Computing PLL coefficients

Whenever I implement a PLL or a similar control loop, I invariably consult the formulas in the paper Controlled-Root Formulation for Digital Phase-Locked Loops, by Stephens and Thomas. Other sources that give formulas for the loop coefficients in terms of the loop bandwidth perform a continuous time analysis and then use a bilinear transform or a similar kind of transform to translate results between continuous time and discrete time. The appeal of the paper by Stephens and Thomas is that they work directly in discrete time, using a beautiful complex contour integral argument to calculate the loop bandwidth in terms of the loop coefficients for a loop of any order. Unfortunately, their method doesn’t give a closed-form formula for the loop coefficients in terms of the loop bandwidth. The loop coefficients can be obtained numerically, and the paper gives tables for common loop bandwidths and orders.

In most of my designs I use a second order loop with supercritical damping, which means that the two loop roots in the z-plane are equal (and hence real). As I was doing a design the other day, I wondered whether in this specific situation, which is much simpler than the general case, a closed-form solution could be obtained. It turns out that this is the case, so I’ll be using this formula from now on. In this short post I explain how this is done and give the formula.

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.