A Collocation-Based Algorithm for Analyzing Bifurcations in Phase Locked Loops with Tanlock and Sawtooth Phase Detectors

Bassam A. Harb, Ahmad Al-Ajlouni and Ali Eyadeh

Mathematical Problems in Engineering, 2018, vol. 2018, 1-7

Abstract:

Analysis of bifurcation of second-order analog phase locked loop (PLL) with tanlock and sawtooth phase detectors is investigated. Both qualitative and quantitative analyses are carried out. Qualitatively, the basin boundaries of the attractors were constructed by plotting the stable and the unstable manifolds of the system. The basin boundaries show that the PLL under consideration for certain loop parameters has a separatrix cycle which terminates the limit cycle (out-of-lock state) and the loop pulls-in. This behavior is known in literature as homoclinic bifurcation and the value of the bifurcation parameter where this process occurs is called the pull-in range. Quantitatively, we propose a collocation-based algorithm to compute the separatrix cycle and the pull-in range. The separatrix cycle is approximated by a finite set of harmonics N with unknown amplitudes and by utilizing the fact that this limit cycle bifurcates from a separatrix cycle, a system of nonlinear algebraic equations is derived. For given values of filter parameters and gain, the algorithm numerically solves for the unknown amplitude of the harmonics and the value of the pull-in range simultaneously by evaluating the system at the collocation points. Results demonstrate that phase locked loop with sawtooth phase detector characteristics has the wider pull-in range followed by tanlock and sinusoidal, respectively.

Date: 2018
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/MPE/2018/8532546.pdf (application/pdf)
http://downloads.hindawi.com/journals/MPE/2018/8532546.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlmpe:8532546

DOI: 10.1155/2018/8532546

Access Statistics for this article

More articles in Mathematical Problems in Engineering from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().