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

In physics and engineering the quality factor or Q factor is a dimensionless parameter that 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, so the oscillations die out more slowly. 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 concept originated in electronic engineering, as a measure of the 'quality' desired in a good tuned circuit or other resonator.

Generally Q is defined to be

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

or, more intuitively,

Q = 2 \pi \times \frac{\mbox{Energy Stored}}{\mbox{Energy dissipated per cycle}} \,

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 1/2 cannot be described as oscillating at all, instead the system is said to be 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 boundary between exponential and oscillatory response. The simplest equal-C, equal-R Sallen?Key filter is critically damped.
• The second-order filter with the flattest passband frequency response (Butterworth filter) has Q = 1/\sqrt{2}.
• The second-order filter with the flattest group delay (Bessel filter) has Q = 1/\sqrt{3}.

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 or equivalently the ratio of the stored energy to the energy dissipated per one radian of the oscillation.^[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 1/e^{2\pi}, 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 high Q tuned circuit in a radio receiver would be more difficult to tune with the necessary precision, but would have more selectivity; it would do a better job of filtering out signals from other stations that lay nearby on the spectrum. The width (bandwidth) 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.

The relationship between Q, the damping ratio ?, and the attenuation ? is ^[3]

\zeta = \frac{1}{2 Q} = { \alpha \over \omega_0 } \,

Q = \frac{1}{2 \zeta} = { \omega_0 \over 2 \alpha } \,

For any 2nd order low-pass filter, the response function of the filter is^[4]

H(s) = \frac{ \omega_c^2 }{ s^2 + \frac{ \omega_c }{Q} s + \omega_c^2 } \,

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 = R \sqrt\frac{C}{L} \,

Complex impedances

For a complex impedance

\tilde{Z} = R + j\Chi \,

the Q factor is the ratio of the reactance to the resistance (or equivalently, the absolute value of the ratio of reactive power to real power), that is:

Q = \left | \frac{\Chi}{R} \right | \,

Thus, one can also calculate the Q factor for a complex impedance by knowing just the power factor of the circuit

Q = \frac{\left | sin \phi \right |}{\left | cos \phi \right |} = \frac{\sqrt{1-PF^2}}{PF} = \sqrt{\frac{1}{PF^2}-1} \,

or just the tangent of the phase angle

Q = \left | tan \phi \right |\,

where \phi is the phase angle and PF is the power factor of the circuit.

Mechanical systems

For a single damped mass-spring system, the Q factor represents the effect of simplified viscous damping or drag, where the damping force or drag force is proportional to velocity. The formula for the Q factor is:^[5]

Q = \frac{\sqrt{M k}}{D} \, ,

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation F_{damping}=-Dv, 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:

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