Rigorous mathematical definitions of the hold-in and pull-in ranges for phase-locked loops

Abstract Various ranges of frequency deviation are widely used by engineers to describe frequency deviations for which the PLL-based circuits achieve lock under some additional conditions. In engineering literature, as a rule, the non-rigorous explanations of such ranges are given. In this paper it is shown that the considered frequency deviation ranges need not be intervals and, strictly speaking, should be defined as sets. Rigorous mathematical definitions of hold-in and pull-in sets are given. Example, showing that the use of the term “range” need additional clarification for high order loop filters, is demonstrated.

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