Citation: Ortigueira, M.D.; Martynyuk, V.; Kosenkov, V.; Batista, A.G. A New Look at the Capacitor Theory. Fractal Fract. 2023, 7, 86. https://doi.org/10.3390/ fractalfract7010086 Academic Editor: Emanuel Guariglia Received: 20 December 2022 Revised: 8 January 2023 Accepted: 10 January 2023 Published: 12 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article A New Look at the Capacitor Theory Manuel Duarte Ortigueira 1,* , Valeriy Martynyuk 2 , Volodymyr Kosenkov 2 and Arnaldo Guimarães Batista 1 1 CTS-UNINOVA and DEE, NOVA School of Science and Technology of NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal; agb@fct.unl.pt 2 Khmelnitsky National University, Instytutska, 11, 29016 Khmelnytskyi, Ukraine; martynyuk.valeriy@gmail.com (V.M.); vladimirkosenkov@ukr.net (V.K.) * Correspondence: mdo@fct.unl.pt Abstract: The mathematical description of the charging process of time-varying capacitors is reviewed and a new formulation is proposed. For it, suitable fractional derivatives are described. The case of fractional capacitors that follow the Curie–von Schweidler law is considered. Through suitable substitutions, a similar scheme for fractional inductors is obtained. Formulae for voltage/current input/output are presented. Backward coherence with classic results is established and generalised to the variable order case. The concept of a tempered fractor is introduced and related to the Davidson–Cole model. Keywords: fractional capacitor; fractional inductor; fractional derivative; fractor; Davidson–Cole model; tempered derivative 1. Introduction Resistors, capacitors and inductors are the fundamental building blocks of basic electric circuits. The laws that underlie their physical behaviour are assumed to be well known. This does not mean that they cannot be called into question when some new result or theory is introduced. This is the recent case concerning the problem of charge storage in capacitors, mainly the fractional ones that follow the Curie–von Schweidler law [1,2]. Two new recently proposed modelling formulae have been subject of some discussion [3–7]. Of course, the two perspectives are clearly different and irreconcilable. A careful reading of both approaches leads us to identify some origins of the different visions and search for an alternative. Firstly, the fractional derivative used in such approaches is not suitable for solving the problem. In fact, the Caputo derivative has several drawbacks [8,9], but the main ones are the confusion between the Heaviside unit step and the constant function, leading to results contradicted by experience [10] and the Caputo derivative of a sinusoid is not a sinusoid [8,11]. Another problem we encounter is the forgetfulness of the past that leads to some mistakes in the use of distribution theory. Here, we tackle the problem and propose a coherent alternative that generalises the classical results. Traditionally, a formula, deduced from the Maxwell equations, relates the charge, q(t), t ∈ R, and voltage, v(t), in a capacitor. It reads q(t)= Cv(t), (1) where C is the capacitance expressed in Farad (F). This formula, obtained under station- ary conditions, expresses a static relation between two physical entities. In reality, the underlying dynamics is not visible. However, it appears in the relationship between q(t) and the current, i (t), as we will see later. In practice, the above relation expresses an approximation that is good enough in many situations. We will assume that it characterises order 1 ideal capacitors. The discussed problem, introduced first by S. Das [3,12], consists Fractal Fract. 2023, 7, 86. https://doi.org/10.3390/fractalfract7010086 https://www.mdpi.com/journal/fractalfract