Q factor

It has been suggested that this article or section be merged into damping. ()

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

In physics and engineering the quality factor or Q 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

<math>

Q = \omega \times \frac{\mbox{Energy Stored}}{\mbox{Power Loss}} \, </math> where <math> \omega </math> 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 in an 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 <math>Q = 1/\sqrt{2}</math>.
• The second-order filter with the best pulse response ( Bessel filter ) has <math>Q = 1/\sqrt{3}</math>.

Physical interpretation of Q

The bandwidth, <math>\Delta f</math>, of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is <math>f_0/\Delta f</math>

Physically speaking, Q is <math>2\pi</math> times the ratio of the total energy stored divided by the energy lost in a single cycle.^[1] 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 <math>1/e^{2\pi}</math>, or about 1/535, of its original energy.^[2] 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

<math>

\Delta f = \frac{f_0}{Q} \, </math>, where <math>f_0</math> is the resonant frequency, and <math>\Delta f</math>, 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

<math> \zeta = \frac{1}{2 Q}.</math>

<math> Q = \frac{1}{2 \zeta}.</math>

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

<math> H(s) = \frac{ \omega_c^2 }{ s^2 + \frac{ \omega_c }{Q} s + \omega_c^2 } </math>

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:

<math>

Q = \frac{1}{R} \sqrt{\frac{L}{C}} \, </math>, where <math>R</math>, <math>L</math> and <math>C</math> 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. <math> Q = \frac {R} {\sqrt\frac{L}{C}} </math>

Complex impedances

For a complex impedance

<math>

\tilde{Z} = R + j\Chi \, </math> the Q factor is the ratio of the reactance to the resistance, that is

<math>

Q = \left | \frac{\Chi}{R} \right | \, </math>

Mechanical systems

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

<math>

Q = \frac{\sqrt{M K}}{R} \, </math>, where M is the mass, K is the spring constant, and R is the mechanical resistance, defined by the equation <math>F_{damping}=-Rv</math>, where <math>v</math> is the velocity.

Optical systems

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

<math>

Q = \frac{2\pi f_o \mathcal{E}}{P} \, </math>, where <math>f_o</math> is the resonant frequency, <math>\mathcal{E}</math> is the stored energy in the cavity, and <math>P=-\frac{dE}{dt}</math> 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 (2005). Foundations of Analog and Digital Electronic Circuits. Morgan Kaufmann. ISBN 1558607358. 

External links

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