IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002
1467
Comprehensive Model of Magnetization Curve,
Hysteresis Loops, and Losses in Any Direction
in Grain-Oriented Fe–Si
F. Fiorillo, L. R. Dupré, Member, IEEE, C. Appino, and A. M. Rietto
Abstract—We report an investigation and theoretical assessment
of the DC magnetic properties of high-permeability grain-oriented
(GO) Fe–Si laminations under variously directed applied fields. We
verified that normal magnetization curves, hysteresis loops, and
energy losses depend on the field direction according to the sample
geometry. This is explainable in terms of specific 180 and 90
domain wall processes and magnetization rotations. We present
a novel phenomenological theory of the magnetization curves and
hysteresis losses in GO laminations, excited along a generic direction; the theory is based on the single crystal approximation and
pre-emptive knowledge of the magnetic behavior of the material
along the rolling (RD) and the transverse (TD) directions. This
approach is consistent with the general structure of Néel’s phase
theory, with the additional consideration of hysteresis and losses.
Epstein and cross-stacked sheet testing methods are the two base
measuring configurations; all the other testing geometries (single
sheet, disk, square) are expected to display intermediate behavior.
The devised model provides, through a direct procedure, thorough
and accurate prediction of magnetization curves and quasi-static
losses in these two basic cases. Its application to the other geometries is equally possible, with only a limited amount of supplementary information.
Index Terms—Iron alloys, magnetic domains, magnetic
hysteresis, magnetic losses, magnetic variables measurement,
magnetization processes.
I. INTRODUCTION
T
HE PROPERTIES of grain-oriented (GO) Fe–Si laminations magnetized along directions different from the
rolling direction need sometimes to be considered in applications. Their knowledge is useful, for instance, when the
calculation of fields and fluxes in joints and corners is required
for the accurate design of transformer cores [1], [2] or when
considering the use of GO laminations in large rotating machines [3]. But such practical circumstances have a connection
with the classical problem of measuring and predicting the
magnetization curve in a single crystal, the subject of celebrated
experiments [4], [5], theoretically assessed by Néel through
the so-called “phase theory” [6]. Néel’s theory, focused on the
magnetization curve of a zero coercivity material (anhysteretic
curve), puts in the right perspective the role of the internal
Manuscript received March 4, 2001; revised October 22, 2001. This work was
partly carried out during a scientific stay of L. R. Dupré at IEN Galileo Ferraris,
supported by FWO-Vlaanderen.
F. Fiorillo, C. Appino, and A. M. Rietto are with the Istituto Elettrotecnico
Nazionale Galileo Ferraris, I-10135 Torino, Italy (e-mail: fiorillo@ien.it).
L. R. Dupré is with the Department of Electrical Engineering, University of
Gent, B-9000 Gent, Belgium.
Publisher Item Identifier S 0018-9464(02)03634-8.
field
, the vectorial sum of applied field
and demagnetizing field
. The domain structure (i.e., the
in a
balance between the different phases) evolves with
way depending on the shape of the crystal and its arrangement
in the magnetic circuit. When talking of intrinsic magnetic
behavior, one should therefore refer to few high-symmetry
directions only. For the specific case of an Fe–Si single crystal
] and
plate with (110) oriented surface, such directions are [
]. In a standard GO lamination they correspond, with a few
[
degrees uncertainty, to the rolling (RD) and transverse (TD)
directions, respectively. Attempts in the recent literature to
theorize the magnetization curve in GO sheets along a generic
direction, without disregarding the specific features of the low
magnetization region (the undifferentiated mode I in Néel’s
phase theory), have indeed focused on the properties along
RD and TD. Basically, these approaches call for some kind
of interpolation between the RD and TD curves, in order to
relationship at a generic angle to
achieve the desired
RD [7]–[9]. Their empirical character, however, poses strong
limitations on their general use and unphysical situations may
occur when the RD and TD properties are considered to behave
independently [2]. It has been suggested that the anhysteretic
might provide, for any direction, sufficient
curve
information for numerical calculations in magnetic cores [10].
This curve, defined as the gradient of a suitable potential
(coenergy), is calculated using the RD
energy function
and TD anhysteretic curves and assuming a suitable analytical
curves, tailored to the
expression for the equipotential
specifically treated cases [10], [11]. Nontrivial numerical
problems may arise, however, during differentiation of
and the use of the anhysteretic curve at low fields may not be
totally appropriate. A further problem with all these models
is the tacit assumption that the prediction regards intrinsic
magnetization curves, where the role of sample geometry
and the associated demagnetizing effects are ignored. But the
experiments in GO laminations made with different sample
configurations and measuring devices (e.g., Epstein frame,
single sheet/strip tester, rotational single sheet tester) provide,
curves [12]–[14], because
for a same material, different
depends on the sample geometry and the magnetic circuit.
In the end, it is not clear which kind of experimental curve
is to be predicted. This compounds with the inability of these
models to predict hysteresis loops and losses.
In the present paper, a novel comprehensive model of magnetization curve, hysteresis loops and energy losses in any direction in GO laminations is presented. This approach is consistent
0018-9464/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002
Fig. 1. Fe–(3 wt%)Si (110) [001] single crystal, cut along the [1
10] direction
(TD). (a) Antiparallel domain structure in the demagnetized state ([001] and
] phases). (b) After application of a sufficiently high-TD directed field, a
[001
0] phase
balanced transition toward the [100] phase (white domains) and the [01
(dark-gray domains) is accomplished. The net magnetization is colinear with the
field .
H
with the observed evolution of the domain structure and takes
at face value the role of sample geometry. Two base measuring
configurations and the related results are discussed in detail. In
the first one, the sample is a narrow strip; in the second, it is
an infinitely extended sheet. The first condition is realized by
means of Epstein strips, tested alone or under parallel stacking
in a conventional frame. As expected and experimentally verified, the high-transversal demagnetizing coefficient makes in
this case applied field and macroscopic magnetization colinear,
whatever the cutting angle of the strip [15]. Wide cross-stacked
sheets conveniently emulate the second condition. It is shown
that in both cases the theoretical prediction can be made exploiting no other data than those obtained in the RD and TD
laminations. Other practical testing conditions (e.g., single sheet
testing, circular sample) can be predicted as well, provided additional information is available (e.g., the sample demagnetizing
coefficient).
II. DOMAINS
AND
IN GO
MAGNETIZATION PROCESS
Fe–Si SHEETS
The conventional longitudinally cut GO Fe–Si sheets are
characterized by a regular pattern of 180 domain walls
(DWs), basically directed along the RD, and subjected to
back-and-forth motion under alternating fields. The domains
evolve instead in a complex fashion when the field is applied
along TD [16]. For a single crystal plate, such an evolution
can be sketched as shown in Fig. 1. The demagnetized state
is characterized, in absence of applied/residual stresses, by
a regular slab-like domain pattern, with magnetization
directed along [
] and [
] (i.e., RD). On increasing the
field, a transition takes place, where the basic 180 domains
transform, through 90 DW processes, into a pattern made
symmetrically directed along
of bulk domains, having
] and [
] (i.e., making an angle of 45 with respect to
[
the lamination plane), and of surface flux closing domains.
When this novel domain structure, having no net magnetization
normal to the sheet plane, occupies a fractional sample volume
, the macroscopic magnetization value is, disregarding the
flux-closing structure
(1)
Fig. 2. Major DC hysteresis loops measured on 0.30-mm-thick
high-permeability grain-oriented Fe–(3wt%)Si laminations (Epstein frame
strips cut along the rolling direction. TD
strips cut along
testing). RD
the transverse direction.
The maximum magnetization value obtainable in the
1.42
Fe–(3wt%)Si plate at the end of this process is thus
T and further increase is obtained by moment rotations. The
rotational contribution can be calculated, as a function of
the applied field, according to standard methods [17]. A
comparison between experimental RD and TD major loops is
shown in Fig. 2. The sigmoid shape of the TD loop reflects the
previously described domain transition, where the applied field
must basically compensate for a magnetoelastic energy term.
On comparing the anhysteretic magnetization curves associated
with these loops, one can estimate such a magnetoelastic energy
around 100 J/m . Looking at these curves from the schematic
perspective of phase theory [6], it is realized that in the RD
curve only mode I exists (two phases). For the TD curve, the
sequence is mode I (four phases), mode III (two phases). For
directions different from RD and TD, mode II (four phases)
and mode IV (one phase) can take place, depending on the
geometry of the sample. In any case, the RD and TD properties
can be defined as intrinsic, in the sense that they are dependent
on the material structure but independent of the sheet sample
geometry, at least within the single crystal approximation
.
Fig. 3 shows a Kerr effect image of domains and their evolution along a major half-loop in an Epstein strip cut at an angle
45 to RD (peak magnetization
1.25 T). Both 180
DW displacements, pertaining to the [
] and [
] phases,
] and [
] phases are
and 90 transitions to and from the [
observed. We conclude that, for a generic direction in the plane
of the lamination, we have a mixture of the two base 180 and
90 processes. The fractional sample volume occupied by the
when
and decreases, be180 domains is
.
cause of the growth of the 90 phases, as
It is therefore natural, in trying to predict the magnetization
curve under a generic applied field direction, to refer to the
RD and TD magnetization curves. This has already been pro-
FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE
1469
X
Fig. 4. Parallel stacking (a) and -stacking (b) of GO Fe–Si laminations. In
the parallel stacked strips, the internal field is
=
( applied
field,
demagnetizing field). With -stacking, it is possible to emulate a
two-dimensionally flux-closed magnetic circuit and
.
and
0], respectively.
are the components of
along [001] and [11
H
Fig. 3. Domain structure along a major half-loop in an Epstein strip cut at 45
to RD. The region shown is a 2 mm wide transverse band (see inset). Starting
from the saturated state, filled with the 90 phases ([100] and [0
10] axes),
transition toward the 180 phases ([001] axis) takes place while proceeding
toward the demagnetized state. This process is reversed on going toward the
oppositely saturated state.
posed in the literature, where, however, the material is normally
treated as a continuum, characterized by a tensorial permeability
[7]–[11]. This results in formal approaches, which, lacking a
direct connection with the domain structure, have limited predicting capabilities.
Since we can speak of intrinsic magnetization curves only for
fields applied along either RD or TD, the measuring configuration must be specified. The generally used methods rely on the
Epstein frame, the single strip, and the single sheet testers [18],
[19]. Recently, the use of the rotational sheet tester, used under
alternating regime, has become popular [12]–[14], [19]–[21].
Whatever the method, the sample shape and the induction con, vectorially comtrol technique, the demagnetizing field
bined with the applied field , always affects magnetization
curve and losses (with the exception of RD and TD testing).
The only practical way to minimize the demagnetizing effects is
to emulate an infinitely extended lamination, by cross-stacking
sufficiently wide sheets ( -stacking). The role of sheet stacking
on losses has been recognized since a long time, but the reason
for the observed differences has never been clarified. It has been
suggested, for example, that different flux transfer mechanisms
H
X
H
H 0H H
H
H H H
at the corner joints in the Epstein frame are responsible for such
differences [18], plainly in contrast with the fact that they persist
in absence of joints (e.g., when using stacked strips on a yoke).
The two base measuring conditions shown in Fig. 4 can then
be envisaged for a generic cutting angle to RD: 1) narrow
, orthogonal to the strip
strip (i.e., Epstein) testing, where
length (i.e., ) is so high to make the transversal magnetizaapproximately zero; 2) -stacked sheets,
tion component
where the two-dimensional (2-D) flux closure makes the demagnegligible, so that internal and applied field
netizing field
. All the other geomeare approximately the same
tries are intermediate cases and can be subjected to modeling
by a further assumption regarding the value of the demagnetizing coefficient. The results discussed in this work have been
obtained on 0.30-mm-thick high-permeability GO Fe–(3wt%)Si
laminations, by means of a digitally controlled setup [22], under
Epstein, -stack and single sheet testing (300 mm 120 mm
sheet) configurations, the latter being taken as an example of intermediate sample geometry.
III. THE MODEL AND ITS APPLICATION TO EXPERIMENTS
It is shown in this section that general and accurate prediction of hysteresis loops, energy losses, and magnetization curves
at any angle to RD can be made exploiting a limited amount
of information. In particular, for the narrow-strip (Epstein) and
-stack measuring conditions the RD and TD properties only
are required. The general framework of the model is Néel’s
phase theory [6], applied to the specific case of Goss oriented
laminations, with the important addition of hysteresis. It is assumed that the material is magnetized at a very low rate, so that
the magnetization process can be considered as quasi-static.
A. Epstein Strips
It is experimentally verified that in the Epstein samples of
conventional thickness (0.27 mm and higher), tested either in
a conventional frame (parallel stacking) or as a single strip inand
serted in a flux-closing yoke, the condition
is satisfied (see Fig. 4). Let us analyze in this case the
magnetization mode I, characterized by DW displacements only.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002
Since all possible phases are not energetically equivalent in the
demagnetized state and coercivity cannot be disregarded, the assumption that in this mode the material is isotropic, as held in
the phase theory, must be dropped. For given volume fractions
and
, the associated net magnetizations
and
are directed along the [
] and [
] axes (thus in the plane of
the lamination), respectively (see Fig. 1), with the value of
related to
by (1). Their components along the transverse direction must be equal and opposite in sign. For a given cutting
and
are related to the values of
angle , the values of
and
by the expressions
(2)
from which
(3)
These magnetization values are referred to the whole sample
and
, we
volume. If we refer to the volume fractions
have the reduced magnetizations
(4)
, having
For an impressed sinusoidal time dependence of
peak amplitude , the behaviors of
are obtained through (1)–(4), as shown in the example
T). As previously
reported in Fig. 5 (
stressed, the magnetization rate is assumed to be so low that
dynamic effects in the domain wall process are negligible (no
dynamic losses). This condition is respected in the present
experiments, performed at frequencies equal to or lower than
1 Hz. Time appears in our equations in order to provide a
parametric representation of the relationship between and .
in the demagnetized state,
Notice, also, that always
according to direct observations (see Fig. 3). It is supposed that
magnetization curves and DC energy losses under RD and TD
excitation are known. The more detailed is such a knowledge,
the more extended is the prediction. Let the DC losses be
and
for peak magnetization values
and
in the RD and TD cut strips, respectively. For
any angle , we have from (3) and (4)
(5)
Fig. 5. GO Fe–Si Epstein strip, cut at 60 to the rolling direction, with
imposed sinusoidal magnetization J (t) and transverse magnetization J = 0.
The magnetization rate J (t) is sufficiently low to make all dynamic effects
negligible. Calculated evolution along an hysteresis loop (J = 1.15 T) of:
(a) sample volume fractions v (t) and v (t) occupied by the 180 and
90 phases, respectively, and (b) corresponding magnetizations J (t) and
J (t). The quantity J (t) is normalized to the volume fraction v (t) (4).
known, the hysteresis loop
, for any angle and
peak magnetization , can be predicted. The relationships (3)
with
and
,
and (4), connecting the imposed
compound with the relationships concerning the applied field
, the internal fields
and
, and the demagne(see Fig. 4). The latter is transversally directed
tizing field
and we can write
The resulting energy loss is then obtained, according to our base
assumptions, by summing up the two contributions deriving
from the 180 and 90 DW processes
(6)
is time averwhere the volume fraction
aged. If the corresponding RD and TD hysteresis loops, para, are
metrically expressed as
(7)
from which
(8)
FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE
Fig. 6. Same sample and peak magnetization as in Fig. 5. (a) Time behaviors
of the fields
( ) and
( ). They are determined from the behaviors of
( ) and
( ) shown in Fig. 5, according to the corresponding RD and
TD loops. (b) Applied field ( ) and transverse demagnetizing field
( ).
They are calculated through (7) and (8).
( ) is conventionally chosen to be
( )
positive when directed as shown in Fig. 4. Its amplitude and sign make
and
( ) balance in such a way that always
0.
J t
J t
H t
J t
H t
Ht
H t
J H t
J t
The time behaviors of
and
, derived from
the known RD and TD loops of peak magnetization values
and
are shown, for the specific case of Fig. 5,
in Fig. 6(a). Fig. 6(b) shows the corresponding time behaviors
and
, calculated through (8). By associating
of
with the imposed
function, the desired hysteresis
loop is eventually obtained, as shown, in comparison with the
experiment, in Fig. 7(a). Another predictive result is provided in
Fig. 7(b). A satisfactory correspondence between predicted and
experimental loops is achieved, in general, in the investigated
induction range 0.15–1.4 T, for all the considered cutting angles
15 30 45 60 75 . The prediction of the hysteresis loop
evidently incorporates that of the normal magnetization curve.
Fig. 8 provides a general overview of theoretical and experimental DC energy loss behaviors versus and . Discrepancies
between theory and experiments are observed when the condition
is not perfectly satisfied (e.g., very low and values)
or rotations become important (e.g., high field values). It is
1471
Fig. 7. Comparison of predicted (dashed lines) and experimental hysteresis
loops in Epstein strip samples with different cutting angles. The quantities
involved in the derivation of the theoretical loop (a) are shown in Figs. 5 and 6.
actually expected from the phase theory that, on increasing the
of the direction
field , mode I will end, for a given set
with the quaternary axes, at the so called
cosines made by
“ideal” remanence value
(9)
(Kaya’s rule [23]), where it is superseded by the magnetization
mode II. At the start of this mode, the two 180 phases have re]) and three phases remain. The
duced to one (directed along [
macroscopic magnetization is, as before, alignedwith the longituis directed along the ternary
dinal field and the internal field
], to guarantee the survival of all three phases. The volaxis [
and
will balance accordingly and, following Néel
umes
[6], we shall write
(10)
is the standard
where is the modulus of the internal field.
expression for the [ ] magnetization curve at high field strength
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002
Fig. 8. Experimental (solid symbols and lines) and theoretical (dashed lines)
quasi-static energy losses in GO Fe–(3wt%)Si Epstein strips (parallel stacking).
Eight different peak induction levels J and five different cutting angles have
been considered. For induction values around and higher than 1.25 T the
prediction may become less accurate, because hysteresis loss modeling does
not account for the rotation of the magnetic moments.
in an Fe–Si crystal [17] and, for the above mentioned symmetry
reasons, the amplitudes of and are related by the equation
(11)
It is assumed that, for the present treatment of Epstein strips,
mode II entirely covers the high field range of technical interest,
excluding the approach to saturation, where it is known that
the three-phase state displays anomalous features [16]. On the
other hand, the transition between mode I and mode II is somewhat blurred in practice. The corresponding theoretical curves,
separately shown in Fig. 12, can provide a convenient fit of the
transition region when interpolated.
B.
-Stacked Laminations
When the laminations are cross-stacked, the applied field and
, and the magnetizathe internal field coincide, because
tion is always two-dimensional. But, as in the narrow strip case,
curve, involving the applied field and the
the measured
longitudinal magnetization component, provides the basic physical information and defines the energy balance of the magnetic
core. Let us, thus, analyze the magnetization process in mode I.
As always, the demagnetized state is occupied by the two 180
phases. In the ideal case of perfect flux closure, single crystal
sheet and absence of stresses, the 90 phases are inhibited for
54.7 (where 54.7 is the angle made by [
] with RD)
. Mode I is thus associated with the 180
and
DW displacements and ends for
. For a given applied
, the RD hysteresis loop
field peak value
peaks at
. The ensuing energy loss is
Fig. 9. Experimental (solid symbols and lines) and theoretical (dashed lines)
quasi-static energy losses in GO Fe–(3wt%)Si cross-stacked sheets versus
cutting angle to RD. The experiments refer to 120-mm-wide, 300-mm-long
sheets, tested in a flux-closing double-C laminated yoke. Notice the surge of
losses for > 45 .
(13)
54.7 all four phases can coexist in mode I, but the 180
If
phases are still energetically favored in the demagnetized state.
along RD and TD are
The components of the applied field
(14)
and the associated DC hysteresis loops, having peak magnetiand
, respectively, are assumed to be known.
zations
Again, the 180 domain wall processes are considered to occur
. The values of the meawithin the volume fraction
and the transverse magnetization
sured magnetization
are given by (2) (evidently now with
) with
. Having thus determined from the RD and TD
loops the correct amplitude and phase relationships between
and
, the macroscopic hysteresis loop
at
of the applied field is obtained, toany given peak value
. Fig. 9 compares the so-calcugether with the loss
T
lated hysteresis losses with experiments in the range
T. Two examples of loop fitting are shown in
54.7 ,
Fig. 10. On increasing the field, mode I ends, for
]
when the 180 DW displacements are completed and the [
] disaturated phase remains. Mode IV follows, where the [
rected magnetization rotates in the (110) plane toward the direc[Fig. 11(a)]. The portion of the normal (i.e., anhystion of
teretic) magnetization curve associated with this mode is calculated by finding, for each value of , the equilibrium direction
of the vector . To this end, the sum of anisotropy energy
(12)
and the associated hysteresis loop
with
is
(15)
FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE
1473
TABLE I
] AXES WITH THE DIRECTION OF THE
ANGLE ' MADE BY THE [100] AND [010
APPLIED FIELD WHEN THIS FORMS AN ANGLE WITH RD [I.E., [001],
SEE Fig. 11(b)]. ' AND ARE RELATED BY (18)
and Zeeman energy
(16)
are the direction cosines of
and is the
where
with the [
] axis, is minimized. From the
angle made by
value of at equilibrium, the component of along the direction of
(17)
is determined.
54.7 , the magnetization process succeeding mode
When
I is the symmetric rotation of the magnetization vectors be] and [
] toward the direction of
longing to the axes [
(mode III) [Fig. 11(b)]. The angle to be covered by the rotation
to reach the saturated state is
of
(18)
and is given, as a function of , in Table I. With reference to
are
Fig. 11, the direction cosines of
Fig. 10. Examples of experimental hysteresis loops in X -stacked sheets (solid
lines) and their theoretical prediction (dashed lines). Notice the ladder-like loop
with = 75 . In this case the 90 transitions start when the 180 DW processes
are largely accomplished.
(19)
and the anisotropy energy
interaction energy is
is provided by (15). The field
(20)
The usual minimization procedure of
the equilibrium angle and the magnetization
provides
(21)
Fig. 11. Rotation of the magnetization vector J in absence of demagnetizing
fields (X -stacking) under (a) mode IV and (b) mode III. The field H is applied
along a direction making an angle to RD and lying in the (110) plane. Process
(a) is accomplished by the J vector belonging to the [001] easy axis and occurs
for < 54.7 . Process (b) involves the J vectors belonging to [0
10] and
[100] and occurs for > 54.7 . The magnetization component along H is
J = J cos(' ), where ' is given by (18) (Table I).
1
0
Fig. 12 provides a general overview of the experimental
normal magnetization curves under the Epstein and -stack
measuring configurations here discussed. A comfortable
agreement between theory and experiment can be observed.
C. Single Sheets/Circular Samples
The variety of measuring configurations suggested for the
measurement of the properties of GO Fe–Si laminations in any
direction lead to a variety of results. They are expected to lie
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002
Fig. 13. Experimental (symbols) and theoretical (lines) normal magnetization
curves for 30 under three measuring configurations. The SST experiments
have been carried out on a 120-mm-wide and 300-mm-long sheet. The applied
field is directed along the sheet length and the flux is closed at the ends by means
of a double laminated C-yoke.
=
The 180 and 90 phases evolve according to the behavior of
the internal fields
Fig. 12. The experimental normal magnetization curves in parallel stacked
Epstein strips and X -stacked sheets (symbols). They are compared with
theoretical predictions (lines) obtained by considering both domain wall
displacements and rotation of the magnetization. Solid symbols and lines:
X -stacked laminations. Open symbols and dashed lines: parallel stacked
Epstein strips. The theoretical curves have been obtained at low and
intermediate induction values (mode I) exploiting the experimental RD and TD
curves. At higher inductions the magnetization rotations have been considered
(modes II, III, IV). Separate fitting lines are provided for these two induction
regions in the Epstein samples.
between the ones obtained in the narrow strips and those obtained in the -stacked sheets, but their prediction requires the
supplementary knowledge of the demagnetizing coefficient. It
is understood that in most experimental arrangements the demagnetizing field is not uniform and so will be the magnetization. We shall consider here the representative cases of a rectangular/square single sheet, cut at an angle to RD, flux-closed
by a yoke along the applied field direction [12] and of a circular
sample, inserted in a 2-D measuring setup [23]. Let be the average (magnetometric) demagnetizing coefficient, transverse to
the applied field in the rectangular/square sheet and isotropic in
the circular specimen. We wish to determine the hysteresis loop
when a field of peak amplitude
, making an angle
with RD, is applied. The parametric equations hold
(22)
is the field meaIn open samples (e.g., circular specimens)
sured with the use of a sensor (e.g., flat -winding, Hall probe)
placed on the sheet surface.
(23)
where the magnetization
given by
, transverse to the applied field, is
(24)
By introducing (24) in (23), we obtain that the corresponding
RD and TD loops have peak magnetization values satisfying the
condition
(25)
is accordingly found through (22). On leaving the
and
magnetization mode I at high fields, the magnetization curve
and
the spemay become difficult to predict, lacking
cial constraints on their direction observed in narrow strips and
-stacked sheets. To simplify the matter, we might adopt in the
high field range a suitable interpolation between the Epstein and
-stack curves, which, as induced by the observation of Fig. 12,
is the less imprecise the larger is the cutting angle . An example
of experimental and predicted normal magnetization curves in a
120-mm-wide, 300-mm-long single sheet, tested in a double-C
flux-closing yoke, is provided in Fig. 13. The corresponding av.
erage demagnetizing factor is estimated to be
FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE
IV. CONCLUSION
The DC magnetization curves, hysteresis loops, and energy
losses in Fe–Si grain-oriented laminations can be predicted for a
generic field direction through a method based on the exploitation of the material properties along the rolling and the transverse directions. Such directions are the only ones for which
properties can be defined independent of the specific sample
shape. For a generic direction, the measuring conditions (i.e.,
sample geometry) must be specified. Two base conditions have
been identified: Epstein strip samples and cross-stacked sheets;
all the other cases (e.g., single sheets, squares, disks) are expected to display intermediate behaviors. While the proposed
interpretative framework relies on the concepts of Néel’s phase
theory, hysteresis loops and losses are theoretically treated here
for the first time and the shortcomings of previous literature
approaches are overcome. Energy losses and hysteresis loops
can be meaningfully predicted in the induction range where the
magnetization process is mostly accomplished by means of domain wall displacements (typically up to around 0.6 –0.7
at intermediate field orientations). The normal magnetization
curve is calculated, starting at the end of the Rayleigh region,
up to applied fields of the order of 10 A/m. The model is
conceptually simple and its application requires elementary calculations. Based on the single crystal approximation, it is expected to display a predicting accuracy dependent on the actual textural quality of the laminations. The present experiments
show that high-permeability laminations are endowed with the
required textural perfection, but it is expected that application of
the model to the conventional grain-oriented alloys, with their
7 misorientation angle of [
] with respect to the rolling direction, should equally provide good results.
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[10] T. Pera, F. Ossart, and T. Waeckerlé, “Numerical representation for
anisotropic material based on coenergy modeling,” J. Appl. Phys., vol.
73, pp. 6784–6786, 1993.
, “Field computation in nonlinear anisotropic sheets using the co[11]
energy model,” IEEE Trans. Magn., vol. 29, pp. 2425–2427, 1993.
[12] T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, “Measurement
of magnetic characteristics along arbitrary directions of grain-oriented
silicon steel up to high flux densities,” IEEE Trans. Magn., vol. 29, pp.
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[13] Y. Tamura, Y. Ishihara, and T. Todaka, “Measurement of magnetic characteristics of silicon steel in any direction by RSST and SST: Method
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[14] O. Benda, J. Bydzovsky, and E. Usak, “Combined influence of shape
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of Si-Fe sheets,” J. Phys. IV (France), vol. 8, pp. 627–630, 1998.
[15] N. J. Layland, A. J. Moses, N. Takahashi, and T. Nakata, “Effects of
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[16] A. Hubert and R. Schäfer, Magnetic Domains, Berlin, Germany:
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[17] F. Brailsford, Magnetic Materials, London, U.K.: Methuen, 1960, p. 54.
[18] A. J. Moses and P. S. Phillips, “Effect of stacking methods on Epsteinsquare power loss measurements,” Proc. Inst. Elect. Eng., vol. 124, pp.
413–416, 1977.
[19] T. Nakata, N. Takahashi, Y. Kawase, and M. Nakano, “Influence of
lamination orientation and stacking on magnetic characteristics of
grain-oriented silicon steel laminations,” IEEE Trans. Magn., vol. 20,
pp. 1774–1776, 1984.
[20] N. Nencib, S. Spornic, A. Kedous-Lebouc, and B. Cornut, “Macroscopic anisotropy characterization of Si-Fe using a rotational single
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Magn. Magn. Mater., vol. 215–216, pp. 112–114, 2000.
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ACKNOWLEDGMENT
The authors wish to thank Mr. S. Rocco and Mr. E. Genova,
who provided substantial help in the experimental activity.
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Fausto Fiorillo was born in Bricherasio, Italy, in 1947. He received the degree
in physics from the University of Torino, Italy, in 1972.
Since then, he has been with the Materials Department at Istituto Elettrotecnico Nazionale Galileo Ferraris (IEN), Torino, where he holds the position of
Research Director. From 1997 to 2000, he was responsible in chief of the whole
scientific activity of IEN. His research activity has covered many aspects of
the properties of magnetic materials, their physical interpretation, and the related measuring techniques, including problems in standardization. He has devoted special attention to the study of the magnetization process, energy losses,
and structural properties of soft magnetic laminations. He is author/co-author
of some 130 scientific papers.
Dr. Fiorillo is a Member of the Italian Physical Society and past-Chairman of
the International Committee of the Soft Magnetic Materials Conference.
Luc R. Dupre (M’01) was born in 1966. He graduated in electrical and mechanical engineering in 1989 and received the Ph.D. degree in applied sciences
in 1995, both from the University of Gent, Belgium.
He joined the Department of Electrical Power Engineering, Laboratory of
Electrical Machines and Power Electronics, University of Gent, in 1989 as a
Research Assistant. Since 1996, he has been a postdoctoral Researcher for the
Fund of Scientific Research-Flanders. His research interests are mainly focused
on numerical methods for electromagnetics, especially in electrical machines,
modeling, and characterization of magnetic materials.
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Carlo Appino was born in Biella, Italy, in 1961. He received the degree in
physics from the University of Torino, Italy, in 1985, and the Ph.D. degree in
physics from the Politecnico of Torino in 1992.
He has carried out his scientific activity at the Materials Department of Istituto
Elettrotecnico Nazionale Galileo Ferraris (IEN), Torino, where he has held the
position of researcher since 1994. His main research activity has been devoted
to magnetization processes in amorphous and crystalline materials.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002
Anna Maria Rietto was born in Moncalieri, Italy, in 1936. She received the
degree in physics from the University of Torino, Italy, in 1959.
She has been with the Istituto Elettrotecnico Nazionale Galileo Ferraris
(IEN), Torino, since 1960. For about 20 years, her research activity has dealt
primarily with dielectric materials and their physical properties. Since the
1980s, she has been involved with research in soft magnetic laminations,
especially magnetic losses and hysteresis under alternating and rotational
magnetic fields.