About critical damping

Having to deal with DSP texts written by engineers, I have sometimes to work a bit to get a good grasp the concepts, which many times are not explained clearly from their mathematical bases. Often, a formula is just used without much motivation. Lately, I've been trying to understand critically damped systems, in the context of PLL loop filters.

The issue is as follows. In a second order filter there is a damping parameter \zeta > 0. The impulse response of the filter is an exponentially decaying sinusoid if \zeta < 1 (underdamped system), a decaying exponential if \zeta > 1 (overdamped system) and something of the form C t e^{-\lambda t} if \zeta = 1 (critically damped system). Critical damping is desirable in many cases because it maximizes the exponential decay rate of the impulse response. However, many engineering texts just go and choose \zeta = \sqrt{2}/2 without any justification and even call this critical damping. Here I give some motivation starting with the basics and explain what is special about \zeta = \sqrt{2}/2 and why one may want to choose this value in applications.

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