## 1. Introduction

Considering the large variety of electric motors, such as asynchronous motors, synchronous motors with variable reluctances, permanent magnets motors with radial or axial flux, the committed firms try to find the best choice of the motor conceived for electric vehicle field.

The electric traction motor is specified by several qualities, such as the flexibility, reliability, cleanliness, facility of maintenance, silence etc. Moreover, it must satisfy several requirements, for example the possession of a high torque and an important efficiency (Zire et al., 2003; Gasc, 2004; Chan., 2004).

In this context, the surface mounted permanent magnets motor (SMPMM) is characterized by a high efficiency, very important torque, and power-to-weight, so it becomes very interesting for electric traction.

In the intension, to ensure the most suitable and judicious choice, we start by an analytical comparative study between two structures of SMPMM which are the permanent magnets synchronous motor with interior rotor (PMSMIR) and the permanent magnets synchronous motor with exterior rotor (PMSMER), then, we implement a methodology of design based on analytical modelling and the electromagnetism laws. Also, in order to understand the thermal behaviour of the motor, we implant a comparative thermal performance of the two structures illustrated with careful attention to the manufacturing techniques used to produce the machine, and the associated thermal resistances and capacitances, to obtain good steady state and transient thermal performance prediction.

## 2. Modelling of two SMPMM structures

### 2.1. Structural data

The structures of motors allowing the determination of the studied geometry are based on three relationships.

The ratio β is the relationship between the magnet angular width

The ratio

The

### 2.2. Geometrical structures of PMSMIR and PMSMER

This part is devoted to an analytical sizing allowing calculation of geometrical sizes of the two SMPMM configurations which are the PMSMER and the PMSMIR.

Figure 1 represents the PMSMER and the PMSMIR with the number of pole pairs is p=4 and a number of principal teeth is 6, between two principal teeth, an inserted tooth is added to improve the wave form and to reduce the leakage flux (Ben Hadj, N. et al., 2007). The slots are right and open in order to facilitate the insertion of coils and to reduce the production cost (Magnussen, F. et al., 2005; Bianchi, N..et al., 2003; Libert, F. et al., 2004).

### 2.3. Analytical sizing of the two SMPMM structures

The analytical study of motor sizing is based on the schedules data conditions parameters (Table 1), the constant characterizing materials (Table 2), the expert data and configurations of the two motors.

Expert data

The expert data are practically represented by three sizes which are, the magnetic induction in the air gap B_{e}, the magnetic induction in the stator yoke B_{sy} and the magnetic induction in the rotor yoke B_{ry}. It should be noted that the zone of variation of these three parameters varies between 0,2 to 1,6T (Ben Hadj et al., 2007).

Structural data

For the two configurations, we adopted the same number of pole pairs P=4, with an air gap thickness equivalent to 2mm, with a relationship β equal to 0,667 and R_{ldla} equal to 1,2.

Data identified by the finite elements method

K_{fu} is the leakage flux coefficient of the PMSMIR which is fixed to 0,95 whereas for the PMSMER, k_{fu} is equal to 0,98. In this context, we define a ratio R_{did} equal to 0,2.

### 2.4. Geometrical sizes

Geometrical parameters of the two structures motors are defined in figure 2. Where:

The magnet height, h

_{m}The slots height h

_{s}and the tooth height h_{tooth}The rotor yoke height, h

_{ry}The stator yoke height, h

_{sy}The air gap thickness, e

In the stator of the PMSMIR, geometrical sizes are defined by:

The slot average width:

The principal tooth section:

Where

The inserted tooth section:

The slot section:

In the stator of the PMSMER, geometrical sizes are defined by:

The slot average width:

The principal tooth section:

The inserted tooth section:

The slot section:

The teeth height

The stator yoke thickness

In the rotor of the two structures, geometrical sizes are defined by:

The expression of the magnet height

Where

Where the magnet induction

The rotor yoke thickness

### 2.5. Electrical sizing

The electromotive force in the two SMPMM structures is expressed by:

The motor electric constant :

The electromagnetic torque :

where

The motor rated current I_{n} is the ratio between the electromagnetic torque and the motor electric constant.

The phase résistance of the motor :

where

## 3. Comparative thermal study between the two SMPMM

In this study, the comparison between the two SMPMM structures consists on the thermal analysis which is based upon lumped-circuit analysis. It represents the thermal problems by using the thermal networks, analogous to electrical circuits. The thermal circuit in the steady state consists of thermal resistances and heat sources connected between motor component nodes. For transient analysis, the heat/thermal capacitances are used additionally to take into account the change in internal energy of the body with time. The thermal resistances for conduction and convection can be obtained by:

Where

The heat capacitance is defined as follow:

Where V is the volume,

The thermal resistances are calculated along the radial direction. The

The different radius for the PMSMIR dimensions are defined as follow:

As described earlier, the thermal resistance values are automatically calculated from motor dimensions and material data.

Figure 5 shows the thermal model in transient behaviour of the PMSMIR.

In this model, the heat sources are respectively the copper losses and iron losses in the stator. The

R_{inso-sy} represents the contact thermal resistance between insolator and the stator yoke (

In the previous expression,

To calculate the outer surface of SMPMM, we considered only the outer surface of the cylinder with radius

The expressions of heat capacities of the PMSMIR are given by the following equations:

Figure 6 shows the thermal model in transient behaviour of the PMSMER.

The expressions of the PMSMER thermal resistances obtained from the resolution of the heat equation at the fields borders.

To calculate the outer surface of SMPMM, we considered only the cylinder outer surface with radius

The expressions of heat capacities of the PMSMER are given by the following equations:

The below table presents the different thermal conductivities of materials (Jérémi, R., 2003)

## 6. Results and simulations

Simulation results with Matlab software allowed us to obtain the curves of temperatures specific to different materials of the PMSMER and PMSMIR structures. The thermal results at steady and transient state is reached by figures 7, 8.

According to the results, we find that the steady state in the PMSMIR is reached after 4000s. However, the steady state in the PMSMER is achieved after 2000s.

By comparing the results in steady and transient state between the two configurations, we note that temperatures of different parts in PMSMIR are higher than temperatures in PMSMER (especially the coil temperature). That’s why, we choose the PMSMER configuration as the best solution in electric traction field.

Moreover, we always look to get a permissible values of coil temperature, based on the proper choice of motors geometric parameters in order to ensure a good compromise between geometric dimensioning and thermal modeling motor.

## 7. Conclusion

In this paper, a thermal model of two SMPMM with interior rotor and exterior rotor was realised, the intension to compare the evolution of the temperatures of different parts of the two motor configurations and especially the modeling of temperature at the coil is made.