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Critical Damping and Related Terms

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

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

we have that the poles are complex only when $ \omega_p >\alpha \;\Leftrightarrow\; Q>1/2$ . 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$ ).


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2018-04-13 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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