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.

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