Recent Advances in Electrical Engineering
Calculation and evaluation of the magnetic field air gap in permanent magnet synchronous machine BELLAL ZAGHDOUD1, SAADOUN ABDALLAH2 Department of electrical engineering University of Annaba, ALGERIA 1
[email protected],
[email protected] Abstract: - The best way to understand the phenomena in any investigated motors is to get inside and to see magnetic field distribution. In this paper we will determine and evaluate the air gap field in a permanent magnet synchronous machine (PMSM) using finite element method (FEM). At first a numerical calculation of the magnetic field distribution is applied. Then a harmonic analysis of the air gap flux density waveform is carried out. The results are presented by diagrams. They discussed and compared with experimentally obtained ones, under no load and full load conditions. They show a very good agreement.
Key-Words: - Finite element, harmonic analysis, magnetic flux density, air gap, permanent magnet synchronous machine.
1 Introduction
2 Notations
Prediction and performance analysis of electrical machines depend mainly on the accuracy in the evaluation of the magnetic field linking the different parts of the machine [1-2]. During the last century several approaches have been used to solve this problem. The formulation of the magnetic field by Maxwell's equations using the vector potential is described by the Poisson differential equation [1-3]. Although its formulation is relatively easy to obtain, resolving the equation is virtually impossible in the case of electrical machines, mainly because of the complexity of the geometry and the nonlinearity of the various media of the domain’s solution. In the case of permanent magnet machines the problem becomes insurmountable because of the lack of an analytical formulation of the magnetomotive force (mmf) magnets. The only alternative to solve this problem is to use numerical methods [4-6]. During the last two decades the finite element method proved to be the most appropriate numerical method in terms of modeling, flexibility and accuracy to solve the nonlinear Poisson’s equation governing the magnetic field in electric machines [5,6]. Currently, several modeling of electrical machines softwares are available. The finite element package FEMM 4.2 (Finite Element Method Magnetic) developed by D. Meeker available for free on its website was used for the modeling of the PMSM model and to solve the Poisson’s equation governing the magnetostatic field.
ISBN: 978-960-474-318-6
A(x,y) Ω(Г) vx vy J Jm v B H M M0 µ µ0
magnetic potential vector domain solution bounded by the contour Г reluctivity of the media in the x direction reluctivity of the media in the y direction current density of the carrying current conductors equivalent current density of the magnets peripheral speed of the rotor flux density field intensity magnetization magnetization constant permeability of medium permeability of free space
3 Poisson equation of magnetic field The formulation for magnetic field in quasi-static regime formulated using the magnetic vector potential ⃗ is represented by Maxwell's equations: ⃗ ⃗⃗ ⃗⃗ ⃗⃗
⃗⃗ ⃗ ⃗⃗
The relation (2) states that the magnetic field ⃗⃗ is solenoidal, while the relationship (3) which represents the Ampere in differential form defines
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the sources of the magnetic field in a medium of permeability. The equation of the magnetic field ⃗⃗ expressed in terms of its sources is using: ⃗⃗
[ ̀ ] ⃗⃗
[ ̀ ] ⃗⃗
⃗
any point of the field can also be determined using interpolation function. The precision of the method depends not only on the dimensions of elements and their number but also the type of the interpolation function. As for the numerical method, the finite element method converges to the exact solution provided to increase the number of subdivisions of the solution domain and to ensure continuity of the interpolation function of its first derivatives along the borders of adjacent elements [5,1]. The main steps for implementation of the finite element method [8] are described below.
[ ̀ ] ⃗⃗⃗ [ ̀ ] ⃗⃗⃗
⃗
⃗
The permanent magnet is represented by a surface current density ⃗ which equivalent intensity is calculated using Stokes' theorem [4,7]. ⃗
∯ ̀
∮
⃗⃗⃗
[ ] ⃗⃗⃗
∮
⃗⃗
4.1 Pre-Processing After the problem geometry is defined we have to complete the entire domain of the electric machine by defining the material properties and boundary conditions. By imposing the vector potential zero at the outside diameter of the machine the magnetic field will be confined to the solution domain, while the periodicity conditions can restrict the solution domain to a double pole pitch. The basic idea of FEM application is to divide that complex domain into elements small enough, under assumption to have linear characteristics and constant parameters. Usually, triangular elements are widely accepted shapes for 2D FE models. After this step is completed, the output is always generation of finite element mesh. It is recommended to make mesh refinements in the regions carrying the interfaces of different materials, or with expected or presumed significant changes in the magnetic field distribution.
̀
In Two-dimensional Poisson equation (9) becomes: [ ̀ ] ⃗⃗
⃗⃗ [ ̀
̀
]
⃗⃗ Finally the formulation of the magnetostatic field expressed using the vector potential is: (
)
(
)
Where and are reluctivity of the medium and take the value of ̀ and ̀ within the permanent magnet.
4 Resolution of the magnetic field problem with FEM
4.2 Processing In order processing part to be executed and output results to be obtained system of Maxwell’s equation should be solved.
The finite element method is a numerical procedure designed to obtain an approximate solution to a variety of field problems governed by differential equations. The solution domain is replaced by the problem of the subdomains of simple geometric shapes, called elements, in order to reconstruct the original domain by their assembly. The unknown variables of the considered field are then expressed by an approximate function called interpolation function [1,2]. These functions are defined on each element using the values that the variable takes from the field on each node. Therefore the knowledge of nodal values and interpolation functions allow to define completely the behavior of the variable field on each element. Once the nodal variables, which are actually the unknown factors of the problem, are calculated, the values of the variables of the field on
ISBN: 978-960-474-318-6
4.3 Post-Processing Using the post-processor software, various results can viewed as the mapping of field lines and the intensity of the magnetic induction in different parts of the machine as shown in Figure 1, however the main result of the simulation is the field distribution in the air gap which is given either in graphical form as shown in figure 2 or in the form of a table that provides the magnetic induction at a point as a function of the air gap of its curvilinear abscissa.
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computed at full load Fig 1: magnetic flux distribution The symmetry of the magnetic flux lines change as a function of the rotor position. The orientations of the flux lines depend on the magnets position witch are the origin of the flux. 1 0.8 0.6
B.n, Tesla
0.4
measured at full load
0.2 0
Figure 3: computed and measured flux density along the air gap at no load and full load
-0.2 -0.4 -0.6 -0.8 -1
0
50
100
150
Length, mm
Examination of these curves shows that rapid variations of the magnetic field are mainly due to the presence of teeth and slots of the stator area. Such variations are predictable because the calculation of the field using an interpolation function of the first order results in the constancy of the field [4,2] at each element of mesh. Moreover, the finite element solution is incomplete because the model is unable to take into account electromagnetic phenomena accompanying the rotation of the rotor. Indeed, in a real machine, the effect of oscillation of the field between the slots and teeth with induced eddy currents in the inner surfaces of the teeth, resulting in a local saturation of the interface zone air gap magnetic circuit [2]. For these reasons the field variations in a real machine are less pronounced.
Fig 2: magnetic field distribution along the air gap
5 Comparison between calculated and measured results Figure 3 shows the measured [9] and computed air gap magnetic field of the permanent magnet synchronous machine at no load and full load.
computed at no load
6 Harmonic analysis of the magnetic flux distribution Like most software on the market do not have specific processing functions for the harmonic analysis of the field distribution: The MATLAB software was used to perform harmonic analysis of the magnetic field from the results obtained by the FEMM software.
measured at no load
ISBN: 978-960-474-318-6
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The following figure (figure 4) shows the measured [9] and computed harmonic analysis of the magnetic field distribution at no load and full load.
between the consecutive pair poles [10,11]. Their amplitudes are relatively small, but they still affect the shape of measured air gap waveform as far as comparison of the computed and measured flux density is concerned. Although the computed and measured value of the fundamental closely, some uncertainties remain concerning the evaluation of the high order harmonics because of the undesirable but inevitable slots effects, field oscillation and local saturation of the teeth [12-14]. These side effects have not only an impact on the amplitude spectrum leading to discrepancies between the measured and computed harmonics but also to tend to introduce an additional phase shift [10]. The later effect provides an indication on the magnetic state of the machine; the greater the phase shift is, the greater is the departure of the measured air gap flux density distribution from the computed ones.
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computed at no load 60
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7 Conclusion
17 18 19 20
As part of the determination of the magnetic field in the gap of permanent magnet synchronous machines, solving the problem of the magnetostatic field formulated using the Poisson equation by the finite element method was presented. FEMM software was used to obtain its numerical solution including the analysis required the development of specific processing functions. Moreover the experimental results obtained confirm the accuracy of the results of the numerical solution. Therefore, the determination of various characteristics and parameters that directly depend on its assessment can be made with confidence
measured at no load
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References: [1] Binns, K.J, Riley, C.P. and Wong, T.M, The efficient evaluation of torque and field gradient in permanent magnet with small air gap, IEEE Transactions on Magnetics, Vol. MAG. 21, N° 6, pp 2435-2438, Nov. 1985. [2] Kostenko, M. and Piotroski, L, Electric machines: Alternating current machines, Vol.2, Mir Publishers, Moscow, 1963. [3] J. Slomczynska, Non linear Analysis of the Magnetic Flux Distribution in the Magnetized Magnet motor Stabilized in Air, IEEE Trans. Magn., vol. 39, no. 5, pp. 3250–3252, Sep. 2003 [4] P. Silvester, Finite elements for electrical engineers, Cambridge University Press, 1990. [5] P. Silvester and M.V. K. Chari, Finite Element Solution of Saturable Magnetic Field Problem,
Harmonic amplitude
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measured at full load Figure 4: measured and computed harmonic analysis of the magnetic field distribution at no load and full load
It can be seen that the flux density waveforms are heavily polluted by the high order spatial harmonic. These harmonics are due to a slight asymmetry
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IEEE Trans. on Power Apparatus and Systems. PAS-89, Vol. 7, pp 1642-1651, 1970. [6] J.K. Binns, T.S. Low and M.A. Jabbar, Computation of the Magnetic Field of Permanent Magnet in Presence of Iron Cores, Proc. IEE, pp. 1377-1381, Dec. 1975. [7] Slemon, G.R. and Strauhent, A., Electric machines, Addison-Wesley Publishing Company, Reading, Massachusetts, 1980. [8] D. Meeker, Finite Element Magnetics, User Manual for FEMM Ver. 4.2, Boston, Massachusets, USA, 2009. [9] A. Saadoun, The analysis of the performance of a permanent magnet synchronous generator, M.Sc Thesis, University of Liverpool, Aug. 1989. [10] N. Bianchi and S. Bolognani, Magnetic models of saturated interior permanent magnet motors based on finite element analysis, in IEEE Ind. Appl. Soc. Annu. Mtg., Conf. Record, St. Louis, MO, pp.27–341, 1998. [11]Robinson, R.B, harmonic in ac rotating machines, IEE Monograph., N° 502, pp 380387, Feb.1962 [12]B. Stumberger, B. Polajzer, M. Toman, and D. Dolinar, Evaluation of experimental methods for determining the magnetically nonlinear characteristics of electromagnetic devices, IEEE Trans. Magn., vol. 41, no. 10, pp.4030– 4032, Oct. 2005. [13]B. Stumberger, G. Stumberger, D. Dolinar, A. Hamler, M. Trlep, Evaluation of Saturation and Cross-Magnetization Effects in Interior Permanent-Magnet Synchronous Motor, .IEEE Trans. on Industry Applications, Vol. 39, No. 5, Oct. 2003. [14]L. Chedot, G. Friedrich, A Cross, Saturation Model for Interior Permanent Magnet Synchronous Machine. Application to a StarterGenerator, 39(th) IAS Annual Meeting, Vol. 1, pp. -70, Seattle, USA, 3–7 Oct. 2004.
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