Phase-Locked Loops (PLLs) are electronic systems that can be used as a synchronized oscillator, a driver or multiplier of frequency, a modulator or demodulator and as an amplifier of phase modulated signals. This book updates the methods used in the analysis of PLLs by drawing on the results obtained in the last 40 years. Many are published for the first time in book form. Nonlinear and deterministic mathematical models of continuous-time and discrete-time PLLs are considered and their basic properties are given in the form of theorems with rigorous proofs. The book exhibits very beautiful dynamics, and shows various physical phenomena observed in synchronized oscillators described by complete (not averaged) equations of PLLs. Specially selected mathematical tools are used - the theory of differential equations on a torus, the phase-plane portraits on a cyclinder, a perturbation theory (Melnikov's theorem on heteroclinic trajectories), integral manifolds, iterations of one-dimensional maps of a circle and two-dimensional maps of a cylinder. Using these tools, the properties of PLLs, in particular the regions of synchronization are described. Emphasis is on bifurcations of various types of periodic and chaotic oscillations. Strange attractors in the dynamics of PLLs are considered, such as those discovered by Roessler, Henon, Lorenz, May, Chua and others.
Introduction: What Is Phase-Locked Loop?; PLL and Differential or Recurrence Equations; Averaging Method; Organization of the Book; The First Order Continuous-Time Phase-Locked Loops: Equations of the System; The Averaged Equation; Solutions of the Basic Frequency; Differential Equation on the Torus; Fractional Synchronization; The System with Rectangular Waveform Signals; The Mapping f(p)=p+2 +a sin(p); The Second Order Continuous-Time Phase-Locked Loops: The System with a Low-Pass Filter; Phase-Plane Portrait of the Averaged System; Perturbation of the Phase Difference; Stable Integral Manifold; The PLL System Reducible to the First Order One; Homoclinic Structures; Boundaries of Attractive Domains; The Smale Horseshoe, Transient Chaos; Higher Order Systems Reducible to the Second Order Ones; One-Dimensional Discrete-Time Phase-Locked Loop: Recurrence Equations of the System; Periodic Output Signals; Rotation Interval and Frequency Locking Regions; Stable Orbits, Hold-In Regions; The Number of Stable Orbits; Bifurcations of Periodic Orbits; Bifurcation of the Rotation Interval; Two-Dimensional Discrete-Time Phase-Locked Loop: Description of the DPLL System by a Two-Dimensional Map; Stable Periodic Orbits; Reduction to a One-Dimensional System: Strange Attractors and Chaotic Steady-States.