Equations Of Phase-locked Loops: Dynamics On Circle, Torus And Cylinder

by Jacek Kudrewicz, Stefan Wasowicz

Overview

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 Rössler, Henon, Lorenz, May, Chua and others.

Product Details

ISBN-13:	9789812770905
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/29/2007
Series:	World Scientific Series On Nonlinear Science Series A , #59
Pages:	236
Product dimensions:	5.80(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface     v
Introduction     1
What is Phase-Locked Loop?     1
PLL and differential or recurrence equations     2
Averaging method     5
Organization of the book     7
The first order continuous-time Phase-Locked Loops     9
Equations of the system     9
The averaged equation     12
Basic properties of solutions     12
Application to Adler's equation     15
Solutions of the basic frequency     18
The Poincare mapping     18
Periodic solutions     20
Asymptotic formulae for periodic solutions     21
Conclusions for the PLL equation     23
Differential equation on the torus     24
Trajectories on the torus     24
Periodic points     26
Rotation number     27
Rotation number as the function of a parameter     28
Fractional synchronization     30
Devil's staircase     30
Constructing of a devil's staircase     31
T-property     34
A fundamental Theorem     36
Consequences for forced oscillators     38
Numerical and analytical approach     39
The system with rectangular waveform signals     43
The Poincare mapping     43
The Arnold's tongues     46
Numerical results and consequences of a symmetry     48
The mapping f(p) = p + 2[pi Mu] + a sin p     50
Small input signal     50
Properties of the rotation number     51
The number of periodic orbits     53
The second order continuous-time Phase-Locked Loops     55
The system with a low-pass filter     55
Phase-plane portrait of the averaged system     57
The phase-plane trajectories     57
The case [Characters not reproducible] > 1. Phase-modulated output signals     59
The case [Characters not reproducible] < 1. Hold-in region     61
Boundary of pull-in region: S[subscript 2] [Characters not reproducible] S[subscript 3]     65
The case [Characters not reproducible] = 1. Boundary of hold-in region     66
The filter with high cut-off frequency     67
The filter with low cut-off frequency     68
Perturbation of the phase difference [phi (omega t)]     70
A basic theorem     71
An approximate formula for periodic solutions     72
Numerical experiments     73
Stable integral manifold     75
The basic notions and motivations     75
An equation of the second order     77
Proof of Theorem 3.3     78
Uniqueness of the manifold     83
The PLL system reducible to the first order one     85
Small values of parameter a = A[Omega]T     85
A neighborhood of the trajectory x = M([phi])     87
Homoclinic structures     89
The Poincare mapping     89
Invariant lines of hyperbolic fixed points     91
Heteroclinic and homoclinic trajectories     93
Melnikov's theorem     96
Boundaries of attractive domains     100
Small values of the parameters: [Delta], a[Characters not reproducible], [epsilon][subscript 0]     101
Large values of a     103
A neighborhood of the line [Characters not reproducible] = H(a)     105
Numerical experiments     106
The Smale horseshoe. Transient chaos     109
Invariant set of the Smale horseshoe     110
Homeomorphism     113
Comments     115
Higher order systems reducible to the second order ones     117
The system with a filter of the higher order     117
Two-dimensional integral manifold     119
Proof of Theorem 3.10     121
The local linearization     123
One-dimensional discrete-time Phase-Locked Loop     127
Recurrence equations of the system     127
Periodic output signals     129
Type of a periodic point     129
Basic properties of periodic points     131
Li and Yorke Theorem     133
Rotation interval and frequency locking regions     137
Definition and properties     137
Selected frequency locking regions     140
Application to the map (4.7)     144
Stable orbits, hold-in regions     145
Stability of periodic points     145
Stable periodic points of the type n/1 and n/2     146
Attractive set of a fixed point     150
Attractive set of a stable periodic orbit     154
The number of stable orbits     155
Schwarzian derivative     156
Application to the map T([tau]) = [tau] + 2[pi Mu]nii + a sin [tau]     158
Bifurcations of periodic orbits     161
Saddle-node bifurcation     161
Period doubling bifurcation     163
The Feigenbaum cascade     165
Invariant measures      168
The Liapunov exponent     171
Skeleton of superstable orbits     173
The Feigenbaum cascade (continuation)     177
Bifurcation of the rotation interval     180
A simplified mapping     182
Superstable periodic orbits of the type 1/k     184
Family of quadratic polynomials     185
Dynamics restricted to the set I[subscript 0]     187
Asymptotic properties for [epsilon left arrow] 0     189
Two-dimensional discrete-time Phase-Locked Loop     191
Description of the DPLL system by a two-dimensional map     191
Stable periodic orbits     195
Periodic points of the type n/1     196
Stability of fixed points     197
Hold-in regions     200
Small values of [lambda]     201
Reduction to a one-dimensional system     202
Existence of an invariant manifold     203
Decay of the invariant manifold     206
Strange attractors and chaotic steady-states     209
Maximal invariant set     209
Attractors     210
Attractive domains     216
Bibliography     221
Index     225