Critical Damping and Related Terms

Since the poles of the transfer function $H(s)$ are [20, p. 624]

$$p_0 \eqsp \sigma_0 \pm j\omega_0 \eqsp -\alpha \pm j\sqrt{\omega_p ^2-\alpha^2} \eqsp -\alpha \pm \sqrt{\alpha^2 - \omega_p ^2}, \protect$$ (E.9)

we have that the poles are complex only when $\omega_p >\alpha \;\Leftrightarrow\; Q>1/2$ . Equivalently, the damping ratio $\zeta\isdeftext 1/(2Q)$ must be less than 1. 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 ($\alpha=0$ and the poles are on the $j\omega$ axis at $\omega=\pm\omega_p =\pm\omega_0$ ).