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

Öz

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.

Anahtar Kelimeler

• 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

Öz

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.

Anahtar Kelimeler

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

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