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Identification of the Equivalent Permeability of Step-Lap Joints of Transformer E.Napieralska Juszczak Universite Artois Artois, France COMPUTATION OF THE EQUIVALENT CHARACTERISTICS OF LAMINATED ANISOTROPIC MAGNETIC CORES Presentation of the structure Layer 1 Layer 2 Layer 3 Layer 4 Air gap 0° 90° Method of establishing equivalent characteristics for various structures z z z Using the homogenization technique makes it possible to replace real 3D structures by simpler homogeneous 2D. In the homogenization technique technique, two concepts are of importance: macrostructure and microstructure. In the considered problem, the macrostructure is a complete assembly of layers in the overlap overlap, while the microstructure is a repeatable structure of two or more layers made of sheets with the different rolling direction direction. The relation between the flux density y vector in the repetitive structures r B= 1 dx ⋅ dy ⋅ H n ∑ i =1 r B i ⋅ dx ⋅ dy ⋅ hi The calculation of components Bix and Biy is based on the total flux penetrating through the limiting surfaces of the volume V. The calculation is based on the total flux penetrating through the limiting surfaces of the volume V. V Bix = φ ’’y φ ’x φ ’’x φ ’y axis x Biy = φ +φ ' ix '' ix 2 ⋅ dyy ⋅ hi φ +φ ' iy '' iy 2 ⋅ dx ⋅ hi The total energy of the macrostructure results from the sum of the magnetic energy Wµi stored in each volume Vi of the layer i Wµi = ( r r Vlayi Bi H i r ν B, Wµ min 2 ) = ( Wµ min Vlayi B ν B , α i 2 i i 2 i r2 2 2 n Feν Fe B Fe + n0ν 0 B0 = 2 nB ) ( r = ∑ Wµi µi , Bi The unknowns are flux densities vectors Bi in layers and the angles ) The starting values for B1x, B1y, B2x and B2y follow an initial assumption that magnetic flux goes entirely p g g y through the laminations avoiding the gaps. Moreover, for small values of induction the flux goes mainly along g y g the rolling direction, whereas for larger values The minimization task was solved using Hook‐Jeeves' method. To calculate the equivalent curve B-H the value of flux density B was changed in the interval 0-Bsat A single point of the curve B B-H H was determined accordmg to the following algorithm: 1) To calculate B1,B2,alfa1,alfa2 B1 B2 alfa1 alfa2 from the minimization task 2) To calculate ν 1(B1, α1) , ν 2(B 2, α 2) 3) To calculate the homogenized reluctivity [ ( ) ) ( ) r ν 1 B12 , α1 ⋅ B12 n1 +ν 2 B22 , α 2 ⋅ B22 n2 ν B, Wμ min = (n1 + n2 ) B 2 ( 4)) To calculate field intensity. y ] min 2D model of the core corner 1 2 3 1 2 5 6 7 9 10 11 5 6 5 3 4 7 10 11 12 14 13 8 3 2 1 7 6 5 5 6 7 Family of characteristic B-H for the directions 0° for the 4 layers . Equivalent static B/H curves for the air-gap (2y;1c;1g) at four different anisotropy angles 1.6 14 1.4 1.2 B [T] 1 0.8 2y;1c;1g 0° 0.6 2y;1c;1g 30° 0.4 y; ; g 2y;1c;1g 60° 0.2 2y;1c;1g 90° 0 0 500 1000 H [A/m] 1500 A two-turn search two-turn The experimental p coil for setA up p search coil for measuring magnetic flux along y axis 1 measuring magnetic flux along x axis 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 y 0 x Layer 1 Measured characteristics for the whole structure (before and after drilling of the holes and installation of the search coils). 1.5 1.12 1 1 11 1.11 B (T) 1.1 1.09 1 08 1.08 1.07 60 0.5 65 70 75 80 85 original lamination (no holes) after drilling of the holes with search coils in position 0 0 200 400 600 H (A/m) 800 1000 1200 3D simulation Eddy current distributions in the x−y cross-section of the lamination Distribution of Bx along g path p 1 0.40 0.35 0.30 140 0.20 3D 130 2D 110 measured path 1 90 y [mm] Bx[T] 0.25 0.15 path 2 70 50 0.10 30 0.05 1T 10 0 0.00 0 20 40 60 80 x[mm] 100 120 140 160 1T 10 30 50 71 91 x [mm] 111 131 151 Distribution of Bx along g path p 2 0.4 140 0.35 130 110 0.3 y [mm] 0.25 Bx[T] path 1 90 path 2 70 50 3D 2D measured 0.2 30 0 0.15 0.1 0 05 0.05 0 0 20 40 60 80 x[mm] 100 120 1T 10 140 160 1T 10 30 50 71 91 x [mm] 111 131 151 Distribution of By y along gp path 1 0.4 0.35 3D 2D measured 0.3 0.25 140 130 110 y [mm] By[T] path 1 90 0.2 path 2 70 50 30 0.15 1T 10 0 0.1 0.05 0 0 20 40 60 80 -0 0.05 05 x[mm] 100 120 140 160 1T 10 30 50 71 91 x [mm] 111 131 151 Distribution of By y along gp path 2 0.5 0.45 140 130 0.4 110 0.35 B By[T] 0.25 y [mm] 3D 2D measured 0.3 path 2 70 50 30 1T 10 0.2 0 0.15 0.1 0.05 0 -0.05 0 05 path 1 90 0 20 40 60 80 x[mm] 100 120 140 160 1T 10 30 50 71 91 x [mm] 111 131 151 Fluxes in three consecutive layers Φ1x Φ1y z x y Φ2y Φ3y Φ1sxy Φ′z Φ2x Φ′2y Φ3x Φ3sxy Φz d1 (conductor) Φ′2x d2 (air gap + insulations) d3 ((conductor) d t ) The flux density components in a sample of dimensions δx×δy×δz may be found as: B ix 1φ +φ = ; 2 δy δz B iy 1φ +φ = ; 2 δx δz B iz 1φ +φ = ; 2 δy δx ' ix ' iy ' iz '' ix '' iy '' iz Models geometry model with two laminations the 90° step‐lap joint Model 1 A series i off simulations i l ti were executed: t d z z z z z For excitations with the flux 42e-6 42e 6 Wb (excitation 2) and 35e-6 35e 6 Wb (excitation 1) the repartitions of the flux density and of the eddy currents were simulated. The real conductivity was taken into account. F the For th excitation it ti with ith th the flflux 42 42e-6 6 Wb and d 35 35e-6 6 Wb th the simulations neglecting the eddy currents were executed. For excitations with the flux 42e-6 Wb and 35e-6WB the anisotropic conductivity was applied applied. Zero conductivity in the normal direction. It permits the analysis of the eddy currents inducted by the normal flux. For excitations with the flux 42e-6 Wb and 35e-6WB the anisotropic i t i conductivity d ti it was applied. li d Z Zero conductivity d ti it iin th the tangential direction. It permits the analysis of the eddy currents inducted by the longitudinal flux. The same simulations were executed for isotropic materials (ThyssenKrupp Electrical Steel Company NO 0.35 mm). Variation of Bx along x in the area of the lamination ((excitation 1and 2)) 1.8 Bx [T] without eddy currents 1.6 with eddy currents 1.4 1.2 1 without eddy currents with eddy currents 0.8 0.6 0.4 0.2 0 x [mm] 80 -0.2 85 90 95 100 105 110 Variation of Bz along x in the area of the lamination (excitation 35e-6 Wb and 42e-6 Wb. 0.16 Bz [T] without eddy currents 0.14 0.12 0.1 with eddy t currents 0.08 0.06 with eddy currents 0.04 0.02 x [mm] 0 80 85 90 95 100 105 110 Variation of Bx along z in the area of the lamination 0.20 Bz [T] 0.18 0 16 0.16 0.14 without eddy currents 0.12 with ith eddy edd currents c rrents 0.10 0.08 0.06 0.04 z [mm] 0.02 -0.05 0.00 0 00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 The repartition of the flux density (left) and eddy currents (right) in the overlapping area ωt=0° ωt 30 ωt=30° The repartition of the flux density (left) and eddy currents (right) in the overlapping area ωt=60° z The repartition of the flux density vectors (left) and eddy currents (right) for the different time instants make it possible to analyze the comportment of the phenomena’s in the overlapping area. All the flux passes there from one lamination to another. This results in a significant normal flux in addition to the longitudinal flux flux. Both cause the eddy currents in the x-y and y-z plans. z The inducted eddy currents are sources of own fluxes. The result is the existence of the flux density components in the y direction and y currents in x direction. Before the overlapping pp g in the lamination eddy only the longitudinal flux exists. Inside the overlapping area longitudinal flux decreases and the normal increases or eddy currents will decrease in the y-z plane and increase in the x-y plane. This results in an excitation of the significant x component of the eddy currents and of the turning of the flux in y direction (Fig. 9). The simulations with the assumption of zero conductivity values in x y in z direction (fig ( g 9)) confirms direction those with zero conductivity the presented conclusions. Eddy current is shifted in phase relative to the flux density of a little less than 90 °. Conclusions z E istence of both normal and longit Existence longitudinal dinal flfluxes es z should be taken into account in the regions where the flux passes from one lamination to another. The eddy y currents induced by y the fluxes modifyy the field distribution in the structure and should be taken into account. The internal air air-gaps gaps greater than 0.1 0 1 mm have a powerful impact on the field distribution, the insulation between the laminations of 0.01mm have a negligible effect ff on the h passing i off the h flflux. z z z z The normal flux p passing g of the one sheet to another does not change direction immediately after the entrance to the lamination, the transition is progressively. z The main purpose of these 3D simulations was to prepare a mathematical model that makes possible calculation of eddy currents distribution in thin ferromagnetic sheet with given flux. The estimation off eddy dd currents di distribution ib i and d their h i iinfluence fl on electromagnetic field distribution, for both the longitudinal and normal fluxes fluxes, have been done done. z We remark that the normal and longitudinal fluxes induce eddy currents. The induced eddy currents modify the field distribution in the structure and should be taken into account account. In the prepared model z existence of both normal and longitudinal fluxes should be taken into the consideration. The analysis y of the step-lap p p structure ((model 2)) permits to conclude that the internal air-gaps higher than 0.1mm have an important influence on the field distribution On the contrary the isolation between distribution. the laminations of 0.01mm has a negligible effect and can be omitted in the mathematical model. z z Very iimportant ffact was concluded, V l d d the h di direction i of the normal flux from one sheet to another one d does nott change h iimmediately di t l after ft th the entrance t z of the lamination, and the transition is progressive. p g Therefore is not p possible to introduce a simple boundary condition stated that flux turns immediatelyy after entering g into sheet. To take into account the eddy currents the superposition of 2D and 1D problem is needed z