## 3 Results  
### Validation  
For validation purposes, the main performance and working parameters of some conventional thrusters (i.e., the discharge chamber diameter from 12 to 30 cm) present in the open literature were compared to those computed by the tool. The thrusters that were selected were: NSTAR [23] in Table 2, XIPS-13 [24] in Table 3, XIPS-25 [25] in Tables 4-6, and Mitsubishi ETS-8 [33] in Table 7. Each thruster has a range of operating conditions, so only some of them were chosen, randomly, for comparison, e.g., for NSTAR, throttle level 15 (TH15) was selected. To ensure the correctness of the thrusters' data listed above, it was verified that these data satisfied, with a good approximation, the Equations (1)-(8). The input given to the model, for comparison, was the thrust and the discharge chamber diameter of the selected ion thrusters in a certain operating condition. The results of the model in Tables 2-7 show an error, compared to the performances and working parameters of the thrusters, in general, which are lower than 10% for most of the characteristics and even lower than 5% for many parameters. The error, in general, is lower in the calculation of the performances of the thrusters and slightly higher in the calculation of the working parameters. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Inputs** &
\begin{tabular}{l} **Degrees of Freedom and** \\ **Assigned Values** \\ \end{tabular} & **Outputs** \\ \hline \(\bullet\) Discharge chamber diameter & \(\bullet\) Discharge voltage \(V_{d}=28\) V & \(\bullet\) Input power \(P_{int}\) (or thrust \(T\)) \\ \(P_{in}\) & \(\bullet\)\(V_{d}-V_{c}+\phi=20\) V & \(\bullet\) Specific impulse \(I_{sp}\) \\ \(\bullet\) Pressure upstream of the injector \(P_{1}\) (only for injector design) & \(\bullet\) Length-to-diameter ratio & \(\bullet\) Propellant utilization efficiency \(\eta_{m}\) \\  & \(\bullet\) Grid geometry (NSTAR) & \(\bullet\) Electrical efficiency \(\eta_{e}\) \\  & \(\bullet\) Total efficiency \(\eta_{T}\) \\  & \(\bullet\) Discharge loss \(\eta_{d}\) \\  & \(\bullet\) Accelerator grid voltage \(V_{acc}\) \\  & \(\bullet\) Net beam voltage \(V_{b}\) \\  & \(\bullet\) Total voltage across \\  & \(\bullet\) accelerator gap \(V_{T}\) \\  & \(\bullet\) Discharge current \(I_{d}\) \\  & \(\bullet\) Ion-beam current \(I_{b}\) \\  & \(\bullet\) Propellant mass flow rate \\ \(\mu_{fp}\) & \(\bullet\) Discharge chamber pressure \\ \(P_{2}\) & \(\bullet\) Discharge chamber pressure \\  & \(\bullet\)\(|B|\) at the cusps on the anode surface \\  & \(\bullet\) Max transverse \(|B|\) between \\  & \(\bullet\) the magnets \\  & \(\bullet\) Magnets width \(w\) \\  & \(\bullet\) Magnets internal diameter \\  & \(D_{int}\) \\  & \(\bullet\) Magnets external diameter \\  & \(D_{ext}\) \\  & \(\bullet\) Anode thickness \(t_{c}\) \\  & \(\bullet\) Injectors diameter \(d_{i}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input, output, and degree of freedom of the developed tool. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **XIPS-25 cm (Mission & **Model** & **Error** \\  & **Level 1) [25]** & & \\ \hline \(T\) (mN) & 92.0 & 92.0 & 0.0\% \\ \(P_{in}\) (kW) & 2.30 & 2.31 & 0.4\% \\ \(I_{sp}\) (s) & 3383 & 3169 & 6.3\% \\ \(\eta_{T}\) & 0.65 & 0.62 & 4.6\% \\ \(V_{b}\) (V) & 1215 & 1244 & 2.4\% \\ \(I_{b}\) (A) & 1.66 & 1.63 & 1.8\% \\ \(V_{T}\) (V) & 1495 & 1555 & 4.0\% \\ \(|V_{accl}\) (V) & 280 & 310 & 10.7\% \\ \(\textit{n}_{fp}\) (mg/s) & 2.78 & 2.96 & 6.5\% \\ \(|B_{cuspl}|\) (G) & Not available & 2339 & / \\ \(|B_{i}|\) (G) & Not available & 35 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between the characteristics and performances of XIPS-25 cm ion thruster (mission level 1) [25] and the results of the developed model.