Q factor
- For other uses of the terms Q and Q factor see Q value.
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In physics and engineering 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.
Generally Q is defined to be
where 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 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.
Physical interpretation of Q
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Bandwidth2.png/350px-Bandwidth2.png)
Physically speaking, Q is 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 , 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
- ,
where is the resonant frequency, and , the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.
Electrical systems
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Bandwidth.svg/350px-Bandwidth.svg.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.
RLC circuits
In a series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is:
- ,
where , and 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.
Complex impedances
For a complex impedance
the Q factor is the ratio of the reactance to the resistance, that is
Mechanical systems
For a single damped mass-spring system, the Q factor represents the effect of mechanical resistance.
where M is the mass, K is the spring constant, and R is the mechanical resistance, defined by the equation , where is the velocity.
Optical systems
In optics, the Q factor of a resonant cavity is given by
- ,
where is the resonant frequency, is the stored energy in the cavity, and 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
- ^ Roger George Jackson (2004). Novel Sensors and Sensing. CRC Press. ISBN 075030989X., p.28
- ^ Benjamin Crowell (2006). "Vibrations and Waves". Light and Matter online text series.
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- Agarwal, Anant (2005). Foundations of Analog and Digital Electronic Circuits. Morgan Kaufmann. ISBN 1558607358.
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External links
- Q to/from 'Bandwith per octave' converter, on audio engineer Eberhard Sengpel's website. Retrieved 2007-10-27.