Q factor

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

The Q factor or quality factor is a measure of the rate at which a vibrating system dissipates its energy into heat. A higher Q indicates a lower rate of heat dissipation. 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. For very strong damping, Q < 1, the system is so strongly damped that it never completes a single oscillation, and in the limit of Q = 0, it simply decays exponentially toward equilibrium.

When the system is driven, 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.

Mathematically, the Q factor is defined as the number of oscillations required for a freely oscillating system's energy to fall off to 1/535 of its original energy, where $535=e^{2\pi}$ . When the system is driven, the relationship to the width of the resonance is given by

Q = \frac{f_0}{\Delta f},

where the resonant frequency is $f_0$ , and $\Delta f$ stands for the bandwidth. On a graph of response (energy of steady-state vibrations) versus frequency, the bandwidth is defined as the "full width at half maximum" or FWHM. This is the width in frequency where the energy falls to half of its peak value.

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 tuned radio frequency receiver (TRF) the Q factor is:

Q = \frac{f_0}{\Delta f} = \frac{1}{R} \sqrt{\frac{L}{C}} where

R

L

and

C

are the resistance, inductance and capacitance of the tuned circuit, respectively.

From the expression for the resonant frequency of a tuned circuit,

\omega = \sqrt{\frac{1}{LC}} the alternative formulation:

Q = \frac{\omega{}L}{R} can be derived.

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 $1/\sqrt{2}$ of the peak (-3 db), rather than 1/2 (-6 db).

Mechanical systems

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

Q = \frac{\sqrt{M K}}{R}

where M is the mass, K is the spring constant, and R is the mechanical resistance.

From the expression for the resonant frequency of a mass-spring system,

\omega = \sqrt{\frac{K}{M}} the alternative formulation:

Q = \frac{\omega{}M}{R} can be derived.

Optical systems

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

Q = \frac{2\pi\nu \mathcal{E}}{P} , where

\nu

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 FWHM 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.

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