Multiobjective Design Optimization of Air-Core Linear ... - IEEE Xplore

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 3, MARCH 2006

Multiobjective Design Optimization of Air-Core Linear Permanent-Magnet Synchronous Motors for Improved Thrust and Low Magnet Consumption Sadegh Vaez-Zadeh and A. Hassanpour Isfahani Department of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran Although air-core linear permanent-magnet (PM) synchronous motors are widely used in precision applications because of their advantages such as fast dynamics, lack of detent force, and negligible iron loss, they basically suffer from low developed thrust, thrust ripple, and excessive use of permanent-magnet materials, all of which lead to undesirable performance and high production cost. In this paper, we analyze performance characteristics of an air-core linear PM synchronous motor by varying motor design parameters in a layer model and a – model of the machine. We propose a multiobjective design optimization to improve thrust, thrust ripple, and consumed magnet volume independently and simultaneously by defining a flexible objective function. A genetic algorithm is employed to search for optimal designs. The results confirm that desirable thrust mean and substantial reduction in magnet volume and thrust ripple can be achieved. We draw several design conclusions from the motor analysis and design optimization. Finally, we carry out a time-stepping finite-element analysis to evaluate the effectiveness of the machine models and the optimization method. Index Terms—Electromagnetic devices, finite-element analysis, linear synchronous motor, modeling, optimization, permanent-magnet motors.

I. INTRODUCTION

L

INEAR MOTORS (LMs) offer many advantages over rotary motors (RMs) where linear and reciprocating motions are required. In particular, absence of mechanical gears and transmission systems provides higher efficiency, higher dynamic performance and improved reliability [1], [2]. Among various linear motor types, linear permanent-magnet synchronous (LPMS) motors enjoy the highest efficiency and force density as well as simplicity of control and high dynamic performance [3], [4]. However, in slotted topologies of LPMS motors, there are detent forces due to slotting and finite length of moving part [5], [6]. A detent force increases thrust ripple and decreases operational accuracy of the motors. On the other hand, in some applications like semiconductor industries and electronics board assembly extra high accuracy is required. Therefore, thrust ripple of slotted topologies make them inefficient for being used in such applications. A solution is to use air-core topologies of LPMS motors. They have no detent force [7]. Therefore, they offer relatively low thrust ripple. Nevertheless, nonsinusoidal distribution of motor flux produced by permanent-magnet (PM) poles results in some thrust ripple. Also, they suffer from low developed thrust which can usually be overcome by excessive use of PM materials in motor poles, leading to a high production cost. Therefore, the enhancement of motor thrust, thrust ripple, and magnet consumption is required to achieve improved motor performance and cost. Nevertheless, improvement of each feature may have an adverse effect on the others, leading to an inappropriate overall motor performance and/or cost. Thus, a wise compromise between these features is required. In order to achieve this goal, optimization of motor structure and dimensions may be performed.

Digital Object Identifier 10.1109/TMAG.2005.863084

Many researchers have been concerned about the design optimization of LPMS motors. Nevertheless, air-core topologies gained less attention. Among limited work on the design optimization of air-core LPMS motors, the optimization of thrust density has been investigated by choosing magnet width and coil width as design variables [8]. A simultaneous optimization of thrust per square of ohmic loss and electrical time constant has been also presented [9]. Improvements of thrust and thrust ripple have been considered separately by deciding the shape and dimensions of coil and PM as design variables [10]. However, in contrast to recent studies on rotary permanent-magnet synchronous motors, the optimization of consumed magnet volume in air-core LPMS motors has not been considered yet. In this paper thrust mean, thrust ripple and magnet volume of a LPMS motor are optimized based on a machine layer model and a - electrical model. Motor length, air gap and magnet dimensions are chosen as design variables. A flexible objective function is defined including thrust mean, thrust ripple and magnet volume. A multi-objective optimization is then carried out by Genetic algorithm (GA) to find out the best set of design variables. Finally, time-stepping finite-element method (FEM) is used to verify the optimization results. II. MACHINE MODEL A. Motor Topology Fig. 1 shows a schematic view of a double sided air-core LPMS motor with a moving short primary. The motor primary is a three-phase air-core winding. Each secondary consists of back iron and permanent-magnet poles facing the primary windings. B. Field Analysis In this section the magnetic field produced by magnets is computed using Maxwell equations. Fig. 2 shows a layer model of the machine assuming that the primary windings are not excited. To simplify the two-dimensional (2-D) analysis, it is assumed that:

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VAEZ-ZADEH AND ISFAHANI: MULTIOBJECTIVE DESIGN OPTIMIZATION OF AIR-CORE LINEAR PMS MOTORS

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Assuming that the permeability of back iron is infinity, the boundary conditions are fulfilled as

and

(5)

From these boundary conditions, constants of (3) and (4) can be found as follows: (6) Fig. 1.

Schematic view of a double sided air-core LPMS motor.

(7) (8) (9)

Fig. 2. A layer model of an air-core LPMS motor.

Flux density can then be achieved by the curl of magnetic vector potential of (3) and (4). As the direction of magnetic vector potential is perpendicular to the - plane, the flux density distribution is obtained as

direction; 1) all regions are extended to infinity in 2) permeability of back iron is equal to infinity; 3) analysis model behaves linearly.

(10) Therefore, the Maxwell equations lead to Laplace and Poisson equations as follows [10]:

Therefore, the flux density in the middle of magnetic air gap has only a normal component given by [11]

for layer I for layer II (1) (11) where is permeability of magnet, vector potential in layers I and II and lent current density given by

and

are magnetic is the PM equivaC. Thrust and Efficiency Calculation

(2) where and are pole pitch and magnet remeance, respectively, and stands for magnet width to pole pitch ratio. The corresponding general solutions of (1) are as follows [11]:

The flux density obtained in the previous subsection is employed to calculate the motor developed thrust. A conventional – electrical model of the machine in a synchronously rotating reference frame can be used in the design optimization. In this model, the flux distribution in air gap is assumed to be sinusoidal and the magnetic saturation is not considered. The motor thrust is then obtained as [3] (12)

(3)

A surface PM type motor with rare earth PM poles having relative permeability close to unity is used in which saturation rarely . As a result, by ignoring very low relucoccurs; thus, tance thrust, (12) is simplified to [3]

(4)

(13)

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If the coil dimensions and current density are known, (13) can be represented by

TABLE I GEOMETRIC PARAMETERS OF THE ORIGINAL LPMS MOTOR

(14) where and stand for coil area per pole per phase, effective current density, and pole pairs, respectively. Also, stands for flux per pole given by [3] (15) where is an essential component of flux density obtained in the previous subsection and is a motor width. In this kind of motor, iron loss is negligible due to lack of iron in moving part and large effective air gap. Therefore, the essential part of electrical loss is the copper loss expressed as [3] (16) and are phase winding resistance, resistivity where of copper, and of length of end connection, respectively. Also, there are additional losses , including mechanical loss and stray loss. Therefore, motor efficiency is given by (17) where

and

is the synchronous speed. Fig. 3. Variations of thrust with magnet dimensions.

III. OPTIMIZATION PROBLEM A multiobjective optimization problem with variables, and constraints is formulated as

objectives,

Maximize (18) . Also, where problem (18) which is described by

is a feasible set of

(19) limit the design variables. The design variThe constraints ables are motor width , air gap , magnet width , and magnet height . The fixed variables are pole pitch , coil height , and primary windings current density . The design objectives in this paper are an increase in the motor developed thrust and reductions in thrust ripple and PM volume. These objectives improve the most important aspects of motor performance and cost. Thrust ripple in an air-core LPMS motor is mainly produced by nonsinusoidal distribution of PM and primary magnetic fields. An analytical representation of thrust ripple is difficult to obtain in the present analytical model. However, with a fixed coil structure, thrust ripple can be reduced by a reduction of PM field harmonics. Therefore, in this paper function is used as an indirect measure of thrust ripple as

(20)

Fig. 4. Variations of magnet volume with magnet dimensions.

is the th harmonics of air gap flux density. The where multiples of the third harmonics components do not influence the machine operation since the motor enjoys a three-phase winding. A two-pole LPMS of the type depicted in Fig. 1 is selected as the basis for optimization. The original geometric parameters of the motor are listed in Table I. Figs. 3–5 show the variations of thrust, magnet volume, and in terms of PM dimensions for constant motor width and air gap. The thrust and magnet volume both increase with the increase of the magnet dimensions. But the former goes toward saturation (Fig. 3) while the latter continues to increase with the increase of PM dimensions (Fig. 4).


Fig. 5. Variations of

H with magnet dimensions.

Fig. 7. Variations of the motor thrust and

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H with air gap.

TABLE II DESIGN CONSTRAINTS

Fig. 6.

Variations of the motor thrust and PM volume with motor width.

The function shows a different pattern of variation with PM dimensions (Fig. 5). It can be concluded that the objectives do not have a simple common optimal point as far as the magnet dimensions are concerned. In fact, meeting an objective may accompany the deterioration of other objectives. Fig. 6 shows the variations of thrust and PM volume with the motor width when the magnet width and height are constant. is constant with motor width variations. The The function motor thrust and are shown in Fig. 7 as functions of air gap for constant magnet dimensions. It is seen that both thrust and reduce with an increasing air gap. The results, presented by 2-D and three-dimensional (3-D) plots, confirm the necessity of a multiobjective design optimization regarding motor thrust mean, thrust ripple, and PM volume. A wide variety of methods exists to compute optimal points for this problem. A widely common approach is based on the reduction of the multiobjective problem of (18) to a single objective one [12], [13]. In order to follow this approach, a new objective function is defined by combining the objectives with cost coefficient or cost powers [13]. An objective function with the general form as (21)

is thus proposed for optimization where and stand for motor thrust, PM volume, and ripple function, respectively. The parameters and are chosen by the designer to determine the relative importance of thrust, PM volume, and thrust ripple in the optimization. A maximization of fulfills simultaneously all objectives of the optimization. Such an objective function provides a higher degree of freedom in selecting appropriate motor parameters. A number of constraints can also be taken into account during the optimization to prevent the possibility of reaching unrealistic optimization results. Magnet height is bounded by demagnetization phenomenon and minimum required thrust. Motor width is limited by an upper bound to prevent low efficiency and by a lower bound to reduce leakage flux effect. Air gap is also limited since a large air gap leads to an increase in PM volume and a reduction in the motor thrust. A small air gap, on the other hand, causes mechanical fault and manufacturing difficulties. Magnet width is also bounded by a lower limit to have an acceptable thrust density. Finally, lower bound of thrust is limited to a minimum required thrust. The limiting values of design variables are listed in Table II. IV. DESIGN OPTIMIZATION Different design optimizations are carried out in this section depending on the selected objectives. The first optimization is aimed toward a maximization of thrust to magnet volume ratio. The next optimization concerned with the minimization of thrust ripple. Finally, all the objectives are integrated to a multiobjective design optimization. The general form of the objective function defined in (21) provides an opportunity to perform all these optimizations according to a same procedure by choosing and . A genetic algorithm (GA) is appropriate values for employed to find an optimal design in each optimization.

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Fig. 9. Variations of thrust to magnet volume in terms of magnet dimensions.

Fig. 8. Flowchart of genetic algorithm. TABLE III GENETIC ALGORITHM PARAMETERS

A. Genetic Algorithm A genetic algorithm provides a random search technique to find a global optimal solution in a complex multidimensional search space [14]. The algorithm consists of three basic operators, i.e., selection, crossover, and mutation. First, an initial population is produced randomly. Then, genetic operators are applied to the population to improve their fitness gradually. The procedure yields in new population at each iteration. Fig. 8 shows the flowchart of genetic algorithm. In this paper, a Roulette wheel method is used for selection and at each step elite individual is sent directly to the next population. Table III shows the GA parameters used in this paper [15].

Fig. 10. Variations of thrust to magnet volume in terms of motor width and air gap.

TABLE IV DIMENSIONS OF OPTIMIZED MOTORS FOR DIFFERENT OBJECTIVE FUNCTIONS

B. Maximization of Thrust to Magnet Volume Ratio In this optimization, the values of and in the objective function of (21) are set to unity and the value of is set to zero. Therefore, a maximization of thrust to magnet volume ratio is aimed with an equal emphasis on thrust and magnet volume. The variations of objective function with magnet dimensions in this case are shown in Fig. 9. It is seen that the magnet height contributes more to the magnet volume rather than to the motor thrust. This is in agreement with the conclusion drawn from Figs. 3 and 4. Therefore, the objective function in this case reduces with the increase of PM dimensions. The variations of objective function in terms of motor width and air gap are also depicted in Fig. 10, showing that it increases with increase of motor width.

This optimization gives a design with a most efficient use of machine PM in thrust production. In other words, keeping the developed thrust constant at 68 N with a current density of A/mm , it gives a design with minimal PM dimensions. In comparison with the original motor with the same current density and thrust, the magnet width and height reduce from 3.9 and 37.8 mm to 3 and 31.5 mm, respectively, as presented in the second column of Table IV. This results in a design with 137 cm or about 22% less magnet consumption as seen in the second column of Table V. The motor efficiency is also calculated for both the optimized motor and the original motor using


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TABLE V DIMENSIONS OF OPTIMIZED MOTORS FOR DIFFERENT OBJECTIVE FUNCTIONS

Fig. 11.

Normalized thrusts of original motor and optimal motor.

(16)–(17), where m/s and . This results in a 2% reduction in the efficiency of the optimized motor as presented in Table V. Also, the design provides a slightly higher thrust ripple as seen in the same table. C. Minimization of Thrust Ripple In this case, the value of and in the objective function of (21) are set to zero and the value of is set to unity in order to take into account the thrust ripple only. The optimized motor in this case and the original motor are compared in term of normalized thrust in Fig. 11, showing an effective ripple reduction of the optimization. In fact, the optimized design experiences a thrust ripple less than about ten times of the one for the original motor, as presented in the third column of Table V. The numerical values of optimal motor parameters are also listed in the third column of Table IV. It is seen that the magnet dimensions and air gap increase while the motor width decreases with respect to the original motor. As a result, the PM volume increases substantially. This may not always be the case due to the special pattern of ripple variations shown in Fig. 5. D. Multiobjective Optimization The two optimizations presented above confirm the necessity for a multiobjective optimization to achieve a high thrust, a low PM volume, and a low thrust ripple. Therefore, thrust, magnet volume, and thrust ripple are simultaneously considered in the objective function by deciding nonzero values for and in (21) as in the fourth column of Tables IV and V. Although

Fig. 12.

Flowchart of FEA.

Fig. 13.

Flux lines in the LPMS motor.

and in general depends on the designer’s the values of will and the requirement of the motor application; here more emphasis is placed on the minimization of PM volume rather than the minimization of thrust ripple by choosing and . This is because the results of the previous optimizations show that the PM volume is less sensitive to the optimization than the thrust ripple. The results of the design optimization are listed in the fourth column of Table IV for this case. It is evident that the PM dimensions reduce, while the air gap and motor width slightly increase in the optimized design with respect to the original motor. The fourth column of Table V shows that the multiobjective optimization provides a design with the same thrust, but with almost 17% less PM volume and 38% less thrust ripple with respect to the original motor. This proves the effectiveness of the proposed optimization in simultaneously meeting all the objectives. V. 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 a limited motor length. Thus, it is necessary to evaluate the extent of model accuracy. In this section, 2-D nonlinear time-stepping FEM is employed to validate the model. It is supposed that the motor is controlled by using a current-controlled inverter. The relative movement is taken into account in the FEM by using time-stepping analysis and Lagrange multiplier method [16]. The forces are then calculated using local virtual work method. A flowchart of the FEM is shown in Fig. 12. The 2-D FEM is carried out by a commercial package and numerical and graphical results are obtained. Fig. 13 shows a graphical

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representation of flux lines in the LPMS motor. Thrusts for different designs are obtained by the FEM and are compared with those obtained by the analytical model as in Table V. It is seen that the results of analytical model are very close to the results of FEM. The maximum error in the case of thrust calculation is less than 5%, which is reasonable. This proves the validity of the design optimization. However, to achieve a more accurate design optimization, a more detailed model of LPMS machine is required. VI. CONCLUSION Different design optimizations are performed on an air-core LPMS motor to achieve high developed thrust, reduced magnet volume, and low thrust ripple. A layer model of the machine is developed and used in defining the optimization problem in each case. Demagnetization of magnet poles is prevented by imposing a lower limit on the magnet height. Also, a low efficiency is avoided by limiting the motor width by an upper bound. PM dimensions in addition to motor width and air gap are chosen as design variables. The analysis of motor performance reveals that the thrust increases with the increase of magnet dimensions but with a slower pace at higher thrust values. In other words, magnet dimensions usually contribute more to magnet volume rather than to motor thrust. Therefore, care must be taken not to decrease the thrust per magnet volume by increasing magnet dimensions too much. Also, thrust per magnet volume is improved by increasing motor width. However, thrust ripple shows a special pattern of variation with magnet dimensions not always decreasing with the increase of magnet dimensions. A genetic algorithm is employed to search for the optimal design variables. Keeping the developed thrust constant, the magnet volume and the thrust ripple are optimized independently to 78% and 10% of those of an original motor, respectively. A multiobjective optimization is defined by applying an appropriate cost power to each objective. This optimization simultaneously reduces the magnet volume and the thrust ripple to 83% and 62% of those of the original motor, respectively, while maintaining a desirable value for the developed thrust. The validity of the performed design optimizations are confirmed by a 2-D nonlinear time-stepping FEM. ACKNOWLEDGMENT This work was supported by the Center of Excellence on Applied Electromagnetic Systems at the University of Tehran, Tehran, Iran. REFERENCES [1] 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. 3262–3277, Sep. 2004. [2] J. G. Gieras and Z. J. Piech, Linear Synchronous Motors. Boca Raton, FL: CRC, 2000. [3] A. Boldea and S. Nasar, Linear Electromagnetic Devices. New York: Taylor & Francis, 2001. [4] S. A. Nasar and I. Boldea, Linear Electric Actuators and Generators. Cambridge, U.K.: Cambridge Univ. Press, 1997.

[5] K. C. Lim, J. K. Woo, G. H. Kang, J. P. Hong, and G. T. Kim, “Detent force minimization techniques in permanent magnet linear synchronous motors,” IEEE Trans. Magn., vol. 38, no. 2, pp. 1157–1160, Mar. 2002. [6] M. Inoue and H. Sato, “An approach to a suitable stator length for minimizing the detent force of permanent magnet linear synchronous motors,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1890–1893, Jul. 2000. [7] N. Fojii and K. Okinaga, “X-Y linear synchronous motor without force ripple and core loss for precision two-dimensional drive,” IEEE Trans. Magn., vol. 38, no. 5, pp. 3273–3275, Sep. 2002. [8] J. Kim et al., “Static characteristics of linear BLDC motor using equivalent magnetic circuit and finite element method,” IEEE Trans. Magn., vol. 40, no. 2, pp. 742–745, Mar. 2004. [9] M. Andriollo et al., “Design optimization of slotless linear PM motors,” in Proc. 4th Int. Symp. Linear Drives for Industry Applications, Birmingham, U.K., Sep. 8–10, 2003. [10] G. H. Kang, J. P. Hong, and G. T. Kim, “A novel design of an air-core type permanent magnet linear brushless motor by space harmonics field analysis,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3732–3736, Sep. 2001. [11] M. J. Chung, M. G. Lee, S. Q. Lee, S. M. Kim, and D.-G. Gweon, “Analytical representation of cogging force in linear brushless permanent magnet motor,” in Proc. ICMT’99, Oct. 1999, pp. 310–314. [12] G. Liuzzi, S. Lucidi, F. Parasiliti, and M. Villani, “Multiobjective optimization techniques for the design of induction motors,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1261–1264, May 2003. [13] S. Vaez-Zadeh and A. R. Ghasemi, “Design optimization of permanent magnet synchronous motors for high torque capability and low magnet volume,” Elect. Power Syst. Res., vol. 74, pp. 307–313, Mar. 2005. [14] E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Reading, MA: Addison-Wesley, 1989. [15] N. Bianchi and S. Bolognani, “Design optimization of electric motors by genetic algorithm,” IEE Proc.—Electr. Power Appl., vol. 145, no. 5, pp. 475–483, Sep. 1998. [16] D. Rodger, H. C. Lai, and P. J. Leonard, “Coupled element for problems involving movement,” IEEE Trans. Magn., vol. 26, no. 2, pp. 548–550, Mar. 1990.

Manuscript received September 12, 2005; revised December 6, 2005 (e-mail: [email protected]).

Sadegh Vaez-Zadeh was born in Mashhad, Iran, in 1959. He received the B.Sc. degree from Iran University of Science and Technology, Tehran, Iran in 1985 and the M.Sc. and Ph.D. degrees from Queen’s University, Kingston, ON, Canada, in 1993 and 1997, respectively, all in electrical engineering. In 1997, he joined the University of Tehran as an Assistant Professor and became an Associate Professor in 2001 and a Full Professor in 2005. He served the university as Head of Power Division from 1998 to 2000 and currently is the Director of Advanced Motion Systems Research Laboratory which he established in 1998 and the Director of Electrical Engineering Laboratory since 1998. He has been an Associate Editor and a Member of Editorial Board of Iranian Journal of Electrical and Computer Engineering since 2001. He has been active in many technical conferences in different capacities, lastly as a technical program committee member of the eighth International Conference on Electrical Machines and Systems held in China in 2005. He has published more than 100 conference and journal papers and translated a book. His research interests include advanced rotary and linear electric machines and drives, motion control, magnetic levitation, electric and hybrid vehicles, power system dynamics, and robust control. Prof. Vaez-Zadeh has received a number of awards domestically, including a best paper award from Iran Ministry of Science, Research and Technology in 2001 and a best research award from the University of Tehran in 2004.

A. Hassanpour Isfahani was born in Isfahan, Iran, in 1980. He received the B.Sc. degree in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in 2002 and the M.Sc. degree in power engineering from the University of Tehran, Tehran, Iran, in 2005. His research interests include design, modeling, and control of electrical machines.