COMPUTING THE FORCE OF LINEAR MACHINES USING FINITE-ELEMENT ANALYSIS

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1 Technical University of Cluj-Napoca, 6 th of May 00 COMPUTING THE FORCE OF LINEAR MACHINES USING FINITE-ELEMENT ANALYSIS Dr. Ferenc Tóth - Norbert Szabó University of Miskolc, Department of Electrical and Electronic Engineering University of Miskolc, H 355 Miskolc-Egyetemvaros Phone: / 59 Fax: elktoth@gold.uni-miskolc.hu elkszabo@gold.uni-miskolc.hu Abstract This paper presents the force of the linear induction motors (LIM) and linear synchronous motor (LSM) calculated on the basis of a two-dimensional Finite-Element Method (FEM). The finite element method (FEM) has been used in electromagnetic field computation for almost 30 years. Such numerical techniques enable the designer to solve problems that are difficult, without considering many empirical factors. The actual force distribution is then found from a given current distribution. There are four basic ways of determining electromagnetic force, namely the Lorenz, Maxwell stress tensor (MTS), classical virtual work, and Coulomb virtual work (CVW) methods. In the first part the paper introduces the possible applications of the four basic methods. In the second part the application of Coulomb virtual work method for determination of asynchronous and synchrony LIM force is presented. The result of the LIM analysis are also shown.. INTRODUCTION The force and torque calculation from finite element analyses becomes an important topic in the domain of the numerical computation of electrical machines. Recently a number of publication have been reported about the accuracy and the reliability of different method [4,5,6,7]. Parallel to the development of the total force calculation is it impotent on the local force density computation, because the local force is required in the design and optimisation of electrical machines. On the one hand a comprehensive understanding of the force distribution is helpful to improve the design. On the other hand the coupling between magnetic and mechanic relates closely with vibration and noise problems in electrical machines. This paper discusses the problem of force calculation where the field vary in time, and where induced on eddy current effects.. THE FEM FORCE COMPUTATION METHODS The electromagnetic force is generally the superposition of the Laplace and the Magnetic Force []: ( J B) dv H grad dv = F L F M F = µ + V V () The Laplace component F L of the electromagnetic force is the result of the iteration between the electric current of density J, and the magnetic field of flux density B. The second component, the magnetic force F M, depends on the magnetisation state of the body studied. The computation of the electromagnetic forces makes use of the following methods: Virtual work method (VWM) The electromagnetic force external on a body placed in the electromagnetic field domain, following a direction defined by the unit vector n u, is the derivative of the magnetic energy W, with respect to the corespondent coordinate u : where the magnetic energy: W F = n u Ψ= con () 4

2 Technical University of Cluj-Napoca, 6 th of May 00 other in D with L length of problem in z-direction: b W Hd B = dv V 0 x y W = L ( H d B)dxdy An alternate expression involves the system magnetic co-energy 0 0 W C and current, I: (3) (4) where the magnetic co-energy: WC F = n u u I = con W h = Bd H dv V 0 C This computation gives the global force, both for conductive, magnetic or non- magnetic bodies, for nonconductive magnetic bodies, and for the permanent magnets. Coulomb s Virtual Work Method (CVWM) Based on Coulomb s virtual work, the force calculation is directly coupled in the system matrix. Coulomb [] and Coulomb and Meunier [3] show that finite element energy functions can be differentiated directly with respect to object position to obtain forces. The component of interest is not displaced physically, the movement in the u direction is virtual. The energy derivatives are based on the virtual deformation, which the element in the air surrounding the object undergo as the object is displaced virtually. It is important to note that only the elements in the surrounding air are virtually deformed, the elements in the object itself and any other ferromagnetic or conducting materials- are not deformed. In mathematical terms the expression for energy in the finite element may be explained as follows: where: ν { A } [ S ] material relativity nodal vector potentials finite element stiffens matrix T { A} [ S ]{ A} W = Lν (7) L length of the problem in z-direction As the finite element undergo virtual deformation, the energy changes. Since the derivative is taken at constants flux linkage, the vector potentials A remain constant. The stiffens matrix, which determined by the shape of the element does change. The derivative of the energy W, witch respect to object position u then becomes: W = Lν Ψ At (8) the force on the object in the u direction is: W Fu = = Lν Ψ T S { A} { A} T S { A} { A} This direct differentiation approach is included in the system of equation describing the finite element model of the motor. So the force is obtained directly from the solution of the system matrix. (5) (6) (8) (9) 4

3 Technical University of Cluj-Napoca, 6 th of May 00 Maxwell Stress Tensor Method ( MSTM) The electromagnetic impulse theorem expresses the equivalence between the evaluation of the force acting on bodies placed inside a closed surface Σ using expression (), and by surface integration of the Maxwell stress tensor T []: The Maxwell stress tensor expression is F = J B H grad µ dv = Td S = T dv V S V (0) T = ( ) B n H H n µ () where n is the unit vector normal to the surface Σ. The surface Σ can be chosen arbitrarily, but it should include the bodies on which the electromagnetic force computed. Thus, if the total field surrounding a body is known, the force exerted on the body in term of the field alone can expressed by: F L µ S 0 ( ) B n B t ds tu + Bn Bt ds nu µ S 0 = where is the surface enveloping the body under force, B is the flux density and n u and t u are the unit vectors in the normal and the tangential direction on the surface S. In -D problems, the surface integral of the force density is reduced to a line integral, making its implementation relatively easy in these problems. The Arkkio s Method (the Average MSTM ) The Arkkio s method [8] is based on the MSTM formulation, but is improves the accuracy of force computation. While the MSTM is based in -D problems on the line integral in the machine air gap, the Arkkio s method based on surface integral in the whole air gap air. In this case it is not necessary to put on a lot of additional air layers in the air gap. For the -D problem the trust force can be described as: F = Bt BndS gµ 0 S where S is a part of the entire air area in the machine, g is the air gap with. 3. THE OBTAINED RESULTS OF ANALYSIS WITH FEM. The magnetic forces by COSMOS/ESTAR may be obtained through two types of method. The first method is only applicable for current carrying conductor and it is based on the Lorenz force equation. The second method the CVWM can be used for obtaining the forces on ferromagnetic objects under externally applied field as well as current carrying conductors. The program can be used time stepping method summary in 33 steps. The results of linear induction motor In the fig.. a linear induction motor model is shown, where the secondary part was standing state (before the starting), can be calculated the magnetic force. () (3) 43

4 Technical University of Cluj-Napoca, 6 th of May 00 Fig.. Geometric model of single-side Linear induction Motor Fig.. shows the value of the local force at the step 30., when the secondary part was standing state, fig.(a) the thrust, and fig.(b) shown the magnetic thrust. The numerical result to refer to that the motor dimension was m width. a) b) Fig.. The local force at the step 30 a) the thrust b) the normal force of single-side LIM Fig 3. shows the average value of the magnetic force in time, in case (a) thrust and in case (b) magnetic thrust (normal direction) force. a) b) Fig. 3. (a) Thrust, and (b) normal force, as a function of time stepping for nominal condition at f=50 Hz. Fig 4. shows a one pole-pair permanent-magnet synchronic motor model, which the magnetic force can be calculated with the CMW method. 44

5 Technical University of Cluj-Napoca, 6 th of May 00 Fig. 4. Geometric model of synchronous Linear Motor with permanent magnet The fig. 5. shown the resolve of the local (a) thrust and (b) magnetic attractive force, when the motor power angle was maximal that δ=90. The numerical result to refer to that the motor width was m. a) b) Fig. 5. The maximal local force of Synchronous Linear Motor a) the thrust b) the normal force In fig 6. shown the calculated (a) thrust and (b) normal force depend on the δ power angel. Here to accent that the numerical result to refer to that motor witch width was m. Fig. 6. (a) Thrust, and (b) normal force of synchronous linear motor, as a function of time period and power angle 45

6 Technical University of Cluj-Napoca, 6 th of May CONCLUSION During linear machine design it is necessary to determine the electromagnetic force. In this work, we have presented the different methods for force calculation in electrical motors. The knowledge of the force variation in term of the length of an linear motor is very impotent for the designer. In this paper the force is calculated during the displacement of the moving part by the Coulomb s virtual work method. 5. REFERENCES [] Dr. Fodor György: Elméleti Elektrotechnika III. Bp.99, pp [] Coulomb J. L. A Methodology for the Determination of Global Electromechanical Quantities from Finite Element Analysis and Its Application to the Evaluation of Magnetic Force, Torques and Stiffens, IEEE Trans. on Magnetics, V9, 983, pp [3] Couloms J.L. and Meunier G.: Finite Element Implementation of Virtual Work Principle for Magnetic or Electric Force and Torque Computation, IEEE Trans. On Magnetics, V0, 984, pp [4] Benhama A., WilliamsonA.C. and Reece A.B. : Force and Torque Computation from -D and 3-D Finite Element Field Solutions, IEE Proc. Electr. Power Appl. Vol. 46. No. I. J January 999, pp [5] Hamler A., Krecsa B. and Hribernik B.: Investigation of the Torque Calculation of a DC PM Motor, IEEE Trans. on Magnetics, Vol 8, no. 5. September 99, pp [6] Sadowski N., Lefévre Y., Cros J.: Finite Element Torque Calculation in Electrical Machines while Considering the Movement, IEEE Trans. on Magnetics, v0l. 8. No. March 99, pp [7] Degtyareva E. L., Potapov L. A. and Richie A:E.: Determination of the Forces and Torque in Electromechanical Devices with the Help of the Maxwell Stress Tensor, ICEM 98, pp [8] Arkkio A. : Time-stepping finite element analysis of induction motors, ICEM-88,988 46