## 3 Results
### On the Best Value for the Ideality Factor
Figure 4: Non-dimensional RMSE, \(\xi\), for a 1-D/2-R equivalent circuit model fitted to the _I-V_ curve of an Emcore ZTJ solar cell (see Figure 1) versus the ideality factor, \(a\). The 1-D/2-R equivalent circuit model was also analytically fitted to the aforementioned \(I\)-\(V\) curves. Using specific values for the ideality factor, \(a\), and the characteristic points of the curve, \(I_{sc}\), \(V_{mp}\), \(I_{mp}\) and \(V_{oc}\), the different parameters of the model were found using the following procedure:  
* \(R_{s}\) was calculated using Equation (8) to (12);
* \(R_{sh}\) was calculated using Equation (5);
* \(I_{pv}\) was calculated using Equation (6);
* \(I_{0}\) was calculated using Equation (7). Once all the parameters were known, the \(I\)-\(V\) curve was found by obtaining the values of the current, \(I\), for each value of the voltage, \(V\), using Equation (13). The ideality factor was initially varied from \(a=0.5\) to \(a=1.8\) for each photovoltaic device (although this range was extended for some devices), and the non-dimensional RMSE, \(x\), was calculated in each case. The results are plotted in Figure 5. Table 3 shows the values of the parameter \(a\) at which the minimum value of \(x\) is reached, with the rest of the parameters for the 1-D/2-R equivalent circuit. Some important conclusions can be drawn based on these results, as follows:  
* The analytical methodology suggested in the present work may give unacceptable results when the ideality factor exceeds a certain value, as complex numbers start to emerge in the calculations. * One of the parameters obtained analytically for the best fit in terms of the non-dimensional RMSE has no physical meaning: this is the negative value of the resistance  
Figure 5: Non-dimensional RMSE, \(\zeta\), corresponding to the analytical approximations for extracting the 1-D/2-R equivalent circuit parameters in relation to the ideality factor, \(a\), selected, for each one of the analyzed solar cells and panels. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Solar Cell/Panel** & \(I_{pp}\) **[A]** & \(I_{0}\) **[A]** & \(a\) & \(R_{s}\) **[\(\Omega\)]** & \(R_{sh}\) **[\(\Omega\)]** & \(\xi\) \\ \hline RTC France & 7.610\(\cdot 10^{-01}\) & 3.200\(\cdot 10^{-07}\) & 1.48 & 3.621\(\cdot 10^{-02}\) & 5.195\(\cdot 10^{+01}\) & 9.33\(\cdot 10^{-03}\) \\ TNJ Spectrolab & 5.262\(\cdot 10^{-01}\) & 9.045\(\cdot 10^{-16}\) & 0.98 & 1.157\(\cdot 10^{-01}\) & 1.845\(\cdot 10^{+02}\) & 4.52\(\cdot 10^{-03}\) \\ ZTI Emcore & 4.634\(\cdot 10^{-01}\) & 5.954\(\cdot 10^{-14}\) & 1.18 & 5.683\(\cdot 10^{-02}\) & 6.026\(\cdot 10^{+02}\) & 3.32\(\cdot 10^{-03}\) \\ Azur Space 3G30C & 5.275\(\cdot 10^{-01}\) & 5.012\(\cdot 10^{-28}\) & 0.56 & 1.322\(\cdot 10^{-01}\) & 1.554\(\cdot 10^{+02}\) & 1.60\(\cdot 10^{-02}\) \\ Photowatt PWP 201 & 1.031\(\cdot 10^{+00}\) & 7.808\(\cdot 10^{-06}\) & 1.44 & 1.273\(\cdot 10^{+00}\) & \(-1.121\cdot 10^{+03}\) & 1.06\(\cdot 10^{-02}\) \\ Kyocera KC200 GT-2 & 8.194\(\cdot 10^{+00}\) & 6.298\(\cdot 10^{-10}\) & 1.02 & 2.447\(\cdot 10^{-01}\) & 1.720\(\cdot 10^{+02}\) & 6.57\(\cdot 10^{-03}\) \\ Selex Galileo SPVS X5 & 5.034\(\cdot 10^{-01}\) & 7.007\(\cdot 10^{-29}\) & 0.56 & 6.884\(\cdot 10^{-01}\) & 7.523\(\cdot 10^{+02}\) & 3.44\(\cdot 10^{-03}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the 1-D/2-R equivalent circuit models analytically fitted to \(I\)-\(V\) curves corresponding to data from RTC France, TNJ Spectrolab, ZT Emcore, and Azur Space 3G30C solar cells, and Photowatt PWP 201, Kyocera KC200 GT-2, and Selex Galileo SPVS X5 solar panels (where \(\zeta\) is the non-dimensional RMSE for the fit). of the shunt resistor shown in Table 3, which is obtained for the Photowatt PWP 201 solar panel. * The best analytical fit gave remarkably low values of a in two cases: the Azur Space 3G30C solar cell, and the Selex Galileo SPVS X5 solar panel (composed of five Azur Space 3G28C series-connected cells), as their values were outside the range [1,1.5] suggested by [PERSON] et al. [74; 75].