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Frequency Characteristics of Dissipative and Generative Fractional RLC Circuits

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Abstract

Equations governing the transient- and steady-state regimes of the fractional series RLC circuits containing dissipative and/or generative capacitor and inductor are posed by considering the electric current as a response to electromotive force. Further, fractional RLC circuits are analyzed in the steady-state regime and their energy consumption/production properties are established depending on the angular frequency of electromotive force. Frequency characteristics of the modulus and argument of transfer function, i.e., of circuit’s equivalent admittance, are analyzed through the Bode diagrams for the whole frequency range, as well as for low and high frequencies employing the asymptotic expansions of transfer function modulus and argument.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. A. Allagui, A.S. Elwakil, M.E. Fouda, A.G. Radwan, Capacitive behavior and stored energy in supercapacitors at power line frequencies. J. Power Sources 390, 142–147 (2018)

    Article  Google Scholar 

  2. A. Allagui, T.J. Freeborn, A.S. Elwakil, M.E. Fouda, B.J. Maundy, A.G. Radwan, Z. Said, M.A. Abdelkareema, Review of fractional-order electrical characterization of supercapacitors. J. Power Sources 400, 457–467 (2018)

    Article  Google Scholar 

  3. A. Allagui, D. Zhang, A.S. Elwakil, Short-term memory in electric double-layer capacitors. Appl. Phys. Lett. 113, 253901-1–5 (2018)

    Article  Google Scholar 

  4. M.C. Bošković, T.B. Šekara, B. Lutovac, M. Daković, P.D. Mandić, M.P. Lazarević. Analysis of electrical circuits including fractional order elements, in 6th Mediterranean Conference on Embedded Computing (MECO), Bar, Montenegro (2017)

  5. A. Buscarino, R. Caponetto, S. Graziani, E. Murgano, Realization of fractional order circuits by a constant phase element. Eur. J. Control 54, 64–72 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Caponetto, S. Graziani, E. Murgano, Realization of a fractional-order RLC circuit via constant phase element. Int. J. Dyn. Control 9, 1589–1599 (2021)

    Article  Google Scholar 

  7. X. Chen, Y. Chen, B. Zhang, D. Qiu, A modeling and analysis method for fractional-order DC-DC converters. IEEE Trans. Power Electron. 32, 7034–7044 (2017)

    Article  Google Scholar 

  8. J.M. Cruz-Duarte, M. Guía-Calderón, J.J. Rosales-García, R. Correa, Determination of a physically correct fractional-order model for electrolytic computer-grade capacitors. Math. Methods Appl. Sci. 44, 4366–4380 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Dzieliński, G. Sarwas, D. Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models. Adv. Differ. Equ. 2011(11), 1–15 (2011)

    MathSciNet  MATH  Google Scholar 

  10. O. Elwy, L.A. Said, A.H. Madian, A.G. Radwan, All possible topologies of the fractional-order Wien oscillator family using different approximation techniques. Circuits Syst. Signal Process. 38, 3931–3951 (2019)

    Article  Google Scholar 

  11. M.E. Fouda, A. Allagui, A.S. Elwakil, S. Das, C. Psychalinos, A.G. Radwan, Nonlinear charge-voltage relationship in constant phase element. Int. J. Electron. Commun. AEÜ 117, 153104-1–4 (2020)

    Article  Google Scholar 

  12. R. Garrappa, E. Kaslik, M. Popolizio, Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial. Mathematics 7, 407-1–21 (2019)

    Article  Google Scholar 

  13. F. Gómez, J. Rosales, M. Guía, \({RLC}\) electrical circuit of non-integer order. Cent. Eur. J. Phys. 11, 1361–1365 (2013)

    Google Scholar 

  14. J.F. Gómez-Aguilar, R. Razo-Hernández, D. Granados-Lieberman, A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response. Revista Mexicana de Física 60, 32–38 (2014)

    MathSciNet  Google Scholar 

  15. M. Guía, J. Rosales, F. Gómez, Analysis on the time and frequency domain for the \({RC}\) electric circuit of fractional order. Cent. Eur. J. Phys. 11, 1366–1371 (2013)

    Google Scholar 

  16. K. Haška, S.M. Cvetićanin, D. Zorica, Dissipative and generative fractional electric elements in modeling \(RC\) and \(RL\) circuits. Nonlinear Dyn. 105, 3451–3474 (2021)

    Article  Google Scholar 

  17. K. Haška, D. Zorica, S.M. Cvetićanin, Fractional \(RLC\) circuit in transient and steady state regimes. Commun. Nonlinear Sci. Numer. Simul. 96, 105670-1–17 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. J.I. Hidalgo-Reyes, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, V.M. Alvarado-Martínez, M.G. López-López, Classical and fractional-order modeling of equivalent electrical circuits for supercapacitors and batteries, energy management strategies for hybrid systems and methods for the state of charge estimation: a state of the art review. Microelectron. J. 85, 109–128 (2019)

    Article  Google Scholar 

  19. J.I. Hidalgo-Reyes, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, V.M. Alvarado-Martínez, M.G. López-López, Determination of supercapacitor parameters based on fractional differential equations. Int. J. Circuit Theory Appl. 47, 1225–1253 (2019)

    Google Scholar 

  20. A. Jakubowska, J. Walczak, Analysis of the transient state in a circuit with supercapacitor. Poznan Univ. Technol. Acad. J. Electr. Eng. 81, 71–77 (2015)

    Google Scholar 

  21. A. Jakubowska, J. Walczak, Analysis of the transient state in a series circuit of the class \({R}{L}_{\beta }{C}_{\alpha }\). Circuits Syst. Signal Process. 35, 1831–1853 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Jakubowska-Ciszek, J. Walczak, Analysis of the transient state in a parallel circuit of the class \({R}{L}_{\beta }{C}_{\alpha }\). Appl. Math. Comput. 319, 287–300 (2018)

    MATH  Google Scholar 

  23. I.S. Jesus, J.A.T. Machado, Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn. 56, 45–55 (2009)

    Article  MATH  Google Scholar 

  24. Y. Jiang, B. Zhang, X. Shu, Z. Wei, Fractional-order autonomous circuits with order larger than one. J. Adv. Res. 25, 217–225 (2020)

    Article  Google Scholar 

  25. D.A. John, K. Biswas, Electrical equivalent circuit modelling of solid state fractional capacitor. Int. J. Electron. Commun. AEÜ 78, 258–264 (2017)

    Article  Google Scholar 

  26. A. Kartci, A. Agambayev, N. Herencsar, K.N. Salama, Series-, parallel-, and inter-connection of solid-state arbitrary fractional-order capacitors: theoretical study and experimental verification. IEEE Access 6, 10933–10943 (2018)

    Article  Google Scholar 

  27. M.M. Khader, J.F. Gómez-Aguilar, M. Adel, Numerical study for the fractional RL, RC, and RLC electrical circuits using Legendre pseudo-spectral method. Int. J. Circuit Theory Appl. 49, 3266–3285 (2021)

    Article  Google Scholar 

  28. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  29. M.S. Krishna, S. Das, K. Biswas, B. Goswami, Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization. IEEE Trans. Electron Devices 58, 4067–4073 (2011)

    Article  Google Scholar 

  30. J.A.T. Machado, A.M.S.F. Galhano, Fractional order inductive phenomena based on the skin effect. Nonlinear Dyn. 68, 107–115 (2012)

    Article  MathSciNet  Google Scholar 

  31. V. Martynyuk, M. Ortigueira, Fractional model of an electrochemical capacitor. Signal Process. 107, 355–360 (2015)

    Article  Google Scholar 

  32. V. Martynyuk, M. Ortigueira, M. Fedula, O. Savenko, Methodology of electrochemical capacitor quality control with fractional order model. Int. J. Electron. Commun. AEÜ 91, 118–124 (2018)

    Article  Google Scholar 

  33. D. Mondal, K. Biswas, Packaging of single-component fractional order element. IEEE Trans. Device Mater. Reliab. 13, 73–80 (2013)

    Article  Google Scholar 

  34. V.F. Morales-Delgado, J.F. Gómez-Aguilar, M.A. Taneco-Hernández, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense. Int. J. Electron. Commun. AEÜ 85, 108–117 (2018)

    Article  Google Scholar 

  35. V.F. Morales-Delgado, J.F. Gómez-Aguilar, M.A. Taneco-Hernández, R.F. Escobar-Jiménez, Fractional operator without singular kernel: applications to linear electrical circuits. Int. J. Circuit Theory Appl. 46, 2394–2419 (2018)

    Article  Google Scholar 

  36. M.A. Moreles, R. Lainez, Mathematical modelling of fractional order circuit elements and bioimpedance applications. Commun. Nonlinear Sci. Numer. Simul. 46, 81–88 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. K. Nosrati, M. Shafiee, On the convergence and stability of fractional singular Kalman filter and Riccati equation. J. Frankl. Inst. Eng. Appl. Math. 357, 7188–7210 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. A.V. Oppenheim, A.S. Willsky, S.H. Nawab, Signals and Systems. Prentice-Hall Signal Processing Series (Prentice-Hall, Hoboken, 1997)

    Google Scholar 

  39. M.D. Ortigueira, D. Valério, Fractional Signals and Systems, volume 7 of Fractional Calculus in Applied Sciences and Engineering (de Gruyter, Berlin, 2020)

    Book  MATH  Google Scholar 

  40. R. Prasad, K. Kothari, U. Mehta, Flexible fractional supercapacitor model analyzed in time domain. IEEE Access 7, 122626–122633 (2019)

    Article  Google Scholar 

  41. R. Prasad, U. Mehta, K. Kothari, Various analytical models for supercapacitors: a mathematical study. Resour. Effic. Technol. 1, 1–15 (2020)

    Google Scholar 

  42. J.J. Quintana, A. Ramos, I. Nuez, Modeling of an EDLC with fractional transfer functions using Mittag-Leffler equations. Math. Probl. Eng. 2013, 807037-1–7 (2013)

    Article  Google Scholar 

  43. A.G. Radwan, Resonance and quality factor of the \({R}{L}_{\alpha }{C}_{\alpha }\) fractional circuit. IEEE J. Emerg. Sel. Top. Circuits Syst. 3, 377–385 (2013)

    Article  Google Scholar 

  44. A.G. Radwan, M.E. Fouda, Optimization of fractional-order \({RLC}\) filters. Circuits Syst. Signal Process. 32, 2097–2118 (2013)

    Article  MathSciNet  Google Scholar 

  45. A.G. Radwan, K.N. Salama, Passive and active elements using fractional \({L}_{\beta }{C}_{\alpha }\) circuit. IEEE Trans. Circuits Syst. I Regul. Pap. 58, 2388–2397 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. A.G. Radwan, K.N. Salama, Fractional-order \({RC}\) and \({RL}\) circuits. Circuits Syst. Signal Process. 31, 1901–1915 (2012)

    Article  MathSciNet  Google Scholar 

  47. A.G. Radwan, A.M. Soliman, A.S. Elwakil, Design equations for fractional-order sinusoidal oscillators: four practical circuit examples. Int. J. Circuit Theory Appl. 36, 473–492 (2008)

    Article  MATH  Google Scholar 

  48. M.S. Sarafraz, M.S. Tavazoei, Realizability of fractional-order impedances by passive electrical networks composed of a fractional capacitor and \({RLC}\) components. IEEE Trans. Circuits Syst. I Regul. Pap. 62, 2829–2835 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. I. Schäfer, K. Krüger, Modelling of coils using fractional derivatives. J. Magn. Magn. Mater. 307, 91–98 (2006)

    Article  Google Scholar 

  50. N. Sene, J.F. Gómez-Aguilar, Analytical solutions of electrical circuits considering certain generalized fractional derivatives. Eur. Phys. J. Plus 134, 260-1–14 (2019)

    Google Scholar 

  51. Z.M. Shah, M.Y. Kathjoo, F.A. Khanday, K. Biswas, C. Psychalinos, A survey of single and multi-component fractional-order elements (FOEs) and their applications. Microelectron. J. 84, 9–25 (2019)

    Article  Google Scholar 

  52. M. Sowa, A subinterval-based method for circuits with fractional order elements. Bull. Pol. Acad. Sci. Tech. Sci. 62, 449–454 (2014)

    Google Scholar 

  53. M. Sowa, “gcdAlpha’’—a semi-analytical method for solving fractional state equations. Poznan Univ. Technol. Acad. J. Electr. Eng. 96, 231–242 (2018)

    Google Scholar 

  54. T.P. Stefański, J. Gulgowski, Electromagnetic-based derivation of fractional-order circuit theory. Commun. Nonlinear Sci. Numer. Simul. 79, 104897-1–13 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  55. T.P. Stefański, J. Gulgowski, Signal propagation in electromagnetic media described by fractional-order models. Commun. Nonlinear Sci. Numer. Simul. 82, 105029-1–16 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  56. R. Süße, A. Domhardt, M. Reinhard, Calculation of electrical circuits with fractional characteristics of construction elements. Forsch. Ingenieurwes. 69, 230–235 (2005)

    Article  Google Scholar 

  57. M.S. Tavazoei, Passively realizable approximations of non-realizable fractional order impedance functions. J. Frankl. Inst. Eng. Appl. Math. 357, 7037–7053 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  58. J. Walczak, A. Jakubowska, Resonance in series fractional order \({R}{L}_{\beta }{C}_{\alpha }\) circuit. Przegląd Elektrotechniczny 90, 210–213 (2014)

    Google Scholar 

  59. S. Westerlund, L. Ekstam, Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1, 826–839 (1994)

    Article  Google Scholar 

  60. B. Zhang, X. Shu, Fractional-Order Electrical Circuit Theory. CPSS Power Electronics Series (Springer, Singapore, 2022)

    Book  Google Scholar 

  61. L. Zhou, Z. Tan, Q. Zhang, A fractional-order multifunctional \(n\)-step honeycomb \(RLC\) circuit network. Front. Inf. Technol. Electron. Eng. 18, 1186–1196 (2017)

    Article  Google Scholar 

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Acknowledgements

This work is supported by the Serbian Ministry of Science, Education and Technological Development under grants 451-03-68/2022-14/200156 (SMC), and 451-03-68/2022-14/200125 (DZ).

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Authors and Affiliations

  1. Department of Power, Electronic and Telecommunication Engineering, Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, Novi Sad, 21000, Serbia

    Kristian Haška & Stevan M. Cvetićanin

  2. Department of Physics, Faculty of Sciences, University of Novi Sad, Trg D. Obradovića 4, Novi Sad, 21000, Serbia

    Dušan Zorica

  3. Mathematical Institute, Serbian Academy of Arts and Sciences, Kneza Mihaila 36, Beograd, 11000, Serbia

    Dušan Zorica

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  1. Kristian Haška
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Correspondence to Stevan M. Cvetićanin.

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Haška, K., Zorica, D. & Cvetićanin, S.M. Frequency Characteristics of Dissipative and Generative Fractional RLC Circuits. Circuits Syst Signal Process 41, 4717–4754 (2022). https://doi.org/10.1007/s00034-022-02025-3

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