Unlock Characteristics of the Optimum Type II Phase-Locked Loop

This paper describes a study of the statistical unlock characteristics of the commonly used optimum type II phase-locked loop taking into account the nonlinearity of the phase detector. The unlock behavior is investigated by simulating the loop with random noise inputs on a digital computer. A criterion of loop unlock is evolved using the phase portrait of the loop and consists of crossing of separatrices by loop error and a measure of loop error rate. Approximately 100 statistically independent trials were made for each of a number of combinations of initial phase error, initial frequency error, and input noise amplitude. From these computer runs, cumulative frequency distributions of unlock time and curves of mean time to unlock were obtained and are presented. The phase portrait of the loop containing time information also is included in the paper, along with a plot of capture characteristics for large initial frequency offsets.