IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006
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Rotor Design on Torque Ripple Reduction for a Synchronous Reluctance Motor With Concentrated Winding Using Response Surface Methodology Jung Min Park1 , Sung Il Kim2 , Jung Pyo Hong2 , and Jung Ho Lee1 Department of Electrical Engineering, Hanbat National University, Yuseong-gu, Daejeon 305-719, Korea Department of Electrical Engineering, Changwon National University, Changwon, Gyeongnam 641-773, Korea This paper deals with the optimum rotor design solution on torque ripple reduction for a synchronous reluctance motor (SynRM) with concentrated winding using response surface methodology (RSM). The RSM has been achieved to use the experimental design method in combination with finite element method (FEM) and well adapted to make analytical model for a complex problem considering a lot of interaction of design variables. Comparisons are given with characteristics of a SynRM according to flux barrier number, flux barrier width variation, respectively. Index Terms—6 slot, concentrated winding, response surface methodology (RSM), synchronous reluctance motor (SynRM), torque ripple.
I. INTRODUCTION
S
YNCHRONOUS reluctance motor (SynRM) has advantages such as low cost and higher efficiency than induction machines. If stator windings of a SynRM are not a conventional distributed one but the concentrated one, the decreasing of copper loss and decreasing of the production cost due to the simplification of winding in factory are obtained. However, the vibration and noise of SynRM caused by torque ripple are relatively greater than other machines. This paper deals with the optimization procedure of a SynRM with concentrated winding to improve torque performance by response surface methodology (RSM). RSM has recently been recognized as an effective optimization approach for modeling performance of electrical devices using statistical fitting methodology [1]–[2]. The focus of this paper is the optimum design relative to torque density, torque ripple, and inductances on the basis of flux barrier number and flux barrier width, in order to improve performance and production cost problems of a SynRM with concentrated winding. Comparisons are given with characteristics of normal distributed winding SynRM and those according to flux barrier number, flux barrier width variation in concentrated winding SynRM, respectively. II. ANALYSIS MODEL
Fig. 1. Design variables and variation direction of the SynRM.
Moreover this optimal design procedure is considered flux barrier width and 3, 4, 5 number of flux barrier. Fig. 1 shows the point variables and variation direction example for the shape change according to the flux barrier width in 3 number of flux barrier. Each pair (W1, W8), (W2, W7), (W3, W6), , move symmetrically on the basis of -axis, and points of P1-P8 move as a condition that flux barrier widths, which are variables according to flux barrier(r) in (1), are varied [3].
A. Design Variables The concentrated winding SynRM has 4 poles and 6 slots. The stack length is 77 mm with a radius of the rotor as 30.1 mm and the air gap length of 0.4 mm. Design variables related to torque performance in the SynRM are slot of stator, air gap, rib of rotor, and flux barrier as shown in Fig. 1.
Digital Object Identifier 10.1109/TMAG.2006.879501
where flux barrier number(r)
(1)
III. CONCEPT OF RESPONSE SURFACE METHODOLOGY The RSM seeks to find the relationship between the design variable and the response through the statistical fitting method, which is based on the the observed data from the system. The response is generally obtained from real experiments or computer simulations. Therefore, finite element analysis (FEA) is
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006
performed to obtain the data of SynRM in this paper. In RSM work, it is assumed that the true functional relationship can be written as
TABLE I ANALYSIS OF VARIANCE
(2) in (2) are in centered and where the variables scaled design units. The form of true response function is unknown and very complicated, so it is approximated. In many cases, the approximating function of the function is normally chosen to be either a first-order or a second-order polynomial model. In order to predict a curvature response more accurately, the second-order model is used in this paper. The model of (3) is the second-order model
(3)
where is regression coefficients and is a random error treated as statistical error. The observation response vector at data point of function may be written in matrix notation as follows: (4) where is a matrix of the levels of the independent variables, is a vector of the regression coefficients, and is a vector of random error. The least squares method, which is to minimize the sum of the squares of the random errors, is used to estimate unknown vector . Therefore, the fitted response vector is given by (5) (5) where is the transpose the matrix . There are many experimental designs for creation of response surface. In this paper, the central composite design (CCD) is chosen to estimate interactions of design variables and curvature properties of response surface in a few times of experiments. The CCD has been widely used for fitting a second-order response surface. Much of the CCD evolves from its use in sequential experimentation. It involves the use of a two-level factorial or fraction combined with 2-k axial or star point. As a result the design involves factorial points, axial points, and center runs. The factorial points represent a variance optimal design for a first-order model or first-order two-factor interaction type model. Center runs clearly provide information about the existence of curvature in the system. If curvature is found in the system, the addition of points allow for efficient estimation of the pure quadratic terms. It is always necessary to examine the fitted model to ensure that it provides an adequate approximation to the true response and verify that none of the least squares regression assumptions are violated. In order to confirm adequacy of the fitted model, analysis-of-variance (ANOVA) table shown in Table I is used in this paper. In Table I, is the total number of experiments and is the number of parameters in the fitted model [4].
Fig. 2.
Flow chart of design procedure.
IV. OPTIMIZATION PROCEDURE Fig. 2 shows the flow chart of total design strategy. The shape coordinates of the rotor have been drawn as a condition of design variables. The ribs have a fixed value due to the mechanical problem. And the new computer-aided design (CAD) file is redrawn with regard to the change of the number of flux barriers automatically as shown in Fig. 1. Next the process of automatic mesh generation follows. In mesh generation, mesh data does not change the node number, element number, region, boundary condition, etc., but only -, -coordinate data of the rotor in the same flux barrier number. In this way, the proposed preprocessor procedure can be performed in a short period of time. This procedure goes on until the mechanical constraint moment of the machine for each design variables is reached and the number of flux barrier will be changed. Design solutions for each flux barrier number will be derived. Finally, the optimization process is performed for minimization of torque ripple. FEA-based simulation for 5 number of flux barrier is used to obtain the responses with respect to the CCD experimental design in Tables II and IV. The torque ripple is simulated FEA in each trial and results, shown in Table IV. Using these experimental results, the secondorder regression model is fitted as (6) and the ANOVA is analyzed in Table III.
PARK et al.: ROTOR DESIGN ON TORQUE RIPPLE REDUCTION FOR A SYNCHRONOUS RELUCTANCE MOTOR
TABLE II LEVEL OF DESIGN VARIABLES (5 NUMBER OF FLUX BARRIER)
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TABLE IV EXPERIMENTAL RESULTS USING CENTRAL COMPOSITE DESIGN
TABLE III ANALYSIS OF VARIANCE
tial model (109.8%) [2]. When the number of flux barrier is 5 and L1 is 2.757 mm, L2 is 1.87 mm, L3 is 1.87 mm, L4 is 2.633 mm, and the torque ripple of concentrated winding SynRM is minimized (63.8%), as shown in Fig. 4 and Table IV. Fig. 3.
Configurations of optimized and initial design for 6 slots.
VI. CONCLUSION This paper presents a RSM optimization technique in order to reduce the torque ripple of the concentrated winding SynRM. For the rapid design, an automatic preprocess that includes an automatic ACAD file drawing and mesh generation with regard to the rotor shape variations have been developed. And appropriateness of RSM in the machines optimization method is verified by the result of optimized the SynRM. Therefore, the RSM approach can be considered as the optimization method for optimum design of SynRM and other machines. ACKNOWLEDGMENT
Fig. 4.
Results of torque analysis.
Since the value
exceeds the value , the null hypothesis that states all coefficients are zeros, is rejected. and indicate that 95.8% of the total variation can be explained by the fitted model (6), and the estimate of the error variance provided by the residual mean square is 4.2% of the error variance estimate using the total mean square. V. RESULT AND DISCUSSION Fig. 3 shows configurations of optimized and initial design for 6 slot. As shown in Fig. 4, the torque ripple of optimized concentrated winding SynRM is larger than those of conventional 24 slot machines but the values are smaller than one of the ini-
This work was supported by MOCIE through the IERC program. REFERENCES [1] Y. K. Kim, Y. S. Jo, and J. P. Hong, “Approach to the shape optimization of racetrack type high temperature superconducting magnet using response surface methodology,” Cryogenics, vol. 41, no. 1, pp. 39–47, 2001. [2] S. J. Park, S. J. Jeon, and J. H. Lee, “Optimum design criteria for a synchronous reluctance motor with concentrated winding using response surface methodology,” J. Appl. Phys., to be published. [3] S. B. Kwon, S. J. Park, and J. H. Lee, “Optimum design criteria based on the rated watt of a synchronous reluctance motor using a coupled FEM & SUMT,” IEEE Trans. Magn., vol. 41, no. 10, pp. 3970–3972, 2005. [4] R. H. Myers and D. C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Design Experiments. New York: Wiley, 1995. Manuscript received June 16, 2006 (e-mail:
[email protected]).
