Q of a Real Second-Order Resonator

The quality factor (Q) of a real two-pole resonator may be defined in terms of its complex one-pole components, where the one-pole components have poles at $p_0=\sigma_0 + j\omega_0$ and $\overline{p_0}=\sigma_0 - j\omega_0$ ; as discussed above, each of these complex resonators has quality factor $Q=\omega_0/(2\vert\sigma_0\vert)$ . However, this is only accurate for their sum when the poles can be treated as fully separated in frequency. In other words, we must assume that the center-frequency and bandwidth of the complex resonators are not significantly affected by summing their frequency responses. In the high-Q case, $\vert\omega_0\vert\gg\vert\sigma_0\vert$ , this assumption is accurate. For greater accuracy, we should calculate where the peak-gain frequency has moved, and what the modified bandwidths may be.

Let's look at what happens when we add a one-pole complex resonator to its complex-conjugate counterpart to form a real second-order resonator:

$$H(s) \eqsp \frac{1}{s-p_0} + \frac{1}{s-\overline{p_0}} \eqsp \frac{s-\overline{p_0} + s-p_0}{(s-p_0)(s-\overline{p_0})} \eqsp 2\frac{s - \sigma_0}{s^2-2\sigma_0s+(\sigma_0^2+\omega_0^2)}.$$

We see that a zero at $s=\sigma_0$ has been induced symmetrically between the poles. This zero, due to phase cancellation from adding the resonators in parallel, symmetrically affects the bandwidth and peak-gain frequency of the two one-pole resonators. For simplicity below, we will let the zero be simply at $s=0$ , which is a typical choice for a real second-order resonator in which no dc gain is desired, and there is little difference when the damping is light ( $\sigma_0\approx 0$ ). This choice also simplifies calculation of the peak-gain frequency $\omega_p$ , leading to the incredibly simple result $\omega_p = \vert p_0\vert$ .^E.1 Moreover, the peak gain at $\omega_p = \vert p_0\vert$ is similarly incredibly simple:

$$H(j\vert p_0\vert) = \frac{1}{2\vert\sigma_0\vert}$$

Since we always have $\sigma_0<0$ for stability, it is convenient to define the positive number

$$\alpha \isdefs -\sigma_0,$$

which is called the damping constant (or damping factor) for real, second-order resonators [20, p. 179]. The damping constant is simply minus the real part of the poles. (We are assuming the poles form a complex-conjugate pair--excluding the case of two different real poles.) A related term from mechanics is the damping ratio $\zeta \isdeftext \alpha/\omega_0$ (see §E.7.3 below).

We also have

$$\omega_p ^2 \eqsp \vert p_0\vert^2 \eqsp \sigma_0^2 + \omega_0^2 \eqsp \alpha^2 + \omega_0^2.$$

That is, $\omega_p = \vert p_0\vert$ is both the pole modulus and the peak-gain frequency for a sum of complex-conjugate one-pole resonators with a zero at $s=0$ separating them. When damping is light, we have $\omega_p \approx\omega_0$ . However, for any nonzero damping we have that the peak gain occurs precisely at frequency $\omega=\omega_p$ , and the gain there is $1/(2\alpha)$ , again when the induced zero has been moved to $s=0$ .

The transfer function of the real second-order resonator with one zero at dc and the other at infinity, with peak gain normalized to $g$ , can be written in terms of our new notation as

$$H(s) \eqsp g\frac{2\alpha\,s}{s^2 + 2\alpha s + \omega_p ^2} \isdefs g\frac{\frac{\omega_p }{Q}s}{s^2 + \frac{\omega_p }{Q}s + \omega_p ^2} \isdefs g\frac{\frac{1}{Q}\tilde{s}}{\tilde{s}^2 + \frac{1}{Q}\tilde{s} + 1}, \quad \tilde{s}\isdef \frac{s}{\omega_p }. \protect$$ (E.7)

This is a good canonical form for a real, second-order resonator [20, p. 179].

In view of the above, it is now reasonable to define the quality factor (Q) of the real second-order resonator Eq.(E.7) by [20, p. 184]

$$\zbox {Q \isdefs \frac{\omega_p }{2\alpha}} \protect$$ (E.8)

where $\omega_p$ and $\alpha$ are defined above as the pole modulus and damping constant, respectively. Remember that for $\omega_p$ to be both the peak-gain frequency and pole-modulus $\vert p_0\vert$ , we require a zero at dc ($s=0$ ), with the other at $s=\infty$ .

The peak-gain frequency $\omega_p$ (pole modulus) coincides with the physical oscillation frequency $\omega_0$ of the resonator impulse response (pole imaginary part) when the damping constant $\alpha$ is zero. The physical oscillation frequency $\omega_0$ of the impulse response remains (twice) the zero-crossing rate of the sinusoidal oscillation underneath the exponential decay of the impulse response. See also §B.6 for a discussion of discrete-time resonators.