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Proceedings

Eddy current losses on Epstein frame overlapped corner sheets J.P.A. Bastos1, N.J. Batistela1, N. Sadowski1 and M. Lajoie-Mazenc2 1

GRUCAD/EEL/CTC, Universidade Federal de Santa Catarina, C.P. 476, Florianópolis, SC, Brazil, 88040-900 2 LEEI de Toulouse, 2, rue Camichel, 31071, Toulouse Cedex, France E-mail: [email protected]

Abstract: Iron losses are commonly divided in eddy currents, hysteresis and anomalous losses. While much work has been accomplished on the two last ones, in this paper we are interested on eddy current losses. Epstein frame devices have overlapped sheets lamination on the corner parts of the iron yoke. In this work we will examine the losses on these regions using a simplified analytical model. Some numerical calculations will be also presented for quantifying such losses. Keywords: lamination, eddy current losses, analytical approach.

I. INTRODUCTION Nowadays the losses have been the object of much attention since electrical devices must be competitive and norms do not accept low efficiency machines. Lately, much effort has been expended to model the hysteretic behavior of soft 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 efficiency of electrical devices plays an important role on saving energy and local or international norms often avoid the commercialization of poorly designed equipment. Therefore the precise modeling of material losses must be obtained by different losses evaluation structures, as the widely used Epstein frame. Because the losses are classically divided in three categories, if the modeling of one type is improved, the whole model will be more accurate. That will allow an efficient characterization of a specific material for its applications, as using it on electrical machine or transformer yokes. The aim of this work is to present a better interpretation of eddy currents losses. The behavior of eddy currents on overlapped sheets present in the corners of the Epstein frame have been often disregarded, observing that the overlapped sheets represent 21.4% of the total iron volume. In this work 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 briefly discuss the model used for the whole losses on the Epstein frame, indicating the difficulties to set the losses parameters. II.

INITIAL CONSIDERATIONS

pt = ph + p f + pe or

α

(1)

pt = K h B + K f B + K e B 2

1.5

(2)

where α , K h , K f and K e have to be determined to describe as well as possible the behavior of a particular material as function of B, the magnitude of magnetic induction applied on it. Below, we will shortly comment the three terms of the right hand side of the equation (2). - the hysteresis power density:

It is given by

ph = K h Bα

(3)

which is the Steinmetz equation [1]. The coefficient K h and the exponent α depend on the material. Using the Epstein frame, a very low frequency magnetic field excitation (typically 1 Hz or lower) is applied to the material and the two variables above are determined by experimental points fitting. 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 of the hysteresis loop surface does not change at the same proportion. Therefore this exponent must vary and, slowly, should decrease. In other words, α can be described as a function of B. Some works [3,4] propose different strategies to model accurately the hysteresis losses. In our previous works [5,6,7], this procedure was not applied yet, and it must be focused in the future. - the eddy current losses power density:

This term is related to the main part of this work. Therefore for didactical purposes we will briefly present its calculation, even though it is well known [1]. Let us consider a single sheet of the Figure 1a and the placement of the required physical quantities in the Figures 1a, 1b and 1c.

Figure 1a: a single sheet of dimensions

lx , l y

and thickness e.

Commonly the power losses are divided in Ph (hysteresis), Pf (eddy currents) and Pe (anomalous or excedent) losses. Let us work with the power volume densities denoted ph , p f and pe , respectively. The following equation gives the total losses density as:

Figure 1b: cross section of a single sheet.

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physical structure of the material. According to some authors [8,9], these losses are originated by the movement of Weiss domain walls when the polarity of the exciting field changes. Bertotti’s work [8] proposes a B exponent equal to 1.5. III. CORNER SHEETS: AN ANALITYCAL APPROACH

In Figures 2a and 2b we can see pictures of the Epstein frame.

Figure 1c: the physical quantities.

Because the sheet thickness e is much smaller than the dimensions l x and l y we consider only the components y of J, the induced current density. E is the electric field and applying rot E = − ∂B ∂t we have: ⎡i ⎢ det ⎢ ∂ x ⎢0 ⎣

k ⎤ ∂B ⎥ ∂z ⎥ = − i ∂t ⎥ 0 ⎦ E ∂E ∂B = or ∂z ∂t Taking into account that at z = 0 (at the middle of the sheet) E and J must be zero for continuity, we arrive to ∂B E ( z, t ) = z ∂t The Joule power loss on a volume is Pf = ∫ σ E 2 dv (4) j ∂y

V

where σ is the electric conductivity; applying this expression for the sheet: + e / 2 lx l y ∂B σ ( )2 z 2 dx dy dz Pf = ∫ − e / 2 ∫0 ∫0 ∂t Pf =

σ ∂B (

12 ∂t

) 2 l x l y e3 =

Figure 2a: Epstein frame

Figure 2b: iron corner sheets.

For a simplified model we will consider that a sheet magnetic flux is divided in two parts, and each one is conducted to an adjacent sheet, as shown in Figure 3a.

σ ∂B

( ) 2 (lx l y e) e 2 12 ∂t

Observing that (lx l y e) is the sheet volume, in a sinusoidal regime of pulsation ω , we have Pf σ = ( Bmω cos ω t ) 2 e 2 lx l y e 12

Figure 3a: Iron sheets at the corner.

And finally, reminding that the average value of cos 2 ω t is 1/2, the expression of average losses power volume density due to eddy currents ( p f ), is: pf =

1 σ Bm 2ω 2 e 2 24

(5)

As mentioned before, this type of losses will be the subject of additional investigation on this work; it will be treated soon. - the anomalous losses power density:

It is given by

pe = K e B1.5 (6) The existence of this equation is based on the fact that the sum of eddy current and hysteresis losses do not correspond to the total losses. It is still a complex subject of research since it needs the consideration of the

Figure 3b: sheets model.

Therefore our model will take into account two halfsheets, as seen in the Figure 3b. This structure is replicated indefinitely. In this figure, for the sheet on the bottom, it is clear that up to the plane A, the flux follows 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) of average losses power volume density due to eddy currents ( p f ) can be applied. The magnetic flux crosses the planes A and B through

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a transversal surface equal to e L / 2 . This magnetic flux necessarily passes through the section between the sheets, whose surface is L2 . Our simplified model considers that the flux is equally distributed on this large interface. As the flux is the same on these two surfaces, we define the “transverse” induction Bt as: B eL / 2 = Bt L2

and Bt = B e / 2 L

(7)

With the above approximations, the corresponding physical situation is shown in the Figure 4 where we can see the induced current density J in the sheet and the term ∂Bt ∂t .

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dBt r (8) dt 2 E depends on r and t. This expression can be also obtained by using the Faraday’s law on its local form. With cylindrical coordinates, we observe that E has only a θ component and it depends only on r. In the other hand, − ∂B ∂t is placed in the z direction and depends E=

only on t. With vector magnitudes, rot E = − ∂B ∂t becomes 1 ∂ ∂B ( rE ) = ∂t r ∂r The above expression is verified with the equation (8). Here one more phenomenon must be considered. It is related to the fact that in a whole sheet the magnetic flux arrives from the two adjacent ones, as seen in the Figures 3 and 6.

Figure 6: magnetic flux entering in the whole sheet.

Figure 4: induced current density in the sheet.

Close to the middle of the sheet, the current loops have circular forms but near its limits, they tend to a square shape. We will consider the circular shape of the loops in a circle with the radius R ≅ 0.56 L , which brings us to similar surfaces ( π R 2 = L2 ). Before proceeding with the calculation, we point out that a similar algebra was performed for square shaped loops and the final results do not differ much from those presented below. In the Figure 5 we apply rot E = − ∂B ∂t in integral form for a circle of a generic radius r, where 0 < r ≤ R .

Observing the magnetic flux variation, the orientations of the current loops must be opposite. Because in the middle of 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 of J with z. It is obviously similar to E = J / σ . Therefore, the equation (8) above is modified to ∂B r z E= t (9) ∂t 2 ( e / 2 ) which is now a function of r, z and t. Applying the equation (4) to the half-sheet, we have: z ⎛ ∂Bt ⎞ r r dr dθ dz ⎟ 2 0 ∫0 ∫0 ⎝ ∂t ⎠ 4 (e / 2) Proceeding with the calculations, we obtain ∂B π P = σ e R 4 ( t )2 ∂t 24 Using R ≅ 0.56 L , ∂Bt ∂t = Bt ω cos ωt and also the P=∫

R

2π

e/2

2

σ⎜

2

2

equation (7) Bt = B e / 2 L , we obtain the following expression

P≅

Figure 5: Applying Faraday’s law.

It gives

960

σ B 2ω 2 e3 L2 cos 2 ωt

or

v∫

L(S )

∂B ⋅ ds S ∂t

E ⋅ dl = − ∫

and

E 2π r = or

π

dBt 2 πr dt

P=

π 480

e 2

σ B 2ω 2 ( L2 ) e2 cos 2 ωt

Considering the average value of cos 2 ωt = 1/ 2 and the volume of half-sheet equal to e L2 / 2 , the average value of the losses power volume density is:

pt =

π

960

σ B 2ω 2 e2

(10)

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or, for comparing with the equation (5), 1 pt = 0.078 σ B 2ω 2 e 2 24

(11)

It is daring to state that the factor 0.078, appearing in the above equation, is exact or very close to it, since, among other approximations, the flux is concentrated around the internal iron corner. Nevertheless, it shows that the losses due to the transverse flux are much smaller than the regular lamination eddy current losses. And there is one more aspect to be considered and it is explained with the help of the Figure 7.

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problems [11,12,13]. We apply the solver using hexahedral edge elements and the ungauged vector potential A formulation, which is well know by its good accuracy as well as its fragility to handle systems with a very large number of unknowns. This last inconvenience does not represent a major trouble here since the domain is simple. Simple but tricky. Our first attempt was to model a domain with two half-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 half-sheets are present, only half a loop can exist. Thus, we opted for the domain shown in Figure 8.

Figure 7: the two magnetic fluxes division.

The sheet on the bottom of the figure has two points to be carefully observed. At the point x = 0 there is no regular lamination flux Φ ; at the point x = L and beyond ( x > L ), only this flux exists. As for the transverse flux

Φ t , at the point x = 0 only this flux is

present while at x = L and beyond it is zero. All in all, it is reasonable to consider that from 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 defined by the equation (5). It is again difficult, based on approximated calculations, to determine how these losses equations are divided in this region. Indeed, since the fields can not be calculated analytically we will perform in the next section some FE analyses to obtain a reasonable approach for losses. We can foresee that the density losses p ft on the overlapped sheets region can be calculated by a expression as p ft = α c

1 σ B 2ω 2 e 2 24

(12)

where

αc < 1

(13) This result seems somewhat strange but, physically, it can be understood by the following: the magnetic induction Bt is much smaller than the longitudinal induction B. Since the magnitude of the induced currents depends on Bt2 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 order to estimate the value of α c .

IV. 3D FE MODELING In order to calculate the magnetic field and the eddy current losses, we used our package FEECAD [2] whose reliability have been demonstrated by several applications, including the solution of TEAM Workshop

Figure 8: the two half sheets domain.

In this figure we have an exciting bar (non conductive for eddy current purposes) where a current density J = 0.08 sin ωt ( A / mm2 ) is applied on the vertical direction. The frequency is 60 Hz and there are no airgaps. A full 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 flux from two half-sheets (thickness equal to 0.25 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 half-sheets, we considered them as non conductive. After careful verification, we noticed that this assumption does not create any considerable perturbation on the final results. Such a verification was done by considering the domain calculated in magnetostatics with J = 0.08 ( A / mm2 ) . The total magnetic flux is only about 1% higher than the magnetodynamic case, showing that the lamination does not represent any significant barrier to it, as one should expect. In this way, the role of the two half-sheets is to “bring” the magnetic flux towards the sheet where the eddy currents are created. We calculate two full cycles and the results were obtained from the second one to avoid major numerical transients. With this current density, the problem is linear and the maximal magnetic induction (observed only on the sheet corners) is close to 2 T. The number of elements and nodes are 12600 and 14415. As graphical result, we present, for a simulated point close to the maximal value of the exciting J, the magnetic induction in Figure 9.

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For the same simulation point, the eddy current density, represented also by arrows (whose size is proportional to induced density J magnitudes), is presented in the Figures 10a, 10b and 10c. In the figures 9 and 10 we can observe the magnetic flux and the eddy currents distribution. As the first one is placed as expected, it is interesting to notice that the distribution of eddy currents follow the proposed hypotheses. It is relatively easy to observe it on the region B. The magnitudes of 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 of eddy current densities are presented. Both quantities were detected close to the internal corner of the sheets. 1,25E+06

2,5

1,00E+06

2 Bmax

1,5

7,50E+05

Jin

1

5,00E+05

0,5

2,50E+05

0 0

0,005

0,01

0,015

t [s]

J [A/mm²]

Figure 9: Magnetic flux distribution in the internal corner.

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B [T]

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0,00E+00 0,02

Figure 11: maximal values of induction and eddy current densities.

To check this simulation we performed, by two different ways, the calculation of 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 of 2 J eddy (14) Pf 1 = ∑ Velem Elem B

Figure 10a: Eddy currents distribution.

σ

for all the FE elements of the conductive iron part. Velem is the element volume. This is a straight result obtained for each time step. The values of Pf 1 are then averaged for the whole cycle. The second way is the expression 2 (15) Pf 2 = σ Bm ω 2 e 2VB / 24 which comes directly from equation (5) and where VB is the total volume of the part B. From the FE calculations, Bm (maximum value of the magnetic induction on the iron part B) is equal to 1.28 T. Then, the averaged Pf 1 and

Figure 10b: Eddy currents distribution close to the internal corner.

Pf 2 can be compared.

They are, respectively,

1.05 × 10−2 and 1.09 ×10−2 W. Although the ways of obtaining these quantities are very distinct, the difference between them does not exceed 4%. This is a very interesting result since it demonstrates that the analytical expression for the eddy current losses, given by the equation (5), is quite accurate and, clearly, it can be used with confidence. As for the purpose of this work the most interesting

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result is presented in the Figure 12, where the total Joule losses are shown for 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. 2,50E-02

P [W]

2,00E-02 1,50E-02 A

1,00E-02

B

5,00E-03 0,00E+00 0

0,005

0,01

0,015

t [s]

0,02

Figure 12: Eddy current losses over the cycle for the regions A and B.

Performing the integration of 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 corresponds to the predictions, originated from 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 of losses density presented in the equation (2). As for the main goal of this work, we will retain its second term, (16) p f = K f B2 We suppose that the number n represents the amount of regular sheets ( 0 < n < 1 ), as ( 1 − n ) corresponds to the quantity of overlapped sheets. For our Epstein frame (see Introduction) we have 21.4% of overlapped sheets and then n = 1 − 0.214 = 0.786 . According our results, the expression (16) can be divided in two parts, as 1 1 p f = n σ ω 2 e 2 B 2 + 0.14 (1 − n) σ ω 2 e 2 B 2 24 24 which can be written 1 (17) p f = (0.86 n + 0.14 ) σ ω 2 e 2 B 2 24 For our frame, we arrive to 1 p f = 0.816 σ ω 2 e 2 B 2 24 showing that the total eddy currents losses are smaller than the classical model prediction.

VI.

the sheets placed on the same level. It is not easy and certainly a matter of investigation. Moreover, manufacturing aspects and possible short-circuits between the sheets increase the difficulty of such a study. The paper [10] treated this subject by using equivalent circuits. Nevertheless, the work here presented can be a starting point for its extension related to magnetic circuits of actual electrical devices. To perform the work here presented, we used an analytical approximation and from it we observed that the iron losses on the overlapped sheets are smaller than the losses on regular part of the lamination. A 3D FE simulation was useful on two aspects. The first one pointed out that, for regular lamination, the classical expression for eddy current losses is quite precise. The second one quantified the difference between the losses in these two distinct areas of the lamination. Finally, we were able to propose a more accurate expression of eddy current losses on the Epstein frame. It can be helpful since it is possible to determine with further precision the other coefficients present in the total losses expression.

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

CONCLUSION

In this work we present a study related to the eddy currents on overlapped sheets of Epstein frames. Such devices are widely used to characterize ferromagnetic materials. As a matter of fact, in the actual devices, the sheet overlapping is not assembled on the same way as on Epstein frames. There is no free space between the sheets and model it is much more complicate, since airgaps must be defined between the overlapped sheets and between

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[11]

[12]

[13]

J.P.A. Bastos, N. Sadowski, "Electromagnetic Modeling by Finite Element Methods”, Marcel Dekker Inc., ISBN: 0-82474269-9, New York, 2003, USA. N. Ida, J.P.A. Bastos, “Electromagnetics and Calculation of Fields”, Springer-Verlag, second edition, ISBN: 0-387-948775, New York, 1997, USA. M. Liwschitz-Garik, C.C. Whipple, “Máquinas de Corrente Contínua” (translated from “Direct Current Machines”), Edições Melhoramentos, 1958, Rio de Janeiro, Brazil. Philip Beckley, “Electrical Steels”,published by the European Electrical Steels, ORB Works, ISBN: 0-9540039-0-X, 2000, UK. J.V. Leite, N. Sadowski, P. Kuo-Peng, N.J. Batistela, J.P.A. Bastos, “The inverse Jiles-Atherton parameters identification”, IEEE Trans. on Magnetics, Vol 39, Number 3, pp 1397-1400, May 2003, USA. N.J. Batistela, F.B.R. Mendes, N. Sadowski, P. Kuo-Peng, J.P.A. Bastos, “A strategy for iron losses separation”, Proceedings in CD, PIERS 2004 – Progress In Electromagnetics Research Symposium, Pisa, March 2004, Italy. C. Simão, N. Sadowski, N.J. Batistela, J.P.A. Bastos, “Analysis of magnetic hysteresis loops under sinusoidal and PWM voltage waveforms”, Proceedings pp 1555-1559, PESC, IEEE 36th Annual Power Electronics Specialists Conference, Recife, June 2005, Brazil. G. Bertotti, “General properties of power losses in soft ferromagnetic material”, IEEE Trans. on Magnetics, Vol 24, Number 1, January 1988, USA. F. Fiorillo, A. Novikov, “An improved approach to power losses in magnetic laminations under nonsinusoidal induction waveform”, IEEE Trans. on Magnetics, Vol 26, Number 5, November 1990, USA. M. Elleuch, M. Poloujadoff, “New transformer model including joint air gaps and lamination anisotropy”, IEEE Trans. on Magnetics, Vol. 34, Number 5, September 1998, USA. J.P.A. Bastos, N. Ida , R.C. Mesquita - "Problem 10: a Solution using Personal Computers" , TEAM workshop, Proceedings pp 63-64, July 1994 - Aix-Les-Bains, France. J.P.A. Bastos, N.Ida, R.C. Mesquita - "Problem 13: a Solution using Personal Computers", TEAM workshop, Proceedings pp 65-66, July 1994 - Aix-Les-Bains, France. J.P.A. Bastos, N. Ida , R.C. Mesquita - "Problem 20: a Solution using Personal Computers", TEAM workshop, Proceedings pp 71-72, July 1994 - Aix-Les-Bains, France.