，，，，

1.Hubei Key Laboratory for High-Efficiency Utilization of Solar Energy and Operation Control of Energy Storage System，School of Electrical and Electronic Engineering，Hubei University of Technology，Wuhan 430068，P.R.China；

2.School of Electrical and Electronic Engineering，Huazhong University of Science and Technology，Wuhan 430074，P.R.China

Abstract:With the increasing requirement for the mechanical vibration and acoustic noise of the permanent magnet synchronous motor（PMSM）drive system，the demand for cogging torque reduction of PMSM has been considerably increased.To solve the problem of oversized cogging torque of axial flux PMSM，a rotor topology with hybrid permanent magnet is proposed to weaken the cogging torque.Firstly，the expression of the cogging torque of the axial flux motor is derived，and the influence of the pole-arc ratio of the permanent magnet on the cogging torque is analyzed.Secondly，the rotor structure with hybrid permanent magnet is adopted to reduce the cogging torque.According to the analytical analysis，the constraints of the size and pole-arc ratio between the hybrid permanent magnets are obtained，and the two permanent magnets related to the minimum cogging torque are determined.And the analysis results are verified by the finite element simulation.Furthermore，the motor performance with and without the hybrid permanent magnet is compared with each other.Finally，the cogging torque is significantly reduced by adopting a rotor structure with hybrid permanent magnets.

Key words：axial flux permanent magnet synchronous motor（AFPMSM）；rotor structure；cogging torque；hybrid permanent magnet；pole-arc ratio

0 Introduction

The axial flux permanent magnet synchronous motor（AFPMSM）has many advantages，such as high power density and short axial length，which is widely used in the application of electric vehicles［1-3］.However，the deteriorate performance of noise，vibration，harshness（NVH）and poor reliability are appeared by the existence of cogging torque of the motor［4］.

In recent years，scholars have made many relevant researches on weakening methods for cogging torque of machine.Several methods are introduced to reduce the cogging torque of variable flux reluctance machines including asymmetrical rotor teeth，circumferential rotor teeth pairing and rotor dummy slots［5］.To reduce the cogging torque for hybrid axial field flux-switching permanent magnet machine，three cogging torque reduction methods by tangential displacement of stators，asymmetric rotor pole shape，and segment-twisted rotor are investigated［6］.Cogging torque optimization of a flux memory polechanging permanent magnet machine is presented by the Taguchi method，the response surface methodology，and the genetic algorithm［7］.The analytical approach used to study cogging torque in axial flux permanent magnet（AFPM）machines can be as efficient as the 3-D finite element analysis（FEA）while saving a huge amount of time and necessitating less initial expertise［8］.The analytical expression of cogging torque of flux reversal permanent magnet motor is derived.The expression considers the small gap between two adjacent magnets belonging to the same sub tooth.Furthermore，the optimal dimensions of the space gap are obtained analytically for cogging torque minimization［9］.The relation between the permanent magnet（PM）and the harmonic order of the cogging has been studied by the analytical method，and the pole-slot coordination with minimum cogging torque has been derived［10］.An improved semi-analytical model is proposed to investigate the cogging torque of AFPM under angular misalignment，and it is more accurate for angular misalignment［11］.The influence of magnet pole-arc ratio on cogging torque of PM motor was studied in Refs.［12-13］，and the cogging torque is effectively weakened by optimizing the pole-arc ratio.

The optimized shape of PM can weaken the cogging torque of PMSM.An asymmetrical V-type rotor configuration is adopted to reduce cogging torque and torque ripple［14］.The application position of skew in order to reduce the cogging torque of a magnetic geared synchronous motor（MGM）is studied［15］.PM is shifted by a certain angle along the radial direction，which reduces the cogging torque and torque ripple［16-17］.However，the manufacturing process of PM is complicated.Dividing the PM into two pieces and only shifting one of them by a certain angle can also effectively suppress the cogging torque without increasing the difficulty of the manufacturing process of PM［18］.

In addition，many scholars use the method of combining magnetic poles to weaken cogging torque of surface-mount radial flux motors.A novel combined rectangle-shaped magnet arrangement was proposed and investigated for cogging torque minimization in Ref.［19］.In Ref.［20］，two materials，NdFeB and samarium cobalt，were mixed along the axial direction，and the best combination of pole-arc ratio was obtained through analytical calculation［20］.In Ref.［21］，two PMs were combined to weaken the harmonics of air-gap magnetic field，and the optimal combination of magnet pole-arc ratio was obtained by zooming optimization algorithm.This paper also mentioned PMs with different thicknesses.The cogging torque is reduced by optimizing the air gap magnetic field.Based on the results of literature survey，the reduction of cogging torque of radial flux permanent magnet motor has been deeply studied［22］.Other scholars have studied the surface mounted radial permanent magnet hybrid motor.Through calculation and simulation analysis，the hybrid permanent magnet motor can not only reduce the permanent magnet cost of the motor，but also weaken the cogging torque of the motor，thus reducing the torque ripple［23］.The above methods proposed by scholars can effectively reduce the cogging torque of the motor.However，these methods either have difficulties in machining and motor assembly，or some methods will weaken the air gap flux density of the motor.Considering the weakening method of cogging torque of radial flux permanent magnet motor，the reduction of cogging torque of axial flux motor is analyzed according to the characteristics of axial flux permanent magnet motor.

In order to weaken the cogging torque without increasing the processing technology and cost，a rotor structure with hybrid permanent magnets is proposed.In this paper，a 7.5 kW yokeless and segmented armature（YASA）axial flux permanent magnet motor is taken as the research object，and the structure and basic parameters of the motor are shown in Fig.1 and Table 1.First，an energy method is used to calculate the cogging torque of AFPMSM.The cogging torque is affected by the magnet pole-arc ratio.Then，the cogging torque with different pole-arc ratios are analyzed by the analytical method when PMs are mixed.The pole-arc ratio is obtained by the graphic method when cogging torque is minimum.Finally，the finite element simulation is used to verify the accuracy of the constraints.

Fig.1 Topology structure of the YASA motor

Table 1 Basic parameters of the YASA motor

1 Analysis of Cogging Torque of Axial Flux Motor

1.1 Calculation of cogging torque

The following assumptions are made for the analytical calculation of cogging torque：（1）The permeability of the core is infinite.（2）The permeability of PM is the same as air.（3）The air gap flux density is uniformly distributed in the radial direction.

Cogging torque is produced by the interaction between PM and iron core when AFPMSM is not powered，and it is the negative derivative of the magnetic field energyWrelative to the position angleα，shown as

The energy stored in the magnetic field can be approximately regarded as the energy in PM and air gap，so it can be expressed as

whereμ0is the permeability of air，αthe angle between the center line of a specified PM and tooth，Brthe remanence density of PM，hmthe length of the magnetization direction of PM，δthe effective air gap length，θthe angle of change in the direction of rotation of the motor andVthe air gap volume，as shown in Fig.2.

If PMs are evenly distributed，the Fourier expansion ofcan be expressed as

whereαpis the pole-arc ratio of PM andpthe number of rotor poles.

The Fourier expansion of［hm（θ）/hm（θ）+can be expressed as

whereGnis the Fourier expansion coefficient of the relative permeability function of motor air gap andZthe number of motor stator slots.

The volume of the air gapVis

whereD1andD2are the machine diameters at inner and outer surfaces，respectively.

Substituting the above analysis results into Eq.（1），the cogging torque of AFPMSM can be expressed as

whereNLis the least common multiple of the number of motor stator slots and the number of rotor poles.

1.2 Influenceof pole-arc ratio on cogging torque

Eq.（6）indicates thatBrnhas a greater influence on the cogging torque，which can be expressed as

wherenis an integer.It can makenZ/2pas an integer.Furthermore，there are infinite integersnthat satisfy above condition.In addition，nis defined to be arranged from small to large asn1，n2，n3，…，nn.pis the number of pole pairs，andαpis the polearc ratio of PM.The number of cycles of cogging torque can be determined（that is，n1can be determined）when the number of poles and slots are determined.The axial flux permanent magnet motor studied in this paper is 24 slot/20 pole.According to the above analysis，the period of cogging torque is 5，i.e.，n1=5，and the obtainednZ/2p（3，6，9，…）times harmonic of the air gap flux density has an influence on cogging torque.According to the graphical method，substitutingn=5，10，15 and 20 into Eq.（7），the curve ofBrnchanging with pole-arc ratioαp，is shown in Fig.3.

Fig.3 Graphical analysis of Brn andαp

Combining Eqs.（6，7），the expression of the cogging torque of AFPMSM can be obtained as

It can be obtained from Fig.3 and Eq.（8）that，asαpchanges in the range of［0，1］，Brnalso changes periodically，andTcog（A）is proportional toBrn，soTcog（A）will also change periodically withαp.

2 Hybrid Permanent Magnet Method

2.1 Analytical analysis

In this paper，a rotor topology with hybrid permanent magnet is adopted to reduce the cogging torque.The structure of the motor with hybrid permanent magnet is shown in Fig.4.This machine uses two kinds of PMs with different materials to excite the motor，and the two PMs are arranged radially，as shown in Fig.5.

Fig.4 Structure of hybrid permanent magnet

Fig.5 Arrangement of hybrid permanent magnets

When two different PMs are mixed in the radial direction，it can be considered that the magnetic circuits generated by the two PMs are in parallel，and the motor with hybrid permanent magnet can be equivalent to two axial flux motors using PM 1 and PM 2，respectively.The cogging torque of the motor with hybrid permanent magnet can be equivalent to the superposition of two axial flux motors［20］.Let the cogging torque of two axial flux motors beTcog（A1）andTcog（A2），shown as

whereD12andD11are the outer diameter and inner diameter of the motor with PM 1，D22andD21the outer diameter and inner diameter of the motor with PM 2，Br1andBr2the remanences of PM 1 and PM 2，andαp1andαp2the pole-arc ratios of PM 1 and PM 2.When the motor with hybrid permanent magnet is in a steady state，the air gap flux density is uniform.Therefore，the air gap flux amplitude of two motors is equal，and it can be obtained as

Assuming that the PM has the same remanence，the cogging torque of two axial flux motors is to be superimposed on each other according to Eq.（10），which can be expressed as

whereBris the equivalent remanence of the motor with hybrid permanent magnet，andDoandDiare the machine diameters at outer and inner surfaces.The cogging torque of this machine can be expressed as

whereBrn（h）is the superposition ofBrnof two motors，which can be expressed as

When the operating point of the PM in the motor with hybrid permanent magnet is unchanged，the air gap flux density and pole-arc ratio of the two PMs should meet

The air gap flux density of surface mount motor is proportional to the remanence of PM.Therefore，the relationship between the pole-arc ratio and remanence of two PMs can be summarized as

Combining Eqs.（10—12），the expression ofBrn（h）can be obtained as

Eq.（16）takesαp1as the only variable.Substituting Eq.（11，16）into Eq.（12），the cogging torque of the motor with hybrid permanent magnet is

It can be obtained by Eq.（17）thatTcog（A）is the sine function ofαp1.The sine function in Eq.（17）is represented byk，shown as

Tcog（A）takes the minimum value whenαp1makeskzero.According to Eq.（18）and the remanences of two kinds of PMs，the curve ofkchanging withαp1can be obtained by the graphical method.In this paper，N38UH and N 35UH are selected for mixing，the remanences of them are 1.25 T and 1.2 T，respectively. Put the remanence into Eq.（18），the curve of the value ofkchanging withαp1under different harmonic orders can be obtained through the graphical method.The results are shown in Fig.6.

Fig.6 Analysis results of k of hybrid permanent magnet motor under different harmonic orders

According to Fig.6，whenBrn1＞0 andBrn2＜0，orBrn1＜0 andBrn2＞0，Brncorresponding to the two PMs in the motor with hybrid permanent magnet weaken each other，and the cogging torque is reduced.The minimum ofTcog（A）can be got whenBrn1=-Brn2andBrnis zero.With the increase ofnandαp，the amplitude ofkshows a tendency of decreasing，which indicates that the rotor topology with hybrid permanent magnet has a very obvious effect on weakening the cogging torque generated by higher harmonics.It can be obtained by analysis thatkis to be zero whenαp1=0.2，0.39，0.5，0.78 and 0.97，meanwhile，Tcog（A）is minimum.

2.2 Finite-element verification

The finite element simulation method is used to verify the correctness of theoretical analysis.Two motors studied in this paper are axial flux motors，i.e.，Nd FeB motor with one kind of Nd FeB and a hybrid permanent magnet motor with a mixture of two Nd FeBs.For Nd FeB motor，the inner diameter and outer diameter of PM are equal to the inner diameter and outer diameter of motor.The size of the PM of hybrid permanent magnet motor needs to be calculated through the constraint conditions in Eq.（11）and the inner and outer diameters of the motor are

The calculation results show thatD12=D21=163 mm.The materials and dimensions of the PMs of the two motors are shown in Table 2.

Table 2 PM sizes of two motors

The cogging torque of NdFeB motor with different magnet pole-arc ratio is simulated by FEM and the results are shown in Fig.7.

Fig.7 Simulation results of cogging torque of NdFeB motor under differentαp

Fig.7 shows that the cogging torque of Nd FeB motor reaches the maximum of 5.7 N·m in the simulation range whenαpis 0.7，and reaches the minimum of 3 N·m whenαpis 0.8.It can be seen from Figs.2，6 that the cogging torques with different magnet pole-arc ratios obtained by the analytical method and the finite element simulation have the same trends，indicating that the graphical method in Fig.6 has a certain accuracy.

Fig.6 shows that the theoretical pole-arc ratio of N 38UH is obtained when the cogging torque of hybrid permanent magnet motor is the smallest.The pole-arc ratio of N35UH is obtained by Eq.（15），and the results are shown in Table 3.

Table 3 Simulation resultsof polar-arc ratio combination

According to Table 3，when the pole-arc ratios of N 38UH and N 35UH are 0.78 and 0.81，respectively，the cogging torque is the smallest.And the greater the pole-arc ratio，the stronger the torque output capacity，which is beneficial to improving torque density of the motor.Therefore，this paper chooses the pole-arc ratios of the two PMs to be 0.78 and 0.81.

Constraints of magnet pole-arc ratio are determined by Eq.（15）.According to the results obtained above，the cogging torque of hybrid permanent magnet motor can obtain the minimum whenαp1andαp2are 0.78 and 0.81，respectively.To verify this conclusion，keepαp1unchanged，letαp2be mixed in the range of［0.75，0.85］，and the cogging torque of each group of hybrid permanent magnet structure is analyzed by the finite element method.

The simulation results of permanent magnet pole-arc ratio constraints are shown in Fig.8.As shown in Fig.8，the FEA results of constraint conditions of magnet pole-arc ratio are consistent with the analytical results，and the accuracy of constraint conditions in Eq.（15）is verified.

Fig.8 Simulation results of pole-arc ratio constraints of PMs

Since the inner and outer diameters and the positions of two kinds of PMs of the motor have been determined，the inner diameter of PM 1D11=120 mm and the outer diameter of PM 2D22=200 mm.Meanwhile，there is no gap in the radial direction of the two PMs.Therefore，the outer diameter of PM 1D12and the inner diameter of PM 2D21are equal.In order to verify the accuracy of constraint conditions，D12is used as a variable to simulate the cogging torque of hybrid permanent magnet motor whenD12changes within the range of［158，167］.The results are shown in Fig.9.

Fig.9 Simulation results of permanent magnet pole-arc ratio constraints

It can be seen from Fig.9 that the simulation result of constraint condition of the permanent magnet size is consistent with the analytical results，which verifies the accuracy of constraint condition in Eq.（11）.

Fig.6 indicates that the cogging torque of hybrid PM motor is the smallest whenαp1=0.78.To verify the conclusions drawn in Fig.6，makeαp1change within the range of［0.7，0.84］，calculateαp2by Eq.（15），and perform finite element simulation analysis on the combination of pole-arc ratio of different PM.The results are shown in Fig.10.

Fig.10 Simulation results ofαp1 and motor cogging torque

It can be seen from Fig.10 that with the increase of PM 1 pole-arc ratio，the cogging torque first decreases and then increases.The cogging torque is minimum whenαp1=0.76 andαp2=0.79.The analytical calculation results show that the cogging torque is the smallest whenαp1= 0.78 andαp2=0.81，there is a 2.6%error between the simulation and calculation results.It is because that the graphical method is based on the derivation of the cogging torque of AFPMSM.Several assumptions are used in the derivation process of cogging torque，for example，the magnetic permeability of the motor is infinite，the internal magnetic flux of the motor will not saturate，and the motor is also assumed to be slotless when calculatingBrn，so a certain error is generated in the calculation result.

Summarizing the above analysis，the size and pole-arc ratio of two kinds of PMs in hybrid permanent magnet motor are shown in Table 4.

Table 4 Parameters of two kinds of PMs in hybrid permanent magnet motor

Table 5 Performance comparison of two motors

3 Analysis and Comparison of Performance

The performance of Nd FeB motor and hybrid permanent magnet motor are compared.The structures of the two motors are shown in Fig.1 and Fig.4，respectively.Due to the pole-arc ratio of N38UH in hybrid permanent magnet motor is 0.76，for ease of comparison，the NdFeB motor uses permanent magnets of the same material and pole-arc ratio.

Fig.11 shows the spatial distribution of the radial air gap magnetic flux density over one pole of the proposed motor.It can be seen that the air gap magnetic flux density amplitude is not the same in thexaxis direction.The magnetic flux density amplitude is larger in the range fromx=60 mm tox=81.5 mm.This part of the magnetic flux density is produced by PM 1，and the remaining part of the air gap magnetic flux density is produced by PM 2.The interaction of different air gap flux densities in the radial direction can reduce cogging torque.

Fig.11 Space distribution of radial air gap flux density over one pole

The magnetic flux density distribution on the rotor with hybrid permanent magnet is shown in Fig.12.It can be seen from Fig.12 that the magnetic flux density on the surface of PM 1 is greater than that on the surface of PM 2，which is also consistent with the above analysis.

Fig.12 Magnetic flux density distribution on rotor with hybrid permanent magnet

The no-load back electromotive force（EMF）and the Fourier analysis waveform before and after hybrid permanent magnet are shown in Figs.13，14.It can be seen from Figs.13，14 that after hybrid permanent magnet the back EMF waveforms remain symmetrical，and the amplitude of the fundamental back EMF is only reduced from 78.8 V to 75 V，reduced by 4.5%，and the total harmonic distortion（THD）is increased from 2.7%to 3.7%.It is because that the application of the hybrid permanent magnet method brings double PMs，resulting in increasing of magnetic flux leakage and decreasing of main magnetic flux.Therefore，the amplitude of the fundamental no-load back EMF decreases，and the back EMF has an effect on output torque.The greater the amplitude of back EMF fundamental wave，the greater the average torque output by the motor.The harmonic content of the back EMF will affect the pulsation of the output torque.The torque pulsation will increase with the increase of the harmonics.

Fig.13 Waveforms of no-load back EMF of two motors

Fig.14 Fourier analysis of two motors with no-load back EMF

The cogging torque waveforms of the two motors are shown in Fig.15.It can be seen from Fig.15 that the cogging torque of the hybrid permanent magnet is reduced from 4.6 N·m to 1.3 N·m，a decrease of 71.7%.It indicates that the method of hybrid permanent magnet can effectively reduce the cogging torque.

Fig.15 Cogging torque waveforms of two motors

The rated torque waveforms of the two motors at 39 A are shown in Fig.16.It can be seen from Fig.16 that the average torque of hybrid permanent magnet motor is reduced from 53.4 N·m to 51.9 N·m，and the torque ripple is decreased from 14.1%to 5.6%.Due to the increase of the magnetic flux leakage after the rotor topology with hybrid permanent magnet is adopted，the torque output capacity is reduced，and the cogging torque is also decreased，thus the torque ripple is reduced.

Fig.16 Rated torque waveforms of two motors

Fig.17 shows the torque ripple curves changing with currentI.As shown in Fig.17，the torque ripple of the two motors shows a downward trend with the increase ofI.And the proposed hybrid permanent magnet motor has the lower torque ripple than the Nd FeB motor at each current.WhenI=4 A，the ripple value of the hybrid permanent magnet motor is 34.2%，which is 57.5%less than that of the Nd FeB one.WhenI=40 A，the proposed hybrid permanent magnet motor is 6.37%，which is 43.37%less than that of the Nd FeB one.This rotor topology is also proved to be effective to reduce the torque ripple，which is consistent with the analysis results in Figs.13，14.

Fig.17 Torque ripple versus current

The peak torque waveforms of two motors are shown in Fig.18.It can be seen from Fig.18 that after the rotor structure with hybrid permanent magnet is adopted，the peak torque is reduced from 103.3 N·m to 101.7 N·m，and the torque ripple is decreased from 9.5%to 4.8%.

Fig.18 Peak torque waveforms of two motors

The cogging torque，average torque，torque ripple and efficiency of two motors are compared in Table 5.For ease of comparison，set the two motors to the same volume.It can be seen from Table 5 that when the rotor structure with hybrid permanent magnet is adopted，the cogging torque drops from 4.6 N·m to 1.3 N·m，a decrease of 71.7%.The torque drops from 53.4 N·m to 51.9 N·m，reduced by 2.8%，and the torque ripple drops from 14.1%to 5.7%，reduced by 59.6%in rated condition.In peak condition，the torque drops from 103.3 N·m to 101.7 N·m，which is reduced by 1.5%，and the torque ripple drops from 9.5%to 4.8%，which is reduced by 49.5%.And the amount of permanent magnets of the two motors is almost the same.Therefore，the method of hybrid permanent magnet can improve the torque performance of the motor.

4 Conclusions

In this paper，a rotor topology with hybrid permanent magnet of AFPMSM is presented and the cogging torque is reduced.The major contributions of this paper are：

（1）The cogging torque can be reduced when the size and pole-arc ratio of the two PMs strictly followed the constraints in this paper.

（2）There exists 2.6%error between the optimal pole-arc ratio obtained by the finite element simulation and analytical results due to the use of hypothetical theory in the analysis of cogging torque.

（3）By using the hybrid permanent magnet motor rotor topology，the magnetic leakage increases，which reduces the no-load back-EMF and the rated torque by 4.5%and 1.9%，respectively.

（4）The cogging torque and torque ripple reductions are 71.7%and 59.6%，respectively，by the topology of hybrid permanent magnet.