On Applications of Elements Modelled by Fractional Derivatives in Circuit Theory

Jacek Gulgowski, Tomasz P. Stefański and Damian Trofimowicz
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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

Energies, 2020, vol. 13, issue 21, 1-17

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.

Keywords: circuit theory; circuit simulation; transmission lines; fractional calculus (search for similar items in EconPapers)
JEL-codes: Q Q0 Q4 Q40 Q41 Q42 Q43 Q47 Q48 Q49 (search for similar items in EconPapers)
Date: 2020
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