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Magnet Pole Shape Design of Permanent Magnet Machine for Minimization of Torque Ripple Based on Electromagnetic Field Theory Seok-Myeong Jang1 , Hyung-Il Park1 , Jang-Young Choi1 , Kyoung-Jin Ko1 , and Sung-Ho Lee2 Department of Electrical Engineering, Chungnam National University, Daejeon 305-764, Korea Gwangju R&D Center, Korea Institute of Industrial Technology, Gwangju 500-480, Korea This paper deals with the magnet pole shape design of permanent magnet machines for the minimization of torque ripple based on electromagnetic field theory. On the basis of a magnetic vector potential and a two-dimensional (2-D) polar system, analytical solutions for flux density due to permanent magnet (PM) and current are obtained. In particular, the analytical solutions for mathematical expressions of magnets with different circumferential thicknesses can be solved by introducing improved magnetization modeling techniques, resulting in accurate calculations of electromagnetic torque. The analytical results are validated extensively by nonlinear finite element method (FEM). Test results such as back-emf measurements are also given to confirm the analyses. Finally, on the basis of derived analytical solutions, a reduction of torque ripple can be achieved. Index Terms—Analytical technique, electromagnetic torque, magnet pole shape, permanent magnet (PM) motors, torque ripple.
I. INTRODUCTION
D
UE TO their high efficiency, high power density, and low maintenance costs, permanent magnet (PM) machines are emerging as a key technology for applications such as home appliances, industrial tools, electric vehicles, etc. [1]. However, PM machines have a cogging torque and a torque ripple, resulting in vibrations and noises. The cogging torque can be easily eliminated by employing a slotless structure for a stator. In recent studies, other methods such as slot/pole combinations and an adjustment of pole arc ratio have been proposed to make a back-emf close to a sine wave form in a synchronous PM machine [2]. In particular, the optimum magnet pole shape design has become the most common practice to reduce harmonic components of back-emf [3], [4]. This method is mainly achieved by finite element (FE) analyses because of their high accuracy and capability of nonlinear computation. However, in the case of optimum magnet pole shape design using FE analyses, time consumption is very severe. Some hybrid models, combining FE in the iron material with analytical solution in the air gap, were proposed in [5], without sacrificing the ability of nonlinear analysis. In addition, it is generally difficult to provide straightforward physical relationships between the performances and parameters. Furthermore, the results obtained by numerical methods are sensitive to the FE meshes, especially for cogging torque, torque ripple, and unbalanced magnetic force. Therefore, the analytical models are useful for understanding the fundamental physics, initial design, and optimization of the machines, while the numerical methods are advantageous for the validation and adjustment of the design [6]. Our work is motivated by the desire to perform optimum magnet pole shape design for the reduction of torque ripple without FE analyses. In previous research, the magnets with different circumferential thickness were not mathematically illus-
Manuscript received February 21, 2011; accepted April 25, 2011. Date of current version September 23, 2011. Corresponding author: H.-I. Park (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2151846
Fig. 1. Manufactured prototype PMSM.
Fig. 2. 2-D analysis model.
trated due to their analytical difficulty. However, by performing mathematical modeling of magnetization taking into account magnet shape shown in Fig. 3, problems stated above can be easily solved, and the reliable optimization results for torque ripple reduction can be obtained rapidly. Thus, this paper deals with the magnet pole shape design of the slotless PM machine for the minimization of torque ripple based on an electromagnetic field theory. At first, on the basis of the magnetic vector potential and the 2-D polar coordinate system, analytical solutions for back-emf and torque are also obtained. Moreover, the magnet pole shape of our analysis model is optimized by using these solutions. II. ANALYSIS MODEL A. Motor Topology Figs. 1 and 2 show a manufactured prototype model and a schematic view of a permanent magnet synchronous
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JANG et al.: MAGNET POLE SHAPE DESIGN OF PERMANENT MAGNET MACHINE FOR MINIMIZATION OF TORQUE RIPPLE
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C. Magnetic Field Distribution by PM In order to obtain the solution of the magnetic fields, the model can be divided into several regions as shown in Fig. 2. In this model, the following assumptions are made [3]. 1) The length of machine is extended to infinity. 2) The permeability of the iron core is constant. 3) The permanent magnets have a linear demagnetization characteristic and are fully and homogeneously magnetized in the direction of magnetization. . Since there is no free current in the PM region, Thus, . The magnetic vector potential is defined as . By the geometry of the rotary machine, the vector potential has only the (axial) components [7]. Therefore, Poisson’s equation, in terms of the coulomb gauge , is given by Fig. 3. Analysis model for the magnetization modeling of PMs taking into account magnet pole shape.
in the air-space/iron (5) in the magnets (6)
TABLE I DESIGN SPECIFICATIONS
where denotes the permeability of the air. In the polar coordinate system, the magnetization is given by (7)
(8)
motor (PMSM) with bread-shape permanent magnets, respectively. The design dimensions and specifications are listed in Table I.
where, denotes the angular speed of rotor. Thus, the governing equations with a magnetic vector potential of (5) and (6) are simplified to the scalar relationship as follows [8]: (9)
B. Magnet Pole Shape Modeling Fig. 3 shows the analysis model for the magnetization modeling of PMs taking into account magnet pole shape. This paper assumes that thin permanent magnets are stacked radially. The pole arc ratio of each permanent magnet piece can be easily adjusted. The pole arc ratio correlates with the dotted line, and if the magnet is assumed to consist of infinite magnet piece, the analysis model can be regarded as the original model
(10) Solving (9) and (10) for
, the results are obtained as (11)
(1) (2) where the subscript denotes th magnet, and the pole arc ratio is given by of th magnet (3) (4) where
is the number of pole pairs.
(12) where and are constants to be determined by applying boundary conditions in the magnetic flux path by the PM rotor. From the magnetic vector potential of (11) and (12), normal and tangential components of the magnetic flux density are given by (13)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 10, OCTOBER 2011
As a consequence, the boundary conditions for the analytical calculation are given by
Fig. 4. Magnetic flux density distribution by PMs at the surface of PM.
(14) D. Magnetic Field Distribution by Winding Current It is assumed that the relative recoil permeability of the PM is unity, and the free current density in the winding region is distributed. Through similar steps as in the PM region case, the governing field equations are given by in the air-space/iron in the coil region where
(15) (16)
Fig. 5. Magnetic flux density distribution by three-phase current source at the surface of PM.
(17)
from (19), the total flux linkage due to PMs for winding pitch can be expressed as
denotes the current density, and it is given by
, and where and are given by
denote the Fourier coefficients.
, (20)
(18) where denotes the peak value of phase current. Figs. 4 and 5 show the comparison between analytical results and FE results for magnetic flux density at the surface of PM, respectively. The predictions are shown to be consistent with FE results.
and are the winding pitch and the length of mawhere can be exchine, respectively. Therefore, the back-EMF pressed as (21) where and are the number of slots per pole per phase and the number of turns per slot, respectively. B. Electromagnetic Torque
III. BACK-EMF AND ELECTROMAGNETIC TORQUE A. Back-EMF The back EMF is given by the product of angular velocity and the rate of change in flux linkage with respect to angular and area , the flux linkage position. From flux density can be expressed as (19)
In permanent-magnet motors, the torque consists of two basic components: electromagnetic torque and cogging torque. In this paper, due to slotless stator, the cogging torque is zero. Once the radial and tangential components of the armature winding flux density in the air gap are known, the electromagnetic torque can be calculated by integrating Maxwell’s stress tensor along a circle with constant radius located inside the air gap. In order to calculate the total torque in, the motor also has
JANG et al.: MAGNET POLE SHAPE DESIGN OF PERMANENT MAGNET MACHINE FOR MINIMIZATION OF TORQUE RIPPLE
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Fig. 6. Back-emf by analytical method, FEA, and experiment.
Fig. 8. 3-D torque ripple table versus pole arc ratio, BL.
torque ripple minimization. Although torque calculation by FEA is more realistic, it should be noted that the slight difference between the analytical and FE calculations due to core saturation does not significantly impact on the optimization results. However, the optimization is based on a flux density calculation, which can be accurately achieved through by an analytical methods. By employing the suggested method, reliable optimization results for torque ripple reduction can be obtained rapidly for not only initial design, but also optimum design. Fig. 7. Electromagnetic torque of initial model by analytical method and FEA.
ACKNOWLEDGMENT to include the flux density components of the permanent magnet field. The total field in the air gap is then [9]
This work was supported by the Human Resources Development and Energy Efficiency & Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) Grant funded by the Korea Government Ministry of Knowledge Economy (No. 20104010100600 and 2006ECM11P101B).
(22) REFERENCES The torque equation can be written as (23) As shown in Figs. 6 and 7, analytical results for the back-emf and the instantaneous torque are validated by the FE results and measurements. IV. TORQUE RIPPLE MINIMIZATION The optimization procedure is employed to minimize the torque ripple. Magnet dimensions are chosen as the optimization variables. Fig. 8 shows the resulting optimum line of the torque ripple data achieved by the varying pole arc ratio and BL. It is seen that a minimized torque ripple section can be obtained by choosing appropriate PM dimensions. V. CONCLUSION This paper proposes an analytical method based on the Maxwell equations to minimize torque ripple in the PMSM with bread-shape PM. Therefore, there is no need to apply the time-consuming FEA to refine the design for satisfactory
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