Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 1
Rotational Magnetization in Transformer Cores – A Review*
Helmut Pfützner1 , Member, IEEE, Edin Mulasalihovic1, Hiroi Yamaguchi2, Damir Sabic1,
Georgi Shilyashki1, Franz Hofbauer1, Student Member, IEEE
1Vienna University of Technology,
Institute of Electrodynamics, Microwave and Circuit Engineering (EMCE), Vienna, Austria 2Steel Research Lab. JFE Steel Corp., Mizushima, Kurashiki, Japan
Usually, rotational magnetization (RM) is associated with rotating machine cores. However, in more restricted ways, it also arises in
3-phase transformer cores. Modern designs of T-joint yield detours of flux, as a source of RM in the T-joint, the middle limb ends, as
well as in the yokes. Simulation of RM is possible by means of so-called rotational single sheet testers which should consider the large
grains of highly grain oriented materials. Their high effective anisotropy yields induction patterns of rhombic or lancet-like type with
maximum values of axis ratio a up to 0.5, and very high angular velocity round the materials hard directions. Compared to elliptic RM
- as arising in non-oriented materials - the corresponding losses are lower due to restricted induction in the hard direction. But they
show strong increase with (i) rising a and (ii) rising angular velocity of the induction vector. The magnetostrictive strain shows a pro-
nounced (negative) maximum in the rolling direction with values up to 10 ppm, the transverse direction and normal direction exhibiting
positive maxima of lower extent. With respect to industrial relevance, significant RM effects are restricted to the vicinities of T-joints.
They represent the location of maximum core loss and also of maximum magnetostrictive strain as a source of audible core noise.
*) Parts of this paper were presented as an invited lecture at the 10th Soft Magnetic Materials Conference (Torino, Italy, September 6-9, 2009).
Index Terms— losses, magnetostriction, rotational magnetization, silicon alloys, transformer cores.
NOMENCLATURE
AM alternating magnetization
CGO conventionally grain oriented
GO grain oriented
HD hard direction
HGO highly grain oriented
MS magnetostriction
ND normal direction
NO non-oriented
OD oblique direction
RD rolling direction
RM rotational magnetization
RSST rotational single sheet tester
SHGO scribed highly grain oriented
TD transverse direction
I. INTRODUCTION
OTATIONAL MAGNETIZATION (RM) is a phenomenon
which has been studied for a hundred years [1]. It has
wide significance for rotating machine cores. SiFe materials of
low effective (i.e. global) anisotropy favour the case that the
induction vector B deviates from the rolling direction (RD)
during the cycle of magnetization. There arise magnetization
patterns of elliptical type (Fig.1), and even of circular type.
This may yield considerably increased losses P. However, the
increase is restricted to the order of 100% if non-oriented
(NO) material is given. This is due to the fact that the easy
axes of small crystals are distributed in a quite random way.
Fig.1. Important definitions at the example of an elliptic magnetization
pattern with axis ratio a = 0.5 (RD rolling direction, TD transverse direction,
HDs hard directions). The magnetization vector B rotates with the
instantaneous angular velocity ω(t) in anti-clockwise direction (or clockwise
direction, respectively). The induction values BRD and BTD concern the
instants when B passes through RD and TD, respectively. They determine the
axis ratio of RM according to a = BTD/BRD. For example, BRD = 1.7 T and
a = 0.5 correspond to BTD = 0.85 T. For a full definition of pattern, the time
function ω(t) is needed.
The evidence of RM in transformer cores was reported in the
early seventies (e.g. [2-4]). Systematic studies have been
started about twenty years ago, regular reports being given
through bi-annual 2-DM Workshops [5]. Intensive work has
been made on simulations of RM. So-called rotational single
sheet testers (RSSTs) were established in several laboratories,
mainly in Europe and in Japan. In most cases, the phenomenon
is studied on a square 80 mm sample which is magnetized in a
defined way. For a review see [6], for a further one - which
however is focussed on early versions of apparatus - see [7].
More detailed descriptions of individual "classical" set-ups
were reported in [8-15], more recent ones in [16-24]. A very
large number of publications report correspondences between
R
TD
HDs RD
BRD
BTD ω
B
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 2
induction patterns B(t) and field patterns H(t), as well as the
corresponding losses P. However, most papers concern the
case of NO SiFe considering that very pronounced RM may
arise in industrial practice for the already mentioned case of
rotating machines (for a critical assessment see [25]).
The present paper reviews the specific case of RM as oc-
curring in a more restricted way in transformer cores. A
presentation of the state-of-the-art is given, complemented by
specific results of the authors which partly have not been
published so far. The main aim of this paper is to present a
review, as a compact information for all three producers of
core material, producers of transformers and scientists which
are involved in the interesting field of RM.
Modern transformer cores tend to be built up from highly
grain oriented (HGO) SiFe - or even more frequently - scribed
highly grain oriented (SHGO) SiFe which disfavour deviations
from the RD. Thus circular RM is without significance here.
And even elliptical patterns do not arise in practice. Instead,
transformer cores show patterns of rhombic – or lancet-like –
shapes which are most pronounced in the vicinity of T-joints
(Fig.2). This has been confirmed by several experimental
studies (e.g., [3,4]; Fig.2b, from [26]).
In spite of regional restrictions of RM, the phenomenon
shows high practical interest due to three reasons:
(i) Transformers are operated in a continuous way.
Thus even smallest increases of the building
factor exhibit relevance due to economic reasons.
(ii) Even smallest increases of audible noise are tried
to be avoided due to increasing environmental
consciousness.
(iii) Compared to other impact factors – e.g. flux
distortion - RM may represent the most signifi-
cant one, as discussed at the end of Section V.
Aiming for industrial relevance, the following is focussed on
practical working conditions. This means that the frequency f
is 50 Hz, qualitatively being representative also for 60 Hz. As
well, focus is put on practically relevant magnetization
patterns.
II. ORIGIN OF ROTATIONAL MAGNETIZATION
Transformer cores show rotations of magnetization with
strong restrictions of location and time. The basic origin of
RM is given by the following two mechanisms:
(a) Most modern core designs use mitred T-joints which
exhibit sharp ends of S-limb laminations, these “V-
elements” (Fig.2a) playing a crucial role.
(b) The high anisotropy of HGO SiFe favours detours of
flux.
Throughout the period of magnetization, the flux tends to
follow the RD. However, for short instants of time, the
induction vector B has to pass through the transverse direction
(TD). Fig.3 illustrates the reason in a schematic way. Fig.3a
depicts the instant of time to when the S-limb does not carry
any total flux. The flux comes from the R-limb in order to flow
into the T-limb. However, this flux take-over is complicated:
The V-element represents an obstacle since being oriented in
vertical direction. The upper half of yoke flux is guided into
the left half of the S-limb. It makes a detour, the length of
which increases with rising anisotropy. And only in a
distributed, continuous way, it "rotates" into the TD, passes
over to the right half and back to the yoke. A second process
of rotation arises when the T-limb shows zero flux at to + T/2
(with T the length of period). This yields a rhombic
magnetization pattern B(t). According to Fig.2b, the pattern
tends to be symmetric, being most pronounced in the ends of
S-limb laminations.
Fig.2. T-joint which is the core region that is most affected by rotational
magnetization. (a) Typical global core geometry. (b) Detail of T-joint region
with distribution of induction patterns B(t) as determined on the model core
(from [26]).
According to Fig.2b, the phenomenon of RM is also given
in the yokes. Again, it is the V-element which favours a detour
of flux. In the instant when the T-limb is without gross flux,
the flux coming from the R-limb is entering into the S-limb as
illustrated by Fig.3b. The lower half II of flux is hindered by
the V-element to pass into the right half of the S-limb. It makes
a detour into the right yoke region and then "rotates", flows
back and enters. Again this yields a rhombic pattern. However,
it tends to be asymmetric due to lacking symmetry of geome-
try. In the T-joint, the pattern axes may show a tilt (see several
examples in Fig.2b). The latter is weak. However, it is relevant
for losses as shown further down.
For the definition of RM patterns, several characteristics
are important (Fig.2):
(i) The axis ratio
a = BTD / BRD (1)
(a)
(b)
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 3
flux path II
(a)
II
I
(b)
global flux
path I
detour
with BRD and BTD the instantaneous induction values when
B passes through the RD and TD, respectively. For exam-
ple, Fig.1 represents a = 0.5 , BTD being reached later than
BRD (by 5 ms for f = 50 Hz), according to anti-clockwise
RM. Highest axis ratios arise close to V-elements, in
Fig.2b up to about amax = 0.45. As a tendency, amax
decreases with increasing grade of effective anisotropy as it
results from technological treatments of the material. The
latter include cold rolling procedures and stress coating of
HGO material as well as scribing of SHGO material. That
is, the grade of effective anisotropy shows a ranking NO -
CGO - HGO - SHGO. High differences are given, in spite
of the fact that the crystal anisotropy is almost constant.
With rising distance from the V-element, the value a
tends to sink in a gradual way, depending on the grade of
anisotropy. For high grade, some RM may spread to the
corners, and possibly even to the ends of outer limbs.
(ii) A significant further quantity is given by BHD, i.e. the
induction in hard direction (HD). As well known, it varies
between 40° to 60° to the RD. The permeability is a mini-
mum here. Thus the flux avoids the HD which explains that
rhombic patterns arise instead of elliptic ones.
(iii) Finally, impact is given by time changes of the angular
velocity ω(t) of the vector B. It tends to be a minimum in
the vicinity of the RD. On the other hand, B passes through
out-of-RD directions with distinctly increased angular
velocity.
It should be stressed that the usual formula ω = 2 π f
(with f the frequency of magnetization) is not valid here.
In fact, 2 π f results as the time mean value of ω(t).
Fig.3. Schematic outline of flux detours as the main basic origin of RM. Two
global flux paths I and II are assumed, the short arrows indicating distributed
instantaneous flux in TD. (a) Instant of zero-flux through the S-limb yielding
almost symmetric induction patterns in the S-limb ends. (b) Instant of zero-
flux in the T-limb yielding asymmetric patterns in the T-joint and quite
symmetric patterns in the yokes.
III. EXPERIMENTAL SIMULATION OF RM
As already mentioned, study on RM is usually performed
on rather small square samples of 80 mm size (see references
in Section I). The application of the corresponding RSST set-
ups is not restricted to NO materials. They can be used for
HGO SiFe as well, as it has been demonstrated in many publi-
cations (see further down). However, HGO materials exhibit
very large crystallites, the typical order of diameter being 10
mm. Due to technical reasons, quasi-homogeneous magne-
tization is restricted to the central sample region which is used
for the arrangement of sensor elements. As a drawback, the
effective size of a sensor may be restricted to that of single
grains which means that the result of measurement is not repre-
sentative for the given type of material.
Fig.4. RSST which uses a large hexagonal sample of about 160 mm diameter
with support of the HD, thus favouring applications for HGO materials in a
specific way [14,27]. (a) Total view with sample. (b) Detail without sample
visualizing a double tangential field coil as commonly applied for RSSTs.
A rotational single sheet tester which has been
specifically designed for the case of transformer core steel is
described in [14,27]. Hexagonal samples of about 160 mm
diameter are applied (Fig.4a). Instead of the usual 4-pole-piece
yoke system, a 3-phase excited 6-pole-piece system is used.
Apart from improved averaging over large grains, the system
supports the HD in a direct way which favours the
investigation of HGO material in a specific way. Software
controlled approximation algorithms allow the generation of
elliptic and rhombic patterns (including tilted ones) with
(a)
(b)
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 4
"natural" course of time, i.e. as arising in practice. A recently
introduced control approximates 4000 instants of time with
0.2% mean square deviation from target values of induction.
Thus, patterns detected on real cores can be simulated in exact
ways, and consequences of pattern modifications on losses and
magnetostriction can be studied systematically.
The field pattern H(t) can be detected by means of a pair of
tangential field coils (Fig.4b). In principle, a variety of alter-
native sensor types would be available, in special Hall sensors,
magneto-resistance (MR) sensors or magneto-impedance (MI)
sensors [28]. However, the latter sensor types are charac-
terized by small dimensions which are disadvantageous in the
case of modern materials due to their large grain size
(compare, e.g., [29]). Even extra-large Hall plates of about 10
mm effective length proved to be ineffective since reflecting
the crystallographic characteristics of individual grains.
The need to average over several grains is also given with
respect to the detection of the induction pattern B(t). Here the
well known needle method can be applied for all types of
material. However, specific studies [30,31] have revealed that
the technique exhibits several sources of error. The sample has
to be arranged in a very precise way to ensure identical
physical status of its upper and lower surface. Further, the HD
may involve high instantaneous values of H as a source of air
flux induction effects which affect the results of the rapid, non-
destructive method. The application of search coils through
holes in the sample proves to be preferable, provided that very
small holes are drilled in a careful way.
For the determination of losses, three options are given:
(A) The needle method in combination with a pair of field
coils.
(B) A pair of search coils in combination with a pair of field
coils.
(C) The calorimetric method, e.g. using a thermistor on the
sample surface in order to detect the initial rate of tempe-
rature rise for a short process of magnetization.
Method (A) is the most simple one apart from being error
prone. Method (B) proves to be more reliable and precise.
Both methods are based on the determination of the Poynting
vector E x H , with E the electric eddy field of the sample
surface. As a specific problem, the measurement of this pro-
duct shows high sensitivity to phase errors if high values of H
are involved. Respective effects of misalignment of the sample
- or of phase errors of electronics - can easily be recognized by
the tendency that different directions of rotation yield different
results for P [20,21,24,32,33]. Small deviations - which apart
from measurement errors may be due to different domain
reconstructions - can be met by averaging over the two results.
But in cases of large deviations, it is preferable to apply
method (C). Experience shows that large deviations are typical
for high H which however tends to be linked with high levels
of P - a feature which favours the application of the thermal
method due to the fact that its drawback of poor resolution is
not relevant here.
Finally, an effective evaluation of multi-directional magne-
tostriction is possible by a set of three traditional strain gauges.
Again, the specific demand for transformer core material is to
average in a sufficient way. It can be fulfilled by extra-long
sensors, e.g. of 40 mm up to 100 mm length [34]. A possible
alternative may result from a method described in [35]: A set
of three 15 mm gauges is arranged on a plastic layer. Ave-
raging could be attainable by restricting the agglutination on
the sample to the outer edges of the layer.
IV. B/H-DEPENDENCIES
The dependence between a given induction pattern B(t)
and the corresponding field pattern H(t) offers effective
information on the multi-directional behaviour of the investi-
gated type of material. In comparison to earlier measurements
on samples cut in different directions to the RD, the RSST
measurement is simple and rapid. But in particular, it gives
more realistic and precise results. And further, the information
is attained in a very compact format that favours the design of
transformer cores.
For non-oriented (NO) materials as applied in rotating ma-
chines, extensive data on B/H-dependencies has been pub-
lished for the case of circulating RM (see [5]). Even for the
NO-case, the H-patterns reveal minimum field intensity in RD,
indicating that some effective anisotropy is given also here.
Apart from lacking industrial relevance, but as a technical
challenge, circular RM of 50 Hz was also tested for CGO and
HGO materials, the field showing values up to 25 kA/m for the
hard directions [36,37]. The results illustrate the high aniso-
tropy of material in an effective way.
As a starting point for the field of transformer cores, let us
have a look on Fig.5 which offers results of RSST measure-
ments for different induction patterns. Fig.5a shows a case of
exactly elliptic B(t) – which however does not arise in prac-
tice, as already mentioned. On the other hand, it is the only
pattern which can be defined in a simple way thus allowing
exact comparisons between different materials. Moreover, it
offers a characterization of texture. For this purpose, an effect-
ive set of parameters was assumed for Fig.5a: BRD is chosen
with 1.7 T, the most usual transformer core induction level.
The axis ratio is chosen as a = 0.5. This is the uppermost limit
for practically usual values. However, 0.5 reveals the given
grade of anisotropy in a pronounced way. The maxima of H
characterize the HDs (Fig.1) in the depicted case of strong
magnetization, while they may deviate from the HDs for lower
magnetization levels. Information on the permeability which is
effective for different directions is offered in a compact way.
The latter is also valid for changes of permeability due to
practically relevant impact factors like mechanical stress [38].
Fig.5b shows a case of exactly rhombic RM where the
maxima of H fall into the TD. In rough approximation, the
corresponding H-pattern is of circular type, the practical rele-
vance of which has been outlined in [39]. And indeed, the
pattern can be classified as being "natural" due to its com-
patibility with the tendency of similar peak values of excitation
for the three phases R, S and T (Fig.2a).
Results of in-situ measurements of B-patterns, such as
given in Fig.2b, reveal the occurrence of "quasi-rhombic" RM
as depicted in Fig.5c. It is characterized by restricted sharpness
and increased induction in hard direction. The asymmetry of
H(t) is due to physical mechanisms like hysteresis and eddy
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 5
currents [27], as indicated by differences for clockwise RM
and anti-clockwise RM, respectively. But also uneven texture
of the investigated sample region and imperfections of experi-
mental modeling may contribute. Finally, Fig.5d depicts a
simulation of tilted B(t) as it can be observed close to V-
elements (compare Fig.2b): Tilts involve highly increased
induction in HD which may yield extreme field values in this
direction.
For a comparison, Fig.6 (from [40]) shows a tilted pattern
which has been detected in the centre of a T-joint region of a
model transformer core in a direct way. Clearly, it illustrates
two tendencies of "natural" patterns, i.e. patterns as actually
arising in real transformer cores:
(i) B(t) tends to show very uneven angular velocity ω (dots
which mark instants of time being concentrated in the vici-
nity of RD), corresponding to minimum out-of-RD time.
(ii) H(t) may be rather chaotic (reflecting strategies of nature
to restrict the field energy; see the corresponding, very
specific literature), in connection to also restricted modulus
of H.
As a conclusion, so far simulations by means of RSST have
been restricted to rough approximations. But the results of
measurement yield very effective information on the grade of
texture of a given type of material as well as on the links
between induction and field. According to the following
Section V, the RM patterns yield losses which depend in
significant ways on the instantaneous changes of magne-
tization. This means that relevance is not restricted to the
shape of pattern, since it is the dynamic which is significant
for eddy current processes.
Fig.5. Different types of induction patterns B(t) and the corresponding field
patterns H(t) for HGO-material magnetized with BRD = 1.7 T (50 Hz) with an
axis ratio a = 0.5 (the uppermost limit in practical cores). 20 instants of time
(spaced by 1 ms with rather low variation of ω) are given, the instant 1 being
marked through a square symbol. (a) Elliptic RM as offering a good charac-
terization of material. (b) Exactly rhombic RM. (c) Quasi-rhombic RM as
being closer to practice. (d) Tilted rhombic RM as arising close to V-elements
in similar ways. In spite of the fact that the axis ratio (according to definition)
is restricted here to a = 0.2, very high values of field (considered by changed
scale for field values) and losses arise here.
In
duct
ion B
(TD
) [T
]
F
ield
H(T
D)
[kA
/m]
Induction B(RD) [T] Field H(RD) [kA/m]
Induction B(RD) [T] Field H(RD) [kA/m]
F
ield
H(T
D)
[kA
/m]
Induction B(RD) [T] Field H(RD) [kA/m]
F
ield
H(T
D)
[kA
/m]
In
duct
ion B
(TD
) [T
]
Indu
ctio
n B
(TD
) [T
]
F
ield
H(T
D)
[kA
/m]
In
duct
ion B
(TD
) [T
]
Induction B(RD) [T] Field H(RD) [kA/m]
(a)
(b)
(c)
(d)
1
3
9
13 19
1
3 9
13 19
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 6
Fig.6. "Natural" induction pattern B(t) of tilted, rhombic type and the
corresponding field pattern H(t) as detected in the centre of a T-joint in a
model transformer core [40]. Twenty instants of time inform about dynamic
variations of the vectors B and H, as a prerequisite for closer interpretations.
B(t) illustrates the - general - tendency of low angular velocity ω of induction
close to the RD.
In principle, a possible complementary option to RSST
simulation is given by numerical modelling. The most
frequently used procedure is to design a so-called vector
hysteresis model that describes the interdependence between
the vectors B and H. In order to describe geometric distri-
butions within the studied transformer core, the vector model
then is implemented into a finite element model (FEM), in
most cases.
With respect to individual B/H-patterns of transformer
cores, the corresponding results of modelling are very rare in
literature. For T-joint regions, single patterns are included in
[41] (see figures 5 and 6), in [42] (figures 9a and 14) as well
as in [43] (figures 2a and 4). Modelling has been performed
also for the close vicinity of overlap regions which exhibit
rotational phenomena of partial rotation in both 3-phase cores
and 1-phase cores. B may show distinct deviations from the
RD, according to results of modelling in [44] (figure 12), [45]
(figure 10), [46] (figure 10a) and [47] (figure 11).
Comparing the above mentioned results of numerical
modelling with measured patterns reveals that - in spite of high
theoretical and numerical effort - restricted industrial relevance
is given. Most results concern materials of low anisotropy, and
also the intensity of magnetization tends to be much lower than
in practice, thus involving less nonlinearity. As a weak point,
the material data which is applied as a basis of modelling tends
to be restricted to results from two measurements, one in RD
and one in TD. This involves magnetization processes -
domain reconstructions - which are not consistent with rota-
tional magnetization, and it does not consider the HD.
In fact, RM of modern core steel involves extremely high
levels of effective anisotropy in complex interaction with non-
linearity, hysteresis and eddy currents. A priori, exact consi-
deration can be assumed to be impossible. In particular, as
pointed out by a recent comparison of methods [48], no
concept is available so far to model the interdependence
between the vectors B and H for the whole range of
technically relevant induction. The problem which has to be
overcome here is illustrated by figure 1b of reference [27]. It
shows that for rising intensity of H in a given direction, the
angle of B may vary by almost 90o. But still, approximate
modelling should be encouraged if performed for practically
relevant conditions. The challenge is to consider modern types
of materials of low thickness (up to 300 µm), magnetized with
BRD between 1.3 T and 1.8 T, for rather small values of axis
ratio a.
V. LOSSES
As a starting point, let us consider circular magnetization,
i.e. a = 1 corresponding to constant momentum B
[20,24,37,49]. As illustrated by Fig.7 (with data from different
labs), losses P tend to increase distinctly with B. Apart from
the fact that the three curves reflect different types of material,
they reveal a general tendency: After a maximum close to
1.5 T, P tends to sink. This can be explained by a decreasing
amount of Bloch wall area, corresponding with decreasing
hysteretic interactions. Let us remind to the fact that losses
approach zero for B = 2 T and f → 0 (as already reported by
Pierre Weiss in 1908 [1]), a case were both walls and eddy
currents are lacking.
As already discussed, circular magnetization which in the
case of GO materials involves very high field values lacks
practical relevance. As demonstrated by Fig.2b, relevance is
restricted to patterns of small a (up to 0.5). The shape of
pattern varies between rhombic and quasi-rhombic type. Most
regions show axis-symmetric patterns. However, the close
vicinity of T-joint overlaps may include tilted patterns.
Unfortunately, results for losses for such "natural" RM-
patterns are very rare in literature.
Fig.7. Losses P as a function of induction B for circular magnetization with
50 Hz. (A) CGO material [20]. (B) HGO 23ZH90 [24]. (C) CGO [49].
(a)
(b)
0 2.0 1.5 1.0 0.5
4
3
A
B
C
B [T]
a = 1
f = 50 Hz
P
[W/kg]
2
1
0
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 7
P
[W/kg]
0
2
1
0 0.4 0.3 0.2 0.1
3
B: SHGO
23ZDKH, 1.7 T
A: GO-SiFe,
1 T
a
f = 50 Hz
0.5
C: 3% HGO
SiFe, 1.7 T
D: 3.2% GO
SiFe, 1.7 T
Fig.8 shows losses P as a function of axis ratio a up to
0.5 based on data for elliptic RM from [9,49,50]. The few
curves are sufficient to visualize the following, most
significant tendencies:
(a) Increasing a yields increases of losses P, the increase beco-
ming more distinct for a > 0.2 (see curves A and B; C and
D representing linear "two-point curves").
(b) The amount of P rises with increasing grade of effective
anisotropy. That is, CGO materials (curves A and D) show
lowest increase, followed by HGO materials (curve C) and
by SHGO materials (curve B).
The latter behaviour has high practical relevance, since it
means that high-grade materials offer the advantage of low
losses for alternating magnetization while they are highly
sensitive with respect to rotational magnetization.
Fig.8. Losses P for BRD = 1.7 T (except A; 1 T) as a function of the axis ratio
a for elliptic RM with 50 Hz. (A) GO SiFe [50]. (B) Laser scribed SHGO.
(C) HGO 3% SiFe [9]. (D) GO 3.2% SiFe [49].
Industrial interest is focussed on minimum impact of RM
on the building factor. As well known, the latter is defined as
the ratio
BF = Pcore / PSST (2)
with Pcore the mean losses of the core and PSST the nominal
losses as determined by SST. Thus - especially with respect to
additional physical impact factors - it is advantageous to pre-
sent RM-losses in an analogous way through the related losses p = Pa > 0 / Pa = 0 . (3) From tests on large hexagonal material samples, typical
data for p as a function of a is summarized in Fig.9. They
demonstrate that the quantity p is an effective source of
information especially in those cases, where the interest is
focussed on a comparison of different physical states of a
given type of material, or of different materials, respectively.
Among others, for the most important impact factors, the
graph illustrates the following tendencies:
(a) Increasing a yields a general tendency of rising p. But - as
already mentioned - the extent depends strongly on the
grade of material. For elliptic RM, a = 0.5 may yield as
little as p = 1.2 - corresponding to 20% increase – for non-
oriented material (curve A). On the other hand, for
transformer core materials, the corresponding orders of p
are 2 for CGO (curves B,C) and up to 3 – i.e. 200% - for
SHGO (curve H). The latter case is characterized by
maximum effective anisotropy which disfavours
magnetization out of the rolling direction.
(b) Increasing BRD yields a decrease of p (explainable by the
fact that Pa = 0 increases with BRD in a non-linear way). A
comparison of curve B with curve C indicates very
pronounced decreases for CGO material. HGO and SHGO
prove to show a weaker impact of BRD.
Fig.9. Related losses p (related to the AM case a = 0) as a function of the axis
ratio a for different impact factors. (A) Elliptic RM of non-oriented material.
(B) and (C) Elliptic RM of CGO material for two different values of BRD .
(D) Rhombic RM of CGO material. (E) Oblique rhombic RM of CGO
material. (F) Elliptic RM of CGO material with an increase of maximum
angular velocity ωmax from 20°/ms up to 40°/ms (notice: the curve is related
to the case 20°/ms). (G) Elliptic RM of HGO material. (H) Elliptic RM of
SHGO material. (I) Elliptic RM of SHGO material with a DC-component of
excitation in RD, approximately 20% compared to AC excitation (notice: the
curve is related to the no-DC case).
(c) The related losses p depend on the shape of the RM-pattern
B(t). Compared to elliptic RM, a rhombic one yields lower
p (compare curve C with curve D). As an explanation,
lower BHD is involved, and thus also much lower HHD (see
Fig.5a,b). On the other hand, strongly increased p may
result from tilted elliptic or rhombic patterns, according to
Fig.5d and Fig.6. As indicated by curve E, p may show
values up to 4, however, depending on details of the
magnetization pattern B(t).
p
1
3
2
0.4 0.3 0.2
D: CGO,
1.8 T, rhombic
a 0.5
B: CGO,
1,3 T
F: CGO,
1.5 T;
high ωmax
G: HGO,
1.7 T
A: NO, 1.5 T
E: CGO, 1.5 T;
oblique
rhombic
H: SHGO,
1.5 T
I: SHGO,
1.5 T,
+DC
hexagonal
samples
(f = 50 Hz)
C: CGO,
1.8 T
0.1 0
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 8
(d) For a given shape of B(t), p depends on the function ω(t),
i.e. on the time-course of the angular velocity of the
induction vector. RSST-simulation of elliptic RM is
usually performed with almost constant spacing ΔB per
time unit (e.g. ranging between 0.3 T and 0.5 T per
millisecond in Fig.5a). However, transformer cores tend to
show much higher dynamics. As visualized by Fig.6a, ΔB
tends to be small close to the RD corresponding to
minimum instantaneous ω . On the other hand, B passes
through the TD and HD with high ΔBmax . In Fig.6a, ΔBmax
is close to 1 T corresponding to large values of maximum
ω. As a consequence, high values of classical eddy current
(EC) losses may yield significantly increased p, especially
for low values of a (see curve F). On the other hand, little
impact is given on anomalous EC losses and on hysteresis
losses. The mechanisms show analogy to distortion of
alternating magnetization (AM). However, it should be
stressed that completely different domain reconstructions
are involved in RM and AM, respectively. This impedes
effective interpretations of RM as a combination of AM in
RD and TD, as striven for by several authors.
(e) Further impact is given by superimposition of DC-
excitement as possible from unbalanced thyristor sets or
from so-called geomagnetically induced currents (GICs).
While the shape of B(t) tends to remain unaffected, a
global shift of the magnetization pattern is given. As indi-
cated by curve I (related to the no-DC case and thus not
originating from the value 1), effects are given especially
for low values of a. They involve highly ordered bar
domains as being typical for mere AC magnetization. DC
yields complex domain reconstructions [51], as an expla-
nation for increased p. The mechanism plays a distinct role
for RM, analogous to the role of distortion for AM.
(f) Finally, further impact will be given by all those factors
which affect the domain configurations of the core mate-
rial. One is temperature which affects through conductivity
changes and modified coating characteristics. A second one
is mechanical stress. Tension proves to yield slight
decreases of p if applied in RD [52], as also observed for
NO-material [53]. Tension in TD yields stronger decreases
[52] since favouring oblique domains.
As a conclusion, induction patterns of transformer cores show
rather low axis ratio a, but the corresponding loss increases of
HGO SiFe approach - or even exceed - those for circular RM
of NO SiFe. The effects depend on many parameters which
complicates effective simulation by means of RSST.
Systematic studies of the most important grades of material
would be desirable in order to estimate consequences on local
values of building factor. However, considerably large
experience exists already from model core experiments. Apart
from moderate RM in the yokes, several studies reveal that
most pronounced RM is to be expected in the T-joint. As
already discussed in [54], the corresponding impact of RM
depends strongly on the joint construction. Reference [55]
discusses simple double overlaps. This case proves to show
rather low impact of RM, the local building factor
BF TJ = PTJ / PSST (4)
reaching the order of 1.3 in the T-joint region, corresponding
to 30% excess loss, as observed also in the inner yoke region.
Considerably higher BFTJ of 1.61 is reported in [56] for a
staggered T-joint design. However, the majority of modern
transformer cores exhibits V-type T-joints. They tend to show
high values of local losses due to pronounced RM. For
example, cases according to Fig.2 for 300 µm thick core
material magnetized with 1.7 T yielded BFTJ = 1.68 for HI-B
steel [56] and BFTJ = 1.7 for CGO 30M5 [51]. Even higher
values are reported in [40] with BFTJ = 1.83 and in [58] with
BFTJ = 1.9. Of course, BFTJ includes contributions also of
several other impacts such as distortion and inhomogeneity of
flux, as well as excess loss at overlaps [59]. Anyhow, high
contributions due to RM are confirmed by data of Fig.8 and
Fig.9: Based on the B-patterns of Fig.2 - and considering that
some RM is also given in the yokes - the mean axis ratio a can
be assumed to be of the order 0.25. Roughly, this corresponds
to related losses p that range between 1.3 and 1.8, depending
on the type of material, the core induction and the B-pattern.
This range is close to that of BFTJ.
The above means that RM represents the dominant impact
on the local building factor BFTJ of modern T-joint designs.
However, the practical relevance for the building factor BF of
the core as a whole depends on the mass portion of T-joint.
The latter tends to be of the order of 10%. Thus, e.g., a local
value BFTJ = 1.8 means that a contribution of 0.08 is given to
the total building factor BF. The latter tends to be within the
order of 1.2 (e.g. [57,60,61,62]), except for very small cores.
Finally, the above yields the conclusion that the loss of T-
joints - and thus also the factor RM - contribute to BF in very
significant ways. A careful estimation indicates that rotational
magnetization and distortion have very similar impact on the
global performance of core. RM may even represent the most
significant impact factor, if a large core is given that is assem-
bled from modern material in a perfect way.
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 9
λRD
[ppm]
-0,5
B [T]
1 2
I I
even domain widths => small ξ
uneven domain widths => rising ξ
approaching saturation => vanishing ξ
VI. MAGNETOSTRICTION
Specific impact of rotational magnetization is also given
for magnetostriction (MS) which tends to be strongly enhanced
in comparison to the case of alternating magnetization (AM).
For a quantitative assessment, it is advantageous to take alter-
nating magnetization (AM) in RD as a starting point. MS-
caused strain λ depends on the crystal orientation and the
resulting directions of magnetic moments within the domains.
Thus, a very specific behaviour results for the pronounced
Goss texture of HGO materials. A key role is given by the
(average) density ξ of moments which are oriented in the
oblique directions (ODs) [100] and [010] in about 45° to the
TD. For AM in RD (Fig.10), ξ can be assumed to be
proportional to the amount of lancet structures due to their
obliquely oriented slot domain. Starting out from the demag-
netized state, ξ increases from increasing B in order to reduce
stray field energy. Due to positive saturation magnetostriction,
a negative value λRD in RD results which is compensated by
positive λTD in TD and λND in normal direction (ND) of lower
intensity (assuming zero volume magnetostriction). For high B,
i.e. approaching saturation, magnetic moments change over to
the [001] (close to the RD). This explains that λRD becomes
positive. In general, AM yields moderate changes of ξ linked
with weak strain of less than 0.5 ppm. However, strong
harmonics result for high B due to the strongly non-linear
function λRD(B) as given in Fig.10.
Fig.10. Typical dependence of MS-caused strain λRD from induction BRD for
HGO SiFe with alternating magnetization in RD. Uneven domain width
favours the formation of lancets and thus high density ξ of magnetic moments
in oblique directions. These moments are withdrawn from the RD as an origin
of negative strain.
The case of RM may be linked with considerably strong
magnetization in TD which is built up by moments in oblique
directions and thus by high ξ. There result high values of strain
which favours effective measurement. For a full assessment,
the latter can be performed by three strain sensors corres-
ponding to three independent unknowns of the strain tensor,
including shear. For example, the measurement can be
performed by means of strain gauges or laser interferometers
[63] which are attached in three different directions of the
sample plane. As closer discussed in [34,35], the results yield
MS-caused strain λθ(ψ,t) for all directions of the plane (with θ
the angle to the RD) as a function of ψ (the angle of B to the
RD) and of time t. Finally, the assumption of constant volume
yields also strain in ND, corresponding to time variations d(t)
of the lamination thickness.
As an example, for an axis ratio of 0.47, Fig.11 includes λθ
for ψ = 0, i.e. for the moment when B passes through the RD.
Compared to the demagnetized state, the RD is shortened by
about 3 ppm, the TD elongated by about 3 ppm while the HD
is not affected. Extreme values result for the instant ψ = 90°.
Here, maximum negative strain of about -18 ppm arises for the
RD. This is due to the fact that a very high amount of moments
is withdrawn from the [001]. It is equally distributed to the two
ODs which explains that the TD is elongated by about 10 ppm.
A somewhat weaker increase of lamination thickness d is to be
expected under the assumption that the volume MS is zero.
Multi-directional measurements allow a deeper understan-
ding of basic mechanisms. But for technical applications, it is
sufficient to apply two sensors for the determination of the
strain values λRD(t) and λTD(t) as resulting in RD and TD,
respectively. For highly grain oriented materials, as illustrated
by Fig.11, this is acceptable due to two reasons:
(i) The main axes of strain remain "frozen-in" for the entire
period (in contrast to NO materials where they are
changing with time; e.g. [64]).
(ii) The HD tends to be almost free of strain in a general way.
Thus a third sensor offers little information if applied
diagonally, according to usual arrangements.
Corresponding results for peak-to-peak values are shown in
Fig.12 for conditions which can be assumed to be relevant for
industrial practice. A high significance for the generation of
vibration and thus also for noise results from the fact that
peak-to-peak strain values may approach 10 ppm, an increase
of one order of magnitude if we compare with AM (Fig.10).
Further, high harmonics arise, especially with respect to the
200 Hz spectral line. Those harmonics depend on both the
shape of the induction pattern and on the angular velocity ω of
the vector B: Changed ω(t) corresponds to changes of B(t) and
thus also of λ(t). On the other hand, specific experiments
revealed that the peak-to-peak strain can be assumed to be a
mere function of RD and a. However, a further impact
quantity is given by mechanical stress. As the best-checked
phenomenon, tension in TD yields distinct decreases of peak-
to-peak values in this direction [52].
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 10
Fig.11. "Snapshots" of magnetostriction-caused strain λθ (in ppm) for all di-
rections of the plane for four instants of time for HGO SiFe elliptically
magnetized with BRD = 1.7 T and a = 0.47 (roughly corresponding to Fig.5a).
Minimum values result for ψ = 0, i.e. the moment when B turns through the
RD with B = 1.7 T. Maximum values result for ψ = 90°, i.e. the moment when
B turns through the TD with B = 0.8 T (from [34]).
Fig.12. MS-caused strain (peak-to-peak values in ppm) for elliptically
magnetized hexagonal samples of SHGO SiFe as a function of axis ratio a
and different values of BRD . (a) λpp,RD in RD. (b) λpp,TD in TD.
According to Section V, rotational magnetization repre-
sents the strongest impact factor for the building factor of T-
joints, and - under certain circumstances - even for the total
building factor of a core. However, for T-joints, the data of
Fig.12 indicates that RM represents the strongest impact factor
also for magnetostrictive strain. On the other hand side, the
global, dynamic expansions of a transformer core are affected
by both magnetostriction and magneto-static forces [65,66].
Due to problems of separation - e.g. involving phase shifts -, it
is difficult to integrate over local strain values in order to
estimate the global effects. Since maximum RM-caused strain
arises in the RD, maximum global vibrations can be expected
in the direction of yoke axes. However, quantitative analyses
by means of accelerometer sensors prove to be complicated by
strong impact of clamping conditions.
Analogous to losses, we finally can expect that the impact
of RM on vibrations increases according to already discussed
anisotropy ranking NO - CGO - HGO - SHGO. However, this
does not mean that a SHGO core exhibits highest noise level.
On the contrary! One reason is that advanced materials show
strong relative RM-caused MS-increases, but low MS for
alternating magnetization. And a second reason is that the axis
ratio a tends to sink with increasing effective anisotropy. As a
conclusion, the impact of RM on transformer core noise needs
further study.
(a)
(b)
R
D
T
D
λ RD
a
a
λ TD
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 11
VII. MAIN CONCLUSIONS
The main conclusions of this review on rotational magne-
tization (RM) of highly grain oriented transformer core steel
are the following:
(1) The origin of RM is mainly given by the V-shaped S-limb
ends which represent a hindrance to the transport of flux. Due
to high effective anisotropy, RM may comprise the whole
yokes.
(2) The simulation of RM through RSSTs should consider
sufficient averaging over the coarse grain structure as well as
the high demand of excitation for hard directions (HDs).
(3) Field patterns H(t) as a function of induction patterns B(t)
offer both detailed information on the texture of material and
numerical data for the quantitative multi-directional descrip-
tion of material.
(4) Circular RM lacks practical relevance. The latter is given
mainly for rhombic patterns (including tilts).
(5) With respect to estimation of the local building factor,
attraction is given for related losses p that are related to AM. p
tends to show (a) strong increases with increasing axis ratio a
and increasing grade of material, (b) slight decreases with
increasing BRD , (c) distinct increases with rising unevenness of
the angular velocity of B and (d) also considerably large
increases with rising DC-components of excitation.
(6) Compared to AM, RM shows drastically increased magne-
tostriction-caused strain. Maximum peak-to-peak values arise
in the RD, lower ones in the TD and ND, zero-values in the
HDs.
(7) In finished cores, pronounced RM is restricted to the T-
joint regions. Here, strongly increased local loss can be
assumed to be the major contribution to the local building
factor. Excessive values of dynamic strain will create a local
source of core vibrations.
The above conclusions demonstrate that worldwide re-
search on rotational magnetization of transformer core mate-
rials has already achieved wide insight in the most relevant
technical aspects. In special, knowledge exists about the major
factors of impact. On the other hand side, theoretical and
numerical models for the prediction of magnetic performances
are rare. Mathematical models are needed which describe the
consequences of dynamic processes on classical eddy current
losses as a function of frequency and instantaneous angular
velocity of rotation. As well, research is needed on the micro-
magnetics of rotation processes. Domain studies should clarify
the mechanisms of both hysteresis and anomalous eddy
currents, as a basis for the estimation of the corresponding
portions of losses.
In particular for highly grain oriented materials, the men-
tioned tasks of research are connected with high expenditure.
However, the latter is justified by strong industrial relevance.
Considering international demand, even single percent
reduction of the building-factor is of commercial interest. As
well, the generation of audible noise gains relevance in the
course of increasing environmental consciousness. As a result,
rotational magnetization exhibits significant industrial rele-
vance, apart from being a challenging academic phenomenon.
ACKNOWLEDGMENT
The authors thank for support from the Austrian Science
Funds FWF (Project No. P21546-N22) and from Siemens
Transformers.
REFERENCES
[1] P.Weiss, V.Planer, "Hysteresis in rotating fields (in French)," J.de Phys.
(Theor.& Appl.), vol. 4/7, p. 5-27, 1908.
[2] A.J.Moses, B.Thomas, "The spatial variation of localized power loss in
two practical transformer T-joints," IEEE Trans.Magn., vol. 9, p. 655-
659, 1973.
[3] K.Narita, T.Yamaguchi, I.Chida, "Distribution of rotating flux and iron
loss in the T-joints of 3-phase 3-limbed model transformer cores," Proc.
Soft Magn.Mater. 2, Cardiff, paper 4.3, 1975.
[4] T.Nakata, Y. Ishihara, K.Yamada, A.Sasano, "Non-linear analysis of
rotating flux in the T-joints of a 3-phase 3-limbed model transformer
core," Proc. Soft Magn.Mater. 2, Cardiff, paper 4.5, 1975.
[5] A very large number of papers in the proceedings of 2DM Workshops,
Braunschweig 1991, Oita 1992, Torino 1993, Cardiff 1995, Grenoble
1997, Bad Gastein 2000, Lüdenscheid 2002, Ghent 2004, Czestochowa
2006 and Cardiff 2008.
[6] F.Fiorillo, Measurement and Characterization of Magnetic Materials.
Amsterdam: Elesevier, 2004.
[7] Y.Guo, J.G.Zhu, J.Zhong, H.Lu, J.X.Jin, "Measurement and modeling of
rotational core losses of soft magnetic materials used in electrical
machines: a review," IEEE Trans.Magn., vol. 44, p. 279-291, 2008.
[8] W.Brix, "Measurements of the rotational power loss in 3% SiFe at
various frequencies using a torque magnetometer," J.Magn. Magn.
Mater., vol. 26, p. 193-195, 1982.
[9] T.Sasaki, M.Imamura, S.Takada, Y.Suzuki, "Measurement of rotational
power losses in silicon-iron sheets using wattmeter method," IEEE
Trans.Magn., vol. 21, p. 1918-1920, 1985.
[10] F.Fiorillo, "On the measurement of rotational losses by means of the
rise of temperature method," IEC Publ. TC68 WG2 N58, 1990.
[11] M.Enokizono, J.Sievert, H.Ahlers, "Optimum yoke construction for
rotational loss measurements apparatus," Anal.Fis. B, vol. 86, p.320-322,
1990.
[12] W.Salz, "A two-dimensional measuring equipment for electrical steel,"
IEEE Trans.Magn., vol.30, p. 1253-1257, 1994.
[13] G.Suard, V.Chaves, J.C.Faugieres, C.L.Ramiarinjaona, J.F.Rialland,
"Rotational power losses in SiFe alloys and amorphous ribbons,"
Anal.Fis. B, vol. 86, p.223-225, 1990.
[14] A.Hasenzagl, B.Weiser, H.Pfützner, "Novel 3-phase excited single sheet
tester for rotational magnetization," J.Magn.Magn.Mater., vol. 160,
p.180-182, 1996.
[15] J.Sievert, H.Ahlers, M.Birkfeld, B.Cornut, F.Fiorillo, K.A.Hempel,
T.Kochmann, A.Kedous-Lebouc, T.Meydan, A.Moses, A.M.Rietto,
"European intercomparison of measurements of rotational power loss in
electrical sheet steel," J.Magn.Magn.Mater., vol. 160, p. 115-118, 1996.
[16] A.Kedous-Lebouc, B.Cornut, "Static and dynamic properties of electrical
steels under 2D excitation: from characterization to modeling," Proc.
1&2-D Magn.Meas.&Test., vol.6, p.172-179, 2000.
[17] D.Makaveev, M.Rauch, M.Wulf, J.Melkebeek, "Accurate field strength
measurement in rotational single sheet testers," J.Magn.Magn.Mater., vol
215-216, p. 673-676, 2000.
[18] D.Makaveev, J.Maes, J.Melkebeek, "Controlled circular magnetization
of electrical steel in rotational single sheet testers," IEEE Trans.Magn.,
vol. 37, p.2740-2742, 2001.
[19] V.Gorican, A.Hamler, M.Jesenik, B.Stumberger, M.Trlep, "Measure-
ment of magnetic properties of grain-oriented silicon steel using round
rotational single sheet tester (RSST)," J.Magn.Magn.Mater., vol 272-
276, p.2314-2316, 2004.
[20] S.Zurek, T.Meydan, “Errors in the power loss measured in clockwise
and anticlockwise rotational magnetisation. Part 1: Mathematical study.”
IEE Proc.Sci.Meas.Technol., vol.153, p.147-151, 2006.
Submitted for publication to IEEE Transactions on Magnetics, 2nd revision 12
[21] S.Zurek, T.Meydan, “Errors in the power loss measured in clockwise
and anticlockwise rotational magnetisation. Part 2: Physical phenomena,”
IEE Proc.-Sci.Meas.Technol., vol.153, p.152-157, 2006.
[22] Z.W.Lin, J.G.Zhu, Y.G.Guo, J.J.Zhong, S.Wang, "Magnetic properties
of soft magnetic composites under 3-dimensional excitations," Appl.El.
Magn.Mech., vol.25, p. 237-241, 2007.
[23] J.Zhong, Y.Guo, Y.Zhu, H.Lu, J.Jin, "Development of measuring
techniques for rotational core losses of soft magnetic materials," Nat.Sci.,
vol.2, p.1-12, 2007.
[24] T.Todaka, Y.Maeda, M.Enokizono, "Counter-clockwise/clockwise rota-
tional losses under high magnetic field," Prz.Elektrot./El.Rev., vol. 85,
p.20-24, 2009.
[25] C.A.Hernandez-Aramburo, T.C.Green, A.C.Smith, “Estimating rota-
tional iron loss in an induction machine,” IEEE Trans.Magn., vol. 39,
p.3527-3533, 2003.
[26] Nippon Steel Corporation (NSC) Catalogue Cat., Electrical steel sheets.
Cat.no.EXE367.
[27] H.Pfützner, “Rotational magnetization and rotational losses of grain
oriented silicon steel sheets – fundamental aspects and theory,” IEEE
Trans.Magn., vol.30, p.2802-2807, 1994.
[28] F.Alves, "Recent advances on 1D-2D magnetic sensors: Contribution of
nanocrystalline materials in MI-sensors," Prz.Elektrot./El.Rev., vol 81, p.
35-39, 2005.
[29] O.Stupakov, R.Wood, Y.Melikhov, D.Jiles, "Measurement of electrical
steels with direct field determination," IEEE Trans. Mag., vol.46, p.298-
301, 2010.
[30] G.Loisos, A.J.Moses, "Critical evaluation and limitations of localized
flux density measurements in electrical steels," IEEE Trans.Magn., vol.
37, p. 2755-2757, 2001.
[31] H.Pfützner, G.Krismanic, "The needle method for induction tests:
sources of error," IEEE Trans.Magn., vol 40, p.1610-1616, 2004.
[32] J.J.Zhong, J.G.Zhu, Y.G.Guo, Z.W.Lin, "Improved measurement with 2-
D rotating fluxes considering the effect of internal field," IEEE
Trans.Magn., vol. 41, p. 3709-3711, 2005.
[33] K.Mori, S.Yanase, Y.Okazaki, S.Hashi, "2-D magnetic rotational loss of
electrical steel at high magnetic flux density," IEEE Trans.Magn., vol.
41, p. 3310-3312, 2005.
[34] A.Hasenzagl, B.Weiser, H.Pfützner, "Magnetostriction of 3% SiFe for
2D magnetization patterns," J.Magn.Magn.Mater., vol.160, p.55-56,
1996.
[35] D.Wakabayashi, Y.Maeda, H.Simoji, T.Todaka, M.Enokizono,
"Measurement of vector magnetostriction in alternating and rotating
magnetic field," Prz.Elektrot./El.Rev., vol 85, p. 34-38, 2009.
[36] S.Zurek, T.Meydan, "Rotational power losses and vector loci under
controlled high flux density and magnetic field in electrical steel sheets,"
IEEE Trans.Magn., vol. 42, p.2815-2817, 2006.
[37] S.Sugimoto, S.Urata, A.Ikariga, H.Shimoji, T.Todaka, M.Enokizono, "A
new measurement device for two-dimensional vector-magnetic property
in high magnetic flux density ranges," Prz.Elektrot./El.Rev., vol 81, p.
27-30, 2005.
[38] V.Permiakov, A.Pulnikov, L.Dupre, J.Melkebeek, "2D magnetic mea-
surements under 1D stress," Prz.Elektrot./El.Rev., vol 81, p. 68-72, 2005.
[39] A.J.Moses, "The case for characterization of rotational losses under pure
rotational field conditions," Abstr. 8th 2DM Worksh. Ghent, p. 10, 2004.
[40] G.Krismanic, C.Krell, H.Pfützner, "Relevance of 2D magnetization for
transformer cores," Proc. 1&2-D Magn.Meas.&Test., vol.6, p.310-316,
2000.
[41] J.V.Leite, A.Benabou, N.Sadowski, M.V.Ferreira da Luz, "Finite element
three-phase transformer modeling taking into account a vector hysteresis
model," IEEE Trans.Magn., vol. 45, p. 1716-1719, 2009.
[42] C.Lee, H.K.Jung, "Nonlinear analysis of the three-phase transformer
considering the anisotropy with voltage source," IEEE Trans.Magn., vol.
36, p.491-499, 2000.
[43] C.Lee, H.K.Jung, "Two-dimensional analysis of three-phase transformer
with load variation considering anisotropy and overlapping stacking,"
IEEE Trans.Magn., vol. 36, p.693-696, 2000.
[44] Y.Zhang, Y.H.Eum, W.Li, D.Xie, C.S.Koh, "An improved modeling of
vector magnetic properties of electrical steel sheet for FEM application
and its experimental verification," IEEE Trans.Magn., vol. 45, p. 1162-
1165, 2009.
[45] H.S.Yoon, Y.H.Eum, Y.Zhang, P.S.Shin, C.S.Koh, "Comparison of
magnetic reluctivity models for FEA considering two-dimensional mag-
netic properties," IEEE Trans.Magn., vol. 45, p.1202-1205, 2009.
[46] M.Enokizono, "Vector magnetic property and magnetic characteristic
analysis by vector magneto-hysteretic E&S model," IEEE Trans.Magn.,
vol. 45, p. 1148-1153, 2009.
[47] H.V.Sande, T.Boonen, I.Podoleanu, F.Henrotte, K.Hameyer, "Simulation
of a three-phase transformer using an improved anisotropy model," IEEE
Trans.Magn., vol. 40, p.850-855, 2004.
[48] T.Tamaki, K.Fujisaki, K.Wajima, K.Fujiwara, "Comparison of magnetic
field analysis methods considering magnetic anisotropy," IEEE
Trans.Magn., vol. 46, p.187-190, 2010.
[49] A.J.Moses, "Rotational magnetization - problems in experimental and
theoretical studies of electrical steels and amorphous magnetic materials,"
IEEE Trans.Magn. vol. 30, p.902-906, 1994.
[50] J.Sievert, "Studies on the measurement of 2-dimensional magnetic
phenomena in electrical sheet steel at PTB," PTB-Berichte, vol. E-43, p
102-116, 1992.
[51] H.Pfützner, G.Shilyashki, F.Hofbauer, D.Sabic, E.Mulasalihovic,
V.Galabov, "Effects of DC-bias on loss distribution of model transformer
core," J.El.Eng., vol.61, p.126-129, 2010.
[52] G.Krell, E.Leiss, H.Pfützner, "Effects of stress on permeability, losses
and magnetostriction of crystalline and amorphous soft magnetic
materials under rotational magnetization," Proc. 1&2-D Magn.Meas.
&Test., vol.6, p.242-247, 2000.
[53] V.Permiakov, A.Pulnikov, L.Dupre, J.Melkebeek, "Rotational magne-
tization in nonoriented Fe-Si steel under uniaxial compressive and tensile
stresses," IEEE Trans.Magn., vol. 40, p. 2760-2762, 2004.
[54] A.J.Moses, B.Thomas: "The spatial variation of localized power loss in
two practical transformer T-joints," IEEE Trans. Mag., vol.9, p.655-659,
1973.
[55] A.J.Moses, B.Thomas, J.E.Thompson: "Power loss and flux density
distributions in the T-joint of a three phase transformer core," IEEE
Trans. Mag., vol.8, pp.785-790, 1972.
[56] Z.Valkovic, A.Rezic, "Flux and loss distribution in three-limb core with
staggered T-joint," Proc. Int.Symp.Elm.Fields, Lodz, 121-124, 1989.
[57] Nippon Steel Corporation (NSC) Catalogue, Technical data on
Orientcore-HI-B, EXE367.
[58] Z.Valkovic, "Recent problems of transformer core design," Phys.Scr.,
vol.T24, p.71-74, 1988.
[59] A.J.Moses, "Prediction of core losses of three phase transformers from
estimation of the components contributing to the building factor,"
J.Magn.Magn.Mater. , vol.254-255, p. 615-617, 2003.
[60] R.S.Girgis, E.G.Nijenhuis, K.Gramm, J.E.Wrethag, "Experimental
investigations on effect of core production attributes on transformer core
loss performance," IEEE Trans.Pow.Del., vol.13, p.526-531, 1998.
[61] M.Ishida, S.Okabe, T.Imamura, M.Komatsubara, "Model transformer
evaluation of high-permeability grain-oriented electrical steels,"
J.Mater.Sci.Technol., vol.16, p. 223-227, 2000.
[62] A.M.A.Haidar, S.Taib, I.Daut, S.Uthman, "Evaluation of transformer
magnetizing core loss," J.Appl.Sci., vol.6, p.2579-2585, 2006.
[63] S.G.Ghalamestani, T.G.D.Hilgert, L.Vandevelde, J.J.J.Dirckx,
J.A.A.Melkebeek, "Magnetostriction measurement by using dual
heterodyne laser interferometers," IEEE Trans.Magn., vol. 46, p. 505-
508, 2010.
[64] S.Somkun, A.J.Moses, P.I.Anderson, P.Klimczyk, "Magnetostriction
anisotropy and rotational magnetostriction of a nonoriented electrical
steel," IEEE Trans.Magn., vol. 46, p.302-305, 2010.
[65] M.Veen, "Measuring acoustic noise emitted by power transformers,"
Proc. 19th Audio Eng.Soc.Conv., Los Angeles, p.1-19, 2000.
[66] G.Shilyashki, H.Pfützner, F.Hofbauer, D.Sabic, V.Galabov, "Magneto-
striction distribution in a model transformer core," J.El.Eng., vol.61,
p.130-132, 2010.
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