表贴式永磁同步电机磁极优化建模与分析

Modeling and Analyzing of Surface-Mounted Permanent-Magnet Synchronous Machines With Optimized Magnetic Pole Shape Zhenfei Chen1,Changliang Xia1,2,Qiang Geng2,and Yan Yan11School of Electrical Engineering and Automation,Tianjin University,Tianjin300072,China2Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy,Tianjin Polytechnic University,Tianjin300387,ChinaTwo types of eccentric magnetic pole shapes for optimizing conventional surface-mounted permanent-magnet(PM)synchronous machines with radial magnetization are presented in this paper.An analytical method based on an exact subdomain model and discrete idea is proposed for obtaining the air-gapflux density distribution in the improved motor.Cogging torque and back EMF analytical models are further built with thefield solution,which provide useful tools for investigating motor performances with unequal thickness magnetic poles.The accuracy and feasibility of the models have been validated by afinite element method.Based on the analytical models,the effects of pole shape parameters on motor performance are investigated.Results show that both pole shapes can perfect magneticfield distribution,decrease harmonic content of back EMF,reduce torque ripples,and improve the utilization of PMs.Index Terms—Exact subdomain model,flux density distribution,magnetic pole shape optimization,surface-mounted permanent-magnet(PM)synchronous machine.I.I NTRODUCTIONT HE surface-mounted permanent-magnet(PM) synchronous machine has been widely used in elevator,wind turbine,and hybrid electric vehicle applications due to its high efficiency,power factor,and torque density [1],[2].The PM pole,as a pivotal part of the PM motor, directly affects motor cost and behavior,such as magnetic field,back EMF,torques,and so on.As a result,magnetic pole design is particularly important in PM motor design and has attracted lots of attention.Studies in[3]–[6]point out that the contributions of different PM parts are not uniform and magnetic pole optimization can not only improve PM material utilization,reduce magnet material cost,but also achieve more sinusoidal magneticfield distribution and lower cogging torque performance.The magneticfield calculation is an important prerequisite for the analysis of PM machines.Many methods have been proposed for magneticfield prediction in past few decades. In[7],the drawbacks and stability of numerical implementa-tion are discussed and a semianalytical framework is presented for solving2-D PM machine models in three different coordi-nates.Nevertheless,analytical modeling is usually much more complex for improved PM motors with optimized magnetic pole configurations,since the radial thickness of magnetic pole changes with the circumferential position,which makes its mathematical modeling more difficult than that of conven-tional magnetic poles.Several analytical methods are given in[8]–[10],which provide valuable theoretical references for magnetic pole design and analysis.Stator slotting is usually neglected or complicated pole boundary is simplified to reduce the difficulty of modeling,which also results in a low accuracy of the models.Manuscript received March3,2014;revised May11,2014;accepted May24,2014.Date of current version November18,2014.Corresponding author:C.Xia(e-mail:motor@).Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TMAG.2014.2327138Fig.1.PM pole shapes.(a)Conventional pole shape S0.(b)Outer arc eccentric pole shape SA.(c)Inner arc eccentric pole shape SB.In this paper,two types of eccentric magnetic pole designs are chosen for pole shape optimization of surface-mountedPM machines with radial magnetization.To solve the problem of unequal thickness magnetic pole modeling,a modifiedsubdomain model method based on discrete idea is proposedto predict magneticfield distribution in the air-gap.With the field solution,cogging torque and back EMF models are built.The effects of magnetic pole dimensions on motor behaviorare further investigated to draw some conclusions.II.A NALYTICAL M ODELINGA.Eccentric Magnetic Pole ShapesCompared with the conventional magnetic pole,two kinds of eccentric magnetic pole shapes for improving thefielddistribution of surface-mounted PM motors are shown inFig. 1.Fig.1(a)is the conventional magnetic pole shape designated as S0,Fig.1(b)is the outer arc eccentric magneticpole shape designated as SA,and Fig.1(c)is the inner surfacearc magnetic pole shape designated as SB.As shown in Fig.1,O is the center of motor and h m isthe magnet thickness at the pole centerline.For conventional pole shape S0,its inner and outer arcs have the same centreO and the radial thickness does not change with position. R r and R m are the radii of magnet inner and outer surfaces,and h m=R m−R r.For the shape SA,the center of its outer arc moves to O and the radius changes to be R o.For the shapeSB,the center of its inner arc moves to O ,and the radius0018-9464©2014IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.Fig.2.General subdomain model for PM motors with unequal thicknessmagnetic poles.changes to be R i .d denotes the distance between points Oand O .B.PM Motor ModelingIt is obvious in Fig.1that mathematical illustration becomes more difficult since the radial magnet thickness of SA and SB configurations changes with circumferential angle;thus,the original exact subdomain model [11]can hardly be used for analytical modeling of the improved PM motor.In this paper,discrete idea and superposition principle are adopted to modify and extend the subdomain model method to PM motors with irregular magnetic pole shapes.In Fig.2,a general subdomain model of a 4-pole and 6-slot motor is taken as an example to illustrate the analytical method.The whole domain is divided into three types of subdomains,viz.,magnet (region 1j ),air-gap (region 2),and slots (region 3i ).In this paper,the magnet region is divided into N equal parts,and the j th part is designated as region 1j (j =1,2,...,N ).The thickness of each part can be considered constant if N is large enough.θj denotes the position of region 1j from the pole centerline.R rj and R mj are the inner and outer radii of region 1j ,respectively.In addition,R s and R sb are the radii of stator inner bore and winding slot bottom,respectively.C.Modified Subdomain ModelBefore the modeling,several assumptions are made to simplify the problem as follows:1)end effect is neglected;2)stator/rotor iron has infinite permeability;3)core saturation and loss are neglected;4)demagnetization characteristic of the magnet is linear;5)nonconductive magnet material is used.In polar coordinates,the scalar potentials in the three regions ϕ1j ,ϕ2,and ϕ3i can be expressed by Laplace’s and Poisson’s equations as follows.1)For Region 1jϕ1j = kA 1k r k +B 1k r −k+M cjk r μr (1−k 2)cos (k α)+ kC 1k r k +D 1k r −k +M sjkr μr (1−k 2)sin (k α)k =1(1)ϕ1j =(A 11r +B 11r−1+M cj 1r ln r /2μr )cos α+(C 11r +D 11r −1+M sj 1r ln r /2μr )sin αk =1.(2)2)For Region 2ϕ2=kA 2k r k +B 2k r −kcos (k α)+ k C 2k r k +D 2k r −k sin (k α).(3)3)For Region 3iϕ3i = mX 3im (r /R sb )m παb −(r /R sb )−m παb×sin [(m π/αb )×(α−αi +αb /2)(4)where μr is the relative recoil permeability of magnet and αb is the mechanical angle of winding slot opening.αb =b 0/R s and b 0is the slot opening width.Q is the stator slot number.αi locates the center of the i th winding slot where i =1,2,3,...,Q .k and m are the harmonic coefficients.A 1k -D 1k ,A 2k -D 2k ,and X 3im are the coefficients to be determined.M cjk and M sjk are variables concerning magnets,which can be given asM cjk = 4pB rk πμ0sin k παpj 2p cos (k ωr t )k /p =1,5,7···0others (5)M sjk = 4pB rk πμ0sin k παpj 2p sin (k ωr t )k /p =1,5,7···0others (6)where μ0is the permeability of air,p is the pole pair number,αpj is the pole arc to pole pitch ratio of the j th PM region,B r is the remanent flux density of magnets,and ωr is the rotor rotational speed.To determine the unknown coefficients and obtain the field distribution,the boundary conditions are defined as follows:H α1|r =Rr =0(7)B r 1|r =R m =B r 2|r =R m (8)H α1|r =R m =H α2|r =R m (9)ϕ2|r =R s =ϕ3i |r =R s (10)B r 2|r =R s =B r 3i |r =R s(11)where subscripts r and αdenote the radial and tangential components of variables,respectively.By applying the boundary conditions given by (7)–(11)to (1)–(6),the radial and tangential components of air-gap flux density can be solved asB r 2=kχrck cos (k α)+kχrsk sin (k α)(12)B α2=k χαck cos (k α)+kχαsk sin (k α)(13)where χrck ,χrsk ,χαck ,and χαck are the harmonic amplitudesobtained using subdomain model method and superposition principle.Based on the analytical model of air-gap flux density in (12)and (13),models of cogging torque T c and phase back EMF E x are built as follows:T c =(πl a r 2/μ0)×k(χrck χαck +χrsk χαsk )(14)E x =(N c l a R s /a )×dxB r 2d α/dt x =A ,B ,C (15)CHEN et al.:MODELING AND ANALYZING OF SURFACE-MOUNTED PM SYNCHRONOUS MACHINES 8102804TABLE IM AIN P ARAMETERS OF P ROTOTYPE MACHINESFig.3.FE and analytical predicted air-gap flux densitywaveforms.Fig.4.FE and analytical predicted cogging torquewaveforms.Fig.5.FE and analytical predicted back EMF waveforms.where l a is the lamination length.x denotes the three-phase stator windings.N c is the number of winding turns per coil and a is the number of parallel-circuits per phase.III.F INITE E LEMENT V ALIDATIONIn this section,the finite element (FE)method is employed to validate the analytical model and the FE software is Ansoft Maxwell 15.4-pole/6-slot surface-mounted PM motors with different pole shapes are taken as examples to test the accuracy of the proposed analytical model.The main motor parameters are listed in Table I.The highest harmonic numberconsideredFig.6.Air-gap flux density variation with discrete magnet number N .TABLE IIC ALCULATION T IME OF THE T WO M ETHODSin analytical model K is 40and the number of assumed parts per pole N is 30.Waveforms of air-gap flux density,cogging torque,and back EMF derived by FE and analytical methods are compared in Figs.3–5.As can be seen,all the analytical predictions well illustrate the effects of pole parameters on performance and have excel-lent agreement with the FE results,which verifies the accuracy and feasibility of the proposed method.Besides,the air-gap length of SA configuration changes with circumferential position,while that of SB is still even.Therefore,under the same motor parameters,the equivalent air-gap length of SA is bigger than that of SB.The former has a much smaller root mean square value of flux density,but a better distribution than the latter,which result in lower cogging torque and more sinusoidal back EMF waveform.Since results of the analytical method are calculated by adding N subdomain models with even thickness magnets,the magnetic field solution is affected by the parameter N .In Fig.6,the air-gap flux density variations are given when N is increased from 1to 50.As can be seen,the results stay unchanged when N is larger than 10,which indicates that the analytical method is stable if N is large enough.In Table II,the time consumed by analytical method and FE method is compared when N is 30.It shows that the analytical method is much faster than FE method,which can improve the efficiency of motor design and analysis.IV.E FFECT OF M AGNET D IMENSIONSON M OTOR B EHAVIORBased on the proposed analytical model in (14)and (15),effects of magnet dimensions on motor behavior are inves-tigated.Variations of cogging torque T c ,fundamental com-ponent amplitude of back EMF E 0,and its total harmonic distortion (THD)with h m and d are shown in Figs.7–9.Some conclusions can be drawn from the above results.1)In Fig.7,cogging torque decreases with either smaller h m or larger d .But the difference is that cogging torque per unit volume continues to decrease when d increases,8102804IEEE TRANSACTIONS ON MAGNETICS,VOL.50,NO.11,NOVEMBER2014Fig.7.Cogging torque variation with h m and d .(a)SA.(b)SB.Fig.8.Back EMF amplitude variation with h m and d .(a)SA.(b)SB.Fig.9.Back EMF THD variation with h m and d .(a)SA.(b)SB.while it increases when h m decreases.This illuminates that cogging torque cannot be effectively eliminated by reducing magnet thickness.Applying the magnet pole shape optimization method would be more useful.2)Like cogging torque,back EMF amplitude gets smaller with the decrease of h m or the increase of d as shown in Fig.8.Fortunately,the contribution of per unit volume increases,while the magnitude of back EMF gets lower.This implies that pole shape optimization can improve the utilization of PM and save the cost of PM machines.3)Apart from the amplitude of back EMF,harmonics is also a focus in PM motor analysis.In Fig.9,the THD of back EMF can hardly vary with h m .But it decreases evidently with the increase of d ,which suggests that higher performance of PM motor could be achieved by employing SA and SB pole configurations since they can not only eliminate cogging torque,but also reduce back EMF harmonics.4)In addition,although SA and SB configurations have similar effects on motor behavior,SA is more sensitive to d than SB.This is because for SB configuration,only pole shape is changed,while its air-gap distribution is still uniform.But for SA configuration,it has not only a sinusoidal magnetic pole,but also an unevenlydistributed air-gap,which will enhance the impact of parameter d .V.C ONCLUSIONEccentric magnetic pole is an effective approach to improv-ing PM motor design.However,uneven thickness of magnetic pole also increases the difficulty of analytical modeling.In this paper,an accurate analytical method is proposed by modifying subdomain model method with discrete idea.With the proposed models,the effects of PM dimensions,h m and d ,on motor behavior are investigated.This paper shows that h m directly affects the magnitude of flux density and further affects the magnitudes of cogging torque and back pared with h m ,parameter d mainly affects the spatial distribution of flux density rather than changing its rger d would produce better flux density distribution,result-ing in smaller cogging torque and more sinusoidal back EMF waveform though it also somewhat decreases its magnitude.Thus,a balance between design requirements,ensuring motor electromagnetic performance and meanwhile reducing back EMF THD and torque ripples,may be achieved by choosing appropriate pole dimensions.A CKNOWLEDGMENTThis work was supported in part by the National Key Basic Research Program of China (973Program)under Grant 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