Optimization Design of a Segmented Halbach Permanent-Magnet Motor Using an Analytical Model

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1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 7, JULY Optimization Design of a Segmented Halbach Permanent-Magnet Motor Using an Analytical Model Miroslav Markovic Yves Perriard Integrated Actuators Laboratory (LAI), Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerl We present an optimization design of a slotless permanent-magnet brushless dc motor (400 W, 6000 rpm), two-segment Halbach arrays. The optimization is based on a motor analytical model which is obtained by solving the magnetic scalar potential using the Fourier series. We show that the iron-cored Halbach configuration is better than the classical one that the air-cored Halbach configuration is even better. This means that the rotor iron can be completely omitted. We very the results by using the finite-element method. Index Terms Brushless dc motor, magnetic field, optimization, permanent magnet. I. INTRODUCTION P ERMANENT-MAGNET (PM) motors Halbach arrays have become more more attractive for many applications [1]. Compared to the magnetic field in the classical configurations ( the parallel or radial magnetization), that created by an Halbach array is more self-shielding, which means that it is not only the back iron which provides the return path of the magnetic flux. By adjusting the number of Halbach array segments, the air gap field can be shaped. However, too many segments signicantly complicate the motor construction [2]. A special type of the Halbach excitation is the ideal one: it creates an ideal sinusoidal field is completely self-shielding, but on the other h it is extremely dficult to industrialize, as it requires a special magnetizer. In the literature, there are several interesting papers in the domain of PM motors Halbach arrays. A complete review of the domain is presented in [3]. The paper compares the segmented the ideal Halbach cylinder machines (which is the main topic of [4] as well), shows a possible field of applications in high-speed, servo, linear drives. The paper [5] presents a completely analytical solution for a PM motor an ideal Halbach array. The field solution contains only one harmonic. An interesting motor design analysis is presented in [6]: it is shown that the Halbach magnetized PM motor has 15% lower volume than the radially magnetized one. However, important details on how the design is obtained are not presented. The paper [7] also analyses advantages of the Halbach magnetization, but no analytical model is given. Finally, an application of Halbach magnetization in a planar motor is presented in [8]. This paper presents a possible good compromise between the classical configuration the ideal Halbach one. It is created by two segments, which means PMs alternately magnetized in the radial the tangential directions. The paper contributions are 1) the magnetic field two-segment Halbach array is solved in a general case, the motor complete mathematical model ready for optimization is generated 2) this model Manuscript received October 08, 2008; revised December 01, 2008 January 19, Current version published June 19, Corresponding author: M. Markovic ( miroslav.markovic@epfl.ch). Color versions of one or more of the figures in this paper are available online at Digital Object Identier /TMAG TABLE I MOTOR INPUT PARAMETERS WITH NUMERICAL VALUES is used for the optimization design for given specications: only an optimization using the complete model, the motor mass as the objective function, can give an answer whether the Halbach magnetization improves the motor performance. The final conclusion is that it does, contrary to [9]. II. PROBLEM DEFINITION The goal is to design a slotless PM brushless dc (BLDC) motor (400 W, 6000 rpm). The maximal ambient temperature is 55, which gives the maximal overtemperature of 100 for the insulation class F. The motor should be connected to a dc source via a six-leg inverter which injects sinusoidal currents in the three motor phases. The motor input parameters (specications the chosen materials properties) are presented in Table I. For this application, a two-segment Halbach array is used (Fig. 1). The radial the tangential PMs cover the fractions of the pole pitch, respectively, ( ), so that they cover the angles, respectively. The empty space between the PMs is. Each PM piece is parallelly magnetized. The classical configuration is a special case of the analyzed one for. The air gaps between the rotor yoke PMs, the stator yoke windings are introduced in order to obtain a general field solution /$ IEEE

2 2956 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 7, JULY 2009 C. Solution The solution of (2) for the three layers is given by (3) (4) (5) Fig. 1. Analyzed motor configuration (only one pole is presented, for 0=(2p) <'<=(2p)). The six unknown integration constants are determined from the boundary conditions [5], as shown in Appendix B. The solution is not written here, but instead the final solution for the flux density is given as follows. III. FIELD SOLUTION A. Assumptions The first step is to determine the magnetic field generated by the PMs. Throughout the analysis, it will be assumed that, which is always the case the modern PM materials. This approximation signicantly simplies solution out introducing an important error (as it will be shown). The following analysis is a generalization of [5], as a nonideal Halbach excitation introduces higher harmonics. The stator rotor iron are treated as ideal, which is achieved by correctly dimensioning both yokes, as we will see later. A multiple pole case is analyzed; the case is analyzed separately [10], as a parallelly magnetized hollow cylinder is an ideal Halbach excitation. The analysis is performed in 2-D, in a polar coordinate system. Knowing the radius the dimensions,,,,,,, the radii are determined as similarly for others (Fig. 1). The electrical frequency is. B. Governing Equation Excitation The configuration in Fig. 1 is divided into three circular layers (1: exterior air, 2: PM, 3: interior air). Both iron yokes will impose boundary conditions. According to [5], the governing equation for the magnetic scalar potential for the three layers is (1) D. Air Field Once the potential determined, its definition gives the flux density consequently:. It gives for the air layers 1 3 (6) (7) (8) (9) (10) The field excitation, which is the layer 2 magnetization, is given in the Fourier series form as, where denotes from now on. The harmonics are given in Appendix A. The solution is given in the form, which substituted in (1) gives where denote, respectively. (2) where (11) (12) (13)

3 MARKOVIC AND PERRIARD: OPTIMIZATION DESIGN OF A SEGMENTED HALBACH PERMANENT-MAGNET MOTOR 2957 IV. ELECTROMAGNETIC TORQUE With the phase instantaneous current density, the rotor mechanical position (Fig. 1 shows ) the motor axial length, the instantaneous electromechanical torque acting on the phase is created by the Laplace force as On the other h, the same flux equals to the mean flux density along the line,. It means (21) (14) an elementary current element an elementary surface area of the current sector in Fig. 1 (defined by ). After substituting here after the integration it follows (15) The flux which should be accepted by the rotor yoke is obtained by integrating the flux density (9) near the rotor for. After the integration, the result is (22) On the other h, the same flux equals to the mean flux density along the line,. It means the useful copper surface in a current sector (16) (17) VI. OTHER QUANTITIES (23) The motor mathematical model for the optimization is completed by adding several additional motor quantities. The equations of the copper, PM, stator iron rotor iron mass (,,, ) are trivial will not be written here. The end-winding length is estimated to (18) The case is singular: the second term (line) in (18) should be substituted by. Furthermore, the current injected in the phase is purely sinusoidal (only the term exists) in phase the corresponding time-variable term in (15): (similarly for the phases ). Concerning the mean value, the product of two cosinus terms has the mean value 1/2, for three phases it gives 3/2. It means that the mean electromagnetic torque is (19) (24), i.e., the wind- (25) (26) From now on, it will be assumed that ings are mounted at the stator surface. A. Losses Efficiency The copper losses are The stator iron losses are V. IRON FIELD Instead of deriving complicated formulas for the iron flux density as it is done for the air, the mean values will be determined. Again, the goal is to have enough iron material to accept the necessary flux. The flux which should be accepted by the stator yoke is obtained by integrating the flux density (6) near the stator for. After the integration, the result is (27) the specic iron losses in W/kg. The rotor iron losses are neglected, as the rotor rotates synchronously the magnetic field generated by the sinusoidal phase currents. The motor mechanical torque is (20) (28)

4 2958 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 7, JULY 2009 Fig. 2. Motor equivalent thermal circuit. The motor efficiency is (29) B. Temperature The winding temperature is determined using a simple thermal model [11]. As it is supposed that the heat is generated only in the stator, the equivalent thermal circuit is presented in Fig. 2. It gives the overtemperature (30) the ambient temperature. Here, is the thermal resistance of the layer windings, given by (31) its equivalent thermal conductivity. Its value is estimated according to [12]: for the filling factor of 0.4, the equivalent thermal conductivity of a bundle of conductors is twice the conductivity of the insulating material between them. As an epoxy is used for potting, the conductivity of 0.22, it gives the estimated value of. is the thermal resistance of the stator yoke, given by (32) the iron thermal conductivity. is the thermal resistance of the natural convection over the stator external surface, given by the convection coefficient [13]. (33) VII. OPTIMIZATION Once the mathematical model completed, the optimization is performed. An advanced optimization software ProDesign is used [14]. A. Optimization Software The software ProDesign uses the motor analytical model presented in the previous sections to determine the optimal design. Beside the analytical model, it also requires the following: values of the fixed input parameters (physical constants); initial values for the free input parameters (which will be the starting point for the optimization process); allowed intervals for the free input parameters (which will define the domain in which the optimal solution will be searched); constraints for the output parameters in the form of equalities non-equalities ( an output is not constrained, it remains free); one output parameter which will be the objective function (for which the maximum or minimum will be searched). The Prodesign is a mono-objective optimization software, which means that it enables the maximization/minimization of only one output parameter. The software uses an advanced Sequential quadratic programming (SQP) method which is proved highly effective for solving constrained optimization problems smooth nonlinear functions in the objective constraints [15]. B. Motor Configurations Three motor configurations are compared: iron-cored classical configuration (Fig. 1 ); iron-cored Halbach configuration (Fig. 1 ); air-cored Halbach configuration (Fig. 1 ). In the configuration, the rotor iron is obviously necessary. With the Halbach array, the idea is to analyze the tangential magnets could completely replace the rotor iron by conducting the necessary return flux, so that the configuration transforms to. C. Optimization Procedure The free parameters are:,,,,,,,,. The objective function is the motor mass. The motor efficiency will take values from a defined set: in such a manner, a Pareto curve (the set of nondominant points [16]) in the plane will be obtained. The imposed constraints are:,,. The right-h side values are shown in Table I. In the configuration, is not a free parameter as it is fixed to zero. In the configuration, the equations for become irrelevant, as the rotor iron does not exist; in addition, becomes a free parameter instead of. VIII. RESULTS A. Pareto Curves The three Pareto curves, the final result of the optimization, are shown in Fig. 3. It is evident that the configuration is

5 MARKOVIC AND PERRIARD: OPTIMIZATION DESIGN OF A SEGMENTED HALBACH PERMANENT-MAGNET MOTOR 2959 Fig. 3. Pareto curve for the analyzed motor configurations,. better than, that is slightly better than. The mass is reduced up to 15%. It means that the Halbach configuration is the best option, that in addition the rotor iron is not necessary. It offers other advantages, such as a simpler production process reduced rotor inertia. It could be noted that the optimization algorithm always finds an optimum out the empty space between the magnets, typical values. As the motor final design, the configuration for is chosen. The criterion is that the motor should be lighter than 800 g (the condition imposed by the industrial partner). The initial values of the free parameters are:,,,,,,. The optimal values of the free parameters are:,,,,,,. The next results are obtained:,,,,,,. B. Results Verication In order to very the results, a commercial FEM software is used. The magnetic flux lines are presented in Fig. 4. At first, the formulas (6) (7) for the air field will be veried. Fig. 5 presents the air field along the circular line for. Obviously, the results match well, the formula gives a slightly stronger field, due to the approximation. The FEM software gives for the stator iron field, the stator iron losses, the electromagnetic torque the mechanical torque which match well the results presented in the previous subsection. The error in the torque calculation is 2.4%. IX. CONCLUSION The paper presented an optimization design for three BLDC motor configurations segmented Halbach array magnets. Fig. 4. Magnetic flux lines. Fig. 5. Air field along the circular line r =19mmfor 0=3 <'<=3. The best configuration in this case is out the rotor iron, as the tangential magnets completely overtake its function of providing the return path for the magnetic flux. However, it cannot be proven that this conclusion is valid in the general case. APPENDIX A PM MAGNETIZATION The field excitation, the layer 2 magnetization, is given by (34) (35)

6 2960 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 7, JULY 2009 After developing (34) (35) in the Fourier series form, the harmonics are obtained as (36) (37) APPENDIX B BOUNDARY CONDITIONS The invariancy of at a boundary gives the condition (38) The invariancy of at a boundary gives the condition (39) The boundary conditions (38) (39) are written for the boundaries The additional two equations are the boundary condition (38) for the boundaries 3 rotor iron 1 stator iron. As the iron is ideal,, which gives the conditions for for. These six equations give the six unknown integration constants. REFERENCES [1] Z. Zhu, Recent development of Halbach permanent magnet machines applications, in Proc. Power Conversion Conf., Apr [2] R. Gupta N. Mohan, A three-phase PM brushless DC motor for low-power low-speed fan applications Optimizing cost efficiency, in Proc. IEEE 42nd IAS Annu. Meeting, Sep [3] Z. Zhu D. Howe, Halbach PM machines applications: A review, Proc. Inst. Elect. Eng., vol. 148, no. 4, pp , Jul [4] Z. Zhu, Z. Xia, D. Howe, Comparison of Halbach magnetized brushless machines based on discrete magnet segments or a single ring magnet, IEEE Trans. Magn., vol. 38, no. 5, pp , Sep [5] Z. Xia, Z. Zhu, D. Howe, Analytical magnetic field analysis of Halbach magnetized PM machines, IEEE Trans. Magn., vol. 40, no. 4, pp , Jul [6] S.-M. Jang et al., Design analysis of high speed slotless PM machine Halbach array, IEEE Trans. Magn., vol. 37, no. 4, pp , Jul [7] X. Yanliang F. Kaijie, Analysis on toothless permanent magnet machine Halbach array, in Proc. Power Electronics Motion Control Conf., Aug. 2006, vol. 3, pp [8] R. Huang, J. Zhou, G. Kim, Minimization design of normal force in synchronous PM planar motor Halbach array, IEEE Trans. Magn., vol. 44, no. 6, pp , Jun [9] K. Davey, Optimization shows Halbach arrays to be non-ideal for induction devices, IEEE Trans. Magn., vol. 36, no. 4, pp , Jul [10] M. Markovic Y. Perriard, Analytical formula for the torque/emf constant of a slotless BLDC motor, in Proc. IEEE 12th Conf. Electromagnetic Field Computation (CEFC), Apr. 2006, pp [11] A. Boglietti et al., A simplied thermal model for variable-speed selfcooled industrial induction motor, IEEE Trans. Ind. Appl., vol. 39, no. 4, pp , Jul./Aug [12] Y. Bertin, Refroidessement des machines électriques tournantes, (in French) Tech. de l Ingénieur, Traité Génie Électrique, p. D [13] M. Markovic, L. Saunders, Y. Perriard, Determination of the thermal convection coefficient for a small electric motor, in Proc. IEEE 41st IAS Annu. Meeting, Oct. 2005, vol. 1, pp [14] ProDesign Optimization Software [Online]. Available: [15] P. Gill, W. Murray, M. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM Rev., vol. 47, no. I, pp , [16] M. Markovic Y. Perriard, Simplied design methodology for a slotless BLDC motor, IEEE Trans. Magn., vol. 42, no. 12, pp , Dec