=-r!! J J-(r',t)ln_! ds' 15th IGTE Symposium HTML version. d 2 r dr. p.j(r,t)+ r!! J(r',t)ln_!_dS'

Transcription

1 15th GTE Symposium HTML version Computation of the Motion of Conducting Bodies Using the Eddy-Current ntegral Equation *Mihai Maricaru, t loan R. Ciric, *Horia Gavrila, *George-Marian Vasilescu and *Florea. Hantila *Department of Electrical Engineering, Politehnica University of Bucharest, Sp!. ndependentei 313, Bucharest, , Romania, mihai.maricaru@upb.ro tdepartrnent of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada Abstract-The analysis of the motion of a system of solid conductors in the presence of magnetic fields is performed by solving the classical mechanics equation of motion under the action of magnetic forces. Application of the eddy-current integral equation and the usage of the local coordinates attached to the bodies in motion allow the determination of electromagnetic field without being necessary to reconstruct the discretization grid at each new position of the conducting bodies. Only the submatrices associated with the coupling between the bodies in relative motion are modified in the global system matrix. A time-domain method of solution is first presented for the.electromagnetic field problem, coupled with the equation of motion, which can be efficiently applied at high frequencies when the time steps are small. The eddy-current integral equation for the derivative of current density contains a term that takes into account the relative motion of the bodies. Since the electromagnetic quantities vary much more rapidly than the mechanical quantities, a second method is also proposed in this paper, where the eddy-current integral equation is solved in the frequency domain by assuming that the bodies are motionless, but by adding supplementary terms due to the actual motion of the bodies. Thus, only the average force over a period of time is now computed. This method is extremely efficient especially at higher frequencies when the time steps are very small. ndex Terms--eddy-current integral equation, electrodynamics of moving conductors, levitation.. NTRODUCTON The equation of translational motion of a solid conducting body of mass m under the action of the magnetic force Fis d 2 r dr m-=f(r,-,'.j)+g dt 2 dt where r is the position vector of a point of the body, for instance of its center of gravity, '.J is a vector representing the imposed current distribution and G is the gravitational force acting on the body. Equation (1) is discretized in time and F is determined at each time step by solving an electromagnetic field problem in the region with moving bodies. The application of the Finite Element Method requires a tremendous amount of computation since it is necessary to reconstruct the discretization mesh at each time step. Moreover, the modifications of the discretization mesh are, usually, accompanied by undesired cumulative errors in the successive solutions of the electromagnetic field. A substantial improvement can be achieved when adopting hybrid Finite Element - Boundary Element Methods [l]. Using the "laboratory" frame of references complicates the field problem solution due to the presence of the motional electric field intensity v 0 x B, where v 0 is the body velocity in this frame of references and B the magnetic induction. This disadvantage is eliminated when employing local frames of reference, attached to the bodies in motion [], [2]. This also allows the usage of the simpler eddy-current integral equation for the bodies at rest, as it has been done in the case the velocities of the bodies are known [2]. However, in many situations the velocities of the bodies are not known, as, (1) for instance, in the case of the electromagnetic levitation, their determination constituting one of the objectives of the present work. n the case of the electromagnetic levitation, to ensure the stability of the solution it is necessary to choose a sufficiently small time step. Since for same accuracy of the results the time period has to be divided practically in the same number of intervals (for example, at 50 Hz in 200 intervals [ l ]), at higher frequencies the time step decreases. Unfortunately, as the time step decreases, the successively computed solutions tend to be very close to each other and the errors in the solution differences increase considerably, the computation procedure becoming inefficient. n the present paper, a new procedure is described for the time-domain solution of the eddy-current integral equation applicable to small time steps. As well, a technique is proposed for accelerating the determination of the trajectory of the moving bodies, based on the frequency-domain solution of the eddy-current integral equation.. TME-DOMAN SOLUTON OF THE EDDY-CURRENT NTEGRAL EQUATON For two-dimensional field problems, the time-domain eddy-current integral equation for motionless conductors is p.j(r,t)+ r!! J(r',t)ln_!_dS' dt n R =-r!! J J-(r',t)ln_! ds' dt R O.; where rand r' are the position vectors of the observation point and of the source point, respectively, n is the (2)

2 15th GTE Symposium HTML version region containing the solid conductors, O; is the region where the imposed current density l; is confined, R = r - r', y = /Jo, µo being the permeability of free 21t space. n the three-dimensional case, the eddy-current integral equation has the form pj(r,t)+~ :r ~J(~,t) dv' =-!.!!_ J l;(r',t) dv'+grad!jj 2 dt n R where!jj is the electric scalar potential. To simplify the formulation, we consider here a two-dimensional structure with a single ~olid conductor. Using a frame of reference attached to the conducting body in motion, the time discretization of (3) leads to p(ji-jo)m +rf V1 -Jo)n_!_dS'' 2 n R (3) with r and r; being the position vectors of the points of n and O;, respectively. The spatial discretization grid in the two-dimensional case is constructed by dividing the region n into polygonal surface elements O>m, with the induced current density considered to be constant through each mm. The region D.; is divided into surface elements m;k, with the imposed current density being constant through each OJ;k ntegrating (5) over each mm yields the following matrix equation for the vector ( f}j) : at!.. 2 M x(lj) (aj.) 1 (-A+yB - =-AJo-rB; - -yc(j;)!.. (8) 2 at!.. a where A is a diagonal matrix with entries Am = PmSm, Pm being the resistivity of the material for mm and Sm its area, and B is a symmetric matrix with its entries corresponding to the elements O)m of n having the form =-p./om-y J 1; 1 n ~ds''-y f J; 0 n ~ds'' (4) n;1 n;o where the subscript "O" indicates the time t and the subscript "l" the time t + M. Dividing ( 4) by M yields ~ a.) M f(oj) - -+r - ln-ds''=-p./ 0 oti2 0t1 R 2 n 2 - Y f (a.;) ln_!_d.s'-y f (n v){j ).!. n_!_ di' (5) ot!_ R "" n; 2 ""'i R where the subscript "_!_" refers to the time t + M, 80; 2 2 is the boundary of O;, v is the velocity of the D.; in the frame of references attached to n, and n is the outward unit vector normal to Ofl;. The last term in (5) is due to the relative motion of n and O;. Solution of (5) gives the current distribution J 1 at the time step t + M in terms of that at the time step t in the form (6) Bm,k = f f n~ d.skd.sm = SmSk (i)"' (i)k -~ J J (nm nk)r 2 n~ dlkdlm orom orok The entries of the matrix B; are defined as in (9), but with the elements (J)k elements m;k matrix Care (9) of n being replaced with the belonging to O;, while the entries of the Cm.it = ~ J J (nm R)(n;k v)ln ~ dl;kdlm. (JO) o<l)m oro;k All integrals in (9) and ( 0) are evaluated by analytic expressions, the entries of the matrix B being calculated only once, but those of the matrices B; and C are to be calculated for each new position of O;. Taking into account the small dimensions of the elements mm, a rapid numerical computation of the force in (7) is performed using the approximation F=rSmJmLP;k (11) m k The magnetic force is evaluated by applying Ampere's force formula, i.e., J J r-r F = -r J(r,t)J;(r;,t) ' 2 ds'; d.s nn. r-r; (7) where the vector P;k is expressed in the form (12)

3 15th GTE Symposium HTML version with rm being the position vector of the center of the element (J)m of n and r;k the position vector of the point of integration on oa\. When the ratio of the linear dimensions to the distance between its center and the center is sufficiently small, P;k can be calculated by in a number of elements Pk in terms of this ratio and by using the summation (13) where rpk is the position vector of the center of Pk and S Pk the area of the element Pk. The same technique is applied for a rapid numerical calculation of the entries in the matrices B; and C in (8), making also use of the relation f the imposed currents are sinusoidal, the phasor representation of (15) is pj(r)+ JA.f J(r')ln_!_dS'' =-1 ). J J.(r')ln_!_dS'' - - R - R n n; (16) where ). = 2;r fr, f being the frequency, and l.. = fe +)Jim is the phasor form of the current density, with j = j::]. The two terms on the right side of (16) show the contribution to the induced current density due to the time variation of the imposed currents and that due to the relative motion. The same technique as in the case of the time-domain analysis is used for the space discretization of (16). One obtains the following algebraic system with complex coefficients: (14) n the case the imposed currents are periodic, the initial distribution of the induced current can be obtained by performing a Fourier expansion and by employing the phasor form of the eddy-current integral equation (see Section V).. SOLUTON OF EQUATON OF MOTON Equation () is solved iteratively. We choose an appropiate time step!lt and assume that the magnetic force F has a linear variation during M. At the time t the body has a position defined by the vector r 0 and a magnetic force Fo is exerted upon it. The iterative process is started by imposing the value fj = Fo at the time t +flt and the position vector "1 results from solving (). The electromagnetic field problem is then solved for the new "1 and a new value of the force F 1 is determined for the time t + M. This operation is repeated until the difference between two successive values of the magnetic force for the time t + M is sufficiently small and, then, we proceed to the next time step. AJim +).BJ re= -A.B fe -yc'jim (17) The.average magnetic force over a period is evaluated using the relation {J J r-r J Fav = -yr l.. (r)~.;(r;) ' ds'; ds' (18) 2 nn r-r; where the asterisk indicates the complex conjugate. For a multiple-conductor systems, one uses local frames of reference attached to each of the conductors. For the conductor q, occupying the region nq c n, q = 1,2,..., ( 16) is written in the form pl.,(r) + j). J n 9 f..(r') n ~ ds''+ j). L J f..(r') n ~ ds'' P*qnp + r f (n. v~q»l..(r') n ~di'= - JA. f l.;(r') n ~ ds'' P*qanp n; V. FREQUENCY-DOMAN SOLUTON OF THE EDDY-CURRENT NTEGRAL EQUATON Since the region n; is moving with the velocity v in the frame of reference attached to n, (2) is written in the form p.j(r t) + rf OJ(r',t) n_!_ds'' = -y f BJ; (r',t) n _!_ds'' at R a1 R n n; -r f (n v)j;(r',t)ln ~di'. 00; (15) where v~q) and vfq) are, respectively, the velocities of the conductor p and of n, with respect to the conductor q. Equation () is always solved separately for each body. V. SOLUTON ACCELERATON FOR THE EQUATON OF MOTON The computation of the motion of conducting bodies can be spectacularly accelerated by using the average value of the force over a period, evaluated using the

4 15th GTE Symposium HTML version l111lml111 l111l111l111l111111l111li11l111l 11il 111 li11li11l111l11il11il111l111l..j' Figure : Discretization of the levitated plate > rm TJill7U "J\N\,..,...,... 1fT O.Q (s) Figure 4: Evolution in time of the coordinate y of the plate for f = 2,000 Hz, with a direct current of 0 A added in the outer coil g L...'-.LL--'-LCC-'-L.>'---'-'L..L_'-L.L.--'-U'-'--.!U-'--'-'-'-'-'-'--''-'-'-''-'-'----'-"~ t (s) Figure 2: Evolution in time of the coordinate y of the plate for /=2,000 Hz g 0.03 > t (s) Figure J: Detail regarding the motion at the beginning for/= 2,000 Hz. phasor representation of current density. Using the algorithm described in Section, the time step is chosen to be a multiple of the period and is adjusted according to the force value, such that when the force decreases the time step is increased and when the force increases it is reduced. V. LLUSTRATVEEXAMPLE A copper plate of width 80 mm, thickness 4 mm (see Fig. 1 ), resistivity 2x10-s nm and of mass density 8.9xl0 3 kg/m 3 is levitated using two coils of200 turns each, of 10 mm x 0 mm in cross section and a distance between the axes of their sides of 70 mm and 30 mm, respectively. The current direction is the same in the outer and inner coils, the current intensity in each tum being i =...fi. sin 2eft, with = 10 A and/= 2,000 Hz e ;: O.Q t(s) Figure 5: Evolution in time of the coordinate y of the plate for /=200Hz. nitially, the conducting plate is located at 1 O mm above the coils. t is assumed that the plate only moves in the vertical direction, but the procedures described in the paper are also applicable when more degrees of freedom are considered. The plate cross section is discretized in 180 rectangular elements, as indicated in Fig.. For the time-domain method presented in this paper, the period was divided in 48 intervals and the motion of the plate was observed during 26,000 periods, i.e. for 1,248,000 time steps. The result is presented in Fig. 2, with the detailed motion at the beginning shown in Fig. 3. The computation took about 6 hours employing a GHz ntel processor notebook. A great reduction in the amount of computation is obtained by approximating the conducting region with a thin strip of thickness equal to the field depth of penetration [3]. The oscillations of the plate can be attenuated by adding a de component in the current coils or by using a permanent magnet. f a direct current of 10 A is added in the outer coil, the plate motion becomes as it is shown in Fig. 4. For a frequency of 200 Hz, the motion of the plate is shown in Fig. 5, the attenuation of the mechanical oscillations being much stronger than for a frequency of 2,000 Hz. t should be remarked that the same results in Figs. 2 and 3 were obtained by using the proposed frequency-domain procedure (see Sections V and V). Only 4,393 variable time steps, of magnitude between a period and 50 periods, were necessary for determining the motion of the plate between t = 0 and t = 13 s. The

5 15th GTE Symposium HTML version required computation time was only 124 s, i.e., about 170 times less than for the time-domain solution. V. CONCLUSON AND REMARKS Two efficient methods are presented for computing the motion of the solid conductors under the action of electromagnetic forces. Practically, for the same accuracy of the results, a tremendous reduction in the amount of computation is achieved when using a frequency-domain procedure. The proposed methods can be extended to nonlinear media. For the time-domain procedure one can utilize the polarization method [4], which allows the formulation of the eddy-current integral equation [2]. For the frequency-domain procedure, one can adopt the method proposed in [5], [6]. Since in some problems, for instance in electromagnetic levitation problems, the air regions are relatively large with respect to the conducting or/and ferromagnetic regions, the weight of the fundamental harmonic in the harmonic spectrum is significant and, thus, for the convergence acceleration in the polarization method one can efficiently employ the technique proposed in [7]. The methods presented can be applied to three-dimensional structures as well, by adopting the eddy-current integral equation proposed in [8] and extended in [2] to nonlinear media and to moving bodies. n this case, the spatial discretization of (16) is done by decomposing the induced current density using functions of the form V x W, where W are edge elements. When edge elements of the first order are used, then the current density is constant inside the tetrahedral volume elements and the relations presented in this paper remain valid. Now, the integrals similar to those in (9) and (10) can only partially be evaluated analytically. Finally, it should be remarked that the procedures in [2], [5], [6] and [7] allow the extension to nonlinear media of the proposed frequency-domain technique which, as illustrated in this paper, could yield a spectacular reduction in the amount of computation needed. ACKNOWLEDGMENT This work was supported in part by the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/89/l.5/S/62557 and by a grant of the Romanian National Authority of Scientific Research, CND-UEFSCD, project number PN--PT-PCCA REFERENCES [1] S. Kurz, J. Fetzer, G. Lehner, and W. M. Rucker, "A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling," EEE Trans. Magn., vol. 34, no. 5, pp , Sep [2] R. Albanese, F. Hantila, G. Preda, and G. Rubinacci, "ntegral formulation for 3-D eddy current computation in ferromagnetic moving bodies," Rev. Roum. Sci. Techn., Electrotechn. et Energ., vol. 41, no. 4, pp , (3]. R. Ciric, F.. Hantila, and M. Maricaru, "Field analysis for thin shields in the presence of ferromagnetic bodies,'' EEE Trans. Magn., vol. 46, no. 8, pp , Aug (4] F. Hantila, "A method of solving magnetic field in nonlinear media," Rev. Roum. Sci. Techn., Electrotechn. et Energ., vol. 20, no. 3,pp , [5]. R. Ciric and F.. Hantila, "An efficient harmonic method for solving nonlinear time-periodic eddy-current problems," EEE Trans. Magn., vol. 43, no. 4, pp , Apr [6]. R. Ciric, F.. Hantila, M. Maricaru, and S. Marinescu, "Efficient analysis of the solidification of moving ferromagnetic bodies with eddy-current control," EEE Trans. Magn., vol. 45, no. 3, pp , Mar [7]. R. Ciric, F.. Hantila, and M. Maricaru, "Convergence acceleration in the polarization method for non! inear periodic fields," COMPEL, vol. 30, no. 6, pp , [8] R. Albanese and G. Rubinacci, "ntegral formulation for 3D eddy-current computation using edge elements," EE Proceedings A, vol.135, no. 7, pp , Sep