## 4 Model Simulations with FEM-3D
### Definition of the Electrical External Circuit Coupling  
To start the simulation with FEM-3D, the electrical circuit coupling necessary to operate with the LIM was set up. To this end, the following steps were necessary:  
1. **Step 1**: To fix the magnetic field density in the airgap, \(B_{g}\), using an analytical method to estimate its value [22]. 2. **Step 2**: To select the configuration of the electrical circuit coupling to supply the device (Y-connected or D-connected system) and to generate the traveling magnetic field defined in Section 2.2. 3. **Step 3**: To use an iterative simulation process to calculate the parameters necessary to define the electrical circuit coupling. The first parameter is the value of the symmetrical line voltage, \(V_{LINE}\). The other parameters are the number of turns per phase and its resistance value: \(N_{phase}\) and \(R_{phase}\)  
Figure 8: Simulation conditions imposed during the simulation. Both are necessary to have the coils defined correctly. The initial value of \(R_{phase}\) is fixed at around 2.6 \(\Omega\) to start the iterative process. In the first step, the magnetic field density in the airgap was considered equal to the average value obtained from Equations (7) and (8), (see reference [22]). Both equations allowed for the obtaining of the value of the first harmonic of \(B_{g}\), called \(B_{g1}\), considering only the geometric parameters of the LIM and not their electrical quantities. So, there were two possibilities to calculate \(B_{g1}\), using Equations (7) or (8). \[B_{g1\_l}=0.464\cdot\tau_{p}^{1/6} \tag{7}\]  
\[\underbrace{B_{g1\_l}=\
icefrac{{B_{g1\_l}}}{{R_{i\_l}}}(1-\
icefrac{{b_{ \infty}}}{{\tau_{\_i}}})\;B_{ts}=5.47\cdot f^{-0.32}}_{B_{g1}=5.47\cdot\
icefrac{{ \infty}}{{R_{i\_l}}}(1-\
icefrac{{b_{\infty}}}{{\tau_{\_i}}})} \tag{8}\]  
where \(B_{ts}\) is the flux density in the primary tooth (\(\Gamma\)); \(k_{is}\) is the primary stacking factor equal to 1; \(b_{\infty}\) is the primary slot opening equal to 8.5 mm; \(\tau_{\_i}\) is the slot pitch equal to 16.5 mm; \(\tau_{\_P}\) is the pole pitch equal to 200 mm, and \(f\) is the frequency equal to 50 Hz. At the end, the value of \(B_{g1}\) is given by Equation (9) and is equal to 0.761 T.  
\[B_{g1}=\frac{B_{g1\_l}+B_{g1\_l}}{2}=\frac{\frac{5.47\cdot f^{-0.32}}{k_{is}} \cdot\left(1-\frac{b_{\infty}}{\tau_{\_i}}\right)+0.464\cdot\tau_{\_p}^{\frac{ 1}{6}}}{2} \tag{9}\]  
In the second step, the coils from the primary winding were supplied by a Y-connected three-phase voltage. This situation is described by the Equation system (10), (11), and (12) (see reference [21]), which generates a traveling magnetic field along the desired direction. \[V_{Phase\_a}=V_{LINE}\cdot\sqrt[]{2}/\sqrt[]{3}\cdot\cos(\omega\cdot t)=V_{ LINE}\cdot\sqrt[]{2}/\sqrt[]{3}\cdot\cos(2\cdot\pi\cdot f\cdot t) \tag{10}\]  
\[V_{Phase\_b}=V_{LINE}\cdot\sqrt[]{2}/\sqrt[]{3}\cdot\cos(\omega\cdot t-2 \cdot\pi/3)=V_{LINE}\cdot\sqrt[]{2}/\sqrt[]{3}\cdot\cos(2\cdot\pi\cdot f \cdot t-2\cdot\pi/3) \tag{11}\]  
\[V_{Phase\_c}=V_{LINE}\cdot\sqrt[]{2}/\sqrt[]{3}\cdot\cos(\omega\cdot t+2 \cdot\pi/3)=V_{LINE}\cdot\sqrt[]{2}/\sqrt[]{3}\cdot\cos(2\cdot\pi\cdot f \cdot t+2\cdot\pi/3) \tag{12}\]  
where \(V_{LINE}\) is the voltage established in a Y-connected symmetrical three-phase system (V); \(V_{Phase\_a},V_{Phase\_b},\ V_{Phase\_c}\) are the supply voltages per phase (V); \(\omega\) is the angular frequency (rad/s) and \(t\) is the instantaneous time of simulation (s). In the third step, \(V_{LINE}\) and \(N_{phase}\) were calculated by the simulations. When the value of \(B_{g}\) reached the value of the magnetic field density established in the first step, \(V_{LINE}\) and \(N_{phase}\) were fixed. The measure of \(B_{g}\) was determined with a plane that crosses the mechanical airgap by its middle section (a detailed study of \(B_{g}\) is made in Section 4.2).