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 tomodel 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.

Fractional order circuit elements are successfully used to model circuits and systems in the last few decades. There are different types of fractional derivatives. Recently, another one named "the conformable fractional derivative" (CFD) has been introduced and shown to give good results for modeling supercapacitors. It is imperative to know how circuit elements behave for different current and voltage waveforms in circuit theory so that they can be exploited at their full potential. A CFD capacitor is not a well-known element, and its usage and circuit solutions are rarely addressed in the literature. In this study, it is examined how a CFD capacitor would behave under DC and AC excitations when it is fed by not only a current source but also a voltage source.

Fractional-order capacitors and memcapacitors have become a major research area in recent decades. Analog applications of both circuit elements are getting more common. In literature, the conformal fractional derivative (CFD) is getting lots of interest due to its easiness to use and to comprehend. Some supercapacitors have already been modeled with the conformal fractional derivative. Two-capacitor problem is an important problem in physics. Recently, a two-capacitor problem with a CFD capacitor and an LTI capacitor has been examined. In this study, a two-capacitor problem, a circuit, which consists of a CFD capacitor and a memcapacitor, has been examined using simulations for the first time in literature. It is found that the circuit is in ever transient state.

Fractional order circuit elements have become important parts of electronic circuits to model systems including supercapacitors, filters, and many more. The conformal fractional derivative (CFD), which is a new basic fractional derivative, has been recently used to model supercapacitors successfully. It is essential to know how electronic components behave under excitation with different types of voltage and current sources. A CFD capacitor that is not a well-known element and its usage in circuits are barely examined in the literature. In this research, it is examined how to calculate the stored energy of a CFD capacitor with a series resistor supplied from a DC voltage source. The solutions given in this study may be used in circuits where supercapacitors are used.