## 2 Methods  
The design tool developed in this work combines a curve-fitting approach with analytical equations in order to obtain the main performances and design parameters of a medium-low-power (\(<\)5 kW) gridded ion thruster. This original allows one to obtain accurate results and reduce the degrees of freedom of the problem considered. For what concerns the curve-fitting approach, some main performance parameters (i.e., thrust, input power, and efficiencies) in some different operating conditions of various  
Figure 1: Gridded ion thruster main components and working principle. engines present in the open literature (T5 [27], T6 [28], T7 [29], NEXT [30], NSTAR [23], XIPS-13 [24], XIPS-25 [25], JPL-15 [31], Mitsubishi ETS 6 [32], and Mitsubishi ETS 8 [33]) were collected to build, using MATLAB software, useful curves from which data can be interpolated. The input parameters for the model are the thrust (or power) and the desired discharge chamber diameter. Given these inputs, from the built curves, it is possible to extract important performances parameters such as efficiencies and input power (or thrust). Figures 2 and 3 show the thrust-power curve and the relation between the thrust to diameter ratio and the total efficiency for GITs. The black points in the figures are the data collected from the thrusters mentioned above, in some operating conditions; the red lines are the interpolating curves, exploited in the model. Figure 3: Thrust-diameter ratio-total efficiency curve. Black points represent the data collected from the ion thrusters T5 [27], T6 [28], T7 [29], NEXT [30], NSTAR [23], XIPS-13 [24], XIPS-25 [25], JPL-15 [31], Mitsubishi ETS 6 [32], and Mitsubishi ETS 8 [33] in some operating conditions; red line represents the interpolating curve, exploited in the model. Figure 2: Thrust-power curve. Black points represent the data collected from the ion thrusters T5 [27], T6 [28], T7 [29], NEXT [30], NSTAR [23], XIPS-13 [24], XIPS-25 [25], JPL-15 [31], Mitsubishi ETS 6 [32], and Mitsubishi ETS 8 [33] in some operating conditions; the red line represents the interpolating curve, exploited in the model. Figures 2 and 3 highlight that there is a clear linear dependence between thrust and power in gridded ion thrusters and that the higher the discharge chamber diameter to produce a certain thrust is, the lower the total efficiency will be. After that, the general equations for electric propulsion and gridded ion thrusters are used to obtain parameters such as mass flow, currents, and voltages from the parameters discussed above. The key equations, valid for GITs, exploited in this tool are:  
\[\eta_{e}=\frac{I_{b}V_{b}}{P_{in}}=\frac{\pi}{9}\epsilon_{0}\sqrt{\frac{2e}{M}} \frac{V_{b}^{\frac{5}{2}}}{P_{in}}\frac{T_{a}}{R^{\frac{3}{2}}\Big{[}\big{(}d+t _{s}\big{)}^{2}+\frac{D_{s}^{2}}{4}\Big{]}}D^{2} \tag{1}\]  
\[\eta_{T}=\gamma^{2}\eta_{e}\eta_{m} \tag{2}\]  
\[\eta_{m}=\frac{gI_{sp}}{\gamma\sqrt{\frac{2 eV_{b}}{M}}} \tag{3}\]  
\[I_{b}=\frac{T}{\gamma\sqrt{\frac{2M}{e}}\sqrt{V_{b}}} \tag{4}\]  
\[\dot{m}_{p}=\frac{I_{b}M}{e\eta_{m}} \tag{5}\]  
\[I_{d}=I_{b}V_{b}\frac{1-\eta_{e}}{\eta_{e}V_{d}} \tag{6}\]  
\[\eta_{d}=\frac{I_{d}V_{d}}{I_{b}} \tag{7}\]  
Figure 3:0  
Equation (1) expresses the electrical efficiency \(\eta_{e}\)[13] as a ratio between the beam power (given by the product of the beam current \(I_{b}\) and the beam voltage \(V_{b}\)) and the input power \(P_{in}\). The numerator is rewritten in a function of grids geometrical parameters (screen-grid thickness \(t_{s}\), grid gap \(d\), screen-grid holes \(D_{s}\), discharge chamber diameter \(D\), optical transparency \(T_{a}\), and the function of the number of the accelerator grid holes and their diameters \(D_{a}\)) [13]. In this equation, \(\epsilon_{0}\) represents the absolute dielectric permittivity, \(e\) the electron charge, \(M\) the ion mass, and \(R\) the ratio of beam voltage to total voltage. Generally, the beam voltage to total voltage ratio is about 0.8 in GITs; this is important for the thruster's life.