Simplified Formula for Mean Cycle-Slip Time of Phase-Locked Loops With Steady-State Phase Error

Previous work has shown that the mean time from lock to a slipped cycle of a phase-locked loop is given by a certain double integral. Accurate numerical evaluation of this formula for the second-order loop has proved extremely vexing because the difference between exponentially large quantities is involved. This article simplifies the general formula to avert this problem, provides a useful approximation to a needed conditional expectation, and produces an asymptotic formula for the mean slip time that is moderately accurate even at low loop SNR (less than 7 dB) and small steady-state phase errors (less than 0.3 rad). The approximations extend to higher order loops, as well.