Filter Q factor. A high filter quality means narrow-band filtering notch , with a large Q factor.

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This results in steep filter flanks with a small bandwidth. A low filter quality means broad-band filtering, with a small Q factor. This results in flat filter flanks with a large bandwidth. The larger the Q the more narrow is the resonance peak. The smaller the Q the more broad is the resonance peak. With "quality" is not meant how valuable the signal is.

It is meant the quality of the filter. If the filter has flat slopes many frequencies are influenced around the cutoff frequency. The filter has therefore a larger bandwidth and the so-called quality factor Q is specified as a low number. If the filter has steep slopes, its bandwidth is smaller. Here a few frequencies below and above its cutoff frequency are affected and the quality factor Q is specified as a high number.

## CrystalWave

Why we always take 3 dB down gain of a filter? Full width at half maximum FWHM. Sound engineers and sound designers "ear people" mostly use the usual sound field quantity. Acousticians and sound protectors "noise fighters" seem to like more the sound energy quantity.

Note: Power gain power amplification is not common in audio engineering.

The quality factor of atomic clocks , superconducting RF cavities used in accelerators, and some high- Q lasers can reach as high as 10 11 [3] and higher. There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the damping ratio , relative bandwidth , linewidth and bandwidth measured in octaves. The concept of Q originated with K. Johnson of Western Electric Company 's Engineering Department while evaluating the quality of coils inductors. His choice of the symbol Q was only because, at the time, all other letters of the alphabet were taken.

The term was not intended as an abbreviation for "quality" or "quality factor", although these terms have grown to be associated with it. The definition of Q since its first use in has been generalized to apply to coils and condensers, resonant circuits, resonant devices, resonant transmission lines, cavity resonators, material Q and spectral lines. In the context of resonators, there are two common definitions for Q , which aren't exactly equivalent.

They become approximately equivalent as Q becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator: [5]. The other common nearly equivalent definition for Q is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes: [8] [9] [5].

In electrical systems, the stored energy is the sum of energies stored in lossless inductors and capacitors ; the lost energy is the sum of the energies dissipated in resistors per cycle. In mechanical systems, the stored energy is the maximum possible stored energy, or the total energy, i. More generally and in the context of reactive component specification especially inductors , the frequency-dependent definition of Q is used: [8] [10] [ failed verification See discussion.

This definition is consistent with its usage in describing circuits with a single reactive element capacitor or inductor , where it can be shown to be equal to the ratio of reactive power to real power. See Individual reactive components. The Q factor determines the qualitative behavior of simple damped oscillators. For mathematical details about these systems and their behavior see harmonic oscillator and linear time invariant LTI system. In negative feedback systems, the dominant closed-loop response is often well-modeled by a second-order system. The phase margin of the open-loop system sets the quality factor Q of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory i.

Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy. The resonant frequency is often expressed in natural units radians per second , rather than using the f 0 in hertz , as. For a two-pole lowpass filter, the transfer function of the filter is [15]. A higher quality factor implies a lower attenuation rate, and so high- Q systems oscillate for many cycles.

### Quality factor definition

For example, high-quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer. For an electrically resonant system, the Q factor represents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals , mechanical friction. In audio, bandwidth is often expressed in terms of octaves. Then the relationship between Q and bandwidth is. The larger the series resistance, the lower the circuit Q. For a parallel RLC circuit, the Q factor is the inverse of the series case: [19] [18].

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## Photonic Crystal Cavity Q-factor - simulation with CrystalWave software

Consider a circuit where R , L and C are all in parallel. The lower the parallel resistance, the more effect it will have in damping the circuit and thus the lower the Q. 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, R , in series with the inductance, L , Q is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improve Q and narrow the bandwidth is the desired result.

The Q of an individual reactive component depends on the frequency at which it is evaluated, which is typically the resonant frequency of the circuit that it is used in. The Q of an inductor with a series loss resistance is the Q of a resonant circuit using that inductor including its series loss and a perfect capacitor. The Q of a capacitor with a series loss resistance is the same as the Q of a resonant circuit using that capacitor with a perfect inductor: [21].

In general, the Q of a resonator involving a series combination of a capacitor and an inductor can be determined from the Q values of the components, whether their losses come from series resistance or otherwise: [21]. 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:. The Q of a musical instrument is critical; an excessively high Q in a resonator will not evenly amplify the multiple frequencies an instrument produces. For this reason, string instruments often have bodies with complex shapes, so that they produce a wide range of frequencies fairly evenly. The Q 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 Q for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higher-Q. An excessively high Q can make it harder to hit a note.

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Q in an instrument may vary across frequencies, but this may not be desirable. Helmholtz resonators have a very high Q, as they are designed for picking out a very narrow range of frequencies. In optics , the Q factor of a resonant cavity is given by.

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.