Limit cycles in bang-bang phase-locked loops

This paper examines the nonlinear dynamics of a model of a second order bang-bang phase-locked loop (BB-PLL). Three distinct steady state dynamical patterns (locking, slew-rate limiting and limit cycles) have been observed for this discrete system. A corresponding continuous model of the BB-PLL is established. This paper focuses on the occurrence and the shape of the limit cycles. In particular, equations for the limit cycle trajectories are determined. The condition for the appearance of limit cycles is then established as a boundary in parameter space. A further theorem transfers this analysis back to the discrete system, where a continuum of cycles is found to occur. A direct relationship between the level of input phase deviation and the occurrence of limit cycles is observed