Nonlinear Analysis of Charge-Pump Phase-Locked Loop: The Hold-In and Pull-In Ranges

Kuznetsov, Nikolay V.; Matveev, Alexey S.; Yuldashev, M. V.; Yuldashev, R. V.

doi:10.1109/tcsi.2021.3101529

Cited by 14 publications

(5 citation statements)

References 36 publications

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“…4 The Chua circuit diagram with the on/off switch K 1 Multistability is a fairly common phenomenon that arises not only in radiophysical systems, but also in various abstract problems of mathematics (see, e.g., the 16th Hilbert problem [26]) and engineering problems (see, e.g., [6,14,[27][28][29][30][31][32][33][34]). Prediction and investigation of coexisting regimes, especially if they are hidden, is a rather difficult task that is important for various engineering applications (see, e.g., phase-locked loops [35][36][37][38][39]). One of the most important questions in the study of multistable systems, especially with hidden attractors, is the choice of the initial conditions to reveal all possible limiting regimes.…”

Section: Schematic Diagram and Mathematical Model Of The Chua Circuitmentioning

confidence: 99%

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

et al. 2022

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After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

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Section: Schematic Diagram and Mathematical Model Of The Chua Circuitmentioning

confidence: 99%

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

et al. 2022

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“…The further development of such systems analysis is connected with consideration of higher-order loop filters and discontinuous phase detector characteristics for revealing hidden oscillations and providing the global stability [Zhu et al, 2020;Kuznetsov et al, 2021c].…”

Section: Discussionmentioning

confidence: 99%

The Gardner problem and cycle slipping bifurcation for type 2 phase-locked loops

Kuznetsov,

Arseniev,

Blagov

et al. 2021

Preprint

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“…Thus, for model (5), the conjecture about global stability by the first approximation is valid. However, in general, this conjecture is not true for a higher-order PLL, second-order type 1 PLL, and Charge-Pump PLL with a proportionalplus-integral filter, where the global stability conditions can be determined by the birth of either self-excited or hidden oscillations [17], [60], [72], [73]. Notice that similar problems are well-known in control theory (see, e.g., the Andronov-Vyshnegradsky results on global stability of the Watt governor and the Aizerman and Kalman conjectures [17], [74]- [76]).…”

Section: A Preliminariesmentioning

confidence: 99%

The Gardner Problem on the Lock-In Range of Second-Order Type 2 Phase-Locked Loops

Kuznetsov

Lobachev

Yuldashev

et al. 2023

IEEE Trans. Automat. Contr.

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Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in telecommunications, computer architecture, gyroscopes, and other applications. One of the key problems of nonlinear analysis of PLL systems has been stated by Floyd M. Gardner as being "to define exactly any unique lock-in frequency." The lock-in range concept describes the ability of PLLs to reacquire a locked state without cycle slipping and its calculation requires nonlinear analysis. The present work analyzes a second-order type 2 phase-locked loop with a sinusoidal phase detector characteristic. Using the qualitative theory of dynamical systems and classical methods of control theory, we provide stability analysis and suggest analytical lower and upper estimates of the lock-in range based on the exact lock-in range formula for a second-order type 2 PLL with a triangular phase detector characteristic, obtained earlier. Applying phase plane analysis, an asymptotic formula for the lock-in range which refines the existing formula is obtained. The analytical formulas are compared with computer simulation and engineering estimates of the lock-in range. The comparison shows that engineering estimates can lead to cycle slipping in the corresponding PLL model and cannot provide a reliable solution for the Gardner problem, whereas the lower estimate presented in this article guarantees frequency reacquisition without cycle slipping for all parameters, which provides a solution to the Gardner problem.

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Nonlinear Analysis of Charge-Pump Phase-Locked Loop: The Hold-In and Pull-In Ranges

Cited by 14 publications

References 36 publications

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

The Gardner problem and cycle slipping bifurcation for type 2 phase-locked loops

The Gardner Problem on the Lock-In Range of Second-Order Type 2 Phase-Locked Loops

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