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THE 8 INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING May 23-25, 2013 Bucharest, Romania
Cogging Torque Investigation in Permanent Magnet Synchronous Motor Used for Electrical Vehicles Steliana Valentina PUSCASU*, Leonard MELCESCU*, Mircea COVRIG* * Electrical Machines Dept., Electrical Engineering Faculty, POLITEHNICA University of Bucharest, Romania e-mail:
[email protected],
[email protected],
[email protected] Abstract- The present paper presents an analysis regarding the influence of the form and arrangement of permanent magnets on the value of the cogging torque for a surface permanent magnet synchronous motor, designed for the direct drive of the wheels of a vehicle. There are taken into consideration influences caused by: the constructive form of the magnets, the ratio between the pole-arc and the pole-pitch, the opening of the stator slots, the introduction of additional slots on the stator teeth, the asymmetrical distribution of the permanent magnets. The study takes into consideration 2D models developed on the basis of the validated 2D model for the built machine. The authors present a few general conclusions valid for the construction type of the analyzed machine. Keywords: permanent magnet synchronous motors (PMSMs), permanent magnet (PM), cogging torque, finite element method
I.
INTRODUCTION
In recent years, permanent magnet synchronous motors (PMSM) have been used in applications that work with low speeds: wind turbines [1] - [5] and in vehicle electrical drive [6] - [9]. One of the most important problems with PMSMs motors is the pulsating torque which is inherent in their design. This ripple is parasitic and can lead to mechanical vibration, acoustic noise, and problems in the drive systems. Minimizing the ripple is a matter of great importance in the design of PMSMs. There are three sources of torque ripple coming from the machine: i) cogging torque, ii) difference between the permeances of the air gap in the d- and q-axis (reluctance torque), and iii) distortion of the magnetic flux density waveform in the air gap [10]. The cogging torque is caused by the interaction between the permanent magnets (PMs) mounted on the rotor and stator anisotropy, due to the number of slots [11]. There are different approaches to reducing the cogging torque. Evaluating the cogging torque during the design stage is a must for low speed electrical machines. The techniques related to the construction of the machine reduce the cogging torque by optimizing the structural parameters of the machine [12]. In [13] the author classified the cogging torque reduction techniques in two categories: techniques that require modifications in the construction of the stator or of the rotor. The most cost effective method for reducing the cogging torque component in PM machines is to make changes in the construction of the rotor or the magnets rather than modifying the stator structure, due to the complexity and the cost of the stator manufacturing [13].
Scientific literature presents numerous methods for reducing the cogging torque to satisfactory values for the application [13], [14-18]. The present paper assesses the effects of cogging torque values when considering construction changes, by using the numerical model of a motor that equips a wheel drive electric vehicle that has been built in the Electrical Machines Laboratory. Because cogging torque reduction methods lead to a change of form in the air gap magnetic field determined by permanent magnets, both effects will be studied together. II.
INITIAL DESIGN OF THE MACHINES
The analyses reported in this paper are based on 21-slot 28pole PMSM. This machine has fractional slot windings and a number of slots per pole and per phase lower than unity (q < 1). The different results given in the following are computed with the aid of the professional software FLUX 2D. The computation domain is represented by a cross section of the machine, Fig. 1. The finite element mesh is presented in Fig. 1. The rotor movement has been considered through a “compressible” air zone placed in the middle of the motor airgap. In order to achieve good numerical accuracy the mesh in that region was thoroughly refined, Fig.1. [19]. The stator and rotor magnetic cores are made of soft magnetic material characterized by saturation magnetization Bs = 2,2 T and initial relative magnetic permeability μr = 1000. The permanent magnets are made of NdFeB with μr = 1,1, and Bs = 1,0446 T.
Fig. 1. Computation domain and detail of the finite-element mesh.
III.
RESULTS
The paper presents the effects of a few construction changes – the magnet geometry, the pole arc to pole pitch ratio, the stator slot opening width and the asymmetrical distribution of the permanent magnets - on cogging torque values and on the fundamental harmonic amplitude values of the magnetic field in the air gap caused by permanent magnets.
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Fig. 3. Cogging torque and fundamental harmonic amplitude values for the case of using a trapezoidal magnet with large base attached to rotor
There is an effective decrease (by 91%) of the cogging torque for angle β = 35°. In the same case, a much slower decrease of the amplitude of fundamental harmonic is obtained - approximately 14%. Fig. 4 presents the results for the trapezoidal magnet with the small base attached to the rotor. cogging torque fundamental harmonic amplitude
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A. The Influence of the Magnet Geometry On The Value Of The Cogging Torque Two different cases were analyzed: the trapezoidal magnet and the steps magnet, both compared to the rectangular magnet used on the studied motor. In the case of the trapezoidal magnet, there are two models: the trapezoidal magnet with the large base attached to the rotor and the trapezoidal magnet with the small base attached to the rotor see Fig. 2.
trapezoidal magnet with trapezoidal magnet with large base attached to rotor small base attached to rotor Fig. 2. Constructive elements of magnets.
In all cases, the thickness of the magnets is equal with the rectangular magnet thickness (hm=6 mm). At the same time, the large base has the same openness as the side of the rectangular magnet. The angle β at the large base of the trapeze was considered as a parameter, this reflecting the reduction of the small base compared to the large base. Fig. 3 presents maximum cogging torque values compared to the maximum torque value obtained for the largest opening slot (Ma_max=9,24 Nm). In the same graph the values are compared to the value of the amplitude of fundamental harmonic for the initial case.
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In all cases the number of stator slots and the number of rotor poles are the same for all studied cases. The cogging torque is calculated by simulating the machine without current in the stator windings for different rotor positions. For each rotor position, the model mesh is automatically recalculated.
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28 0,9 mm 21 3 NdFeB 294 mm 224,4 mm 3,6 mm 215,4 mm 200 mm
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PARAMETERS OF ANALYZED MACHINE Pole number Airgap Number of slots Number of phases PM material Stator outer diameter Stator inner diameter Magnet thickness Rotor outer diameter Rotor inner diameter
cogging torque fundamental harmonic amplitude
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TABLE I
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The machine parameters are given in Table I.
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Fig. 4. Cogging torque and fundamental harmonic amplitude values for the case of using a trapezoidal magnet with small base attached to rotor.
The cogging torque decrease is slower than in the previous version; also, the decrease of the fundamental harmonic amplitude is smaller. Building trapezoidal magnets has a rapid effect on decreasing maximum cogging torque and slow effects on decreasing the amplitude of the fundamental harmonic, but both solutions face technological problems. The authors also investigated the solution of using different models of two steps magnets, while varying the height of the steps, as well as their length - the magnet dimensions are represented by the parameters h1 and h2 - see Fig. 5. All variants were compared with the initial case, in which the magnet is rectangular and all the steps magnets analyzed have the same maximum thickness - H1 and the same length H2.
Fig. 5. Analyzed magnet geometry.
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cogging torque fundamental harmonic amplitude
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Fig. 6. Cogging torque and fundamental harmonic amplitude values for the case of magnet steps - h2=0,25H2.
It must be noted that the cogging torque has a minimum value relative to α. The construction of narrow magnets has rapid effects on decreasing the maximum cogging torque and significant effects on decreasing the amplitude of the fundamental harmonic. The most advantageous solution is the one with a magnet opening of αm = 9,7°, when the cogging torque decreases by 80%. This version also implies a lower cost for the magnet and a significant decrease of the flux density in the air gap. C. The Influence of the Stator Slot Opening Width on the Value of the Cogging Torque The effect of the slot opening width on the cogging torque, for the previously shown motor, is investigated in this section. The initial value of the opening slot is c = 6,7 mm and will be reduced to a minimum to allow the theoretical winding of the machine – see Fig. 9.
The efficiency of this method, due to a low minimum value, is obvious. This method leads to a significant reduction of the fundamental harmonic amplitude value, namely approximately 8% of the magnetic field in the air gap. B. The Influence of the Pole-Arc to Pole-Pitch Ratio on the Value of The Cogging Torque The polar coverage factor α is defined as the ratio between the pole arc αm and the pole pitch τ , as in (1): α α= m. (1) τ Fig. 7 presents the permanent magnet.
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Fig. 9. Slot opening width.
Fig. 10 presents the maximum cogging torque values, compared to the maximum value of the cogging torque (Ma_max = 9,24 Nm) obtained for the largest opening. In the same graph are included the values of fundamental harmonic amplitude registered at the initial amplitude of the fundamental harmonic, depending on the value of slot opening. cogging torque fundamental harmonic amplitude
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Fig. 7. Definition of pole–arc to pole–pitch ratio.
The authors analyze the basic machine with a rectangular magnet and a pole arc value of αm = 12,7 ° and the polar pitch τ = 12.85°. The interval α m ∈[7,7 o;12,7 o ]and, respectively, α ∈[0,59;0,98]is taken into consideration for the study. The results are presented in Fig. 8. cogging torque fundamental harmonic amplitude
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value relative to α. 0
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Fig. 10. Cogging torque and fundamental harmonic amplitude values for the case of modify stator slot opening width.
This shows the important effect of decreasing the slot opening on the cogging torque and on the fundamental spatial harmonic amplitude. B [%]
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Fig. 6 presents the results for the following situation: the thickness of the magnet is H1, the step width is constant (h2 = 0,25 H2) and the variable h1 is the thickness from the top side of the step.
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Fig. 8. Cogging torque and fundamental harmonic amplitude values for the case of optimization the ratio between the pole-arc and the pole-pitch of the magnet.
D. The Influence Caused by Asymmetrical Distribution of the Permanent Magnets on the Value of the Cogging Torque For the studied motor, we suppose that 27 out of the 28 adjacent magnets have an equal opening angle (γ), while the remaining one is different (φ=360º- 27γ). Starting from the initial configuration which has the opening angle γ = 12.85 °, we simulated several options for reducing the cogging torque, taking into consideration a reduced decrease of the value of the fundamental wave amplitude - see Fig.13. The purpose was to find the magnet
opening which has the lowest cogging torque value. The result was this angle position: γ = 12,73 °. The complete results are presented in Fig. 11. cogging torque fundamental harmonic amplitude
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Fig. 11. Cogging torque and fundamental harmonic amplitude values for the case of asymetrical distribution of the PM.
A rapid decrease of the cogging torque occured, while the maximum flux density in the air gap declined slowly. III. MODEL VALIDATION The model validation has been done by comparing experimentally and numerically computed electromagnetic torque values for the functioning of the machine under load and for its start. Table II presents comparative numerical and experimental results, while also showing the current values for which measurements were made. TABLE II NUMERICAL AND EXPERIMENTAL RESULTS
Operating mode
If [A]
load operation Starting mode
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M [Nm] experimental 31 200
numeric 35 198.5
To be noted: the numerical model neglects friction torque, the supplementary losses in the stator due to the harmonics produced by PWM inverter, and hysterezis losses in the rotor. Given the presented results, the numerical models developed by the authors may be taken into consideration as highly trustworthy. IV.
CONCLUSIONS
Various techniques for minimizing the cogging torque of PMSMs have been presented in this paper. Several constructive variants for magnets were taken into account: trapezoidal magnets, steps magnets, magnets with different openings, with asymmetrical distribution of the permanent magnets and with constructive changes in the machine stator: decreasing opening slots. Based on the presented analyses, in order to reduce the cogging torque for the studied machine, we propose the following structural changes: reducing the size of slot opening to a technological minimum (c = 2,7 mm) and the asymmetrical distribution of the 28 permanent magnets. Both changes can be performed with minimum costs. Other options
taken into consideration have little effect or require higher expenses. REFERENCES [1]
A. Grauers, “Design of direct-driven permanent-magnet generators for wind turbines”, Doctoral thesis, Chalmers University of Technology, Goteborg, Sweden, 1996. [2] T. Hartkopf, M. Hofmann, S. Jockel, “Direct-drive generators for megawatt wind turbine”, European Union Wind Energy Conference, EWEC’97, Dublin, 1997. [3] B. Chalmers , W. Wu, E. Spooner, “An axial-flux permanent-magnet generator for a gearless wind energy system”, IEEE Transactions on Energy Conversion, Vol. 14, No. 2, pp. 251-257, 1999. [4] O. Carlson , A. Grauers, A.Williamson, S. Engstrom, E. Spooner, “Design and test of a 40 kW directly driven permanent-magnet generator with a frequency converter”, EuropeanWind Energy Conference and Exhibition, EWEC’99, Nice, France, 1999. [5] J. Chen, C. Nayar , L. Xu, “Design and finite-element analysis of an outerrotor permanent-magnet generator for directly coupled wind turbines”, IEEE Transactions on Magnetics, Vol. 36, No. 5, pp. 38023809, 2000. [6] R. Wrobel, P. Mellor, “Design considerations of a direct drive brushless machine with concentrated windings”, IEEE Trans. on Energy Conversion, vol. 23, no. 1, pp. 1-8, 2008 [7] F. Magnussen, D. Svechkarenko, P. Thelin, C. Sadarangani, “Analysis of a PM machine with concentrated fractional pitch windings”, NORPIE, 2004 [8] J. Wang, Z. Xia, D. Howe, “Three-phase modular permanent magnet brushless machine for torque boosting on a downsized ICE vehicle”, IEEE Trans. On Vehicular Technology, vol. 54, no. 3, pp. 809- 816, 2005 [9] A. EL-Refaie, T. Jahns, P. McCleer, J. McKeever, “Experimental verification of optimal flux weakening in surface PM machines using concentrated windings”, IEEE Trans. on Industry Applications, vol. 42, no. 2, pp. 443-453, 2006 [10] J.A. Guames, P.M. Garcia, A.M. Iraolgoitia, J.J. Ugartemendia, „Influence of slot opening width and rotor pole radius on the torque of PMSM”, International Conference on Renewable Energies and Power Quality, Valencia, April, 2009 [11] N. Bianchi, M. Dai Prè, L. Alberti, E. Fornasiero, “Theory and design of fractional- slot PM machines”, tutorial course notes, 23 Sept. 2007 [12] K.Abbaszadeh, F. Rezaee Alam, M. Teshnehlab, “Slot opening optimization of surface mounted permanent magnet motor for cogging torque reduction”, Energy Conversion and Management 55, 2012,108 [13] M. Aydin, “Magnet Skew in Cogging Torque Minimization of Axial Gap Permanent Magnet Motors”, Proceedings of the 2008 International Conference on Electrical Machines, 2008 [14] L. Dosiek, P. Pillay, “Cogging Torque Reduction in Permanent Magnet Machines”, Ieee Trans. on Industry Applications, vol. 43, no. 6, 2007, pp 1565-1571 [15] C. Bianchini, F. Immovilli, E. Lorenzani, A. Bellini, M. Davoli, “Review of Design Solutions for Internal Permanent Magnet Machines Cogging Torque Reduction, 2011, Ieee Transactions on Magnetics, vol. 48, no.10, Oct. 2012, pp 2685-2693 [16] A. Kumar, S. Marwaha, A. Marwaha, “Comparison of methods of minimization of cogging torque in wind generators using FE analysis”, J. Indian Inst. Sci., July–Aug. 2006, 86, 355–362 [17] E. Muljadi, J. Green, “Cogging torque reduction in a permanent magnet wind turbine generator, 21st American Society of Mechanical Engineers Wind Energy Symposium Reno, Nevada January 2002 [18] T. Tudorache, L. Melcescu, M. Popescu, “Methods for Cogging Torque Reduction of Directly Driven PM Wind Generators”, OPTIM, 2010 [19] S. Uncuta, L. Melcescu, M. Covrig, O.Craiu, T. Ursu, “Numerical Analysis of a PM Synchronous Motor Used in Electrical Vehicles”, ATEE, May 2011 [20] S.V. Uncuţă (Puscasu), ” Contributions on increasing the performance of electromechanical converters with permanent magnets, the PhD Thesis (available only in Romanian), December 2012
