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AbstractThis paper presents an efficient finite-element-based

approach for a cogging torque analysis of pancake-type motors

with inherent three-dimensional (3-D) flux distributions. For

such cases, 3-D finite element analysis (FEA) that accounts for all

3-D effects is often required for convincing and reliable results.

However, to obtain sufficient information, a considerable number

of models may need to be created and result in a time-consuming

analysis process using 3-D FEA. Therefore, a method named the"2-D Plus" is proposed to significantly improve the time

efficiency of analysis based on two-dimensional techniques

without sacrificing accuracy. Complete and smooth cogging

torque waveforms can then be rapidly produced with the aid of

only two extra 3-D models. The results from the presented

technique agree well with that from 3-D simulations. The

effectiveness of the proposed method in cogging torque analysis is

thus verified.

Index Termscogging torque, 3-D flux distribution, finite

element analysis

I. INTRODUCTIONERMANENT-MAGNET (PM)brushless motors have beenmore and more widely applied in various occasions due to

their high power and torque density, high efficiency, and

maintenance free. However, the inherent cogging torque is

considered as a critical issue for some applications such as

hard disk or DVD-ROM spindle motors, where considerable

effort has been made for cogging torque reduction [1,2]. As

shown in Fig. 1, conventional spindle motors have a typical

configuration of 3 phases, 12 poles and 9 slots, which are

often criticized for the high cost and complexity of

manufacturing as well as the difficulty to miniaturize [3,4].

Research has been conducted for application of pancake-type

motors to DVD-ROM or hard disk drives with an emphasis on

elimination of the above disadvantages [1-4].Unlike conventional spindle motors, pancake-type motors

possess a 3-dimensional (3-D) flux distribution, which

requires 3-D finite element analysis (FEA) for characteristic

Manuscript received March 13, 2006. This work was supported by

Sunonwealth Electric Machine Industry Co., Ltd.

M. F. Hsieh is with the Department of Systems and Naval Mechatronic

Engineering, National Cheng Kung University, Tainan 701, Taiwan ROC

(phone: +886-6-2747018 Ext. 216; fax: +886-6-2747019; e-mail:

mfhsieh@mail.ncku.edu.tw).

M. C. Tsai and Y. C. Lai are with the Department of Mechanical

Engineering, National Cheng Kung University, Tainan 701, Taiwan ROC (e-

mail: mctsai@mail.ncku.edu.tw).

A. Horng is with Sunonwealth Electric Machine Industry Co., Ltd.,Kaohsiung, Taiwan ROC (e-mail: sunon@email.sunon.com.tw).

simulation. Prieto et al [5] point out the drawbacks of using

3D FEA, including complicated problem definitions,

convergence problem, and very long simulation time due to a

large number of elements. Hence, the authors develop a

simplified method called the "double 2-D" for analyses of 3-D

components. However, the method mainly focuses on static

magnetic components without permanent magnets, e.g.,

transformers. Mao and Tsai [6] also apply a simplified methodfor analysis of a switched reluctance motor with 3-D flux

distribution.

Fig. 1 Conventional spindle motors [4]

To overcome the disadvantages of 3-D simulations, this

paper proposes an approach named the "2-D Plus," which

applies FEA for cogging torque analysis of pancake-type

motors and that with similar configurations. The finite element

package ANSOFT EM Field Simulator isemployed here [7].

To demonstrate the effectiveness of the "2-D Plus," a spindle

motor [3,4] and a fan motor, as shown in Fig. 2(a) and (b),

respectively, are investigated. The "2-D Plus" analysis first

superposes two separate 2-D simulation results directly for

each motor so that a smooth back EMF waveform is rapidly

plotted. This is useful for qualitative prediction but would be

rough quantitatively as the 3-D effects are entirely neglected.

A procedure is further employed to determine a modification

factor using only two extra 3-D models so that the waveform

obtained by the 2-D superposition is modified to an accurate

representation. Therefore, with the "2-D Plus," the simulation

of a motor with 3-D flux distribution can be performed using

2-D analyses "plus" two 3-D models instead of 3-D analysis.

This would be significantly time-efficient for research such as

cogging torque reduction by shaping of stator salient poles or

magnet pole arcs, where iterative calculation is required.

This paper is organized as follows. Section 2 introduces the

motors investigated, followed by presentation of the analysis

and results in Section 3. Conclusion is given at the end.

An Efficient Approach for Cogging Torque Analysis

of Motors with Three-Dimensional Flux DistributionMin-Fu Hsieh,Member, IEEE, Mi-Ching Tsai, SeniorMember, IEEE, Alex Horng,

and Yi-Chi Lai

P

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(a) Pancake-type spindle motor [4]

Magnet

One Stator

Layer

Magnet

One Stator

Layer(b) Fan motor and stator dimension (only one stator layer is displayed)

Fig. 2 Investigated pancake-type (a) spindle and (b) fan motors

II. ANALYSISThe single-phase spindle motor shown in Fig. 2(a) contains

two layers of stator yoke, with 6 salient poles on each layer.

There is a 30 degree shift in mechanical angle between the two

layers, which are linked by a shaft. The single-phase winding

is simply wound around the shaft and in between the two

stator layers. The details about the dimensions and materialproperty of the single-phase, 12-slot, 12-pole motor can be

found in [3] and does not recur here. From the flux indicated

in Fig. 3, it is apparent that the flux is in 3-dimensional

distribution. Also, a cogging torque cycle of 30 mechanicalangle is predicted. To obtain a smooth waveform for a cycle, it

is typical to run around 30 modeling increments, i.e., one

model for each degree. This would be easy for 2-D FEA with

the function of automatic parameter regulation in the simulator.However, the function is not available to the 3-D process, and

each modeling increment will be handled manually.

Fig. 3 3-D flux distribution of the pancake-type spindle motor

By observing the interaction between the rotor and each

stator layer with respect to a rotor position, it is found that thetwo layers behave identically in terms of torque direction and

magnitude. This implies the feasibility to produce the overall

cogging torque waveforms by directly superposing the

individual result of each layer with 2-D analysis. Nevertheless,the effect in the axial direction is omitted, and it is expected

that the direct superposition may give good qualitative but not

quantitative results. As shown in Fig. 4, although the 2-D case

takes the thickness into account, the flux does not flow in theaxial direction. In contrast, the actual case has the shaft and

the rotor back iron as its axial flux passage. Therefore, a

modification is necessary for accurate quantitative analysis

and will be detailed in the following section. Fordemonstration, the results obtained by the "2-D Plus" will be

compared with that from 3-D modeling on this pancake-type

spindle motor.

The fan motor shown in Fig. 2(b) has a similar layout to thespindle motor but the number of salient poles and magnetic

poles is reduced to half. Also, the shape and dimension of the

salient poles is different. The major parameters of the fan

motor are listed in Table 1 (dimension shown in Fig. 2(b)).

upper

lower

only thickness

of iron core is

considered

modification

is required

magnet iron core magnet

single

layer

actual

layout

upper

lower

only thickness

of iron core is

considered

modification

is required

magnet iron core magnet

single

layer

actual

layout

Fig. 4 Contrast between the single layer (2-D) and actual layout (3-D)

TABLEI

Major parameters of fan motor

Symbol Value and unit

Magnet coercivity -121000 A/m

Remanence 0.16 Tesla

Stator and rotor back iron property H23 (Kawasaki Steel Corporation)

III. RESULTSA. Qualitative ComparisonThe resultant cogging torque waveform from the direct

superposition of 2-D analyses (without modification factor

applied) is shown in Fig. 5(a) for the spindle motor. Incomparison with the result using 3-D FEA shown in Fig. 5(b),it can be seen that these two waveforms are very similar in

shape but not in magnitude, as expected.

Fig. 5 Results from (a) 2-D superposition and (b) 3-D FEA (spindle motor)

(a)

(b)

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The analysis for the fan motor is shown in Fig. 6, again the

two methods agree well with each other qualitatively. Thisdemonstrates that the proposed approach should be sufficient

for qualitative prediction without modification.

Fig. 6 Comparison between 2-D superposition and 3-D FEA for fan motor

B. Quantitative ModificationFor quantitative accuracy, modification is made according

to two extra 3-D simulations. From the waveform obtained by

2-D superposition shown in Fig. 5(a), a point on the negative

slope except the zero-torque one is first selected to perform a3-D FEA so that a cogging torque is obtained. A modification

factor 0.25 is determined as the ratio of this cogging torque to

that from the 2-D superposition at the same angular position.

Secondly, every point on the curve in Fig. 5(a) is multipliedby the factor 0.25 to produce a new cogging torque waveform,

which is compared with the 3-D FEA, as shown in Fig. 7(a).

Repeating the process for the positive slope, another factor

0.18 is thus computed. Again the comparison is presented inFig. 7(b). Apparently, the two comparisons shown in Fig. 7(a)

and (b) both present that the two waveforms on one slope

match better than that on the other slope. Finally, taking theaverage of these two factors, the overall modification factor

0.215 is determined. The final predicted cogging torque

waveform and its comparison with the 3D case are presented

in Fig. 8, where both waveforms agree well. The result for thefan motor is not presented

IV. CONCLUSIONAn efficient and fast simplified approach named "2-D Plus"

has been successfully developed for analysis of pancake-type

motors and those of similar configurations with three-dimensional flux distributions for a reduction in simulation

time. Without the aid of any 3-D simulations, the method with

direct 2-D superposition is capable of predicting qualitative

information such as the cycle and trend of the cogging torque.

Quantitatively, two extra 3D models are employed for

calculation of modification factors to obtain sufficiently

accurate waveforms. Therefore, the proposed "2-D Plus"

method is time efficient and useful in further relevant research

such as cogging torque reduction using salient pole or magnet

shaping. To sum up, significant time can be saved for analyses

where a large number of models need to be run.

Fig. 7 Comparing 3D result to 2D results with modification factor (a) 0.25

and (b) 0.18

Fig. 8 Result comparison for 3D and 2D with overall modification factor

ACKNOWLEDGMENT

The support of Sunonwealth Electric Machine Industry Co.,

Ltd. on this research is highly acknowledged.

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