齿槽转矩计算

Development of Analytical Equations to Calculate the Cogging Torquein Transverse Flux MachinesM. V. Ferreira da Luz (1), P. Dular (2), N. Sadowski (1), R. Carlson (1) and J. P. A. Bastos (1)(1) GRUCAD, Dept. of Electrical Engineering, Federal University of Santa Catarina, Brazil.(2) Dept. of Electrical Engineering and Computer Science, F.N.R.S., ULG, Belgium.Po. Box 476, 88040-900, Florianópolis, Santa Catarina, Brazil.E-mail of Corresponding Author: mauricio@grucad.ufsc.brAbstract - Cogging torque is produced in a permanent magnetmachine by magnetic attraction between the rotor permanentmagnets and the stator teeth. It is an undesirable effect thatcontributes to torque ripple, vibration and noise of the machine. In this paper, the resultant cogging torque values are computed using a three-dimensional (3D) finite element analysis. For this, the rotor movement is modeled by means of the moving bandtechnique in which a dynamic allocation of periodic or anti-periodic boundary conditions is performed. The 3D finite element method is the most accurate tool to carry out cogging torque. However, it does not easily allow a parametric study. For this reason, an analytical model was developed in order to predict the cogging torque. The tools are intended to be used for the study of transverse flux machines.I.I NTRODUCTIONAlthough permanent magnet (PM) machines are high performance devices, there are torque variations that affect their output performance. These variations during one revolution arise from factors as: commutation of the phase currents; ripple in the current waveform caused by chopping; variations in the reluctance of the magnetic circuit due to slotting as the rotor rotates. This last effect is called cogging [1]. Cogging torque arises from the interaction between permanent magnets and slotted iron structure and occurs in almost all types of PM motors. It manifests itself by the tendency of a rotor to align in a number of stable positions even when the machine is unexcited, and results in a pulsating torque, which does not contribute to the net effective torque. Therefore, one major task in developing PM machines is to minimize the cogging torque. Several methods are known. Some researchers minimize the cogging torque by skewing, an asymmetric distribution of the magnets or pole shifting [2]. Others works consider the relative air-gap permeance by modeling the shape of slots, the tooth width, or using teeth pairing, extra slots or notches in the teeth [3]. Others works control the function of the magnetization manipulating the shape of the magnets, the magnetization of the magnets themselves, the pole arc to pole pitch ratio, and the shape of the iron core [4].To verify the effects of machine geometry on the cogging torque is important to determinate its waveform. The electromagnetic torque can be calculated analytically or numerically in a variety of ways, such as Maxwell Stress and co-energy methods. However, they require very accurate global and local field solutions, particularly for the determination of cogging torque. In other words, a high level of mesh discretisation is required in a finite element method (FEM) calculation, whilst a reliable physical model is essential to an analytical prediction. A lot of work has been done on prediction of cogging torque in PM motors. They are divided into three groups. The first group uses analytical approaches [4, 5]. The second group uses the bi-dimensional (2D) and three-dimensional (3D) FEM simulation [6] and the third one uses a combined numerical and analytical method [7].In the last years we have developed a set of numerical tools for efficiently studying PM machines with FEM. A 3D magnetodynamic formulation, using the magnetic vector potential as the main unknown, discretized with edge finite elements, has been developed with adapted techniques for considering stranded conductors, periodicity and anti-peridodicity boundary conditions, moving band connection conditions and moving parts. The rotor displacement is modeled by means of a layer of finite elements placed in the air gap [8]. This method, named Moving Band Method, uses an automatic relocation of periodicity or anti-periodicity boundary conditions allowing the simulation of any displacement between stationary and moving parts of an electrical machine. The 3D FEM is the most accurate tool to carrying out cogging torque. However, it does not easily allow a parametric study. Moreover, the 3D simulation demands a high computation time. Hence, the purpose of this paper is to develop an analytical model and to compare it with 3D FEM for a transverse flux machine. This comparison allows finding an analytical model fast and precise to study the cogging torque behavior in order to satisfy some industrial design constraints for machines.The contribution of this paper could be divided in two aspects: the first one is the cogging torque calculation using the Moving Band Method for a 3D problem considering two moving bands in the same motor. The second aspect is the development of the analytical model to the transverse flux permanent magnet (TFPM) machine.TFPM machines have been found to be highly viable candidates in electric and hybrid propulsion applications [6]. Of particular interest are the double-sided topologies where high energy permanent magnets are mounted in the rotor rims in a flux concentration arrangement, yielding high air gap flux densities. The topology of such a machine requires 3D finiteelement analyses to accurately predict the machine parameters [6]. II. M AGNETODYNAMIC F ORMULATIONA bounded domain Ω of the two or three-dimensionalEuclidean space is considered. Its boundary is denoted Γ. Theequations characterising the magnetodynamic problem in Ω are[9]:j h = curl , b e t curl ∂−=, 0 div =b , (1a-b-c) r b h b +μ=, e j σ=, (2a-b)where h is the magnetic field, b is the magnetic flux density, b r is the permanent magnet remanent flux density, e is the electric field, j is the electric current density, including source currents j s in Ωs and eddy currents in Ωc (both Ωs and Ωc are included in Ω), μ is the magnetic permeability and σ is the electric conductivity.The boundary conditions are defined on complementary parts Γh and Γe , which can be non-connected, of Γ,0h =×Γh n , 0 . e =Γb n , 0e=×Γe n , (3a-b-c) where n is the unit normal vector exterior to Ω. Furthermore, global conditions on voltages or currents in inductors can be considered [8]. The a -formulation, with a magnetic vector potential a and an electric scalar potential v, is obtained from the weak form of the Ampère equation (1a) and (2a-b) [9], i.e.,0)' ,()' ,grad v ( )' , ( ',)' curl , ()' curl , url c (s h s c c t s r =−σ+∂σ+>×<+ν−νΩΩΩΓΩΩa j a a a a h n a b a a),(F 'a Ω∈∀awhere s h n × is a constraint on the magnetic field associated with boundary Γh of the domain Ω and μ=ν/1 is themagnetic reluctivity.F a (Ω) denotes the function space defined on Ω which contains the basis and test functions for both vector potentials a and a'. (. , .)Ω and <. , .>Γ denote a volume integral in Ω and a surface integral on Γ of products of scalar or vector fields.Using edge finite elements for a , a gauge condition associated with a tree of edges is generally applied.III. P ERIODICITY C ONDITIONS AND M OVING B AND M ETHOD Another important point is the simulation of the rotor movement. The applied technique permits the use of only one mesh for the calculation.Generally, to model electrical machines not presenting fractional windings, the calculation domain can be reduced to one or two poles using anti-periodic or periodic boundary conditions [9]. The discretisation of these boundaries is performed in a similar way, linking all their geometricalentities (nodes, edges and facets) by pairs. These boundaries are denoted ΓA and ΓB , respectively the reference boundary(which contains all the degrees of freedom) and its associatedboundary [8].For the a -formulation, periodicity conditions are split up into a strong relation on the normal component of b and a weakrelation on the tangential component of the magnetic field h .When edge finite elements are used for a , the strongperiodicity (anti-periodicity, with the other sign) relation for apair of equally oriented edges on ΓA and ΓB is a B = ± a A , (5) where a A and a B are the circulations of a along the considered edges on ΓA and ΓB . In 3D, periodicity conditions have to be consistent with gauge conditions (when used) associated with trees of edges [8].The periodicity boundary conditions can be directly applied to the moving band [8] connection (Fig. 1). The connection between the moving and the stationary regions (both being separately meshed), through the moving band, is similar to a periodicity connection (direct identification of the degrees offreedom; Fig. 1, boundaries b-b'). When (anti-) periodicity conditions are considered on both sides of the band (Fig. 1,boundaries a-a'), a complementary part of this band has to be connected through the same conditions to the moving region (Fig. 1, boundaries c-c') [8].Such connection conditions have to be updated for each position during the movement. When the calculation domain angle is exceeded, the moving part must be relocated in front of the stationary part, while inverting the connection conditions (i.e., inverting the rotor field sources) if anti-periodicity conditions are used.The movement is considered using the Lagrangian approach, i.e. with a moving coordinate system [10]. This approach is easily and implicitly considered with the a -formulation because no deformation is done in the domains involving the time derivative, i.e., in the conducting regions.IV. N UMERICAL P REDICTION OF C OGGING T ORQUE The cogging torque is computed at each angular position by means of 3D FEM analysis, integrating the Maxwell stress tensor on a surface containing the rotor, with null stator currents.To the aim of reducing the numerical errors, the cogging torque should be computed as the mean value of the Maxwellstress tensor on the whole airgap volume V g [9], i.e.∫∫∫∧=gV cogging dv )d (T F r , (6)where F is the Maxwell stress and the r is the dummy radius.V. A NALYTICAL P REDICTION OF C OGGING T ORQUE The cogging torque experienced by all estator teeth has the same shape, but are offset from each other in phase by the angular slot pitch [11]. The cogging torque experienced by the k th stator tooth can be written as the Fourier series()∑∞=ϕ+−+=θ1n n s n o ck )θn(θ2 cos T 2 T )(T , (7)where θ is the mechanical angular position of the rotor and ϕn is the phase angle of the k th harmonic component. T n are the Fourier series coefficients and they are determined by the magnetic field distribution around each tooth, the air gap length, and the size of the slot opening between teeth [11]. The method is based on the derivation of the flux density distributions in airgaps as a function of the machine design parameters. θs is the angular slot pitch calculated by sms N N π=θ, (8) where N m is the number of stator slots and N s is the number of magnet poles.Since the cogging torque of each tooth adds to create the net cogging torque of the motor, the motor cogging torque can be written as ∑−==θ1N 0k ck 2n cogging s )(θT S )(T , (9)where S 2n is the skew factor, which is given by ⎟⎟⎠⎞⎜⎜⎝⎛απαπ=s sk m sk m sn 2N N n sin N n N S , (10) where αsk is the slot pitches.In the analytical approach the assumptions used supposed that the end effects and the iron saturation are negligible.VI. R ESULTSThe analyzed TFPM machine as shown in Fig. 2 has 90 poles, a rated power of 10 kW, a rated voltage 220 V, and a rated speed of 200 rpm. This motor was manufactured by WEG Industries - Brazil. Fig. 3 shows a CAD model of the TFPM machine. Fig. 4 and Fig. 5 show the assembly details of the inner and outer stator for one phase of the TFPM machine. In this doubled-sided construction, the rotor is arranged between an inner and an outer stator.Figure 2. The TFPM machine manufactured by WEG Industries - Brazil.Figure 3. A CAD model of the TFPM machine.Figure 4. Assembly details of the inner and outer stator - one phase of theTFPM machine.Fig. 6 shows the ring-shaped windings of the TFPM machine.The Nd-Fe-B permanent magnets in the rotor are magnetized with an alternating polarity in circumferential direction. Therefore, the flux concentrating elements in the rotor increase the magnetic flux density in the airgaps beyond the remanent flux density of the Nd-Fe-B magnets. Fig. 7 shows the magnetic flux distribution due to the Nd-Fe-B magnets to the one-phase of TFPM machine.Figure 5. Assembly details of the inner stator - one phase of the TFPMmachine.Figure 6. Ring-shaped windings of the TFPM machine.Figure 7. Magnetic flux distribution to the one phase of the TFPM machine. The typical feature of TFPM machine is the magnetic flux path which has sections where the flux is transverse to the rotation plane and the ring-shaped winding in the stator in which the direction of the current corresponds to the movement direction of the rotor. This design leads to a structure in which the design of the magnetic circuit becomes almost independent from the design of the electrical circuit. Hence, there is the possibility to achieve higher torque values by increasing the number of pole pairs without affecting the electrical circuit parameters [12]. Also the absence of end-turns in stator winding which results in reduced copper losses is one of the major advantages of this machine structure.Considering the electromagnetic symmetries and using periodic boundary conditions, the smaller domain of study consists of an 8-degree sector of the whole structure. The 3D mesh without the air elements is shown in Fig. 8. In this figure we can see the stator, the coils, the rotor with the permanent magnets and the two moving bands (one inner and another external to the rotor). Each air gap was divided in three equal layers, being the moving band located in the central layer. Hexahedra in the moving band and prisms elsewhere have been used. The mesh of the structure has 40 divisions along the moving band.Figure 8. The studied domain and 3D mesh for TFPM machine. Results are presented for a speed of 200 rpm and when the machine operates at no-load condition, i.e. only the permanent magnet excitation is considered. Fig. 9 shows the cogging torque produced by both outer and inner parts of one phase.Figure 9. The cogging torque (normalized) produced by both outer and inner parts of one phase versus angle for TFPM machine.VII.C ONCLUSIONSIn this paper, the cogging torque was calculated with a 3D magnetodynamic formulation and with adapted techniques for considering stranded conductors, periodicity and anti-peridodicity boundary conditions, moving band connection conditions and moving parts. The Moving Band Method was implemented for 3D problems considering one or more moving bands in the same motor.The 3D FEM is the most accurate tool to carrying out cogging torque. However, it does not easily allow a parametric study. For this reason, an analytical model was developed in order to predict the cogging torque of TFPM machine. The comparison of the results between the analytical model and the 3D FEM simulation was satisfactory.Consequently, the developed analytical model allows fast and precise study of the influence of rotor permanent magnet distribution as well as the opening of stator auxiliary poles on the cogging torque behaviour in order to satisfy some industrial design constraints for machines. The skewing of the stator slots or, alternatively, of the permanent magnets also is taken into account with the analytical model.A CKNOWLEDGMENTThe authors thank the cooperation of the WEG Industries - Brazil. This work was supported by National Council for Scientific and Technological Development (CNPq) of Brazil.R EFERENCES[1] J. R. Hendershot Jr. and T. J. E. Miller, Design of Brushless Permanent-Magnet Motors, Magna Physics Publishing and Clarendon Press - Oxford, 1994.[2] N. Bianchi and S. 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