two simple models

IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 2245 Two Simple Models for Analytical Calculation of Eddy Currents in Thin Conducting Plates Nicolaos J. Siakavellas Abstract— Two simple models are proposed for the analytical calculation of eddy currents induced by a time-varying magneticfield in thin conducting plates of various shapes. In the firstmodel, it is assumed that the current paths are determinedexclusively by the shape of the plate. This assumption makesevident a shape factor that characterizes the shape of the plateunder investigation and is independent of the dimensions of theplate. Thus, several important parameters, such as the equivalentresistance of the plate, the total circulating current, etc., areexpressed analytically as a function of this shape factor. Theapplication of the model makes possible the estimation of the total eddy current, circulating in plates of any shape, but the accuracy of this estimate is strongly affected by the shape of theplate. This model is then improved for plates having symmetricalshapes by taking into consideration the principle of minimumenergy dissipation. The results, obtained by the improved modelfor the total circulating current, are in good agreement with theresults obtained numerically as well as with analytical resultsobtained by the use of variational methods. Index Terms— Eddy currents, modeling, plates, resistance, shape. I. INTRODUCTION DURING the last two decades, electromagnetic field com- putation and modeling of eddy currents have received increasing attention. Eddy currents are involved in variousdomains of applied electricity. They play an important rolein many industrial applications [1], for example in the design of electromagnetic devices, magnetic fusion devices, magnetic levitation vehicles, etc. The phenomena concerning an electromagnetic field prob- lem are described mathematically by Maxwell equations. Con- sequently, the problem is defined in general by a partial differential equation with appropriate boundary and initialconditions. In most cases, a numerical method is adopted forthe solution of such a problem. Since the computer power has increased impressively, the numerical methods become more and more attractive. It is possible with the help of these meth-ods to solve complicated electromagnetic problems even inthree dimensions, for which some decades ago, it was difficult to obtain a solution. There are, of course, many limitations and inconveniencesassociatedwith numericalmethods.Ingeneral,theyconsumeconsiderablecentralprocessingunit(CPU)time.Besides, the solution process of any problem must be repeated for any set of input data. That may be required even if one set Manuscript received June 10, 1996; revised October 9, 1996. The author is with the Department of Mechanical Engineering, University of Patras, 265 00 Patras, Greece. Publisher Item Identifier S 0018-9464(97)03111-7. Fig. 1. Schematic diagram of the plate. is a minor perturbation on a previous set. Thus, an analytical solution to an electromagnetic field problem always posesa great challenge being powerful and rapid. It is also veryimportantsinceithelpsunderstandbetterthephysicalbehavior of electromagnetic fields. However, to obtain an analytical solution for a three-dimensional (3-D) application may be ex-tremely difficult and in many cases, impossible. Nevertheless,a large class of problems of practical interest exists for which a two-dimensional (2-D) electromagnetic analysis is sufficient to achieve quite satisfactory results and interpretations. In the present paper, we propose to solve analytically a 2-D problem: Calculate the eddy current induced by an uniform time-varying magnetic field, either harmonic or transient, in thin conductive plates of various shapes. Our intention is tocombine the simplest possible modeling approach with themaximum attainable accuracy. II. D ESCRIPTION OF THE PROBLEM A conductive plate of thickness and faces with arbitrary shape of surface is placed in a time-varying magnetic field (Fig. 1). This field, of magnetic induction is uniform in space and perpendicular to the faces of the plate. Its time variation may be either harmonic, according to (1) or transient, for example (2) where is the magnetic induction at ,is the angular frequency and is a decay time constant. The magnetic flux variation through the surface of the plate induces eddy currents in the plate. We assume that the plate lies in the plane, while the magnetic field is parallel to the axis. In the following, the index will be omitted for simplicity. The objective is to calculate analytically the total circulating current, , as well as some characteristic electro- 0018–9464/97$10.00 1997 IEEE 2246 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 magnetic parameters of the problem. For this purpose, the following assumptions are made. 1) The applied pulse of flux passing through the plate is not attenuated by the eddy-currents, i.e., the penetration depthof the time-varying magnetic field is sufficiently greater than the plate thickness . This penetration depth is given as (3) for the time-harmonic field and as (4) for the transient field where andare the perme- ability and conductivity of the material of the plate,respectively. We then consider the current uniformly distributed over the thickness of the plate. 2) The magnetic field produced by the eddy current is small compared to the applied magnetic field, and con-sequently, it may be neglected. The present analysis is, therefore, restricted to the so-called resistance limited case. This choice will be justified below. Thegoverningideaistoadoptamodelassimpleaspossible. In a first approach, it is assumed that the current paths are determined exclusively by the geometry, i.e., the shape of theplate. This idea has also been used in previously publishedwork [2] where we treated the problem of a thin rectangular plate with length , width , and thickness placed in a time-varying magnetic field according to (2). This problem isresolved by assuming that the eddy current circulates aroundthe center of the plate in rectangular paths, one within the other. From the above assumptions [1) and 2)], in the above work we adopted only the first one, concerning the penetrationdepth. Therefore, the magnetic field produced by the eddycurrent is not neglected in this work [assumption 2)]. So, we have also considered the conductive plate as an equivalent electric circuit with an inductance and resistance . This can be described by (5) where is the electromotive force, induced by the mag- netic flux variation through the surface of the plate. The following analytical expressions for the equivalent resistance,inductance, and emf were derived and given in [2]: (6) (7) (8)The analytical solution of (5) yields the total circulating current in the plate as a function of time (9) The assumption of perfect rectangular paths instead of rectangular paths with rounded corners results in a slightoverestimation of and. However, we verify from (9) that the overestimations in andare almost mutually compensated. So, the agreement between analytical and ex- perimental results for the current are very satisfactory [2]. Themaximum discrepancy of 4% is observed when the currentreaches its maximum value. At that instant, the term in (5) vanishes, so there is no more compensation betweenthe terms. That leads us to consider in the present work theresistance limited case [assumption 2)] in order to make clearthe deficiencies of the adopted model, since in that case we have no more mutual compensations of discrepancies between theseterms.Oncetheprecisionofourmodelhasbeenclarified,we then would proceed to the following step: To solve theproblem without adopting assumption 2), i.e., by taking into consideration the magnetic field produced by the eddy current. Obviously, now we need to calculate analytically, besides theequivalent resistance of the plate, an equivalent inductance aswe have done in the rectangular plate mentioned above [2]. That is possible for plates of various shapes if we approach the current path by straight segments, but this requires increasedtime and effort. The analytical calculation of the equivalentinductance of plates having relatively simple shapes could be the subject of future work. The present analysis will be restricted to the resistance limited case of a thin plate. III. T HEGEOMETRIC MODEL In general, the plate may have an arbitrary shape with its perimeterforminganarbitrarycurve(Fig.1).Wecanapproachthis curve by a broken line, so that we have an irregularpolygon with sides. Since the time-varying magnetic field is uniform in space and the plate is thin, we may assume that the eddy current circulates around the center of mass of the plate (Fig. 2). If we join now the point with the vertices of the polygon, we divide the plate into triangular elements. As wehavealreadyreported,weconsiderinafirstapproachofthe problem that the current distribution in the plate is determinedexclusively by the geometry, i.e., the shape of the plate. Thecurrent flow is, therefore, parallel to the sides of the plate, i.e., the current circulates in paths that form polygons, one inside the other as illustrated in Fig. 2. Obviously, the shape of eachpolygon is similar to the shape of the plate. We note by the basis of the triangular element ,b yits height, by the distance of a given current path from the point , and bythe length of the portion of this current path within the element (Fig. 3). The Maxwell equation, is written for a given current path (10) SIAKAVELLAS: EDDY CURRENTS IN THIN CONDUCTING PLATES 2247 Fig.2. Thedivisionoftheplateintotriangularelements,showingtwotypical current paths. Fig. 3. Characteristic dimensions and angles in a triangular element. In the above equation the implicit assumption, as we have already reported, is that is spatially uniform. If we use Ohm’s law, , we may rewrite (10) into the form (11) The total current circulating the plate (around ) is given by (12) where . Considering now two adjacent ele- ments, we verify from Fig. 3 that (13) For any elements andof arbitrary position, we obtain from (13) (14) Taking into consideration (14), (12) is written (15)Equation(15)yieldsthefollowingrelation betweenthecurrent densities and: (16) We can express the current densities in (11) as function of a given current density by means of (16). Furthermore, the distances from the pointmay be written as according to (14). So, we verify from Fig. 3 and (14) that (17) Equation (11) is then written as (18) From (18), we obtain an expression for any current density at a given instant as a function of the distance from the point (19) Inserting the above expression for in (15), we deduce the total current circulating in the plate (20) Let us calculate now the total power dissipated in the plate at a given instant . The elementary power dissipated in an elementary volume of the triangular element is (21) By inserting expression (19) for and expression (17) for into (21) and integrating from to, we obtain the power dissipated in the triangular element (22) 2248 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 Obviously, the total power dissipated in the plate is (23) or (24) The equivalent resistance of the plate is now determined by the fact that the total power dissipated in the plate is alsogiven by or using (20) for the total current by (25) Comparing (24) and (25) yields (26) From (26) we observe that the equivalent resistance of the plate is independent of the surface of the face of the plate. Itdepends only on the resistivity of the material, the thicknessof the plate, and a dimensionless factor that may be defined as (27) This characteristic parameter depends exclusively on the shape of the face exposed to the variable magnetic flux, which isindependent of the dimensions of the plate and may be consid- eredasashapefactor .Consequently,plateswithgeometrically similar faces are characterized by the same shape factor. Thus,(26) may be written (28) From the above equation we conclude that, plates with similar faces have the same equivalent resistance, if they have thesame thickness and consist from the same material. The shape factor appears in all the above formulas, con- cerning the current density, the total current, and the powerdissipated in the plate. Furthermore, we note that, the surfaceof the face of the plate also appears in these formulas (29) Therefore, the current density (19), the total circulating eddy current (20), and the power dissipated in the plate (24) may all be expressed as functions of the plate surface and the shape factor (30) (31) (32) For the equivalent electromotive force, ,w e obtain from (28) and (31) (33) So, we verify (8) which was derived through a different formulation in [2]. We can apply and test now the model to some conducting plates with simple geometric shapes. A. Circular Plate Letbe the radius of the plate. From (27), the shape factor is (34) Consequently, the equivalent resistance will be (35) According to (30)–(32) and taking into consideration (34), we obtain, for the current density, the total circulating eddy current, and the power dissipated in the plate, the followingexpressions: (36) (37) (38) SIAKAVELLAS: EDDY CURRENTS IN THIN CONDUCTING PLATES 2249 B. Regular Polygon We consider now that the faces of the plate have the shape of a regular polygon with sides ( ). Letbe the side of the polygon and its distance from the point , that coincides with the geometric center. Equation (27) yieldsthe shape factor for a regular polygon (39) Consequently, the equivalent resistance is (40) while for the current density, we obtain from (30) (41) The total current and the power dissipated in the plate are given as a function of the surface of the plate by (31) and (32) if we use the value of the shape factor given by (39). These quantities, may be expressed also as a function of one of the characteristic dimensions of the polygon, for examplethe side or its distance from the center. We then have (42) (43) We may remark that, if we consider the circular plate as a regular polygon with an infinite number of sides ( ), we again have formulas (34)–(38). C. Rectangular Plate Let usconsidernowthattheshapeoftheplateisrectangular with length and width . If we note by the ratio (i.e., the aspect ratio of the rectangle), we obtain from (27) for theshape factor (44) Therefore, the equivalent resistance of the plate is given by (45) i.e., we find again (6). From (30)–(32), we obtain the formulas derived in [2] for the current densities, the total circulatingcurrent, and the power dissipated in the plate, respectively (46) (47) (48) In the present model, we have taken into consideration only the geometry, i.e., the shape of the conductor, and we haveexpressed some electromagnetic parameters of interest as afunctionofpurelygeometricparameters.However,inthisway, we introduce an error which depends on whether or not the current path suggested by the shape of the conductor is alsoenergetically the shortest one. The only case where that is trueis the case of the circular plate. Consequently, the formulas derived for a circular plate, (34)–(38) are exact. In any other case, we will have a discrepancy that strongly depends on theconductor shape. We will clarify this in Section V, where wecompare the analytical results with the numerical ones. IV. T HEIMPROVED MODEL Up to now, we have assumed that the current paths at any position of the plate have the shape of the plate. In order to obtain an exact analytical solution, we must determine at any position the exact shape of the current path. Thatis a sufficiently difficult task and requires too much timeand effort. Furthermore, it is not certain that the resulting analytical formulas will be sufficiently simple, so that they have a practical interest. The shape of the current paths as afunction of the position for plates with various shapes havebeen approached in [3] by preselecting the form of the stream function and by using variational methods. We will refer to that procedure more extensively in Section V. We propose now a method to improve the geometric model developed in the previous section, so that we arrive to a sufficiently precise and, at the same time, simple model. This improvement is possible, for plates with symmetric shapeunder the following assumptions. 1) Take into consideration the fact that, the current distri- bution in a conductor is such that the energy dissipatedis a minimum. 2) Retain the idea adopted in the geometric model for current paths parallel to the sides of the plate. This has two advantages: i) The analytical computationsare performed easier, and the formulas maintain their 2250 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 simplicity. ii) In the case where the secondary magnetic field, produced by the eddy current is not negligible, itis possible although quite tedious to calculate analyti-cally the equivalent inductance of the plate. We have performed this calculation for a rectangular plate in [2] as we have already mentioned in Section II and derived(7) for the inductance. On the contrary, if we considerthe real-curved current paths, it is prohibitively difficult and, in many cases impossible, to calculate analytically the inductance of the plate. The above assumptions will permit us to improve the precisionof thepreviousformulasandatthe sametimeexploit their simplicity. However, our analysis will be limited toconductors of symmetrical shape. A. Regular Polygon Through a given area of the plate, the flux variation is . If the shape of this area is a regular polygon with sides, a unit charge moving onthe circumferenceto encircle thisarea must cover a distance equal to the perimeter of the polygon (49) Instead, if the shape of the surface is circular, the unit charge moving on the circumference in order to encircle the samearea (i.e., a flux variation ) covers a distance equal to the perimeter of the circle (50) It is, of course, known that, between all the possible paths that cover the same area , the circular is the shortest one. Taking into consideration these remarks, the above assump- tions 1) and 2) seem to be incompatible because they imply minimum energy dissipation and at the same time, polygonal, i. e., not circular, current paths. In order to compromisethese assumptions, we introduce an effective conductivity having the following property: the real conductivity multiplied by the length of the idealized current path, i.e., the perimeter of the polygon that surrounds the area is equal to the product of the effective conductivity by the length of the minimum current path , i.e., the perimeter of an equivalent circle. So we have or (51) If we use now (49) and (50), we obtain from (51) the effective conductivity for a plate, having the shape of a regular polygon (52) It is obvious that, if we consider for the plate the effective conductivity given by (52) instead of the real one and repeat the procedure of the previous section, we again find, forthe parameters of interest, formulas that are similar to thoseobtained by the geometric model, but, with instead of . Equation (41) for the current density at a distance from the center of the polygon, may then be written (53) We observe that we may preserve the real conductivity in the formulas if we introduce an improved shape factor , so that or . Thus, taking into consideration (39), (49), and (50) we obtain the improved shape factor (54) Consequently, the equivalent resistance is (55) The formulas for the total current and the power dissipated in the plate may be written (56) (57) We note again that for we obtain the corresponding formulas for a circular plate. B. Rectangular Plate We consider now the case of a rectangular plate with length andwidth .Throughagivenarea ,wehaveafluxvariation . If the shape of the area is rectangular, a unit charge moving on the circumference in order to surround this areamustcoveradistanceequaltotheperimeter oftherectangle, i.e., (58) where is the aspect ratio. In the case of the regular polygon, the shortest path that a unit charge may travel in order to surround a given area is theperimeterofacircle.Thatisalwaysvalidbut,inthepresent case, we must also take into consideration the elongated shape of the rectangle. Therefore, we consider the perimeter of anequivalent ellipse instead of the perimeter of an equivalentcircle. We suppose that a unit charge, in order to surround a given area and at the same time have the dissipated energy minimum, must cover a distance equal to the perimeter of an ellipse of surface .I fis the ratio of the semi-axis of SIAKAVELLAS: EDDY CURRENTS IN THIN CONDUCTING PLATES 2251 that ellipse, we have [4] or, approximately (59) In order to compromise between rectangular current paths and minimum energy dissipation, we introduce, as we have done in the case of the regular polygon, an effective conductivity so that or (60) where andare given by (58) and (59), respectively. However, in order to calculate by means of (59), we must determine previously the ratio of the semi-axis of that ellipse. Obviously, the perimeter must be greater than the perimeter of an equivalent circle, but smaller than the perimeter of an equivalent ellipse with ratio of semi-axis equal to, i.e., the aspect ratio of the rectangle under investigation. Thus, it is evident that if we consider the circle as an ellipsewith ratio of semi-axis equal to one, then lies between and, i.e., . A method to determine is given in detail in the Appendix, where is obtained as one of the roots of a quadratic equation. Using now (58) and (59),we obtain from (60) the following expression for the effectiveconductivity of a rectangular plate: (61) The formulas for the parameters of interest derived in Section III-C are still valid if we replace by. However, we may introduce an improved shape factor , as in the case of the regular polygon, so that we preserve in our formulas thereal conductivity instead of the effective one. This improved shape factor, , where is given by (44), is expressed as (62) The equivalent resistance of the rectangular plate is now given by (63) The expressions yielding the total circulating current and the power dissipated in the plate may be written (64)(65) Before we proceed to the evaluation of the models (the geo- metric and the improved one), we should note the following. 1) A square plate may be treated either as a polygon with or as a rectangle with . We observe that the formulas derived for a regular polygon with coincide perfectly with the corresponding formulas derived for a rectangle with . 2) Theshapefactorofacircularplate seemstobe,under the assumptions introduced in our models, a physical limit. Therefore, it is not possible for a plate of any shape to have a shape factor less than . V. EVALUATION OF THE MODELS We are ready now to evaluate our models. We compare the analytical results obtained for plates of various shapes with: a) numerical results; b) analytical results, obtained by the use of variational methods; c) the current distribution obtained numerically and analyt- ically. A. Numerical Solution and Results To obtain a numerical solution of the problem, we proceed as follows. We transform Faraday’s law, , by using Ohm’s law, , into (66) Since div , we introduce the current vector potential , defined as (67) Taking into consideration that the plate is thin and combining (66) and (67), we finally obtain (68) where is the component of normal to the plate. Thus instead of a vector quantity, we consider a scalar one, and the3-D problem is transformed into a 2-D problem. This scalarquantity is the so-called stream function, frequently used for thin plates [5], [6]. Equation (68) has been solved numerically for plates of various shapes. For the numerical analysis, a finite-differencemesh has been considered. The discretization of (68) on the mesh has been performed by using central difference approach for the second derivatives. The resulting set of algebraicequations has been solved with the Gauss–Seidel method [7]by taking into consideration the boundary condition at the edge of the plate. The number of grid points used inall computations was larger than 800, and it can get as big as8000 in some of the cases we have examined. This is more 2252 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 than adequate to practically ensure independence of the results from the exact number of grid points. The data for our sample problem are the following. A conductive plate, with electrical conductivity S/m, surface m, and thickness mm, is placed in a uniform time-varying magnetic field, either harmonic (1) ortransient (2) perpendicular to the faces of the plate. For the computations, we have considered T/s. This would be the maximum value that can take, if we consider for example T and rad/s in the case of the time-harmonic field (1), or T and ms in the case of the transient field (2). In any case, the percentagediscrepancy between numerical and analytical results for any set of ( )o r( ) is not affected by the time evolution ofandisexactly thesame atanyinstant.Thus,theresults presented in Table II are valid for any combination of ( ) or ( ) that satisfy T/s or T/s, respectively, provided that the values of orare such that the resulting penetration depth is sufficiently greater than the plate thickness. For example, for rad/s we have from (3) mm, while for ms we have from (4) mm. We performed our computations for conductive plates with the following shapes: a) circular, b) regular polygons with (equilateral triangle), (square), (hexagon), (octagon), (dodecagon), and c) rectangles with aspect ratio (square), ,,, and . The dimensions of each plate have been adjusted so that thesurface of the face subjected to the variable magnetic flux is min all cases. In Table I, we give the shape factors for the plates con- sidered, resulting from both models, the geometric and the improved one, i.e., by (34), (39), (44), (54), and (62). The total circulating current in each plate has been cal- culated i) by using the geometric model, ii) by using theimproved model, and iii) numerically, and the results aregiven in Table II. We also give the absolute discrepancy betweennumericalandanalyticalresultsforeachcase,i.e., numerical-analytical /numerical (%). Obviously the numerical results depend on the finite-difference mesh considered andwe, therefore, include here the maximum discrepancy. As we observe from Table II in the case of the geometric model, the discrepancy between analytical and numericalresults depends strongly on the shape of the plate. So, we have a discrepancy of 25% for the equilateral triangular plate, but only 1.5% for the dodecagon and 3.8% for a rectangle withaspect ratio . On the contrary, the application of the improved model yields a maximum discrepancy of about4%. As far as the power dissipated in the plate is concerned,the application of the geometric model yields a discrepancy between analytical and numerical results of about 17% for the equilateral triangular plate or a rectangular plate withaspect ratio . An 11% discrepancy is observed for a square plate and less than 2.5% for an octagonal plate. Theapplication of the improved model yields for the power lossesa discrepancy of 12.5% in the case of the more elongated rectangle ( ) but less than 0.5% in the case of a square or an octagonal plate. A more detailed comparisonTABLE I SHAPEFACTORS TABLE II TOTALEDDYCURRENT(INA)ANDDISCREPANCIES (%) between analytical and numerical results for the dissipated power in rectangular plates is given in Table IV, where wecompare also with the analytical results obtained by the useof variational methods. In conclusion, the geometric model gives an estimation of the total eddy current circulating in a conductive plate of anyshape as well as of the corresponding power losses. In somecases, these estimations may have a sufficient precision. On thecontrary, the improved model gives asatisfactory precision foranycase,but itislimitedtoconductiveplates ofsymmetricshape. In any case, the estimations of the eddy current and thepower losses given by the geometric model provide a lower bound forthe total circulatingcurrent and thepower dissipated in the conductor, respectively. Obviously, the estimate forthe equivalent resistance, obtained by the geometric model, SIAKAVELLAS: EDDY CURRENTS IN THIN CONDUCTING PLATES 2253 provides an upper bound for the equivalent resistance of the plate. B. Comparison with Analytical Formulas Obtained by Using Variational Methods In an earlier publication [3], the eddy currents in thin conducting plates were calculated by the use of variational methods. The procedure minimizes a functional, related to the power losses, so that the stream function can be determined.A detailed description of this procedure is also given in [8]. Two variational methods have been used in [3], namely the Ritz and the Kantorovich methods. The Ritz method preselects the form of the stream function for a plate of given shape,i.e., chooses a trial function of multiplied by appropriate coefficients. These coefficients are then determined by mini- mizing the functional. The Kantorovich method separates the streamfunctiontotheproductoftwofunctions.Onedependingonly on , the other only on . The form of the function depending on is preselected. The function depending on is then determined by solving the Euler equation, resulting from the minimization of the functional. We compare now the formulas obtained by the improved model for the total current and the power dissipated in rectan- gular plates with the corresponding formulas obtained in [3] by the use of variational methods. We have derived in Section IV the following formula for the total current circulating in a rectangular plate (64): (64) If we use our notation, we obtain from [3] the following formulas for the total current: (69) (70) where the indexes VR and VK indicate Variational-Ritz and Variational-Kantorovich, respectively. We observe from (64), (69), and (70) that the total current is expressed by the term multiplied by a coefficient depending on the method considered and the aspect ratio of the rectangle. The values of these coefficients for , and are compared in the Table III, where we have reported also the corresponding values resulting from the numericalcomputations, i.e., the ratio of the total current divided by theterm . We follow now the same procedure for the power losses in rectangular plates. In Section IV, we have derived thefollowing formula (65): (65)TABLE III COMPARISON OF “CURRENT”COEFFICIENTS TABLE IV COMPARISON OF “POWER”COEFFICIENTS If we use our notation, the corresponding formulas derived in [3] are written (71) (72) We observe again from (65), (71), and (72) that the power is expressed by the term multiplied by a coefficient depending on the method considered and the ratio . The values of these coefficients for , 2.0, 3.0, 4.0, and 5.0 arecomparedintheTableIVwiththecorrespondingnumericalvalues, i.e., the ratio of the dissipated power divided by theterm . From Tables III and IV, we may observe the following. Our analytical results for the total current, obtained by applicationof the improved model, are substantially better than the resultsobtained by the Ritz method, except for the case of the square plate. However, they are less accurate than the results obtained by the Kantorovich method (Table III). As far as the powerdissipated in the plate is concerned, our results are in verygood agreement with those obtained by the two variational methods or numerically for small values of . For ,wehaveadiscrepancythatincreaseswith .Theresults obtained by the Ritz method manifest the same behavior,although the increased discrepancy with is smoother. Only the Kantorovich method yields results in very good agreement with the numerical ones for any value of considered (Table IV). 2254 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 (a) (b) (c) Fig. 4. Geometrical characteristics of the current distribution computations. C. Comparison of Current Distributions Obtained Numerically and Analytically The analytical models provide idealized current paths hav- ing the shape of the plate under investigation. In reality, thecurrent flow inside the plate does not follow exactly the shape of the plate. Therefore, it is interesting to compare the current distribution inside the plate given by the improved model withthat obtained numerically. The eddy current circulates around the center of mass of the plate that coincides with the geometric center for plateshaving symmetrical shapes. Therefore, the current circulatingthroughanysectionwitharea astheshadedoneinFig. 4(a) is independent of the position of the point at the edge of the plate and equal to the total eddy current. Furthermore,the current is uniformly distributed over the thickness of the plate, according to the assumption 1) in Section II. The emerging question now has to do with the current distributionbetween the center and the edge of the plate. If we considera point on the line between and[Fig. 4(a)], the current circulating through the section with area depends on two parameters: The distance of the point from the center of the plate and the orientation of the line . The current distribution over sections lying in some characteristic directions has been computed analytically (improved model) and numerically. In fact, we have computed the current circulating between the points and, i.e., through the section with area asafunctionofthedistance .Inordertoreportonthesame diagram, the current distribution over some characteristic ori-entations for a given plate, we have considered the normalizeddistance instead of . Thus, varies from zero to one as moves from to. In the case of plates having the shape of a regular poly- gon, we have considered the current distribution over two characteristic directions [Fig. 4(b)]: One perpendicular to the side of the polygon and the other in the direction joining thecenter of the plate with a given vertex. For rectangularplates, we have computed the current distribution over three characteristicdirections[Fig.4(c)]:Perpendiculartothelongerside, perpendicular to the shorter side, and lying on thediagonal. The results of the computations are shown in Fig. 5 for plates having the shape of a regular polygon and in Fig. 6for rectangular plates. The current distribution given by theimproved model (solid line in Figs. 5 and 6) is the same over anydirectionconsidered.Thisisanexpectedresultbecausewe have assumed current paths similar to the shape of the plate.Therefore, the current density depends on one space variableonly. On the other hand, the current distributions obtained numerically (dashed lines in Figs. 5 and 6) are different for each of the directions considered. This is so because in reality,the current density depends on two space variables. However,the current distribution obtained by the improved model is in very good agreement with the numerical results for the current distribution over the shortest distance between the center andthe edge of the plate and represents in most cases, a meanvalueforthecurrentdistributionoveralldirections.Inthecase of elongated rectangles, the numerical results for the current distribution over the direction perpendicular to the shorterside of the plate have a special characteristic which is worthmentioning. The fraction of the current circulating between the center and the mid-distance from the shorter side is practically negligible. As we can verify from Fig. 6 (with ), the current circulating between the center and a normalized distance0.6islessthanthe1/20ofthetotalcirculatingcurrent. Thus, the main fraction of the current circulates in the outer part(between0.6and1.0),whereweobservearapidriseofthecurve.Thiscanexplainwhyforrectangleswith wehave the maximum discrepancy (12.5%) for the dissipated power, between improved model and numerical results (Section V-A). The same istrue forcomparisons between the improved modeland analytical results obtained by the use of a variationalmethod (Section V-B). VI. C ONCLUSIONS In the present work, we have proposed two simple models for the analytical calculation of eddy currents in thin con-ductive plates. In a first step, we have developed a purelygeometric model, i.e., we have assumed that the current paths in the plate are determined exclusively by the shape of the plate. This assumption made apparent a dimensionless shape factorthatischaracteristicfortheshapeoftheplateconsidered and is independent of the dimensions of the plate. Thus, the parameters of interest, i.e., current densities, equivalent resistance of the plate, total circulating current, and powerdissipatedintheplate,areexpressedasafunctionofthisshapefactor. By using this model, we can estimate the total eddy current circulating in plates of any shape, but the precision of the estimation depends strongly on the shape of the plate.However, theprecision may besatisfactory for some particularcases. Anyway, this estimation constitutes a lower bound for the total circulating current. In a second step, we have improved the geometric model by taking into consideration the fact that the current distribution SIAKAVELLAS: EDDY CURRENTS IN THIN CONDUCTING PLATES 2255 Fig. 5. Current distribution in plates having the shape of regular polygon. ______ Improved model. — — — Numerical in diagonal direction. ...... Numerical in direction perpendicular to the side. in the plate is such that the dissipated energy is minimum. The application of this improved model yields a maximum discrepancy between analytical and numerical results for the total circulating current of the order of 4%, but the model is restricted to plates with symmetrical shape. We have compared also our analytical results for plates with rectangular shape to analytical results obtained by us- ing variational methods, namely the Ritz and Kantorovich methods. We have verified that our results concerning the total circulating current are better than those obtained by the Ritz method but are less accurate than the results obtained by the Kantorovich method. As far as the power losses are Fig. 6. Current distribution in rectangular plates. ______ Improved model. — — — Numerical in diagonal direction. - - - - Numerical in direction perpendicular to the shorter side. ...... Numerical in direction perpendicular to the longer side. concerned, our results are accurate for regular polygons or less elongated rectangles but less accurate than those obtained bythe variational methods for elongated rectangles. This is duemainly to the fact that, in our models, we have approximatedthe real-curved current paths by straight segments. Thus, weconsider in each current path segment a mean value for thecurrent density, i.e., we assume that it depends on only onespace variable ( ,o r), instead of two. Consequently, the mean current densities are less accurate when the current pathsegments considered are strongly unequal as in the case ofelongatedrectangles.Inthatcase,ourmodelsyieldanestimate 2256 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 3, MAY 1997 for the current densities less accurate than the variational methods where curved current paths have been considered thatbetter represent the real current paths. This deficiency in ourmodelsislessevidentinthecaseofthetotalcirculatingcurrent thatdependslinearlyonthecurrentdensitiesbutmoreapparent in the case of the power losses that are proportional to thesquare of the current densities. However, the use of variationalmethods requires guessing a different stream function for each shape of plate and needs lengthier calculations. On the contrary, our method is simpler and needs fewer calculations.It is rather more general than the variational, since the need toanalyze each shape of plate separately is avoided, and we may treatagroupofshapes(forexamplepolygons)simultaneously. Despite the fact that our models are based on geometrical assumptions rather than physical ones, they achieve in mostcases quite satisfactory results and at the same time, combine effectively electromagnetics and geometry. This geometric character of our models offers a further advantage: it allows anextension (obviously under some supplementary assumptions)to cases where the secondary magnetic flux is not negligible. This is due to the fact that we approach the curved current paths by idealized ones consisting of straight segments. Thus,wehave thepossibilityofcalculatinganalyticallythemagneticflux through the surface of the plate due to the eddy current. Consequently, we may calculate analytically the equivalent inductance of the plate. This is planned for some plates ofchosen shape in future work. Another extension of the presentpaper would be to modify the improved model that is now limited to plates with symmetrical shape so that it is applicable to plates of any shape. A PPENDIX We seek the ratio of the semi-axis of an ellipse having the following property. Its perimeter , given by (59) as a function of , must approximate the length of the shortest (energetically) current path among all the possible current paths surrounding the same area in a rectangle with aspect ratio. If we consider an ellipse with area and ratio of semi-axis , its perimeter represents a probable current path in the rectangle under investigation, but this path is not energetically the shortest one. On the other hand, if weconsider a circle with area , its perimeter has the shortest path, but it is not appropriate as current path in the rectangle because of the elongated shape of the rectangle. Consequently, the perimeter must be greater than given by (50) but smaller than given as a function of by a relation similar to (59), i.e., (73) Thus, we assume that is given by the relation (74) whereand are weighting factors, determined with the following grounds. The required ellipse with ratio of semi-axis will be closer to an ellipse with ratio of semi-axis , i.e., equal to theTABLE V aspect ratio of the rectangle under investigation, dependingon whether this rectangle is elongated or not. On the other hand, this ellipse will be again closer to an ellipse with ratio of semi-axis , i.e., a circle, depending on how well this rectangle approaches a square ( ). Thus, we assume that a proportionality exists between andand similarly between and, i.e., and ( ),o r (75) From (75), we obtain (76) The fact that allows us to write the above relations as (77) Consequently, (74) is now written (78) If we take into consideration (50), (59), and (73), we put (78) into the form (79) From (79), we obtain the following relation between and: (80) Equation (80) results to a quadratic equation for (81) where, for brevity, we set (82) The solution of the quadratic (81) yields (83) SIAKAVELLAS: EDDY CURRENTS IN THIN CONDUCTING PLATES 2257 The root with the minus sign is rejected because the resulting value for is less than unity. So, we retain only the root with the plus sign. In Table V, we give, for , 2.0, 3.0, 4.0, and 5.0, the resulting values for andas well as the ratio . We remark that, for , the ratio is approximately equal to (0.8862). Thus, we may use the relationship for a rapid estimation of instead of solving (81). REFERENCES [1] Y. R. Crutzen, G. Molinari, and G. Rubinacci, eds., Industrial Applica- tionofElectromagneticComputerCodes. , vol.1. Dordrecht,Germany: Kluwer, 1990. [2] N. J. Siakavellas, “Analytical modeling of eddy currents induced by a time-varying magnetic field in a conductive plate,” COMPEL-Int. J. Comput.Math.Electr.Electron.Eng., vol. 13, no. 3, pp. 497–508, 1994. [3] R.Sikora,J.Purczynski,W.Lipinski,andM.Gramz,“Useofvariational methods to the eddy currents calculation in thin conducting plates,” IEEE Trans. Magn., vol. MAG-14, no. 5, pp. 383–385, Sept. 1978. [4] J. J. Tuma, EngineeringMathematicsHandbook. New York: McGraw- Hill, 1970. [5] A. Krawczyk and J. A. Tegopoulos, Numerical Modeling of Eddy Currents. Oxford, U.K.: Clarendon, 1993. [6] T. Morisue, “Analysis of a coupled problem: The FELIX cantilevered beam,”IEEE Trans. Magn., vol. 26, no. 2, pp. 540–543, 1990.[7] M. L. James, G. M. Smith, and J. C. Wolford, Applied Numerical Methods for Digital Computation. New York: Harper Collins, 1993. [8] J. A. Tegopoulos and E. E. Kriezis, EddyCurrentsinLinearConducting Media. Amsterdam, The Netherlands: Elsevier, 1985. Nicolaos J. Siakavellas was born in Missolonghi, Greece, on July 5, 1950. He received the diploma degree in physics from the National KapodistrianUniversity, Athens, Greece, in 1973 and the diplome d’etudes approfondies degree in gas and plasma physics and the Ph.D. degree from Universite de Paris-Sud, Centre d’Orsay, France, in 1975 and 1978, respectively. From September 1975 to June 1978, he was a member of the T.F.R. Group at the Nuclear Research Centre of Fontenay-aux-Roses. During the academic years 1974 to 1975, 1975 to 1976, 1976 to 1977, and 1977 to 1978, he was a Grand-Holder of the French government. From September1978 to March 1979 and from February 1980 to June 1981, he was a post doctoral researcher in Electricite de France (EDF) at the Nuclear Reactors and Exchangers Service, Clamart. Since 1982, he has worked at the Department of Mechanical Engineering of the University of Patras, Patras, Greece, where he holds the position of Assistant Professor. During short periods in 1984, 1985, and 1988, he was a Visiting Researcher at the Nuclear Reactors and Exchangers Service of EDF. From September 1989 to October 1990, he was a Visiting Scientist at the Institute of Systems Engineering and Informatics, Commission of the European Communities, Joint Research Centre, Ispra,Italy. His main research interests are numerical and analytical methods in electromagnetic field computation and modeling of coupled electromagnetic- mechanical and electromagnetic-thermal systems. DrSiakavellasisafoundermemberoftheInternationalCompumagSociety.