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. The fusion energy gain factor, usually expressed with the symbol Q, is the ratio of fusion power produced in a reactor to the power required to maintain the plasma in steady state. ...

Generally Q is defined to be

where ω is defined to be the angular frequency of the circuit (system), and the Energy saved and Power lost are considered from the formulations of specific systems.

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. Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ...

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.^[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]

The bandwidth, Δf, of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is f₀ / Δf

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 Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... This article is about resonance in physics. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...

,

where f₀ is the resonant frequency, and Δf, the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value. This article is about resonance in physics. ... This article or section does not cite its references or sources. ...

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. Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... A crystal oscillator is an electronic circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency. ... Friction is the force that opposes the relative motion or tendency toward such motion of two surfaces in contact. ...

In a series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is: An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ... The Tuned Radio Frequency Receiver (short TRF) was invented by Edwin Armstrong in 1918. ...

,

where R, L and C are the resistance, inductance and capacitance of the tuned circuit, respectively. Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ...

In a parallel LRC circuit, Q is equal to the reciprocal of the above expression. An RLC circuit is a kind of electrical circuit composed of a resistor (R), an inductor (L), and a capacitor (C). ...

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 of the peak (approximately –3 dB), rather than 1/2 (–6 dB). A point where this occurs is known as a corner frequency, a cut-off frequency, a half-power point, or a break point.^[3]

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 F_damping = − Rv, where v is the velocity.

Optical systems

In optics, the Q factor of a resonant cavity is given by Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... A cavity resonator uses resonance to amplify a wave. ...

,

where f_o 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. The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, infrared radiation, microwaves, radio waves, and visible light are all forms of light. ... Experiment with a laser (likely an argon type) (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: light amplification by stimulated emission of radiation. ... For other uses, see Pulse (disambiguation). ... Q-switching, sometimes known as giant pulse formation, is a technique discovered circa 1962 by R.W. Hellwarth and F.J. McClung using electrically switched Kerr cell shutters and is a technique by which a laser can be made to produce a pulsed output beam. ...

References

General:

• Foundations of Analog and Digital Electronic Circuits, by Anant Agarwal and Jeffrey Lang

External links

• Resonance - a chapter from an online textbook
• Conversion: Quality factor Q to 'bandwidth per octaves' and 'bandwidth per octaves' N to quality factor Q