# Q factor

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Q factor

:"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

:$Q = omega imes frac\left\{mbox\left\{Energy Stored\left\{mbox\left\{Power Loss ,$

or, more intuitively,

:$Q = 2 pi imes frac\left\{mbox\left\{Energy Stored\left\{mbox\left\{Energy dissipated per cycle ,$

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.
*The second-order filter with the flattest passband frequency response (Butterworth filter) has $Q = 1/sqrt\left\{2\right\}$
*The second-order filter with the flattest group delay (Bessel filter) has $Q = 1/sqrt\left\{3\right\}$.

Physical interpretation of Q

Physically speaking, "Q" is $2pi$ 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. [cite book | title = Novel Sensors and Sensing | author = Roger George Jackson | url = http://books.google.com/books?id=6CZZE9I0HbQC&pg=PA28&ots=N230HguQKA&dq=%22q+factor%22+energy&sig=V5twxCWlAEz5bpwKEG06WY0jido | year = 2004 | publisher = CRC Press | isbn = 075030989X , p.28]

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 $1/e^\left\{2pi\right\}$, or about 1/535, of its original energy. [cite web | title = Vibrations and Waves | work = Light and Matter online text series | author = Benjamin Crowell |date=2006 | url = http://www.lightandmatter.com/html_books/3vw/ch02/ch02.html | , Ch.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

:$Delta f = frac\left\{f_0\right\}\left\{Q\right\} ,$,

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 &zeta;, and the attenuation α is [cite book | title = Circuits, Signals, and Systems | author = William McC. Siebert | publisher = MIT Press ]

:$zeta = frac\left\{1\right\}\left\{2 Q\right\} = \left\{ alpha over omega_0 \right\} ,$

:$Q = frac\left\{1\right\}\left\{2 zeta\right\} = \left\{ omega_0 over 2 alpha \right\} ,$

For any 2nd order low-pass filter, the response function of the filter is [cite book | title = Circuits, Signals, and Systems | author = William McC. Siebert | publisher = MIT Press ]

:$H\left(s\right) = frac\left\{ omega_c^2 \right\}\left\{ s^2 + frac\left\{ omega_c \right\}\left\{Q\right\} s + omega_c^2 \right\} ,$

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.

RLC circuits

In a series RLC circuit, and in a tuned radio frequency receiver (TRF) the "Q" factor is:

:$Q = frac\left\{1\right\}\left\{R\right\} sqrt\left\{frac\left\{L\right\}\left\{C ,$,

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.:

:$Q = R sqrtfrac\left\{C\right\}\left\{L\right\} ,$

Complex impedances

For a complex impedance

:$ilde\left\{Z\right\} = R + jChi ,$

the "Q" factor is the ratio of the reactance to the resistance, that is

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

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

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

or just the tangent of the phase angle

:$Q = left | tan phi ight |,$

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: [http://www.physics.uwa.edu.au/__data/page/115450/lecture5_(amplifier_noise_etc).pdf] :$Q = frac\left\{sqrt\left\{M k\left\{D\right\} ,$,

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation $F_\left\{damping\right\}=-Dv$, where $v$ is the velocity.

Optical systems

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

:$Q = frac\left\{2pi f_o,mathcal\left\{E\left\{P\right\} ,$,

where $f_o$ is the resonant frequency, $mathcal\left\{E\right\}$ is the stored energy in the cavity, and $P=-frac\left\{dE\right\}\left\{dt\right\}$ 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:
*Cite book|last=Agarwal|first=Anant|coauthors=Lang, Jeffrey|title=Foundations of Analog and Digital Electronic Circuits|date=2005|publisher=Morgan Kaufmann|isbn=1558607358|url = http://books.google.com/books?id=83onAAAACAAJ&dq=intitle:%22Foundations+of+Analog+and+Digital+Electronic+Circuits%22&as_brr=0&ei=Pt4kR8-MDqK8pgKcntndAg

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