*2.2. Electromagnetic Theory and Simulation Software*

The model of an electromagnetic actuator has to take into account many non-linearities such as:


**Figure 10.** Example of magnetic field saturation [15].

Considering this, it is diffficult to find a theoretical solution to calculate the strength of the force applied to the rod. For taking non-linearities into account, the model used is a finite elements one obtained using an open-source simulation tool called *FEMM 4.2*. It has been developed by D.C. Meeker. This tool calculates force and inductance values under different conditions [16]. More precisely, magnetic field −→*<sup>B</sup>* and potential vector −→*<sup>A</sup>* are calculated everywhere using a successive approximation finite element solver on an axisymmetric model with a spherical boundary as shown in Figure 11.

The mesh used for this computation is determined using an heuristic approach having the following characteristics: a maximum allowable mesh size is then computed as 1% of the length of the diagonal of the bounding box of any region, leading to generate a default mesh with about 4200 elements in an empty square region as shown in Figure 12. Fine meshing is also forced in all corners and a five-degree default discretization is used for arc segments.

**Figure 11.** *FEMM 4.2* model: flux density.

**Figure 12.** *FEMM 4.2* mesh with its boundary.

Evaluating force and inductance value which are integral values on the mesh is also done by *FEMM 4.2* on pre-defined specific parts of the system such as the iron plunger or the inductance of the coil. Computation needs approximately 5 s on a standard *Intel Core I7* processor.

In order to compute force and inductance for all combinations of currents and plunger position, a *LUA* script is used in *FEMM 4.2*. Simulations have been done for 30 different positions of the plunger and 6 different currents for each coil: 0 A, 40 A, 80 A, 120 A, 160 A and 200 A. Simulation times are as follows:


Considering the high computational cost of the electromagnetic simulations, our study has been limited to four coils, but we will show later that it is not necessary to go further.
