Quality Factor (Q)

The quality factor (Q) of a two-pole resonator is defined by [20, p. 184]

$$Q \isdef \frac{\omega_0}{2\alpha} \protect$$ (E.7)

where $\omega_0$ and $\alpha$ are parameters of the resonator transfer function

$$H(s) = g\frac{s}{s^2 + 2\alpha s + \omega_0^2}. \protect$$ (E.8)

Note that Q is defined in the context of continuous-time resonators, so the transfer function $H(s)$ is the Laplace transform (instead of the z transform) of the continuous (instead of discrete-time) impulse-response $h(t)$. An introduction to Laplace-transform analysis appears in Appendix D. The parameter $\alpha$ is called the damping constant (or ``damping factor'') of the second-order transfer function, and $\omega_0$ is called the resonant frequency [20, p. 179]. The resonant frequency $\omega_0$ coincides with the physical oscillation frequency of the resonator impulse response when the damping constant $\alpha$ is zero. For light damping, $\omega_0$ is approximately the physical frequency of impulse-response oscillation ($2\pi$ times the zero-crossing rate of sinusoidal oscillation under an exponential decay). For larger damping constants, it is better to use the imaginary part of the pole location as a definition of resonance frequency (which is exact in the case of a single complex pole). (See §B.6 for a more complete discussion of resonators, in the discrete-time case.)

By the quadratic formula, the poles of the transfer function $H(s)$ are given by

$$p = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} \isdef -\alpha \pm \alpha_d . \protect$$ (E.9)

Therefore, the poles are complex only when $Q>1/2$. Since real poles do not resonate, we have $Q>1/2$ for any resonator. The case $Q=1/2$ is called critically damped, while $Q<1/2$ is called overdamped. A resonator ($Q>1/2$) is said to be underdamped, and the limiting case $Q=\infty$ is simply undamped.

Relating to the notation of the previous section, in which we defined one of the complex poles as $p\isdef \sigma_p+j\omega_p$, we have

$$\sigma_p = -\alpha$$ (E.10)

$$\omega_p = \sqrt{\omega_0-\alpha^2}.$$ (E.11)

For resonators, $\omega_p$ coincides with the classically defined quantity [20, p. 624]

$$\omega_d \isdef \omega_p = \sqrt{\omega_0^2 -\alpha^2} = \frac{\alpha_d}{j}.$$

Since the imaginary parts of the complex resonator poles are $\pm\omega_d$, the zero-crossing rate of the resonator impulse response is $\omega_d/\pi$ crossings per second. Moreover, $\omega_d$ is very close to the peak-magnitude frequency in the resonator amplitude response. If we eliminate the negative-frequency pole, $\omega_d/\pi$ becomes exactly the peak frequency. In other words, as a measure of resonance peak frequency, $\omega_d$ only neglects the interaction of the positive- and negative-frequency resonance peaks in the frequency response, which is usually negligible except for highly damped, low-frequency resonators. For any amount of damping $\omega_d/\pi$ gives the impulse-response zero-crossing rate exactly, as is immediately seen from the derivation in the next section.