Mechanism of the Cycle Slips in the Stochastic Analog Phase-Locked Loops of the First and Second Orders

A posteriori broadband input noise characteristics of the analog phase locked loop (PLL) system along the cycle slip trajectory were studied on the basis of Markov stochastic model. It was demonstrated that the cycle slips were caused by an unlikely event - existence of an extended period of time, in which a random process, which describes the noise, mainly preserves its sign. A mathematical model of the most probable cycle slip trajectory in the stochastic analog of PLL of the first and second orders in the form of the ordinary differential equations with respect to the coordinate system on the phase plane was presented. There was demonstrated equivalence ofthe approaches to search the most probable cycle slip trajectory: the search for the maximum of the probability density function of the points of coordinates along the trajectory and the solution of the variational problem. A model of abnormal noise associated with the cycle slips effect was presented. An approximated formula for description of the spectral power density of the abnormal noise was offered. The process of the cycle slips in the PLL with the non-linear element in the feedback loop was investigated. It was demonstrated that such systems have improved characteristics of the average time before a cycle sleep in comparison with the traditional ones. An explanation for an increase of time before a tracking failure, when using a non-linear element in the feedback loop, was presented. Approaches to selection of the type of nonlinearity were considered. Numerical values for the average time before a cycle sleep for various types of non-linearity and parameters of PLL on the basis of computer simulation were presented.

фазовая автоподстройка частоты, срыв слежения, аномальный шум, фазовая ошибка, нелинейный элемент, phase-locked loop, cycle slip, abnormal noise, phase error, nonlinear element

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