Modeling and Design Optimization of Permanent Magnet Linear Synchronous Motor with Halbach Array

Transcription

1 Modeling and Design Optimization of Permanent Magnet Linear Synchronous Motor with Halbach Array N. Roshandel Tavana, and A. Shoulaie and Department of Electrical Engineering, Iran University Of Science and Technology, Tehran, Iran Abstract: In this paper, an analytical analysis is presented to calculate air gap flux density distribution, thrust and efficiency in air-core permanent magnet linear synchronous motor with Halbach array based on Maxwell equations. In order to improve mean thrust, thrust ripple, magnet and copper consumption, the main design parameters of analyzed machine are optimized by using genetic algorithm in an appropriate objective function. The results show an enhancement in motor performance. Finally, we have used 2-D nonlinear time stepping finite element method to demonstrate validity of the analytical analysis and optimization method. Keywords: Optimization, finite-element method, analytical model, linear synchronous motor, genetic algorithm, Halbach array., Nomenclature Magnetic field intensity vector Unit vectors in and direction Motor width Air-gap length Winding factor Coil packing factor Number of pole pairs Magnet height Magnet width Pole pitch Horizontal magnetized magnet width Winding pitch Output power Copper loss Mechanical loss Remanence Synchronous speed Current density Equivalent magnetization current Load angle 1. Introduction Nowadays, linear motors, which can provide thrust force directly, are more and more used in factory automation and numerical control systems. They offer numerous advantages over rotary-to-linear systems, in regard to their simplicity, efficiency, positioning accuracy and dynamic performance, in terms of both their acceleration capability and bandwidth [1], [2]. Of the various types of linear motor, permanent magnet linear synchronous motor (PMLSM) with Halbach magnetized topology exhibits an essentially sinusoidal air-gap field distribution and a sinusoidal back-emf waveform, as well as negligible cogging force, without employing skew or distributed winding [3], [4]. Hence, in this paper an aircore PMLSM with Halbach array (HA) which provides extra high accuracy operation is employed for design optimization. Proper performance of PMLSMs requires optimizations of their design and control. Design of PMLSMs has so for been presented based on different modeling techniques including magnetic equivalent circuit with lumped elements, analytical method using Maxwell equations, and finite-element method (FEM) [5]. This paper uses analytical method for design optimization because of it has some advantages over other methods for preliminary design of PMLSMs which lie in its accuracy and suitability for optimization. Design optimization of permanent magnet (PM) machines with vertical magnetization [6], has been considered in many researches so far in which different objective have been studied. Nevertheless, Halbach topologies gained less attention. Among limited work on the design optimization of this type of motors, a simultaneous optimization of weight and torque has been investigated by choosing inner and outer stator width, height, and angle for angular direction magnetization [7]. J. Choi et al. have applied a magnet array maximizing the tangential force to a torsional spring composed of twoand three magnet rings [8]. Reduction of detent force has also been studied in [9]. R. Huang et al. have minimized the normal force by using genetic algorithm in synchronous PM planner motor with HA [10]. Unfortunately, in recent studies, the optimization of air-core PMLSM with HA has not been considered yet. Therefore, in this paper, mean thrust, thrust ripple, magnet and copper volume of an air-core PMLSM with HA are optimized. Usually, improvement in one feature 441

2 might have an adverse effect on the other one. Therefore, a compromise is needed between these features. In order to achieve this goal, a multi-objective optimization is employed. First, analytical method is presented for PMLSM with HA. An effective objective function regarding mean thrust, thrust ripple, magnet and copper volume is then proposed and the genetic algorithm is used to optimize design parameters. Finally, 2-D nonlinear time stepping FEM is carried out to evaluate the design optimization. 2.1 Motor Topology 2. Analysis Model Fig. 1 shows a schematic view of a double sided aircore PMLSM with HA. The moving short primary of motor is a three-phase air core winding. Each secondary consists of back iron and PMs facing the primary windings. Therefore, the governing field equations, in terms of magnetic vector potential lead to Laplace and Poisson equations as follows [11]: 0 in region 1 in region 2 where and is magnetization vector of PMLSM with HA and is given by (1) (2) where and denote the components of in and directions, respectively, and may be expressed as Fourier series sin 2 cos,, cos 2 sin,, (3) (4) where and. The boundary conditions to be satisfied by the solution to (1) are Fig. 1: Topology of a double sided PMLSM with HA. 2.2 Field Distribution Due to PM Source In order to establish analytical solutions for the magnetic field distribution in the foregoing machine topology, the following assumptions are made: 1) the length of machine is extended to infinity. 2) the permeability of iron core is infinite. 3) linear behavior of analysis model can be assumed. 4) the permeability of PM material is equal to the permeability of free space. Consequently, the magnetic field analysis is confined to two regions, viz, the airspace/winding and permanent magnet region. Fig. 2 shows simplified model of machine. ; 0; (5) By solving (1), the tangential and normal components ( and ) of the flux density produced by the PMs in the air gap is provided from the curl of as follows:, cos,,, sin,, where and are given by sin cos 2 sin sin cos (7) (6) (8) Fig. 2: Simplified model of machine and equivalent magnetization current distribution. 2.3 Thrust and Efficiency Prediction The thrust force exerted on the armature, resulting from the interaction between the winding current and the PM field, is given by 442

3 Assuming that each coil side on the armature occupies areas bounded by 2, 2, and. The total thrust force exerted on the one coil side may be obtained from the following integration [12]: 2 which may be written as where is given by sin,, 4 sin 2 (12) (9) (10) (11) Therefore, the total force exerted on a phase winding comprising a number of series connected coils, is obtained as cos 2,, (13) where is defined as thrust constant of the th harmonic, and is given by 2 sin 2 (14) For a three-phase machine carrying balanced sinusoidally time-varying currents, the total thrust force is obtained from sin (15),,,, cos cos (16) As will be evident from (16), the mean thrust and normalized total thrust ripple is given by,,, (18) In this kind of motor, iron loss is negligible due to lack of iron in moving part. The essential part of electrical loss is the copper loss. Therefore, motor efficiency is given by where. (19) 3. Design Using Genetic Algorithm Some of the main design parameters of motor are selected as variables whose values are determined through an optimization procedure. A two-pole PMLSM with HA shown in Fig. 1 is selected as the basis for optimization. The geometric parameters of the motor are listed Table I. In this paper, design variables are motor width, magnet height and width of horizontal magnetized magnet in PMLSM with HA. The fixed variables are pole pitch, pole pairs, primary windings current density and coil width. To have a more realistic design, some constraints are applied to design variables listed in Table II. Table I. Specification of Initial Motor. Parameter Unit Value Number of - 1 Coil/Phase/Pole - 1 mm 42 mm 14 mm 7 mm 1.6 mm 3.5 mm 8 mm 72 m/s 4.2 w 0.04 Table II. Design Variables and Constrained Conditions. Parameter Unit Min Max mm 2 5 mm 0 21 mm 5 10 mm mm N To obtain an optimal design considering motor thrust force, PM volume, copper volume and normalized force ripple, the fitness function is defined as follows: 3 2 sin (17),,,,,,,, (20) 443

4 where,, are design variables. The parameters,, are chosen by the designer to determine the relative importance of thrust, PM volume, copper volume and thrust ripple in optimization. Maximization of fulfills simultaneously all objectives of optimization. Such an objective function provides a higher degree of freedom in selecting appropriate design variables. The genetic algorithm is employed to search for maximum value of. The genetic algorithm provides a random search technique to final a global optimal solution in a complex multidimensional search space. A genetic algorithm with parameters listed in Table III is employed to search for optimal design. Fig. 3 shows the flowchart of genetic algorithm. In this paper, the Roulette wheel method is used for selection and at each step elite individual is sent directly to the next population [13], [14]. Table IV. Specification of Designed Motors. Variable Unite motor 1 motor 2 motor mm mm mm mm mm Table III. Genetic Algorithm Parameters. Parameter Value Mutation rate 0.2 Selection rate 0.5 Population size 50 Number of generation 500 Fig. 4. Improvement of fitness function. Fig. 3. Flowchart of genetic algorithm. The values of,, in general dependent on the designer's will and the requirement of the motor application, here three sets of power coefficient are used to optimize the motor. In the first step, thrust force to thrust ripple ratio is maximized by using 1, 1 and 0. In the second step, due to magnet is more expensive than copper, 1, 0, 1.5 and 1.1 are chosen for optimization. Eventually, in the third step, more emphasis is placed on the minimization of copper and PM volume rather than the minimization of thrust ripple by choosing 1, 0.5, 1.1 and 1.6. The results of optimization for dimensions of motors are listed in Table IV. Fig. 4 shows the enhancement of fitness function during the optimization process in these three optimization steps. Fig. 5. Flowchart of FEM. 4. Design Evaluation The design optimizations in this work were carried out based on the analytical model of the machine presented in Section II. Therefore, validity of the design optimizations greatly depends on the accuracy of the model. However, the model is obtained by some simplifications such as ignoring saturation and considering an infinite motor length. Thus, it is necessary to evaluate the extent of model accuracy. In this Section, 2-D nonlinear timestepping FEM is employed to validate the model. It is supposed that the motor controlled by using a currentcontrolled inverter. The relative movement is taken into account in the FEM by using time-stepping analysis and 444

5 Fig. 6. Magnetic flux lines in PMLSM motor. Fig. 7. Flux density distribution in PMLSM motor. Lagrange multiplier method [6], [15]. The forces are then calculated using local virtual work method. A flowchart of the FEM is shows in Fig. 5. A graphical representation of flux lines and flux density distribution in the analyzed motor are depicted in Figs. 6 and 7, respectively. Comparison between analytically predicted and FE calculated open-circuit distributions of the normal flux density component as function of position is depicted in Fig 8. It is seen that the FEM accurately verifies the analytical method. The results of mean thrust, thrust ripple, efficiency PM and copper volume for initial and optimal motors are shown in Table V. It is seen that good agreement is achieved in mean thrust for results of FEM and analytical prediction. Table V. performance of Initial and Motors. Performance (Unit) Initial motor motor 1 motor 2 motor 3 Thrust (N) Analytical FEM Thrust ripple (N) FEM PM volume (Cm 3 ) Copper volume (Cm 3 ) Efficiency The optimized motors thrust value obtained by FEM is shown in Fig. 9. It is seen that thrust ripple reduces effectively in optimized motor 1. In fact, the optimized motor 1 experiences a thrust ripple less than about six times of one for the initial motor, while the mean thrust increases almost 2.6 N. But as shown in Table V, with the same magnet volume, copper volume increase 22% in optimized motor 1 with respect to the initial motor. In the optimized motor 2, the PM and copper are used effectively in thrust production. Therefore, in comparison with the initial motor, the PM and copper consumption reduce about 14% and 8%, respectively. But with almost the same thrust mean, thrust ripple increases 105%. A multi-objective optimization is aimed to achieve high thrust, low magnet and copper volume and low thrust ripple in optimized motor 3. The results presented in table II show that multi-objective optimization provides a Fig. 8. Normal component of flux density distribution as a function of position. Fig. 9. Thrust force as a function of armature position at rated current. design with almost the same thrust, 79% decrease in thrust ripple and 8% decrease both in PM and copper volume, with respect to the initial motor. 5. Conclusion A multi-objective design optimization method was applied on air-core permanent magnet linear synchronous motor with Halbach array to achieve high developed thrust, reduced magnet and copper volume, and low thrust ripple simultaneously. The analytical analysis based on Maxwell equations is derived to predict air gap magnetic flux density distribution, thrust and efficiency. Motor dimensions were optimized using the genetic algorithm. It is seen that in the first optimization step, the thrust ripple decreases up to 81%. in the second optimization step, the magnet and copper volume decrease 14% and 8%, respectively. Eventually, in the third optimization step, thrust ripple, magnet and copper volume decrease 79%, 8% and 8%, respectively. The results of design optimizations are then verified by a 2-D nonlinear time stepping finite element method. References [1] A. boldea and S. Nasar, Linear Electromagnetic Devised. New York: Taylor & Francis, [2] J. Wang and D. Howe, Design optimization of radially magnetized, iron-cored, tubular permanent-magnet machines and drive systems, IEEE Trans. Magn., vol. 40, no. 5, pp , Sep

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