Analysis of Parallel Resonance Circuit Consisting of a Capacitor Modelled Using Conformal Fractional Order Derivative Using Simulink

Öz

Fractional order circuit elements are becoming an essential part of modelling electronics and mechanic systems in the last few decades. Recently, conformable fractional derivate (CFD) has been introduced in literature. It is easier to understand and use than the other Fractional order derivatives and it is already being used to model circuit components. CFD has already been used to model supercapacitors in literature. It is important to know how circuits behave for different types of current and voltage waveforms. In this paper, a parallel circuit which consists of a resistor, an inductor and a CFD capacitor fed by a current source has been examined for DC and sinusoidal signals using SimulinkTM toolbox of MatlabTM since it was not possible to find analytical solutions of the circuit. Simulink provides an easier way to examine such circuits.

Anahtar Kelimeler

• Fractional Order Derivative
• Conformable Fractional Derivative
• Circuit Analysis
• Circuit Theory
• Circuit Modeling

Öz

Kesirli dereceli devre elemanları, son birkaç on yılda, elektronik ve mekanik sistemlerin modellenmesinin önemli bir parçası haline gelmeye başlamıştır. Son zamanlarda, uyumlu kesirli türev (CFD) literatürde ortaya çıkmıştır. Anlaşılması ve kullanılması diğer kesirli dereceli türevlerden daha kolaydır ve devre bileşenlerini modellemek için çoktan kullanılmaktadır. CFD literatürde süper kondansatörleri modellemek için hâlihazırda kullanılmıştır. Devrelerin farklı akım ve gerilim dalga biçimleri için nasıl davrandığını bilmek önemlidir. Bu makalede, bir akım kaynağı tarafından beslenen bir direnç, bir indüktör ve bir CFD kapasitörün paralel bağlanmasından oluşan devre, devrenin analitik çözümünü bulmak mümkün olmadığından dolayı, MatlabTM’in SimulinkTM alt programı ile DC ve sinüzoidal sinyaller kullanılarak incelenmiştir. Simulink bu devreleri incelemek için kolay bir yol sağlamaktadır.

Anahtar Kelimeler

• Kesirli dereceli Türev
• Uyumlu Kesirli Türev
• Devre Analizi
• Devre Teorisi
• Devre Modellemesi

Kaynakça

1. [1] 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.
2. [2] Yang, X. J. (2019). General fractional derivatives: theory, methods and applications. Chapman and Hall/CRC.
3. [3] Ross, B. (1977). The development of fractional calculus 1695–1900. Historia Mathematica, 4(1), 75-89.
4. [4] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006.
5. [5] Babiarz, A., Czornik, A., Klamka, J., & Niezabitowski, M. (2017). Theory and applications of non-integer order systems. Lecture Notes Electrical Engineering, 407.
6. [6] 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.
7. [7] Moreles, M. A., & Lainez, R. (2016). Mathematical modelling of fractional order circuits. arXiv preprint arXiv:1602.03541.
8. [8] Adhikary, A., Khanra, M., Pal, J., & Biswas, K. (2017). Realization of fractional order elements. Inae Letters, 2(2), 41-47.

9. [9] 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.
10. [10] 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.
11. [11] 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.
12. [12] Alagoz, B. B., & Alisoy, H. Z. (2014). On the Harmonic Oscillation of High-order Linear Time Invariant Systems. Scientific Committee.
13. [13] Tariboon, J., & Ntouyas, S. K. (2016). Oscillation of impulsive conformable fractional differential equations. Open Mathematics, 14(1), 497-508.
14. [14] 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.
15. [15] Khalil, R.; al Horani, M.; Yousef, A.; Sababheh, M. A new definition of fractional derivatuive. J. Comput. Appl. Math. 2014, 264, 65–70.
16. [16] Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
17. [17] Zhao, D., & Luo, M. (2017). General conformable fractional derivative and its physical interpretation. Calcolo, 54(3), 903-917.
18. [18] Sikora, R. (2017). Fractional derivatives in electrical circuit theory–critical remarks. Archives of Electrical Engineering, 66(1), 155-163.
19. [19] 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.
20. [20] Kopka, R. (2017). Estimation of supercapacitor energy storage based on fractional differential equations. Nanoscale research letters, 12(1), 636. Data alaınan makale
21. [21] 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.
22. [22] 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.
23. [23] 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.
24. [24] 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.
25. [25] 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).
26. [26] 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.
27. [27] 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.
28. [28] PALAZ U., MUTLU R., Analysis of a Capacitor Modelled with Conformable Fractional Derivative Under DC and Sinusoidal Signals. Submitted to Sakarya University Journal of Science.