ELECTRIC actuators are proving to be an alternative to

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1 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER Material-Efficient Permanent-Magnet Shape for Torque Pulsation Minimization in SPM Motors for Automotive Applications Wenliang Zhao, Thomas A. Lipo, Life Fellow, IEEE, and Byung-Il Kwon, Senior Member, IEEE Abstract This paper focuses on the design and analysis of a novel material-efficient permanent-magnet (PM) shape for surface-mounted PM (SPM) motors used in automotive actuators. Most of such applications require smooth torque with minimum pulsation for an accurate position control. The proposed PM shape is designed to be sinusoidal and symmetrical in the axial direction for minimizing the amount of rare earth magnets as well as for providing balanced axial electromagnetic force, which turns out to obtain better sinusoidal electromotive force, less cogging torque, and, consequently, smooth electromagnetic torque. The contribution of the novel PM shape to motor characteristics is first estimated by 3-D finite-element method, and all of the simulation results are compared with those of SPM motors with two conventional arched PM shapes: one previously reported sinusoidal PM shape and one step skewed PM shape. Finally, some finite-element analysis results are confirmed by experimental results. Index Terms Electrical machines, electromagnetic force, finite-element analysis (FEA), finite-element method (FEM), permanent-magnet (PM) machines, sinusoidal electromotive force (EMF). λ N c k w B g τ p L st p ω r e φ f f e θ e T e NOMENCLATURE Flux linkage of a phase winding. Number of coil turns per phase. Winding factor. Airgap flux density. Magnet pole pitch. Motor stack length. Number of magnet pole pairs. Mechanical angular speed. Induced phase back electromotive force (EMF). Magnetic flux. Electrical frequency. Electrical rotor position angle. Electromagnetic torque. Manuscript received June 26, 2013; revised October 10, 2013; accepted December 2, Date of publication January 21, 2014; date of current version May 2, This work was supported by the BK21PLUS program through the National Research Foundation of Korea funded by the Ministry of Education. W. Zhao and B.-I. Kwon are with the Department of Electronic Systems Engineering, Hanyang University, Ansan , Korea ( zhaowenliang. kr@gmail.com; bikwon@hanyang.ac.kr). T. A. Lipo is with the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI USA ( lipo@engr. wisc.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIE E 0 e a, e b, e c i a, i b, i c P iron N element P hi (B,f e ) P ei (B,f e ) B I rms R P copper η N c n Q HCF Fundamental amplitude of back EMF. Phase back EMF. Phase current. Iron loss. Element number. Hysteresis loss of each element. Joule loss of each element. Magnetic flux density of each element. RMS value of the armature current. Stator resistance. Copper loss. Motor efficiency. Cogging torque periods during a slot pitch. Step skewing number. Number of stator slots. Highest common factor. I. INTRODUCTION ELECTRIC actuators are proving to be an alternative to hydraulic types due to their reliability, energy efficiency, precise controllability, and environmental considerations [1], [2]. The main automotive applications include electric power steering, electromechanical brakes, active suspensions, damping and stabilization actuators, clutch and shift actuators, air conditioning, and ventilation systems [3]. High-performance permanent-magnet (PM) motors combining high power density and good efficiency by using rare earth magnets are favored for these applications. However, rare earth materials included in the rare earth PM motors have the problem of high cost and limited supply. Therefore, the development of high-performance motors with less or no rare earth magnets is needed. There exists a wealth of literature about designing traction motors for high torque/power density with less or no rare earth magnets [4], [5]. As to automotive actuators, the major trend is to design the motors to be free of vibration and acoustic noise, to obtain smooth torque with minimum pulsation for an accurate position control, and to improve the drive comfort. Thus, the research and development of machines free of torque pulsation with less or no rare earth magnets may be considered as an important research direction. Two components of torque pulsation can be defined as follows: 1) cogging torque, which arises from the interaction between the rotor PMs and stator slotted iron structure and 2) torque ripple, which occurs as a result of the field distribution and the armature magnetomotive force (MMF). In SPM IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 5780 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014 motors, torque ripple is mainly due to the interaction of the MMF caused by the stator windings and the MMF caused by the rotor magnets, which is closely related to the harmonics in the back EMF. There exists an extensive literature with various techniques for minimizing torque pulsation. Some researchers deal with the torque pulsation problem from the control side [6] [9], while some others rely on machine design concepts [11] [28]. Among the various approaches, modification of the PM shape has been recognized as an effective method for reducing torque pulsation [12] [17], [19] [28]. One of the most common techniques is skewing, which can be either continuous or stepwise [13], [21] [23], [26] [28]. Skew can reduce the cogging torque to zero theoretically with one slot pitch skewing and can improve the back EMF waveform as well. However, the skewing technique has some drawbacks such as reducing the useful magnet flux linking the stator windings as well as increasing the leakage inductance and stay losses [25]. Moreover, conventional continuous and step skewing techniques exhibit unbalanced axial electromagnetic force, inevitably leading to some vibration and acoustic noise as well as damage on bearing systems resulting from the axially asymmetrical structure. In [28], the unbalanced axial electromagnetic force can be literally eliminated by the alternative herringbone rotor skewing technique, while it leads to more complex structure and ineffectiveness of improving back EMF waveform. Based on the fact that the PM shape substantially affects the back EMF waveform and consequently the cogging torque, a sinusoidal PM shape was designed and verified by two preliminary models through 3-D finite-element analysis (FEA), and it has been proposed for obtaining smooth output torque in [15]. However, those models still exhibit unbalanced axial electromagnetic force. Thus, there is no literature providing a technique which gives an overall consideration of minimized torque pulsation, sinusoidal back EMF waveform, and balanced axial electromagnetic force. This paper presents a novel PM shape designed to be sinusoidal and symmetrical in the axial direction for SPM motors. Due to the symmetrical structure of a sinusoidal PM shape, the unbalanced axial electromagnetic force is totally eliminated. Meanwhile, the proposed PM shape achieves a combination of both reducing PM material to a minimum and also reducing the harmonics in the back EMF, which consequently reduces the cogging torque and realizes a smooth electromagnetic torque. In order to highlight the contribution of the proposed PM shape to motor characteristics, the analysis results are compared with those of SPM motors having two conventional arched PM shapes: one previously reported sinusoidal PM shape and one step skewed PM shape. Section II discusses the modeling of SPM motors in detail. Sections III and IV show the comparison of both 3-D FEA results and experimental results. Finally, concluding remarks are given in Section V. II. MODELING OF SPM MOTORS A. Basic Model-SPM Motor With Arched PM Shape The two conventional SPM motors, shown in Fig. 1, are referred to as the basic models in this paper. They have a very simple structure that is compatible with its commercial use in automotive and other low-cost applications. The motors share Fig. 1. Configuration of the basic models. (a) Stator and windings. (b) Rotor of basic model 1 with 180 magnet span. (c) Rotor of basic model 2 with 120 magnet span. 1-Stator core. 2-Windings. 3-Rotor core. 4-Magnets. TABLE I SPECIFICATIONS OF THE BASIC MODELS the same stator with six slots, as shown in Fig. 1(a), and threephase concentrated-coil windings are placed in the slots. The rotor is mounted with radially magnetized NdFeB PMs. The motor with PMs which cover 180 electric degrees per pole is referred to as basic model 1, shown in Fig. 1(b), and the motor with PMs which cover 120 electrical degrees per pole is referred to as basic model 2, shown in Fig. 1(c). The specifications for the two basic models are listed in Table I. B. Proposed Model SPM Motor With Sinusoidal PM Shape 1) Design Principle: The conventional SPM motor is usually adopted with the arched PM shape, as shown in Fig. 1(b) and (c). In this paper, this PM shape is regarded as rectangular due to the radial magnetization, as shown in Fig. 2(a). As is discussed in [10], the SPM motor with arched PMs generates a rectangular magnetic flux distribution in the airgap, which results in a rectangular back EMF waveform in a full pitch winding as shown in Fig. 2(b). Fig. 2(c) shows the resulting harmonics of the back EMF. As is known, these harmonics produce torque ripple and have a detrimental effect on efficiency due to the iron loss [29], [30]. Moreover, magnets spanning 180 electrical degrees result in magnet material waste because the flux at the transitions between the North and South poles does not contribute materially to the torque. It has been shown that the PM shape can be designed to be sinusoidal, which eliminates the harmonics of back EMF and saves on magnet material as illustrated in Fig. 3(a) [15]. A novel improved PM shape [Fig. 3(b)] is the subject of this paper. The PM shapes in Fig. 3(a) and (b) follow the same principle of producing sinusoidal back EMF. The flux linkage of a phase winding is equal to λ = N c k w B g τ p 0 ( ) π L st sin x pw r t dx (1) τ p = 2 π N ck w B g L st τ p cos(pw r t). (2)

3 ZHAO et al.: MATERIAL-EFFICIENT PM SHAPE FOR TORQUE PULSATION MINIMIZATION IN SPM MOTORS 5781 Fig. 4. Design sketch of practical PM shapes and rotor topologies. (a) Previous sinusoidal PM shape and rotor. (b) Proposed axially symmetrical sinusoidal PM shape and rotor. 1-Rotor core. 2-PMs. From the Lenz s law, the induced back EMF is e = dλ dt = 2 π N ck w B g L st τ p sin(pw r t) (3) =2πf e N c k w φ f sin(θ e ) (4) Fig. 2. PM shape and back EMF waveform of basic model 1. (a) Rectangular PM shape and airgap flux distribution. (b) Back EMF waveform. (c) FFT analysis of back EMF. where φ f =(2/π)B g L st τ p and f e = p(w r /2π). Based on (4), the back EMF waveform varies in a sinusoidal manner due to the sinusoidal PM shape as shown in Fig. 3(c). The instantaneous torque for a three-phase SPM motor without magnetic saturation is given by T e =(e a i a + e b i b + e c i c )/w r (5) when three-phase balanced sinusoidal currents are injected into the stator coils, smooth output torque production with virtually no torque ripple can be obtained. 2) Topology of the Proposed Model: Because of the production volume, an actuator motor should necessarily be easy to manufacture. Obviously, the ideal sinusoidal PM shape of Fig. 3(b) introduces difficulty in manufacturing. Instead, in this paper, the magnets are segmented into stepwise stacks. Fig. 4(a) and (b) shows a relatively cost-effective quasi-sinusoidal PM shape design for practical use, and the two models with sinusoidal PM shape for investigation in this paper are named as model 3 and model 4, respectively. It is evident that more steps lead to progressive improvement in obtaining a sinusoidal back EMF waveform. The selection of step numbers depends on a compromise between accuracy and complexity/cost considerations. It is noted that model 4 adopts more stack segments for producing a more sinusoidal PM shape, which is used for estimating the effects of the sinusoidal quality of PM shape on the back EMF waveform when compared with model 3. Fig. 3. Proposed PM shapes and back EMF waveform. (a) Sinusoidal PM shape and airgap flux distribution. (b) Axially symmetrical sinusoidal PM shape and airgap flux distribution. (c) Ideal back EMF waveform. C. Analysis Condition for Comparison In order to verify the contribution of proposed model 4, two basic models and one previously reported model 3 have been

4 5782 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014 TABLE II DESIGN PARAMETERS OF THE ANALYSIS MODELS first adopted for comparison, as listed in Table II. The analysis conditions for comparison are as follows. 1) The four models have the same specification; the same grade PMs and iron materials. Basic model 1 has a larger amount of PM material, while the other three models contain similar amounts of PM material. 2) In order to obtain reasonable comparative results, the rms values of back EMF are kept the same by adjusting the number of turns per coil in the four models. The copper wire was selected with the proper diameter to obtain similar stator copper weight for the four models. Consequently, the current density was increased from 4.1 to 4.8 Arms/mm 2 in models ) Due to the axial geometry design of sinusoidal PM shape for the proposed models, 3-D finite-element method (FEM) is utilized to analyze all of the models for obtaining relatively accurate results for comparison. The back EMF and cogging torque are analyzed for the noload case. The electromagnetic torque and iron loss are obtained by feeding with a three-phase balanced sinusoidal current source for the sake of a simple performance comparison. Fig. 5. Open circuit magnetic field distribution. (a) Model 1. (b) Model 2. (c) Model 3. (d) Model 4. Fig. 6. Axial electromagnetic force at the no-load case. III. THREE-DIMENSIONAL FEM ANALYSIS A. Magnetic Field Distribution Fig. 5 shows the open circuit magnetic field distribution for the four models. The red rectangular shape shows the leakage flux distribution between two poles. As shown in Fig. 5(a), model 1 with rectangular PMs which cover 180 electrical degrees contains more leakage flux than model 2 which covers 120 electric degrees in Fig. 5(b). Although spaces also exist between North and South poles in model 3, the PM shape produces significant leakage flux as shown in Fig. 5(c). Proposed model 4 with an axially symmetrical PM shape shows good ability to reduce the leakage flux in Fig. 5(d). Fig. 7. Axial electromagnetic force at the load case. B. Axial Electromagnetic Force In many applications, the axial electromagnetic force is an important issue especially in those which cannot tolerate any vibration and acoustic noise or in cases where precise position control is necessary. Fig. 6 shows the axial electromagnetic force without load for the four models. As with the skewing method in [26], model 3 exhibits an inherent drawback of un- balanced axial electromagnetic force, while model 4 combining the two basic models contains nearly zero axial electromagnetic force due to the symmetrical PM structure. When the motors are operated with load, the drawback of model 3 is enlarged as shown in Fig. 7, which demonstrates that a symmetrical structure for machines will be necessary in some applications with stringent operating conditions.

5 ZHAO et al.: MATERIAL-EFFICIENT PM SHAPE FOR TORQUE PULSATION MINIMIZATION IN SPM MOTORS 5783 Fig. 8. Phase back EMF of the four models. Fig. 10. Cogging torque of the four models. Fig. 9. FFT analysis of phase back EMFs. C. Back EMF and Cogging Torque The phase back EMFs of the four models are shown in Fig. 8. Due to the adjustment of the number of coil turns, the rms values of back EMF are kept the same in four cases for a reasonable comparison of EMF harmonics and torque pulsations. The two basic models show rectangular back EMF waveforms, while model 3 and model 4 show good sinusoidal back EMF waveforms. In order to evaluate the sinusoidal quality of the back EMF, a fast Fourier transform (FFT) analysis of back EMFs is shown in Fig. 9, and the dominant back EMF harmonics of the fifth and seventh orders are enlarged. Model 3 and model 4 contain almost only a fundamental component of back EMF by directly utilizing the proposed magnet shape design and without any complex optimization procedures. The harmonic distortions (THDs) of the back EMF of the four models are 23.6%, 22.8%, 3.3%, and 2.8%, respectively, which are calculated by E 2 THD = 1 + E2 2 + E (6) E 0 It is noted that model 4 shows a better sinusoidal back EMF waveform than model 3 because model 4 adopts more stack steps for a better sinusoidal PM shape. Hence, an even better sinusoidal back EMF waveform is expected when the PM step number and step size are designed to be more sinusoidal, consistent with an ease of manufacture issue. Fig. 10 shows the cogging torque comparison for the four models. Proposed model 4 shows the least peak-to-peak value of cogging torque as N m, which is reduced by 86.1%, Fig. 11. Electromagnetic torque of the four models. 70.3%, and 20.9%, as compared with those of model 1, model 2, and model 3, respectively. It should be mentioned here that model 1 has the largest amount of magnets, while model 2 has the least amount of magnets, and it can be concluded that the PM shape is the main contributor to the reduction of cogging torque. D. Electromagnetic Torque In this paper, sinusoidal current excitation has been utilized to evaluate the torque ripple of the four models. The electromagnetic torque at the rotational speed of 5000 r/min is shown in Fig. 11. The average output torque of the four models is approximately the same, resulting in the similar output power due to the adjustment of stator winding turns. It is noted that basic model 2 is regarded as the preferred reference model rather than basic model 1 since it has the same current density with model 3 and model 4, providing the same operating point comparison. The torque ripple factor defined as the ratio of the peak-topeak torque value to the average torque value is adopted for torque ripple calculation, which has the form K T = T max T min T AVG. (7) The torque ripple factor of model 4 is 5.6%, which is decreased by 74.7%, 70.1%, and 5.4%, respectively, as compared with model 1, model 2, and model 3. The comparison results are summarized in Table III.

6 5784 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014 TABLE III THREE-DIMENSIONAL FEM ANALYSIS RESULTS Fig. 12. SPM motor with step skewed PMs. (a) Stator. (b) Rotor. E. Iron Loss and Efficiency As stated in [17], the harmonics of the back EMF have a significant influence on the iron loss. In order to evaluate the iron loss accurately considering nonlinear phenomena, 3-D FEM modeling was used, employing the commercial software JMAG which is based on the equation P iron = N element i [ Phi (B,f e )+P ei (B,f e ) ]. (8) The iron loss follows the same variation with the THD of the back EMF as shown in Table III. The reduction in iron loss is obtained in spite of the fact that model 3 and model 4 contain a slightly larger amount of PM than basic model 2. The copper loss can be found by P copper =3I 2 rmsr. (9) Fig. 13. Comparison of cogging torque. Although the copper loss of model 4 is the highest, it still maintains high efficiency as the other three models due to reduction of iron loss. The efficiency is herein defined as η = P out P out + P copper + P iron (10) where the output power P out is obtained by P out = T e 2πω r 60. (11) F. Quantitative Comparison With Step Skewing Method In order to highlight the advantage of proposed model 4, an SPM motor with step skewed PMs keeping the same specification as model 4 has been introduced for comparison, as shown in Fig. 12. Since the conventional skewing method improves the back EMF waveform by reducing the areas of the back EMF trapezoid, thus reducing the machine performance [22], [25], each step of skewed magnets in the comparative model is designed to cover 120 electrical degrees per pole to eliminate the effects of two adjacent poles. The mechanical skewing angle between two adjacent steps is given as θ ss = 2kπ, k =1, 2, 3... (12) nn c Q Fig. 14. Comparison of phase back EMFs. Normally, k is chosen as unity so that the machine torque performance is prevented from degradation. The cogging torque period over a slot pitch is given by N c = 2p HCF(2p, Q). (13) The total SPM motor with step skewed magnets was modeled and analyzed by 3-D FEM. Fig. 13 shows the comparison of cogging torque. Based on the skewing principle for eliminating the cogging torque, the model with step skewed magnets indeed contains less cogging torque compared with proposed model 4. However, proposed model 4 shows better sinusoidal back EMF waveform with lower harmonics as indicated in Figs. 14 and 15, which results in less torque ripple as compared with the model with step skewed magnets as shown in Fig. 16. Moreover, the unbalanced axial electromagnetic force inevitably occurs in the model with step skewed magnets due to the skewed axial geometry as shown in Fig. 17, while it is eliminated in model 4 by obtaining a significant cogging torque and torque ripple reduction as well as back EMF improvement. The comparison data between the two models are summarized in Table IV.

7 ZHAO et al.: MATERIAL-EFFICIENT PM SHAPE FOR TORQUE PULSATION MINIMIZATION IN SPM MOTORS 5785 Fig. 15. FFT analysis of phase back EMFs. Fig. 18. Manufactured prototypes of model 1, model 3, and model 4. Fig. 16. Comparison of electromagnetic torque. Fig. 17. Comparison of axial electromagnetic force. TABLE IV THREE-DIMENSIONAL FEM ANALYSIS RESULTS IV. EXPERIMENTAL VALIDATION The 3-D FEM analysis results at no-load case for model 1, model 3, and model 4 have been confirmed by experimental measurements. Fig. 18 shows the prototypes of the manufactured SPM motor models. The back EMF was tested as a generator at 1000 r/min. In order to obtain an accurate measurement of cogging torque, it was measured by the commercial cogging torque analyzer (ATM-50MN, SUGAWARA Laboratories Fig. 19. Back EMF waveforms of model 1, model 3, and model 4 at 1000 r/min. (a), (c), and (e) Simulated waveforms. (b), (d), and (f) Measured waveforms. Inc.), which measures the torque per angle by rotating the rotor at 1 rad/min and then uploads the angle-torque characteristics to a computer running windows. A comparison of simulated and measured back EMF waveforms is shown in Fig. 19. Fig. 19(a), (c), and (e) shows the simulated results, while Fig. 19(b), (d), and (f) shows the corresponding measured results, respectively. The measured and simulated results show good accordance in back EMF waveform except that the measured rms value of model 1 has a slight difference with the simulated rms value due to the manufacturing tolerance, as shown in Table V. The cogging torque under a no-load condition compared by FEM simulation and measured results is given in Fig. 20. Fig. 20(a) shows the comparison of simulated results, and Fig. 20(b) shows the comparison of measured results. The measured cogging torque of model 4 is decreased by 72.0% and 16.5%, as compared with that of model 1 and model 3,

8 5786 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014 TABLE V COMPARISON OF SIMULATED AND MEASURED RESULTS for avoiding extra vibration, acoustic noise, and bearing losses compared with the previously reported sinusoidal shape and the conventional step skewed magnets, especially in the automotive applications requiring precise position control, such as active steering and brake systems. 3) The SPM motors with a sinusoidal PM shape can effectively minimize the torque pulsation. In the proposed model, not only cogging torque but also torque ripple is significantly reduced compared with the models with the conventional arched PM shape, the previously reported sinusoidal PM shape, and the conventional step skewed magnets. 4) The SPM motors with a sinusoidal PM shape sustain less iron loss due to the minimization of harmonics in the back EMF, which contributes to high efficiency especially when the motors are operated in the high-speed region. Fig. 20. Comparison of simulated and measured cogging torque. (a) Simulated waveform. (b) Measured waveform. respectively. The measured results retain the same cogging torque reduction trends as with the simulated results. However, the measured peak-to-peak value of the cogging torque has some differences with the simulated value. This is presumably because the simulation models assume perfect manufacture and assembly of the prototype motors, while there are inevitably mechanical tolerances in manufacture and assembly difficulties in practice. In particular, each step of the proposed model was designed by special angles for finally sinusoidal PM pole shape, which makes it more difficult for magnets to be cut by designed tolerance. Still, it is satisfied that the cogging torque is highly reduced in the proposed model in both the simulation and experimental results. V. C ONCLUSION In this paper, a novel sinusoidal rotor PM shape has been proposed to be axially symmetric for the purpose of eliminating unbalanced axial electromagnetic force and minimizing torque pulsation. The proposed PM shape was designed for a fundamental waveform of back EMF, which realizes a good combination of PM material reduction and back EMF harmonic minimization. In order to facilitate the ease of manufacturing, the proposed sinusoidal PM shape was designed to have stepwise stacks approximating a sine wave. A detailed 3-D FEA for the five models to predict the main characteristics was illustrated by comparison results. Finally, some results are confirmed by experimental measurements. The following conclusion can be obtained from these results. 1) The SPM motors with a sinusoidal PM shape in the axial direction can produce nearly a pure sinusoidal back EMF even with full pitched stator windings. The proposed axially symmetric sinusoidal PM shape appears to have a better sinusoidal back EMF waveform with less harmonic components than any known designs previously reported. 2) The proposed model with a symmetrically sinusoidal PM shape can effectively eliminate the unbalanced axial electromagnetic force, which is a prominent advantage REFERENCES [1] L. Hao and C. Namuduri, Electromechanical regenerative actuator with fault-tolerance capability for automotive chassis applications, IEEE Trans. Ind. Appl., vol. 49, no. 1, pp , Jan./Feb [2] N. Bianchi, M. D. Pre, and S. Bolognani, Design of a fault-tolerant IPM motor for electric power steering, IEEE Trans. Veh. Technol., vol. 55, no. 4, pp , Jul [3] D. Iles-Klumpner, Automotive Permanent-Magnet Brushless Actuation Technologies, Ph.D. dissertation, Faculty Elect. Eng., Univ. Politechnica Timisoara, Timisoara, Romania, [4] K. Kiyota and A. 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