Q factor

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

In physics and engineering the Q factor or quality factor 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. 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.

Generally Q is defined to be

$Q=\omega \times {\frac {\mbox{Energy Stored}}{\mbox{Power Loss}}}\,$

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 or equal to 1/2 cannot be described as oscillating at all, instead the system is said to be in an overdamped (Q < 1/2) or critically damped (Q = 1/2) state. However, if Q > 1/2, the system's amplitude oscillates, while simultaneously decaying exponentially. This regime is referred to as underdamped.

Physical interpretation of Q

The bandwidth, $\Delta f$, of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is $f_{0}/\Delta f$

Physically speaking, Q is $2\pi$ 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 $1/e^{2\pi }$, 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

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

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.

Electrical systems

A graph of a filter's gain magnitude, illustrating the concept of -3 dB at a 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.

RLC circuits

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

$Q={\frac {1}{R}}{\sqrt {\frac {L}{C}}}\,$,

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. $Q={\frac {R}{\sqrt {\frac {L}{C}}}}$

Complex impedances

For a complex impedance

${\tilde {Z}}=R+j\mathrm {X} \,$

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

$Q=\left|{\frac {\mathrm {X} }{R}}\right|\,$

Mechanical systems

For a single damped mass-spring system, the Q factor represents the effect of mechanical resistance.

$Q={\frac {\sqrt {MK}}{R}}\,$,

where M is the mass, K is the spring constant, and R is the mechanical resistance, defined by the equation $F_{damping}=-Rv$, where $v$ is the velocity.

Optical systems

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

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

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 (2005). Foundations of Analog and Digital Electronic Circuits. Morgan Kaufmann. ISBN 1558607358. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Q to/from 'Bandwith per octave' converter, on audio engineer Eberhard Sengpel's website. Retrieved 2007-10-27.