Q factor

From Free net encyclopedia

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

The Q factor or quality factor is a measure of the "quality" of a resonant system. Resonant systems respond to frequencies close to their natural frequency much more strongly than they respond to other frequencies. The Q factor indicates the amount of resistance to resonance in a system. Systems with a high Q factor resonate with a greater amplitude (at the resonant frequency) than systems with a low Q factor. Damping decreases the Q factor.

1 Electrical Systems
2 Mechanical Systems
3 Optical Systems
4 External links

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Electrical Systems

Image:Bandwidth.png

For an electrically resonant system, the Q factor represents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals, mechanical friction. The Q factor is defined as the resonant frequency (center frequency) f₀ or f_c divided by the bandwidth Δf or BW:

<math>

Q = \frac{f_c}{f_2 - f_1} = \frac{f_c}{\Delta f} </math>

Bandwidth BW or Δf = f₂ - f₁, where f₂ is the upper and f₁ the lower cutoff frequency. On a graph of response versus frequency, the bandwidth is defined as the 3 dB change in level (voltage, current, or power) on either side of the center frequency. For the case of power, the bandwidth is the same as the "full width at half maximum" or FWHM. This is the width in frequency where the power falls to half of its peak value. When dealing with voltage or current, the bandwidth is the width where the level falls to <math>1/\sqrt{2}</math> of its peak value and thus is not equal to the FWHM.

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

<math>

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

<math>

\omega = \sqrt{\frac{1}{LC}} </math> the alternative formulation:

<math>

Q = \frac{\omega{}L}{R} </math> can be derived.

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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.

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

<math>

\omega = \sqrt{\frac{K}{M}} </math> the alternative formulation:

<math>

Q = \frac{\omega{}M}{R} </math> can be derived.

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Optical Systems

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

<math>

Q = \frac{2\pi\nu \mathcal{E}}{P} </math>, where <math>\nu</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 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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