Stability and Transient Dynamics of PLLs in Theory and Experiments

This paper presents a generalization of the classical phase-locked loop (PLL) theory. It includes the effects of time-delays and mutual coupling between PLLs. Two methods for finding stable solutions to locked states and their transient dynamics are discussed. The theoretical predictions of these methods are verified by experimental measurements obtained of a classical PLL entrained by a clock. For entrainment the generalized and classical PLL theory overlap. The analysis correctly predicts the phase-relations of phase-locked states, the loop-gain dependency on the component characteristics and time-delays and the transient dynamics, i.e., perturbation decay rate and the frequency of perturbation decay. Thus the generalized theory allows a deeper understanding of a PLL’s response. The model covers PLLs of arbitrary order and number of inputs.