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

Year 2023, , 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

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