MODELING surface-mounted permanent-magnet (PM)

Transcription

1 Modeling of Axial Flux Permanent-Magnet Machines Asko Parviainen, Markku Niemelä, and Juha Pyrhönen Abstract In modeling axial field machines, three dimensional (3-D) finite-element method (FEM) models are required in accurate computations. However, 3-D FEM analysis is generally too time consuming in industrial use. In order to evaluate the performance of the axial flux machine rapidly, an analytical design program that uses quasi-3-d computation is developed. In this paper the main features of the developed program are illustrated. Results given by the program are verified with two-dimensional and 3-D finite element computations and measurements. According to the results, it is possible to evaluate the performance of the surface-mounted axial flux PM machine with reasonable accuracy via an analytical model using quasi-3-d computation. Index Terms Analytical modeling, axial flux permanentmagnet (PM) synchronous machine, PM machines. I. INTRODUCTION MODELING surface-mounted permanent-magnet (PM) machines can be done via analytical, two-dimensional (2-D) finite-element method (FEM) or three dimensional (3-D) FEM analysis. In industrial use, an analytical approach or 2-D FEM is preferably used in computations due to their speed compared to the 3-D FEM. In modeling axial flux PM machines, the requirements related to the speed and accuracy of the computations are contradictory. An axial flux machine is an inherent 3-D geometry from the point of modeling. Thus, analytical or 2-D FEM analysis, usually performed on the average radius of the machine, do not generally yield sufficient accuracy in computations. However, the 3-D FEM analysis is usually too time consuming from the point of view of engineers working in R&D departments in the electrical machine industry, especially if a preliminary design of a machine is an objective. To improve this situation, an analytical design tool for axial flux surface-mounted PM machines is developed in cooperation with industry. The analytical design tool uses quasi-3-d computation described in [1] and [2], nonlinear reluctance network [3], [4], and analytical design methods presented, for example, in [5] during the design of surface-mounted axial flux PM machines. In this paper the main features of the developed design program are presented and the results given by the program are compared to the results given by the 2-D and 3-D finite-element analysis (FEA) and measurements made for the prototype machine. Fig. 1. A principle how to transform the 3-D geometry of an axial flux machine to a 2-D geometry, which can be used in quasi-3-d computation. II. QUASI-3-D COMPUTATION From the point of quasi-3-d modeling, the axial flux PM machine can be considered to be composed of several linear machines. The overall performance of the axial flux machine is obtained by summing the performance of individual linear machines. The approach allows taking into account different magnet shapes and variation of tooth width in the direction of the machine radius. A principle of how to transform the 3-D geometry of an axial flux machine to a corresponding 2-D model, which is used in quasi-3-d computation, is illustrated in Fig. 1. In quasi-3-d computation the average diameter of a particular computation plane, starting from the external diameter of the machine, is given by the equation (1)

2 TABLE I PARAMETERS OF A PROTOTYPE MACHINE Fig. 2. Definitions for the magnet length and pole pitch. where is the external diameter of the stator, ( 1 for the first computation plane, 3 for the second plane, 5 for the third plane, etc.), is the number of computation planes used in computation, and is the total length of the stator. It is defined as Fig. 3. One of the stators of the machine and the rotor of the machine during magnet installation. where is the internal diameter of the axial flux machine stator. The pole pitch for each computation plane is given by (2) (3) where is the number of poles. In general, the pole shoe width-to-pole-pitch ratio vary along the machine radius. Thus, it is defined as may where is the length of the magnet at an individual computation plane (Fig. 2). A. Overview III. ANALYTICAL DESIGN PROGRAM The design program is developed in the Matlab environment. The decision to use Matlab as a code development environment was based on the fact that Matlab includes a remarkable amount of built-in functions, which alleviate the programming work compared, for example, to direct C/C++ code. A Matlab-independent version of software is also developed. In the design program, the user can define the initial design values of an axial flux machine via a graphical user interface or alternatively via an initialization file. Depending on the given initial values, there are two possible ways to perform the computation. 1) If the machine main dimensions are not given in the initialization of the computation the program calculates the main dimensions of the machine, based on axial flux machine sizing equations presented in [5] and electrical and magnetic loadings allowed. Performance computations are then based on the calculated main dimensions of the machine. (4) Fig. 4. The magnet shape and the outlines of a rotor pole used in the prototype machine. 2) If the machine main dimensions are given in the initialization, the program uses the given values in the computation if the values given are acceptable. The machine performance, including torque and backelectromotive-force (EMF) waveforms, is displayed via graphs. The design results, such as phase resistance, material consumption needed in a design, and several other parameters are presented and saved to a result file. B. Reference Machine Used in Computations The functionality of the developed analytical design program, as well as the performed FE computations, are verified via measurements made for a 5-kW prototype machine. The prototype machine consists of a single rotor and two stators. The stators are operating in parallel in star connection. In this construction, there may arise problems due to unbalanced axial force under load if the currents in the stator windings are not equal. However, this solution is preferred, because the machine still can work if one of the stators is electrically disconnected. The main parameters of the prototype machine are given in Table I. Computations, reported in this paper, are based on the design given in Table I. The rotor structure and one of the stators of the machine are illustrated in Fig. 3. The magnet shape used is sinusoidal, shown in Fig. 4, and is described more accurately in Appendix I.

3 C. Analytical Model for Air-Gap Flux Density Distribution The analytical expression for the air-gap flux density distribution in the case of a nonslotted stator can be solved from Poissons s and Laplace s equations via magnetic vector potential by taking into account the symmetry conditions and by assuming that the relative permeability of the stator and rotor iron is infinite [7]. For radial flux machines, an analytical expression for the air-gap flux density distribution, produced only by the PM, is given in [6]. For axial flux machines, handled as a linear motor, it is more convenient to use Cartesian coordinates. Such an expression is given in [7] and shown by (5), at the bottom of the page, where is the remanence flux density of the PM material, is the pole pitch defined in (3), is the pole shoe width-to-pole-pitch ratio defined in (4), is the physical length of the air gap multiplied by Carter s coefficient,, is the PM relative recoil permeability multiplied by, and is the thickness of the PM. The influence of the slot openings on the waveform of the is taken into account in the model by introducing a relative permeance function described in [8]. The air-gap flux density distribution, which includes the effect of the stator slot openings, is given by the equation Fig. 5. Air-gap flux density distribution obtained from the analytical model and 2-D FEA. The pole shoe width-to-pole-pitch ratio is 0.26 in this computation. (6) In the dimensioning of the machine winding the information related to the fundamental component of the air-gap flux density is an essential parameter. The quasi-3-d computation method used evaluates the amplitude of the resulting fundamental component of the air-gap flux density based on (6) and the equation where is the number of the computation planes used in the analytical model and is the peak value of the air-gap flux density fundamental component of on computation plane produced by the PM. The fundamental components of the air-gap flux density are obtained via an FFT-algorithm, which is performed for all computation planes used in the computation. The effect of the armature reaction on the air-gap flux density distribution is evaluated iteratively during the design. During the first sizing loop of the phase winding the stator currents are set to zero and the winding is designed based on the allowed electrical loading and required EMF. Then, the machine performance is computed including direct- and quadrature-axis currents, and, giving the corresponding flux density components in the air gap. During the second iteration loop the effect of stator currents is taken into account and the phase winding is redesigned. The (7) Fig. 6. A comparison between the flux density values given by the analytical, 2-D FEM and 3-D FEM models for five different pole shoe width-to-pole-pitch ratio values. The fundamental component of the air-gap flux density is presented. performance of the machine is computed again and new currents are calculated. Iteration is continued until the required accuracy is reached. A comparison between the air-gap flux density values given by different modeling methods under no-load condition is given in Figs As a conclusion, there appears only a small difference between the values given by the analytical approach and 2-D and 3-D FEA. Values given by the analytical approach are slightly higher compared to the FEM solutions, which can be explained via iron saturation effect. However, the difference is negligibly small, so from the point of view of machine design, the analytical model gives an accurate enough result for. The calculated air-gap flux density level was confirmed with measurement. The air-gap flux density was measured with a flux (5)

4 Fig. 8. Leakage flux paths in a situation where the magnets are partly short circuited via stator iron. These flux lines act as an effective flux for the point of the air-gap flux density distribution, but actually this part of flux does not contribute to the EMF. Fig. 7. Actual no-load flux density distribution in the air gap obtained from the 3-D FEA. Note that the visualization plane is extended over the actual air-gap area in the direction of the machine radius. density probe giving the flux density of 0.75 T under a tooth in the air gap. The measured value is in good agreement with the calculated value of 0.76 T given by the FEA and (5). D. Computation of No-Load Phase Voltage The no-load phase voltage produced by the magnets only is first evaluated from the air-gap flux density distribution for each computation plane as where is the number of coil turns per phase, is the winding factor for the fundamental wave, and is the air-gap flux which is obtained by integrating numerically the air-gap flux density distribution given by (6). The no-load phase voltage for the whole machine is then computed as By neglecting the leakage fluxes, which mainly flow through tooth tips according to Fig. 8, a notable difference in the amplitudes of the no-load phase voltages is expected between the analytical approach and FEA. The comparison between the analytical approach, time transient 3-D FEA, and measurement is shown in Fig. 9. According to 3-D FEA, the amplitude of the obtained no-load voltage is about 11% less compared to the voltage amplitude given by the analytical method in a situation where it is assumed that the flux produced by the magnet is going fully (8) (9) Fig. 9. Waveform of no-load phase voltage obtained via the analytical method by using 20 computation planes and the waveform of the no-load phase voltage obtained from time transient 3-D FEA and measurement. through the phase winding. However, the obtained waveform of the phase voltage is very similar in both computations. Thus, it is possible to introduce a leakage factor, which can be used in the analytical computation. The drawback of this method is that an FEA is required in order to find the correct leakage factor for each design if high accuracy is needed. However, in a preliminary design of a machine this uncertainty may be tolerated and it is possible to introduce a leakage factor, which is based on previous design experiences or reluctance network approach, Fig. 12. A comparison between the measured and calculated no-load phase voltage reveals that the curvatures are very similar except on the very top of the measured voltage. The difference is explained via magnet manufacturing; each magnet was composed of two independent blocks and the joint is parallel to the radius of the machine. The joint reduces the flux produced by the PM causing the observed flattened top for the measured no-load phase voltage.

5 Fig. 10. The effect of the numbers of computation planes on the obtained waveform of no-load phase voltage. The leakage factor used in the computations is kept as a constant for all computation planes. In Fig. 10 the effect of the number of the computation planes on the waveform of the no-load phase voltage is illustrated. The number of the computation planes required to achieve an accurate result is typically between If more planes are used, the trend is that changes in a waveform are so small that there is no practical benefit to increase the number of the computation planes. However, the computation of cogging torque is more sensitive to the number of the computation planes used. For the sinusoidal magnet, presented in Fig. 4, the number of computation planes required to achieve a constant waveform for cogging torque lies between Thereby, the computation of the cogging torque determines what is the required number of the computation planes for a particular magnet shape. E. Computation of Cogging Torque and Load Torque Cogging torque is computed analytically based on the no-load air-gap flux density distribution. The torque is obtained for each computation plane via the virtual work method as presented in [9] (10) where is the volume of air gap and is the angular position of the rotor. The total cogging torque produced by the machine is obtained in a similar way as the no-load voltage Fig. 11. A comparison between 3-D FEA, analytical model and measurement in a computation of the cogging torque. In the analytical model, the number of computation planes used is 20. The magnet joint influences the measured cogging torque by increasing its amplitude, therefore it is reasonable to compare directly only the results given by the FEA and the analytical method. As a conclusion, it is possible to evaluate the cogging torque with sufficient accuracy via the analytical method. This is an advantageous result since we may conclude that the analytical method is suitable in the computation of the cogging torque, which opens a possibility to compare reliably and rapidly different magnet shapes. The electrical torque produced by the machine is obtained from the general electromagnetic torque equation for any electrical machine (12) where is the number of phases, is the order of the machine phase, is the back EMF of the phase, and is the current of phase. is the angular speed of the rotor [10]. In (12) it is assumed that the phase currents are sinusoidal. The instantaneous value of the total torque is achieved by adding the instantaneous value of the cogging torque to the instantaneous value of the electromagnetic torque (13) (11) The analytical computation of the cogging torque is verified via a 3-D FEA and measurement. A set of magnetostatic problems was solved in order to evaluate the cogging torque produced by the machine. Fig. 11 shows the comparison between the computations and measurement. It can be noticed that the analytical model gives higher amplitude for the cogging torque as compared to 3-D FEA whereas the measurement gives the highest amplitude but the curvatures are in good agreement. F. Computation of Motor Performance Motor performance computations are obtained by using classical electrical machine design methods. These computation methods are defined in the literature, for example, in [5], and therefore are not repeated here. Table II compares the values obtained from the analytical design program and from the measurements made for the prototype machine. The main difference in a computation arises from the computation of the fundamental component of the air-gap flux density, defined in Section III-C, and from the computation of the iron

6 TABLE II CALCULATED AND MEASURED VALUES TABLE III FLUX DENSITY COMPARISON BETWEEN USED RELUCTANCE NETWORK AND 2-D FEM TABLE IV COMPUTATION TIME COMPARISON After the rotor angle is obtained from the measurement the inductances are calculated from the equations of flux linkages (15) Fig. 12. A part of reluctance network used. Circles are MMF sources and boxes are reluctances. losses. The analytical model uses a nonlinear reluctance network during the computation. Such a network is illustrated in Fig. 12. A nonlinear reluctance network allows evaluating the flux density levels and iron losses for each computation plane separately as was previously done in [11] by using 2-D FEM (14) where is the calculated iron loss on computation plane. The reluctance network used does not take into account the 3-D behavior of the flux and it is too coarse to take into account all leakage flux paths. Due to the simplifications used, the flux density levels obtained are slightly higher compared to the values given by 2-D or 3-D FEM in different parts of the machine, as shown in Table III. The values given by the 2-D FEM in Table III are average values corresponding to a particular section of the machine stator. In Table II, the direct-axis reactance and the quadrature axis reactance are reactances, which the frequency converter estimated during the prototype machine identification run. The load angle is then calculated based on the motor model used with parameters obtained from the identification run. The direct torque control (DTC) inverter used estimates the inductances as follows [14]. Estimate the flux linkage in the stator coordinates. Measure the rotor angle. Transform the estimated flux linkage to the rotor coordinates. Calculate the inductances and reactances. where is the direct axis inductance, is the PMs flux linkage estimated by the inverter, is the estimated direct-axis flux linkage. and is the direct-axis current. The quadratureaxis inductance is calculated in a similar way as (16) where is the estimated quadrature-axis flux linkage and is the quadrature-axis current [14]. Based on (15) and (16), the reactances and are calculated on a known frequency. A thermal lumped-parameter model for interior-type axial flux machines is under development and will be added to the design program. This will further improve the computation accuracy. G. Computation Time Comparison The 3-D FE model takes into account the 3-D effects in the machine electromagnetic behavior, thus, the model is evidently the most accurate one if the mesh used is a proper one. As a drawback, the working time needed to build up a 3-D FE-model and to solve it is usually longer if compared to analytical or 2-D FE-models. However, in the quasi-3-d method used, the situation may be different since 2-D-FEA must be performed separately for all computations planes. Table IV summarizes the time needed to prepare the models as well as to perform the computations. The time given for the preparation work of the models is based on the experience of the authors with the FE software used and the analytical design program described in this paper. Despite the quasi-3-d method used, the preparation work in 2-D FEA has to be done only once since it is possible to parameterize the model. The required computation time in 2-D and 3-D FEA is given for one computation step. The number of surface elements (2-D FEA) or volume elements (3-D FEA)

7 The coordinate of the magnet outline in any point is calculated in the coordinate frame according to the equation (A.2) where is the total length of stator defined in (2). The magnet outline described by (A.2) is modified because the width of the pole is a function of the machine radius in the case of an axial flux machine. The modification is obtained by introducing a radius ratio factor for each sector, used in the computation. The radius ratio is defined as (A.3) Fig. 13. Forming a magnet outline, which produces sinusoidal flux linkage. The magnet outline is first formed in Cartesian coordinate system x y, which is fixed to the corner of the magnet. The origin of the polar coordinate system and Cartesian coordinate system X -Y, used to form the final shape of magnet outline, is in the center of the arc. On the right, the initial and the developed shapes of the magnet. is reported. It can be recognized from the reported computation times that the 2-D FEA tends to be more time consuming than the 3-D FEA if the number of computation planes is high. For the reported FE models this situation appears if there are more than 200 planes, which is practically too many. IV. CONCLUSION An analytical design program using quasi-3-d computation was developed in order to model surface-mounted axial flux PM machines rapidly. The main features of the design program are introduced. The advantage of the analytical design tool is the essentially shorter computation time compared to FE models. Compared to the finite-element computations and measurements, the analytical design tool gives sufficiently accurate results. In order to improve the accuracy of the analytical computation a thermal lumped-parameter model for an interiortype axial flux machine is under development. APPENDIX In axial flux machines the movement of the magnet appears along a circular path toward the phase winding and the slot openings at any radius. By introducing a magnet shape that produces a sinusoidal flux linkage it is possible to reduce torque pulsations produced by the electrical machine [12], [13]. The shape also offers a good rejection capacity against cogging torque as shown in Fig. 11. To form a sinusoidal flux linkage in the case of an axial flux machine with sinusoidal magnet shape, the magnet outline must be defined via the following procedure. The initial width of the magnet base at the outer radius of the machine is selected to be equal to the length of the pole arc at radius (Fig. 13) where is the number of poles. (A.1) at radius. is a constant defining the initial pole shoe widthto-pole-pitch ratio on radius. The final shape of the magnet outline is achieved by multiplying the length of the arc formed by the sinusoidal magnet at radius with the factor. The obtained magnet shape is shown on the right-hand side of Fig. 13. In practice, leakage fluxes, and the fact that it is not useful to set the pole shoe widthto-pole-pitch ratio equal to 1 on the outer radius of the machine, cause the waveform of the no-load phase voltage to be not totally harmonic free. REFERENCES [1] S. Gair, A. Canova, J. F. Eastham, and T. Betzer, A new 2D FEM analysis of a disc machine with offset rotor, in Proc. Int. Conf. Power Electronics, Drives and Energy Systems for Industrial Growth, vol. I, 1995, pp [2] G. Cvetkovski, L. Petkovska, M. Cundev, and S. Gair, Quasi 3D FEM in function of an optimization analysis of a PM disk motor, in Proc. Int. Conf. Electrical Machines, vol. IV, Helsinki, Finland, Aug. 2000, pp [3] C. B. Rasmussen and E. Ritchie, A magnetic equivalent circuit approach for predicting PM motor performance, in Conf. Rec. IEEE-IAS Annu. Meeting, 1997, pp [4] J. Perho, Reluctance network for analysing induction machines, Ph.D. dissertation, Dept. elect. Eng., Helsinki Univ. Technol., Helsinki, Finland, [5] J. F. Gieras and M. Wing, Permanent Magnet Motor Technology-Design and Applications New York, [6] Z. Q. Zhu, D. Howe, E. Bolte, and B. Ackermann, Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part I: Open-circuit field, IEEE Trans. Magn., vol. 29, pp , Jan [7] M.-J. Chung and D.-G. Gweon, Modeling of the armature slotting effect in the magnetic field distribution of a linear permanent magnet motor, in Arch. Elektrotech., 2002, vol. 84, pp [8] Z. Q. Zhu and D. Howe, Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors, IEEE Trans. Magn., vol. 28, pp , Mar [9] G. Barakat, T. El-meslouhi, and B. Dakyo, Analysis of the cogging torque behavior of a two-phase axial flux permanent magnet synchronous machine, IEEE Trans. Magn., vol. 37, pp , July [10] S. Huang, M. 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8 Asko Parviainen was born in Kiuruvesi, Finland, in He received the M.Sc. degree in electrical engineering in 2000 from Lappeenranta University of Technology, Lappeenranta, Finland, where he is currently working toward the Ph.D. degree in electrical engineering. He is also currently a Research Assistant at Lappeenranta University of Technology. His research interests include design and modeling of electrical machines, in particular, low-speed axial and radial flux PM machines. Markku Niemelä received the B.Sc. degree in electrical engineering from Helsinki Institute of Technology, Helsinki, Finland, in 1990, and the M.Sc. and Ph.D. degrees from Lappeenranta University of Technology, Lappeenranta, Finland, in 1995 and 1999, respectively. He is currently a Senior Researcher in the Laboratory of Electrical Drives Technology, LUT. His current interests include control of line converters, sensorless control, and design of synchronous machines. Juha Pyrhönen received the M.Sc. degree in electrical engineering, the Licentiate of Science (Technology) degree, and the Ph.D. degree (Technology) from Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 1982, 1989, and 1991, respectively. He became an Associate Professor of Electric Engineering at LUT in 1993 and a Professor of Electrical Machines and Drives in He is currently Head of the Department of Electrical Engineering. He is engaged in electric motor and electric drive research and development. He is also the leader of the Carelian Drives and Motor Centre which in cooperation with Finnish ABB Company develops new electric motors and drives. Synchronous motors and drives, switched reluctance motors and drives, induction motors and drives, solid-rotor high-speed induction machines and drives, as well as active network bridge control, are included in his current interests. He is leading the research work of several postgraduate research groups working in these target areas.