## 3 Modified LIM Models and Implementation with FEM-3D  
In this section, we describe the modifications made in model 1 to obtain model 2, model 3, and model 4. In addition, we describe the main issues that we have to take into account to simulate them with FEM-3D. The TFLIM prototype (model 1), described in Section 2, will be modified to improve the thrust force. To this end, three different new models were developed:  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Electrical and Magnetic Properties of the** **Materials Seed in FEM-3D** \\ \hline \multicolumn{2}{c}{**Aluminum**} \\ \hline Conductivity (Sm\({}^{-1}\)) & \(3.73\times 10^{7}\) \\ Relative Permeability & 1 \\ \hline
**Steel** & \\ \hline Conductivity (Sm\({}^{-1}\)) & 0 \\ Relative Permeability & 2500 \\ \hline
**Copper** & \\ \hline Resistivity (\(\Omega\)mm\({}^{2}\)/m) & \(2.37\times 10^{-2}\) \\ Relative Permeability & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Electrical and magnetic properties.  
* Model 2, where the moving part was redesigned. More concretely, the aluminum layer was modified with the introduction of two narrow slots of air or diamagnetic material (\(\mu_{r}=1\)). Each slot has a width of 1 mm and they are located symmetrically at 108 mm from the edge of the aluminum layer (see Figure 6). The new regions, insulated electrically, allowed us to have three regions available inside the aluminum layer with new paths for the eddy currents, so that each zone of aluminum was located above lateral and central teeth. Consequently, the thrust force originated over the lateral teeth developed along the direction of the motion.
* Model 3, where the primary part included a ferromagnetic yoke located under the central teeth. The high of this ferromagnetic block was 50 mm, its width was 83 mm, and it was extended along 503.5 mm that corresponded with the length of the LIM. With this new block, the motor produced a longitudinal flux that operated through the main magnetic circuit with the transversal flux. So, the combined action of these fluxes provided a higher thrust force. Figure  
For the implementation of the models with FEM-3D, the time domain selected corresponds to a transitory state. In all simulations, the frequency \(f\) is fixed at 50 Hz. The computation process time was calculated using three types of times: \(T_{cycle}\) (called cycled time) is the period of the AC voltage signal employed during the simulations [s]; \(T_{simulation}\) (s) (called simulation time), is the number of cycles necessary to achieve the stationary state which allows us to evaluate the advantages of the proposed changes without increasing the computation effort; \(T_{step}\) (called step time simulation) is the fraction of the \(T_{cycle}\) necessary to converge to the solution (s). The \(T_{simulation}\) is defined as three times the \(T_{cycle}\), which is divided into 10 steps to fix \(T_{step}\) (see Equation (6)). These times were selected experimentally, after several simulations that achieved the correct model.  
\[T_{cycle}=\
icefrac{{1}}{{f}}=0.02\;s.\rightarrow\left\{\begin{array}{l}T_{ simulation}=3\cdot T_{cycle}=0.06\;s.\\ T_{step}=T_{\text{\tiny{scit}}}/_{10}=0.002\;s.\end{array}\right. \tag{6}\]  
Finally, Figure 8 resumes all necessary conditions to simulate correctly all models proposed: