Analysis of a Capacitor Modelled with Conformable Fractional Derivative Under DC and Sinusoidal Signals

Yıl 2021, Cilt: 17 Sayı: 2, 193 - 198, 28.06.2021

Utku Palaz , Reşat Mutlu

https://doi.org/10.18466/cbayarfbe.757813

Cited By: 5

Öz

Fractional order circuit elements are successfully used to model circuits and systems in the last few decades. There are different types of fractional derivatives. Recently, another one named “the conformable fractional derivative” (CFD) has been introduced and shown to give good results for modeling supercapacitors. It is imperative to know how circuit elements behave for different current and voltage waveforms in circuit theory so that they can be exploited at their full potential. A CFD capacitor is not a well-known element and its usage and circuit solutions are rarely addressed in literature. In this study, it is examined how a CFD capacitor would behave under DC and AC excitations when it is fed by not only a current source but also a voltage source.

Anahtar Kelimeler

Fractional Order Derivatives, Circuit Analysis, Circuit Theory, Energy Analysis, Circuit Modeling

Kaynakça

• Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.
• Yang, X. J. (2019). General fractional derivatives: theory, methods and applications. Chapman and Hall/CRC.
• Ross, B. (1977). The development of fractional calculus 1695–1900. Historia Mathematica, 4(1), 75-89.
• Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006
• Babiarz, A., Czornik, A., Klamka, J., & Niezabitowski, M. (2017). Theory and applications of non-integer order systems. Lecture Notes Electrical Engineering, 407.
• Moreles, M. A., & Lainez, R. (2016). Mathematical modelling of fractional order circuits. arXiv preprint arXiv:1602.03541.
• Freeborn, T. J. (2013). A survey of fractional-order circuit models for biology and biomedicine. IEEE Journal on emerging and selected topics in circuits and systems, 3(3), 416-424.
• Adhikary, A., Khanra, M., Pal, J., & Biswas, K. (2017). Realization of fractional order elements. Inae Letters, 2(2), 41-47.
• Tsirimokou, G., Kartci, A., Koton, J., Herencsar, N., & Psychalinos, C. (2018). Comparative study of discrete component realizations of fractional-order capacitor and inductor active emulators. Journal of Circuits, Systems and Computers, 27(11), 1850170.
• Kartci, A., Agambayev, A., Herencsar, N., & Salama, K. N. (2018). Series-, parallel-, and inter-connection of solid-state arbitrary fractional-order capacitors: theoretical study and experimental verification. IEEE Access, 6, 10933-10943.
• Sotner, R., Jerabek, J., Kartci, A., Domansky, O., Herencsar, N., Kledrowetz, V., ... & Yeroglu, C. (2019). Electronically reconfigurable two-path fractional-order PI/D controller employing constant phase blocks based on bilinear segments using CMOS modified current differencing unit. Microelectronics Journal, 86, 114-129.
• Podlubny, I., Petráš, I., Vinagre, B. M., O'leary, P., & Dorčák, Ľ. (2002). Analogue realizations of fractional-order controllers. Nonlinear dynamics, 29(1-4), 281-296.
• Alagoz, B. B., & Alisoy, H. Z. (2014). On the Harmonic Oscillation of High-order Linear Time Invariant Systems. Scientific Committee.
• Alagöz, B. B., & Alisoy, H. Estimation of Reduced Order Equivalent Circuit Model Parameters of Batteries from Noisy Current and Voltage Measurements. Balkan Journal of Electrical and Computer Engineering, 6(4), 224-231.
• Khalil, R.; al Horani, M.; Yousef, A.; Sababheh, M. A new definition of fractional derivatuive. J. Comput. Appl. Math. 2014, 264, 65–70.
• Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
• Zhao, D., & Luo, M. (2017). General conformable fractional derivative and its physical interpretation. Calcolo, 54(3), 903-917.
• Sikora, R. (2017). Fractional derivatives in electrical circuit theory–critical remarks. Archives of Electrical Engineering, 66(1), 155-163.
• Lewandowski, M., & Orzyłowski, M. (2017). Fractional-order models: The case study of the supercapacitor capacitance measurement. Bulletin of the Polish Academy of Sciences Technical Sciences, 65(4), 449-457.
• Kopka, R. (2017). Estimation of supercapacitor energy storage based on fractional differential equations. Nanoscale research letters, 12(1), 636. Data alaınan makale
• Freeborn, T. J., Elwakil, A. S., & Allagui, A. (2018, May). Supercapacitor fractional-order model discharging from polynomial time-varying currents. In 2018 IEEE International Symposium on Circuits and Systems (ISCAS) (pp. 1-5). IEEE.
• Freeborn, T. J., Maundy, B., & Elwakil, A. S. (2013). Measurement of supercapacitor fractional-order model parameters from voltage-excited step response. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 367-376.
• Tariboon, J., & Ntouyas, S. K. (2016). Oscillation of impulsive conformable fractional differential equations. Open Mathematics, 14(1), 497-508.
• Piotrowska, E., & Rogowski, K. (2017, October). Analysis of fractional electrical circuit using Caputo and conformable derivative definitions. In Conference on Non-integer Order Calculus and Its Applications (pp. 183-194). Springer, Cham.
• Piotrowska, Ewa. "Analysis of fractional electrical circuit with sinusoidal input signal using Caputo and conformable derivative definitions." Poznan University of Technology Academic Journals. Electrical Engineering (2019).
• Morales-Delgado, V. F., Gómez-Aguilar, J. F., & Taneco-Hernandez, M. A. (2018). Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense. AEU-International Journal of Electronics and Communications, 85, 108-117.
• Martínez, L., Rosales, J. J., Carreño, C. A., & Lozano, J. M. (2018). Electrical circuits described by fractional conformable derivative. International Journal of Circuit Theory and Applications, 46(5), 1091-1100.
• Piotrowska, E. (2018, October). Analysis the conformable fractional derivative and Caputo definitions in the action of an electric circuit containing a supercapacitor. In Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2018 (Vol. 10808, p. 108081T). International Society for Optics and Photonics.
• Morales-Delgado, V. F., Gómez-Aguilar, J. F., & Taneco-Hernandez, M. A. (2018). Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense. AEU-International Journal of Electronics and Communications, 85, 108-117.
• Gómez-Aguilar, J. F. (2018). Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels. The European Physical Journal Plus, 133(5), 197.
• https://www.wolframalpha.com/input/?i=cos%28omega*t%2Bphi%29*%28t%5E%28a-1%29%29