Phase-Locked Loop Pull-In Frequency

Abstract

A computerized procedure for obtaining the pull-in frequency of a phase-locked loop is described. The procedure, which consists of solving for the limit cycle of the out-of-lock loop and finding the frequency offset below which no solution exists, is quite general with respect to phase detector function and loop filter. It is applied in this study to the case of a second-order loop with a sinusoidal phase detector. The results of the study give normalized pull-in frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> as a function of the damping factor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\zeta</tex> and normalized natural frequency β of the linearized closed loop. A review of several previously published investigations of this case is given, and the limitations identified point the way to the more exact solution described here. In addition, an experimental study is described which produces excellent agreement with the new results, and a curve-fitting procedure is outlined which leads to a highly accurate functional description of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> in terms of β and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\zeta</tex> .