Pull-in time in second-order phase-locked loops with a sawtooth comparator

A phase-locked loop incorporating a sawtooth phase comparator and an or phase-lag filter is considered in a noisefree environment. The pull-in time problem during initial synchronization or abrupt change of the input frequency is studied in detail and equivalently the out-of-lock time, when the loop is "knocked out of lock and then brought back to its normal operation." An exact and convenient graphical method is presented for the evaluation of the number of "skipped" cycles and the pull-in time (or out-of-lock time) for any input mistuning within the pull-in range and any initial state (i.e., any initial phase and frequency offset) of the system. For large initial mistuning, a simple approximation that avoids the graphical procedure is given. These graphical methods provide qualitative insight and quantitative information as to the pull-in performance and beat mode of operation.