Transient analysis of the first-order phase-locked loop (Corresp.)

The problem of obtaining the transient solution to a class of Fokker-Planck equation derived from the first-order nondetuning phase-locked loop is considered. Two methods are proposed to obtain the eigenvalues and the eigenfunctions of the spatial operator for the Fokker-planck equation. One is a perturbation method in which the signal-to-noise ratio is adopted as a parameter. The other is a method similar to the Ince-Goldstein method for solving the Mathieu equation. Numerical calculations show that the latter is superior to the former in accuracy and in covering a wider range of signal-to-noise ratios.