Q factor

The bandwidth, $\Delta f$ , of a damped oscillator is shown on a graph of energy versus frequency. The Q factor of the damped oscillator, or filter, is $f_0/\Delta f$ . The higher the Q, the narrower and 'sharper' the peak is.

In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is,^[1] or equivalently, characterizes a resonator's bandwidth relative to its center frequency.^[2] Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low one. Oscillators with high quality factors have low damping so that they ring longer.

Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the bandwidth. Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High Q oscillators oscillate with a smaller range of frequencies and are more stable. (See oscillator phase noise.)

The quality factor of oscillators varies substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q = ¹⁄₂. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability need high quality factors. Tuning forks have quality factors around Q = 1000. The quality factor of atomic clocks, superconducting RF cavities used in accelerators, and some high-Q lasers can reach as high as 10¹¹^[3] and higher.^[4]

There are many alternate quantities used by physicists and engineers to describe how damped an oscillator is and that are closely related to the quality factor. Important examples include: the damping ratio, relative bandwidth, linewidth and bandwidth measured in octaves.

The concept of Q factor originated in electronic engineering, as a measure of the 'quality' desired in a good tuned circuit or other resonator.

1 Definition of the quality factor
2 Q factor and damping
- 2.1 Quality factors of common systems
3 Physical interpretation of Q
4 Electrical systems
- 4.1 RLC circuits
- 4.2 Complex impedances
5 Mechanical systems
6 Optical systems
7 See also
8 References
9 Further reading
10 External links

Definition of the quality factor

There are two separate definitions of the quality factor that are equivalent for high Q resonators but are different for strongly damped oscillators.

Generally Q is defined in terms of the ratio of the energy stored in the resonator to the energy being lost in one cycle:

$Q = 2 \pi \times \frac{\mbox{Energy Stored}}{\mbox{Energy dissipated per cycle}}. \,$

The factor of 2π is used to keep this definition of Q consistent (for high values of Q) with the second definition:

$Q = \frac{f_r}{\Delta f} = \frac{\omega_r}{\Delta \omega}, \,$

where f_r is the resonant frequency, Δf is the bandwidth, ω_r is the angular resonant frequency, and Δω is the angular bandwidth.

The definition of Q in terms of the ratio of the energy stored to the energy dissipated per cycle can be rewritten as:

$Q = \omega \times \frac{\mbox{Energy Stored}}{\mbox{Power Loss}} \,$

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.

Q factor and damping

Main articles: damping and linear time invariant (LTI) system

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.)

A system with low quality factor (Q < ½) is said to be overdamped. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by exponential decay, approaching the steady state value asymptotically. It has an impulse response that is the sum of two decaying exponential functions with different rates of decay. As the quality factor decreases the slower decay mode becomes stronger relative to the faster mode and dominates the system's response resulting in a slower system. A second-order low-pass filter with a very low quality factor has a nearly first-order step response; the system's output responds to a step input by slowly rising toward an asymptote.

A system with high quality factor (Q > ½) is said to be underdamped. Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude of the signal. Underdamped systems with a low quality factor (a little above Q = ½) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. More generally, the output of a second-order low-pass filter with a very high quality factor responds to a step input by quickly rising above, oscillating around, and eventually converging to a steady-state value.

A system with an intermediate quality factor (Q = ½) is said to be critically damped. Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Like an underdamped response, the output of such a system responds quickly to a unit step input. Critical damping results in the fastest response (approach to the final value) possible without overshoot. Real system specifications usually allow some overshoot for a faster initial response or require a slower initial response to provide a safety margin against overshoot.

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.e., has a higher quality factor).

Quality factors of common systems

A unity gain Sallen–Key filter topology with equivalent capacitors and equivalent resistors is critically damped (i.e., $Q = 1/2$ ).
A Butterworth filter (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped $Q = 1/\sqrt{2}$ .^[5]
A Bessel filter (i.e., continuous-time filter with flattest group delay) has an underdamped $Q = 1/\sqrt{3}$ .

Physical interpretation of Q

Physically speaking, Q is $2\pi$ 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.^[6]

It 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.

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 $1/e^{2\pi}$ , or about 1/535, of its original energy.^[7]

The width (bandwidth) of the resonance is given by

$\Delta f = \frac{f_0}{Q} \,$ ,

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 factors Q, damping ratio ζ, and attenuation α are related such that^[8]

$\zeta = \frac{1}{2 Q} = { \alpha \over \omega_0 }.$

So the quality factor can be expressed as

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

and the exponential attenuation rate can be expressed as

$\alpha = \zeta \omega_0 = { \omega_0 \over 2 Q }.$

For any 2^nd order low-pass filter, the response function of the filter is^[8]

$H(s) = \frac{ \omega_c^2 }{ s^2 + \underbrace{ \frac{ \omega_c }{Q} }_{2 \zeta \omega_c = 2 \alpha }s + \omega_c^2 } \,$

For this system, when $Q > 0.5$ (i.e, when the system is underdamped), it has two complex conjugate poles that each have a real part of $\alpha$ . That is, the attenuation parameter $\alpha$ represents the rate of exponential decay of the oscillations (e.g., after an impulse) of the system. A higher quality factor implies a lower attenuation, and so high Q systems oscillate for long times. For example, high quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer.

Electrical systems

A graph of a filter's gain magnitude, illustrating the concept of -3 dB at a gain of 0.707 or half-power bandwidth. The frequency axis of this symbolic diagram can be linear or logarithmically scaled.

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

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

where $R$ , $L$ and $C$ are the resistance, inductance and capacitance of the tuned circuit, respectively. For a parallel RLC circuit, the Q factor is the inverse of the series case:^[9]

$Q = R \sqrt{\frac{C}{L}} \,$ ,

In a parallel RLC circuit where the R is in series with the L, Q is the same. This is the most common circumstance because, for resonators, limiting the resistance of the inductor to improve Q and narrowing the bandwidth is the desired result.

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. In this case the X and R are interchanged. This is useful in filter design to determine the bandwidth.

Complex impedances

For a complex impedance

$\tilde{Z} = R + j\Chi \,$

the Q factor is the ratio of the reactance to the resistance (or equivalently, the absolute value of the ratio of reactive power to real power), that is:

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

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

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

or just the tangent of the phase angle

$Q = \left | \tan \phi \right |\,$

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:

$Q = \frac{\sqrt{M k}}{D} \,$ ,

where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation $F_{\text{damping}}=-Dv$ , where $v$ is the velocity.^[10]

Optical systems

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

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

where $f_o$ is the resonant frequency, $\mathcal{E}$ is the stored energy in the cavity, and $P=-\frac{dE}{dt}$ 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.