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

Abstract

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.

Limitations of PLL simulation: Hidden oscillations in MatLab and SPICE

“…It is shown that the use of default simulation parameters for the study of twophase PLL in MatLab and SIMULINK can lead to wrong conclusions concerning the operability of the loop, e.g. the pull-in (or capture) range (see discussion of rigorous definitions in [17], [18]). …”

Robust stability analysis and improved design of phase‐locked loops with non‐monotonic nonlinearities: LMI‐based approach

“…Unlike several recent works where rigorous analyses are established for low order PLLs and PLLs with no integrators in the filters [10,12,16,34], the analysis presented in this paper is more general in the sense that not only it is applicable to both PLLs with analog mixer and XOR gate but also to PLLs without any order or type restriction. Unlike several recent works where rigorous analyses are established for low order PLLs and PLLs with no integrators in the filters [10,12,16,34], the analysis presented in this paper is more general in the sense that not only it is applicable to both PLLs with analog mixer and XOR gate but also to PLLs without any order or type restriction.…”

“…The tracking behavior as well as the stability of such a system is largely dependent on the phase detector (PD) characteristics. While this is convenient during the initial phases of modern PLL designs, the linear assumption of the PD is only valid when the phase difference between the output and input remains sufficiently small [9], [10]. While this is convenient during the initial phases of modern PLL designs, the linear assumption of the PD is only valid when the phase difference between the output and input remains sufficiently small [9], [10].…”

Hidden oscillations in SPICE simulation of two-phase Costas loop with non-linear VCO

“…4 Next section demonstrates that simulation of two-phase Costas loop in SPICE with default simulation parameters may lead to wrong conclusions concerning the pull-in range and lock-in range (see rigorous definitions in Kuznetsov et al (2015b); Leonov et al (2015b)). …”