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Microsyst Technol (2014) 20:1497–1504DOI 10.1007/s00542-014-2153-4

TechnIcal PaPer

Robust optimal design of a magnetizer to reduce the harmonic components of cogging torque in a HDD spindle motor

Changjin Lee · Gunhee Jang

received: 11 September 2013 / accepted: 3 april 2014 / Published online: 20 april 2014 © Springer-Verlag Berlin heidelberg 2014

magnet (PM) in BlDc motors generates additional slot harmonics to the cogging torque. In small BlDc motors, such as 2.5″ hDD spindle motors, the effect of additional slot harmonics is as serious as that of the fundamental har-monics of the cogging torque. a coil-positioning error in the magnetizer is a major source of uneven magnetization of a ring-shaped PM in a hDD spindle motor (lee et al. 2011). however, it is difficult to estimate and reduce the coil-positioning error when manufacturing a magnetizer. also, each magnetizer has a different range of coil-posi-tioning error.

Many researchers have investigated the development of a magnetizer for the BlDc motor to reduce cogging torque. lin et al. (2000) proposed a magnetizing fix-ture design for a ring-shaped PM with the finite element method in order to reduce the cogging torque of BlDc motors. Jewell and howe (1992) proposed a design for post-assembly impulse magnetizing fixtures to magnetize a ring-shaped PM for BlDc motors using the finite ele-ment method. Koh (2003) proposed a method for reduc-ing the cogging torque of BlDc motors using a new mag-netization pattern of a PM. however, these studies did not consider the cogging torque due to uneven magnetization, such as the generation of slot harmonics of the cogging torque. lee and Jang (2011) proposed a new magnetiz-ing fixture to reduce both the fundamental harmonic and additional slot harmonics of the cogging torque using a back-yoke with a notch. however, their research focused on only one case of coil-positioning error and was lim-ited to the parametric study of several design variables of a magnetizer.

This research proposes a robust optimal design meth-odology for the magnetizer due to coil-positioning error to reduce considerable harmonic components of the cog-ging torque in a BlDc motor. The proposed robust design

Abstract This research proposes a robust optimal design methodology to reduce the cogging torque of a hard disk drive (hDD) spindle motor due to the coil-positioning error of the magnetizer. The design optimization prob-lem of the magnetizer is formulated with an objective function of the cogging torque and the constraints of the torque constant. The coil-positioning errors measured by computerized tomography are considered as the random variables of the robust optimal design problem. additional design variables of the magnetizer are chosen in the opti-mization problem, such as back-yoke thickness, notch depth, etc. Magnetic finite element analysis of the hDD spindle motor is also performed to calculate the cogging torque and torque constant. The cogging torque and torque constant of the optimal design are compared with those of the conventional design, demonstrating that the proposed method effectively reduces the cogging toque of the hDD spindle motor.

1 Introduction

The cogging torque of a spindle motor is a source of vibra-tion and noise in a hard disk drive (hDD), and the exci-tation frequencies of cogging torques are the harmonics of the least common multiple of the poles and slots in ideal brushless Dc (BlDc) motors. however, manufacturing errors generate additional harmonics to cogging torque. In particular, the uneven magnetization of the permanent

c. lee · G. Jang (*) Department of Mechanical engineering, hanyang University, 17 haengdang-dong, Seongdong-gu, Seoul 133-791, republic of Koreae-mail: ghjang@hanyang.ac.kr

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optimization of the magnetizer is formulated with an objec-tive function of cogging torque and the constraints of the torque constant. The coil-positioning errors of a real mag-netizing fixture in a 2.5″ hDD spindle motor were meas-ured using computerized tomography, and the ranges of the coil-positioning error are considered as random variables in

the robust optimal design problem. The design variables of the magnetizer (such as back-yoke thickness, notch depth, and magnetizing voltage) are chosen in the optimization problem. The cogging torque and torque constant of the optimal design are compared with those of the conventional design.

2 Method of analysis

Figure 1 shows the procedure of the proposed robust opti-mal design problem. The first step is the finite element analysis (Fea) of the magnetizer in Fig. 2a. The magneti-zation of the PM is determined by simultaneously solving the nonlinear transient electromagnetic field equations and the differential electric circuit equations for the capacitor-discharge magnetizer as follows:

where μ, A, J0, Mx, My, σ, Φ, R, L, C, and Q0 are the mag-netic permeability, magnetic vector potential, current den-sity, x and y components of magnetization and conductivity, flux linkage, equivalent resistance, inductance, capacitance, and initial charge of the capacitor, respectively. The mag-netization of the PM is determined using the magnetization and demagnetization curves (lee et al. 2011; nakata and Takahashi 1986). In the second step, finite element analy-sis of the BlDc motor with 12 poles and 9 slots was per-formed in order to calculate the cogging torque and torque constant using the Maxwell stress tensor. The residual mag-netic flux density and magnetization direction of the PM

(1)∂

∂x

(

1

µ

∂A

∂x

)

+∂

∂y

(

1

µ

∂A

∂y

)

= −J0 +1

µ

(

∂My

∂x−

∂Mx

∂y

)

+ σ∂A

∂t

(2)dΦ

dt+ R i(t) + L

di(t)

dt−

1

C(Q0 −

∫

i(t)dt) = 0

Fig. 1 Procedure of the proposed optimal design problem

Fig. 2 Magnetizing fixture of the permanent magnet

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calculated in the first step are imported into the second step to calculate cogging torque and torque ripple.

The robust design optimization of the magnetizer to reduce the cogging torque is formulated as follows:

Minimize

where f(Xi) is an objective function of the cogging torque, and w1, w12, w12, w2, w21, w22, w3, w31, and w32 are the weighting factors. In addition, µcog/µ

0cog is the normalized

mean of the cogging torque, and µcog/µ0cog is the normal-

ized standard deviation of the cogging torque. The subscripts add and fund denote the additional harmonic and fundamen-tal harmonic values of the cogging torque, respectively. In this research, the additional harmonic is the 9th harmonic, and the fundamental harmonic is the 36th harmonic. k is the reliability index, and σKT

is the standard deviation of the torque constant. Xi represents the design variables of the magnetizer, such as back-yoke thickness, notch depth, and magnetizing voltage. eleven design variables of the magnet-izer were chosen in this optimization problem, as shown in Table 1, including eight uncontrollable and random variables (the positioning errors of the magnetizing coils) and three controllable and deterministic variables (back-yoke thick-ness, notch depth, and magnetizing voltage).

The coil-positioning errors are chosen as random vari-ables because the coil-positioning errors are a dominant source of uneven magnetization of a ring-shaped PM in

(3)

f (Xi) = w1(w11

µcog_add

µ0

cog_add

+ w12

σcog_add

σ 0

cog_add

)

+ w2(w21

µcog_fund

µ0

cog_fund

+ w22

σcog_fund

σ 0

cog_fund

)

(4)

Subject to µKT+ kσKT

≤ (KT )upper_limit

µKT− kσKT

≥ (KT )lower_limit

(Xi)lower_limit ≤ Xi ≤ (Xi)upper_limit

a hDD spindle motor (lee et al. 2011). The magnetizer is designed to magnetize the ring-shaped PM of a BlDc motor with 12 poles and 9 slots. It has 12 teeth, and the magnet-izing coil is wound 2 turns at each tooth. The total number of magnetizing coils in the cross-section of the magnetizer is

Table 1 Design variables of the robust design optimization of a magnetizer

Variable Type Variables lower Initial Upper Unit

Uncontrollable and random variable

coil coil_upper_1_radial −0.09 0 0.1 mm

coil_upper_2_radial −0.09 0 0.1 mm

coil_lower_1_radial −0.09 0 0.22 mm

coil_lower_2_radial −0.09 0 0.22 mm

coil_upper_1_tangential −0.1 0 0.18 mm

coil_upper_2_tangential −0.1 0 0.18 mm

coil_lower_1_tangential −0.1 0 0.1 mm

coil_lower_2_tangential −0.1 0 0.1 mm

controllable and deterministic variable

Magnetizing yoke notch depth 0 0 0.4 mm

Back-yoke thickness 0 0 2 mm

circuit Magnetizing voltage 700 800 900 V

Fig. 3 histograms of the displacement of the magnetizing coils

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48. It is difficult to consider the positioning errors of all coils because of computation time, so this research considers eight random variables, which are the radial and circumferential displacements of the four magnetizing coils around a tooth, as shown in Fig. 2b. The effect of overall uneven magnetiza-tion can be estimated by investigating the case with uneven magnetization of one pole (akihiro and Shinichi 1992).

The coil-positioning errors of a real magnetizing fixture in a 2.5″ hDD spindle motor are measured using com-puterized tomography. every coil has positioning error of 10–240 μm with respect to the ideal coil position. Figure 3 shows the distribution of the radial and circumferential dis-placements of the magnetizing coils, and the coil-position-ing errors seem to have a normal distribution. In the reli-ability analysis, the coil-positioning errors are assumed to have a normal distribution. The mean of the normal distri-bution is 0, and the standard deviation of the normal distri-bution is one-third of the maximum displacement.

Since the robust design optimization involves repeated performance of the reliability analysis at each design point, it requires too much computation time to calculate the cogging

torque and torque constant using finite element analysis. The metamodel-based design optimization technique is applied to reduce computation time in solving the robust design opti-mization. The metamodel for the cogging torque and torque constant is obtained by finite element analysis, which is per-formed at the experimental points specified by an orthogo-nal array (Oa) as a design of experiments (DOe) technique. In order to generate metamodels as accurately as possible, four types of Kriging models and two types of regression models are constructed. Table 2 shows r2 and predicted r2 values of the metamodels for the cogging torque and torque constant. These values are close to one in the most accu-rate metamodel. as shown in Table 2, the regressive radial basis function (rBF) model is chosen for the robust design optimization of the 9th harmonic of cogging torque, and the polynomial regression full quadratic model is chosen for the robust design optimization of the 36th harmonic of the cog-ging torque and torque constant.

reliability analysis (ra) is performed using the enhanced dimension reduction (eDr) method, which is faster than other sampling methods such as latin hypercube sampling

Table 2 r2 and predicted r2 values of metamodels

Kriging model Predicted r2

cogging torque Torque constant

9th harmonic 36th harmonic

constant

exponential 0 0.786 0.862

Gaussian 0 0.770 0.840

General exponential 0 0.797 0.863

linear

exponential 0 0.712 0.866

Gaussian 0 0.776 0.875

General exponential 0 0.727 0.871

Simple quadratic

exponential 0 0.631 0.863

Gaussian 0 0.707 0.866

General exponential 0 0.684 0.872

Full quadratic

exponential 0 0.773 0.800

Gaussian 0 0.714 0.795

General exponential 0 0.792 0.797

regression model r2

cogging torque Torque constant

9th harmonic 36th harmonic

Polynomial regression

linear 0.103 0.349 0.825

Simple quadratic 0.261 0.605 0.991

Full quadratic 0.883 0.998 0.999

radial basis function 0.913 0.979 0.825

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(lhS). To find the optimal solution, an evolutionary algo-rithm (ea) is used as an optimization method. The meta-model for the optimal design is constructed, and it is solved

using commercial process integration and design optimiza-tion (PIDO) software called PIanO (PIDOTech Inc. PIanO user’s manual version 3.3).

Table 3 comparison of the cogging torque and torque constant of the initial, robust optimal (rO), and deterministic optimal (DO) models

Weighting factors a Optimal model 1: w1 = 0.1 w2 = 0.9, b Optimal model 2: w1 = 0.9 w2 = 0.1 w11 = w12 = w21 = w22 = 0.5

lower limit Initial model rO model 1a rO model 2b DO model Upper limit

notch depth (mm) 0 0 0.001 0.079 0.39 0.4

Back-yoke thickness (mm) 0 0 0.019 1.151 2.0 2

Magnetizing voltage (V) 700 800 825.94 700.99 701.07 900

Objective function 1 0.773 0.717 –

normalized mean (9th) 1 0.946 0.747 0.716

normalized SD (9th) 1 1.153 0.716 1.061

normalized mean (36th) 1 0.989 1.233 1.445

normalized SD (36th) 1 0.949 1.762 1.517

normalized mean (pk–pk) 1 0.860 0.816 0.688

normalized SD (pk–pk) 1 1.091 0.808 1.120

Upper constraint (KT) 5.880 5.862 6.113 6.098 6.2

lower constraint (KT) 5.7 5.708 5.711 5.997 6.049

Fig. 4 histograms of the 9th harmonic of the cogging torque

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3 Simulation model

The full finite element model of the magnetizing fixture has 28,567 triangular elements and 14,318 nodes. The plastic bonded ndFeB magnet has a residual flux density of 0.695 T and a relative recoil permeability of 1.2. The full finite ele-ment model of the BlDc motor with 12 poles and 9 slots in the 2.5″ hDD is developed in order to calculate cogging torque using the Maxwell stress tensor. It has 24,356 trian-gular elements and 12,217 nodes. The PM region is divided into eight layers in both finite element models, and the resid-ual magnetic flux density and magnetization direction of each element of the PM in the magnetizer are imported into this finite element model in the BlDc motor in order to cal-culate the cogging torque and torque constant.

4 Results and discussion

Table 3 compares the objective function and constraints of the conventional and optimal models. The object func-tion in eq. (3) has six weighting factors. The optimal

design point varies according to weighting factors of the harmonic component of the cogging torque. rO model 1 is developed to reduce the 36th harmonic of cogging torque, which is the fundamental frequency of cogging torque in the ideal BlDc motor with 12 poles and 9 slots. It has weighting factors of w1 = 0.1 and w2 = 0.9 in the objective function. rO model 2 is developed to reduce the 9th harmonic of cogging torque due to uneven mag-netization. It has weighting factors of w1 = 0.9 w2 = 0.1 and w11 = w12 = w21 = w22 = 0.5 in the objective func-tion. The initial model is the conventional model of a 2.5″ hDD spindle motor, and its magnetizer does not include the back-yoke and notch. The DO model is obtained by a deterministic optimization of the magnetizer, which is per-formed to minimize the peak–peak value of the cogging torque. In the deterministic optimization, coil positions are fixed to the locations measured using computerized tomog-raphy, and they are not random variables.

reliability analysis is performed in the rO models using the latin hypercube sampling method in order to compare them with the initial and deterministic models. The number of samples is 100,000 for each model.

Fig. 5 histograms of the 36th harmonic of the cogging torque

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Figure 4 shows the histograms of the 9th harmonic of cogging torque. as shown in Fig. 4, rO model 2 effectively reduces the mean and standard deviation of the 9th har-monic of cogging torque. Though the mean of the 9th har-monic of the DO model is smaller than that of rO model 2, the variation of the 9th harmonic is greater than that in rO model 2. The DO model and rO model 2 include the back-yoke and decrease the uneven magnetization of the PM by increasing the fully-magnetized region and decrease the 9th harmonic of the cogging torque due to uneven magnetiza-tion (lee et al. 2011).

Figure 5 shows the histograms of the 36th harmonic of the cogging torque. rO model 1 slightly decreases the mean and standard deviation of the 36th harmonic of the cogging torque. Because the initial model is designed to minimize the fundamental frequency of the cogging torque, the reduction of the 36th harmonic in rO model 1 is smaller than that of the 9th harmonic in rO model 2. In rO model 2 and the DO model, the distribution of the 36th harmonic is moved toward an increasing mean and stand-ard deviation. The back-yoke in rO model 2 and the DO model changes the magnetization pattern of the PM from

sinusoidal to trapezoidal, so the fundamental frequency of cogging torque increases.

Figure 6 shows the histograms of the peak-to-peak value of cogging torque. rO model 2 and the DO model effectively reduce the mean of the peak-to-peak value of the cogging torque, demonstrating that the design opti-mization for reducing the 9th harmonic due to uneven magnetization is more effective than that for reducing the 36th harmonic. The DO model can most effectively reduce the mean of the peak-to-peak value of the cogging torque, but the variation of the cogging torque is greater than that in rO model 2. Deterministic optimization can reduce the cogging torque for only one case of coil-posi-tioning errors. however, when coil positioning error is unknown and varies in each magnetizer, robust optimiza-tion reduces cogging torque more effectively than deter-ministic optimization.

The robust optimal model varies according to the weight-ing factors of the objective function. Figure 7 shows the change of the design variables according to the weighting factor. as the weighting factor of the 9th harmonic (w1) increases, the thickness of the back-yoke increases in order

Fig. 6 histograms of the peak-to-peak value of the cogging torque

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to reduce uneven magnetization by increasing the fully-mag-netized region. also, the depth of the notch increases from the point of w1 = 0.5 and w2 = 0.5 in order to suppress the excessive increase of the 36th harmonic, which increases rapidly due to the change of the magnetization pattern by the back-yoke. On the other hand, the magnetizing voltage decreases as w1 increases. as the thickness of the back-yoke increases, the fully-magnetized region of the PM increases, and a high torque constant can be obtained at the same mag-netizing voltage. Therefore, the magnetizing voltage has to

be reduced in order to satisfy the constraint of the torque constant as the thickness of the back-yoke increases.

5 Conclusions

This research proposes a methodology to develop a robust optimal design of the magnetizer due to the coil-position-ing error to reduce the considerable harmonic components of cogging torque in a BlDc motor. The cogging torque of the robust optimal models is compared with those of conventional and deterministic optimal models. The robust optimal design to reduce additional slot harmonics due to uneven magnetization of the magnetizer effectively reduces the mean and standard deviation of the cogging torque compared with other models. The back-yoke reduces additional slot harmonics, and the notch suppresses the excessive increase of the fundamental harmonic of cog-ging torque. The proposed method can be utilized not only to develop a robust design for the magnetizer, but also to reduce magnetically-induced vibration and noise generated from a hDD spindle motor.

Acknowledgments This work was supported by a national research Foundation of Korea (nrF) grant funded by the Korean government (MeST) (no. 2012r1a2a1a01).

References

akihiro D, Shinichi Y (1992) cogging torque investigation of PM motors resulting from asymmetry property of magnetic poles: influence of performance variation between permanent magnets. electr eng Jpn 163:95–102

hong KJ, choi Dh, Kim MS (2000) Progressive quadratic approxi-mation method for effective constructing the second-order response surface models in the large scaled system design. Trans Korean Soc Mech eng a 24:3040–3052

Jewell GW, howe D (1992) computer-aided design of magnetiz-ing fixtures for the post-assembly magnetization of rare-earth permanent magnet brushless Dc motors. Ieee Trans Magn 28:3036–3038

Koh cS (2003) new cogging-torque reduction method for brushless permanent-magnet motors. Ieee Trans Magn 39:3503–3506

lee cJ, Jang Gh (2011) Development of a new magnetizing fixture for the permanent magnet brushless Dc motors to reduce the cogging torque. Ieee Trans Magn 47:2410–2413

lee cJ, lee cI, Jang Gh (2011) Source and reduction of uneven magnetization of the permanent magnet of a hDD spindle motor. Ieee Trans Magn 47:1929–1932

lin YK, hu Yn, lin TK, lin hn, chang Yh, chen cY, Wang SJ, Ying TF (2000) a method to reduce the cogging torque of spindle motors. J Magn Magn Mater 209:180–182

nakata T, Takahashi n (1986) numerical analysis of transient mag-netic field in a capacitor-discharge impulse magnetizer. Ieee Trans Magn MaG-22: 526–528

PIDOTech Inc. PIanO user’s manual version 3.3Wang GG, Shan S (2007) review of metamodeling techniques in

support of engineering design optimization. aSMe J Mech Des 129:370–380Fig. 7 Design variables according to the weighting factors