Q factor

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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 $ω$ 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.

1 Usefulness of 'Q'
- 1.1 special values of Q
2 Physical interpretation of Q
3 Electrical systems
- 3.1 RLC circuits
- 3.2 Complex impedances
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 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.

[edit] special values of Q

critically damped (Q = 1/2); the simplest equal-C, equal-R Sallen Key filter.
The second-order filter with the flattest passband frequency response ( Butterworth filter ) has $Q = 1/\sqrt{2}$ .^{[citation needed]}
The second-order filter with the best pulse response ( Bessel filter ) has $Q = 1/\sqrt{3}$ .^{[citation needed]}

[edit] Physical interpretation of Q

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

Physically speaking, Q is $2π$ 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 $1 / e 2π$ , 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

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

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.

The relationship between Q and the damping ratio is

$\zeta = \frac{1}{2 Q}.$ ^{[citation needed]}

$Q = \frac{1}{2 \zeta}.$ ^{[citation needed]}

For any 2nd order low-pass filter, the response function of the filter is

$H(s) = \frac{ \omega_c^2 }{ s^2 + \frac{ \omega_c }{Q} s + \omega_c^2 }$ ^{[citation needed]}

[edit] 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.

[edit] 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. $Q = \frac {R} {\sqrt\frac{L}{C}}$

[edit] Complex impedances

For a complex impedance

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

the Q factor is the ratio of the reactance to the resistance, 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 $φ$ is the phase angle and $P F$ is the power factor of the circuit.

[edit] 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:

$Q = \frac{\sqrt{M k}}{D} \,$ ^[3],

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation $F d a m p i n g = - D v$ , 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.