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On Applications of Elements Modelled by Fractional Derivatives in Circuit Theory

Author

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  • Jacek Gulgowski

    (The Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-308 Gdansk, Poland)

  • Tomasz P. Stefański

    (The Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 80-233 Gdansk, Poland)

  • Damian Trofimowicz

    (SpaceForest Ltd., 81-451 Gdynia, Poland)

Abstract

In this paper, concepts of fractional-order (FO) derivatives are reviewed and discussed with regard to element models applied in the circuit theory. The properties of FO derivatives required for the circuit-level modeling are formulated. Potential problems related to the generalization of transmission-line equations with the use of FO derivatives are presented. It is demonstrated that some formulations of FO derivatives have limited applicability in the circuit theory. Out of the most popular approaches considered in this paper, only the Grünwald–Letnikov and Marchaud definitions (which are actually equivalent) satisfy the semigroup property and are naturally representable in the phasor domain. The generalization of this concept, i.e., the two-sided fractional Ortigueira–Machado derivative, satisfies the semigroup property, but its phasor representation is less natural. Other ideas (including the Riemann–Liouville and Caputo derivatives—with a finite or an infinite base point) seem to have limited applicability.

Suggested Citation

  • Jacek Gulgowski & Tomasz P. Stefański & Damian Trofimowicz, 2020. "On Applications of Elements Modelled by Fractional Derivatives in Circuit Theory," Energies, MDPI, vol. 13(21), pages 1-17, November.
    • Handle: RePEc:gam:jeners:v:13:y:2020:i:21:p:5768-:d:439748
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    References listed on IDEAS

    as
    1. P. V. Shah & A. D. Patel & I. A. Salehbhai & A. K. Shukla, 2014. "Analytic Solution for the Electric Circuit Model in Fractional Order," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-5, June.
    2. Jakubowska-Ciszek, A. & Walczak, J., 2018. "Analysis of the transient state in a parallel circuit of the class RLβCα," Applied Mathematics and Computation, Elsevier, vol. 319(C), pages 287-300.
    3. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    4. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
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