Q factor
In physics and engineering, the quality factor or factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. Higher indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high, while a pendulum immersed in oil has a low one. Resonators with high quality factors have low damping, so that they ring or vibrate longer.
Explanation
The factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator. Sinusoidally driven resonators having higher factors resonate with greater amplitudes 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- 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- oscillators oscillate with a smaller range of frequencies and are more stable.The quality factor of oscillators varies substantially from system to system, depending on their construction. Systems for which damping is important have near. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability have high quality factors. Tuning forks have quality factors around 1000. The quality factor of atomic clocks, superconducting RF cavities used in accelerators, and some high- lasers can reach as high as 1011 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 originated with K. S. Johnson of Western Electric Company's Engineering Department while evaluating the quality of coils. His choice of the symbol 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.
Definition
The definition of since its first use in 1914 has been generalized to apply to coils and condensers, resonant circuits, resonant devices, resonant transmission lines, cavity resonators, and has expanded beyond the electronics field to apply to dynamical systems in general: mechanical and acoustic resonators, material and quantum systems such as spectral lines and particle resonances.Bandwidth definition
In the context of resonators, there are two common definitions for, which are not exactly equivalent. They become approximately equivalent as becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator:where is the resonant frequency, is the resonance width or full width at half maximum 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.
Stored energy definition
The other common nearly equivalent definition for is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes:The factor makes expressible in simpler terms, involving only the coefficients of the second-order differential equation describing most resonant systems, electrical or mechanical. 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 sum of the potential and kinetic energies at some point in time; the lost energy is the work done by an external force, per cycle, to maintain amplitude.
More generally and in the context of reactive component specification, the frequency-dependent definition of is used:
where is the angular frequency at which the stored energy and power loss are measured. This definition is consistent with its usage in describing circuits with a single reactive element, where it can be shown to be equal to the ratio of reactive power to real power.
-factor and damping
The -factor determines the qualitative behavior of simple damped oscillators.Starting from the stored energy definition for, it can be shown that, where is the damping ratio. There are three key distinct cases:
- A system with low quality factor 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 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 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 is said to be critically damped. Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output. Like an underdamped response, the output of such a system responds quickly to a unit step input. Critical damping results in the fastest response 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.
Some examples
Physical interpretation
Physically speaking, is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation; or nearly equivalently, at high enough values, 2 times the ratio of the total energy stored and the energy lost in a single cycle.It is a dimensionless parameter that compares the exponential 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. More precisely, the frequency and period used should be based on the system's natural frequency, which at low values is somewhat higher than the oscillation frequency as measured by zero crossings.
Equivalently, the factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to, or about or 0.2%, of its original energy. This means the amplitude falls off to approximately or 4% of its original amplitude.
The width of the resonance is given by :
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, rather than using the in hertz, as
The factors, damping ratio, natural frequency, attenuation rate, and exponential time constant are related such that:
and the damping ratio can be expressed as:
The envelope of oscillation decays proportional to or, where and can be expressed as:
and
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
For this system, when , 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 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 sinusoidal tone for a long time after being struck by a hammer.
| Filter type | Transfer function |
| Lowpass | |
| Bandpass | |
| Notch | |
| Highpass |