Q factor

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

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.

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.

Physically speaking, the Q factor is approximately (for large values of Q) 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. Or equivalently, Q is <math>2\pi</math> times the ratio of the total energy stored divided by the energy lost in a single cycle.

Image:Bandwidth2.png

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.

1 Electrical systems
2 Mechanical systems
3 Optical systems
4 References
5 External links

[edit] 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.

In a series or parallel LRC 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.

Often for an electrical system the response can most easily be measured as an amplitude (voltage or a current), rather than energy or power. Since power and energy are proportional to the square of the amplitude of the oscillation, the bandwidth on an amplitude-frequency graph should be measured to <math>1/\sqrt{2}</math> of the peak (approximately -3 db), rather than 1/2 (-6 db).

[edit] 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.

[edit] 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.