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

Transcription

1 Microsyst Technol (2014) 20: DOI /s 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 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 problem 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 optimization 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 vibration and noise in a hard disk drive (HDD), and the excitation 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 , Republic of Korea ghjang@hanyang.ac.kr 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 harmonics 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-positioning 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 fixture 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 element method. Koh (2003) proposed a method for reducing the cogging torque of BLDC motors using a new magnetization 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 magnetizing 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 limited to the parametric study of several design variables of a magnetizer. This research proposes a robust optimal design methodology for the magnetizer due to coil-positioning error to reduce considerable harmonic components of the cogging torque in a BLDC motor. The proposed robust design

2 1498 Microsyst Technol (2014) 20: optimization of the magnetizer is formulated with an objective function of cogging torque and the constraints of the torque constant. The coil-positioning errors of a real magnetizing fixture in a 2.5 HDD spindle motor were measured 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 optimal design problem. The first step is the finite element analysis (FEA) of the magnetizer in Fig. 2a. The magnetization of the PM is determined by simultaneously solving the nonlinear transient electromagnetic field equations and the differential electric circuit equations for the capacitordischarge magnetizer as follows: ( 1 x µ ) A + ( 1 x y µ ) A = J ( My y µ x M ) x + σ A y t (1) Fig. 1 Procedure of the proposed optimal design problem dφ dt + Ri(t) + L di(t) dt 1 C (Q 0 i(t)dt) = 0 where μ, A, J 0, M x, M y, σ, Φ, R, L, C, and Q 0 are the magnetic permeability, magnetic vector potential, current density, x and y components of magnetization and conductivity, flux linkage, equivalent resistance, inductance, capacitance, and initial charge of the capacitor, respectively. The magnetization 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 analysis of the BLDC motor with 12 poles and 9 slots was performed in order to calculate the cogging torque and torque constant using the Maxwell stress tensor. The residual magnetic flux density and magnetization direction of the PM (2) Fig. 2 Magnetizing fixture of the permanent magnet

3 Microsyst Technol (2014) 20: Table 1 Design variables of the robust design optimization of a magnetizer Variable Type Variables Lower Initial Upper Unit Uncontrollable and random variable Controllable and deterministic variable Coil Coil_upper_1_radial mm Coil_upper_2_radial mm Coil_lower_1_radial mm Coil_lower_2_radial mm Coil_upper_1_tangential mm Coil_upper_2_tangential mm Coil_lower_1_tangential mm Coil_lower_2_tangential mm Magnetizing yoke Notch depth mm Back-yoke thickness mm Circuit Magnetizing voltage V 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 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 magnetizing coil is wound 2 turns at each tooth. The total number of magnetizing coils in the cross-section of the magnetizer is f (X i ) = w 1 (w 11 µ cog_add µ 0 cog_add + w 2 (w 21 µ cog_fund µ 0 cog_fund σ cog_add + w 12 σcog_add 0 ) σ cog_fund + w 22 σcog_fund 0 ) (3) Subject to µ KT + kσ KT (K T ) upper_limit µ KT kσ KT (K T ) lower_limit (X i ) lower_limit X i (X i ) upper_limit (4) where f(x i ) is an objective function of the cogging torque, and w 1, w 12, w 12, w 2, w 21, w 22, w 3, w 31, and w 32 are the weighting factors. In addition, µ cog /µ 0 cog is the normalized mean of the cogging torque, and µ cog /µ 0 cog is the normalized standard deviation of the cogging torque. The subscripts add and fund denote the additional harmonic and fundamental 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 σ KT is the standard deviation of the torque constant. X i represents the design variables of the magnetizer, such as back-yoke thickness, notch depth, and magnetizing voltage. Eleven design variables of the magnetizer 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 thickness, notch depth, and magnetizing voltage). The coil-positioning errors are chosen as random variables because the coil-positioning errors are a dominant source of uneven magnetization of a ring-shaped PM in Fig. 3 Histograms of the displacement of the magnetizing coils

4 1500 Microsyst Technol (2014) 20: Table 2 R 2 and predicted R 2 values of metamodels Kriging model Predicted R 2 Cogging torque Torque constant 9th harmonic 36th harmonic Constant Exponential Gaussian General exponential Linear Exponential Gaussian General exponential Simple quadratic Exponential Gaussian General exponential Full quadratic Exponential Gaussian General exponential Regression model R 2 Cogging torque Torque constant 9th harmonic 36th harmonic Polynomial regression Linear Simple quadratic Full quadratic Radial basis function 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 magnetization 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 computerized tomography. Every coil has positioning error of μm with respect to the ideal coil position. Figure 3 shows the distribution of the radial and circumferential displacements of the magnetizing coils, and the coil-positioning errors seem to have a normal distribution. In the reliability analysis, the coil-positioning errors are assumed to have a normal distribution. The mean of the normal distribution is 0, and the standard deviation of the normal distribution 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 optimization. The metamodel for the cogging torque and torque constant is obtained by finite element analysis, which is performed at the experimental points specified by an orthogonal 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 R 2 and predicted R 2 values of the metamodels for the cogging torque and torque constant. These values are close to one in the most accurate 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 cogging 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

5 Microsyst Technol (2014) 20: Table 3 Comparison of the cogging torque and torque constant of the initial, robust optimal (RO), and deterministic optimal (DO) models Lower limit Initial model RO model 1 a RO model 2 b DO model Upper limit Notch depth (mm) Back-yoke thickness (mm) Magnetizing voltage (V) Objective function Normalized mean (9th) Normalized SD (9th) Normalized mean (36th) Normalized SD (36th) Normalized mean (pk pk) Normalized SD (pk pk) Upper constraint (K T ) Lower constraint (K T ) Weighting factors a Optimal model 1: w 1 = 0.1 w 2 = 0.9, b Optimal model 2: w 1 = 0.9 w 2 = 0.1 w 11 = w 12 = w 21 = w 22 = 0.5 Fig. 4 Histograms of the 9th harmonic of the cogging torque (LHS). To find the optimal solution, an evolutionary algorithm (EA) is used as an optimization method. The metamodel for the optimal design is constructed, and it is solved using commercial process integration and design optimization (PIDO) software called PIAnO (PIDOTECH Inc. PIAnO user s manual version 3.3).

6 1502 Microsyst Technol (2014) 20: 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 T and a relative recoil permeability of 1.2. The full finite element 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 triangular elements and 12,217 nodes. The PM region is divided into eight layers in both finite element models, and the residual 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 calculate 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 function 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 w 1 = 0.1 and w 2 = 0.9 in the objective function. RO model 2 is developed to reduce the 9th harmonic of cogging torque due to uneven magnetization. It has weighting factors of w 1 = 0.9 w 2 = 0.1 and w 11 = w 12 = w 21 = w 22 = 0.5 in the objective function. 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 performed to minimize the peak peak value of the cogging torque. In the deterministic optimization, coil positions are fixed to the locations measured using computerized tomography, 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

7 Microsyst Technol (2014) 20: 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 harmonic of cogging torque. Though the mean of the 9th harmonic 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 backyoke 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 magnetization (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 standard deviation. The back-yoke in RO model 2 and the DO model changes the magnetization pattern of the PM from 1503 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 optimization 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-positioning errors. However, when coil positioning error is unknown and varies in each magnetizer, robust optimization reduces cogging torque more effectively than deterministic optimization. The robust optimal model varies according to the weighting 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 (w 1 ) increases, the thickness of the back-yoke increases in order Fig. 6 Histograms of the peakto-peak value of the cogging torque

8 1504 Microsyst Technol (2014) 20: to reduce uneven magnetization by increasing the fully-magnetized region. Also, the depth of the notch increases from the point of w 1 = 0.5 and w 2 = 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 w 1 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 magnetizing 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-positioning 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 cogging 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 Fig. 7 Design variables according to the weighting factors 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: Hong KJ, Choi DH, Kim MS (2000) Progressive quadratic approximation method for effective constructing the second-order response surface models in the large scaled system design. Trans Korean Soc Mech Eng A 24: Jewell GW, Howe D (1992) Computer-aided design of magnetizing fixtures for the post-assembly magnetization of rare-earth permanent magnet brushless DC motors. IEEE Trans Magn 28: Koh CS (2003) New cogging-torque reduction method for brushless permanent-magnet motors. IEEE Trans Magn 39: 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: 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: 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: Nakata T, Takahashi N (1986) Numerical analysis of transient magnetic field in a capacitor-discharge impulse magnetizer. IEEE Trans Magn MAG-22: PIDOTECH Inc. PIAnO user s manual version 3.3 Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. ASME J Mech Des 129: