Lineer Zamanla Değişmeyen Endüktörlü ve Uyumlu Kesirli Dereceli Türev ile Modellenmiş Kondansatörlü Bir Salınım Devresinin Analizi

Yıl 2022, Cilt: 5 Sayı: 1, 22 - 28, 31.07.2022

Mendi Arapi , Reşat Mutlu

https://doi.org/10.55581/ejeas.1126234

Cited By: 1

Öz

Kesirli dereceli devre elemanları araştırmacılar tarafından aralıksız olarak incelenmektedir. Literatürde giderek daha popüler hale gelmekteler. Son on yılda Uyumlu Kesirli Türev önerilmiş ve oldukça önem kazanmıştır. Bir LC tank devresinin veya bir LC osilatörünün incelenmesi, neredeyse tüm lisans fizik kitaplarında bulunabilir. Kesirli mertebeden kondansatör devreleri üzerine çok sayıda çalışma var olmasına rağmen, bildiğimiz kadarıyla, doğrusal zamanla değişmeyen bir endüktörden ve Uyumlu Kesirli Türev ile modellenmiş bir süper kondansatörden yapılmış bir osilatörün incelenmesi literatürde bulunmamaktadır. Bu makalede, literatürde ilk defa Uyumlu Kesirli Türev ile modellenmiş bir süper kondansatör ve lineer zamanla değişmeyen bir endüktör içeren kayıpsız bir osilatör devresi incelenmiştir. Devrenin doğal tepkisi analitik olarak bulunmuştur. Devrenin davranışı, farklı başlangıç koşulları için benzetimlerle gösterilmiştir.

Anahtar Kelimeler

Devre Analizi, Doğal Yanıt, Kesirli Dereceden Devreler, Salınım devresi, Süperkondansatör, Uyumlu Kesirli Türev.

Analysis of an Oscillation Circuit with a Linear Time-invariant Inductor and a Capacitor Modelled with Conformal Fractional Order Derivative

Öz

Fractional order circuit elements are being examined by researchers unremittingly. They are ever becoming more popular in the literature. The Conformable Fractional Derivative has been proposed and gained importance in the last decade. Examination of an LC tank circuit or an LC oscillator can be found in almost all undergrad physics books. There’s a considerable number of studies on fractional-order capacitor circuits but, to the best of our knowledge, examination of an oscillator made of a linear time-invariant inductor and a supercapacitor modeled with Conformable Fractional Derivative has not been found in literature. In this paper, a lossless oscillator circuit containing a linear time-invariant inductor and a supercapacitor modeled with Conformable Fractional Derivative is examined for the first time in the literature. Natural response of the circuit has been found analytically. Its behavior has been illustrated with simulations for different initial conditions.

Anahtar Kelimeler

Circuit analysis, Conformable Fractional Derivative, Fractional Order Circuit, Natural response, Oscillation circuit, Supercapacitor.

Kaynakça

• Mainardi. F. (2018). Fractional Calculus: Theory and applications. Mathematics ,6(9),145.
• Oldham. B. K., Spainer. J. (1974). Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press.
• Ross.B. B. (1977). “The development of fractional calculus 1695–1900”, Historia Mathematica, vol. 4, no.1, pp. 75-89.
• 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. J. X. (2019). General fractional derivatives: theory, methods, and applications, Chapman and Hall/CRC.
• Kilbas. A . A., H. M. Srivastava, J. J. Trujillo. (2006). Theory and Applications of Fractional Differential Equations, Elsevier.
• Hilfer. R. (2000). Application of Fractional Calculus in Physics. World Scientific Publishing.
• Amirian. M., Jamali. Y. (2017). The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A primer.
• Vosika. B. Z., Lazovic M. G., Misevic. N. G., SIMIC. B. J. (2013) Fractional Calculus Model of Electrical Impedance Applied to Human Skin, Vol. 8, Issue 4.
• Atangana. A. (2018). Fractional Operators with constant and variable order with Application to Geo Hydrology, (2018).
• Khalil. R.,Horani. A. M.,Yousef. A.,Sababheh. M.(2014). “A new definition of fractional derivative,” J. Comput. Appl. Math., vol. 264, pp. 65–70.
• Abdeljawad. T .(2015). “On conformable fractional calculus,” Journal of computational and Applied Mathematics, vol. 279, pp. 57-66.
• Kahouli. O., Elloumi. M., Naifar. O., Alsaif. H., Kahouli. B., Bouteraa. Y. (2022). Electrical Circuits Described by General Fractional Conformable Derivative. Vol. 10 Article 851070.
• Piotrowska. E. (2019). Analysis of fractional Electrical Circuit with sinusoidal input signal using Caputo and Conformable Derivative definitions, No. 97, 155-167.
• Piotrowska. E. (2018). “Analysis the conformable fractional derivative and Caputo definitions in the action of an electric circuit containing a supercapacitor,” Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, vol. 10808, p. 108081T, International Society for Optics and Photonics.
• Banchuin. R., Tan. W. K. A. (2021). On The Fractional Domain Analysis of HP TiO2 Memristor Based Circuits with Fractional Conformable Derivative. VOL. 8, NO. 1.
• 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.
• 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.
• Delgado. M. V. F., Aguilar J. F. G.,Hernandez. M. A. T. (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.
• Palaz. U., Mutlu. R. (2021). “Analysis of a Capacitor Modelled with Conformable Fractional Derivative Under DC and Sinusoidal Signals,” Celal Bayar University Journal of Science, vol. 17, no. 2, pp. 193-198.
• Palaz. U., Mutlu. R. (2021) “Two Capacitor Problem with an LTI Capacitor and a Capacitor Modelled Using Conformal Fractional Order Derivative,” European Journal of Engineering and Applied Sciences, vol. 4, no. 1, pp. 8-13.
• Mohammed. A. A. H. A. Kandemir. K., Mutlu. R. (2020). “Analysis of Parallel Resonance Circuit Consisting of a Capacitor Modelled Using Conformal Fractional Order Derivative Using Simulink,” European Journal of Engineering and Applied Sciences, vol. 3, no. 1, pp 13-18.
• Aguliar. G., Hernandez. R., Garcia. R., Calderon. (2014). Fractional RC and LC Electrical circuit, Vol.15, 311-319.
• Halliday. D., Resnick. R., Walker. J. (2013). Fundamentals of physics, John Wiley & Sons.
• Yener. Ş. Ç., Mutlu. R. (2017).“Small signal model of memcapacitor-inductor oscillation circuit,” in Electric Electronics, Computer Science, Biomedical Engineerings’ Meeting (EBBT), pp. 1-4.
• “L*C*(x^(1-a))y’’+L*C(1-a)*(x^(-a))*y’+y=0 - Wolfram|Alpha.” https://www.wolframalpha.com/input/?i=L*C*%28x%5E%281-a%29%29*y%27%27%2BL*C*%281-a%29*%28x%5E%28-a%29%29*y%27%2By%3D0 (accessed Jun. 04, 2022).
• Koepf. W. (2017). Solving Differential Equations in Terms of Bessel Functions (pp. 39-46).
• Davis, P. J. (1959). "Leonhard Euler's Integral: A Historical Profile of the Gamma Function". American Mathematical Monthly. 66 (10): 849–869. doi:10.2307/2309786. JSTOR 2309786. Retrieved 3 December 2016.