*3.2. FE Methods Applied for Linear Induction Motors*

In recent decades, LIM modeling and analysis started relying more on FEA simulations instead of analytical solutions [55,65–69]. Electromagnetic FEA calculations are crucial to optimizing LIM system performance as they can provide results necessary for predicting the end-effect shaped mechanical characteristic—force versus speed. This characteristic is crucial in designing efficient LIM controls as well as traction vehicle functionality. To simplify the FEA model and to minimize the time to numerical solution, the symmetrical three-phase current is typically used. In addition, the non-linear magnetizing characteristics of the LIM primary and back iron are simplified by linearization.

The most typical LIM analysis is the static analysis. One of the challenges that must be solved in the numerical FEA calculation is the proper evaluation of the penetration of the electromagnetic field into the moving and conducting reaction rail. Such modeling and analysis can be extremely difficult and time consuming as it requires a proper choice of the FE mesh, which depends on the velocity of the LIM and slip [70–72].

Because of non-linearity of the magnetizing characteristics and the continuous quasi steady state of LIM operation, time domain (transient) analysis must be performed to achieve a steady state. A transit LIM is a large machine, more than 2 m in length and over 60 cm wide, and even for 2D calculation it demands an extremely high number of mesh nodes. At high speeds, to achieve satisfactory computation accuracy, the time step of a transient analysis must necessarily be small and with the addition of a large distance the LIM must traverse before the steady state has been achieved, which significantly increases the solution space, the transient solution often becomes impractical. To overcome this problem, a recently developed feature of the Maxwell2D software, the translational motion periodic Master-Slave boundary, has been used to make the necessary calculations to render the LIM performance characteristics [73]. A time decomposition method, patented by ANSYS, is yet another attempt to improve on a solution time in electromagnetic transient analysis but even with these advances the time to solution for large a LIM is prohibitively long.

The analytically calculated forces acting on a coil in the horizontal and vertical directions as a function of frequency for different velocities are shown in Figure 6. The comparison between the analytical results and the FEA simulations, performed with COM-SOL and ANSYS Maxwell2D software, shows an exceptionally good agreement.

**Figure 6.** Average horizontal force density (**a**) and average vertical force density (**b**) acting on the coil as a function of frequency for different LIM velocities [61].

LIM 2D models are shown in Figure 7. The analytical approach presented in [62] applies to a simplified 2D LIM model, as shown in Figure 7a. The source vector potential is obtained by summing vector potentials of all current-carrying wires of the LIM winding. The real coils of the motor are modeled to retain their position and currents and to form a

complete three-phase, six-pole winding. The modeled winding is a two-layer type, but the analytical model treats the respective top and bottom layer currents as positioned at the same distance from the conducting plate. This was done to make sure that the magnetic reluctance for currents corresponding to two different layers but located in the same slot are identical, which closely approximates the conditions of the real motor.

**Figure 7.** Evolution of 2D models of the LIM used for the numerical field evaluation. Idealized coils of a three-phase, six-pole machine [62] (**a**). Full model of the LIM [70] (**b**).

To further investigate and evaluate the applicability of the analytical solution of a simplified model as a LIM performance prediction tool, a 2D LIM model with teeth and a finite primary, as shown in Figure 7b, was developed and calculated using the FEA simulations.

For comparison, as shown in Figure 8, the performance characteristics obtained by the analytical approach were overlaid with the results generated by FEA simulations (COMSOL and ANSYS Maxwell2D). The agreement between these different calculation methods confirms the accuracy of the applied analytical and numerical methods and models.

The electric vector potential formalism was chosen for the calculations of the back iron power loss [29,74,75] (see Section 4.3). The same formalism was used to determine the magnetic field in the end regions of the induction motors as well as the motors' impedances [74,75]. This approach was also used for the formulation of the 3D equation for the scalar potential describing distribution of the electromagnetic field. The equations were solved analytically (separation of variables method) and numerically (FEA), which made it possible to determine the impedance of the windings for different boundary conditions defined on the surface of the region of analysis.

**Figure 8.** Average thrust per unit depth versus slip frequency for a LIM from Figure 7 [62].

#### **4. Selected Problems of LIM Applications**

The following section presents some of the LIM problems that were addressed and solved by the authors. The subject transit LIM is a six-pole, double-layer back wound motor with a pole pitch of 45 cm, a 9 mm mechanical gap, and a 13.5 mm magnetic gap. The reaction rail is made of a 4.5 mm thick aluminum screen over an inch-thick back iron.
