Finite Element Analysis of Hybrid Excitation Axial Flux Machine for Electric Cars

this paper presents a FEM analysis of a hybrid excitation brushless axial flux machine (HEBAFM) for traction electric vehicle purposes in order to compare with the results from an analytical method used to determine the flux densities in each part of the machine. The magnetic quantities of the proposed topology were investigated in order to obtain a satisfactory level of flux densities avoiding a possible saturation of the material. Keeping the magnetic induction under the saturation point, it will be feasible to increase the speed beyond the rated speed. The results from using the analytical method as well as via FEM simulation analysis presented a good approximation and are shown at the end of this paper.


I. INTRODUCTION
DC motors haves been used for almost two centuries. This is easily explained not only because it can operate at the flux weakening region but also for its excellent torque response. In order to solve the problem of the exhaustive electromechanical maintenance in this type of machine, during the 1960s, permanent magnet brushless dc machines were developed. The drawback of these machines is the difficulty in controlling the speed, especially when flux weakening region operation is required.
Hence the main purpose of this paper is a comparative study of a Hybrid Excitation Brushless Axial Flux Machine [1] topology proposed in Figure 1, using the finite element analysis via 3D simulation software and the analytical method to make its operation at the constant power region possible keeping the flux density level in the critical parts of the machine under the saturation point.

II. THE AXIAL FLUX MOTOR EQUATION TORQUE
The first step during the development of the topology was to predict the developed torque upon its main dimensions and magnetic variables of the machine. The developed torque equation of the double side axial flux motor can be calculated as follows [2]: (1) Finite Element Analysis of Hybrid Excitation Axial Flux Machine for Electric Cars Pelizari, A. In (1), R OUT is the outer radius of the disc in meters, A is the rms linear current density, i.e., Am/2 -1/2 in Ampère.turns/meter, B 1AVG the fundamental air gap average flux density in Tesla, K d is the diameter factor of the disc. The constant Am is the peak of the linear current density and it can be calculated as: In (2), m 1 represents the number of phases of the stator, N 1 the number of turns of the armature winding and I A the phase current of the stator winding. The air gap average flux density in (1) can be obtained in terms of the air gap maximum flux density B g , as in (3) ( The term  in (3) is the relationship between coil pitch and pole pitch. Therefore, once the average flux density has already been defined, the developed torque behavior can be viewed as in Figure 2 as a function of both linear current density and average flux density.

III. ARMATURE DESIGN
In this step, the sizing of armature was based on its main dimensions obtained through the equations in section II as well as on its rated data. The general data and the rated characteristics of the hybrid excitation axial flux machine topology proposed are shown in Table I. In this type of machine, due the greater disc diameter, the ratio of speed of the topology proposed was 600/1200 revolutions per second to avoid vibration of the machine, i.e., ratio of speed of 1:2 and therefore the air gap flux density ratio adopted was 2:1. Hence, the number of poles can be determined as: In (4), f is the rated frequency in (Hz), Ns the synchronous speed in rpm. After calculating the number of poles, the next step was to obtain the average flux density B 1AVG as a function of peak air gap flux density calculation. Hence, the fundamental flux density per pole can be calculated as in (5): Based on average flux density per pole, the magnetic flux per pole is determined as (6) Thus, replacing (6) in (5), results in (7) Since K d is a diameter factor it can be obtained as  The Figure 3 illustrates the dependence of the flux per pole as a function of D OUT and pair of poles.
K D is the dimension factor as a function of K d , which can be calculated as (10) Hence, the outer diameter of the axial flux motor disc can be determined as follows [4] (11) Referring to (11) the outer diameter variation can be seen as in Figure 4 since the K D , K 1 , n s factors are constant. And the inner diameter is consequently The armature data and its main characteristics are presented in Table II.   Since both sides of the armature windings were in series wye connection (Y), the electric current of the armature can be determined by (14) Thus, Table III summarizes the main data of the armature winding. Using (1), the three-phase calculated developed torque in both conditions can be viewed in Table   IV.

IV. HYBRID EXCITATION DESIGN
To avoid disc vibration, a ratio of speed of 600/1200 rpm was selected, thus, a maximum air gap flux density of 0.65 T produced by hybrid excitation for 600 rpm was adopted herein. Consequently, at the flux weakening region, half the maximum air gap flux density, for a maximum speed of 1200 rpm, only PM excitation was considered. Basically, the hybrid excitation [5,6] proposed is composed of two systems as in Figures 5a and 5b: one system consists of 2 pairs of coils with 875 turns allocated at the outer and the inner part of the armature and fixed at the cover, which produces up to 0.325 T controlled by a d.c. source with maximum voltage of 100 V. The latter is a permanent magnet system as in Figure 5b composed of 24 NdFeB-35 permanent magnets fixed to the rotor which produces 0.325 T. The main concern is to prevent saturation from occurring in the ferromagnetic cores, for instance, teeth, yokes or even in the cover, since that not only one, but two armature windings provide magnetic flux through the rotor.
Since the flux per pole is produced by two stators, the flux density at the pole as a function of a minimum area as in Figure 11, becomes:  Fig.11. View of the magnetic flux pathway in the rotor and the minimum area of the pole.
The flux per pole at the inner yoke of the rotor can be determined by (18) Hence the flux density at the inner yoke of the rotor becomes The flux density at the outer yoke of the rotor is (20) And the flux density at the teeth of the armature can be calculated as In (21), the term S ARM_TEETH represents the area of the armature teeth, K STACK is the stacking factor.
Lastly, the flux density at the yoke of the armature was calculated as in (22), i.e.
Using the magnetization curve given in figure 12 and     In summary, the flux densities calculated in the motor as well as the FEM results are presented in Table VIII and Table IX, respectively   TABLE VIII - The constant ,  and  are respectively the magnetic reluctivity, electric conductivity and constant speed in steady state regime. In the same equation, V is the scalar electric potential of the source, Hc is the coercive field intensity of the permanent magnets.   The transient regime simulations with constant speed were carried out for two flux conditions, both of them an external electric circuit coupled. Figure 17a shows the electric circuit with a three-phase source connected to the armature winding and a dc rectifier converter feeding the electric excitation At the end of the simulation, the harmonics of field at steady state were extracted up to the 40 th harmonic, as illustrated in Figure 18 with double excitation at 600 rpm. The developed torque calculated from the analytical method over the regions is presented in Figure   19 and the torque obtained through the simulation at rated speed and flux weakening region are presented in Figures 20a and 20b.

VI. CONCLUSION
The armature core and the rotor disc were designed based on the analytical method varying the dimensions such as area of the pole, area of the yokes and area of the teeth, thus preventing a critical situation which is magnetic saturation, since the saturation point of the ferromagnetic core used in this project was 2,1 Tesla. Through the analytical method and FEM simulations, the results from Table VII and Table VIII show that there was no saturation in any part of the machine, despite the high level of flux density at the minimum area of the pole. In spite of the greater losses at 120 Hz, the efficiency achieved is satisfactory making the topology proposed possible. The torque results obtained analytically and simulated presented good approximation.