# Quality Factor

Quality factor is an extremely important fundamental concept in electrical and mechanical engineering. An oscillator (active) or resonator (passive) can be described by its Q-factor, which is inversely proportional to bandwidth. For these devices, the Q factor describes the damping of the system. In some instances, it is better to have either a lower or higher quality factor. For instance, with a guitar you would want to have a lower quality factor. The reason is because a high Q guitar would not amplify frequencies very evenly. To lower the quality factor, complex or strange shapes are introduced for the instrument body. However, the soundhole of a guitar (a Helmholtz resonator) has a very high quality factors to increase its frequency selectivity.

A very important area of discussion is the Quality Factor of a filter. Higher Q filters have higher peaks in the frequency domain and are more selective. The Quality factor is really only valid for a second order filter, which is based off of a second order equation and contains both an inductor and a capacitor. At a certain frequency, the reactances of both the capacitor and inductor cancel, leading to a strong output of current (lower total impedance). For a tuned circuit, the Q must be very high and is considered a “Figure of Merit”.

In terms of equations, the quality factor can be thought of in many different ways. It can be thought of as the ratio of “reactive” or wasted power to average power. It can also be thought of as the ratio of center frequency to bandwidth (NOTE: This is the FWHM bandwidth in which only frequencies that are equal to or greater than half power are part of the band). Another common equation is 2π multiplied by the ratio of energy stored in a system to energy lost in one cycle. The energy dissipated is due to damping, which again shows that Q factor is inversely related to damping, in addition to bandwidth.

Q can also be expressed as a function of frequency: The full relationship between Q factor and damping can be expressed as the following:

When Q = 1/2, the system is critically damped (such as with a door damper). The system does not oscillate. This is also when the damping ratio is equal to one. The main difference between critical damping and overdamping is that in critical damping, the system returns to equilibrium in the minimum amount of time.

When Q > 1/2 the system is underdamped and oscillatory. With a small Quality factor underdamped system, the system many only oscillate for a few cycles before dying out. Higher Q factors will oscillate longer.

When Q < 1/2 the system is overdamped. The system does not oscillate but takes longer to reach equilibrium than critical damping.