Q factor

# Q factor

Overview
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

the quality factor or Q factor is a dimensionless
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

parameter that describes how under-damped
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

an oscillator
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

or resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...

is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency.
Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

the quality factor or Q factor is a dimensionless
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

parameter that describes how under-damped
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

an oscillator
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

or resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...

is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency.
Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; 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. Oscillators with high quality factors have low damping
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

so that they ring longer.

Sinusoidally
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

driven resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...

s having higher Q factors resonate
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

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
RLC circuit
An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively...

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. (See oscillator phase noise
Oscillator Phase Noise
Oscillators inherently produce high levels of phase noise. That noise increases at frequencies close to the oscillation frequency or its harmonics. With the noise being close to the oscillation frequency, it cannot be removed by filtering without also removing the oscillation signal...

.)

The quality factor of oscillators varies substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have . Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability need high quality factors. Tuning forks have quality factors around Q = 1000. The quality factor of atomic clock
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...

s, superconducting RF
Superconducting Radio Frequency science and technology involves the application of electrical superconductors to radio frequency devices. The ultra-low electrical resistivity of a superconducting material allows an RF resonator to obtain an extremely high quality factor, Q...

cavities used in accelerators, and some high-Q lasers
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...

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 and that are closely related to the quality factor. Important examples include: the damping ratio
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

, relative bandwidth, linewidth
Oscillator linewidth
The concept of a linewidth is borrowed from laser spectroscopy. The linewidth of a laser is a measure of its phase noise. The spectrogram of a laser is produced by passing its light through a prism. The spectrogram of the output of a pure noise-free laser will consist of a single infinitely thin...

and bandwidth measured in octave
Octave (electronics)
In electronics, an octave is a doubling or halving of a frequency. The term is derived from the musical octave which similarly describes such frequency ratios, but the prefix octa-, denoting eight, has no significance in physics...

s.

The concept of Q factor originated in electronic engineering, as a measure of the 'quality' desired in a good tuned circuit or other resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...

.

## Definition of the quality factor

In the context of resonators, Q is defined in terms of the ratio of the energy stored in the resonator to the energy supplied by a generator, per cycle, to keep signal amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

constant, at a frequency (the resonant frequency), fr, where the stored energy is constant with time:

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 and capacitors; the lost energy is the sum of the energies dissipated in resistors per cycle. In mechanical systems, the stored energy is the maximum possible stored energy, or the total energy, i.e. the sum of the potential
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

and kinetic
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

energies at some point in time; the lost energy is the work done by an external conservative force
Conservative force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.It is possible to define a numerical value of...

, per cycle, to maintain amplitude.

For high values of Q, the following definition is also mathematically accurate:

where fr is the resonant frequency, Δf is the bandwidth, ωr = 2πfr is the angular
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

resonant frequency, and Δω is the angular bandwidth.

More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition of Q is used:

where ω is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

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 Complex impedances
.)

## Q factor and damping

The Q factor determines the qualitative behavior of simple damped
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

oscillators. (For mathematical details about these systems and their behavior see harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

and linear time invariant (LTI) system.)
• A system with low quality factor (Q < ½) 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 asymptotically. It has an impulse response
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

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
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

with a very low quality factor has a nearly first-order step response; the system's output responds to a step
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

input by slowly rising toward an asymptote
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 they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

.

• A system with high quality factor (Q > ½) 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 = ½) 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
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut 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 = ½) 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 safety margin
Factor of safety
Factor of safety , also known as safety factor , is a term describing the structural capacity of a system beyond the expected loads or actual loads. Essentially, how much stronger the system is than it usually needs to be for an intended load...

against overshoot.

In negative feedback
Negative feedback
Negative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...

systems, the dominant closed-loop response is often well-modeled by a second-order system. The phase margin
Phase margin
In electronic amplifiers, phase margin is the difference between the phase, measured in degrees, of an amplifier's output signal and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted,...

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).

### Quality factors of common systems

• A unity gain Sallen–Key filter topology with equivalent capacitors and equivalent resistors is critically damped (i.e., ).
• A Butterworth filter
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...

(i.e., continuous-time filter with the flattest passband frequency response) has an underdamped .
• A Bessel filter
Bessel filter
In electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay . Bessel filters are often used in audio crossover systems...

(i.e., continuous-time filter with flattest group delay
Group delay
Group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component...

) has an underdamped .

## Physical interpretation of Q

Physically speaking, Q is times the ratio of the total energy stored divided by the energy lost in a single cycle or equivalently the ratio of the stored energy to the energy dissipated over one radian of the oscillation.

It is a dimensionless parameter that compares the time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

for decay of an oscillating physical system's amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

to its oscillation period
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy.

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 , or about 1/535, of its original energy.

The width (bandwidth) of the resonance is given by
,

where is the resonant frequency, and , the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.

The factors Q, damping ratio
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

ζ, and attenuation
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

α are related such that

So the quality factor can be expressed as

and the exponential attenuation rate can be expressed as

For any 2nd order low-pass filter, the response function of the filter is

For this system, when (i.e., when the system is underdamped), it has two complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

poles that each have a real part of . That is, the attenuation parameter represents the rate of exponential decay of the oscillations (e.g., after an impulse
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

) of the system. A higher quality factor implies a lower attenuation, and so high Q systems oscillate for long times. For example, high quality bells have an approximately pure sinusoidal tone
Pure tone
A pure tone is a tone with a sinusoidal waveshape.A sine wave is characterized by its frequency, the number of cycles per second—or its wavelength, the distance the waveform travels through its medium within a period—and the amplitude, the size of each cycle...

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
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

and, for electromechanical resonators such as quartz crystals
Crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...

, mechanical friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

.

### RLC circuits

In an ideal series RLC circuit
RLC circuit
An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively...

A tuned radio frequency receiver is a radio receiver that is usually composed of several tuned radio frequency amplifiers followed by circuits to detect and amplify the audio signal. Prevalent in the early 20th century, it can be difficult to operate because each stage must be individually tuned...

(TRF) the Q factor is:
,

where , and are the resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

, inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

and capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

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:,

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 inductor is:

Where:
• is the frequency.
• is the inductance.
• is the inductive reactance.
• is the resistance of the inductor.

The Q of a capacitor is:

Where:
• is the frequency.
• is the capacitance.
• is the capacitive reactance.
• is the resistance of the capacitor.

## Mechanical systems

For a single damped mass-spring system, the Q factor represents the effect of simplified viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

damping or drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

, where the damping force or drag force is proportional to velocity. The formula for the Q factor is:

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation , where is the velocity.

## Optical systems

In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, the Q factor of a resonant cavity is given by

where is the resonant frequency, is the stored energy in the cavity, and 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
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

in the cavity is proportional to the cavity's Q. If the Q factor of a laser's
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

cavity is abruptly changed from a low value to a high one, the laser will emit a pulse
Pulse
In medicine, one's pulse represents the tactile arterial palpation of the heartbeat by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed against a bone, such as at the neck , at the wrist , behind the knee , on the inside of the elbow , and near the...

of light that is much more intense than the laser's normal continuous output. This technique is known as Q-switching
Q-switching
Q-switching, sometimes known as giant pulse formation, is a technique by which a laser can be made to produce a pulsed output beam. The technique allows the production of light pulses with extremely high peak power, much higher than would be produced by the same laser if it were operating in a...

.

• Damping ratio
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

• Attenuation
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

• Phase margin
Phase margin
In electronic amplifiers, phase margin is the difference between the phase, measured in degrees, of an amplifier's output signal and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted,...

• Bandwidth
• Q meter
Q meter
A Q meter is a piece of equipment used in the testing of radio frequency circuits. It has been largely replaced in professional laboratories by other types of impedance measuring device, though it is still in use among radio amateurs. It was developed at Boonton Radio Corporation in Boonton, New...

• Dissipation factor
Dissipation factor
In physics, the dissipation factor is a measure of loss-rate of energy of a mode of oscillation in a dissipative system. It is the reciprocal of Quality factor, which represents the quality of oscillation....