# Sizing Methodology of a Fast Charger for Public Service Electric Vehicles Based on Supercapacitors
## 2 Materials and Methods
### Calculation of the Value of the Variables Involved in the Charging Process
#### 2.2.2 Selection of the Properties of the Inductor: Inductance and Internal Resistance  
Since the internal resistances of the two banks of SCs are known, the value of the internal resistance of the inductance, R\({}_{\rm L}\), can be directly calculated from (2). From the definition of \(\gamma\) in (17), two curves, \(\beta(\gamma)\), can be obtained for each of the imposed conditions: maximum charging time, t\({}_{\rm charge}\), and peak current, I\({}_{\rm max}\). The first one is extracted from (13) and is designated by \(\beta_{1}(\gamma)\) as follows:  
\[\beta_{1}(\gamma)=\frac{7}{t_{\rm charge}\cdot(\gamma-1)}. \tag{30}\]The second function, \(\beta_{2}(\gamma)\), is calculated from (16), as a function of the maximum admissible current. Considering how the parameter \(\beta\) has been defined, where according to expression (7) it is satisfied  
\[\beta^{2}=\alpha^{2}-\frac{1}{\text{L}\cdot\text{C}_{\text{eq}}}, \tag{31}\]  
the value of the inductance can be obtained as a function of \(\gamma\) and \(\beta\) from (31) as follows:  
\[\frac{1}{\sqrt{\text{L}}}=\beta\cdot\sqrt{\text{C}_{\text{eq}}\cdot(\gamma^{2}- 1)}. \tag{32}\]  
Substituting this value in (16) and considering the definition of \(\gamma\), a second function, \(\beta_{2}(\gamma)\), which depends on the peak current, will be obtained:  
\[\beta_{2}(\gamma)=\frac{\text{I}_{\text{max}}}{\Delta\text{U}\cdot\text{C}_{ \text{eq}}\cdot\sqrt{(\gamma^{2}-1)}}\cdot\left(\frac{\gamma+1}{\gamma-1} \right)^{\frac{\gamma}{2}}. \tag{33}\]  
\(\beta_{1}(\gamma)\) and \(\beta_{2}(\gamma)\) are positive continuous and decreasing functions (Figure 4). They are defined for values of \(\gamma\) belonging to the open interval (1, \(\infty\)). Both have the particularity that if \(\gamma\) tends to 1, \(\beta\) tends to infinity and that if \(\beta\) tends to zero, \(\gamma\) tends to infinity. In order not to exceed the maximum charging time, \(t_{\text{charge}}\), the pair (\(\gamma\), \(\beta\)) must lie above the curve \(\beta_{1}(\gamma)\). Similarly, for the maximum current value, \(\text{I}_{\text{max}}\), not to be exceeded, any pair of values (\(\gamma\), \(\beta\)) must be below the curve \(\beta_{2}(\gamma)\). The solution must therefore be a pair of (\(\gamma\), \(\beta\)) values that lies between the \(\beta_{1}(\gamma)\) and \(\beta_{2}(\gamma)\) curves, thus fulfilling both constraints. Figure 4: Functions \(\beta_{1}(\gamma)\), \(\beta_{2}(\gamma)\), \(\beta_{3}(\gamma)\) and \(\beta_{4}(\gamma)\) represented for different cases; (**a**) case with no possible solution; (**b**–**d**), different cases where it is possible to find a solution (striped area). By choosing a pair of values (\(\gamma\), \(\beta\)), the value of the inductance, L, and the minimum resistance of the whole circuit, R\({}_{\rm{Tmin}}\), can be obtained as follows:  
\[\rm{L}=\frac{1}{\beta^{2}\cdot C_{eq_{max}}\cdot(\gamma^{2}-1)}, \tag{34}\]  
\[\rm{R_{Tmin}}=2\cdot\beta\cdot\gamma\cdot L=\frac{2\cdot\gamma}{\beta\cdot C_{ eq_{max}}\cdot(\gamma^{2}-1)}. \tag{35}\]  
Since the internal resistances of the charger's SC bank and the vehicle's accumulator, R\({}_{\rm{1}}\) and R\({}_{\rm{2}}\), are known, the value of the internal resistance of the smoothing inductor, R\({}_{\rm{L}}\), will be directly calculated. As the charging station is going to handle very high currents, the most suitable option is to use air-core inductors with large cross-section conductors. In addition to the above restrictions for \(\gamma\) and \(\beta\), it is necessary to add two further conditions: the values of L and R\({}_{\rm{T}}\) must be calculated for the lowest value of the voltage in the vehicle's accumulator, U\({}_{\rm{02 min}}\), which corresponds to the maximum equivalent capacitance, C\({}_{\rm{eq\_max}}\), and the minimum equivalent resistance, R\({}_{\rm{Tmin}}\).