Markov Chains-Based Derivation of the Phase Detector Gain in Bang-Bang PLLs

Due to the presence of a binary phase detector (BPD) in the loop, bang-bang phase-locked loops (BBPLLs) are hard nonlinear systems. Since the BPD is usually also the only nonlinear element in the loop, in practical applications, BBPLLs are commonly analyzed by first linearizing the BPD and then using the traditional mathematical techniques for linear systems. To the author's knowledge, in the literature, the gain of the linearized BPD (Kbpd) is determined neglecting the effect of the BBPLL dynamics on the effective jitter seen by the BPD. In this brief, we develop an approach to the determination of Kbpdwhich takes into consideration also this effect. The approach is based on modeling the dynamics of a BBPLL as a Markov chain. This approach gives new insights into the behavior of the BBPLL and leads to an expression for the Kbpd, which is more general than the one currently known in literature