For other uses of the terms Q and Q factor see Q value.

In physics and engineering the quality factor or Q factor is a dimensionless parameter that compares the 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. A higher Q indicates a lower rate of energy dissipation relative to the oscillation frequency, so the oscillations die out more slowly. For example, a pendulum suspended from a high-quality bearing, oscillating in air, would have a high Q, while a pendulum immersed in oil would have a low one.

The concept originated in electronic engineering, as a measure of the 'quality' desired in a good tuned circuit or other resonator.

Generally Q is defined to be

or, more intuitively,

where $omega$ is defined to be the angular frequency of the circuit (system), and the energy stored and power loss are properties of a system under consideration.

Usefulness of 'Q'

The Q factor is particularly useful in determining the qualitative behavior of a system. For example, a system with Q less than 1/2 cannot be described as oscillating at all, instead the system is said to be overdamped (Q < 1/2), gradually drifting towards its steady-state position. However, if Q > 1/2, the system's amplitude oscillates, while simultaneously decaying exponentially. This regime is referred to as underdamped.

special values of Q

• critically damped (Q = 1/2); the simplest equal-C, equal-R Sallen Key filter.
• The second-order filter with the flattest passband frequency response (Butterworth filter ) has $Q\; =\; 1/sqrt\{2\}$.
• The second-order filter with the best pulse response (Bessel filter ) has $Q\; =\; 1/sqrt\{3\}$.

Physical interpretation of Q

Physically speaking, Q is $2pi$ times the ratio of the total energy stored divided by the energy lost in a single cycle.

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 $1/e^\{2pi\}$, or about 1/535, of its original energy.

When the system is driven by a sinusoidal drive, its resonant behavior depends strongly on Q. Resonant systems respond to frequencies close to their natural frequency much more strongly than they respond to other frequencies. A system with a high Q resonates with a greater amplitude (at the resonant frequency) than one with a low Q factor, and its response falls off more rapidly as the frequency moves away from resonance. Thus, a radio receiver with a high Q would be more difficult to tune with the necessary precision, but would do a better job of filtering out signals from other stations that lay nearby on the spectrum. The width of the resonance is given by

where $f\_0$ is the resonant frequency, and $Delta\; f$, the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.

The relationship between Q and the damping ratio is

$zeta\; =\; frac\{1\}\{2\; Q\}.$

$Q\; =\; frac\{1\}\{2\; zeta\}.$

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

$H(s)\; =\; frac\{\; omega\_c^2\; \}\{\; s^2\; +\; frac\{\; omega\_c\; \}\{Q\}\; s\; +\; omega\_c^2\; \}$

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.

RLC circuits

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

where $R$, $L$ and $C$ are the resistance, inductance and capacitance of the tuned circuit, respectively.

In a parallel RLC circuit, Q is equal to the reciprocal of the above expression. $Q\; =\; frac\; \{R\}\; \{sqrtfrac\{L\}\{C\}\}$

Complex impedances

For a complex impedance

the Q factor is the ratio of the reactance to the resistance, that is

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:

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.

Optical systems

In optics, the Q factor of a resonant cavity is given by

where $f\_o$ is the resonant frequency, $mathcal\{E\}$ is the stored energy in the cavity, and $P=-frac\{dE\}\{dt\}$ 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.

References

General:

• Agarwal, Anant; Lang, Jeffrey Foundations of Analog and Digital Electronic Circuits. Morgan Kaufmann.

External links

• Q to/from 'Bandwidth per octave' converter of audio engineer Eberhard Sengpiel. Retrieved 2007-10-27.