>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

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</msub>

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</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q={\frac {1}{{\frac {1}{Q_{L}}}+{\frac {1}{Q_{C}}}}}}</annotation>

</semantics>

</math><img src="./40f036f796829d4a323449e4f3f4f5e6e70c8eec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:15.137ex; height:7.176ex;" alt="{\displaystyle Q={\frac {1}{{\frac {1}{Q_{L}}}+{\frac {1}{Q_{C}}}}}}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Mechanical_systems">Mechanical systems</h2></div>

For a single damped mass-spring system, the Q factor represents the effect of simplified <a href="Viscosity" title="Viscosity">viscous</a> damping or <a href="Drag_(physics)" title="Drag (physics)">drag</a>, where the damping force or drag force is proportional to velocity. The formula for the Q factor is:

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q={\frac {\sqrt {Mk}}{D}},\,}</annotation>

</semantics>

</math>

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation Fdamping = −Dv, where v is the velocity.<a href="#cite_note-24">[24]</a>

<div class="mw-heading mw-heading2"><h2 id="Acoustical_systems">Acoustical systems</h2></div>

The Q of a musical instrument is critical; an excessively high Q in a <a href="Resonator" title="Resonator">resonator</a> 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 Q of a <a href="Brass_instrument" title="Brass instrument">brass instrument</a> or <a href="Wind_instrument" title="Wind instrument">wind instrument</a> needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed.

By contrast, a <a href="Vuvuzela" title="Vuvuzela">vuvuzela</a> is made of flexible plastic, and therefore has a very low Q for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higher Q values. An excessively high Q can make it harder to hit a note. Q in an instrument may vary across frequencies, but this may not be desirable.

<a href="Helmholtz_resonator" class="mw-redirect" title="Helmholtz resonator">Helmholtz resonators</a> have a very high Q, as they are designed for picking out a very narrow range of frequencies.

<div class="mw-heading mw-heading2"><h2 id="Optical_systems">Optical systems</h2></div>

In <a href="Optics" title="Optics">optics</a>, the Q factor of a <a href="Resonant_cavity" class="mw-redirect" title="Resonant cavity">resonant cavity</a> is given by

<mfrac>

<mrow>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle Q={\frac {2\pi f_{o}\,E}{P}},\,}</annotation>

</semantics>

</math>

where fo is the resonant frequency, E is the stored energy in the cavity, and P = −⁠dE/dt⁠ is the power dissipated. The optical Q is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance. The average lifetime of a resonant <a href="Photon" title="Photon">photon</a> in the cavity is proportional to the cavity's Q. If the Q factor of a <a href="Laser" title="Laser">laser</a>'s cavity is abruptly changed from a low value to a high one, the laser will emit a <a href="Pulse_(physics)" title="Pulse (physics)">pulse</a> of light that is much more intense than the laser's normal continuous output. This technique is known as <a href="Q-switching" title="Q-switching">Q-switching</a>. Q factor is of particular importance in <a href="Plasmonics" title="Plasmonics">plasmonics</a>, where loss is linked to the damping of the <a href="Surface_plasmon_resonance" title="Surface plasmon resonance">surface plasmon resonance</a>.<a href="#cite_note-25">[25]</a> 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.<a href="#cite_note-26">[26]</a>

<ul><li><a href="Acoustic_resonance" title="Acoustic resonance">Acoustic resonance</a></li>

<li><a href="Attenuation" title="Attenuation">Attenuation</a></li>

<li><a href="Chu%E2%80%93Harrington_limit" title="Chu–Harrington limit">Chu–Harrington limit</a></li>

<li><a href="List_of_piezoelectric_materials" title="List of piezoelectric materials">List of piezoelectric materials</a></li>

<li><a href="Phase_margin" title="Phase margin">Phase margin</a></li>

<li><a href="Q_meter" title="Q meter">Q meter</a></li>

<li><a href="Q_multiplier" title="Q multiplier">Q multiplier</a></li>

<li><a href="Dissipation_factor" title="Dissipation factor">Dissipation factor</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHickman2013" class="citation book cs1">Hickman, Ian (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l-AgBQAAQBAJ&pg=PA42">Analog Electronics: Analog Circuitry Explained</a>. Newnes. p. 42. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781483162287</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFTooley2006" class="citation book cs1">Tooley, Michael H. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rwWWvtQYEO0C&pg=PA78">Electronic circuits: fundamentals and applications</a>. Newnes. pp. 77–78. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-6923-8</bdi>. <a rel="nofollow" class="external text" href="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">Archived</a> from the original on 2016-12-01.</cite>

</li>

<a rel="nofollow" class="external text" href="http://www.rp-photonics.com/q_factor.html">Encyclopedia of Laser Physics and Technology: Q factor</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090224211703/http://www.rp-photonics.com/q_factor.html">Archived</a> 2009-02-24 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

<a rel="nofollow" class="external text" href="http://tf.nist.gov/general/enc-q.htm">Time and Frequency from A to Z: Q to Ra</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080504160852/http://tf.nist.gov/general/enc-q.htm">Archived</a> 2008-05-04 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

<cite id="CITEREFGreen1955" class="citation journal cs1">Green, Estill I. (October 1955). <a rel="nofollow" class="external text" href="http://www.collinsaudio.com/Prosound_Workshop/The_story_of_Q.pdf">"The Story of Q"</a> (PDF). <a href="American_Scientist" title="American Scientist">American Scientist</a>. 43: 584–594. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121203044200/http://www.collinsaudio.com/Prosound_Workshop/The_story_of_Q.pdf">Archived</a> (PDF) from the original on 2012-12-03. Retrieved 2012-11-21.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> B. Jeffreys, Q.Jl R. astr. Soc. (1985) 26, 51–52

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li><cite id="CITEREFAgarwalLang2005" class="citation book cs1"><a href="Anant_Agarwal" title="Anant Agarwal">Agarwal, Anant</a>; Lang, Jeffrey (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=83onAAAACAAJ&q=intitle:%22Foundations+of+Analog+and+Digital+Electronic+Circuits%22">Foundations of Analog and Digital Electronic Circuits</a>. Morgan Kaufmann. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-55860-735-8</bdi>.</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Quality_factor" class="extiw external" title="commons:Category:Quality factor">Quality factor</a>.</div></div>

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<ul><li><a rel="nofollow" class="external text" href="http://www.sengpielaudio.com/calculator-cutoffFrequencies.htm">Calculating the cut-off frequencies when center frequency and Q factor is given</a></li>

<li><a rel="nofollow" class="external text" href="http://www.techlib.com/reference/q.htm">Explanation of Q factor in radio tuning circuits</a></li></ul>

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