COMPARISON OF DISPLACEMENT DUE TO MAXWELL FORCES AND MAGNETOSTRICTION IN BLDC MOTOR STATIC DISPLACEMENT

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1 Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych Nr 62 Politechniki Wrocławskiej Nr 62 Studia i Materiały Nr magnetic force, magnetostriction, finite element method, brushless DC motor Jerzy PODHAJECKI*, Adrian MŁOT*, Mariusz KORKOSZ** COMPARISON OF DISPLACEMENT DUE TO MAXWELL FORCES AND MAGNETOSTRICTION IN BLDC MOTOR STATIC DISPLACEMENT In the paper authors calculated the relative importance of magnetostriction deformation compared to deformation due to reluctance forces in the brushless direct current motor with permanent magnet (BLDC). The computed results concerning static displacement due to Maxwell forces and magnetostriction by using a 2-D numerical program are based on the finite element method (FEM). The numerical results show increase the deformation due to magnetostriction in the stator iron. 1. INTRODUCTION Noise emitted from rotating electrical machines is derived from aero dynamical (cooling-air flow), mechanical (the friction between different parts of machine in bearings and brushes) and electromagnetic sources (Maxwell forces and magnetostriction) [1, 6, 7]. Magnetostriction is a property of ferromagnetic materials that makes them to change their shape in the same direction as the applied magnetic field. In the case of non-rotating machinery without air-gaps (transformers, inductors) magnetostriction is the major cause of noise. In rotating electrical machines, apart from Maxwell forces, also magnetostriction leads to vibrations of the magnetic core that directly contribute to acoustic noise [1 3, 6 7]. In the modelling of deformation due to Maxwell forces and magnetostriction, there are two types of calculations: the magnetic calculation and the mechanical one [2, 3, 6]. For both the magnetic as well as the mechanical calculation the finite element method is used. Maxwell forces calculated in the magnetic analysis are loads in mechanical analysis. * Department of Electrical Engineering, Automatic Control and Computer Science, Technical University of Opole, ul. Luboszycka 7, , Opole. ** Department of Electrical Engineering, Technical University of Rzeszow, ul. W. Pola 2, Rzeszów.

2 320 The magnetostrictive strains are represented in the mechanical analysis by a set of temperatures that induce mechanical strains and the deformation of the same size as those caused by magnetostriction. The magnetostriction strain was calculated by using: λ = 10 B 6 2 [2]. 2. PROTOTYPE OF MOTOR FOR THE MAGNETIC AND MECHANIC CALCULATION An effective two-dimensional analysis of the magnetic field distribution for a BLDC motor with integral slot-windings has been presented in detail by the authors [4, 5]. A typical BLDC motor with surface mounted magnets is shown in Figure 1. Table 1 includes the technical specifications of the considered prototype of the BLDC motor. Table 1. The parameters of the BLDC motor Air-gap length 1.5 mm PM material Sintered NdFeB PM radial thickness 3 mm Stator outer diameter 60 mm Stator and rotor stack length 59 mm Residual flux density 1.21 T Coercive magnetizing intensity 892 ka/m 3 Rated Power 0.55 kw Rated nominal current 10 A Max rotational speed 1500 rpm Fig.1. Prototype of 6-pole motor with surface mounted magnets The analysis of the magnetic field in 2-D space is performed under the following assumptions: the field is considered to be magnetostatic; the current density within the cross section of the coils is uniform; the rotational speed of the motor is constant. The current density in the coils equals A/m 2 for the analyzed BLDC motor. The stator fabricated from the limited iron (isotropic steel type EP with 6 magnetostrictive coefficient 10 ) was manufactured by Besel Company on the basis of the Sh80-6 type induction motor. The rotor was made of solid iron with a permament magnet glued onto its surface. The permanent magnets have the shape of cylindrical sectors [4, 5]. The second quadrat of the B H characteristic of the permanent magnet type Nd2 Fe14 B is used to provide the flux source of the motor. The residual flux density B is 1.21T, and the coercive magnetizing intensity H is of -892 ka/m. r c

3 321 The analysis of the mechanical field in 2-D space is performed under the following assumptions: the effect of mechanical stress on magnetization and magnetostriction strain has been neglected, and the effect of the mounting of the machine on deformation has not been taken into account. In the mechanical analysis the stator is modelled by material electrical steel 11 E = 2 10 Pa Young Modulus, ν = 0. 3 Poisson modulus and ρ = 7830 kg/m 3 10 density. The windings are modelled by E =10 Pa Young Modulus, ν = 0. 3 Poisson modulus and ρ = 5000 kg/m 3 density [7]. Displacement boundary conditions in a structural analysis are chosen, namely the nodes of the outer edge of the stator core are allowed to move in the radial direction but are prohibited from moving in the tangential direction [1]. 3. MODELING MAGNETOSTRICTION In the modeling of magnetostriction in 3-D for plane material with isotropic magnetostriction the strains in the local frame are given by: λx = λ λy = λ / 2 λz = λ / 2 The local xy axis is rotated in such a way that the flux density vector B coincides with the local x axis [2]. The magnetostrictive strain in the direction of B (x direction) 6 2 is described as λ = λ( B) = 10 B. The magnetostrictive strain is found using the element s flux density B and the λ(b) characteristic of the material [2]. In a 2-D plane strain analysis the strains λ x and λ y in the local frame are then given by: λx = λ + νλt = 0.85λ λy = λt + νλt = 0.65λ λz = where: λ = λ( B) = 10 B λ = λ / 2, ν = 0. 3 Poisson modulus. t The magnetostrictive strains are represented in the mechanical analysis by a set of temperatures that induce mechanical strains of the same size as strains caused by magnetostriction. (1) (2)

4 RESULTS The equations of the magnetomechanical problem in FEM are described by the quations of the magnetic (4) and mechanical field (5) [1, 6]. S ][ A] = [ J ] (4) [ e [ K ][ U ] = [ F] (5) where: [S] electromagnetic stiffness matrix, [A] magnetic potential matrix, [ J e ] current density excitation matrix, [K] mechanical stiffness matrix, [U ] displacement matrix, [F] Maxwell forces vector calculated in the magnetic analysis are loads in the mechanical analysis. Fig. 1 shows the distribution of the magnetic flux lines in the motor cross-section ( J = A/m 2 e ) while Fig. 2 gives the stator deformation and displacement only from magnetostriction effects. The shape of deformation of the stator has been multiplied by the factor equal k = original deformed Fig. 1. Flux lines for particular position of rotor.137e e e e e e e e e-07 Fig. 2. Stator deformation (30000 x magnified) and displacement (m) from magnetostriction The figures below compare the stator deformation and displacement which occurred without (Fig. 3) and with (Fig. 4) magnetostriction effects.

5 323 original original deformed deformed.101e e e e e e e e e e e e e e e e e e e-06 Fig. 3. Stator deformation (30000 x magnified) and displacement (m) only from Maxwell forces Fig. 4. Stator deformation (30000 x magnified) and displacement from Maxwell forces and magnetostriction effects Fig. 5 below shows the calculation of radial displacement at the teeth shank at the inner radius (0.040 m) with and without magnetostriction for a particular position of the rotor. In Fig. 6 radial displacement at the back of iron at the outer radius (0.060 m) is calculated with and without magnetostriction for a particular position of the rotor. Fig.5. Radial displacement at the teeth shank at inner radius (0.040 m) with and without magnetostriction effects for particular position of rotor. Fig.6. Radial displacement at back of iron at outer radius (0.060 m) with and without magnetostriction effects for particular postion of rotor. 5. CONCLUSIONS The results of the analysis presented in the paper show increase in the magnitude of the deformation due to magnetostriction in the stator iron. The maximum value of

6 324 magnetostriction displacement is equal approximately 20% of displacement from Maxwell forces. This deformation is one of the reasons why electric machinery emits acoustic noise. LITERATURE [1] BELAHCEN A., Magnetoelasticity, magnetic forces and magnetostriction in electrical machines, 2004, Helsinky University of Technology (PhD research). [2] DELAERE K., W.Heylen, R. Belmans, and K. Hameyer, Comparision of Induction Machine Stator Vibration Spectra Induced by Reluctance Forces and Magnetostriction, IEEE TRANSACTIONS ON MAGNETICS, Vol.38, No.2, March [3] LAFTMAN L., The Contribution to noise from Magnetostriction and PWM Inverter in a Induction Machine, August 1995, Lund Institute of Technology (PhD research). [4] ŁUKANISZYN M., MŁOT A., Numerical Analysis of the Brushless DC Motor Electromagnetic Torque and Pulsations-Components of Torque (in polish), XLI Międzynarodowe Sympozjum Maszyn elektrycznych, SME 2005, Jarnołtówek 2005r, s [5] ŁUKANISZYN M., MŁOT A., Three dimensional analysis of magnetic field of PM BLDC motor (in polish), Materiały XII Sympozjum Podstawowe Problemy Energoelektroniki I Elektromechaniki, PPEE 2005, s [6] MOHAMMED O., Implementation of Coupled Magnetomechanical Analysis Including Magnetostrictive Effects in Electrical Machinery, LACCET 2004, Miami, Florida, USA. [7] WITCZAK P., Wyznaczanie drgań mechanicznych silnika indukcyjnego wywołanych siłami magnetycznymi, 1995, Politechnika Łódzka (praca habilitacyjna). PORÓWNANIE PRZEMIESZCZEŃ POCHODZĄCYCH OD SIŁ MAXELLA ORAZ ZJAWISKA MAGNETOSTRYKCJI W SILNIKU BLDC ANALIZA STATYCZNA W blachach magnetycznych poddanych działaniu zmiennego pola magnetycznego powstaje zjawisko magnetostrykcji. Zjawisko magnetostrykcji oraz siły Maxwella są źródłem dodatkowego hałasu w maszynach elektrycznych. W artykule opisano sposób obliczania przemieszczeń statycznych wywołanych siłami Maxwella oraz zjawiskiem magnetostrykcji z wykorzystaniem metody elementów skończonych. Obliczenia przeprowadzano na modelu bezszczotkowego silnika prądu stałego z komutacją elektroniczną. Pokazano przemieszczenia wybranych punktów stojana silnika wywołane siłami Maxwella oraz zjawiskiem magnetostrykcji. Dokonano porównania wpływu obu zjawisk na wartości przemieszczeń.