Nonlinear Analysis of Phase-locked Loop-Based Circuits
Main problems of simulation and mathematical modeling of high-frequency signals for analog Costas loop and for analog phase-locked loop (PLL) are considered. Two approachers which allow to solve these problems are considered. In the first approach, nonlin
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Nonlinear Analysis of Phase-locked Loop-Based Circuits R.E. Best, N.V. Kuznetsov, G.A. Leonov, M.V. Yuldashev, and R.V. Yuldashev
Abstract Main problems of simulation and mathematical modeling of highfrequency signals for analog Costas loop and for analog phase-locked loop (PLL) are considered. Two approachers which allow to solve these problems are considered. In the first approach, nonlinear models of classical PLL and classical Costas loop are considered. In the second approach, engineering solutions for this problems are described. Nonlinear differential equations are derived for both approaches. Keywords Phase-locked loop • Nonlinear analysis • Dynamical model • Simulation
The Phase-locked loop (PLL) is a classical circuit widely used in telecommunication and computer architectures. PLL was invented in the 1930s–1940s [5] and then intensive studies of the theory and practice of PLL were carried out [11, 33, 40]. One of the first applications of PLL is related to the problems of wireless data transfer. In radio engineering, PLL-based circuits (e.g., Costas Loop, PLL with squarer) are used for carrier recovery, demodulation, and frequency synthesis (see, e.g., [6, 14, 35]).
R.E. Best Best Engineering, Oberwil, Switzerland e-mail: [email protected] N.V. Kuznetsov () • G.A. Leonov • M.V. Yuldashev • R.V. Yuldashev Saint Petersburg State University, St Petersburg, Russia University of Jyväskylä, Jyväskylä, Finland e-mail: [email protected]; [email protected]; [email protected]; [email protected] J.A.T. Machado et al. (eds.), Discontinuity and Complexity in Nonlinear Physical Systems, Nonlinear Systems and Complexity 6, DOI 10.1007/978-3-319-01411-1__10, © Springer International Publishing Switzerland 2014
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Although the PLL is essentially a nonlinear control system, in modern literature, devoted to the analysis of PLL-based circuits, the main direction is the use of simplified linear models, the methods of linear analysis, empirical rules, and numerical simulation (see plenary lecture of Abramovich at American Control Conference 2002 [1]). Rigorous nonlinear analysis of PLL-based circuit models is often a very difficult task [4, 9, 10, 37], so for analysis of nonlinear PLL models, in practice, in numerical simulation is widely used (see, e.g., [6]). However for high-frequency signals, complete numerical simulation of physical model of PLL-based circuit in signals/time space, which is described by a nonlinear nonautonomous system of differential equations, is a very challenging task [2, 3] since it is necessary to observe simultaneously very fast time scale of the input signals and slow time scale of signal’s phases. Here the relatively small discretization step in numerical procedure does not allow one to consider phase locking processes for high-frequency signals in reasonable time. Here two approaches, which allow one to overcome these difficulties, are considered. The first idea is traced back to the works of [40] and consists in construction of mathematical models of PLL
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