**1. Introduction**

Modern technologies such as portable electronic devices, electrical transportation, communication systems, and smart medical equipment need efficient energy storage systems [1,2]. Electrical energy storage devices are also used for smart grid control, grid stability, and peak-power saving as well as for frequency and voltage regulation [3–5]. Electricity generated from renewable sources (e.g., solar power, wind energy) can hardly deliver an immediate response to demand because of fluctuating power supply [6–8]. Hence, it has been suggested to preserve the harvested electrical energy for future requirements. The present status of electrical energy storage technologies is quite far away from the needed demand. These circumstances motivate us to continue scientific research for the improvement in the parameters of existing storage devices and to develop new storage machinery.

Currently, ultracapacitors (UCs) are considered as an efficient energy storage system for electrical devices [9]. Electric double-layer capacitors (EDLCs) or symmetric double-layer UCs have attracted attention as plausible electrical energy storage devices [10–14]. EDLCs are a complex of two identical porous electrodes, electrolytic solution, and a separator, which is used as an ion conductor. The negative electrode attracts the cations during the charging process, and the anions are collected at the pores of the positive plate. The EDLCs are characterized by longer life cycle due to the absence of chemical reactions, efficient charge-discharge cycles, ability to discharge at higher current density, fast charging-discharging ability, and the lack of heavy metals, which make the device environmentally friendly [11,14]. The increasing popularity of UCs has been directed toward a better understanding of dynamic behavior and ultimately improved performance. The parameters responsible for the dynamic behavior of UCs have not been studied for the applicable range. In very recent work, we studied the dynamic behavior of a double-layer UC [13]. The capacitance remained nearly constant for a wide range of temperatures (+25 ◦C to −40 ◦C), but the internal resistance increased ~1.5 times as the temperature decreased to −40 ◦C [13]. The equivalent electrical circuit models are required to simulate the device parameters for the development and design of electrical appliances. Previously, different types of equivalent circuit models have been proposed in order to understand the dynamic characteristics of UCs [13,15–20]. Importantly, the electrochemical analysis-based modeling approaches have also been employed to study the performance of UCs [21–24].

A mathematical model should demonstrate model precision, robustness, and ease of application in the well-known software platforms (e.g., MATLAB and others). The functionality of UCs is defined by the movement of charged particles (ions) from positive to negative electrodes in the electrolyte. Hence, the correct description of UCs should be based on partial differential equations (PDE). These equations have to describe the continuum flow of ions, which determine the internal resistance and the capacitance of UC. In the electrolyte, the particle movement is related to the diffusion of ions, which is linearly dependent on the concentration difference in adjacent points of a space. The electrical potential describes the charge distributions in the electrolyte, and electrical resistance defines the diffusion movement of ions. Thus, this information should be used to fix the constraints of the equivalent electrical circuit and to characterize the internal resistance and capacitance. In previous studies, this principle is employed to design an equivalent circuit using a finite number of resistors and capacitors [15]. However, the application of a finite number of resistances and capacitances cannot describe UC parameters with high precision. The requirement to simplify the equivalent circuit prevents the use of a significant number of elements. Moreover, the complicated equivalent circuits create substantial mathematical difficulties to determine the magnitudes of equivalent electrical components correctly. Several artificial mathematical operators (fractional impedance [16], Warburg impedance [25], and constant phase element [26]) have been employed to analyze the equivalent circuit. Previous studies did not describe the precise physical phenomena responsible for the electrical properties of UCs. As a result, these equivalent circuits require permanent matching of circuit parameters depending on applied voltage and current. These methods could not explain exact changes in the UC parameters during their functionality since equivalent circuits do not have a rigorous physical base. In a previous article, we proposed that the infinite r–C chains-based equivalent circuit model could describe the behavior of double layer UCs [13]. The multibranch r–C circuit modeling approach was also studied by other researchers [27–33]. The frequency-domain models comprise the best overall performance in terms of complexity, correctness, and robustness [31–33]. Logerais et al. proposed the multibranch r–C circuit model for the analysis of UC [28], but the proposed model did not provide rigorous closed-form analytical solutions and did not consider the inductance of connecting cables and electrodes. Navarro et al. considered the inductance of connecting cables and an infinite number of r–C chains. However, the reverse Fourier transform can be difficult to apply for the prediction of voltage alterations during charge-discharge due to the lack of a closed-form analytical solution.

This work aimed to study the dynamic behavior of symmetric double layer UC and develop an adequate equivalent circuit model. The novelty of the proposed work was the application of the reverse Fourier transforms to get a time-domain response of UC parameters such as voltage and current. The reverse Fourier transforms analysis was based on the rigorous analytical solution for the frequency-domain impedance spectroscopy. The rest of this paper is organized as follows. First, the internal resistance and capacitance are measured for repetitive charge-discharge cycles. Second, the impedance, which includes reactance and resistance, is measured at different applied voltage and temperatures using electrochemical impedance spectroscopy (EIS). Third, the impedance data are analyzed using an infinite r–C chain equivalent circuit model. Fourth, dynamic parameters of UC are represented using the reverse Fourier transform analysis. Finally, the proposed equivalent circuit model is simulated using the PSIM simulating package.
