JOURNAL OF APPLIED PHYSICS 97, 10E511 共2005兲
Magnetic shielding properties of sheet metal products taking into account
hysteresis effects
Peter Sergeant,a兲 Luc Dupré, and Lode Vandenbossche
Department of Electrical Energy, Systems and Automation, Ghent University, Sint-Pietersnieuwstraat 41,
B-9000 Gent, Belgium
Marc De Wulf
Arcelor—Ocas, JF. Kennedylaan 3, B-9060 Zelzate, Belgium
共Presented on 11 November 2004; published online 10 May 2005兲
Analytical expressions are presented to find the shielding effectiveness and the losses of a shield
consisting of ferromagnetic, isotropic, nonlinear, and hysteretic material, characterized by the
Preisach distribution function in the Rayleigh region. The nonlinear shield is divided into a sufficient
number of piecewise linear sublayers with a permeability that is constant 共space independent兲 and
complex 共to model hysteresis兲. Simulations of an infinitely long cylindrical shield in transverse
sinusoidal flux show that the shielding of perfectly linear material is better than the one of nonlinear
metal sheets. More hysteresis and nonlinearity deteriorate the shielding factor, as eddy current losses
decrease. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1850380兴
I. INTRODUCTION
In order to design a good magnetic shield for a given
shielding problem, one wants to compare several materials
concerning their shielding effectiveness and electromagnetic
losses. In this article, fast analytical expressions1 are presented for the shielding effectiveness of multilayered shields
consisting of nonlinear material with given Preisach distribution. Simulation results are shown to reveal the influence of
nonlinearity and hysteresis on a cylindrical steel shield with
an experimentally determined Preisach distribution.
In Ref. 2, transfer relations connect electromagnetic values at side ␣ of a layer to the values at the other side ␤.
冋 册冋
H
គ ␣x
Bគ ␣y /␮0
a兲
Electronic mail: peter.sergeant@ugent.be
0021-8979/2005/97共10兲/10E511/3/$22.50
T11 T12
T21 T22
册冋 册
H
គ ␤x
Bគ ␤y /␮0
,
共1兲
where the coefficients Tij, 1 艋 i , j 艋 2 are dependent on the
geometry and the material properties of the shield. Starting
from an initial field H
គ 0, we obtain expressions for the tangential component of H̄ and for the normal component of B̄
共y = 0 on surface ␤兲
H
គx=
II. SHIELDING FACTOR OF LINEAR MATERIALS
In this article, a bar on top of a symbol means a space
vector X̄ and an underlined symbol indicates a time phasor, a
complex representation of a quantity that varies sinusoidally
in time: Xគ = X sin共␻t兲.
គ 0兩
The shielding factor in a point is defined as 兩H
គ s兩 / 兩H
with H
គ s the magnetic field with the shield present and H
គ 0 the
magnetic field in the same point without shield. To calculate
the shielding factor, we use the analytical expressions presented in Ref. 2. We apply them to an infinitely long cylindrical shield in a transverse magnetic field.3 Figure 1共a兲
shows Hoburg’s geometry of the shield that consists of several layers of linear materials. Far from the shield, the magnetic field is the uniform transverse imposed field H
គ 0. Layer
l has thickness dl, conductivity ␴l, and constant permeability
␮l. Its outer surface is labeled ␤ and its inner surface is ␣.
To obtain easier expressions, the problem is converted to
Hoburg’s planar geometry in Fig. 1共b兲. In this “unwrapped”
version of Fig. 1共a兲, the magnetic field source is a sheet of
surface current that varies sinusoidally with wavelength ␭
= 2␲ri = 2␲ / k. This conversion avoids the use of Bessel
functions—see expressions in Ref. 2—and the corresponding
numerical overflow.
=
␩
关A cosh共␩y兲 + Ac sinh共␩y兲兴cos共kx兲,
␮ s
Bគ y = k关As sinh共␩y兲 + Ac cosh共␩y兲兴sin共kx兲,
共2兲
with ␩ = 冑k2 + ␥2 with ␥ = 冑␻␮␴ j. As and Ac are constants
គx
depending on H
គ 0 and the material properties of the layer. H
and Bគ y were chosen because they are continuous in adjacent
material layers.
FIG. 1. 共a兲 A multilayered cylindrical shield. The magnetic field is a uniform
transverse field and 共b兲 a multilayered planar shield. The magnetic field is
imposed by a sinusoidally distributed source.
97, 10E511-1
© 2005 American Institute of Physics
10E511-2
J. Appl. Phys. 97, 10E511 共2005兲
Sergeant et al.
FIG. 2. Hysteresis loops for several hm, calculated 共solid lines兲 and measured 共markers兲. For hm = 35 A / m, the “approximated” loop is shown, obtained by using only the fundamental component ␮I of the Fourier analysis
of b共t兲.
The shielding factor for the whole shield is calculated by
splicing together the transfer relations 共1兲 of all layers as
explained in Ref. 2. As all materials are linear, the shielding
factor does not depend on the imposed field H
គ 0.
FIG. 4. Shielding factor, eddy current 共ec兲 and hysteresis 共hy兲 losses of 1
and 5 mm thick shields. Analytic model 共an兲 is compared with finite element
共FE兲 model for 1 mm thickness. ri = 0.3 m and H
គ 0 = 10 A / m.
2
bdescend关h共t兲兴 = c1h共t兲 + c2hm
−
c2
关h共t兲 − hm兴2 ,
2
共3兲
III. SHIELDING FACTOR OF NONLINEAR HYSTERETIC
MATERIALS
A. Preisach model
In many shielding situations, the magnetic field in 共a
major part of兲 the shield is weak enough to assume that the
working conditions are in the Rayleigh area. For materials
exhibiting hysteresis, b共t兲 is not merely a function of h共t兲,
but of its history as well. Usually, b共t兲 is split into a reversible part brev关h共t兲兴 and an irreversible part birr关h共t兲 , hpast共t兲兴.
To present the latter, the classical Preisach model is used.4 In
this article, we choose brev共t兲 = c1h共t兲 and the Preisach distribution function P共␣ , ␤兲 = c2. Hence, the ascending and descending branch of the hysteresis loop are given by
c2
2
+ 关h共t兲 + hm兴2 ,
bascend关h共t兲兴 = c1h共t兲 − c2hm
2
between the extremal values hm and −hm. The quadratic expressions 共3兲 represent repeatable hysteresis loops in the
Rayleigh area. As in this article only shielding situations
with weak fields in the Rayleigh region are considered, this
choice for brev and P共␣ , ␤兲 is acceptable if we start with
demagnetized shielding material: See the experimental verification in Fig. 2.
For a sinusoidal magnetic field, the hysteresis loop described by Eq. 共3兲 gives rise to a non-sinusoidal time variation of b共t兲. In order to evaluate the shielding efficiency of
the nonlinear hysteretic material, we consider only the fundamental component BI of the nonsinusoidal b共t兲 by using a
Fourier analysis 共Fig. 2兲. BI and the corresponding component ␮I of the permeability are
冋
Bគ I = 共c1 + c2兩H
គ 兩兲 − j
册
4
c2兩H
គ兩 H
គ,
3␲
4
Bគ I
␮I = = 共c1 + c2兩H
គ 兩兲 − j c2兩H
គ 兩 = ␮r + j ␮i .
3␲
H
គ
共4兲
B. Procedure
FIG. 3. Shielding map for several thicknesses of the steel cylinder with
0.3 m radius. The solid line indicates nonlinear material with hysteresis, the
dash-dot line is hypothetic nonlinear material without hysteresis and the
dotted line represents linear material with constant ␮I.
In order to take into account the nonlinear hysteretic
material behavior, the shield is divided into n fictitious linear
sublayers 关Fig. 1共b兲兴. Each fictitious sublayer p has a conគ 共y p兲兴, calculated with Eq. 共4兲. c1
stant but complex ␮ p = ␮I关H
and c2 in Eq. 共4兲 are the same for all sublayers and H
គ 共y p兲 is
the magnetic field at the source side of sublayer p.
J. Appl. Phys. 97, 10E511 共2005兲
Sergeant et al.
10E511-3
␮ must be determined in every sublayer. It is replaced by
␮ and calculated by using Eq. 共4兲, which needs H
គ as input
value. Here, some approximations are unavoidable, because
the analytical algorithm accepts only one ␮I per sublayer
while the real ␮I differs from point to point. 共1兲 The x dependence of H
គ x ⬃ cos共kx兲 and Bគ y ⬃ sin共kx兲—see Eq. 共2兲—is
taken into account by taking the rms value 共divide by 冑2兲.
共2兲 The y dependence is “discretized” by the sublayers, each
having its own ␮Ip, p = 1 , . . . , n. 共3兲 The space vector H̄ in the
x , y plane is taken into account by calculating both
␮I共H
គ x / 冑2兲 and ␮I共H
គ y / 冑2兲 and averaging them.
I
គ are known in the sublayers, the problem
Since ␮ p nor H
has to be solved iteratively. First, ␮Ip is calculated for every
sublayer p = 1 , . . . , n by evaluating Eq. 共4兲 with H
គ 0 as input
argument. After calculating the shielding factor using the calគ in the shield is found from
culated ␮Ip, the distribution of H
គ and the ␮Ip
Eq. 共2兲, and the ␮Ip are updated. Iteratively, the H
are updated until the procedure converges. Contrary to the
linear case, the shielding factor depends on H
គ 0.
I
C. Eddy current losses
With H̄ known from Eq. 共2兲 in each sublayer p, we find
¯ ⫻ H̄. In the cylindrical shield,
the current density J̄ by J̄ = ⵜ
the eddy current losses Pec per meter length in the z direction
are
n
1
Phy =
2 p=1
兺
n
=
兺
p=1
冕 冕
␭
y p+1
dx
0
yp
␻␲ sin共⬔ ␮Ip兲
8k兩␮Ip兩
n
1
Pec =
2 p=1
兺
n
=
冕 冕
␭
y p+1
dx
0
yp
兩 ␥ 兩 4␲
兺 p I2
p=1 8k␴ p兩␮ 兩
p
Jគ 共x,y兲 · Jគ *共x,y兲
dy
␴p
再 冋
*
AspAsp
冋 冉
*
+ 2 Re AspAcp
sinh共2␩rpd p兲
␩rp
−
sinh共2j␩ipd p兲
j␩ip
册
cosh共2␩rpd p兲 − 1
+
cosh共2j␩ipd p兲 − 1
j␩ip
+
sinh共2j␩ipd p兲
j␩ip
册冎
冊册
␩rp
冋
*
+ AcpAcp
sinh共2␩rpd p兲
␩rp
共5兲
.
Here, the asterisk symbolizes the complex conjugate, ␩rp is
the real part of ␩ in sublayer p while ␩ip is the imaginary
part. Every shield sublayer p ranges in the y direction from
y p to y p+1, having thickness d p. For every shield sublayer, Asp
and Acp can be calculated by Eq. 共2兲, as H
គ x and Bគ y are known
once Sec. III B is executed.
D. Hysteresis losses
The power dissipated by hysteresis per meter length is
Re关H
គ 共x,y兲 · j␻Bគ I*共x,y兲兴dy
再 冋
*
2 Re AspAcp
共k2 − 兩␩ p兩2兲
+ 共兩kAsp兩2 + 兩␩ pAcp兩2兲
冋
sinh共2␩rpd p兲
␩rp
−
cosh共2j␩ipd p兲 − 1
cosh共2␩rpd p兲 − 1
*
+ AspAcp
共k2 + 兩␩ p兩2兲
j␩ip
␩rp
册
冋
册
sinh共2j␩ipd p兲
sinh共2␩rpd p兲 sinh共2j␩ipd p兲
+ 共兩␩ pAsp兩2 + 兩kAcp兩2兲
+
j␩ip
␩rp
j␩ip
IV. SIMULATION RESULTS
Figure 3 shows the shielding factor of a cylinder in steel
with ri = 0.3 m radius, ␴ = 8.5⫻ 106 / ⍀m and complex permeability in the Rayleigh area 共4兲 with c1 = 168.3␮0 and c2
= 18.4␮0 共values of Magnetil–Arcelor兲. The imposed source
field H
គ 0 is at a large distance from the shield 10 A / m. This
low field ensures that the magnetization is in the Rayleigh
គ 兲, the
area. By neglecting the imaginary part of ␮I = ␮r共H
shielding factor is slightly worse. Choosing a constant ␮I
= ␮r共H
គ 0兲 in Eq. 共4兲 results in an overestimation of the shielding efficiency.
Figure 4 shows the shielding factor and the total losses
គ 0, and
共5兲 + 共6兲 for a 1 mm thick cylinder with the same ri, H
material properties as in Fig. 3. In order to evaluate this
analytical approach, the results for the presented model are
compared with CPU time consuming finite element 共FE兲 cal-
册冎
·
共6兲
culations. The correspondence with the FE calculations is
acceptable up to 10 kHz. Here, the FE mesh becomes too
coarse to accurately model the small skin depth.
When varying the parameters c1 and c2 in Eq. 共4兲 such
គ 0兲 is constant, more hysteresis and nonlinearity 共c1
that ␮r共H
lower and c2 higher兲 in the Rayleigh area deteriorate the
shielding factor, but cause less total electromagnetic losses
共increase of Phy and decrease of Pec兲. The reason is that with
c1 lower and c2 higher, 兩␮I兩 decreases faster inside the shield.
1
R. B. Schulz, V. C. Plantz, and D. R. Brush, IEEE Trans. Electromagn.
Compat. 30, 187 共1988兲.
2
J. F. Hoburg, IEEE Trans. Electromagn. Compat. 38, 92 共1996兲.
3
M. Öktem and B. Saka, IEEE Trans. Electromagn. Compat. 43, 170
共2001兲.
4
I. D. Mayergoyz, Mathematical Models of Hysteresis 共Springer-Verlag,
Berlin, 1991兲.