Cycle Slipping in Phase-Locked Loops

This article shows that the expected first-slip time of a phase-locked loop of arbitrary order can be found as the solution of a first-order linear differential equation, to which formal solutions are easily written. Computation of an exact solution involves being able to evaluate a certain conditional expectation which, for the first-order loop, is readily done and yields the exact known result. For higher order loops, an approximate evaluation of the expectation is presented and compared with experimentally obtained data of the secondorder loop. The first-slip time based on this approximation also compares favorably with measured times.