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 . The impulse response of the filter is an exponentially decaying sinusoid if (underdamped system), a decaying exponential if (overdamped system) and something of the form if (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 without any justification and even call this critical damping. Here I give some motivation starting with the basics and explain what is special about and why one may want to choose this value in applications.