PLL was invented in 1930s-1940s and was used in radio and TV (synchronization, demodulation and frequency synthesis). Nowadays PLL can be produced as a single integrated circuit. There are several types of PLL (classical analog PLL, ADPLL, DPLL, and others) and its modifications (Costas loop, PLL with square, and others) which are used widespread in a great amount of modern electronic applications (telecommunications, computers architectures and others). Various methods for analysis of phase-locked loops are well developed by engineers, but the problems of construction of adequate nonlinear models and nonlinear analysis of such models are still far from being resolved. As was remarked in a plenary lecture at ACC-2002, the main direction in modern literature, devoted to the analysis of stability and synthesis of PLL, is the use of simplified linear models, the methods of linear analysis, empirical rules, and simulation. However it is well known that the application of the methods of linearization and linear analysis without justification can lead to wrong results. Numerical simulation of PLL in signals space is, as a rule, rather laborious because a simulation step, which must be sufficiently small to distinctly observe the dynamics of phase detector, makes difficult the observation of the dynamics of all systems. The simulation in phase-frequency space permits one to overcome these difficulties but requires the construction of the corresponding models of PLL and also can lead to untrue results. It was shown showed analytically the possibility of the existence of hidden oscillations in two-dimensional model of PLL: with the computational point of view in the considered system all the trajectories tend to equilibrium, but, in fact, a domain of attraction of equilibria is bounded. In this survey, it is described the general approach to nonlinear analysis and design of analog phase locked loop, which are based on the construction of nonlinear mathematical models in signal and phase-frequency space and applying rigorous mathematical the methods of nonlinear analysis of high-frequency oscillations.
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