## 3 Results
### Validation
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **XIPS-25 cm (Mission Level 2)** [25] & **Model** & **Error** \\ \hline \(T\) (mN) & 79.9 & 79.9 & 0.0\% \\ \(P_{in}\) (kW) & 2.00 & 2.01 & 0.5\% \\ \(I_{sp}\) (s) & 3320 & 3087 & 7.0\% \\ \(\eta_{T}\) & 0.64 & 0.61 & 4.7\% \\ \(V_{b}\) (V) & 1215 & 1160 & 4.5\% \\ \(I_{b}\) (A) & 1.44 & 1.46 & 1.3\% \\ \(V_{T}\) (V) & 1495 & 1449 & 2.3\% \\ \(|V_{acc}|\) (V) & 280 & 290 & 5.5\% \\ \(\dot{m}_{p}\) (mg/s) & 2.46 & 2.64 & 7.3\% \\ \(|B_{cusp}|\) (G) & Not available & 2169 & / \\ \(|B_{l}|\) (G) & Not available & 15 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison between the characteristics and performances of XIPS-25 cm ion thruster (mission level 2) [25] and the results of the developed model. For what concerns the XIPS-13 ion thruster, the error is quite high for the beam voltage \(V_{b}\), as shown in Table 3. \(V_{b}\) in the tool is a function of the total efficiency \(\eta_{T}\), the specific impulse \(I_{sp}\), the input power \(P_{in}\), and the grid geometry. The error obtained is probably due to the different grid geometry exploited by this engine because the error on the other parameters is quite small. There is also a slightly high error for the beam current \(I_{b}\), the total voltage \(V_{T}\), and the accelerator grid voltage \(V_{acc}\) because the calculation of these parameters \(V_{b}\) has been exploited. For what concerns the comparison with the XIPS-25 ion thruster, the error is small for almost all of the parameters of each operating condition, as shown in Tables 4 and 5. The error is slightly higher only in the calculation of the accelerator grid voltage \(V_{acc}\). The error of the tool increases, on average changing the operating condition from 1 to 3, and this can be related to the calculation of the total efficiency \(\eta_{t}\) through the function shown in Figure 3. Moving away from the nominal condition, the error of the function increases. The total efficiency \(\eta_{T}\) affects all of the values calculated in the tables, except for the input power. For what concerns the ETS-8 ion thruster, as in XIPS-13 ion thruster, the error on \(V_{b}\) is quite high, as shown in Table 7. Additionally, in this case, even if the error on the specific impulse \(I_{sp}\), and the total efficiency \(\eta_{T}\), is not negligible, the error may be associated mainly with a different grid geometry. In fact, in the calculation of \(V_{b}\), the total efficiency \(\eta_{T}\) is at the numerator and the specific impulse \(I_{sp}\) at the denominator. Having obtained a higher specific impulse and a lower total efficiency, with respect to the thruster, a lower value of \(V_{b}\) should have been obtained with respect to the real performances of the thruster. The other errors obtained are due to the errors in the estimation of \(V_{b}\). The model has been further used to reproduce the characteristics of a miniaturized GIT (8 cm of discharge chamber and 100 W as input power) developed by [PERSON] [42; 49], as shown in Table 8. In Ref. [42], the performance targets of the miniaturized ion thruster developed by [PERSON] are listed, and in Ref. [49] the actual performances are listed. In this case, to allow for a more effective comparison, it was decided to give the model, as an input parameter, no longer the thrust but the power (100 W). For what concerns the comparison between the model results and the [PERSON] miniaturized gridded ion thruster characteristics, the results in Table 8 show that the performances and working parameter values obtained from the model are between the values of the performance targets parameters [42] and the effective performances of the selected engine [49]. Thus, the model predictions are closer to the real performances respect to Ref. [42]. Furthermore, the error made by the model, compared to the real performances of the miniaturized ion thruster developed by [PERSON], is almost below 10%, and below 5% for half of the parameters.