## 5 Conclusions  
The main goal of this paper was to develop a new configuration for a more efficient LIM, based on a prototype model called model 1. To this end, we proposed three changes to the geometric parameters of the fixed and moving parts and we defined a new model for each proposed change. The first change was incorporated into model 2, including three different regions into the aluminum layer through two slots of dielectric, in order to separate the loops of the induced currents. Thus, the thrust force generated over the lateral teeth was developed in the desired direction. The second change was proposed in model 3, where we included a ferromagnetic yoke in the primary part to produce a longitudinal magnetic flux. Finally, the third change was included in model 4 and implied the optimization of the width of the transversal slots. In standstill conditions (slip = 1), model 4 achieved the maximum thrust force of all models, around 75% with respect to the prototype model (model 1). However, this model presented a great disadvantage because it produces a loss of levitation force which begins to operate inside an attraction zone. With motion and velocities below 10 m/s (slip > 0.5), model 4 also developed the maximum thrust force. However, for velocities between 10 and 20 m/s (slip < 0.5), the changes added into the geometry did not contribute to increasing the thrust force. In this case, model 1 developed the maximum thrust force. Besides, for the whole range of analyzed velocities, model 4 operated inside the attraction zone. Figure 34: Electric current phase C versus slips. Hence, for velocities below 10 m/s (slip \(>\) 0.5), model 3 presented a similar thrust force as model 4 and operated inside the levitation zone for velocities below 6 m/s (slip \(>\) 0.7). Regarding the consumed electric currents, model 3 presented the most efficient configuration for all slip values because it achieved two main magnetic circuits operating inside the motor--transversal and longitudinal. In brief, for velocities below 10 m/s, model 3 represents an improvement in respect to model 1, due to the high thrust force developed and the low electric current consumed. To simulate all proposed models with FEM-3D, we presented a method that allows one to analyze the magnetic field density along the airgap in an LIM. The method is composed of two steps. First, according to an analytical method, the first harmonic of the magnetic field density in the airgap was computed. This is the starting point of the simulations with FEM-3D. Second, with an iterative process, the external circuit coupling to supply the LIM was fixed. To this end, we used two electrical parameters: the number of turns of the coils per phase, \(N_{phase}\), and the level of voltage in an AC three-phase system, \(V_{LINE}\). Using FEM-3D, the coupling between the primary winding of the LIM and the voltage sources has been defined completely. In addition, the selected transitory state in FEM-3D gives us valuable information to design a control strategy of a LIM. In the transitory state, the use of FEM-3D allows us to analyze the unbalanced system originated by the longitudinal end-effect. It is a phenomenon that cannot be neglected and presents the same distribution for all models and values of slip: \(I_{Phase,a}<I_{Phase,c}<I_{Phase,b}\). This behavior implies a great advantage of using FEM-3D for two reasons. Firstly, the highest phase (\(l_{b}\)) will be a reference signal to design the insulated scheme and the electrical protection of the LIM. Secondly, the unbalanced electric system supposes high variations in the forces during the transitory state. This effect implies that it is necessary to carry out an analysis of the vibration modes using FEM when we want to design the structural support of an LIM in mechanical engineering. All authors have collaborated in the writing of the paper. [PERSON] has made the design and simulation of all experiments and prepared the original draft presentation.