## 2 Methods
The capacitance of ordinary linear capacitors \(C\) does not depend on the voltage of their charge \(U\), and the energy accumulated in them under the condition of zero initial voltage is determined by the expression:  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]1  
If the SC charge started at zero initial conditions (\(t_{\text{i}}=0\) voltage \(U_{\text{i}}=0\) and the magnitude of the charge \(Q_{\text{i}}=0\)), then taking into account expressions (13)-(14) we can obtain an expression for the energy \(W_{\text{SC}}\) accumulated in the SC at a certain voltage final voltage \(U_{\text{f}}=U\):  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]2  
The dose of energy taken from the battery (AB) \(W_{\text{AB}}\) can be found by the formula:  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]3  
where \(U_{\text{AB}}\)--the battery voltage. The energy transfer coefficient \(\eta\) is equal to the ratio of the energy received in the LC or in the SC to the energy taken from the AB for the entire charge time:  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]4  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]5  
where \(W_{\text{LC}}(t_{i})\), \(W_{\text{SC}}(t_{i})\), \(W_{\text{LC}}\Big{(}t_{f}\Big{)}\), \(W_{\text{SC}}\Big{(}t_{f}\Big{)}\) are energies that have been accumulated in the LC and SC in accordance with the switching and after the completion of the transient process of charge from the AB; \(W_{\text{AB}}=W_{\text{AB}}(t_{i})-W_{\text{AB}}\Big{(}t_{f}\Big{)}\) i.e., the energy given by AB during the transition process. According to expressions (17)-(19) we obtain the energy of losses in the charge circuit of the LC and SC from the battery. The energy of losses is the difference between the energy given by the AB and the energy received during the charge of the LC or SC:  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]6  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]7  
When modeling, the SC cannot be limited to one constant value of capacitance, as in the case of LC. If it is necessary to specify one value of capacitance, for SC it is necessary to calculate the equivalent capacitance, \(C_{\text{ekv}}(U_{1})\), for a specific value of voltage \(U_{1}\). According to expressions (17) and (18), the SC accumulates more energy than the LC is charged to the same voltage. In a previous study [22], it was shown that when discharged and then charged to the same voltage for SC and LC with equal internal resistances (\(R_{\text{LC}}=R_{\text{SC}}\)), the supercapacitor has a load resistance \(R_{\text{n}}\) with more energy and with greater pulse power than LC. The expression for the capacitance of the LC, \(C_{W}\), which is equivalent to the capacitance of the SC, provided the equality of the accumulated energies (\(W_{\text{LC}}=W_{\text{SC}}\)) and the voltage on their plates (\(U_{\text{LC}}=U_{\text{SC}}=U\)), can be obtained from (18):  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]8  
To study and compare the energy characteristics of the circuit when charging the SC and LC from the battery, mathematical modeling was performed in the Matlab application package. The time constant was calculated by the formula \(\tau=RC\). The parameters of the SC and lithium-ion battery [11; 20] used in the study are given in \[C(U)=C_{1}+k\ |U|. \tag{3}\]9  
We assume that the initial voltage on the LC and SC was zero. The studies were performed under the condition that the capacitance of the linear capacitor, \(C_{W}\), is equivalent to the capacitance of the supercapacitor, provided that the energy stored in them is equal (\(W_{\rm LC}=W_{\rm SC}=2196\) (J) at \(U_{\rm LC}=U_{\rm SC}=0.99U_{\rm n}\)) calculated according to expression (24) and is equal to \(C_{\rm W}=846.84\) F. The internal supports of the LC and the first (instantaneous) branch of the SC are the same: \(R_{\rm LC}=R_{1}=0.0025\) Ohms.