Q factor

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In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is, [1] and characterizes a resonator's bandwidth relative to its centre frequency. [2] Higher Q indicates a lower rate of energy loss relative to the stored energy of the resonator; 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, so that they ring or vibrate longer.

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Engineering is the use of scientific principles to design and build machines, structures, and other things, including bridges, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application. See glossary of engineering.

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the SI of one unit that is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

Explanation

Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher Q factors resonate 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. (See oscillator phase noise.)

In mechanical systems, resonance is a phenomenon that only occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts. This causes the system to oscillate with larger amplitude than when the force is applied at other frequencies.

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:

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

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 12. 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 clocks, superconducting RF cavities used in accelerators, and some high-Q lasers can reach as high as 1011 [3] and higher. [4]

An atomic clock is a clock device that uses a hyperfine transition frequency in the microwave, or electron transition frequency in the optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element. Atomic clocks are the most accurate time and frequency standards known, and are used as primary standards for international time distribution services, to control the wave frequency of television broadcasts, and in global navigation satellite systems such as GPS.

An optical cavity, resonating 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 oscillators and some interferometers. Light confined in the cavity reflects multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam.

There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the damping ratio, relative bandwidth, linewidth and bandwidth measured in octaves.

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes.

Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to passband bandwidth or baseband bandwidth. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth applies to a low-pass filter or baseband signal; the bandwidth is equal to its upper cutoff frequency.

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 line. If the laser exhibits phase noise, the line will have non-zero width. The greater the phase noise, the wider the line. The same will be true with oscillators. The spectrum of the output of a noise-free oscillator has energy at each of the harmonics of the output signal, but the bandwidth of each harmonic will be zero. If the oscillator exhibits phase noise, the harmonics will not have zero bandwidth. The more phase noise the oscillator exhibits, the wider the bandwidth of each harmonic.

The concept of Q originated with K. S. Johnson of Western Electric Company'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. [5] [6] [7]

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, material Q and spectral lines. [5]

Resonant devices

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: [5]

${\displaystyle Q\ {\stackrel {\mathrm {def} }{=}}\ {\frac {f_{r}}{\Delta f}}={\frac {\omega _{r}}{\Delta \omega }},\,}$

where fr is the resonant frequency, Δf 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 = 2πfr is the angular resonant frequency, and Δω is the angular half-power bandwidth.

Coils and condensers

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: [8] [9] [5]

${\displaystyle Q\ {\stackrel {\mathrm {def} }{=}}\ 2\pi \times {\frac {\text{energy stored}}{\text{energy dissipated per cycle}}}=2\pi f_{r}\times {\frac {\text{energy stored}}{\text{power loss}}}.\,}$

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 and kinetic energies at some point in time; the lost energy is the work done by an external conservative 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: [8] [10] [ failed verification (See discussion.)] [9]

${\displaystyle Q(\omega )=\omega \times {\frac {\text{maximum energy stored}}{\text{power loss}}},\,}$

where ω is the angular frequency 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 < 12) 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 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.
• A system with high quality factor (Q > 12) 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 = 12) 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 = 12) 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 against overshoot.

In negative feedback systems, the dominant closed-loop response is often well-modeled by a second-order system. The phase margin 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).

Physical interpretation

Physically speaking, Q is 2π 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. [12]

It is a dimensionless parameter that compares the exponential time constant τ for decay of an oscillating physical system's amplitude to its oscillation period. 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 e−2π, or about 1535 or 0.2%, of its original energy. [13] This means the amplitude falls off to approximately eπ or 4% of its original amplitude. [14]

The width (bandwidth) of the resonance is given by

${\displaystyle \Delta f={\frac {f_{0}}{Q}},\,}$

where f0 is the resonant frequency, and Δf, the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.

The resonant frequency is often expressed in natural units (radians per second), rather than using the f0 in hertz, as

${\displaystyle \omega _{0}=2\pi f_{0}.}$

The factors Q, damping ratio ζ, attenuation rate α, and exponential time constant τ are related such that: [15]

${\displaystyle Q={\frac {1}{2\zeta }}={\omega _{0} \over 2\alpha }={\tau \omega _{0} \over 2},}$

and the damping ratio can be expressed as:

${\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{0}}={1 \over \tau \omega _{0}}.}$

The envelope of oscillation decays proportional to eαt or et/τ, where α and τ can be expressed as:

${\displaystyle \alpha ={\omega _{0} \over 2Q}=\zeta \omega _{0}={1 \over \tau }}$

and

${\displaystyle \tau ={2Q \over \omega _{0}}={1 \over \zeta \omega _{0}}={1 \over \alpha }.}$

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 transfer function of the filter is [15]

${\displaystyle H(s)={\frac {\omega _{0}^{2}}{s^{2}+\underbrace {\frac {\omega _{0}}{Q}} _{2\zeta \omega _{0}=2\alpha }s+\omega _{0}^{2}}}\,}$

For this system, when Q > 12 (i.e., when the system is underdamped), it has two complex conjugate poles that each have a real part of −α. That is, the attenuation parameter α represents the rate of exponential decay of the oscillations (that is, of the output after an impulse) 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.

Filter type (2nd order)Transfer function [16]
Lowpass${\displaystyle H(s)={\frac {\omega _{0}^{2}}{s^{2}+{\frac {\omega _{0}}{Q}}s+\omega _{0}^{2}}}}$
Bandpass${\displaystyle H(s)={\frac {{\frac {\omega _{0}}{Q}}s}{s^{2}+{\frac {\omega _{0}}{Q}}s+\omega _{0}^{2}}}}$
Notch (Bandstop)${\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+{\frac {\omega _{0}}{Q}}s+\omega _{0}^{2}}}}$
Highpass${\displaystyle H(s)={\frac {s^{2}}{s^{2}+{\frac {\omega _{0}}{Q}}s+\omega _{0}^{2}}}}$

Electrical systems

For an electrically resonant system, the Q factor represents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals, mechanical friction.

Relationship between Q and bandwidth

The 2-sided bandwidth relative to a resonant frequency of F0 Hz is F0/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

${\displaystyle Q\ =\ {\frac {2^{\frac {BW}{2}}}{2^{BW}-1}}\ =\ {\frac {1}{2\sinh \left({\frac {\ln(2)}{2}}BW\right)}},}$

where BW is the bandwidth in octaves. [17]

RLC circuits

In an ideal series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is: [18]

${\displaystyle Q={\frac {1}{R}}{\sqrt {\frac {L}{C}}}={\frac {\omega _{0}L}{R}}={\frac {1}{\omega _{0}RC}}}$

where R, L and C are the resistance, inductance and capacitance 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: [19] [18]

${\displaystyle Q=R{\sqrt {\frac {C}{L}}}={\frac {R}{\omega _{0}L}}=\omega _{0}RC}$ [20]

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. [21]

${\displaystyle Q_{L}={\frac {X_{L}}{R_{L}}}={\frac {\omega _{0}L}{R_{L}}}}$

where:

• ω0 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: [21]

${\displaystyle Q_{C}={\frac {-X_{C}}{R_{C}}}={\frac {1}{\omega _{0}CR_{C}}}}$

where:

• ω0 is the resonance frequency in radians per second,
• C is the capacitance,
• XC is the capacitive reactance, 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: [21]

${\displaystyle Q={\frac {1}{{\frac {1}{Q_{L}}}+{\frac {1}{Q_{C}}}}}}$

Mechanical systems

For a single damped mass-spring system, the Q factor represents the effect of simplified viscous damping or drag, where the damping force or drag force is proportional to velocity. The formula for the Q factor is:

${\displaystyle Q={\frac {\sqrt {Mk}}{D}},\,}$

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. [22]

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 or wind instrument needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed. By contrast, a vuvuzela 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 resonators 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 is given by

${\displaystyle Q={\frac {2\pi f_{o}\,E}{P}},\,}$

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 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, where loss is linked to the damping of the surface plasmon resonance [23] . 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 [24] .

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