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In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how
underdamped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
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. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. 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
damping Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
, so that they ring or vibrate longer.


Explanation

The Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher ''Q'' factors
resonate Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscill ...
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'' tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High-''Q'' oscillators 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 ''Q'' near . 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 An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s, superconducting RF cavities used in accelerators, and some high-''Q''
lasers A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
can reach as high as 1011 and higher. There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
, relative bandwidth,
linewidth A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...
and bandwidth measured in octaves. The concept of ''Q'' originated with K. S. Johnson of
Western Electric Company The Western Electric Company was an American electrical engineering and manufacturing company officially founded in 1869. A wholly owned subsidiary of American Telephone & Telegraph for most of its lifespan, it served as the primary equipment ma ...
'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.


Definition

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, 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.


Bandwidth definition

In the context of resonators, there are two common definitions for ''Q'', which aren't 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: :Q \mathrel\stackrel \frac = \frac, where ''fr'' is the resonant frequency, 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, ''ωr'' = 2π''fr'' is the angular resonant frequency, and Δ''ω'' is the angular half-power bandwidth. Under this definition, ''Q'' is the reciprocal of fractional bandwidth.


Stored energy definition

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: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. :Q \mathrel\stackrel 2\pi \times \frac = 2\pi f_r \times \frac. 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
inductors An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
and
capacitors A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
; 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 and
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ent ...
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 ''Q'' is used: :Q(\omega) = \omega \times \frac, where ''ω'' is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
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.)


''Q'' factor and damping

The ''Q'' factor determines the qualitative behavior of simple damped oscillators. (For mathematical details about these systems and their behavior see harmonic oscillator and linear time invariant (LTI) system.) * A system with low quality factor (''Q'' < ) is said to be
overdamped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples i ...
. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
, approaching the steady state value
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
ally. It has an impulse response that is the sum of two 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 step input by slowly rising toward an
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
. * A system with high quality factor (''Q'' > ) is said to be
underdamped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
. 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'' = ) 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 (''Q'' = ) is said to be
critically damped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
. 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 safety margin against overshoot. In
negative feedback Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by othe ...
systems, the dominant closed-loop response is often well-modeled by a second-order system. The
phase margin Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
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).


Some examples


Physical interpretation

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. It is a dimensionless parameter that compares the exponential time constant for decay of an oscillating physical system's
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of ampl ...
to its oscillation
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
. 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 or 0.2%, of its original energy. This means the amplitude falls off to approximately ''e''−π or 4% of its original amplitude. The width (bandwidth) of the resonance is given by (approximately): :\Delta f = \frac, \, where ''f''N is the
natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all part ...
, and Δ''f'', the
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
, 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 ''f''N in hertz, as :\omega_\mathrm = 2\pi f_\mathrm. The factors ''Q'',
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
ζ,
natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all part ...
ωN, attenuation rate α, and exponential time constant are related such that: :Q = \frac = \frac = \frac, and the damping ratio can be expressed as: :\zeta = \frac = = . The envelope of oscillation decays proportional to ''e''−α''t'' or ''e''−''t''/, where α and can be expressed as: :\alpha = = \zeta \omega_\mathrm = and :\tau = = = \frac. The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as ''e''−2α''t'' or ''e''−2''t''/. For a two-pole lowpass filter, the transfer function of the filter is :H(s) = \frac \, For this system, when ''Q'' >  (i.e., when the system is underdamped), it has two
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
poles that each have a
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of −α. That is, the attenuation parameter α represents the rate of
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
of the oscillations (that is, of the output after an
impulse Impulse or Impulsive may refer to: Science * Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time * Impulse noise (disambiguation) * Specific impulse, the change in momentum per uni ...
) 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 pure sinusoidal tone for a long time after being struck by a hammer.


Electrical systems

For an electrically resonant system, the ''Q'' factor represents the effect of
electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...
and, for electromechanical resonators such as
quartz crystals Quartz is a hard, crystalline mineral composed of silica ( silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical for ...
, mechanical friction.


Relationship between ''Q'' and bandwidth

The 2-sided bandwidth relative to a resonant frequency of ''F''0 Hz is ''F''0/''Q''. 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 octaves. Then the relationship between ''Q'' and bandwidth is :Q = \frac = \frac, where ''BW'' is the bandwidth in octaves.


''RLC'' circuits

In an ideal series ''RLC'' circuit, and in a tuned radio frequency receiver (TRF) the ''Q'' factor is: :Q = \frac \sqrt = \frac = \frac where ''R'', ''L'' and ''C'' are the resistance,
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
and
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
of the tuned circuit, respectively. The larger the series resistance, the lower the circuit ''Q''. For a parallel ''RLC'' circuit, the ''Q'' factor is the inverse of the series case: :Q = R \sqrt = \frac = \omega_0 R C Consider a circuit where ''R'', ''L'' and ''C'' are all in parallel. The lower the parallel resistance, the more effect it will have in damping the circuit and thus the lower the ''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.


Individual reactive components

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. :Q_L = \frac=\frac where: * ''ω''0 is the resonance frequency in radians per second, * ''L'' is the inductance, * ''XL'' is the inductive reactance, and * ''RL'' is the series resistance of the inductor. 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: :Q_C = \frac=\frac where: * ''ω''0 is the resonance frequency in radians per second, * ''C'' is the capacitance, * ''XC'' is the
capacitive reactance In electrical circuits, reactance is the opposition presented to alternating current by inductance or capacitance. Greater reactance gives smaller current for the same applied voltage. Reactance is similar to resistance in this respect, but does ...
, and * ''RC'' is the series resistance of the capacitor. 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: : Q = \frac


Mechanical systems

For a single damped mass-spring system, the ''Q'' factor represents the effect of simplified
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the in ...
damping or drag, where the damping force or drag force is proportional to velocity. The formula for the Q factor is: :Q = \frac, \, where ''M'' is the mass, ''k'' is the spring constant, and ''D'' is the damping coefficient, defined by the equation ''F''damping = −''Dv'', where ''v'' is the velocity.


Acoustical systems

The ''Q'' of a musical instrument is critical; an excessively high ''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 ''Q'' of a
brass instrument A brass instrument is a musical instrument that produces sound by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips. Brass instruments are also called labrosones or labrophones, from Latin a ...
or
wind instrument A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitch ...
needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed. By contrast, a
vuvuzela The vuvuzela is a horn, with an inexpensive injection-molded plastic shell about long, which produces a loud monotone note, typically around B♭ 3 (the first B♭ below middle C). Some models are made in two parts to facilitate storage, 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. 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.
Helmholtz resonator Helmholtz resonance or wind throb is the phenomenon of air resonance in a cavity, such as when one blows across the top of an empty bottle. The name comes from a device created in the 1850s by Hermann von Helmholtz, the ''Helmholtz resonator'', wh ...
s have a very high Q, as they are designed for picking out a very narrow range of frequencies.


Optical systems

In optics, the ''Q'' factor of a
resonant cavity A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonat ...
is given by :Q = \frac, \, where ''fo'' is the resonant frequency, ''E'' is the stored energy in the cavity, and ''P'' = − 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 photon in the cavity is proportional to the cavity's ''Q''. If the ''Q'' factor of a laser's cavity is abruptly changed from a low value to a high one, the laser will emit a pulse of light that is much more intense than the laser's normal continuous output. This technique is known as ''Q''-switching. ''Q'' factor is of particular importance in
plasmonics Plasmonics or nanoplasmonics refers to the generation, detection, and manipulation of signals at optical frequencies along metal-dielectric interfaces in the nanometer scale. Inspired by photonics, plasmonics follows the trend of miniaturizing opt ...
, where loss is linked to the damping of the surface plasmon resonance. 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.


See also

*
Acoustic resonance Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration (its ''resonance frequencies''). The term "acoustic resonance" is sometimes used to nar ...
*
Attenuation In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable at ...
* Chu–Harrington limit *
Piezoelectric material properties This page lists properties of several commonly used piezoelectric materials. Piezoelectric materials (PMs) can be broadly classified as either crystalline, ceramic, or polymeric. The most commonly produced piezoelectric ceramics are lead zirconate ...
*
Phase margin Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
* Q meter * Q multiplier * Dissipation factor


References


Further reading

*


External links


Calculating the cut-off frequencies when center frequency and ''Q'' factor is given


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