Control of Bifurcation and Chaos in a Class of Digital Tanlock Loops with Modified Loop Structure

Sarkar, B. C.; Sarkar, Saumendra Sankar De; Banerjee, Tanmoy

doi:10.1142/s0218127414300146

Cited by 1 publication

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“…The idea of synchronization in coupled phase locked loop was investigated by Buckwalter et al [21] using harmonic balance technique. Digital phase locked loop with tanlock phase detector characteristics has gained a great attention in literature but less effort has been paid to analog PLL with tanlock and sawtooth phase detectors [22][23][24][25][26][27]. In this paper, a collocationbased algorithm is used here to analyze bifurcations and computing the pull-in range of a second-order, Type II phase locked loop with tanlock and sawtooth phase detector characteristics.…”

Section: Introductionmentioning

confidence: 99%

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

Harb

Al-Ajlouni

Eyadeh

2018

Mathematical Problems in Engineering

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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.

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