Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31

Study of Magnetostriction Effects in Electrical Machinery

O. A. Mohammed, Fellow IEEE, S. Liu, member IEEE, and S. Ganu, student member IEEE Energy Systems Lab., Electrical and Computer Engineering Department, Florida International University, Miami, FL 33174, USA e-mail: [email protected]

ABSTRACT Two types of magnetostriction effects are important in the study of noise and vibration in electrical machines. The first one is the dependence of dimension and elastic properties on magnetic property and the second is dependence of the magnetization characteristics on stress. The first one, usually referred as the direct effect and is described with butterfly magnetostriction curve l(H ) , while the second, referred as the inverse effect, is expressed with magnetization curves at various stress levels. The effect of the inverse magnetostriction (IME) on stator iron deformation is studied by using the magneto- elastic coupled FE analysis. The procedure of including IME in the coupled FE analysis is proposed. The inclusion of IME in magnetic field equation reflects in the formation of the stiffness matrix. The permeability of each ferromagnetic element is considered as the function not only of the flux density but also the stress. The IME is taken into account in magnetic force calculation. The force formula based on the virtual work principle is proposed. Besides the reluctance forces, the force due to the IME is added to the excitation matrix of elastic field equations. The proposed idea is implemented on a PM synchronous motor. The stator deformations as well as the force acting on the stator iron with and without IME are compared.

I. INTRODUCTION Reduction of noise radiation from electrical machines has been pursued by researchers and engineers. The noise generated due to magnetic related reasons is referred as magnetic noises. The magnetic forces and magnetostriction effects are considered as two main sources of magnetic noises in electrical machines. In rotating machines, the large difference in magnetic permeability between the air gap and the stator teeth causes considerable reluctance forces. When the rotor rotates, each stator tooth is alternately pulled and released. As a result, the stator teeth vibrate and generate noises. Besides the reluctance forces, the force due to the magnetostriction effects is the other source of noise generation. There are various kinds of magnetostriction effects existing in ferromagnetic materials. Volume magnetostriction refers to the anomalous volume change associated with the magnetism. It is mainly the function of temperature. Joule magnetostriction effects describes the deformation occurred by the external applied magnetic field. It is usually described with the butterfly magnetostriction curve l(H ) [1]. Villari effects describe the stress dependence of Joule magnetostriction. Mechanical stress existing in ferromagnetic materials may modify the shape of the MH() curve and hence of the l ()H curves. It is usually described by different magnetization curves at various stress levels. In addition, there are so-called form effects existing in materials having finite dimensions and definite shape. In general, magnetostriction effects are classified into two categories: the direct effect and the inverse effect. The direct magneto-elastic effects mean the dimensions and elastic properties of the magnetic material depend on its magnetic state. The inverse magneto-elastic effects mean that the magnetic properties of the magnetic materials are influenced by the applied and internal mechanical stresses. In electrical engineering, the Joule magnetostriction effects are considered as the main factor producing transformer noise by elongating and shrinking the laminated steel sheets. Some researchers made use of the method of thermal stress calculation to incorporate the magneto-elastic effects in mechanical field analysis. The expansion of the free body due to magnetostriction is calculated based on the magnetic flux distribution, and the MS forces are found as the forces needed to deform the expanded body back into its original shape. From the theoretical point of view, the Villari effect must have its contributions to magnetic forces, which may cause deformations too. This has been neglected by designers and was not studied by researchers before. This paper studies the magnetic force originated from the inverse magnetostriction effects. The formula to evaluate the force due to the inverse magnetostriction effects is proposed[2]. For studying the magnetostriction effects on the stator deformation, the inverse magnetostriction effect is included in magneto-elastic coupled FE analysis. The inverse magnetostriction is considered in both magnetic field and mechanical field analysis. In magnetic field equation ([SAF ][]= [ ]), it is considered in forming the stiffness matrix[S] . In elastic field equation ([KUF ][]= [ ]), it is taken into account in forming the excitation matrix[F] .

759 Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31

II. INCLUDING THE INVERSE MAGNETOSTRICITON IN FE MAGNETIC FIELD EQUATION The inverse magnetostriction effect shows that the magnetization property changes with the stress inside the ferromagnetic objects. In FE analysis without the inverse magnetostriction effect considered, the nonlinear material property iteration for all the elements of a ferromagnetic object is performed on one BH curve. With the consideration of the inverse magnetostriction effects, different elements of a ferromagnetic object may present different magnetization property due to the different stress level existing on them. At the same time, the stress generated by magnetic forces depends on the magnetization status. The mutual dependence between the magnetization and the stress are considered while forming the stiffness matrix [S] of magnetic field equation. Besides performing the iteration of the nonlinear BH curve for determining the magnetic operating point of each element, the iteration for finding the stress level as well as the corresponding magnetization BH curve of each element is performed. The magnetization iteration can be considered as an internal loop; while the stress and magnetization property iteration is an external loop. This is the way of including the inverse magnetostriction effects in FE magnetic field analysis. Below is the detailed procedure: first using initial magnetization property BH0, perform nonlinear magneto-mechanical FE analysis to calculate the stress of every element. According the element stress value, determine which BH curve should be used for the next step material property iteration. Then perform nonlinear magneto-elastic FE analysis till the stored energy of stator iron keeps unchanged.

III. MAGNETIC FORCE CONSIDERING THE INVERSE MAGNETOSTRICITON EFFECT In the case of using scalar potential , the magnetic field co-energy stored in region R is: æöH Wco =×òòç÷ BdHdR (1) R èø0 According to the virtual work principle, the magnetic force can be calculated as follows:

H ¶¶W æöæö F=-co =-ç÷ç÷ BdHdR × (2) ¶¶UUç÷òòç÷ èøR èø0 Where, U represents the virtual displacement of the object. Without considering the magnetostriction effects, the magnetic force acting on one element can be expressed as[3]: æöH e æö¶H ¶ ()det ()G F=-×[det]ç÷ B() G +×ç÷ B dH dRe òòèø¶¶UUç÷ (3) Re èø0

Where, Re represents the element region. G is the local Jacobian derivative matrix and det ()G is its determinant. With the effect of stress s on the permeability considered, one more force term will be added to the above formula: æöæöHH eTTTæö¶H ¶ ()det ()G ¶¶s F=-×+×+×[ç÷ B det() Gç÷ç÷ B dH B dH det() G ] dRe (4) òòòèø¶Uç÷ç÷ ¶¶¶ UUs ( ) Re èøèø00

In the case of using vector potential, the force formula becomes:

æöæöBB eTTTæö¶B ¶ ()det ()G ¶¶s F=-×+×+×[ç÷ H det() Gç÷ç÷ H dB H dB det() G ] dRe (5) òòòèø¶Uç÷ç÷ ¶¶¶ UUs ( ) Re èøèø00 The magnetic force term due to the inverse magnetostriction effects is rewritten below:

eTéùB ¶¶s (6) Fime=- H ××× dB G dR e òòëûêú0 ¶¶s () U Re e Where, Fime is the elemental magnetic force due to IME.

Re : region of element e , described with local coordinate system.

In 2D plane element, dRdde = zh, in 3D volume element, dRe = dddzhx. s , B , and H : stress, flux density and field strength of element e . U : virtual displacement in any direction. G : determinant of the local Jocabian derivative matrix G .

Since rotating machines are working under rotating magnetic fields, anisotropic reluctivity model should be adopted as the magnetic properties [4]. In 2D case, for the diagonal reluctivity tensor model, we have:

760 Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31

e1 '2'2 (7) Fime =- {[uxxuxyyuy()() sB+ u s B ] ´( E /() 1 +- u (1 2 u )) | Gdd |} zh 2 ò Re Using the full reluctivity tensor, one has:

e11 '2' ' '2 (8) Fime =- {[usxxux()()()B+++´+- us xyuxyyxuxy BB us BB us yyuy() B ]( E /() 1 u (1 2 u )) | G |} d zh d ò 22 Re ' ' ' ' Where E is the Young’s modulus; u is the Poisson’s ratio.n xx ,n xy ,n yx and n yy are the derivatives of the corresponding components of the tensor reluctivity to the element stress in u direction. These derivatives are calculated according to the measured BH curves at various stress levels [5].

IV. PM MOTOR IMPLEMENTATION AND RESULTS The proposed procedure for including the inverse magnetostriction effects (IME) in the coupled magneto- elastic FE analysis are implemented on a PM surface mounted synchronous motor. The rotor and stator iron is made of M19. The magnetic force distribution in stator iron is investigated. The results are shown for the initial rotor position. Forces along three representative circles are evaluated. They are at the teeth surface (radius=0.069723m), the circle going through the middle of the teeth shank (radius=0.084624m), and the circle in the back iron(radius=0.100053m). Both cases with and without magnetostriction effects are studied. Fig. 1(a) and Fig. 2(a) are the force profiles along the teeth surface without and with IME. The force in the case of considering IME is the summation of the reluctance forces and the magnetostriction at teeth surface. The reluctance forces exist only at stator iron surface when virtual work principle is used for force calculation. At teeth surface, the reluctance force is larger than the force contributed by the IME. This give the reason why the force profiles with and without IME looks similar at teeth surface. Fig. 1(b) and Fig. 2(b) show the force information at the stator teeth shank. There are reluctance forces along the shank surface. The force profile repeats itself along the air gap periphery. Their values are small due to the very small value of the leakage fluxes passing through the elements next to the teeth shank. From Fig. 8.8, it can be seen that the including the IME results larger magnetic forces at the teeth shank. This is due to the highest flux density at the center of the teeth shank and the proportional relation between the flux density and the magnetostriction force. The unrepeated force profile is due to the use of the anisotropic reluctivity tensor. Fig. 1(c) and Fig. 2(c) show the forces in the back iron. There are no reluctance forces inside the ferromagnetic region when using the virtual work principle for the force calculation. With IME considered, the magnetostriction force occurs inside the region. Therefore, we have no force in the back iron without IME, seen Fig. 1(c). Comparing Fig. 2(b) and Fig. 2(c), one can find that the magnetostriction force values in the back iron are much smaller than the magnetostriction forces at the teeth shank. This is because the flux density in the back iron is much smaller than in teeth shank. The stator deformation under the consideration of IME does not repeat any more and is larger compared to without IME. The mesh on the deformed stator iron in the case of IME considered can be seen in Fig. 3. Fig. 3 shows the 2D mesh of the deformed stator iron at three rotor positions in the case of IME considered. Fig. 4 gives the 3D flux density distribution on the deformed stator iron.

V. CONCLUSION The method of including the inverse magnetostriction effects in magneto-elastic coupled FE analysis is presented. The effect is considered in both magnetic and elastic analysis. The formula of calculating the magnetic force due to IME is proposed. From the implementation on a PM motor, one can conclude that the inverse magnetostriction has significant effect on the stator deformation. It should be considered in the design stage.

REFERENCES [1]. Tremolet de Lacheisserie,Etienne du, Magnetostriction Theory and Applications of Magnetoelasticity, CRC Press, 1993. [2]. O. A. Mohammed, “Coupled magnetoelastic finite element formulation including anisotropic reluctivity tensor and magnetostriction effects for machinery applications,” IEEE Transactions on Magnetics, vol. 37, No. 5, pp. 3388-3392, September 2001. [3]. J. L. Coulomb, “A Methodology for the determination of global electromechanical quantities from finite element analysis and its application to the evaluation of magnetic forces, torque and stiffness,” IEEE Transactions on Magnetics, vol. 19, No.6, pp.2514-2519, November 1983. [4]. M. Enokizono, T. Todaka, and S. Kanao, “Two-dimensional magnetic properties of silicon steel sheet subjected to a rotating field,” IEEE Transactions on Magnetics, vol. 29, no. 6, pp.3550-3552, November 1993. [5]. O. A. Mohammed, S. Liu, and S. Ganu, “ Inverse magnetostrictive effects and its inclusion in magneto-mechanical modeling of electric machines,” proceedings of MEPCON2003, December, 2003, Egypt.

761 Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31

4.E+02 4.E+02

3.E+02 3.E+02

3.E+02 3.E+02 2.E+02 2.E+02 2.E+02 2.E+02

2.E+02 Force (N)

Force (N) 2.E+02

1.E+02 1.E+02 8.E+01 8.E+01

4.E+01 4.E+01 0.E+00 0.E+00 0 23 46 69 93 115 138 162 185 207 231 254 276 299 323 345 0 23 46 69 93 115 138 162 185 207 231 254 276 299 323 345 Angle (degree) Angle (degree) (a) (a) 1.E+02 1.0E+00 1.E+02 8.0E-01 1.E+02 8.E+01 6.0E-01 7.E+01

4.0E-01 6.E+01 Force(N) 5.E+01 2.0E-01 4.E+01 2.E+01 0.0E+00 1.E+01 2 28 52 78 102 128 152 178 202 228 252 278 302 328 352 0.E+00

An gle ( degr ee) 2 28 52 78 102 128 152 178 202 228 252 278 302 328 352 Angle (degree) (b) (b) 1.E+00 2.E+01

9.E-01 2.E+01 8.E-01 2.E+01 7.E-01 6.E-01 2.E+01 5.E-01 1.E+01 Force (N) Force (N) 4.E-01 9.E+00 3.E-01 6.E+00 2.E-01 1.E-01 3.E+00 0.E+00 0.E+00 3 27 53 78 102 127 153 177 203 227 253 277 303 327 353 3 27 53 78 102 127 153 177 203 227 253 277 303 327 353 Angle (degree) Angle (degree) (c) (c) Figure 1 Magnetic force without IME Figure 2 Magnetic force with IME (a) at teeth surface (b) at teeth shank (c)in the back iron (a) at teeth surface (b) at teeth shank (c)in the back iron

0 degree 10 degree 20 degree

Figure 3 2D mesh of the deformed stator iron with IME considered

0 degree 10 degree 20 degree

Figure4 3D Flux density distribution on the deformed stator iron with IME considered

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