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.

1 Usefulness of 'Q'
2 Physical interpretation of Q
3 Electrical systems
4 Mechanical systems
5 Optical systems
6 References
7 External links

[edit] 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.

[edit] Physical interpretation of Q

Physically speaking, Q is $2π$ times the ratio of the total energy stored divided by the energy lost in a single cycle.

Also, (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 $1 / e 2π$ , or about 1/535, of its original energy.

The bandwidth,

Δ 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 / Δ f

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

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

where $f 0$ is the resonant frequency, and $Δ f$ , the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.

[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.

BW or Δf = bandwidth

In a series LRC 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 LRC circuit, Q is equal to the reciprocal of the above expression.

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 (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.

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

where M is the mass, K is the spring constant, and R is the mechanical resistance, defined by the equation $F d a m p i n g = - R v$ , where $v$ is the velocity.

[edit] 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.