quantity

quality factor

A measure of a resonator's sharpness of resonance, indicating energy loss rate.
18 related articles · 2 formulas

Description

The quality factor, often denoted as Q, is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the energy stored in the resonator to the energy lost per cycle, multiplied by 2π. A high Q factor indicates that the resonator loses energy slowly, meaning it has a sharp resonance peak and a narrow bandwidth. In physics and engineering, Q is crucial for characterizing the performance of systems ranging from mechanical pendulums to optical cavities and superconducting microwave resonators.

History

The concept of quality factor was introduced by the electrical engineer K. S. Johnson in the 1920s while working on radio frequency coils at Bell Labs. It was later adopted widely in physics and engineering.

How it works

The quality factor is mathematically expressed as Q = 2π × (energy stored) / (energy lost per cycle). For a damped harmonic oscillator, Q is inversely proportional to the damping coefficient. In practice, a resonator with high Q will ring for many cycles after being excited, and its frequency response will be sharply peaked at the resonant frequency. Losses can arise from various mechanisms, such as internal friction in materials, electrical resistance, radiation, or coupling to the environment.

In practice

High-Q resonators are essential in technologies such as quartz crystal oscillators for precise timekeeping, superconducting cavities for particle accelerators, and optical microcavities for laser and sensor applications.

Fun fact

The quality factor of a superconducting microwave resonator can exceed one billion, meaning it can store energy for millions of cycles before losing it, which is crucial for quantum computing applications.

Concept graph →

Formulas

$$S_a^{1/2} = \sqrt{\frac{4 k_B T k}{m^2 \omega Q}}$$
Spectral density of thermal acceleration noise depends on temperature and resonator properties.
What is in the formula
S_a^{1/2}variableSpectral density of acceleration noise (square root) — acceleration noise spectral density metre per second squared per root hertz
TvariableTemperature — temperature kelvin
kvariableSpring constant of the resonator — spring constant newton per metre
mvariableMass of the resonator — mass kilogram
ωvariableAngular resonance frequency — angular frequency radian per second
QvariableQuality factor of the resonator — quality factor
k_BconstantBoltzmann constant, relates thermal energy to temperature — boltzmann constant = 1.380649e-23 joule per kelvin
$$G \propto Q_m P \Re[G_n] / (M \omega_m^2)$$
Gain in optomechanics scales with quality factor, power, and optical response.
What is in the formula
Gvariableoptomechanical gain
Q_mvariablemechanical quality factor — quality factor
Pvariableoptical power — power watt
G_nvariableoptical response function (complex)
Mvariableeffective mass of mechanical mode — mass kilogram
ω_mvariablemechanical angular frequency — angular frequency radian per second
Reoperatorextracts the real component of a complex number — real part

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