Eddy current losses on Epstein frame overlapped corner sheets J.P.A. Bastos 1, N.J. Batistela 1, N. Sadowski 1 and M.

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1 1th International IGTE Symposium Proceedings Eddy current losses on Epstein rame overlapped cner sheets J.P.A. Bastos 1, N.J. Batistela 1, N. Sadowski 1 and M. Lajoie-Mazenc 1 GRUCAD/EEL/CTC, Universidade Federal de Santa Catarina, C.P. 476, Flianópolis, SC, Brazil, LEEI de Toulouse,, rue Camichel, 31071, Toulouse Cedex, France jpab@grucad.usc.br Abstract: Iron losses are commonly divided in eddy currents, hysteresis and anomalous losses. While much wk has been accomplished on the two last ones, in this paper we are interested on eddy current losses. Epstein rame devices have overlapped sheets lamination on the cner parts o the iron yoke. In this wk we will examine the losses on these regions using a simpliied analytical model. Some numerical calculations will be also presented quantiying such losses. eywds: lamination, eddy current losses, analytical approach. I. INTRODUCTION Nowadays the losses have been the object o much attention since electrical devices must be competitive and nms do not accept low eiciency machines. Lately, much et has been expended to model the hysteretic behavi o sot materials, which is, indeed, a complex task. In our research group, iron characterization and, particularly, hysteresis models have been put together as well as their application on FE codes[1]. We remark that nowadays, the eiciency o electrical devices plays an imptant role on saving energy and local international nms oten avoid the commercialization o poly designed equipment. Theree the precise modeling o material losses must be obtained by dierent losses evaluation structures, as the widely used Epstein rame. Because the losses are classically divided in three categies, i the modeling o one type is improved, the whole model will be me accurate. That will allow an eicient characterization o a speciic material its applications, as using it on electrical machine transmer yokes. The aim o this wk is to present a better interpretation o eddy currents losses. The behavi o eddy currents on overlapped sheets present in the cners o the Epstein rame have been oten disregarded, observing that the overlapped sheets represent 1.4% o the total iron volume. In this wk we will investigate how the eddy current losses act on these parts. To do so, an analytical approach is employed. The results are compared with a 3D Finite Elements (FE) analysis. Previously, we will briely discuss the model used the whole losses on the Epstein rame, indicating the diiculties to set the losses parameters. II. INITIAL CONSIDERATIONS Commonly the power losses are divided in (hysteresis), P (eddy currents) and P e (anomalous excedent) losses. Let us wk with the power volume densities denoted ph, p and p e, respectively. The ollowing equation gives the total losses density as: P h where α, pt = ph + p + p e (1) α 1.5 p = B + B + B () t h e h, and e have to be determined to describe as well as possible the behavi o a particular material as unction o B, the magnitude o magnetic induction applied on it. Below, we will shtly comment the three terms o the right hand side o the equation (). - the hysteresis power density: It is given by ph = hb α (3) which is the Steinmetz equation [1]. The coeicient h and the exponent α depend on the material. Using the Epstein rame, a very low requency magnetic ield excitation (typically 1 Hz lower) is applied to the material and the two variables above are determined by experimental points itting. The main problem here is α. As example suppose that α = 1.8. When B increases the losses power density will increase too. It is valid up to a certain limit, since when the material is close to its saturation level, the increment o the hysteresis loop surace does not change at the same proption. Theree this exponent must vary and, slowly, should decrease. In other wds, α can be described as a unction o B. Some wks [3,4] propose dierent strategies to model accurately the hysteresis losses. In our previous wks [5,6,7], this procedure was not applied yet, and it must be ocused in the uture. - the eddy current losses power density: This term is related to the main part o this wk. Theree didactical purposes we will briely present its calculation, even though it is well known [1]. Let us consider a single sheet o the Figure 1a and the placement o the required physical quantities in the Figures 1a, 1b and 1c. Figure 1a: a single sheet o dimensions l x, l y and thickness e. Figure 1b: cross section o a single sheet.

2 Proceedings th International IGTE Symposium physical structure o the material. Accding to some auths [8,9], these losses are iginated by the movement o Weiss domain walls when the polarity o the exciting ield changes. Bertotti s wk [8] proposes a B exponent equal to 1.5. Figure 1c: the physical quantities. Because the sheet thickness e is much smaller than the dimensions l x and we consider only the components y l y o J, the induced current density. E is the electric ield and applying rot E= we have: i j k det x y z = i 0 E 0 E = z Taking into account that at z = 0 (at the middle o the sheet) E and J must be zero continuity, we arrive to E(,) z t = z t The Joule power loss on a volume is P V E dv = σ (4) where σ is the electric conductivity; applying this expression the sheet: + e/ lx ly P = σ ( ) z dx dy dz e / 0 0 III. CORNER SHEETS: AN ANALITYCAL APPROACH In Figures a and b we can see pictures o the Epstein rame. Figure a: Epstein rame Figure b: iron cner sheets. F a simpliied model we will consider that a sheet magnetic lux is divided in two parts, and each one is conducted to an adjacent sheet, as shown in Figure 3a. σ 3 σ P = ( ) lxlye = ( ) ( lxly ) 1 1 e e Observing that ( lle) is the sheet volume, in a x y sinusoidal regime o pulsation ω, we have P σ = ( Bmω cos ωt) e lle 1 x y And inally, reminding that the average value o cos ω t is 1/, the expression o average losses power volume density due to eddy currents ( p ), is: 1 p = σbm ω e (5) 4 As mentioned bee, this type o losses will be the subject o additional investigation on this wk; it will be treated soon. - the anomalous losses power density: It is given by 1.5 pe = eb (6) The existence o this equation is based on the act that the sum o eddy current and hysteresis losses do not crespond to the total losses. It is still a complex subject o research since it needs the consideration o the Figure 3a: Iron sheets at the cner. Figure 3b: sheets model. Theree our model will take into account two halsheets, as seen in the Figure 3b. This structure is replicated indeinitely. In this igure, the sheet on the bottom, it is clear that up to the plane A, the lux ollows the (here denoted) regular way. It is similar to the sheet on the top, beyond the plane B. On these parts, the classical equation (5) o average losses power volume density due to eddy currents ( p ) can be applied. The magnetic lux crosses the planes A and B through

3 1th International IGTE Symposium Proceedings a transversal surace equal to el/. This magnetic lux necessarily passes through the section between the sheets, whose surace is L. Our simpliied model considers that the lux is equally distributed on this large interace. As the lux is the same on these two suraces, we deine the transverse induction as: and B t BeL/= BL t B = Be/L t With the above approximations, the cresponding physical situation is shown in the Figure 4 where we can see the induced current density J in the sheet and the term t. (7) dbt r E = (8) dt E depends on r and t. This expression can be also obtained by using the Faraday s law on its local m. With cylindrical codinates, we observe that E has only a θ component and it depends only on r. In the other hand, is placed in the z direction and depends only on t. With vect magnitudes, rot E= becomes 1 ( re) = r r The above expression is veriied with the equation (8). Here one me phenomenon must be considered. It is related to the act that in a whole sheet the magnetic lux arrives rom the two adjacent ones, as seen in the Figures 3 and 6. Figure 4: induced current density in the sheet. Close to the middle o the sheet, the current loops have circular ms but near its limits, they tend to a square shape. We will consider the circular shape o the loops in a circle with the radius R 0.56L, which brings us to similar suraces ( π R = L ). Bee proceeding with the calculation, we point out that a similar algebra was permed square shaped loops and the inal results do not dier much rom those presented below. In the Figure 5 we apply rot E= in integral m a circle o a generic radius r, where 0 < r R. It gives and Figure 5: Applying Faraday s law. L( S) E dl = ds S db Eπ r = t π r dt Figure 6: magnetic lux entering in the whole sheet. Observing the magnetic lux variation, the ientations o the current loops must be opposite. Because in the middle o the sheet the current continuity is respected, the current density J is necessarily equal to zero at z=0. Then we admit a linear variation o J with z. It is obviously similar to E = J / σ. Theree, the equation (8) above is modiied to t r z E = (9) t ( e/) which is now a unction o r, z and t. Applying the equation (4) to the hal-sheet, we have: R π e/ t r z P = σ r θ t 4( e/) drd dz Proceeding with the calculations, we obtain π 4 P = σ er ( t ) 4 Using R 0.56L, t t = Btω cosωt and also the equation (7) Bt = Be/L, we obtain the ollowing expression π 3 P σ B ω e L cos ωt 960 π e P = σ B ω ( L ) e cos ωt 480 Considering the average value o cos ω t = 1/ and the volume o hal-sheet equal to el /, the average value o the losses power volume density is: π pt = σ B ω e (10) 960

4 Proceedings th International IGTE Symposium, comparing with the equation (5), 1 pt = σb ω e (11) 4 It is daring to state that the act 0.078, appearing in the above equation, is exact very close to it, since, among other approximations, the lux is concentrated around the internal iron cner. Nevertheless, it shows that the losses due to the transverse lux are much smaller than the regular lamination eddy current losses. And there is one me aspect to be considered and it is explained with the help o the Figure 7. problems [11,1,13]. We apply the solver using hexahedral edge elements and the ungauged vect potential A mulation, which is well know by its good accuracy as well as its ragility to handle systems with a very large number o unknowns. This last inconvenience does not represent a maj trouble here since the domain is simple. Simple but tricky. Our irst attempt was to model a domain with two hal-sheets (as in Figure 3b). But as the results were plotted, we noticed that the eddy currents had a complete loop on the sheets. Because only hal-sheets are present, only hal a loop can exist. Thus, we opted the domain shown in Figure 8. Figure 7: the two magnetic luxes division. The sheet on the bottom o the igure has two points to be careully observed. At the point x = 0 there is no regular lamination lux Φ ; at the point x = L and beyond ( x > L ), only this lux exists. As the transverse lux Φt, at the point x = 0 only this lux is present while at x = L and beyond it is zero. All in all, it is reasonable to consider that rom the point x = 0 to the point x = L, there is a transition between the transverse losses, given by the equation (11) to the classical losses deined by the equation (5). It is again diicult, based on approximated calculations, to determine how these losses equations are divided in this region. Indeed, since the ields can not be calculated analytically we will perm in the next section some FE analyses to obtain a reasonable approach losses. We can esee that the density losses p t on the overlapped sheets region can be calculated by a expression as 1 p t = αc σb ω e (1) 4 where α c < 1 (13) This result seems somewhat strange but, physically, it can be understood by the ollowing: the magnetic induction B t is much smaller than the longitudinal induction B. Since the magnitude o the induced currents depends on B t it is clear that the losses due to such currents are smaller than regular losses on the laminations. Now, we present the numerical calculations in der to estimate the value o α. IV. 3D FE MODELING In der to calculate the magnetic ield and the eddy current losses, we used our package FEECAD [] whose reliability have been demonstrated by several applications, including the solution o TEAM Wkshop c Figure 8: the two hal sheets domain. In this igure we have an exciting bar (non conductive eddy current purposes) where a current density J = 0.08 sin ωt ( A/ mm ) is applied on the vertical direction. The requency is 60 Hz and there are no airgaps. A ull iron sheet (thickness equal to 0.5 mm) is divided in two parts (with the same physical characteristics): the part A is receiving the transverse lux rom two hal-sheets (thickness equal to 0.5 mm) and the part B is acting as a regular lamination. In this way we can compare the losses in the parts A and B. To avoid eddy currents in the two hal-sheets, we considered them as non conductive. Ater careul veriication, we noticed that this assumption does not create any considerable perturbation on the inal results. Such a veriication was done by considering the domain calculated in magnetostatics with J = 0.08 ( A/ mm ). The total magnetic lux is only about 1% higher than the magnetodynamic case, showing that the lamination does not represent any signiicant barrier to it, as one should expect. In this way, the role o the two hal-sheets is to bring the magnetic lux towards the sheet where the eddy currents are created. We calculate two ull cycles and the results were obtained rom the second one to avoid maj numerical transients. With this current density, the problem is linear and the maximal magnetic induction (observed only on the sheet cners) is close to T. The number o elements and nodes are 1600 and As graphical result, we present, a simulated point close to the maximal value o the exciting J, the magnetic induction in Figure 9.

5 1th International IGTE Symposium Proceedings Figure 9: Magnetic lux distribution in the internal cner. F the same simulation point, the eddy current density, represented also by arrows (whose size is proptional to induced density J magnitudes), is presented in the Figures 10a, 10b and 10c. In the igures 9 and 10 we can observe the magnetic lux and the eddy currents distribution. As the irst one is placed as expected, it is interesting to notice that the distribution o eddy currents ollow the proposed hypotheses. It is relatively easy to observe it on the region B. The magnitudes o J in the region A are small, as previewed. Figure 10c: Eddy currents distribution in the regular Region B. In the Figure 11 the domain maximal induction values (in magnitude) and the maximal values o eddy current densities are presented. Both quantities were detected close to the internal cner o the sheets. B [T],5 1,5 1 0,5 Bmax Jin 1,5E+06 1,00E+06 7,50E+05 5,00E+05,50E+05 J [A/mm²] 0 0,00E ,005 0,01 0,015 t [s] 0,0 Figure 11: maximal values o induction and eddy current densities. Figure 10a: Eddy currents distribution. Figure 10b: Eddy currents distribution close to the internal cner. To check this simulation we permed, by two dierent ways, the calculation o the power losses on the part B, where the iron sheet is playing its regular role. In the FE calculation the losses are calculated by the sum o Jeddy P1 = V elem (14) Elem B σ all the FE elements o the conductive iron part. V elem is the element volume. This is a straight result obtained each time step. The values o P 1 are then averaged the whole cycle. The second way is the expression P (15) = σbm ω e VB /4 which comes directly rom equation (5) and where V B is the total volume o the part B. From the FE calculations, B m (maximum value o the magnetic induction on the iron part B) is equal to 1.8 T. Then, the averaged P 1 and P can be compared. They are, respectively, and W. Although the ways o obtaining these quantities are very distinct, the dierence between them does not exceed 4%. This is a very interesting result since it demonstrates that the analytical expression the eddy current losses, given by the equation (5), is quite accurate and, clearly, it can be used with conidence. As the purpose o this wk the most interesting

6 Proceedings th International IGTE Symposium result is presented in the Figure 1, where the total Joule losses are shown the region A (overlapped sheets) and the region B (regular sheets). These curves and losses power density have similar shapes since A and B volumes are identical. P [W],50E-0,00E-0 1,50E-0 1,00E-0 5,00E-03 0,00E ,005 0,01 0,015 0,0 t [s] Figure 1: Eddy current losses over the cycle the regions A and B. Perming the integration o the losses on the cycle and calculating the relationship between the losses on the overlapped sheet (region A) and the regular one (region B), we obtain α c = 0.14 which cresponds to the predictions, iginated rom the analytical approach, that the losses in the overlapped sheets region are smaller than the losses in the regular lamination. V. APPLYING THE RESULTS In this session, we go back to the expression o losses density presented in the equation (). As the main goal o this wk, we will retain its second term, p = B (16) We suppose that the number n represents the amount o regular sheets ( 0 < n < 1), as (1 n ) cresponds to the quantity o overlapped sheets. F our Epstein rame (see Introduction) we have 1.4% o overlapped sheets and then n = = Accding our results, the expression (16) can be divided in two parts, as 1 1 p = n σω e B (1 n) σω e B 4 4 which can be written 1 p = (0.86n+ 0.14) σω e B (17) 4 F our rame, we arrive to 1 p = σω e B 4 showing that the total eddy currents losses are smaller than the classical model prediction. VI. CONCLUSION In this wk we present a study related to the eddy currents on overlapped sheets o Epstein rames. Such devices are widely used to characterize erromagnetic materials. As a matter o act, in the actual devices, the sheet overlapping is not assembled on the same way as on Epstein rames. There is no ree space between the sheets and model it is much me complicate, since airgaps must be deined between the overlapped sheets and between A B the sheets placed on the same level. It is not easy and certainly a matter o investigation. Meover, manuacturing aspects and possible sht-circuits between the sheets increase the diiculty o such a study. The paper [10] treated this subject by using equivalent circuits. Nevertheless, the wk here presented can be a starting point its extension related to magnetic circuits o actual electrical devices. To perm the wk here presented, we used an analytical approximation and rom it we observed that the iron losses on the overlapped sheets are smaller than the losses on regular part o the lamination. A 3D FE simulation was useul on two aspects. The irst one pointed out that, regular lamination, the classical expression eddy current losses is quite precise. The second one quantiied the dierence between the losses in these two distinct areas o the lamination. Finally, we were able to propose a me accurate expression o eddy current losses on the Epstein rame. It can be helpul since it is possible to determine with urther precision the other coeicients present in the total losses expression. REFERENCES [1] J.P.A. Bastos, N. Sadowski, "Electromagnetic Modeling by Finite Element Methods, Marcel Dekker Inc., ISBN: , New Yk, 003, USA. [] N. Ida, J.P.A. Bastos, Electromagnetics and Calculation o Fields, Springer-Verlag, second edition, ISBN: , New Yk, 1997, USA. [3] M. Liwschitz-Garik, C.C. Whipple, Máquinas de Crente Contínua (translated rom Direct Current Machines ), Edições Melhamentos, 1958, Rio de Janeiro, Brazil. [4] Philip Beckley, Electrical Steels,published by the European Electrical Steels, ORB Wks, ISBN: X, 000, U. [5] J.V. Leite, N. Sadowski, P. uo-peng, N.J. Batistela, J.P.A. Bastos, The inverse Jiles-Atherton parameters identiication, IEEE Trans. on Magnetics, Vol 39, Number 3, pp , May 003, USA. [6] N.J. Batistela, F.B.R. Mendes, N. Sadowski, P. uo-peng, J.P.A. Bastos, A strategy iron losses separation, Proceedings in CD, PIERS 004 Progress In Electromagnetics Research Symposium, Pisa, March 004, Italy. [7] C. Simão, N. Sadowski, N.J. Batistela, J.P.A. Bastos, Analysis o magnetic hysteresis loops under sinusoidal and PWM voltage wavems, Proceedings pp , PESC, IEEE 36 th Annual Power Electronics Specialists Conerence, Recie, June 005, Brazil. [8] G. Bertotti, General properties o power losses in sot erromagnetic material, IEEE Trans. on Magnetics, Vol 4, Number 1, January 1988, USA. [9] F. Fiillo, A. Novikov, An improved approach to power losses in magnetic laminations under nonsinusoidal induction wavem, IEEE Trans. on Magnetics, Vol 6, Number 5, November 1990, USA. [10] M. Elleuch, M. Poloujado, New transmer model including joint air gaps and lamination anisotropy, IEEE Trans. on Magnetics, Vol. 34, Number 5, September 1998, USA. [11] J.P.A. Bastos, N. Ida, R.C. Mesquita - "Problem 10: a Solution using Personal Computers", TEAM wkshop, Proceedings pp 63-64, July Aix-Les-Bains, France. [1] J.P.A. Bastos, N.Ida, R.C. Mesquita - "Problem 13: a Solution using Personal Computers", TEAM wkshop, Proceedings pp 65-66, July Aix-Les-Bains, France. [13] J.P.A. Bastos, N. Ida, R.C. Mesquita - "Problem 0: a Solution using Personal Computers", TEAM wkshop, Proceedings pp 71-7, July Aix-Les-Bains, France.