The terms hold-in, pull-in (capture), and lock-in ranges are widely used byengineers for the concepts of frequency deviation ranges within which PLL-basedcircuits can achieve lock under various additional conditions. Usually onlynon-strict definitions are given for these concepts in engineering literature.After many years of their usage, F.~Gardner in the 2nd edition of hiswell-known work, Phaselock Techniques, wrote "There is no natural way to defineexactly any unique lock-in frequency" and "despite its vague reality, lock-inrange is a useful concept." Recently these observations have led to thefollowing advice given in a handbook on synchronization and communications "Werecommend that you check these definitions carefully before using them." Inthis survey it is shown that, from a mathematical point of view, in some casesthe hold-in and pull-in "ranges" may not be the intervals of values but a unionof intervals and thus their widely used definitions require clarification.Rigorous mathematical definitions for the hold-in, pull-in, and lock-in rangesare given. An effective solution for the problem on the unique definition ofthe lock-in frequency, posed by Gardner, is suggested.
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