A Novel Surface Inset Permanent Magnet Synchronous Motor for Electric Vehicles

: Aiming to successfully meet the requirements of a large output torque and a wide range of ﬂux weakening speed expansion in permanent magnet synchronous motors (PMSM) for electric vehicles, a novel surface insert permanent magnet synchronous motor (SIPMSM) is developed. The method of notching auxiliary slots between the magnetic poles in the rotor and unequal thickness magnetic poles is proposed to improve the performance of the motor. By analyzing the magnetic circuit characteristics of the novel SIPMSM, the notching auxiliary slots between the adjacent magnetic poles can a ﬀ ect the q -axis inductance, and the shape of magnetic pole e ﬀ ects the d-axis inductance of the motor. The combined action of the two factors not only weakens the cogging torque, but also improves the ﬂux weakening capability of the motor. In this paper, the response surface methodology (RSM) is used to establish a mathematical model of the relationship between the structural parameters of the motor and the optimization objectives, and the optimal design of the motor is completed by solving the mathematical model. Experimental validation has been conducted to show the correctness and e ﬀ ectiveness of the proposed SIPMSM.


Introduction
With the rapid development of the automobile industry, the two global problems of environmental pollution and energy shortage are becoming more and more serious. Under such a severe situation, many countries have begun to formulate plans to ban the sale of fuel vehicles. Electric vehicles are powered by electricity and have the advantages of zero emission, low noise and energy saving. Therefore, the development and promotion of electric vehicles is highly valued by governments all over the world [1].
As the core component of electric vehicle, the performance of the driving motor directly affects the performance of electric vehicles. The research and development of high-performance electric vehicle drive motors has become one of the important factors restricting the development of electric vehicles [2]. The main types of drive motors for electric vehicles are brushless DC motors, induction motors, switched reluctance motors and permanent magnet synchronous motors (PMSM). The PMSM Symmetry 2020, 12, 179 3 of 14

Structure Design
The shape of magnetic pole effects the output characteristics of PMSM directly. The permanent magnet of typical SIPMSM is a tile shape with inner and outer arc centers at the same point-as shown in Figure 1. A PMSM with this kind of magnetic pole usually has the disadvantages of large cogging torque, large leakage and poor flux weakening capability [16]. Therefore, a novel SIPMSM is developed, as shown in Figure 2. The permanent magnet in the novel SIPMSM is an unequal thickness magnetic pole with different inner and outer radians, which results in the uneven distribution of the radial air-gap flux density and remarkable magnetic congregate effect. In order to reduce the leakage flux and the high harmonic content in the air-gap, an auxiliary slot is notched in the rotor, as shown in Figure 3.

Structure Design
The shape of magnetic pole effects the output characteristics of PMSM directly. The permanent magnet of typical SIPMSM is a tile shape with inner and outer arc centers at the same point-as shown in Figure 1. A PMSM with this kind of magnetic pole usually has the disadvantages of large cogging torque, large leakage and poor flux weakening capability [16]. Therefore, a novel SIPMSM is developed, as shown in Figure 2. The permanent magnet in the novel SIPMSM is an unequal thickness magnetic pole with different inner and outer radians, which results in the uneven distribution of the radial air-gap flux density and remarkable magnetic congregate effect. In order to reduce the leakage flux and the high harmonic content in the air-gap, an auxiliary slot is notched in the rotor, as shown in Figure 3.

Structure Design
The shape of magnetic pole effects the output characteristics of PMSM directly. The permanent magnet of typical SIPMSM is a tile shape with inner and outer arc centers at the same point-as shown in Figure 1. A PMSM with this kind of magnetic pole usually has the disadvantages of large cogging torque, large leakage and poor flux weakening capability [16]. Therefore, a novel SIPMSM is developed, as shown in Figure 2. The permanent magnet in the novel SIPMSM is an unequal thickness magnetic pole with different inner and outer radians, which results in the uneven distribution of the radial air-gap flux density and remarkable magnetic congregate effect. In order to reduce the leakage flux and the high harmonic content in the air-gap, an auxiliary slot is notched in the rotor, as shown in Figure 3.    According to the performance requirements of PMSM for electric vehicles, the structure parameters of the novel SIPMSM are determined by using the empirical formula (see Table 1).

Infulence on Cogging Torque
Compared with the traditional SIPMSM, the shape of the auxiliary slot and the unequal thickness of the permanent magnet between magnetic poles changes the harmonic content in the air gap flux density, which inevitably affects the cogging torque. In this paper, a mathematical model of the cogging torque of the novel SIPMSM is established based on the energy method, and the influence of auxiliary slot in the rotor on the cogging torque is analyzed.
The cogging torque is defined as the negative derivative of the magnetic field energy, relative to the position angle when the armature winding does not turn on the current [17]. The cogging torque of the permanent magnet motor can be expressed as where E is the energy of the magnetic field in the air-gap; B is the air-gap flux density; V is the volume of air-gap between stator and rotor; α is the relative position angle between stator and rotor. The energy of the air-gap magnetic field can be expressed as According to the performance requirements of PMSM for electric vehicles, the structure parameters of the novel SIPMSM are determined by using the empirical formula (see Table 1).

Infulence on Cogging Torque
Compared with the traditional SIPMSM, the shape of the auxiliary slot and the unequal thickness of the permanent magnet between magnetic poles changes the harmonic content in the air gap flux density, which inevitably affects the cogging torque. In this paper, a mathematical model of the cogging torque of the novel SIPMSM is established based on the energy method, and the influence of auxiliary slot in the rotor on the cogging torque is analyzed.
The cogging torque is defined as the negative derivative of the magnetic field energy, relative to the position angle when the armature winding does not turn on the current [17]. The cogging torque of the permanent magnet motor can be expressed as where E is the energy of the magnetic field in the air-gap; B is the air-gap flux density; V is the volume of air-gap between stator and rotor; α is the relative position angle between stator and rotor. The energy of the air-gap magnetic field can be expressed as where θ is the angle between the air-gap magnetic density and the central line of the magnetic pole; µ 0 is vacuum permeability. The mathematical model of air-gap density distribution along the rotor surface with the unequal thickness magnetic pole is expressed as where, B r (θ) is the distribution of the remanence of permanent magnet along the circumferential direction; δ(θ, α) is the distribution of effective air-gap length along the circumferential direction; h m (θ) is the distribution of the direction of magnetization along the circumferential direction at the minimum thickness of permanent magnet. Substituting (2) into the E, we obtain Perform Fourier decomposition of B 2 r (θ) without considering the influence of the relative position of stator and rotor.
where, p is the number of pole pairs and α p is polar arc coefficient.
where L a is the axial length of the motor; R 1 is the outer arc radius of unequal thickness magnetic poles; h is the vertical distance from the center of the outer arc to the permanent magnet; n is an integer that makes (nz/2p) an integer. According to the triangle similarity theorem, one can get where h max is the maximum thickness of permanent magnet; h m is the minimum thickness of permanent magnet; d is the width of permanent magnet. Substitute (8) into (7), then the cogging torque expression of the SIPMSM with unequal thickness magnetic poles can be described as Symmetry 2020, 12, 179 6 of 14 When the auxiliary slot is notched between the magnetic poles, the back EMF and magnetic field distribution of the motor will be greatly affected, and the high-order harmonic content in the air-gap flux density will be reduced, so the cogging torque of the motor will be weakened [18].
When the number of auxiliary slots is k, the Fourier decomposition coefficients of [ h m h m +δ(θ,α) ] in the − π z , π z can be expressed as: where θ s0 is the width of auxiliary slot in the rotor, X d is auxiliary slot in rotor, and k = 1. By substituting (10) into (9), the expression of cogging torque with slots between magnetic poles of unequal thickness can be obtained as From (11), it can be seen that the structural parameters affecting the cogging torque include the axial length of the motor, the number of pole pairs, the maximum thickness of permanent magnet, the minimum thickness of permanent magnet, the width of permanent magnet, the width of slotting, the depth of the auxiliary slot and the number of stator slots. This paper mainly studies the influence of the notching auxiliary slot between adjacent magnetic poles.

Influence of Notching Auxiliary Slots
The auxiliary slots will increase the magnetic reluctance of the q axis magnetic circuit of SIPMSM. On the one hand, the difference of inductance between the d and q axis will produce reluctance torque. On the other hand, increasing the d axis inductance or reducing the q axis inductance can improve the flux weakening capability of the motor [19,20]. This section mainly studies the influence of the shape of unequal thickness magnetic poles and size of auxiliary slots on the flux weakening capability of the motor.
When the speed of PMSM exceeds the base speed, the phase current and phase voltage will reach the maximum value. In order to ensure that the limiting voltage does not exceed the limit voltage of the controller, the flux weakening control of PMSM is needed [21,22]. When the motor reaches the maximum speed, the stator current is used to weaken the magnetic field. The voltage amplitude is equal to the voltage limit of the controller. It can be seen that the d and q axis voltage in the rotating coordinate system can be expressed as: where u d is the voltage of d axis; u q is the voltage of q axis; R s is the resistance of armature winding; i d is the current of d axis; i q is the current of q axis; Ψ d is the flux linkage of d axis; Ψ q is the flux linkage of q axis; ω is the angular speed of rotor rotation.
The change of current and flux linkage is zero when the motor is in steady state, so the steady state d and q axis voltage equation of the motor is obtained as: When the motor is controlled by flux weakening control, the base speed of the motor can be expressed as: where u lim is the limit voltage; ψ PM is the flux linkage of permanent magnet; L d is the d axis inductance of motor; L q is the q axis inductance of motor.
When the output current of the inverter is all d axis demagnetizing current, that is i q = 0. The ideal maximum speed of the motor can be expressed as According to (14), the expansion of the weakening speed of the magnetic flux can be improved by reducing the permanent magnetic flux chain and increasing the d-axis inductance or reducing the q-axis inductance. According to (15), the expansion of the weakening speed of the magnetic flux can also be improved by increasing the limiting current and limiting voltage. Therefore, this paper mainly changes the inductance of the d and q-axis magnetic circuits by optimizing the design of unequal thickness magnetic poles and the slot size of the rotor. Changing these inductances can improve the performance of magnetic flux weakening speed. The ratio of the maximum speed to the base speed is the flux weakening expansion rate [23]. The expression of the flux weakening speed expansion can be expressed as: Inductance is a physical quantity to measure the ability of a coil to produce electromagnetic induction, and it is the flux linkage produced by the unit current [24]. According to the definition, we can get the inductance of SIPMSM where F c is the magnetomotive of permanent magnet; R M is the magnetic resistance of magnetic circuit through which the self-induction flux; N is the number of turns of conductor. The expressions for calculating the d axis inductance of the novel SIPMSM as: where, R r is the magnetic resistance at the maximum thickness of permanent magnet steel; R z is the rotor core magnetic resistance between the permanent magnet steel and the air-gap; R a is the magnetic resistance of the air-gap; R s is the magnetic resistance of stator; R y is the magnetic resistance of the stator yoke. The expressions for calculating the q axis inductance of the novel SIPMSM as: where, R d is the magnetic resistance of auxiliary slot. The developed SIPMSM adopts the asymmetric magnetic circuit; the d axis flux linkage passes through unequal thickness magnetic poles, rotor core and air-gap, stator teeth and stator yoke; the q Symmetry 2020, 12, 179 8 of 14 axis flux linkage passes through the slot of slot, air-gap, stator boots and stator yoke. Due to large magnetic resistance of permanent magnet, the thickness of magnetic poles is approximately equal to the air-gap. When the auxiliary slot is notched between the magnetic poles in the rotor, the inductance in the q axis decreases with the increase in the depth of the auxiliary slot.
In summary, the depth of the auxiliary slot and maximum thickness of the permanent magnet influence the cogging torque and flux weakening speed expansion. This paper will optimize these two parameters.

Optimization Model
In the proposed SIPMSM, the complicated rotor topology and different saturation degree of the iron core make it difficult to solve the relationship between the object and the independent variables by analytic method. Because of the flexibility of the real response surface and the interaction between factors, the second order or higher order model is usually used to approximate the response objective function. The relationship between the maximum thickness of permanent magnet h max , the depth of auxiliary slot X d , the cogging torque T cog and the flux weakening expansion rate ρ can be obtained by the combination of RSM and finite element method. The experimental arrangements and finite elements results are shown in Table 2, where X 1 and X 2 are the factor coded values, Y 1 is the cogging torque, and Y 2 is the flux weakening expansion rate. The second-order model can be expressed as follows: where x i is the coded variables; β i is the linear effect of x i ; β ij is the interaction between x i and x j ; β ii is the quadratic effect of x i . Because of the space limitation of rotor, the parameter level is limited in a certain range. Therefore, the central composite surface design (CCF) is adopted in this paper. All test points do not exceed the requirements of the cube and meet the design requirements [25][26][27].
The regression models of Y 1 and Y 2 are analyzed using Design-Expert software. According to the regression model equation, the response surface and contour map of the interaction factors to peak value of cogging torque and flux magnetic speed expansion can be established, as shown in  The regression models of Y1 and Y2 are analyzed using Design-Expert software. According to the regression model equation, the response surface and contour map of the interaction factors to peak value of cogging torque and flux magnetic speed expansion can be established, as shown in   It can be seen from Figure 4, when x1 varies from 5 to 7 mm, the peak value of the cogging torque decreases first and then increases. If x1 is fixed at a certain level, the peak value of the cogging torque decreases first and then increases. Therefore, according to the analysis of partial regression equation and contour plot, the effect of the depth of auxiliary slot in rotor on the flux magnetic speed expansion is greater than the maximum thickness of the permanent magnet.
It can be seen from Figure 5, when x1 changes from 5 to 7 mm, the value of flux weakening expansion ratio of the motor increases first and then decreases. If x1 is fixed at a certain level, with the increase in x2, the peak value of cogging torque trends to decrease. Therefore, according to the analysis of partial regression equation and the contour plot, the effect of the maximum thickness of the permanent magnet is greater than the depth of auxiliary slot in the rotor.
In the motor design, the cogging torque is always expected to be small, while the flux weakening speed expansion is always expected to be large. So, a quasi-function is constructed using RSM. The relationship between independent variables and response variables in the motor and the nonlinearity is solved by solving the mathematical model for multi-objective optimization. Thus, the global solution of optimization calculation is obtained: (hmax, Xd) = (6.40, 7). The optimal response is (Tcog, ρ) = (1.16, 2.08).

Experimental Validation
In order to verify the correctness of the optimization design and the superiority of the novel SIPMSM. In this paper, two SIPMSMs with the same-rated parameters were trial-produced, as shown in Figures 6 and 7. The optimum design structures of traditional SIPMSM and the novel SIPMSM were tested and verified, and the performance of the test results was compared. The structure of the stator, winding layout, outer diameter of rotor, volume of permanent magnet and pole arc coefficient of the two motors are the same-the main difference between the two motors is that the maximum thickness of the rotor pole and slotting structure are different. It can be seen from Figure 4, when x 1 varies from 5 to 7 mm, the peak value of the cogging torque decreases first and then increases. If x 1 is fixed at a certain level, the peak value of the cogging torque decreases first and then increases. Therefore, according to the analysis of partial regression equation and contour plot, the effect of the depth of auxiliary slot in rotor on the flux magnetic speed expansion is greater than the maximum thickness of the permanent magnet.
It can be seen from Figure 5, when x 1 changes from 5 to 7 mm, the value of flux weakening expansion ratio of the motor increases first and then decreases. If x 1 is fixed at a certain level, with the increase in x 2 , the peak value of cogging torque trends to decrease. Therefore, according to the analysis of partial regression equation and the contour plot, the effect of the maximum thickness of the permanent magnet is greater than the depth of auxiliary slot in the rotor.
In the motor design, the cogging torque is always expected to be small, while the flux weakening speed expansion is always expected to be large. So, a quasi-function is constructed using RSM. The relationship between independent variables and response variables in the motor and the nonlinearity is solved by solving the mathematical model for multi-objective optimization. Thus, the global solution of optimization calculation is obtained: (h max , X d ) = (6.40, 7). The optimal response is (T cog , ρ) = (1.16, 2.08).

Experimental Validation
In order to verify the correctness of the optimization design and the superiority of the novel SIPMSM. In this paper, two SIPMSMs with the same-rated parameters were trial-produced, as shown in Figures 6 and 7. The optimum design structures of traditional SIPMSM and the novel SIPMSM were tested and verified, and the performance of the test results was compared. The structure of the stator, winding layout, outer diameter of rotor, volume of permanent magnet and pole arc coefficient of the two motors are the same-the main difference between the two motors is that the maximum thickness of the rotor pole and slotting structure are different.  The two prototypes were tested on the eddy current dynamometer, as shown in Figure 8. With the increase in the motor speed, the load increases continuously, so that the motor current reaches the limit value. Then the sensor detects and collects signals. Finally, the output characteristic curve and efficiency map of the two SIPMSMs are shown in Figures 9 and 10.   The two prototypes were tested on the eddy current dynamometer, as shown in Figure 8. With the increase in the motor speed, the load increases continuously, so that the motor current reaches the limit value. Then the sensor detects and collects signals. Finally, the output characteristic curve and efficiency map of the two SIPMSMs are shown in Figures 9 and 10.  The two prototypes were tested on the eddy current dynamometer, as shown in Figure 8. With the increase in the motor speed, the load increases continuously, so that the motor current reaches the limit value. Then the sensor detects and collects signals. Finally, the output characteristic curve and efficiency map of the two SIPMSMs are shown in Figures 9 and 10.  The two prototypes were tested on the eddy current dynamometer, as shown in Figure 8. With the increase in the motor speed, the load increases continuously, so that the motor current reaches the limit value. Then the sensor detects and collects signals. Finally, the output characteristic curve and efficiency map of the two SIPMSMs are shown in Figures 9 and 10.   Compared with Figures 9 and 10, it can be seen that the maximum efficiency of the novel SIPMSM is 91.12%, and the maximum speed that can be achieved is only 4750 rpm, while the maximum efficiency of traditional SIPMSM is 93.45%, and the maximum speed is 6250 rpm. Moreover, the novel SIPMSM has a wider range of high efficiency working area, which shows that the novel SIPMSM has a better performance in terms of flux magnetic speed expansion, thus verifying the veracity of theoretical analysis.
At the same time, PMSM for electric vehicles needs a variable speed drive and wide speed range, because electric vehicles need to start and stop frequently at low speeds under traffic congestion conditions, and good acceleration performance is required when traffic is smooth or running on highways [28,29]. Obviously, the novel SIPMSM has this output characteristic.
Finally, the experiment on the torque ripple was tested on the test platform from startup to related load, and the test results are shown in Figure 11.  Compared with Figures 9 and 10, it can be seen that the maximum efficiency of the novel SIPMSM is 91.12%, and the maximum speed that can be achieved is only 4750 rpm, while the maximum efficiency of traditional SIPMSM is 93.45%, and the maximum speed is 6250 rpm. Moreover, the novel SIPMSM has a wider range of high efficiency working area, which shows that the novel SIPMSM has a better performance in terms of flux magnetic speed expansion, thus verifying the veracity of theoretical analysis.
At the same time, PMSM for electric vehicles needs a variable speed drive and wide speed range, because electric vehicles need to start and stop frequently at low speeds under traffic congestion conditions, and good acceleration performance is required when traffic is smooth or running on highways [28,29]. Obviously, the novel SIPMSM has this output characteristic.
Finally, the experiment on the torque ripple was tested on the test platform from startup to related load, and the test results are shown in Figure 11. Compared with Figures 9 and 10, it can be seen that the maximum efficiency of the novel SIPMSM is 91.12%, and the maximum speed that can be achieved is only 4750 rpm, while the maximum efficiency of traditional SIPMSM is 93.45%, and the maximum speed is 6250 rpm. Moreover, the novel SIPMSM has a wider range of high efficiency working area, which shows that the novel SIPMSM has a better performance in terms of flux magnetic speed expansion, thus verifying the veracity of theoretical analysis.
At the same time, PMSM for electric vehicles needs a variable speed drive and wide speed range, because electric vehicles need to start and stop frequently at low speeds under traffic congestion conditions, and good acceleration performance is required when traffic is smooth or running on highways [28,29]. Obviously, the novel SIPMSM has this output characteristic.
Finally, the experiment on the torque ripple was tested on the test platform from startup to related load, and the test results are shown in Figure 11. The novel SIPMSM The tranditional SIPMSM Figure 11. Torque ripple comparison between motor startup and rated load.
From Figure 11, it can be seen that the output torque of the two SIPMSMs reaches a stable state at 77 s, owing to the same control mode and rated parameters. However, the output torque of the novel SIPMSM is clearly higher than the traditional SIPMSM. This is because the reasonable design of the permanent magnets with unequal thickness makes the air-gap magnetic field have a significant magnetic congregate effect and enhances the output torque. It is not difficult to see that the torque ripple of the novel SIPMSM is lower than the traditional SIPMSM when it reaches steady state.
Torque ripple of PMSM is composed of cogging torque and ripple torque [30]. However, ripple torque is mainly caused by the motor control mode. In order to control a single variable, this paper uses the same controller for two motors. In this case, torque ripple is only determined by the cogging torque. Compared with Figures 10 and 11, it can be seen that the torque ripple of the novel SIPMSM is significantly lower than the traditional SIPMSM. Therefore, notching auxiliary slots in the rotor reduces the high order harmonic content in the air-gap flux density, weakens the cogging torque, and verifies the correctness of the design. The veracity of the theoretical analysis and optimization design is confirmed.

Conclusions
A novel SIPMSM by the method of notching auxiliary slots between the magnetic poles in the rotor and unequal thickness magnetic poles is developed. Compared with the traditional SIPMSM, it has the characteristics of large output torque, a wide range of speed regulation and a low cogging torque. The findings of the experimental results are described as follows: (1) The mathematical model of cogging torque of the novel SIPMSM is established, and the influence factors of the cogging torque and flux weakening speed expansion of the novel SIPMSM were deduced. Combining the RSM and finite element method, the multi-objective optimization design of the influencing factors was carried out. The optimization results show that when the maximum thickness of magnetic poles is 6.4mm and the depth of rotor slot is 7mm, the peak value of cogging torque is 1.16 nm and the flux weakening speed rate is 2.08. (2) The prototype test shows that-compared with the traditional SIPMSM-the new SIPMSM not only enhances the output torque and reduces the torque ripple, but also improves the performance of flux weakening speed expansion. At the same time, the high efficiency range of the constant power operation is widened, and more in line with the performance requirements of PMSM for electric vehicles. Therefore, the novel SIPMSM is more suitable for electric vehicles. From Figure 11, it can be seen that the output torque of the two SIPMSMs reaches a stable state at 77 s, owing to the same control mode and rated parameters. However, the output torque of the novel SIPMSM is clearly higher than the traditional SIPMSM. This is because the reasonable design of the permanent magnets with unequal thickness makes the air-gap magnetic field have a significant magnetic congregate effect and enhances the output torque. It is not difficult to see that the torque ripple of the novel SIPMSM is lower than the traditional SIPMSM when it reaches steady state.
Torque ripple of PMSM is composed of cogging torque and ripple torque [30]. However, ripple torque is mainly caused by the motor control mode. In order to control a single variable, this paper uses the same controller for two motors. In this case, torque ripple is only determined by the cogging torque. Compared with Figures 10 and 11, it can be seen that the torque ripple of the novel SIPMSM is significantly lower than the traditional SIPMSM. Therefore, notching auxiliary slots in the rotor reduces the high order harmonic content in the air-gap flux density, weakens the cogging torque, and verifies the correctness of the design. The veracity of the theoretical analysis and optimization design is confirmed.

Conclusions
A novel SIPMSM by the method of notching auxiliary slots between the magnetic poles in the rotor and unequal thickness magnetic poles is developed. Compared with the traditional SIPMSM, it has the characteristics of large output torque, a wide range of speed regulation and a low cogging torque. The findings of the experimental results are described as follows: (1) The mathematical model of cogging torque of the novel SIPMSM is established, and the influence factors of the cogging torque and flux weakening speed expansion of the novel SIPMSM were deduced. Combining the RSM and finite element method, the multi-objective optimization design of the influencing factors was carried out. The optimization results show that when the maximum thickness of magnetic poles is 6.4mm and the depth of rotor slot is 7 mm, the peak value of cogging torque is 1.16 nm and the flux weakening speed rate is 2.08. (2) The prototype test shows that-compared with the traditional SIPMSM-the new SIPMSM not only enhances the output torque and reduces the torque ripple, but also improves the performance of flux weakening speed expansion. At the same time, the high efficiency range of the constant power operation is widened, and more in line with the performance requirements of PMSM for electric vehicles. Therefore, the novel SIPMSM is more suitable for electric vehicles.