Finite element analysis of hybrid excitation axial flux machine for electric cars

Finite element analysis of hybrid excitation axial flux machine for electric cars

Pelizari, A.^I; Chabu, I.E^II

^IUniversity of Sao Paulo - [e-mail]

^IIUniversity of Sao Paulo - [e-mail]

ABSTRACT

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.

Index terms: Electric traction motor; electric vehicle application; hybrid excitation axial flux machine; simulation of electric motors.

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]:

In (1), ROUT 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), m₁ represents the number of phases of the stator, N₁ 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 α 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

Thus, replacing (6) in (5), results in

Since K_d is a diameter factor it can be obtained as

Substituting the K_d factor in (8), after some mathematical treatment results

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

Hence, the outer diameter of the axial flux motor disc can be determined as follows [4]

Referring to (11) the outer diameter variation can be seen as in Figure 4 since the K_D, K_ω1, n_s factors are constant.

And the inner diameter is consequently

The armature data and its main characteristics are presented in Table II.

As the topology is a double side stator machine, the number of turns per phase per stator can be calculated as

Since both sides of the armature windings were in series wye connection (Y), the electric current of the armature can be determined by

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.

A magnetic equivalent circuit was designed in order to calculate the flux densities in the most relevant parts of the topology. Figures 6 and 7 illustrate the pathway flux and Figures 8 and 9 the equivalent circuit with both excitation systems.

Referring to Figure 10, average flux density B_1AVG becomes:

From (15), the flux per pole can be determined since S_P is the area of the pole

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:

The flux per pole at the inner yoke of the rotor can be determined by

Hence the flux density at the inner yoke of the rotor becomes

The flux density at the outer yoke of the rotor is

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 the formulas from (15) up to (22), the flux density in each part of the machine can be calculated analytically. The calculated values are shown in Tables V, ^VI and ^VIII with double excitation, electric excitation and permanent magnet excitation only, respectively.

V. FEM ANALYSIS

This section presented the results of the finite element analysis in magnetostatic and transient analysis simulation via 3D software compare them with those of the analytical solution. The problem formulation in a magnetostatic study is

In (23), J is the current density, A is the magnetic vector potential and µ is the magnetic permeability of the medium. The current sources were imposed with their rated values which are 15 A in the armature winding and in electric excitation coils 1.6 A respectively according to the magnetomotive force calculated from ^{Table VI}. In this type of solution, the solver assumes that the electric current is uniformly distributed over the cross section of the coil and the direction of the current is determined by the arrows as in Figures 13a and 13b.

Figures 14 and 15 illustrate the color map simulations of the topology in both conditions.

In summary, the flux densities calculated in the motor as well as the FEM results are presented in ^{Table VIII} and Table IX, respectively

A transient simulation analysis with constant speed imposed was carried out in both working conditions. The formulation of constant speed magnetic transient analysis is

The constant ν, σ and ν 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. Table IX presents the quantities used in the simulation

For convenience, due the memory size of the computer, only ¼ of the geometry was simulated as in Figure 16.

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 circuit. However, Figure 17 b presents only three-phase armature windings since there is no dc current at 1200 rpm.

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. Table XI summarizes the torque results obtained from the analytical method and simulation.

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.

• [1] Caricchi, F.; Crescimbini, F., "Axial-flux permanent-magnet machine with water-cooled ironless stator", proc. IEEE Power Tech Conf., pp. 98-103, 1995.
• [2] Gieras, J.F., R. Wang, and M. J. Kamper, "Axial flux permanent magnet brushless machines", 2004.
• [3] Wang, S.; Xia, Y.; Wang, X., "State of the art of hybrid excitation permanent magnet synchronous machines", Electrical Machines and Systems (ICEMS), International Conference on, pp.1004 - 1009, 2010.
• [4] Xia, Y. ; Wang, S. ; Ma, M. ; Hao L.; Qiu A. ; Huang, S.; "Basic principles of hybrid excitation PM synchronous generator utilizing harmonic for excitation Electrical Machines and Systems (ICEMS), International Conference on, pp. 1010 - 1013, 2010.
• [5] Kefsi, L.; Touzani, Y.; Gabsi, M., "Hybrid Excitation Synchronous Motor control with a new flux weakening strategy", VPPC, IEEE, p.p. 1 - 5, 2010.
• [6] Kamper, M.; Wang, R., "Analysis and performance evaluation of axial flux air-cored stator permanent magnet machine with concentrated coils", IEEE electric machines & drives conference, 2007.

Publication Dates

• Publication in this collection
  27 Jan 2015
• Date of issue
  Dec 2014