Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large

IEEE Trans. Circuits Syst. I

Yuldashev

et al. 2015

113

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.

Section: Classical Nonlinear Mathematical Models Ofmentioning

confidence: 99%

“…4 There are examples of non odd PD characteristics, where (9) does not hold true (see, e.g. BPSK Costas loop with sawtooth signals [18], [19] and others).…”

Section: Classical Nonlinear Mathematical Models Ofmentioning

confidence: 99%

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

IEEE Trans. Circuits Syst. I

Yuldashev

et al. 2015

113

“…The averaging under certain conditions [6], [7], [9]- [11] and approximation ϕ(t) ≈ sin(θ 1 (t) − θ 2 (t)) allow one to proceed from the analysis of the signal space model to the study of PLL model in the signal's phase space. Rigorous consideration of this point is often omitted (see, e.g.…”

Section: Accepted To Ieee 7th International Congress On Ultra Modern mentioning

confidence: 99%

Limitations of PLL simulation: Hidden oscillations in MatLab and SPICE

Bianchi

2015 7th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)

et al. 2015

Abstract-Nonlinear analysis of the phase-locked loop (PLL) based circuits is a challenging task, thus in modern engineering literature simplified mathematical models and simulation are widely used for their study. In this work the limitations of numerical approach is discussed and it is shown that, e.g. hidden oscillations may not be found by simulation. Corresponding examples in SPICE and MatLab, which may lead to wrong conclusions concerning the operability of PLL-based circuits, are presented.

“…While Assumption 1 is reasonable from a practical point of view, its use in the analysis of Costas loop requires further consideration (see, e.g., [10], [11]). Various averaging methods (see, e.g., [12]) allow one to justify Assumption 1 and obtain conditions under which it can be used rigorously (see [13], [14]). …”

Section: Classical Engineering Consideration Of the Costas Loop Omentioning

confidence: 99%

“…To overcome these problems, in place of using Assumption 2 one can apply averaging methods [12], [14] and consider a simplified mathematical model in signal's phase space. However, this requires the consideration of a constant data signal (Assumption 3) and constant frequency of input carrier (7), i.e.…”

Section: Nonlinear Models Of Costas Loopmentioning

confidence: 99%

A short survey on nonlinear models of the classic Costas loop: Rigorous derivation and limitations of the classic analysis

Best¹,

2015 American Control Conference (ACC)

et al. 2015

Abstract-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 physical model.