# Identification of the Equivalent Permeability of Step

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&deg;
90&deg;
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 W&micro;i stored in each volume
Vi of the layer i
W&micro;i =
(
r r
Vlayi Bi H i
r
ν B, W&micro; min
2
)
=
(
W&micro; 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
= ∑ W&micro;i &micro;i , 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
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&deg; 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&deg;
0.6
2y;1c;1g 30&deg;
0.4
y; ; g
2y;1c;1g 60&deg;
0.2
2y;1c;1g 90&deg;
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&times;δy&times;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&deg; 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&deg;
ωt 30
ωt=30&deg;
The repartition of the flux density (left) and eddy currents
(right) in the overlapping area
ωt=60&deg;
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 &deg;.
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