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Understanding Injection Locking and Synchronization with Van der Pol-Like Self-sustained Oscillators

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Abstract

The exchange of energy between real systems can be better understood by the analysis of their associated differential equations. While numerical simulations offer limited insight of such real systems or interconnected self-sustained oscillators, representative electronic circuits can provide additional intuitive understanding. Representative electronic circuits allow the manipulation and analysis of various parameters in the system involved. Here, we present such a prototype consisting of a nonlinear Wien oscillator. It was tested by analyzing the frequency near synchronization during injection locking, the formation of a dead band and the synchronization between oscillators by using the envelope amplitude and instantaneous frequency formalism. The width of the dead band grows when either the injected current is increased or the oscillator stored energy is decreased. The sympathy between self-sustained oscillators was analyzed using the same tool to produce a general condition for energy transfer controlled by the frequency of the oscillators and the stored energy of each oscillator. Additionally, a finite network of different oscillators with adjustable coupling was explored to identify unlocked (different frequency), completely locked (same frequency and phase ratio), partially locked (same frequency but phase ratio changing with time) and mixed states. This analysis can be used to explore detailed interactions between a system and its peculiar manifestation near symmetry breaking.

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Acknowledgements

This paragraph of the first footnote will contain the date on which you submitted your paper for review. This work was supported by the program UNAM-DGAPA-PAPIIT, Grant number IN112017.

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Authors and Affiliations

  1. Centro de Física Aplicada y Tecnología Avanzada, UNAM Juriquilla, Boulevard Juriquilla 3001, 76230, Querétaro, Mexico

    Luis Enrique Beltran-Gonzalez & Rafael Quintero-Torres

  2. Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ontario, M5S 3G4, Canada

    Rafael Quintero-Bermudez

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  1. Luis Enrique Beltran-Gonzalez
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  2. Rafael Quintero-Bermudez
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  3. Rafael Quintero-Torres
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Correspondence to Rafael Quintero-Torres.

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Beltran-Gonzalez, L.E., Quintero-Bermudez, R. & Quintero-Torres, R. Understanding Injection Locking and Synchronization with Van der Pol-Like Self-sustained Oscillators. Circuits Syst Signal Process 39, 4775–4791 (2020). https://doi.org/10.1007/s00034-020-01403-z

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