Q factor

 ( Electrical Parameters ) 

$$Q\; \backslash mathrel\backslash stackrel\{\backslash text\{def\}\}\{=\}\; \backslash frac\{f\_\backslash mathrm\{r\}\}\{\backslash Delta\; f\}\; =\; \backslash frac\{\backslash omega\_\backslash mathrm\{r\}\}\{\backslash Delta\; \backslash omega\},$$

where is the resonant frequency, is the resonance width or full width at half maximum (FWHM) i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, is the angular resonant frequency, and is the angular half-power bandwidth.

Under this definition, is the reciprocal of fractional bandwidth.

The width (bandwidth) of the resonance is given by (approximately): $$\backslash Delta\; f\; =\; \backslash frac\{f\_\backslash mathrm\{N\}\}\{Q\},\; \backslash ,$$ where is the natural frequency, and , the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.

The resonant frequency is often expressed in natural units (radians per second), rather than using the in hertz, as $$\backslash omega\_\backslash mathrm\{N\}\; =\; 2\backslash pi\; f\_\backslash mathrm\{N\}.$$

The factors , damping ratio , natural frequency , attenuation rate , and exponential time constant are related such that:

$$Q\; =\; \backslash frac\{1\}\{2\; \backslash zeta\}\; =\; \backslash frac\{\; \backslash omega\_\backslash mathrm\{N\}\; \}\{2\; \backslash alpha\; \}\; =\; \backslash frac\{\; \backslash tau\; \backslash omega\_\backslash mathrm\{N\}\; \}\{\; 2\; \},$$

and the damping ratio can be expressed as:

$$\backslash zeta\; =\; \backslash frac\{1\}\{2\; Q\}\; =\; \{\; \backslash alpha\; \backslash over\; \backslash omega\_\backslash mathrm\{N\}\; \}\; =\; \{\; 1\; \backslash over\; \backslash tau\; \backslash omega\_\backslash mathrm\{N\}\; \}.$$

The envelope of oscillation decays proportional to or , where and can be expressed as:

$$\backslash alpha\; =\; \{\; \backslash omega\_\backslash mathrm\{N\}\; \backslash over\; 2\; Q\; \}\; =\; \backslash zeta\; \backslash omega\_\backslash mathrm\{N\}\; =\; \{1\; \backslash over\; \backslash tau\}$$ and $$\backslash tau\; =\; \{\; 2\; Q\; \backslash over\; \backslash omega\_\backslash mathrm\{N\}\; \}\; =\; \{1\; \backslash over\; \backslash zeta\; \backslash omega\_\backslash mathrm\{N\}\}\; =\; \backslash frac\{1\}\{\backslash alpha\}.$$

The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as or .

For a two-pole lowpass filter, the transfer function of the filter is

$$H(s)\; =\; \backslash frac\{\; \backslash omega\_\{\backslash mathrm\; N\}^2\; \}\{\; s^2\; +\; \backslash underbrace\{\; \backslash frac\{\; \backslash omega\_\{\backslash mathrm\; N\}\; \}\{Q\}\; \}\_\{2\; \backslash zeta\; \backslash omega\_\{\backslash mathrm\; N\}\; =\; 2\; \backslash alpha\; \}s\; +\; \backslash omega\_\{\backslash mathrm\; N\}^2\; \}\; \backslash ,$$

For this system, when (i.e., when the system is underdamped), it has two complex conjugate poles that each have a real part of . That is, the attenuation parameter represents the rate of exponential decay of the oscillations (that is, of the output after an impulse response) into the system. A higher quality factor implies a lower attenuation rate, and so high- systems oscillate for many cycles. For example, high-quality bells have an approximately pure tone for a long time after being struck by a hammer.

For example, an antenna tuned to have a value of 10 and a centre frequency of 100 kHz would have a 3 dB bandwidth of 10 kHz.

In audio, bandwidth is often expressed in terms of . Then the relationship between and bandwidth is

$$Q\; =\; \backslash frac\{2^\backslash frac\{BW\}\{2\}\}\{2^\{BW\}\; -\; 1\}\; =\; \backslash frac\{1\}\{2\; \backslash sinh\backslash left(\backslash frac\{1\}\{2\}\backslash ln(2)\; BW\; \backslash right)\},$$ where is the bandwidth in octaves.

$$Q\; =\; \backslash frac\{1\}\{R\}\; \backslash sqrt\{\backslash frac\{L\}\{C\}\}\; =\; \backslash frac\{\backslash omega\_0\; L\}\{R\}\; =\; \backslash frac\; \{1\}\; \{\backslash omega\_0\; R\; C\}$$

where , , and are the resistance, inductance and capacitance of the tuned circuit, respectively. Larger series resistances correspond to lower circuit values.

For a parallel RLC circuit, the factor is the inverse of the series case:

$$Q\; =\; R\; \backslash sqrt\{\backslash frac\{C\}\{L\}\}\; =\; \backslash frac\{R\}\{\backslash omega\_0\; L\}\; =\; \backslash omega\_0\; R\; C$$ Frequency Response: Resonance, Bandwidth, Q Factor (PDF)

Consider a circuit where , , and are all in parallel. The lower the parallel resistance is, the more effect it will have in damping the circuit and thus result in lower . This is useful in filter design to determine the bandwidth.

In a parallel LC circuit where the main loss is the resistance of the inductor, , in series with the inductance, , is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improve and narrow the bandwidth is the desired result.

$$Q\_L\; =\; \backslash frac\{X\_L\}\{R\_L\}=\backslash frac\{\backslash omega\_0\; L\}\{R\_L\}$$

where:

• is the resonance frequency in radians per second;
• is the inductance;
• is the inductive reactance; and
• is the series resistance of the inductor.

The of a capacitor with a series loss resistance is the same as the of a resonant circuit using that capacitor with a perfect inductor:

$$Q\_C\; =\; \backslash frac\{-X\_C\}\{R\_C\}=\backslash frac\{1\}\{\backslash omega\_0\; C\; R\_C\}$$

where:

• is the resonance frequency in radians per second;
• is the capacitance;
• is the capacitive reactance; and
• is the series resistance of the capacitor.

In general, the of a resonator involving a series combination of a capacitor and an inductor can be determined from the values of the components, whether their losses come from series resistance or otherwise:

$Q\; =\; \backslash frac\{1\}\{\backslash frac\{1\}\{Q\_L\}\; +\; \backslash frac\{1\}\{Q\_C\}\}$

The of a brass instrument or wind instrument needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed. By contrast, a vuvuzela is made of flexible plastic, and therefore has a very low for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higher values. An excessively high can make it harder to hit a note. in an instrument may vary across frequencies, but this may not be desirable.

Helmholtz resonators have a very high , as they are designed for picking out a very narrow range of frequencies.

• Acoustic resonance
• Attenuation
• Chu–Harrington limit
• List of piezoelectric materials
• Phase margin
• Q meter
• Q multiplier
• Dissipation factor

• (2025). 9781558607354, Morgan Kaufmann. . ISBN 9781558607354

• Calculating the cut-off frequencies when center frequency and Q factor is given
• Explanation of Q factor in radio tuning circuits