research report 2014-08 material-efficient permanent ...lipo.ece.wisc.edu/2014pubs/2014-08.pdf>...

Research Report

University of Wisconsin-MadisonCollege of Engineering

Wisconsin Power Electronics Research Center2559D Engineering Hall1415 Engineering DriveMadison WI 53706-1691

© Confidential

2014-08

Material-Efficient Permanent Magnet Shape forTorque Pulsation Minimization in SPM Motors

for Automotive Applications

W. Zhao, T. A. Lipo*, B. Kwon

Department of Electronic SystemsEngineering

Hanyang UniversityAnsan, 426-791, Korea

*Department of Electrical and ComputerEngineering

University of Wisconsin-MadisonMadison, WI 53706 USA

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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 providing balanced axial electromagnetic force, which turns out to obtain better sinusoidal electromotive force (EMF), less cogging torque and consequently smooth electromagnetic torque. The contribution of the novel PM shape to motor characteristics is firstly estimated by 3-D finite element method (FEM), and all 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 (FEA) results are confirmed by experimental results.

Index Terms—Electrical machines, electromagnetic force, finite element method (FEM), finite element analysis (FEA), permanent magnet (PM) machines, sinusoidal electromotive force (EMF).

NOMENCLATURE

λ Flux linkage of a phase winding. Nc Number of coil turns per phase. kw Winding factor. Bg Airgap flux density. τp Magnet pole pitch. Lst Motor stack length. p Number of magnet pole pairs. ωr Mechanical angular speed. e Induced phase back EMF. ϕf Magnetic flux.

Manuscript received June 26, 2013; revised October 10, 2013; accepted

December 2, 2013. Copyright (c) 2013 IEEE. Personal use of this material is permitted. However,

permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].

This research was supported by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R33-2008-000-10104-0).

Wenliang Zhao and Byung-Il Kwon are with the Department of Electronic Systems Engineering, Hanyang University, Ansan-si, Gyeonggi-do, 426-791 Korea (e-mail: [email protected]; [email protected]).

Thomas A. Lipo is with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691 USA (e-mail: [email protected]).

fe Electrical frequency. θe Electrical rotor position angle. Te Electromagnetic torque. E0 Fundamental amplitude of back EMF. ea, eb, ec Phase back EMF. ia, ib, ic Phase current. Piron Iron loss. Nelement Element number. Phi (B,fe) Hysteresis loss of each element. Pei (B,fe) Joule loss of each element. B Magnetic flux density of each element. Irms The rms value of armature current. R Stator resistance. Pcopper Copper loss. η Motor efficiency. Nc Cogging torque periods during a slot pitch. n Step skewing number. Q The number of stator slots. HCF The highest common factor.

I. INTRODUCTION

LECTRIC actuators are proving to be an alternative to hydraulic types due to their reliability, energy efficiency,

precise controllability and environmental considerations [1]-[3]. 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 [4].

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 permanent magnets is needed. There exist a wealth of literature about designing traction motors for high power density with less or no rare earth permanent magnets [5]-[8]. 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 as well as to improve the drive comfort. Thus, the research and development of machines free of torque pulsation with less or no permanent magnets may be considered as an important

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

E


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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; 2) torque ripple, which occurs as a result of the field distribution and the armature magnetomotive force (MMF). In SPM 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 [9]-[17], while some others rely on machine design concepts [19]-[45]. Among the various approaches, modification of the PM shape has been recognized as an effective method for reducing torque pulsation in SPM motors [20]-[25], [28]-[43].

One of the most common techniques is skewing which can be either continuous or stepwise [21], [31]-[33], [40]-[43]. Skew can reduce the cogging torque to zero with one slot pitch skewing, and 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 [36]. 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 [43], 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 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 FEA and has been proposed for obtaining smooth output torque in [23]. 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, 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. Section III and Section IV show the comparison of both 3-D FEA results and experimental results. Finally, concluding remarks are given in Section V.

(a)

(b) (c)

Fig. 1. Configuration of basic models. (a) Stator and windings. (b) Rotor of basic model 1-180o magnet span. (c) Rotor of basic model 2-120o magnet span. 1- Stator core. 2- Stator windings. 3- Rotor core. 4- Permanent magnets.

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 compatible with its commercial use in automotive and other low cost applications. The motors share the same stator with six slots as shown in Fig. 1(a), and three phase 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 the 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 as 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 Fig. 1(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 [18], the SPM motor with arched PMs generates a rectangular magnetic flux distribution in the air gap, which results in a rectangular back EMF waveform in a full pitch winding as

TABLE I SPECIFICATIONS OF BASIC MODELS

Item Unit Value

Number of pole pairs - 2 Number of slots - 6

Stator outer diameter mm 88 Stator inner diameter mm 51 Rotor outer diameter mm 50 Rotor inner diameter mm 15

Airgap length mm 0.5Permanent magnet thickness mm 3.5

Lamination axial length mm 52


3

(a)

(b)

(c)

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.

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 [45]-[48]. Moreover, magnets spanning 180 electrical degrees result in magnet material waste because the flux at the transitions between the North and South poles do 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) [23]. A novel improved PM shape, Fig. 3(b), is the subject of this paper. The PM shapes in Fig. 3(a) and Fig. 3(b) follow the same principle of producing sinusoidal back EMF. The flux linkage of a phase winding is equal to

0

p

c w g st rp

N k B L sin( x pw t)dx

(1)

2c w g st p rN k B L cos(pw t)

(2)

From the Lenz’s law, the induced back EMF is 2

c w g st p r

de N k B L sin(pw t)

dt

(3)

2 e c w f ef N k sin( )

(4)

where flux 2f g st pB L

, and frequency

2r

e

wf p

.

(a)

(b)

(c)

Fig. 3. PM shapes and back EMF waveform of the proposed model. (a) Sinusoidal PM shape. (b) Axially symmetrical sinusoidal PM shape. (c) Ideal back EMF waveform.

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

e a a b b c c rT (e i e i e i ) / w (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 Fig 4(c) show a relatively cost-effective quasi-sinusoidal PM shape design for practical use. Fig. 4 (b) and Fig. 4 (d) show the rotor topologies of sinusoidal PM models for investigation in this paper, which are named model 3 and model 4, respectively. It is evident that more steps lead to progressive improvement in obtaining a sinusoidal back EMF. 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 when compared with model 3.


4

(a) (b)

(c) (d)

Fig. 4. Design sketch of practical PM shapes and rotor topologies of sinusoidal PM shape models. (a) Previous sinusoidal PM shape. (b) Rotor of model 3. (c) Proposed axially symmetrical sinusoidal PM shape. (d) Rotor of model 4. 1- Rotor core. 2- Permanent magnets.

C. Analysis Condition for Comparison

In order to verify the contribution of the proposed model 4 to motor characteristics, two basic models and one previously reported model 3 have been firstly adopted for comparison as listed in Table II. The analysis conditions for comparison are as follows. 1) The four models have the same outside diameter and axial

length; the same grade PMs and iron materials. The basic model 1 has a larger amount of PM material, while the other three models contain similar amounts of PM material.

2) The paper focuses on the estimation of PM shape effects on motor characteristics such as back EMF, cogging torque and torque ripple. 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.

3) Since the stator weight will be increased by increasing the number of turns per coil for basic model 2, model 3 and model 4, 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 Arms/mm2 to 4.8 Arms/mm2 in models 2-4.

4) Due to the axial geometry design of sinusoidal PM shape for proposed models, 3-D FEM is utilized to analyze all the models for obtaining relatively accurate results for comparison. The back EMF and cogging torque are analyzed for the no load 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.

(a) (b)

(c) (d)

Fig. 5. Open circuit magnetic field distribution. (a) Model 1. (b) Model 2. (c) Model 3. (d) Model 4.

III. 3-D FEM ANALYSIS

A. Magnetic Field Distribution and PM Flux Linkage

Figure 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 magnet shape produces significant leakage flux as shown in Fig. 5(c). The proposed model 4 with an axially symmetrical PM shape shows good ability to reduce the leakage flux as shown in Fig. 5(d).

Fig. 6 shows the PM flux linkage of the four models. As mentioned in the prior section, the arched PM shape leads to a rectangular magnetic flux distribution in the air gap. Therefore, the flux linkage waveforms of two basic models are trapezoidal having straight line segments in Fig. 6(a) and Fig. 6(b). Model 3 and model 4 show nearly an exact sinusoidal PM flux linkage waveform due to the sinusoidal magnetic flux distribution in the air gap benefitting from the sinusoidal PM shape as shown in Fig. 6(c) and Fig. 6(d).

TABLE II DESIGN PARAMETERS OF ANALYSIS MODELS

Item Unit Model

1 Model

2 Model

3 Model

4

Rotor topology

-

Magnet volume cm3 26.0 17.8 19.8 18.2 Rotor volume cm3 94.4 86.2 88.2 86.6

No. of coil turns mm 125 146 146 154 Wire diameter mm 0.6 0.55 0.55 0.55

Phase resistance mm 2.5 3.5 3.5 3.8 Current density A/mm2 4.1 4.8 4.8 4.8


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(a) (b)

(c) (d) Fig. 6. PM flux linkage. (a) Model 1. (b) Model 2. (c) Model 3. (d) Model 4.

Fig. 7. Axial electromagnetic force at no load case.

Fig. 8. Axial electromagnetic force at 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. Figure 7 shows the axial electromagnetic force without load for the four models. As with the skewing method in [40], model 3 exhibits an inherent drawback of unbalanced 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. 8, which demonstrates a symmetrical structure for machines will be necessary in some applications with stringent operating conditions.

Fig. 9. Phase back EMF of the four models.

Fig. 10. FFT analysis of phase back EMFs.

C. Back EMF and Cogging Torque

The phase back EMFs of the four models are shown in Fig. 9. Due to the adjustment of 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 a rectangular back EMF waveform, while model 3 and model 4 show good sinusoidal back EMF. In order to evaluate the sinusoidal quality of the back EMF, an FFT analysis of back EMFs is shown in Fig. 10, and the dominant back EMF harmonics of the 5th and 7th 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 distortion (THD) of the back EMF for the four models is 23.6%, 22.8%, 3.3%, 2.8%, respectively, which is calculated by

2 2 2

1 2 3

0

E E E ...THD

E

(6)

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. 11 shows the cogging torque comparison for the four models. Model 4 shows the least peak to peak value of cogging torque as 0.0072 Nm, which is reduced by 86.1%, 70.3%, 20.9%, as compared with 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 cogging torque reduction.


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Fig. 11. Cogging torque of the four models.

Fig. 12. Electromagnetic torque of the four models.

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 5000 rpm is shown in Fig. 12. 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 peak to peak torque value to average torque value is adopted for torque ripple calculation, which has the form as

max minT

AVG

T TK

T

(7)

The torque ripple factor of model 4 is 5.6%, which is decreased by 74.7%, 70.1%, 5.4%, respectively, as compared with model 1, model 2 and model 3. The comparison results are summarized in Table III.

E. Iron Loss and Efficiency

As stated in [25], the harmonics of the back EMF have a significant influence on the iron loss. In order to evaluate iron loss accurately considering nonlinear phenomena, 3-D FEM modeling was used employing the commercial software JMAG which is based on the equation

[ ( ) ( )]element

i i

N

iron h e e ei

P P B, f P B, f (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.

(a) (b)

Fig. 13. SPM motor with step skewed permanent magnets. (a) Stator. (b) Rotor with step skewed magnets.

The copper loss can be found by 23copper rmsP I R (9)

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

out

out copper iron

P

P P P

(10)

where the output power Pout is obtained by 2

60r

out eP T

(11)

F. Quantitative Comparison with Step Skewing Method

In order to highlight the advantage of the proposed model 4, a SPM motor with step skewed permanent magnets keeping the same specification as model 4 has been introduced for comparison as shown in Fig 13. Since the conventional skewing method improves the back EMF waveform by reducing the areas of the back EMF trapezoid, thus reducing the machine performance [31], [36], 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

2ss

c

k

nN Q

k=1, 2, 3… (12)

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

2

2c

pN

HCF ( p,Q) (13)

TABLE III 3-D FEM ANALYSIS RESULTS

Item Unit Model

1 Model

2 Model

3 Model

4

Phase back EMF V 15.3 15.5 15.4 15.3

THD of back EMF % 23.6 22.8 3.3 2.8

Cogging torque Nm 0.0518 0.0241 0.0091 0.0072

Electromagnetic torque Nm 0.4911 0.4971 0.5069 0.5053

Ripple torque Nm 0.109 0.093 0.031 0.028

Torque ripple factor % 22.1 18.7 5.9 5.6

Output power W 257.1 260.3 265.4 264.4

Copper loss W 9.8 13.7 13.7 14.4

Iron loss W 8.4 7.2 7.1 6.8

Efficiency % 93.4 92.6 92.7 92.6


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Fig. 14. Comparison of cogging torques.

Fig. 15. Comparison of phase back EMFs.

Fig. 16. FFT analysis of phase back EMFs.

The total SPM motor with step skewed magnets was modeled and analyzed by 3-D FEM. Figure 14 shows the comparison of cogging torque. Based on the skewing principle for eliminating the cogging torque, the model with step skewed magnets indeed contain less cogging torque compared with the proposed model 4. However, the proposed model 4 shows better sinusoidal back EMF waveform with lower harmonics as indicated in Fig. 15 and Fig. 16, which results in less torque ripple as compared with the model with step skewed magnets when the models are fed with sinusoidal current excitation in Fig. 17. Moreover, an unbalanced axial electromagnetic force inevitably occurs in the model with step skewed magnets due to the skewed axial geometry as shown in Fig. 18. The unbalanced axial electromagnetic force will lead to axial vibration and hence potential damage the bearing systems, while it is eliminated in model 4 with obtaining a significant cogging torque and torque ripple reduction as well as back EMF improvement. The comparison data between two models is summarized in Table IV.

Fig. 17. Comparison of electromagnetic torque.

Fig. 18. Comparison of axial electromagnetic force.

IV. EXPERIMENTAL VALIDATION

The 3-D FEM analysis results at no load for model 1, model 3 and model 4 have been confirmed by measurements. Fig. 19 shows the prototypes of the manufactured motor models. Fig. 20 shows the experimental setup for back EMF and cogging torque measurement. The back EMF was tested as a generator at 1000 rpm. In order to obtain an accurate measurement of cogging torque, it was measured by the commercial cogging torque analyzer (ATM-50MN, SUGAWARA Laboratories Inc.), which measures torque per angle by rotating the rotor at 1 rad/min, then uploads the angle-torque characteristics to a computer running Windows.

A comparison of simulated and measured back EMF waveforms is shown in Fig. 21. Fig. 21(a), (c) and (e) are simulated results, while Fig. 21 (b), (d) and (f) are 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 value which is due to the manufacturing tolerance.

TABLE IV 3-D FEM ANALYSIS RESULTS

Item Unit Step skewing Model 4

Phase back EMF V 15.4 15.3 THD of back EMF % 4.3 2.8

Cogging torque Nm 0.0043 0.0072 Electromagnetic torque Nm 0.5022 0.5053

Ripple torque Nm 0.046 0.028 Torque ripple factor % 9.2 5.6


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(a) Model 1 (b) Model 3 (c) Model 4

Fig. 19. Manufactured prototypes of model 1, model 3 and model 4.

(a) Back EMF measurement (b) Cogging torque measurement

Fig. 20. Experiment setup for back EMF and cogging torque.

(a) (b)

(c) (d)

(e) (f)

Fig. 21. Back EMF waveforms of model 1, model 3 and model 4 at 1000 rpm. (a), (c) and (e) Simulated waveforms. (b), (d) and (f) Measured waveforms.

The cogging torque under a no load condition compared by FEM simulation and measured results is given in Fig. 22. Figure 22 (a) shows the comparison of simulated results, and Fig. 22 (b) shows the comparison of measured results. The measured cogging torque of model 4 is decreased by 72.0%, 16.5%, as compared with model 1 and model 3, respectively. The measured results retain the same cogging torque reduction trends as with the simulated results. However, the

(a) Simulated waveform

(b) Measured waveform

Fig. 22. Comparison of simulated and measured cogging torque of model 1, model 3 and model 4.

measured peak to peak value of cogging torque has some differences with the simulated value as shown in Table V. 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 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. CONCLUSION

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 pulsations. Based on an FFT analysis of back EMF in basic model 1, 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 ease of manufacturing, the proposed sinusoidal PM shape was designed to have stepwise stacks approximating a sine wave. A detailed 3-D FEA for five models to predict the main

TABLE V COMPARISON OF SIMULATED AND MEASURED RESULTS

Item Unit Model 1 Model 3 Model 4

Back EMF

Simulated V 15.3 15.4 15.3 Measured V 14.5 15.3 15.1

Cogging torque

Simulated Nm 0.0518 0.0091 0.0072

Measured Nm 0.0396 0.0133 0.0111


9

characteristics was illustrated by comparison results. Finally, some results are confirmed by experimental results. The following conclusions can be obtained from these results: 1) 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 shape appears to have a better sinusoidal back EMF with less harmonic components than any known designs previously reported.

2) The proposed model with a symmetrically sinusoidal PM shape can effectively eliminate unbalanced axial electromagnetic forces, which is a prominent advantage for avoiding extra vibration and acoustic noise as well as bearing losses compared with the previously reported sinusoidal shape [23], especially in the automotive applications requiring precise position control, such as active steering and brake systems.

3) SPM motors with 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 as well as the previously reported sinusoidal PM shape [23].

4) Conventional step skewing rotor magnets are effective for eliminating cogging torque, however the technique has negative effects on machine performance by producing unbalanced axial electromagnetic forces. The proposed axial-symmetric sinusoidal PM pole shape can be a good alternative for obtaining less cogging torque and better sinusoidal back EMF with no unbalanced axial electromagnetic force.

5) SPM motors with a sinusoidal PM shape sustain less iron loss due to the minimization of harmonics in the back EMF, which contributes to its high efficiency especially when motors operate in the high speed region.

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Wenliang Zhao received the B.S. degree in control science and engineering from Harbin Institute of Technology, Weihai, Shandong, China, in 2011.

He is currently a Ph.D. student in electronic systems engineering at Hanyang University, Ansan, Korea. His research interests include design, analysis and optimization of electric machines

with analytical method and finite element method.

Thomas A. Lipo (M’64–SM’71–F’87–LF’06) was born in Milwaukee, WI in 1938. He received the B.E.E. and M.S.E.E. degrees from Marquette University, Milwaukee, in 1962 and 1964, respectively, and the Ph.D. degree in electrical engineering from the University of Wisconsin, Madison, in 1968.

From 1969 to 1979, he was an Electrical Engineer with the Power Electronics Laboratory, Cooperate Research and Development, General Electric Company, Schenectady, NY. In 1979, he joined Purdue University West Lafayette IN as a Professor of electrical engineering. In 1981, he joined the Department of Electrical and Computer Engineering, University of Wisconsin, as a Professor. He has been an Emeritus Professor since January 1, 2009. He has published over 550 technical papers, five books, and 40 patents and has received numerous awards for his work.

Byung-Il Kwon (M’87–SM’13) was born in 1956. He received the B.S and M.S degrees in electrical engineering from Hanyang University, Ansan, Korea, and the Ph.D. degree in electrical engineering from the University of Tokyo, Tokyo, Japan, in 1989.

He was a Visiting Researcher with the Faculty of Science and Engineering

Laboratory, University of Waseda, Tokyo, from 1989 to 2000, a Researcher with the Toshiba System Laboratory in 1990, a Senior Researcher with the Institute of Machinery and Materials Magnetic Train Business in 1991, and a Visiting Professor with the University of Wisconsin-Madison, from 2001 to 2002. He is currently a Professor at Hanyang University. His current research interests are analytical methods (MEC and FEM), applications in electromagnetic fields, especially electric machine design and motor control.

research report 2014-08 material-efficient permanent ...lipo.ece.wisc.edu/2014pubs/2014-08.pdf>...

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