Approximations to the Pull-In Frequency of High Order Linear and Nonlinear Loops

Two methods of approximating the pull-in frequency cp of linearphase-locked loops are presented. Both methods are based on thephysically plausible, but not mathematically apparent, contention thatthe response of the phase error to a step change in mistuning, ϕϵ(t),can be used to predict ωp. The first method uses the ratio of the steadystatetate to peak valuϕϵ(t) as a measure of ωp. In the second method,the approximation is based on the shape of the time response of thephase error. This second technique is also employed to obtain ameasure of ωp for a nonlinear loop. Two other ways of approximatingthe wp of a nonlinear phase-locked loop, based on the concept ofobtaining an equivalent linear loop, are presented. These approximationsions represent a more easily implemented alternative to computersimulation as a method of obtaining a measure for ωp.