A Comparative Thermal Study of Two Permanent Magnets Motors Structures with Interior and Exterior Rotor

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 $$L_m$$and the pole-pitch$$L_p$$. This relationship is used to adjust the magnet angular width according to the motor pole-pitch.

$$\beta =\frac{L_m}{L_p}$$E1

$$L_p=\frac{\pi }{p}$$E2

The ratio $$R_{ldla}$$is the relationship between the angular width of a principal tooth and the magnet angular width. This ratio is responsible for the regulation of the principal tooth size which has a strong influence on the electromotive force form.

$$R_{ldla}=\frac{A_{tooth}}{L_m}$$E3

The $$R_{did}$$ratio is the relationship between the angular width of the principal tooth and the angular width of the inserted tooth$$A_{toothi}$$. This relationship fixes the inserted tooth size.

$$R_{did}=\frac{A_{tooth}}{L_m}$$E4

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).

Figure 1.

Permanent magnets motors with exterior rotor and interior rotor

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.

Definition	Symbol	Value
Electric vehicle mass	M	1000 kg
Angle of starting	ad	3°
Time of starting	td	4 s
Outside temperature	Tout	40°C
Maximum motor power	Pmmax	21,635 kW
Winding temperature	Tw	95°C
Base speed of the vehicle	Vb	30 km/h
Maximum Speed of the vehicle	Vmax	100 km/h
Slots load factor	kr	0,44
Current density in the slots	δ	7 A/mm^2

Table 1.

The schedules data conditions

Definition	Symbol	Value
Remanent magnetic induction of the magnets	Bm	1,175 T
Demagnetization Induction	Bc	0,383 T
Magnetic induction in teeth	Btooth	0,9 T
Magnets permeability	μa	1,05
Mechanical losses coefficient	km	1%
Copper resistivity at 95°C	Rcu	17,2 10^-9 Ωm
The copper resistivity variation coefficient	α	0,004
Density of the electrical sheets	Mvt	7850 kg
Density of magnets	Mva	7400 kg
Density of copper	Mvc	8950 kg
Sheets quality coefficient	Q	1,1

Table 2.

Specific constants of materials

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:

1. The magnet height, h_m

2. The slots height h_s and the tooth height h_tooth

3. The rotor yoke height, h_ry

4. The stator yoke height, h_sy

5. The air gap thickness, e

Figure 2.

PMSMER and PMSMIR parameters

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

The slot average width:$$W_s$$

$$W_s=\frac{D_m+e+h_{tooth}}{2}{A}_s$$E5

The principal tooth section:$$S_{tooth}$$

$$S_{tooth}=\frac{D_m+e}{2}{A}_{tooth}{l}_m$$E6

Where$$l_m$$, $$D_m$$are the average motor length and the average motor diameter.

The inserted tooth section:$$S_{toothi}$$

$$S_{toothi}=\frac{D_m+e}{2}{A}_{toothi}{l}_m$$E7

The slot section: $$S_s$$

$$S_s=\frac{1}{2}\left[\frac{2\pi }{N_{tooth}}-A_{tooth}-A_{toothi}\right]\frac{D_m+e}{2}{l}_m$$E8

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

The slot average width:$$W_s$$

$$W_s=\frac{D_m-e-h_{tooth}}{2}{A}_s$$E9

The principal tooth section:$$S_{tooth}$$

$$S_{tooth}=\frac{D_m+e}{2}{A}_{tooth}{l}_m$$E10

The inserted tooth section:$$S_{toothi}$$

$$S_{toothi}=\frac{D_m+e}{2}{A}_{toothi}{l}_m$$E11

The slot section: $$S_s$$

$$S_s=\frac{1}{2}\left[\frac{2\pi }{N_{tooth}}-A_{tooth}-A_{toothi}\right]\frac{D_m-e}{2}{l}_m$$E12

The teeth height $$h_{tooth}$$of the PMSMIR and the PMSMER are expressed by equation Eq. 13 and Eq. 14 where $$N_{sph}$$is the number of turns per phase,$$I_n$$ is the rated current and $$N_{teeth}$$in the number of teeth.

$$h_{tooth}={\sqrt{\frac{N_{sph}.I_n}{N_{teeth}\delta K_r{A}_s}+\left(\frac{Dm+e}{2}\right)}}^2-\frac{Dm+e}{2}$$E13

$$h_{tooth}={\sqrt{\frac{N_{sph}.I_n}{N_{teeth}\delta K_r{A}_s}+\left(\frac{Dm-e}{2}\right)}}^2-\frac{Dm-e}{2}$$E14

The stator yoke thickness $$h_{sy}$$is obtained by the application of the flux conservation theorem, where $$B_{tooth}$$is the magnetic induction in the tooth.

$$h_{sy}=\frac{B_{tooth}{S}_{tooth}}{2l_m{B}_{sy}}$$E15

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

The expression of the magnet height $$h_m$$ is the same one in the two structures. It is obtained by the application of the Ampere theorem.

Where $${\mu }_a$$is the air permeability and $$k_{fu}$$is the flux leakage coefficient.

$$h_m=\frac{{\mu }_a{B}_{e}e}{M(Ta)-\frac{B_e}{k_{fu}}}$$E16

Where the magnet induction $$M\left(Ta\right)$$at Ta°C is defined by:

$$M(Ta)=M\left[1+\alpha m(Ta-20)\right]$$E17

The rotor yoke thickness $$h_{ry}$$is defined:

$$h_{ry}=\frac{B_e{S}_{tooth}}{2k_{fu}{l}_m{B}_{ry}}$$E18

2.5. Electrical sizing

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

$$EM{F}_i(t)=\frac{8}{\pi }{N}_{sph}{l}_m{D}_m{B}_e\sin \left(\frac{\pi \beta }{2}\right)\sin \left(\frac{\pi }{2}\beta R_{ldla}\right){\Omega }_m\sin \left(p{\Omega }_{m}t\right)$$E19

The motor electric constant : $$K_e$$

$$K_e=\frac{12}{\pi }{N}_{sph}{l}_m{D}_m{B}_e\sin \left(\frac{\pi \beta }{2}\right)\sin \left(\frac{\pi }{2}\beta R_{ldla}\right)$$E20

The electromagnetic torque :$$T_{em}$$

$$T_{em}(t)=\frac{1}{\Omega }{\displaystyle \sum _{i=1}^{3}EM{F}_i(t)i_i\left(t\right)}$$E21

where$$EM{F}_i^{}$$, $$i_i$$and $${\Omega }_m$$represent respectively the electromotive force, the current of the i phase and the angular speed of the motor.

The motor rated current I_n is the ratio between the electromagnetic torque and the motor electric constant.

$$I_n=\frac{T_{em}}{K_e}$$E22

The phase résistance of the motor : $$R_{ph}$$

$$R_{ph}=R_{co}(T_w)\frac{N_{sph}\delta L_{sp}}{I_n/\sqrt{2}}$$E23

where $$R_{co}(T_w)$$is the copper receptivity at the temperature of winding $$T_w$$and $$L_{sp}$$is the spire average length (Ben Hadj et al., 2007).

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:

$$R_{convection}=\frac{l}{Ak}\left[K/W\right]$$E24

$$R_{conduction}=\frac{l}{A_{c}h}\left[K/W\right]$$E25

Where $$l$$is the distance between the point masses and $$A$$is the interface area, k is the heat conductivity, $$A_c$$is the cooling cross section between the two regions and h is the convection coefficient calculated from proven empirical dimensionless analysis algorithms.

The heat capacitance is defined as follow:

$$C=V\rho c\left[Ws/K\right]$$E26

Where V is the volume,$$\rho$$ is the density and c is the heat capacity of the material. The simplified stator for the thermal study of the two SMPMM structure are given by figures 3 and 4, also the thermal model for the two structures are implemented in MATLAB simulater where the different radius for the PMSMER dimensions are defined as follow:

$$R_{carter}=R_1=R_f+\frac{e}{2}+h_m+h_{ry}+e_{carter}$$E27

$$R_{rotoryoke}=R_2=R_f+\frac{e}{2}+h_m+h_{ry}$$E28

$$R_{magnet}=R_3=R_f+\frac{e}{2}+h_m$$E29

$$R_4=R_f+\frac{e}{2}$$E30

$$R_{slot}=R_5=R_f\frac{}{}\frac{e}{2}$$E31

$$R_{insolator}=R_6=R_5\frac{}{}{h}_s$$E32

$$R_7=R_5\frac{}{}{h}_s\frac{}{}{t}_{insolator}$$E33

$$R_8=R_7\frac{}{}{h}_{sy}$$E34

The thermal resistances are calculated along the radial direction. The $$R_i$$radius are calculated from dimensions of motor, where $$R_f$$is the Bore raduis and $$t_{insolator}$$ is the thickness insulator.

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

$$R_1=R_f+\frac{e}{2}$$E35

$$R_2=R_f+\frac{e}{2}+h_{tooth}$$E36

$$R_3=R_f+\frac{e}{2}+h_{tooth}+t_{insolator}$$E37

$$R_4=R_f+\frac{e}{2}+h_{tooth}+t_{isnolator}+h_{sy}$$E38

$$R_5=R_f+\frac{e}{2}+h_{tooth}+t_{insolator}+h_{sy}+t_{carter}$$E39

Figure 3.

Simplified stator for the thermal study in the PMSMER

Figure 4.

Simplified stator for the thermal study in the PMSMIR

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.

Figure 5.

Thermal model of the PMSMIR in transient behaviour

In this model, the heat sources are respectively the copper losses and iron losses in the stator. The $$T_i$$variables are the temperatures in various points of the motor. The expressions of thermal resistances of the PMSMIR result from the resolution of the heat equation at the fields borders.

$$R_{coil}$$represents the coil thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{coil}=\frac{1}{4\pi l_m{\lambda }_{coil}}[1-2\frac{R_1^2}{R_2^2-R_1^2}\ln \frac{R_2}{R_1}]$$E40

$$R_{inso}$$represents the isolator thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{inso}=\frac{\ln \frac{R_3}{R_2}}{2\pi {\lambda }_{inso}{l}_m}$$E41

R_inso-sy represents the contact thermal resistance between insolator and the stator yoke ($$K.W^{\frac{}{}1}$$).

$$R_{inso-sy}=\frac{\frac{1}{300}}{2\pi R_3{l}_m}$$E42

$$R_{jco}$$represents the thermal resistance of stator yoke ($$K.W^{\frac{}{}1}$$).

$$R_{jco}=\frac{\ln \frac{R_4}{R_3}}{2\pi {\lambda }_{iron}{l}_m}$$E43

$$R_{fco}$$represents the thermal resistance of conduction in the stator yoke ($$K.W^{\frac{}{}1}$$).

$$R_{fco}=\frac{1}{4\pi l_m{\lambda }_{iron}}[1-2\frac{R_3^2}{R_4^2-R_3^2}\ln \frac{R_4}{R_3}]$$E44

$$R_{sy-ca}$$represents the thermal resistance between stator yoke and the carter ($$K.W^{\frac{}{}1}$$).

$$R_{sy-ca}=\frac{\frac{1}{1500}}{2\pi R_4{l}_m}$$E45

$$R_{ca}$$represent the thermal resistance of carter ($$K.W^{\frac{}{}1}$$).

$$R_{ca}=\frac{\ln \frac{R_5}{R_4}}{2\pi {\lambda }_{ca}{l}_m}$$E46

$$R_{ext}$$represents the convection thermal resistance between the carter and ambient air ($$K.W^{\frac{}{}1}$$).

$$R_{ext}=\frac{l}{h{S}_{ext}}$$E47

In the previous expression, $$S_{ext}$$represents the outer surface of the motor and $$h$$is the heat transfer coefficient between the carter and the ambient air. It can be between 20 and 40 $$K.W^{\frac{}{}1}{m}^{\frac{}{}2}$$for a motor with natural ventilation, and may exceed 80 $$K.W^{\frac{}{}1}{m}^{\frac{}{}2}$$for the motor forced air.

To calculate the outer surface of SMPMM, we considered only the outer surface of the cylinder with radius $$R_5$$and height $$l_m$$ (Gasc, 2004; Chan., 2004).

$$S_{ext}=2\pi R_5{l}_m$$E48

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

$$C_{coil}$$represents the heat capacity of coil ($$J{K}^{\frac{}{}1}$$).

$$C_{coil}={\rho }_{co}{V}_{co}{C}_{th-co}$$E49

$$C_{inso}$$represents the heat capacity of insolator ($$J{K}^{\frac{}{}1}$$).

$$C_{inso}={\rho }_{inso}{V}_{inso}{C}_{th-inso}$$E50

$$C_{sy}$$represents the heat capacity of stator yoke ($$J{K}^{\frac{}{}1}$$).

$$C_{sy}={\rho }_{iron}{V}_{sy}{C}_{th-iron}$$E51

$$C_{ca}$$represents the capacity of carter ($$J{K}^{\frac{}{}1}$$).

$$C_{ca}={\rho }_{alu}{V}_{ca}{C}_{th-alu}$$E52

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

Figure 6.

Thermal model of the PMSMER in transient behaviour

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

$$R_{sy}$$represents the stator yoke thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{sy}=\frac{1}{4\pi l_m{\lambda }_{iron}}\left[1-2\frac{R_8^2}{R_7^2-R_8^2}\ln \frac{R_7}{R_8}\right]$$E53

$$R_{inso}$$represents the insolator thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{inso}=\frac{1}{4\pi l_m{\lambda }_{inso}}\left[2\frac{R_7^2}{R_6^2-R_7^2}\ln \frac{R_6}{R_7}\right]$$E54

$$R_{coil}$$represents the coil thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{coil}=\frac{1}{4\pi l{\lambda }_{co}}\left[1-2\frac{R_6^2}{R_5^2-R_6^2}\ln \frac{R_5}{R_6}\right]$$E55

$$R_{mag}$$represents the magnet thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{mag}=\frac{\ln \frac{R_1}{R2}}{2\pi {\lambda }_{mag}{l}_m}$$E56

$$R_{ry}$$represents the rotor yoke thermal resistance ($$K.W^{\frac{}{}1}$$).

$$R_{ry}=\frac{\ln \frac{R_2}{R_3}}{2\pi {\lambda }_{iron}{l}_m}$$E57

$$R_{ca}$$represents the thermal resistance of carter. ($$K.W^{\frac{}{}1}$$)

$$R_{ca}=\frac{\ln \frac{R_1}{R_2}}{2\pi {\lambda }_{ca}{l}_m}$$E58

To calculate the outer surface of SMPMM, we considered only the cylinder outer surface with radius $$R_1$$and the height$$l_m$$.

$$S_{ext}=2\pi R_1{l}_m$$E59

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

$$C_{mag}$$represents the heat capacity of magnet ($$J{K}^{\frac{}{}1}$$).

$$C_{mag}={\rho }_{mag}{V}_{mag}{C}_{th-mag}$$E60

$$C_{ry}$$represents the heat capacity of rotor yoke ($$J{K}^{\frac{}{}1}$$).

$$C_{ry}={\rho }_{iron}{V}_{ry}{C}_{th-iron}$$E61

$$C_{coil}$$represents the heat capacity of coil ($$J{K}^{\frac{}{}1}$$).

$$C_{coil}={\rho }_{co}{V}_{co}{C}_{th-co}$$E62

$$C_{inso}$$represents the heat capacity of insolator ($$J{K}^{\frac{}{}1}$$).

$$C_{inso}={\rho }_{inso}{V}_{inso}{C}_{th-inso}$$E63

$$C_{sy}$$represents the heat capacity of stator yoke ($$J{K}^{\frac{}{}1}$$).

$$C_{sy}={\rho }_{iron}{V}_{sy}{C}_{th-iron}$$E64

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

Material	Conductivities; (Wm-1K-1)	Mass heat capacity (Jkg-1K-1)	Density; (Kgm-3)
Copper (Coil)	λco=5	Cth-co= 398	ρco = 8953
Insolator	λinso =0.25	Cth-inso= 1250	ρinso = 1200
(Iron)Stator Yoke	λiron =25	Cth-iron= 460	ρiron = 7650
Magnet	λmag =6.5	Cth-mag= 420	ρmag = 7400
aluminium (Carter)	λca =180	Cth-alu = 883	ρalu = 2787

Table 3.

The thermal conductivities of materials

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.

Figure 7.

Various temperatures in different parts of the PMSMER in transient state.

Figure 8.

Various temperatures in different parts of the PMSMIR in transient state.

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.