Q factor
[1] Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force.
[2] Higher Q indicates a lower rate of energy loss and the oscillations die out more slowly.
Resonators with high quality factors have low damping, so that they ring or vibrate longer.
Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q near 1⁄2.
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] There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is.
The concept of Q originated with K. S. Johnson of Western Electric Company's Engineering Department while evaluating the quality of coils (inductors).
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.
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]
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.
The Q factor determines the qualitative behavior of simple damped oscillators.
Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy.
The resonant frequency is often expressed in natural units (radians per second), rather than using the fN in hertz, as
The factors Q, damping ratio ζ, natural frequency ωN, attenuation rate α, and exponential time constant τ are related such that:[17][page needed]
The envelope of oscillation decays proportional to e−αt or e−t/τ, where α and τ can be expressed as:
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.
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.
[19] In an ideal series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is:[20]
For a parallel RLC circuit, the Q factor is the inverse of the series case:[21][20]
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.
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.
For this reason, string instruments often have bodies with complex shapes, so that they produce a wide range of frequencies fairly evenly.
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 values.
Helmholtz resonators have a very high Q, as they are designed for picking out a very narrow range of frequencies.
where fo is the resonant frequency, E is the stored energy in the cavity, and P = −dE/dt is the power dissipated.
[25] 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.
