Two Capacitor Problem with a LTI Capacitor and a Capacitor Modelled Using Conformal Fractional Order Derivative

Abstract

Fractional order circuit elements have been started to model different types of circuit elements, circuits and systems in the last decades. There are different types of fractional derivatives. Recently, a new simple fractional derivative method called“conformable fractional derivative” has been brought out. It is simpler than other fractional derivatives and has already been used to
model supercapacitors. It is important to model the new circuit elements and analyze the circuits containing them so that they can be exploited at their full potential. Two capacitor problem is a famous problem in physics and circuit theory. In this study, a new two capacitor problem a circuit which consists of an LTI capacitor and a supercapacitor which has been modelled with conformable fractional derivative have been examined. The differential equations which describe the circuit have been derived. The circuit current is found explicitly however the voltages of the capacitors do not have analytical solutions. That’s why they are solved numerically.

Keywords

• Circuit Analysis
• Circuit Modelling
• Circuit Theory
• Energy Analysis
• Fractional Order Derivatives

LTI Kapasitor ve Konformal Kesirli Mertebeden Türev Kullanılarak Modellenmiş Kapasitör ile İki Kapasitör Problemi

Abstract

Kesirli mertebeden devre elemanları, son yıllarda farklı tipteki devre elemanlarını, devreleri ve sistemleri modellemeye başlanmıştır. Farklı kesirli türev türleri vardır. Son zamanlarda, "uyumlu kesirli türev” adı verilen yeni bir basit kesirli türev yöntemi ortaya çıkmıştır. Diğer kesirli türevlerden daha basittir ve süperkapasitörleri modellemek için zaten kullanılmıştır. Yeni devre
elemanlarını modellemek ve onları içeren devreleri analiz etmek, böylece tam potansiyellerinde kullanılabilmeleri için önemlidir. İki kapasitör problemi, fizikte ve devre teorisinde ünlü bir problemdir. Bu çalışmada, bir LTI kondansatör ve bir süperkapasitörden oluşan ve uyumlu fraksiyonel türev ile modellenen yeni bir iki kondansatör problemi incelenmiştir. Devreyi tanımlayan diferansiyel denklemler türetilmiştir. Devre akımı açıkça bulunur, ancak kapasitörlerin voltajlarının analitik çözümleri yoktur. Bu yüzden sayısal olarak çözülürler.

Keywords

• Devre Analizi
• Devre Modelleme
• Devre Teorisi
• Enerji Analizi
• Kesirli Mertebeden Türev
• Kesirli Mertebeden Türev

References

1. Weilbeer M. Efficient numerical methods for fractional differential equations and their analytical background. Papierflieger, 2005.
2. Oldham K, Spanier J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
3. Aguilar JFG, Baleanu D. Solutions of the telegraph equations using a fractional calculus approach. Proc. Romanian Acad. A 2014; 15: 27-34.
4. Tarasov VE. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer Science & Business Media, 2011.
5. Mitkowski W, Kacprzyk J, Baranowski J, eds. Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland. Springer Science & Business Media, 2013.
6. Moreles MA, Lainez R. Mathematical modelling of fractional order circuits. arXiv preprint arXiv:1602.03541, 2016.
7. Freeborn TJ. A survey of fractional-order circuit models for biology and biomedicine. IEEE Journal on emerging and selected topics in circuits and systems 2013; 3: 416-424.
8. Adhikary A, Khanra M, Pal J, Biswas K. Realization of fractional order elements. Inae Letters 2017; 2: 41-47.

9. Tsirimokou G, Kartci A, Koton J, Herencsar N, Psychalinos C. Comparative study of discrete component realizations of fractional-order capacitor and inductor active emulators. Journal of Circuits, Systems and Computers 2018; 27: 1850170.
10. Kartci A, Agambayev A, Herencsar N, Salama KN. Series-, parallel-, and inter-connection of solid-state arbitrary fractional-order capacitors: theoretical study and experimental verification. IEEE access 2018; 6: 10933-10943.
11. Sotner R, Jerabek J, Kartci A, Domansky O, Herencsar N, Kledrowetz V, Yeroglu C. 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 2019; 86: 114-129.
12. Podlubny I, Petráš I, Vinagre BM, O'leary P, Dorčák Ľ. Analogue realizations of fractional-order controllers. Nonlinear dynamics 2002; 29: 281-296
13. Alagoz B, Ali̇soy H . H.Z. Alisoy; On the Harmonic Oscillation of High order Linear Time Invariant Systems. Balkan Journal of Electrical and Computer Engineering. 2014; 2: 113 121.
14. Alagöz BB, 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 2018; 6: 224-231.
15. Atangana A, Secer A. A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstract and applied analysis 2013; 1-8.
16. Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. Journal of Computational and Applied Mathematics 2014; 264: 65-70.
17. Abdeljawad T. On conformable fractional calculus. Journal of computational and Applied Mathematics 2015; 279: 57-66.
18. Zhao D, Luo M. General conformable fractional derivative and its physical interpretation. Calcolo 2017; 54: 903-917.
19. Sikora R. Fractional derivatives in electrical circuit theory–critical remarks. Archives of Electrical Engineering 2017; 66: 155-163
20. Morales-Delgado VF, Gómez-Aguilar JF, Taneco-Hernandez MA. Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense. AEU-International Journal of Electronics and Communications 2018; 85: 108-117.
21. Martínez L, Rosales JJ, Carreño CA. Electrical circuits described by fractional conformable derivative. International Journal of Circuit Theory and Applications 2018; 46: 1091-1100.
22. Gómez-Aguilar JF. Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels. The European Physical Journal Plus 2018; 133: 197.
23. Lewandowski M, Orzyłowski M. Fractional-order models: The case study of the supercapacitor capacitance measurement. Bulletin of the Polish Academy of Sciences: Technical Sciences 2017; 449-457.
24. Kopka R. Estimation of supercapacitor energy storage based on fractional differential equations. Nanoscale research letters 2017; 12: 636.
25. Freeborn TJ, Elwakil AS, Allagui A. Supercapacitor fractional-order model discharging from polynomial time-varying currents. In: IEEE International Symposium on Circuits and Systems (ISCAS); 27-30 May 2018; Florence, Italy. IEEE, 2018. p. 1-5.
26. Freeborn TJ, Maundy B, Elwakil AS. Measurement of supercapacitor fractional-order model parameters from voltage-excited step response. IEEE Journal on Emerging and Selected Topics in Circuits and Systems 2013; 3: 367-376.
27. Piotrowska E. Analysis the conformable fractional derivative and Caputo definitions in the action of an electric circuit containing a supercapacitor. Proc. SPIE 10808, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2018; 108081T.
28. Halliday D, Resnick R. Walker J. Fundamentals of physics. John Wiley & Sons, 2013.
29. McDonald KT. A Capacitor Paradox. arXiv preprint physics/0312031, 2003.
30. Powell RA. Two‐capacitor problem: A more realistic view. American Journal of Physics 1979; 47: 460-462.
31. Al-Jaber SM, Salih SK. Energy consideration in the two-capacitor problem. European Journal of Physics 2000; 21: 341.
32. O'Connor WJ. The famous ' lost' energy when two capacitors are joined: a new law? Physics Education 1997; 32: 88.
33. Sommariva AM. Solving the two capacitor paradox through a new asymptotic approach. IEE Proceedings-Circuits, Devices and Systems 2003; 150: 227-231.
34. Choy TC. Capacitors can radiate: Further results for the two-capacitor problem. American Journal of Physics 2004; 72: 662-670.
35. Mutlu R, AKIN OÇ. The memcapacitor-capacitor problem. In: 2nd International Conference on Computing in Science And Engineering Proceedings; 1-4 June 2011; İzmir, Turkey.
36. Adams RA, Essex C. Calculus: a complete course. Boston: Addison-Wesley, 1999.