New Results on Mean Time-to-First-Slip for a First-Order Loop

Approximate results are derived from a new expression for the mean time-to-first-slip for a first-order loop with arbitrary periodic equivalent phase detector characteristic in the presence of frequency detuning. They are accurate for loop signal-to-noise ratios greater than 3 dB or less than 0 dB. The approximations found serve to provide insight into the mean first slip time dependence on the shape of the equivalent phase detector characteristic and the frequency detuning, and dependence on loop parameters. The new expression itself is also of theoretical interest, since it allows one to explicitly express the stationary phase error probability density function and probability current in terms of the mean first slip time. This sheds further light on the interconnections between the conventional modulo-2π approach and the renewal theory approach to the Fokker-Planck method. Extensions to nonperidic equivalent phase detector characteristics are also considered.