The Poincaré–Andronov–Hopf bifurcation theory and its application to nonlinear analysis of RC phase‐shift oscillator
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Published:2023-09-25
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Volume:
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ISSN:0098-9886
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Container-title:International Journal of Circuit Theory and Applications
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language:en
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Short-container-title:Circuit Theory & Apps
Author:
Georgiev Zhivko D.1ORCID,
Uzunov Ivan M.23,
Todorov Todor G.4,
Trushev Ivan M.1
Affiliation:
1. Department of Theoretical Electrical Engineering Technical University of Sofia Sofia Bulgaria
2. Department of Applied Physics Technical University of Sofia Sofia Bulgaria
3. Institute of Mechanics Bulgarian Academy of Sciences Sofia Bulgaria
4. Department of Electronics Technical University of Sofia Sofia Bulgaria
Abstract
SummaryIn the paper, a nonlinear analysis of RC phase‐shift oscillator with operational amplifier is done. Using Kirchhoff's laws, a nonlinear system of three differential equations that describes the behavior of the oscillator is obtained. This system is analyzed using the Poincaré–Andronov–Hopf bifurcation theory. The basic principles of the Poincaré–Andronov–Hopf bifurcation theory are presented in advance, including the Center Manifold Theory and the Theory of Normal Forms. It was found that, under certain conditions, the oscillator system has a limit cycle, which means that the oscillator under consideration generates periodic oscillations. The period, amplitude, and stability of the generated periodic oscillations are determined theoretically. The obtained analytical results are confirmed by a numerical analysis of the nonlinear system of differential equations performed by MATLAB.
Subject
Applied Mathematics,Electrical and Electronic Engineering,Computer Science Applications,Electronic, Optical and Magnetic Materials