Having to deal with DSP texts written by engineers, I have sometimes to work a bit to get a good grasp of 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.