Non-Local Methods for Pendulum-Like Feedback Systems

Abstract-The terms hold-in, pull-in (capture), and lock-in ranges are widely used by engineers for the concepts of frequency deviation ranges within which PLL-based circuits can achieve lock under various additional conditions. Usually only non-strict definitions are given for these concepts in engineering literature. After many years of their usage, F. Gardner in the 2nd edition of his well-known work, Phaselock Techniques, wrote "There is no natural way to define exactly any unique lock-in frequency" and "despite its vague reality, lock-in range is a useful concept". Recently these observations have led to the following advice given in a handbook on synchronization and communications "We recommend that you check these definitions carefully before using them" [1, p.49]. In this survey an attempt is made to discuss and fill some of the gaps identified between mathematical control theory, the theory of dynamical systems and the engineering practice of phase-locked loops. It is shown that, from a mathematical point of view, in some cases the hold-in and pull-in "ranges" may not be the intervals of values but a union of intervals and thus their widely used definitions require clarification. Rigorous mathematical definitions for the hold-in, pullin, and lock-in ranges are given. An effective solution for the problem on the unique definition of the lock-in frequency, posed by Gardner, is suggested.Index Terms-Phase-locked loop, nonlinear analysis, analog PLL, high-order filter, local stability, global stability, stability in the large, cycle slipping, hold-in range, pull-in range, capture range, lock-in range, definition, Gardner's problem on unique lock-in frequency, Gardner's paradox on lock-in range.

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“…Analytical tools for estimating the number of cycle slips depending on the parameters of the loop can be found, e.g. in [11], [58], [64].…”

Section: Definition 5 Ifmentioning

confidence: 99%

Hold-In, Pull-In, and Lock-In Ranges of PLL Circuits: Rigorous Mathematical Definitions and Limitations of Classical Theory

Леонов

Kuznetsov

Yuldashev

et al. 2015

IEEE Trans. Circuits Syst. I

123

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“…Various generalizations of this theorem and numerous examples of applying the method of nonlocal reduction, including the applying to synchronous machines, can be found in the works (Leonov, 1975;Leonov, 1976;Gelig et al, 1978;Leonov et al, 1992;Leonov et al, 1996a). Various criteria for the existence of circular solutions and second-kind cycles were also obtained within the framework of this method (Gelig et al, 1978;Leonov et al, 1992;Leonov et al, 1996a;Yakubovich et al, 2004).…”

Section: Methods Of Nonlocal Reductionmentioning

confidence: 98%

“…Various criteria for the existence of circular solutions and second-kind cycles were also obtained within the framework of this method (Gelig et al, 1978;Leonov et al, 1992;Leonov et al, 1996a;Yakubovich et al, 2004).…”

Section: Methods Of Nonlocal Reductionmentioning

confidence: 99%

“…They were invented in the 1930s-1940s (De Bellescize, 1932Wendt & Fredentall, 1943) and then intensive studies of the theory and practice of PLLs were carried out (Viterbi, 1966;Lindsey, 1972;Gardner, 1979;Lindsey and Chie, 1981;Leonov et al, 1992;Encinas, 1993;Nash, 1994;Brennan, 1996;Stensby, 1997). One of the first applications of phase-locked loop (PLL) is related to the problems of data transfer by radio signal.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear Analysis and Design of Phase-Locked Loops

Леонов¹,

Kuznetsov²,

Seledzhi³

2009

Automation Control - Theory and Practice

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“…The non-local reduction method was originally developed to study the global stability and oscillations of differential equations for automatic control systems and synchronous electric machines (Gelig et al, 1978;Leonov et al, 1992;Yakubovich et al, 2004). For alternating current induction machines the elements of this method were introduced in (Leonov, 2004(Leonov, , 2006aLeonov and Kondrat'eva, 2009), where the limit load problem was considered.…”

Section: Introduction and The Structure Of The Workmentioning

confidence: 99%