A Coenergy-Based Model
of Anisotropic Magnetic Properties
K. Chwastek, J. Szczygłowski1, W. Wilczyński2
1
Czestochowa University of Technology, Czestochowa, Poland
2
Institute of Electrical Engineering, Warsaw, Poland
ABSTRACT
Anisotropy affects magnetic properties of steels used in electrical engineering.
Designers of magnetic circuits require proper modelling tools to take this
phenomenon into account. The concept of coenergy offers an approach, which is
thermodynamically coherent and simple to be implemented numerically. The model
advanced
recently
by
Biró
et
al.
B

f
(
H
)
is applied to describe anisotropic m
m dependencies at arbitrary angles for
grain-oriented steel, grade 097-30N5. The modelled dependencies are in an acceptable
agreement with experimental data for engineering purposes. The description makes it
possible to predict the magnetization curves at arbitrary angles using a limited number
of measurements. It allows to cut down time and efforts related to sample processing
for measurements.
Introduction
The concept of coenergy [1,2] has been recently the subject of renewed interest in
the scientific community. Many useful descriptions of complex mechatronic devices,
e.g. those based on bond graph theory [3], can be easily developed using this
formalism [4,5].
The great potential behind the coenergy-based models (their coherence with the
principles of thermodynamics) has been noticed by B. Cornut [6]. A practical
approach to describe the anisotropic magnetic properties in soft magnetic materials
has been advanced by T. Péra et al. [7]. Recently, the description has been further
extended by O. Biró et al. [8]. Their proposal is the subject of study in the present
paper.
Even the so-called isotropic steels used in electrical engineering can reveal a
significant anisotropy [6,9-11]. One of the indicators in common use is the ratio
P1.5  
P10.5  P190
.5
 100 %,
P10.5  P190
.5
where the symbols P10.5 , P190
.5 denote loss density measured either in the rolling direction
or in the transverse one, respectively, which is determined in quasi-static conditions at
Bm  1.5 T. For example, the value of P1.5 for the isotropic electrical steel used for
alternators, grade M 330-35A (basic properties: maximum loss density at 1.5 T, 50 Hz
– 3.30 W/kg, thickness 0.35 mm, minimum flux density for H  2500 A/m is 1.49 T,
whereas for H  5000 A/m is 1.6 T) may reach 17 %, as declared by the producer.
Yet, the true values obtained in practical measurements are smaller, about 11 %, for
this grade [11].
Other macroscopic anisotropy measures in common use are defined as the
X
0
X
 B2500
 B2500
differences B2500
and H1X.5  H10.5  H1X.5 , where X is arbitrary angle
1-92
(usually either 90o or 55o / 60o ), B2500 denotes magnetic induction measured at
H  2500 A/m, whereas H1.5 denotes magnetic field strength measured at B  1.5 T.
The graphs like the one depicted in Fig. 1 are also useful for the designers of magnetic
circuits.
Fig. 1. Induction anisotropy for electrical steel, grade 097-30N5.
Source: own measurements
The effect of anisotropy is yet more pronounced for anisotropic steels used in
transformers. It is directly related to grain orientation within magnetic material, cf.
Fig. 2.
Fig. 2. Magnetic domains in grain-oriented electrical steel as seen in domain viewer.
The arrow shows rolling direction of the steel. Black curves outline the grain
structure. Source: Wikimedia, Creative Commons license, authors: Stanislaw Zurek,
Harshad V. Patel, Wolfson Centre for Magnetics, Cardiff
The knowledge of directional magnetic properties of electrical steel allows us to
tailor and optimize the operation point of magnetic circuits. On the other hand, it is a
1-93
tedious and laborous task to prepare samples for magnetic measurements, which are
cut at different angles from the rolling direction. The deteriorating influence of
processing technologies on magnetic properties of electrical steel has been examined
in detail by one of the co-authors of the present paper [12, 13].
a)
b)
c)
d)
Fig. 3. Microstructure of a stator tooth in an electric machine: a) highly deformed
edge of the tooth, b) middle part of the tooth, c) at the edge after classical thermal
treatment of the whole stator sheet pack, d) at the edge after induction treatment of the
tooth-slot zone [12]
Taking the abovegiven into account, it is thus desirable to focus on descriptions,
which make it possible to predict the directional dependencies between the vectors:


magnetic field strength H and magnetic flux density B for arbitrary angles, from a
limited number of magnetic measurements, carried out for well-specified directions,
i.e. for the rolling and the transverse one. The solution is to avail of the coenergybased approach. On the one hand, it seems much easier to work with scalar coenergy

based relationships than to consider vector B  f (H ) dependencies. On the other one,
some approaches advanced in literature, e.g. those based on linear interpolation
between many data, are incorrect from the point of thermodynamics, as pointed out by
B. Cornut [6]. Some first-order derivatives of induction and field are in fact secondorder derivatives of a state variable and should be equal, what is neglected by some
model developers. The coenergy-based models, like the one described in Ref. [14],
are thermodynamically coherent, what makes them useful and important.
Model description
Manufacturers of electrical steel usually provide the dependencies between the
amplitudes Bm  f ( H m ) just for two orthogonal directions: when the samples are
subjected to magnetization along the rolling direction and for the transverse


orientation. For these directions the vectors H and B are collinear. The
measurements in these two directions provide the basis for formulation of the model,
which is capable to predict Bm  f ( H m ) for intermediate directions.
1-94
Whereas the best magnetization conditions are obtained for the rolling direction
(angle   0o , cf. Fig. 4, the opposite is not true for the transverse direction.
Actually, the worst magnetization conditions are obtained for the angle   60o.
Fig. 4. The considered geometry. Index I =1, 2 denotes the rolling
and the transverse direction, respectively.
In order to describe the contours traced by flux density vector for any fixed value
of H , Biró et al. [8] have proposed to use the relationship
  n
  n
 B  e1   B  e2 

  
  1 ,
B
(
H
)
B
(
H
)
 1
  2

(1)
 
where e1,e2 are unit vectors, whereas B1 ( H ) and B2 ( H ) denote nonlinear scalar
dependencies in the rolling (i = 1) and the transverse direction (i = 2) , respectively,
whereas n is a model parameter. Biró et al. [8] suggest assuming n  1.4.
The relationship (1) has to be supplemented with another one, which determines
the angle  . In the aforementioned Reference it is assumed, that static permeability
  B / H in the rolling direction is solely the function of flux density in that direction,
what allows us to write
 
  B
B  e1  H  e1
(2)
H1 ( B )
or, in equivalent form H1 ( B) cosB  H cosH .
The international standard IEC 60404 prescribes the conditions for magnetic
measurements, e.g. the shape of flux density waveform should be controlled to be
sine. This is achieved by the use of additional feedbacks in measurement setups.
The choice of flux density as the input variable in the developed model is a logical
consequence of the aforementioned regulations. The sought dependencies are
B  B ( H projected ) for given angle  . The design of measurement setup implies B  0o.
Modelling
Measurements were carried out for samples of anistropic steel, grade 097-30-N5
using the Epstein device. The basic properties of the examined steel are given in Table
I.
1-95
Table 1 Basic magnetic properties of the examined steel
Grade
Thickness Maximum loss
Minimum polarization
density
J
@ 1,7 T, 50 Hz
@ 800 A/m
097-30-N5 0,30 mm
1,50
1,75
The samples were cut into stripes 305  0,5 mm  30  0,2 mm at different angles
from the rolling direction. The measurements were carried out in accordance with the
international standard IEC 404 – 2 part 2 1996 – 03: Methods of measurement of the
magnetic properties of electrical steel sheet and strip by means of an Epstein frame.
The coenergy-based model was implemented in a popular spreadsheet software. In
order to interpolate the measurement data for two orthogonal angles   0o ,90o ,
the freeware add-on [15] was found useful. The exponent n was kept fixed to 1.4.
The results of modelling the Bm  f ( H m ) dependencies for chosen values of angle
 are depicted in Figure 5. The values of angle H varied in the range 64o ;89o for
the depicted cases and were lower for lower values of flux density B and angle  .
Fig. 5. The results of modelling for grain-oriented steel 097-30-N5.
On the basis of the presented results it can be stated, that the coenergy-based model is
able to reproduce the directional Bm  f ( H m ) dependencies in anisotropic steel
reasonably well. The discrepancies between the model and experimental data, in
particular for   30o , could be the result of arbitrary choice of the exponent n = 1.4.
Conclusions
A simple description of anisotropic magnetic properties of electrical steel based on
coenergy concept has been presented. The model makes it possible to predict the
Bm  f ( H m ) dependencies for arbitrary values of angle  on the basis of a limited
number of measurements, carried out in two orthogonal directions. It can be easily
implemented in a popular spreadsheet software. The model reproduces correctly the
1-96
typical behaviour of anisotropic steel – the worst magnetic properties are obtained for
closeto
The proposed description is easier in implementation than the existing models of
rotational losses. It also allows us to avoid troublesome measurements for
intermediate directions of samples.
References
[1] Mawardi O.B., “On the concept of coenergy”, Journal of the Franklin Institute
1957, 264, No. 4, 313-332.
[2] Woodson H.H.,
Melcher J.R., “Electromechanical dynamics”, 3 vols.
(Massachusetts Institute of Technology: MIT OpenCourseWare). http://mit.ocw.edu.
Accessed March 2010.
[3] Borutzky W., “Bond graph modelling and simulation of multidisciplinary systems
– an introduction”, Simulation Modelling Practice and Theory 2009,17, 3-21.
[4] Baliño J.L., Lattereguy A.E., Gandolfo Gaso E.F., “A general bond graph
approach for computational fluid dynamics”, Simulation Modelling Practice and
Theory 2006, 14, 884-908.
[5] Donaire A., Junco S., “Energy shaping, interconnection and damping assignment,
and integral control in the bond graph domain”, Simulation Modelling Practice and
Theory 2009, 17, 152-154.
[6] Cornut B., Kedous-Lebouc A., Waeckerlé Th., “From metallurgy to modelling of
electrical steels: a multiple approach to their behaviour and use based on physics and
experimental investigations (invited paper)”, Journal of Magnetism and Magnetic
Materials 1996, 160, 102-106.
[7] Péra T., Ossart F., Waeckerlé Th., “Numerical representation for anisotropic
materials based on coenergy modeling”, Journal of Applied Physics 1993, 73 (10),
6784-6786.
[8] Biró O., Außerhofer S., Preis K., Chen Y., “A modified elliptic model of
anisotropy in nonlinear magnetic materials”, In: Proceedings of the 8th International
Symposium on Electric and Magnetic Fields (EMF'2009), Mondovi (Italy) 26th-29th
May 2009, Institute Montefiore, University of Liége, Book of Abstracts pp. 25-26,
full text on CD.
[9] Tumański S. , “Investigations of the anisotropic behaviour of SiFe steels”, Journal
of Magnetism and Magnetic Materials 2003, 254-255, 50-53.
[10] Fryskowski B., “Experimental evaluation of magnetic anisotropy in electrical
steel sheets”, Journal of Magnetism and Magnetic Materials 2008, 320, 515-522.
[11] Chwastek K., PhD Dissertation “The application of effective field theory in
modelling of dynamic hysteresis loops” – in Polish, Czestochowa University of
Technology, 2007.
[12] Wilczyński W., “The influence of technology on magnetic properties of cores in
electric machines” – in Polish, Prace Instytutu Elektrotechniki nr 215, Instytut
Elektrotechniki 2003.
[13] Wilczyński W., “Influence of magnetic circuit production for their magnetic
properties”, Journal of Materials Science 2003, 38, 4905-4910.
[14] Dupré L., Van Keer R., Melkebeek J.A.A., Numerical modeling of anisotropy
and eddy current effects in ferromagnetic laminations using a co-energy formulation,
International Journal of Numerical Modelling: Electronic Networks, Devices and
Fields 2001, 14, 423-438.
[15] http://www.netrax.net/~jdavita/XlXtrFun/XlXtrFun.htm. Accessed March 2010.
1-97