The half-plane pull-in range of a second-order phase-locked loop
type: Item
instance of: scholarly article
title: The half-plane pull-in range of a second-order phase-locked loop
zbMATH Open document ID: 0870.34042
DOI: 10.1016/0016-0032(96)00010-5
author: Bassam A. Harb
author: John L. Stensby
published in: Journal of the Franklin Institute
publication date: 1997-09-08T00:00:00Z
review text: The behaviour of an analog phase-locked loop (PLL), which is commonly used in instrumentation and communication systems, is governed by nonlinear differential equations. Being a frequency (phase) controlled (synchronised) system, the PLL is, among others, characterised by its pull-in range defined as the maximum value of frequency detuning \(\Delta \omega\) for which the phenomena of pull-in occurs. In this paper a definition is given for the PLL's half-plane pull-in range \(\Omega_2\). This definition implies that pull-in is guaranteed from anywhere in the phase plane's lower-half if \(0<\Delta \omega< \Omega_2\). It was shown that the PLL under consideration (second order type I) has an unstable separatrix cycle for \(\Delta \omega= \Omega_2\). A Galerkin-based algorithm was developed for computing the half-plane pull-in frequency. The algorithm determines the value \(\Delta \omega= \Omega_2\), at which the limit cycle bifurcates from the unstable separatrix cycle.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 902668
zbMATH Keywords: PLL
zbMATH Keywords: phase-locked loop
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: unstable separatrix cycle
zbMATH Keywords: limit cycle
MaRDI profile type: Publication
full work available at URL: 0016-0032(96)00010-5
OpenAlex ID: W2061212285
cites work: False Lock and Bifurcation in the Phase Locked Loop
cites work: A Floquet theory of the linear synchronous machine
cites work: Saddle node bifurcation at a nonhyperbolic limit cycle in a phase locked loop
cites work: Q3997917
cites work: Numerical computation of nonlinear forced oscillations by Galerkin's procedure
description: scientific article; zbMATH DE number 902668