## 2 Methods  
Energy Processes in the Aperiodic Charge Circuit of a Supercapacitor and an Equivalent Linear Capacitor  
The dependence of the capacitance of supercapacitors on the voltage at its terminals was taken into account during the research. In studies conducted in [2; 3; 13; 14; 15; 16; 17; 18; 19; 20; 21] it was noted that the physics of the SC involves the presence of part of the capacitance, which depends on the applied voltage to its terminals. Experimental measurements of charge from a DC voltage source were performed for SCs with a capacitance of 470, 1500 and 2600 F [18; 20; 21]. The total capacitance of the SC consists of a series connection of many capacitors, and nanoelectrodes of primary capacitors, which are due to a double electric layer of charges on the surface of the pores of activated carbon of the main electrodes of the SC. One of the possible explanations for the reason for the increase in the total capacitance of the SC with increasing voltage is connection with the increase in the dielectric constant of the electrolyte or with a decrease in the thickness of the electric double layer [18]. The total capacitance of the IC can be represented by the expression:  
\[C=\frac{\varepsilon\epsilon_{0}S}{d}, \tag{1}\]  
where \(\varepsilon\) is the dielectric constant of the electrolyte, \(\varepsilon_{0}\) is the dielectric constant of vacuum, \(S\) is the total surface area of the developed surface of the nanoporous electrode, and \(d\) is the thickness of the electric double layer (dielectric between the charges of the electric double layer). The explanation of the nonlinear increase in the capacitance of the SC with increasing voltage on it is that the area of accumulation of charges in the electrodes of the SC increases with the application of higher voltages. Under such conditions, the process occurs due to the penetration of charges into the developed nanopores of electrodes of smaller diameter [3; 18]. The process causes an increase in the charge accumulated in the SC, nonlinear relative to the increase in voltage at the terminals of the SC. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Device** & **Supercapacitor** & **Capactor** & **Lithium-Ion Battery** \\ \hline Charge time & 1–30 s & \(10^{-3}<t<10^{-6}\), s & \(1<t<5\), h. \\ \hline Discharge time & 1–30 s & \(10^{-3}<t<10^{-6}\), s & \(t>0.3\), h. \\ \hline Number of cycles & 1,000,000 & 1,000,000 & \(>\)500 \\ \hline Specific energy, W h/kg & \(<\)30 & \(<\)0.1 & \(<\)200 \\ \hline Specific power, W/kg & \(<\)91,000 & \(>\)1,000,000 & \(<\)3900 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of supercapacitor, capacitor and lithium-ion battery. Under the capacitance C between two bodies, which have equal and opposite in sign charges, the absolute value of the ratio of charge on one of the bodies to the voltage between the bodies is given by  
\[C=Q/U. \tag{2}\]  
From the definition of capacitance follows the unit of its dimension 1 C/V = 1 F. The capacitance is between two conductive bodies that are separated by a dielectric. In an electrostatic field, the voltage between two bodies U can be linearly expressed by the charge \(Q\); then the ratio \(Q/U\) is independent of neither the value of \(Q\) nor the value of \(U\). Capacitance depends only on the configuration of bodies, their size, distance between bodies, and dielectric properties. This is also true for linear capacitors (LC). Exceptions are, for example, devices that use ferroelectrics, i.e., substances in which the dielectric constant is a function of \(U\). For supercapacitors, the magnitude of the capacitance C is a function of the voltage \(U\)[2; 3; 8; 9; 10]. Expression (2) is valid for both LC and SC, because it establishes the relationship between the charge accumulated on each of the plates \(Q\) and the voltage between them \(U\). Under the conditions of the experiment, the SC was charged from a DC voltage source [15]. Small ripples with a value of 10 mV at a frequency of 1 MHz were superimposed on the DC voltage component.