Electromagnetic-based derivation of fractional-order circuit theory - Publication - Bridge of Knowledge

Search

Electromagnetic-based derivation of fractional-order circuit theory

Abstract

In this paper, foundations of the fractional-order circuit theory are revisited. Although many papers have been devoted to fractional-order modelling of electrical circuits, there are relatively few foundations for such an approach. Therefore, we derive fractional-order lumped-element equations for capacitors, inductors and resistors, as well as Kirchhoff’s voltage and current laws using quasi-static approximations of fractional-order Maxwell’s equations. The proposed approach is not limited by the geometry of the considered lumped elements and employs the concepts of voltage and current known from the circuit theory. Finally, the proposed theory of circuit elements is applied to interpretation of Poynting’s theorem in fractional-order electromagnetism.

Citations

  • 2 2

    CrossRef

  • 0

    Web of Science

  • 2 5

    Scopus

Cite as

Full text

download paper
downloaded 279 times
Publication version
Accepted or Published Version
License
Creative Commons: CC-BY-NC-ND open in new tab

Keywords

Details

Category:
Articles
Type:
artykuł w czasopiśmie wyróżnionym w JCR
Published in:
Communications in Nonlinear Science and Numerical Simulation no. 79, pages 1 - 13,
ISSN: 1007-5704
Language:
English
Publication year:
2019
Bibliographic description:
Stefański T., Gulgowski J.: Electromagnetic-based derivation of fractional-order circuit theory// Communications in Nonlinear Science and Numerical Simulation. -Vol. 79, (2019), s.1-13
DOI:
Digital Object Identifier (open in new tab) 10.1016/j.cnsns.2019.104897
Bibliography: test
  1. J. R. Carson, Electromagnetic theory and the foundations of electric circuit theory, The Bell System Technical Journal 6 (1) (1927) 1-17. doi:10.1002/j.1538-7305.1927.tb00189.x. open in new tab
  2. H. A. Haus, J. R. Melcher, Electromagnetic fields and energy, Prentice Hall Englewood Cliffs, N.J, 1989.
  3. P. Russer, Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition (Artech House Anten- nas and Propagation Library), Artech House, Inc., Norwood, MA, USA, 2006. open in new tab
  4. G. Carlson, C. Halijak, Approximation of fractional capacitors (1/s) (1/n) by a regular newton process, IEEE Transactions on Circuit Theory 11 (2) (1964) 210-213. doi:10.1109/TCT.1964.1082270. open in new tab
  5. K. Steiglitz, An rc impedance approximant to s(-1/2), IEEE Transactions on Circuit Theory 11 (1) (1964) 160-161. doi:10.1109/TCT.1964.1082252. open in new tab
  6. T. Kaczorek, Positive linear systems consisting ofnsubsystems with different fractional orders, IEEE Transactions on Cir- cuits and Systems I: Regular Papers 58 (6) (2011) 1203-1210. doi:10.1109/TCSI.2010.2096111. open in new tab
  7. A. Shamim, A. G. Radwan, K. N. Salama, Fractional smith chart theory, IEEE Microwave and Wireless Components Letters 21 (3) (2011) 117- 119. doi:10.1109/LMWC.2010.2098861. open in new tab
  8. M. S. Sarafraz, M. S. Tavazoei, Realizability of fractional-order impedances by passive electrical networks composed of a frac- tional capacitor and rlc components, IEEE Transactions on Cir- cuits and Systems I: Regular Papers 62 (12) (2015) 2829-2835. doi:10.1109/TCSI.2015.2482340. open in new tab
  9. M. S. Sarafraz, M. S. Tavazoei, Passive realization of fractional-order impedances by a fractional element and rlc components: Conditions and procedure, IEEE Transactions on Circuits and Systems I: Regular Papers 64 (3) (2017) 585-595. doi:10.1109/TCSI.2016.2614249. open in new tab
  10. M. A. Moreles, R. Lainez, Mathematical modelling of fractional order circuit elements and bioimpedance applications, Communications in Nonlinear Science and Numerical Simulation 46 (2017) 81 -88. doi:https://doi.org/10.1016/j.cnsns.2016.10.020. URL http://www.sciencedirect.com/science/article/pii/S1007570416303598 open in new tab
  11. G. Liang, J. Hao, D. Shan, Electromagnetic interpretation of fractional- order elements, Journal of Modern Physics 8 (2017) 2209-2218. doi:doi: 10.4236/jmp.2017.814136. open in new tab
  12. R. Sikora, S. Paw lowski, Fractional derivatives and the laws of elec- trical engineering, COMPEL -The international journal for computa- tion and mathematics in electrical and electronic engineering 37 (4) (2018) 1384-1391. arXiv:https://doi.org/10.1108/COMPEL-08-2017- 0347, doi:10.1108/COMPEL-08-2017-0347. URL https://doi.org/10.1108/COMPEL-08-2017-0347 open in new tab
  13. K. J. Latawiec, R. Stanis lawski, M. Lukaniszyn, W. Czuczwara, M. Ry- del, Fractional-order modeling of electric circuits: modern empiricism vs. classical science, in: 2017 Progress in Applied Electrical Engineering (PAEE), 2017, pp. 1-4. doi:10.1109/PAEE.2017.8008998. open in new tab
  14. A. S. Elwakil, Fractional-order circuits and systems: An emerging inter- disciplinary research area, IEEE Circuits and Systems Magazine 10 (4) (2010) 40-50. doi:10.1109/MCAS.2010.938637. open in new tab
  15. M. D. Ortigueira, An introduction to the fractional continuous-time lin- ear systems: the 21st century systems, IEEE Circuits and Systems Mag- azine 8 (3) (2008) 19-26. doi:10.1109/MCAS.2008.928419. open in new tab
  16. V. E. Tarasov, Fractional vector calculus and fractional maxwell's equations, Annals of Physics 323 (11) (2008) 2756 -2778. doi:https://doi.org/10.1016/j.aop.2008.04.005. URL http://www.sciencedirect.com/science/article/pii/S0003491608000596 open in new tab
  17. M. D. Ortigueira, M. Rivero, J. J. Trujillo, From a generalised helmholtz decomposition theorem to fractional maxwell equations, Communications in Nonlinear Science and Numerical Simulation 22 (1) (2015) 1036 -1049. doi:https://doi.org/10.1016/j.cnsns.2014.09.004. URL http://www.sciencedirect.com/science/article/pii/S1007570414004481 open in new tab
  18. R. Ismail, A. G. Radwan, Rectangular waveguides in the fractional-order domain, in: 2012 International Conference on Engineering and Technol- ogy (ICET), 2012, pp. 1-6. doi:10.1109/ICEngTechnol.2012.6396151. open in new tab
  19. E. K. Jaradat, R. S. Hijjawi, J. M. Khalifeh, Maxwell's equations and electromagnetic lagrangian density in fractional form, Journal of Mathematical Physics 53 (3) (2012) 033505. arXiv:https://doi.org/10.1063/1.3670375, doi:10.1063/1.3670375. URL https://doi.org/10.1063/1.3670375 open in new tab
  20. D. Baleanu, A. K. Golmankhaneh, A. K. Golmankhaneh, M. C. Baleanu, Fractional electromagnetic equations using fractional forms, International Journal of Theoretical Physics 48 (11) (2009) 3114-3123. doi:10.1007/s10773-009-0109-8. URL https://doi.org/10.1007/s10773-009-0109-8 open in new tab
  21. N. Engheta, On fractional calculus and fractional multipoles in electro- magnetism, IEEE Transactions on Antennas and Propagation 44 (4) (1996) 554-566. doi:10.1109/8.489308. open in new tab
  22. N. Engheia, On the role of fractional calculus in electromagnetic the- ory, IEEE Antennas and Propagation Magazine 39 (4) (1997) 35-46. doi:10.1109/74.632994. open in new tab
  23. A C C E P T E D M A N U S C R I P T
  24. A. N. Bogolyubov, A. A. Potapov, S. S. Rehviashvili, An approach to introducing fractional integro-differentiation in classical electrody- namics, Moscow University Physics Bulletin 64 (4) (2009) 365-368. doi:10.3103/S0027134909040031. URL https://doi.org/10.3103/S0027134909040031 open in new tab
  25. V. E. Tarasov, Fractional integro-differential equations for electromag- netic waves in dielectric media, Theoretical and Mathematical Physics 158 (3) (2009) 355-359. doi:10.1007/s11232-009-0029-z. URL https://doi.org/10.1007/s11232-009-0029-z open in new tab
  26. V. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer-Verlag Berlin Heidel- berg, 2011. open in new tab
  27. H. Nasrolahpour, A note on fractional electrodynamics, Communica- tions in Nonlinear Science and Numerical Simulation 18 (9) (2013) 2589 -2593. doi:https://doi.org/10.1016/j.cnsns.2013.01.005. URL http://www.sciencedirect.com/science/article/pii/S1007570413000312 open in new tab
  28. K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, Academic Press, New York, 1974. open in new tab
  29. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. open in new tab
  30. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, 2006. open in new tab
  31. C. Li, F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, 2015. open in new tab
  32. S. Thevanayagam, Dielectric dispersion of porous media as a frac- tal phenomenon, Journal of Applied Physics 82 (5) (1997) 2538-2547. arXiv:https://doi.org/10.1063/1.366065, doi:10.1063/1.366065. URL https://doi.org/10.1063/1.366065 open in new tab
  33. B. Tellegen, A general network theorem, with applications, Philips Res. Rep. 7 (1952) 259-296. open in new tab
  34. A C C E P T E D M A N U S C R I P T
  35. P. Penfield, R. Spence, S. Duinker, A generalized form of tellegen's theorem, IEEE Transactions on Circuit Theory 17 (3) (1970) 302-305. doi:10.1109/TCT.1970.1083145. open in new tab
  36. J. Osiowski, J. Szabatin, Fundamentals of Circuit Theory (in Polish), Scientific and Technical Publishing House, Warsaw, 1995. open in new tab
  37. T. Itoh, Numerical techniques for microwave and millimeter-wave pas- sive structures / edited by Tatsuo Itoh, Wiley, New York, 1989.
  38. M. Fouda, A. Elwakil, A. Radwan, A. Allagui, Power and energy analysis of fractional-order electrical energy storage devices, Energy 111 (2016) 785 -792. doi:https://doi.org/10.1016/j.energy.2016.05.104. URL http://www.sciencedirect.com/science/article/pii/S036054421630723X open in new tab
Verified by:
Gdańsk University of Technology

seen 211 times

Recommended for you

Meta Tags