Two implementations of fractional-order relaxation oscillators

Abstract

This work proposes general formulas for designing two different topologies of fractional-order relaxation oscillators. One topology contains an Operational Amplifier and the other one relies on an Operational Trans-Resistance Amplifier. The design procedure hinges on the general fractional-order natural and step responses of RC, which is proved in this work depending on Mittag Leffler function. The proposed topologies can be controlled to generate symmetrical and non-symmetrical square wave signals. They also benefit from the employment of fractional-order capacitors (FOCs), which makes it possible to obtain higher frequencies using simple components. Furthermore, these topologies are verified through numerical solutions, circuit simulations, and experimental implementations. This encourages the authors to build circuit emulators for the FOC, which is achieved by applying the Foster-I synthesizing technique to the Matsuda’s approximation of \(s^\alpha\).

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Acknowledgements

A preliminary version of this work is published in ICM 2018 conference [Fractional-Order Relaxation Oscillators Based on Op-Amp and OTRA]. Authors would like to thank Science and Technology Development Fund (STDF) for funding the project \(\#\) 25977 and Nile University for facilitating all procedures required to complete this study.

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Correspondence to Ahmed G. Radwan.

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Elwy, O., AbdelAty, A.M., Said, L.A. et al. Two implementations of fractional-order relaxation oscillators. Analog Integr Circ Sig Process 106, 421–432 (2021). https://doi.org/10.1007/s10470-020-01640-x

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Keywords

  • Relaxation oscillator
  • Fractional-order capacitor
  • Approximation
  • Matsuda
  • Fractional-order circuits