Q factor

thumb|upright=1.5| A damped oscillation. A low factor – about 5 here – means the oscillation dies out rapidly.

In physics and engineering, the quality factor or factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. Resonators with high quality factors have low damping, so that they ring or vibrate longer. For example, a pendulum suspended from a high-quality bearing, oscillating in air, has a high , while a pendulum immersed in oil has a low one.

There are two definitions of Q that give numerically similar, but not identical, results. The more general definition is the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. An alternative definition of factor, more applicable to high oscillators, is as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force.

Explanation

The factor is a parameter that describes the resonance behavior of an damped harmonic oscillator (resonator). Sinusoidally driven resonators having higher 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- 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- oscillators oscillate with a smaller range of frequencies and are more stable.

The quality factor of devices varies substantially from system to system, depending on their function and design. Systems for which damping is important (such as dampers keeping a door from slamming shut) have 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 clocks, superconducting RF cavities used in accelerators, and some high- lasers can reach 10<sup>11</sup> 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, relative bandwidth, linewidth and bandwidth measured in octaves.

The concept of originated with K. S. Johnson of Western Electric Company's Engineering Department while evaluating the quality of coils (inductors). His choice of the symbol 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 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 pendulum's -factor is: , where is the mass of the bob, is the pendulum's radian frequency of oscillation, and is the frictional damping force on the pendulum per unit velocity.

| The design of a high-energy (near terahertz) gyrotron considers both diffractive Q-factor, <math display="inline">Q_D \approx 30 \left(\frac{L}{\lambda}\right)^2</math> as a function of resonator length , wavelength , and ohmic -factor (–modes)

<math display="block">Q_\Omega = \frac{R_\mathrm{w{\delta} \frac{1 - m^2}{v^2_{m,p,</math>

where is the cavity wall radius, is the skin depth of the cavity wall, is the eigenvalue scalar ( is the azimuth index, is the radial index; in this application, skin depth is </math>)

| In medical ultrasonography, a transducer with a high -factor is suitable for doppler ultrasonography because of its long ring-down time, where it can measure the velocities of blood flow. Meanwhile, a transducer with a low -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.

Physical interpretation

Physically speaking, is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation; or nearly equivalently, at high enough values, 2 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 to its oscillation period. 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 values is somewhat higher than the oscillation frequency as measured by zero crossings.

Equivalently (for large values of ), the factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to , or about or 0.2%, of its original energy. This means the amplitude falls off to approximately or 4% of its original amplitude.

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

<math display="block">\Delta f = \frac{f_\mathrm{N{Q}, \,</math>

where is the natural frequency, and , 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 in hertz, as

<math display="block">\omega_\mathrm{N} = 2\pi f_\mathrm{N}.</math>

The factors , damping ratio , natural frequency , attenuation rate , and exponential time constant are related such that:

<math display="block">Q = \frac{1}{2 \zeta} = \frac{ \omega_\mathrm{N} }{2 \alpha } = \frac{ \tau \omega_\mathrm{N} }{ 2 },</math>

and the damping ratio can be expressed as:

<math display="block">\zeta = \frac{1}{2 Q} = { \alpha \over \omega_\mathrm{N} } = { 1 \over \tau \omega_\mathrm{N} }.</math>

The envelope of oscillation decays proportional to or , where and can be expressed as:

<math display="block">\alpha = { \omega_\mathrm{N} \over 2 Q } = \zeta \omega_\mathrm{N} = {1 \over \tau}</math>

and

<math display="block">\tau = { 2 Q \over \omega_\mathrm{N} } = {1 \over \zeta \omega_\mathrm{N = \frac{1}{\alpha}. </math>

The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as or .

For a two-pole lowpass filter, the transfer function of the filter is

! scope="row" | Lowpass

| <math>\frac{ \omega_\mathrm{N}^2 }{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>

! scope="row" | Bandpass

| <math>\frac{ \frac{\omega_\mathrm{N{Q}s}{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>

! scope="row" | Notch (bandstop)

| <math>\frac{ s^2 + \omega_\mathrm{N}^2}{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>

! scope="row" | Highpass

| <math>\frac{ s^2 }{ s^2 + \frac{ \omega_\mathrm{N} }{Q}s + \omega_\mathrm{N}^2 }</math>

Electrical systems

upright=1.2|thumb|A graph of a filter's gain magnitude, illustrating the concept of −3 dB at a voltage gain of 0.707 or half-power bandwidth. The frequency axis of this symbolic diagram can be linear or [[logarithmically scaled.]]

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 and bandwidth

The 2-sided bandwidth relative to a resonant frequency of (Hz) is <math>\frac{F_0}{Q}</math>.

For example, an antenna tuned to have a 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 and bandwidth is

<math display="block">Q = \frac{2^\frac{BW}{2{2^{BW} - 1} = \frac{1}{2 \sinh\left(\frac{1}{2}\ln(2) BW \right)},</math>

where is the bandwidth in octaves.

RLC circuits

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

<math display="block">Q = \frac{1}{R} \sqrt{\frac{L}{C = \frac{\omega_0 L}{R} = \frac {1} {\omega_0 R C}</math>

where , , and are the resistance, inductance and capacitance of the tuned circuit, respectively. Larger series resistances correspond to lower circuit values.

For a parallel RLC circuit, the factor is the inverse of the series case:

Consider a circuit where , , and 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 . 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, , in series with the inductance, , is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improve and narrow the bandwidth is the desired result.

Individual reactive components

The 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 of an inductor with a series loss resistance is the of a resonant circuit using that inductor (including its series loss) and a perfect capacitor.

<math display="block">Q_L = \frac{X_L}{R_L}=\frac{\omega_0 L}{R_L}</math>

where:

is the resonance frequency in radians per second;
is the inductance;
is the inductive reactance; and
is the series resistance of the inductor.

The of a capacitor with a series loss resistance is the same as the of a resonant circuit using that capacitor with a perfect inductor:

Acoustical systems

The of a musical instrument is critical; an excessively high 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 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 for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higher values. An excessively high can make it harder to hit a note. in an instrument may vary across frequencies, but this may not be desirable.

Helmholtz resonators have a very high , as they are designed for picking out a very narrow range of frequencies.

Optical systems

In optics, the 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 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 . If the 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 -switching. factor is of particular importance in plasmonics, 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.

References

External links

Calculating the cut-off frequencies when center frequency and Q factor is given
Explanation of Q factor in radio tuning circuits