Solution of the Cauchy Problem in Fourier Series for a Linear RC Circuit with a Resistor and Two Capacitors

Year 2023, Volume: 13 Issue: 2, 163 - 173, 31.12.2023

Numan Özkutlu , Hafiz Alisoy , Gülizar Alisoy

https://doi.org/10.54370/ordubtd.1294863

Abstract

As it is known, solutions obtained in terms of Fourier series are widely used for more detailed analysis of mathematical models defined for different engineering problems. In this study, for the mathematical analysis of transient events in active RC based linear electronic circuits, the Cauchy problem is defined and solved in terms of Fourier series. For this reason, first of all, the mathematical model needed for the mathematical definition of the investigated problem was created. Based on the defined mathematical model, a quadratic differential equation for the linear RC electric circuit is obtained and the voltage and current changes are analysed for simple special cases using the Fourier series. As a result, mathematical modelling of transient processes occurring in linear RC circuits and mathematical problems defined as the Cauchy problem for such circuits have been solved analytically. It is thought that the results of the research will contribute theoretically and practically to the solution of the problems that arise in the study and design automation of different electronic circuits containing nonlinear circuit elements.

Keywords

Electronics, Differential Equations, Fourier Series, Mathematical Model, RC Circuit

Bir Direnç ve İki Kapasitörlü Lineer Bir RC Devresi için Cauchy Probleminin Fourier Serisi Cinsinden Çözümü

Abstract

Bilindiği üzere, değişik mühendislik problemleri için tanımlanan matematiksel modellerin daha ayrıntılı analizi için Fourier serisi cinsinden elde edilen çözümler yaygın olarak kullanılmaktadır. Bu çalışmada, aktif RC tabanlı lineer elektronik devrelerde gerçekleşen transient olayların matematiksel analizi için Cauchy problemi tanımlanmış ve Fourier serisi cinsinden çözülmüştür. Bu nedenle öncelikle, incelenen problemin matematiksel tanımı için ihtiyaç duyulan matematiksel model oluşturulmuştur. Tanımlanan matematiksel modele dayalı olarak lineer RC elektrik devresi için ikinci dereceden bir diferansiyel denklem elde edilmiş ve Fourier serisi kullanılarak basit özel durumlar için gerilim ve akım değişimleri analiz edilmiştir. Sonuç olarak, lineer RC devrelerinde gerçekleşen geçici süreçlerin matematiksel modellenmesi ile bu tür devreler için Cauchy problemi biçiminde tanımlanmış matematiksel problemler analitik çözülmüştür. Elde edilen araştırma sonuçlarının, doğrusal olmayan devre elemanları içeren değişik elektronik devrelerin çalışma ve tasarım otomasyonunda ortaya çıkan problemlerin çözümüne teorik ve pratik katkı sağlayacağı düşünülmektedir.

Keywords

Elektronik, Diferansiyel Denklemler, Fourier Serileri, Matematiksel Model, RC Devre

References

• Bruton, L. T. (1980). RC active circuits theory and design. Praentice-Hell, Inc.
• Chua, L. O., & Ng, C. Y. (1979). Frequency domain analysis of nonlinear systems: General theory. IEE Journal on Electronic Circuits and Systems, 3(4), 165-185. https://doi.org/10.1049/ij-ecs.1979.0030
• Chua, L. O., & Lin, P.-M. (1975). Computer-aided analysis of electronic circuits: Algorithms and computational techniques. Prentice-Hall.
• Dumitriu, L., Iordache, M., & Voicu, N. (2007, August). Symbolic hybrid analysis of nonlinear analog circuits. In 2007 18th European Conference on Circuit Theory and Design (pp. 970-973). IEEE. https://doi.org/10.1109/ECCTD.2007.4529760
• Ionkin P. A. (1976). Fundamentals of linear cirtuit theory. McGraw-Hill Education.
• Manthe A., Li Z., Shi C. J. R., & Mayaram K. (2003). Symbolic analysis of nonlinear analog circuits. In Proceedings of the 40th annual Design Automation Conference (pp. 542-545). https://doi.org/10.1145/775832.775969
• Polyanin, A. D., & Manzhirov, A. V. (2006). Handbook of mathematics for engineers and scientists. CRC Press.
• Rabiner, L. (1971). Techniques for designing finite-duration impulse-response digital filters. IEEE transactions on communication technology, 19(2), 188-195. https://doi.org/10.1109/TCOM.1971.1090625
• Sánchez-López, C., Fernández, F. V., Tlelo-Cuautle, E., & Tan, S. X. D. (2011). Pathological element-based active device models and their application to symbolic analysis. IEEE Transactions on Circuits and Systems I: Regular Papers, 58(6), 1382-1395. https://doi.org/10.1109/TCSI.2010.2097696
• Sanchez-Lopez, C. (2013). Pathological equivalents of fully-differential active devices for symbolic nodal analysis. IEEE Transactions on Circuits and Systems I: Regular Papers, 60(3), 603-615. https://doi.org/10.1109/TCSI.2013.2244271
• Soliman, A. M., & Saad, R. A. (2010). The voltage mirror–current mirror pair as a universal element. International Journal of Circuit Theory and Applications, 38(8), 787-795. https://doi.org/10.1002/CTA.596
• Sun, Y., & Fidler, J. K. (1997). Companent value rengers of tunable impedance matching networks in RF communications systems. IEEE Conference Publication, 2(441), 185-189. https://doi.org/10.1049/cp:19970786
• Teor C. (1984). Translated from teoreticheskaya matematicheskaya fizika. PN Lebedev Physics Institute, USSR Academy of Sciens, 60 (9), 395-403. https://doi.org/10.1049/cp:19970786
• Troster, G., & Langheinrich, W. (1985). An optimal design of active distributed RC networks for the MOS technology. In Proceedings of the conference on Design, Automation and Test in Europe (pp. 1431-1434). https://dl.acm.org/doi/abs/10.5555/789083.1022876
• Yerzhan, A. A., & Zauytbek, K. (2015). An approach to the problem solving of sensitivity determining of electronic circuitry. International Journal of Management, Information Technology and Engineering, 3(9), 83-94. https://paper.researchbib.com/view/paper/51700