0000000000105956
AUTHOR
M. V. Yuldashev
Lock-in range of classical PLL with impulse signals and proportionally-integrating filter
In the present work the model of PLL with impulse signals and active PI filter in the signal's phase space is described. For the considered PLL the lock-in range is computed analytically and obtained result are compared with numerical simulations.
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, 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.
Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture
The Aizerman and Kalman conjectures played an important role in the theory of global stability for control systems and set two directions for its further development – the search and formulation of sufficient stability conditions, as well as the construction of counterexamples for these conjectures. From the computational perspective the latter problem is nontrivial, since the oscillations in counterexamples are hidden, i.e. their basin of attraction does not intersect with a small neighborhood of an equilibrium. Numerical calculation of initial data of such oscillations for their visualization is a challenging problem. Up to now all known counterexamples to the Kalman conjecture were const…
Lock-in range of PLL-based circuits with proportionally-integrating filter and sinusoidal phase detector characteristic
In the present work PLL-based circuits with sinusoidal phase detector characteristic and active proportionally-integrating (PI) filter are considered. The notion of lock-in range -- an important characteristic of PLL-based circuits, which corresponds to the synchronization without cycle slipping, is studied. For the lock-in range a rigorous mathematical definition is discussed. Numerical and analytical estimates for the lock-in range are obtained.
Nonlinear analysis of charge-pump phase-locked loop: the hold-in and pull-in ranges
In this paper a fairly complete mathematical model of CP-PLL, which reliable enough to serve as a tool for credible analysis of dynamical properties of these circuits, is studied. We refine relevant mathematical definitions of the hold-in and pull-in ranges related to the local and global stability. Stability analysis of the steady state for the charge-pump phase locked loop is non-trivial: straight-forward linearization of available CP-PLL models may lead to incorrect conclusions, because the system is not smooth near the steady state and may experience overload. In this work necessary details for local stability analysis are presented and the hold-in range is computed. An upper estimate o…
Nonlinear analysis of classical phase-locked loops in signal's phase space
Abstract Discovery of undesirable hidden oscillations, which cannot be found by the standard simulation, in phase-locked loop (PLL) showed the importance of consideration of nonlinear models and development of rigorous analytical methods for their analysis. In this paper for various signal waveforms, analytical computation of multiplier/mixer phase-detector characteristics is demonstrated, and nonlinear dynamical model of classical analog PLL is derived. Approaches to the rigorous nonlinear analysis of classical analog PLL are discussed.
Hidden attractors in dynamical models of phase-locked loop circuits : limitations of simulation in MATLAB and SPICE
During recent years it has been shown that hidden oscillations, whose basin of attraction does not overlap with small neighborhoods of equilibria, may significantly complicate simulation of dynamical models, lead to unreliable results and wrong conclusions, and cause serious damage in drilling systems, aircrafts control systems, electromechanical systems, and other applications. This article provides a survey of various phase-locked loop based circuits (used in satellite navigation systems, optical, and digital communication), where such difficulties take place in MATLAB and SPICE. Considered examples can be used for testing other phase-locked loop based circuits and simulation tools, and m…
Tutorial on dynamic analysis of the Costas loop
Abstract Costas loop is a classical phase-locked loop (PLL) based circuit for carrier recovery and signal demodulation. The PLL is an automatic control system that adjusts the phase of a local signal to match the phase of the input reference signal. This tutorial is devoted to the dynamic analysis of the Costas loop. In particular the acquisition process is analyzed. Acquisition is most conveniently described by a number of frequency and time parameters such as lock-in range, lock-in time, pull-in range, pull-in time, and hold-in range. While for the classical PLL equations all these parameters have been derived (many of them are approximations, some even crude approximations), this has not…
A Survey on Dynamic Analysis of the Costas Loop
This survey is devoted to the dynamic analysis of the Costas loop. In particular the acquisition process is analyzed in great detail. Acquision is most conventiently described by a number of frequency and time parameters such as lock-in range, lock-in time, pull-in range, pull-in time, and hold-in range. While for the classical PLL equations for all these parameters have been derived (many of them are approximations, some even crude approximations), this has not yet been carried out for the Costas loop. It is the aim of this analysis to close this gap. The paper starts with an overview on mathematical and physical models (exact and simplified) of the different variants of the Costas loop, c…
A short survey on nonlinear models of the classic Costas loop: rigorous derivation and limitations of the classic analysis
Rigorous nonlinear analysis of the physical model of Costas loop --- a classic phase-locked loop (PLL) based circuit for carrier recovery, is a challenging task. Thus for its analysis, simplified mathematical models and numerical simulation are widely used. In this work a short survey on nonlinear models of the BPSK Costas loop, used for pre-design and post-design analysis, is presented. Their rigorous derivation and limitations of classic analysis are discussed. It is shown that the use of simplified mathematical models, and the application of non rigorous methods of analysis (e.g., simulation and linearization) may lead to wrong conclusions concerning the performance of the Costas loop ph…
Non-linear analysis of a modified QPSK Costas loop
A Costas loop is one of the classical phase-locked loop based circuits, which demodulates data and recovers carrier from the input signal. The Costas loop is essentially a nonlinear control system and its nonlinear analysis is a challenging task. In this article for a modified QPSK Costas loop we analyze the hold-in, pull-in and lock-in ranges. New procedure for estimation of the lock-in range is considered and compared with previously known approach. peerReviewed
Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large
The analysis of the stability and numerical simulation of Costas loop circuits for high-frequency signals is a challenging task. The problem lies in the fact that it is necessary to simultaneously observe very fast time scale of the input signals and slow time scale of phase difference between the input signals. To overcome this difficult situation it is possible, following the approach presented in the classical works of Gardner and Viterbi, to construct a mathematical model of Costas loop, in which only slow time change of signal?s phases and frequencies is considered. Such a construction, in turn, requires the computation of phase detector characteristic, depending on the waveforms of th…
Hold-in, Pull-in and Lock-in Ranges for Phase-locked Loop with Tangential Characteristic of the Phase Detector
In the present paper the phase-locked loop (PLL), an electric circuit widely used in telecommunications and computer architectures is considered. A new modification of the PLL with tangential phase detector characteristic and active proportionally-integrating (PI) filter is introduced. Hold-in, pull-in and lock-in ranges for given circuit are studied rigorously. It is shown that lock-in range of the new PLL model is infinite, compared to the finite lock-in range of the classical PLL. peerReviewed
On the Gardner Problem for the Phase-Locked Loops
This report shows the possibilities of solving the Gardner problem of determining the lock-in range for multidimensional phase-locked loops systems. The development of analogs of classical stability criteria for the cylindrical phase space made it possible to obtain analytical estimates of the lock-in range for third-order system.
Nonlinear analysis of phase-locked loop
Abstract New method for the rigorous mathematical nonlinear analysis of PLL systems is suggested. This method allows to calculate the characteristics of phase detectors and carry out a rigorous mathematical analysis of transient process and stability of the system.
Stability and Chaotic Attractors of Memristor-Based Circuit with a Line of Equilibria
This report investigates the stability problem of memristive systems with a line of equilibria on the example of SBT memristor-based Wien-bridge circuit. For the considered system, conditions of local and global partial stability are obtained, and chaotic dynamics is studied. peerReviewed
Analytical methods for computation of phase-detector characteristics and PLL design
An effective analytical methods for computation of phase detector characteristics are suggested. For high-frequency oscillators new classes of such characteristics are described. Approaches to a rigorous nonlinear analysis of PLL are discussed.