Mathematical Physics Models of RLC Electrical Circuit with Generalized $\mathcal{M}$ Derivative

Abstract

In this article, we develop a mathematical model of an RLC electrical circuit using the generalized truncated $\mathcal{M}$-derivative, which extends the concept of the classical $\mathcal{M}$-series. The resulting generalized $\mathcal{M}$-derivative RLC model is analyzed via the Laplace transform, allowing the determination of the circuit's total impedance and dynamic response. Kirchhoff’s current and voltage laws are incorporated to ensure consistency with fundamental circuit principles. The model in our study offers strong potential for capturing memory related dynamics compared to classical models. Additionally, the model's behavior changes depending on the values of the fractional parameters we use, which provides an advantage. The proposed model has been validated using MATLAB based numerical simulations. Analytical expressions for current and voltage in the circuit are derived and the behavior of the system is illustrated through detailed graphical simulations using MATLAB. However, experimental validation using experimental measurement data is outside the scope of this paper. The study demonstrates the effectiveness of the generalized $\mathcal{M}$-derivative in capturing complex transient dynamics of RLC circuits and provides a framework for further analysis of fractional order electrical systems.