## 2 Methods
\[\mu=\frac{B}{H(B)} \tag{26}\]  
The software FEMM solves the problem finding a field that satisfies Equations (23)-(25) through a magnetic vector potential approach. A vector potential, \(A\), is defined, as shown in Equation (27). \[B=\
abla\times A \tag{27}\]  
For what concerns Equation (24), it is always satisfied by the definition of \(B\) shown in Equation (27). Equation (23) can be rewritten as shown in Equation (28). \[\
abla\times\left(\frac{1}{\mu(B)}\
abla\times A\right)=J \tag{28}\]  
In the 2-D planar and axisymmetric cases, the \(A\) vector has only one non-zero component, the one in an out-of-plane direction. All of the conditions reduce to a single equation, exploited to find \(A\). Then, \(B\) and \(H\) can be deduced by differentiating \(A\). To solve the problem, a semicircle-shaped computational domain has been created around the engine, with the straight edge along the engine's axis of symmetry. The problem faced is axisymmetric, and for this kind of problem the Dirichlet condition \(A=0\) on the boundary is enforced by the software. Figure 6: Flowchart of the tool with the main outputs. For what concerns the mesh, the software FEMM discretizes the problem domain using triangular elements. Over each triangular element, the solution is approximated by a linear interpolation of the values of potential at the vertices of the triangle. In Figure 7, there is an example of a mesh generated by the software. Obviously, in order to execute the calculation, the materials have to be chosen. The sizing of magnet's width and anode thickness has been executed through an iterative procedure represented in the flowchart shown in Figure 8.  
The iterative procedure consists of imposing first attempt values regarding the dimensions of the magnets and the thickness of the anode. Once these values are set, the values of the maximum transverse magnetic field between the magnets and the magnetic field at the cusps on the anode surface are estimated. After that, the initial values (the dimension of the magnets and anode thickness) are then modified to obtain magnetic field values closer to those to be obtained. The procedure continues until the the maximum transverse magnetic field between the magnets and the magnetic field at the cusps on the anode surface to be obtained are reached. The size of the magnetic rings is influenced mainly by the maximum transverse magnetic field; instead, the value of the magnetic field at the anode surface influences the distance between the magnet surface and the anode surface that faces the plasma in the chamber. The tool can be used in two ways: to determine performances of a ion thruster given in terms of input thrust (or power) or size (diameter of the discharge chamber), or to determine the minimum size, knowing the thrust (or power) requirements that the engine must meet, through an iterative procedure. By determining a minimum size to obtain a certain thrust (or power), performances and geometrical and working parameters (e.g., mass flow rate) can be obtained. Figure 7: Computational domain and mesh generated by the software FEMM to solve the axysimmetric magneto-static problem. The model has some degrees of freedom that have been fixed according to consideration about operative life, efficiency, and complexity: the discharge voltage \(V_{d}\), the cathode voltage drop \(V_{c}\), the length of the discharge chamber \(L\), and the grid geometry. For the discharge voltage \(V_{d}\), a maximum value of 28 V has been imposed according to Ref. [42] to minimize screen-grid erosion; the value of the cathode voltage drop \(V_{c}\) is imposed to maintain the quantity \((V_{d}-V_{c}+\phi)\) under 21.2 V (the second ionization potential) but still maintain a high value in order to increase the ionization to excitation ratio [43, 44]; \((V_{d}-V_{c}+\phi)\approx 20\) V is a good compromise. This quantity, multiplied by the electron charge \(e\), represents the energy of the electrons emitted by the cathode. In the above expression, \(\phi\) represents the sheath potential.