Equations of phase-locked loops : dynamics on circle, torus and cylinder / Jacek Kudrewicz, Stefan Wasowicz.

Format:

Book

Author/Creator:

Kudrewicz, Jacek.

Contributor:

Wąsowicz, Stefan.

Series:

World Scientific series on nonlinear science. Monographs and treatises ; Series A, v. 59.

World Scientific series on nonlinear science. Series A ; v. 59

Language:

English

Subjects (All):

Phase-locked loops--Mathematics.

Phase-locked loops.

Nonlinear oscillators--Mathematics.

Nonlinear oscillators.

Physical Description:

1 online resource (236 p.)

Edition:

1st ed.

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, c2007.

Language Note:

English

Summary:

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 dyn

Contents:

Preface; Contents; 1. Introduction; 1.1 What is Phase-Locked Loop?; 1.2 PLL and differential or recurrence equations; 1.3 Averaging method; 1.4 Organization of the book; 2. The first order continuous-time Phase-Locked Loops; 2.1 Equations of the system; 2.2 The averaged equation; 2.2.1 Basic properties of solutions; 2.2.2 Application to Adler's equation; 2.3 Solutions of the basic frequency; 2.3.1 The Poincare mapping; 2.3.2 Periodic solutions; 2.3.3 Asymptotic formulae for periodic solutions; 2.3.4 Conclusions for the PLL equation; 2.4 Differential equation on the torus

2.4.1 Trajectories on the torus2.4.2 Periodic points; 2.4.3 Rotation number; 2.4.4 Rotation number as the function of a parameter; 2.5 Fractional synchronization; 2.5.1 Devil's staircase; 2.5.2 Constructing of a devil's staircase; 2.5.3 T-property; 2.5.4 A fundamental Theorem; 2.5.5 Consequences for forced oscillators; 2.5.6 Numerical and analytical approach; 2.6 The system with rectangular waveform signals; 2.6.1 The Poincare mapping; 2.6.2 The Arnold's tongues; 2.6.3 Numerical results and consequences of a symmetry; 2.7 The mapping f(p) = p + 2 + asinp; 2.7.1 Small input signal

2.7.2 Properties of the rotation number2.7.3 The number of periodic orbits; 3. The second order continuous-time Phase-Locked Loops; 3.1 The system with a low-pass filter; 3.2 Phase-plane portrait of the averaged system; 3.2.1 The phase-plane trajectories; 3.2.2 The case > 1 . Phase-modulated output signals; 3.2.3 The case < 1. Hold-in region; 3.2.4 Boundary of pull-in region: S2 = S3; 3.2.5 The case = 1. Boundary of hold-in region; 3.2.6 The filter with high cut-off frequency; 3.2.7 The filter with low cut-off frequency; 3.3 Perturbation of the phase difference (wt); 3.3.1 A basic theorem

3.3.2 An approximate formula for periodic solutions3.3.3 Numerical experiments; 3.4 Stable integral manifold; 3.4.1 The basic notions and motivations; 3.4.2 An equation of the second order; 3.4.3 Proof of Theorem 3.3; 3.4.4 Uniqueness of the manifold; 3.5 The PLL system reducible to the first order one; 3.5.1 Small values of parameter a = A T; 3.5.2 A neighborhood of the trajectory x = M ( ); 3.6 Homoclinic structures; 3.6.1 The Poincare mapping; 3.6.2 Invariant lines of hyperbolic fixed points; 3.6.3 Heteroclinic and homoclinic trajectories; 3.6.4 Melnikov's theorem

3.7 Boundaries of attractive domains3.7.1 Small values of the parameters:; 3.7.2 Large values of a; 3.7.3 A neighborhood of the line = H ( ); 3.7.4 Numerical experiments; 3.8 The Smale horseshoe. Transient chaos; 3.8.1 Invariant set of the Smale horseshoe; 3.8.2 Homeomorphism; 3.8.3 Comments; 3.9 Higher order systems reducible to the second order ones; 3.9.1 The system with a filter of the higher order; 3.9.2 Two-dimensional integral manifold; 3.9.3 Proof of Theorem 3.10; 3.9.4 The local linearization; 4. One-dimensional discrete-time Phase-Locked Loop; 4.1 Recurrence equations of the system

4.2 Periodic output signals

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 221-224) and index.

ISBN:

9786611911638

9781281911636

1281911631

9789812770912

9812770917

OCLC:

826660294