1D and 2D Magnetization in Electrical Steels under Uniaxial Stress

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1D and 2D Magnetization in Electrical Steels under Uniaxial Stress
Viatcheslav Permiakov
Promotoren: prof. dr. ir. J. Melkebeek, prof. dr. ir. L. Dupré
Proefschrift ingediend tot het behalen van de graad van
Doctor in de Toegepaste Wetenschappen
Vakgroep Elektrische Energie, Systemen en Automatisering
Voorzitter: prof. dr. ir. J. Melkebeek
Faculteit Ingenieurswetenschappen
Academiejaar 2004 - 2005
ISBN 90-8578-022-5
NUR 961, 967
Wettelijk depot: D/2005/10.500/22
Acknowledgement
1
Acknowledgement
The author kindly thanks the two promoters, Prof. Jan Melkebeek and
Prof. Luc Dupré, for a chance to obtain an internationally recognized
degree from Ghent University, Belgium, and for a great help with
improving my English writing skills when preparing the Ph.D thesis.
The author would like to thank the following people for their
contribution to the current research and many discussions we had:
ƒ Jehudi Maes, engineer at PICANOL BELGIUM
ƒ Dimitre Makaveev, engineer at BOMBARDIER BELGIUM
ƒ Marc De Wulf, researcher at OCAS, ARCELOR GROUP
ƒ Josef Ghijselen, engineer at INVERTO BELGIUM
ƒ Jan Declercq, head R&D at PAUWELS INTERNATIONAL
ƒ Tony Moses, head of Wolfson Center, Cardiff, UK
ƒ Fausto Fiorillo, research director at IEN, Torino, ITALY
ƒ Martin J. Sablik, senior researcher in San Antonio, TX, USA
ƒ Johannes Sievert, senior researcher at PTB, GERMANY
ƒ Philip Beckley, senior researcher at European Steels, UK
Special thanks to a former colleague from Urals Technical University,
Alexander Pulnikov, for his contribution to this study.
Voor de administratieve taken kon ik steeds terecht bij Ingrid Dubois. Bij
technische problemen stonden Christian Vervust en Fernand De Boever
steeds paraat.
This work was financially supported by the STWW-project IWT 980357,
the Interuniversity Attraction Poles project IAP-P5/34, the GOA-project
99-200/4, the FWO-project G.0322.04. The author would like to thank the
project partners, participating in the above mentioned R&D projects.
The work is devoted to my parents, as it is them who gave me the birth
and the education for life. Спасибо моим родителям за то, что дали
мне возможность получить лучшее образование и путевку в жизнь.
2
Summary
Summary
The aim of this study is to expand the knowledge about electromagnetic
behaviour of non-oriented and grain-oriented electrical steels, used in all
parts of Great Chain of Energy Transformation. This study deals with
the effects of uniaxial stress on magnetic properties of electrical steels.
The work consists of two major parts. The first part defines the
framework of the present study. The second part starts from a 1D study,
followed by a more complex 2D magnetization in electrical steels.
Chapter 1 is an introduction, positioning the subject of this study.
A better understanding of various production procedures leads to a
more accurate prediction of energy losses and the "building factor".
Chapter 2 deals with the analysis of the working conditions,
affecting the behaviour of electrical steel in actual machines. A detailed
study is performed for 2D magnetic conditions in induction machines,
from theoretical and experimental point of view. The iron energy losses
depend on magnetic flux patterns as well as on mechanical stresses.
Chapter 3 presents the art of magnetic measurements, followed
by the choice of the magnetic measurement technique employed in the
present study. The advanced magnetomechanical setup created in
EELAB allows to control and maintain sinusoidal induction and to carry
out an unique range of 2D magnetic measurements under 1D stress.
Evidently, the extensive know-how in magnetic measurements
created by the team of EELAB, where the author had the chance to
reside for five years, has contributed to this doctoral research.
Chapter 4 is an investigation of the simplest 1D case of uniaxial
magnetization at uniaxial stress. A physically sound explanation is
presented for the stress effects. The loss separation is applied for a
statistical characterization of electrical steels under magnetomechanical
conditions. Although some effects of stresses are known in literature, a
few new aspects are presented.
Chapter 5 deals with the complex 2D magnetic measurements
under uniaxial compression and tension. This chapter of the study
includes both alternating magnetization and rotational magnetization.
Chapter 6 concludes this study with the recommendations on
how the energy loss in various electrical steels can behave under stress,
as well as what other subjects are related to the present study.
The author hopes that the work has extended the know-how of
EELAB as well as has consolidated the position of EELAB worldwide.
Contents
3
CONTENTS
Acknowledgement
1
Summary
2
Contents
3
List of Symbols
7
1. INTRODUCTION.
1.1. Energy transformation.
1.2. Energy losses and magnetism.
1.2.1. The history of magnetism.
1.2.2. Magnetic materials and domain theory.
1.2.3. Application of magnetic materials in energy
transformation.
1.3. Electrical steels.
1.3.1. Lamination.
1.3.2. Resistivity and various alloys.
1.3.3. Grain size and orientation of grains.
1.3.4. Production of electrical steels.
1.4. Stress effects in electrical steels.
1.4.1. Production stress in electrical steels.
1.4.2. The effect of punching.
1.4.3. The effect of core building.
1.5. Engineering choice of electrical steels.
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2. WORKING CONDITIONS OF ELECTRICAL STEELS.
2.1. Standard characteristics of electrical steel.
2.2. Target research of working conditions in electrical steels.
2.2.1. Target research in grain-oriented steels.
2.2.2. Target research in non-oriented steels.
2.3. 2D magnetic working conditions in induction machines.
2.4. 2D mechanical working conditions in induction machines.
2.5. Conclusions.
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56
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29
4
Contents
3. MAGNETIC MEASUREMENTS UNDER STRESS.
3.1. Standard magnetic measurements in electrical steels.
3.1.1. Epstein frame.
3.1.2. Single sheet tester.
3.1.3. DC magnetic measurements of iron and steel.
3.1.4. Magnetic measurement techniques used in standard
methods.
3.2. State of the art in 2D magnetic measurements.
3.2.1. 2D single sheet testers.
3.2.2. Shift to a horizontal rotational single sheet tester.
3.2.3. Typical operational algorithm for magnetic
measurements.
3.3. Magnetic measurements under stress.
3.4. 2D magnetic measurement system under stress.
3.4.1. The mechanical subsystem of the 2D setup.
3.4.2. The magnetic subsystem of the 2D setup.
3.4.3. Magnetic measurement technique.
3.4.4. Waveform control.
3.5. Magnetic measurement procedure. Operational software.
3.6. Sample preparation.
3.7. Conclusions.
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65
4. 1D MAGNETIC MEASUREMENTS UNDER STRESS.
4.1. Uniaxial measurements in electrical steels.
4.1.1. Uniaxial magnetization in grain-oriented steels.
4.1.2. Uniaxial magnetization in non-oriented steels.
4.1.3. Uniaxial magnetization at tensile plastic
deformation.
4.1.4. Uniaxial magnetization in semi-processed and
annealed steels.
4.1.5. Uniaxial distorted magnetization under stress.
4.1.6. Experimental results of 1D measurements:
preliminary tendencies.
4.2. The magnetomechanical effect: theoretical approach.
4.2.1. Energy theory and domain structures.
4.2.2. Applied stress and domain theory.
4.2.3. Levels of understanding: microscopic, domain,
macroscopic.
4.2.4. Explanation of various effects: the cross points in
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Contents
two quadrants.
Explanation of various effects: the critical tensile
stress.
4.2.6. Explanation of various effects: the plastic
deformation.
4.3. Statistical theory of energy losses.
4.3.1. Classical loss.
4.3.2. Hysteresis loss.
4.3.3. Excess loss.
4.3.4. Loss separation technique.
4.4. Separation of losses under applied stress.
4.5. Conclusions.
4.2.5.
5
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153
5. 2D MAGNETIC MEASUREMENTS UNDER STRESS.
5.1. Vector magnetization under uniaxial mechanical load.
5.2. 2D magnetic measurements in electrical steels
5.2.1. Justification of the 1D measurements in Chapter 4.
5.2.2. Alternating magnetization at 0 and 90 degrees with
uniaxial stress.
5.2.3. Circular rotational magnetization.
5.2.4. Energy loss at 2D magnetization under stress.
5.2.5. 2D magnetization under tensile plastic deformation.
5.3. 2D magnetomechanical effect in literature.
5.4. Ratios between rotational and alternating losses.
5.5. Conclusions.
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6. STRESS EFFECTS IN ELECTRICAL STEELS.
6.1. Conclusions of the present study.
6.1.1. The art of 2D magnetomechanical measurements.
6.1.2. The 1D magnetomechanical measurements.
6.1.3. The 2D alternating magnetomechanical
measurements.
6.1.4. The 2D rotational magnetomechanical
measurements.
6.1.5. The parameters and the stress effects.
6.2. Extensions of the present study.
6.2.1. Residual stresses.
6.2.2. NDE: non-destructive evaluation.
6.2.3. Magnetostriction.
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Contents
6.2.4. Biaxial stresses.
6.3. Applications of the present research.
6.3.1. Magnetomechanical measurements in soft magnetic
materials.
6.3.2. Recommendations to manufacturers of electrical
steels.
6.3.3. Recommendations to manufacturers of power
transformers.
6.3.4. Recommendations to manufacturers of rotating
machines.
6.4. Which engineering choice of section 1.5 is better?
6.5. Conclusions.
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List of International Publications
195
List of Attended Conferences and Awards
197
Bibliography
199
List of Symbols
7
LIST OF SYMBOLS
General notations:
•
•
•
•
•
Scalars: A.
Vectors: A (the symbol in bold).
Components of the vectors, or their projections on main axes: Ax.
Time variations of the scalars: A(t).
Various parameters: V0, Wh.
Symbols:
Symbol
α
ρ
Φ
µ
µ0
µ0M
a
B
Br
D
E
f
G
H
Hc
I
J
lm
M
N
NS
P
S
t
T
Tc
Definition
Any angle between vectors [deg]
Mass density [kg/m3]
Magnetic flux [Wb]
Magnetic permeability
Permeability of free space equal to 4π10-7 [H/m]
Magnetic polarization [T]
Axis ratio of ellipse, varies from 0 to 1
Magnetic flux density [T]
Remanence [T]
Electric flux density [C/m2]
Electrical field [V/m]
Magnetizing frequency [Hz]
Parameter equal to 0.1357
Magnetic field [A/m]
Coercive field, or coercive force [A/m]
Electric current [A]
Electric current density [A/m2]
Mean magnetic path length [m]
Magnetization [A/m]
Number of turns of a magnetizing winding
Effective turns of a search coil, e.g. the H coil
Power loss [W/kg]
Cross sectional area [m2]
Time [sec]
Period [sec]
Curie temperature [K]
8
List of Symbols
U
V
V0
Vsensor
W
Induced emf in a search coil [V]
Voltage of the excitation windings [V]
Parameter related to microstructure [A/m]
Voltage measured by the needle probes [V]
Energy loss [J/ m3]
Abbreviations:
emf
rms
1D
2D
AC
DC
GO
NO
RD
RSST
SST
TD
Electromotive force
Effective value, stands for "root-mean-square"
One-dimensional
Two-dimensional
Alternating current
Direct current
Grain-oriented steel
Non-oriented steel
Rolling direction
Rotational single sheet tester
Single sheet tester
Transverse direction
Introduction
9
There are no such things as applied sciences, only applications of science.
Louis Pasteur (1822 - 1895)
Engineering is the science of economy, of conserving the energy,
kinetic and potential, provided and stored up by nature for the use of man.
The task is to utilize this energy so that there may be the least possible waste.
William A. Smith, 1908.
CHAPTER 1. INTRODUCTION.
1.1. Energy transformation.
One of the fundamental features of energy is its ability to be transformed
or converted from one form of energy into another. Different forms of
energy have been discovered during the modern history of mankind. It
is also believed that some forms of energy are yet to be discovered.
The curiosity of human nature was a driving force towards the
practice of how to employ different forms of energy. One of the causes of
the industrial revolution was the knowledge of how to transform energy
and how to control that transformation. The next natural step is making
knowledge useful for individual and society needs.
One of the universal forms of energy is electrical energy. It is
easy to produce, easy to transport, and most important, easy to convert
from one form of energy into another. The Russian engineer DolivoDobrovolsky was the first to create the complete chain of AC electrical
energy transformation, including a three-phase induction machine. In
1891, he presented the AC system at the World Electrotechnical Show in
Frankfurt. Today, the majority of energy systems are built based on a
similar chain of AC energy transformation as shown in Fig. 1.1.
Fig. 1.1. Energy transformation in Great Chain of Energy Transportation.
10
Chapter 1.
A schematic chain of electrical energy transportation consists of a
generator, which transforms mechanical energy into electrical energy, a
step-up transformer, which converts the AC energy from low voltage to
high voltage, a line or a grid, which transports electrical energy, a series
of step-down transformers, and finally, an electrical energy consumer, as
shown in Fig. 1.1. The major consumers of electrical energy are electrical
motors, which transform electrical energy into mechanical work.
The schematic energy chain shown in Fig. 1.1 looks fine,
however, it hides a side effect of energy transformation. Whether
electrical energy is transformed into mechanical energy or mechanical
energy into electrical energy or electrical energy into electrical energy,
there is always some part of the energy which is lost during the
transformation. In electromagnetic energy conversion, an important part
of this energy loss is due to the magnetic core of the converting device.
Here, the questions of efficiency and economical value arise.
How crucial is that part of energy for the transformation process? What
is the cost of the energy loss? What are the ways to improve the system
efficiency? How to predict what happens in case of new designs or when
new materials are used? What will be the energy loss when the existing
technology of electrical machine production is modified?
1.2. Energy losses and magnetism.
When considering a typical chain of electrical energy transformation,
two types of electrical machines can be distinguished. One type
transforms mechanical energy into electrical energy, or vice versa, i.e.
generators and motors. Another type transforms AC electrical energy of
one value into another, i.e. transformers. Here and later, both rotating
and stationary electromagnetic energy converters, i.e. motors and
transformers, are called "machines". Due to the fact, that those two types
of electrical machines participate in the majority of electrical energy
transformation, new knowledge about these machines is very valuable
to the industrial and scientific society.
The energy conversion in transformers and induction machines
is based on electromagnetic principles. In order to understand why this
principle was chosen in the majority of electrical machines, one should
take a brief look at the historical development of scientific ideas, which
were the foundation of all electromagnetic applications.
Introduction
1.2.1.
11
The history of magnetism.
"Once upon a time some sheep shepherds in Magnesia region of Turkey found
that their iron-shod staffs became attracted to certain rocks. A regional name
was attached to the effect and “magnetism” entered the world vocabulary"
[Beckley02].
The effect was first used in a primitive navigation compass,
which consisted of a piece of natural rock called lodestone, or “leading
stone”. Later it was observed that steel needles rubbed by a natural
magnet themselves became magnetized and formed a much more
convenient compass component.
Since then the art of magnetism expanded into a science starting
from the studies of Gilbert (1544–1603) and Galileo (1564–1642), and
continued by Faraday (1791–1867). As in most disciplines, the theory of
magnetism started from statistical observations and a lot of experiments.
Gilbert’s book, published in 1600, was the first work on magnetism in
Europe. His experiments with some magnets led to the idea of magnetic
poles. He discovered that like poles repel each other and unlike attract.
He likened the polarity of the magnet to the polarity of the Earth. A
“north seeking” pole is the end of a magnet that points to the Earth’s
north magnetic pole when freely suspended, for example, in a compass.
In 1819 Oersted (1777–1851) discovered the deflection of a
compass needle by electric current flowing in a conductor. The
magnitude and direction of this influence is related to the size and the
direction of the current involved. This discovery of a fundamental
connection between electricity and magnetism engaged the scientific
community and led to a brainstorm in electrodynamics research by
scientists such as Ampere (1775–1836). His experiments showed that the
deflection of a compass relative to an electrical current obeyed the right
hand rule. He also showed that parallel wires with current in the same
direction attract, those with current in opposite directions repel.
A series of experiments by Faraday showed that time-varying
magnetism would induce electric current in a nearby electric circuit.
Furthermore, Faraday first suggested that a time-varying current would
induce electromotive forces, or emfs, in a separate conductor system. He
first proposed that magnetism was a circular force and invented the
term “lines of force”. Faraday’s laws expressed the electromagnetic
environment, as it is known today. First, when a conductor moves with
respect to a magnetic field, an emf is developed in that conductor.
12
Chapter 1.
Second, the magnitude of the developed emf is proportional to the rate of
change of the magnetic flux. He also believed in the possibility of “the
production of any one power from another, or the conversion of one into
another.” Faraday concepts of relating magnetism and electricity were
then used to make the first transformers. The invention of the dynamo in
1865 naturally followed and began the era of electricity and
electromagnetism.
In 1860 mathematician Maxwell (1831–1879) from the Cavendish
laboratory formulated the fundamental principles of electromagnetic
fields. In 1856 he wrote "On Faraday's Lines of Force", in which he
translated Faraday's theories into mathematical form, presenting the
lines of force as imaginary tubes containing an incompressible fluid. In
1861 he published "On Physical Lines of Force", in which he treated the
lines of force as real entities, based on the movement of iron filings in a
magnetic field and using the analogy of an idle wheel. In 1865, he
published "On a Dynamical Theory of the Electromagnetic Field".
Maxwell's formulation of electricity and magnetism was published later
in "A Treatise on Electricity and Magnetism", which included the
formulas today known as the Maxwell Equations. Those principles
became the backbone of further developments in theoretical and applied
electromagnetism. Here, a brief evaluation is presented.
Macroscopic electromagnetic phenomena are often described by
the following vector fields, which can be space and time dependent:
E
- electrical field [V/m]
D
- electric flux density [C/m2]
H
- magnetic field [A/m]
B
- magnetic flux density [T]
These vector fields are not independent, but interconnected by
Maxwell’s equations:
∂B
( Faraday ' s
∂t
∂D
∇× H = J +
( Ampere' s
∂t
∇ ⋅ D = γ (electric Gauss
∇ ⋅ B = 0 ( magnetic Gauss
∇× E = −
(1.1)
law)
law)
(1.2)
law)
law)
(1.3)
(1.4)
Here, J is the current density vector [A/m ], while γ [C/m ] equals the
charge density. It is well known that these vectors are connected through
the constitutive laws.
2
3
Introduction
13
To recall some terminology necessary for this work, one must
define the magnetic field and magnetization. The phenomenon of
magnetic field can be described by its action [Jiles91]. The torque on a
compass needle, which is an example of a magnetic dipole, is the most
familiar property of a magnetic field. The strength of the field of force is
called the magnetic field strength H. It has direction as well as strength.
The direction is that in which a north pole, subjected to magnetic field,
tends to move, or that indicated by the north-seeking end of a small
compass needle placed at the point.
The Faraday’s "lines of induction" pass from a magnetized material
into the air at a north pole, enter again at a south pole, and pass through
the material from the south pole back to the north to form a closed loop.
The total number of lines crossing a given area S is a measure for the
magnetic flux Φ in that area. According to Faraday's theory, the rate of
change of the magnetic flux Φ generates an emf in a closed circuit
through which the flux passes. The magnetic flux per unit area is the
flux density, or magnetic induction B.
B = µ0 H
(1.5)
Here, the factor µ0 = 4π10-7 [H/m] is called permeability of free space.
In order to describe the magnetic properties of materials, one
must have a quantitative measure of magnetization M, which is the
material response to the field. Both H and M contribute to the induction
lines, which is defined by the following relation:
B = µ0 (H + M)
(1.6)
Here, H and M are generally vectors. Electrical currents outside the
material generate the magnetic field H. The magnetization M is
generated by the uncompensated spin and orbital momentum of
electrons within the material [Jiles91].
In the technical literature, the vector expression (1.6) is often
reduced to a scalar one, assuming the three vectors being parallel.
When a piece of unmagnetized iron is brought near a magnet or
is subjected to a time-varying magnetic field, the magnetization in the
iron due to the magnetic field is described by a magnetization curve, as
shown in Fig. 1.2. This curve is obtained by plotting the magnetization
M or the magnetic induction B against the field strength H. Such curves
are of fundamental importance for describing the magnetic properties of
materials, especially for considering electrical steels.
14
Chapter 1.
For most materials, if the field strength is first increased from
zero to a given value and then decreased again, as indicated by the
arrows of Fig. 1.3, the original curve is not retraced. The induction “lags
behind” the field and follows a characteristic curve, shown by the
broken line in Fig. 1.3. This phenomenon was named “hysteresis” by
Ewing (1855-1935) in 1891, and the characteristic curve was called a
hysteresis loop.
Fig. 1.2. Initial magnetization curve and permeability for iron [Jiles91].
Fig. 1.3. Initial magnetization curve (solid) and hysteresis loop (dotted).
1 gauss = 10-4 T, and 1 oersted = (1000/4π)A/m = 79.58 A/m. [Bozorth51].
Introduction
15
When considering the initial magnetization curve, the ratio B/H
is called the permeability µ. For ferromagnetic materials the ratio µ/µ0 is
much larger than 1 [Borzorth51].
For a BH loop symmetrical to the origin, the absolute value of the
induction at H = 0 is called the remanence Br for that loop. It represents
the magnetization obtained after applying a field to the specimen and
then removing it. The absolute value of H for which B = 0 is called the
coercive field Hc. The coercive field is the field needed to bring the
magnetization from the remanence to zero. It represents the magnitude
of the field that must be applied to a material in order to reverse its
magnetization.
Coercive field, remanence and permeability are often used as a
measure of quality of the material. The ferromagnetic materials
(hereafter, magnetic materials) have been divided into soft and hard
magnetic materials, having a low or high value of Hc, respectively. Soft
magnetic materials are easy to magnetize due to a low coercive field. An
additional parameter of basic importance in the characterization of soft
magnetic materials is the power loss P. For electrical steels studied here,
the electromagnetic power loss P [W/kg] is proportional to the area of
the BH loop [Sievert84] obtained at a time periodic flux:
1 T
dB (t )
(1.7)
P=
H (t )
dt
dt
ρT ∫0
Here, ρ is the density of the material, T is the period of the cycle, f is the
magnetizing frequency. For simplicity, the electromagnetic power loss P
is also called "power loss" or "total power loss".
1.2.2.
Magnetic materials and domain theory.
The scientific explanation for mechanisms behind the magnetism was
developed historically. A ferromagnetic material has long been regarded
as an assemblage of small permanent magnets. Ewing together with
Weber (1804–1891) assumed that each atom was a permanent magnet
free to turn in any direction about its center. When the material is
unmagnetized, the magnets are arranged with random orientation.
When it is magnetized, they are lined up with their axes approximately
parallel. The nature of this small magnet has been the subject of a
number of studies during many years.
In 1907, based on earlier works carried out by Ampere, Weber
and Ewing, physicist Weiss (1865–1940) concluded that ferromagnetic
16
Chapter 1.
materials must be fully magnetized at all times. Below a critical
temperature, or the Curie temperature Tc, the magnetic moments
spontaneously attain long-range order and the material acquires a
substantial spontaneous magnetization Ms. In case of iron, where Tc ~ 103
K, the molecular field is estimated to be 109 A/m. Thus, well below Tc,
the moments are nearly perfectly aligned. In iron, at room temperature,
this gives a spontaneous polarization µ0M of order of 2 T. The molecular
field hypothesis explains the main aspects of the temperature
dependence of the spontaneous magnetization. However, it seemed if
there exists such an enormous internal field, one could expect the
material to be spontaneously magnetized to saturation under all
circumstances. How can a hysteresis loop exist at all? It seems impossible
that the external field of 100 A/m can reduce the magnetization of the
material to zero, if it confronted with the internal fields of 109 A/m.
Weiss proposed to resolve this difficulty by dividing the material
into regions of many atoms, called “magnetic domains”. In each domain,
the molecular field dictates the degree of magnetic moment alignment.
However, the orientation of spontaneous magnetization can vary from
domain to domain. Therefore, when magnetization is averaged over
large volumes with many domains, the result, mainly determined by the
relative orientation and volume of domains, can be very different from
the spontaneous magnetization, and even close to zero [Bozorth51].
The history of the comprehension of magnetic materials in the
twentieth century has been the history of the elaboration of Weiss’ ideas.
It took many years before quantum mechanics could give some
microscopic interpretation of the molecular fields. In fact, magnetic
domains are the result of the complex balance of several competing
energy terms, a balance by no means trivial to treat theoretically
[Bertotti98]. However, a number of modern techniques allow producing
magnetic domain images to prove their physical existence in magnetic
materials. In this study, a simplified interpretation of the domain theory
is used to investigate other phenomena. The evolution of magnetic
domains is shown in Fig. 1.4.
Generally, when the field is applied externally, a rearrangement
of the domain structure takes place, mainly due to domain wall
movement. The domains with magnetization approximately pointing in
the direction of the applied field are energetically favoured. After the
magnetic field changes its sign from negative to positive, the domains
with magnetization pointing in the direction of the applied field expand
Introduction
17
by consuming unfavourable surrounding domains, as shown in Fig. 1.4.
There are four kinds of magnetization processes, namely,
reversible and irreversible wall movements and reversible and
irreversible domain rotations.
Considering a hysteresis loop obtained for an electrical steel,
three mechanism are usually present: reversible and irreversible wall
movement and reversible rotation.
Fig. 1.4 depicts only domain wall movement under the
application of a positive external field, starting from the appearance and
the growth of the favourable domains, reaching the coercive field and
growing up to the disappearance of unfavorable domains. At high field,
a single domain practically occupies the whole space.
Fig. 1.5 depicts the simple case of domains aligned with the
external field direction. When a strong external magnetic field appears,
the magnetic domains with the same orientation as the external field
grow in size, while opposite 180-degree domains shrink. It moves the socalled “domain walls” to enlarge the favourable domains that are coaxial
with the external field H, see Fig.1.5.
Fig. 1.4. BH loop. Domains changes due to magnetization [Bertotti98].
18
Chapter 1.
Fig. 1.5. Domain theory: a) energy stored in a field, reduction of energy by
subdivision into lower energies, b) optimal energy storage in a group of domains
in iron, c) domain wall movement at strong external field H [Beckley02].
1.2.3.
Application of magnetic materials in energy transformation.
The unique property of ferromagnetic materials lies in the ability to
develop a large magnetic flux by comparatively small applied magnetic
field.
In soft magnetic materials the crystal structure facilitates a
response to applied fields and makes it reversible. Consequently, the
magnetization process requires a small amount of energy. Those
properties have been successfully exploited in electromagnetic devices,
such as transformers and rotating machines.
In the nineteenth century there was an enormous engineering
activity in the US and Europe, surrounding the invented induction
machine. The knowledge of machine design and performance prediction
progressed rapidly. At the end of the 19th century, Edison laboratory
developed a range of electromagnetic applications. Started with Edison,
Tesla (1856–1943) was the first to use AC current in a two-phase
electrical generator, a power transformer and MHz frequency devices.
Twenty years after the Tesla patents, in 1907, Smith surveyed the loss
separation methods used today in motor design [Glew98].
Taking a brief look back to section 1.1, it was stated that the
energy transformation is inevitably related to some energy losses.
Introduction
19
Indeed, a century ago, engineers have named the main sources of energy
losses in induction machines. Generally, there are two main components
of energy losses in induction machines and transformers: “copper”
losses in the windings and “iron” losses in the magnetic core. To answer
the questions of section 1.1, the designer of a machine must make a
choice about the ratio between the copper and iron loss. In many cases,
conventional techniques are applied to help engineers make that choice
and to compromise between the two losses. This book is about the
“iron” energy losses only.
The iron energy loss is an inevitable inefficiency of the
electromagnetic energy transformation, which is present in all parts of
the Great Chain of Energy Transformation, such as generators,
transformers or electrical motors.
Without magnetic material present, the energy transformation
would become hundreds and thousands times more difficult and costly.
In other words, the performance of magnetic material in the machine is
crucial for energy conversion with optimal energy loss.
Another choice the engineer should make is the choice of design
technology. For many years, engineers applied various computational
parameters and coefficients, based on years of experience (empirical)
rather than on pure science. The design methods have been developed to
a major extent since the first machine was built. The computer era made
the calculations easier. However, many assumptions of older design
techniques have survived the changes. As a result, it happens sometimes
that the machine shows worse characteristics when built in comparison
with the theoretical design. Iron energy loss are entered into the design
model from standard properties of electrical steels.
This is one of the reasons for the incorrect prediction of (iron)
energy losses, i.e. the so-called “building factor” [Dupre03]. The building
factor defines the ratio of the actually measured iron losses in the
machine and the energy losses calculated under standard conditions.
The building factor is usually larger than 1. The main reasons for that are
discussed in Chapter 2 of this book.
The last choice, or in fact, the first step in every design, is the
choice of the electrical steel grade to be used in the device, see section
1.5. For a century, scientists and material engineers did their best to
produce cost-effective materials for various purposes. Today, Fe-Si steels
are applied in the majority of electromagnetic devices. According to
[Schoppa00], about 80% of the produced soft magnetic materials are
20
Chapter 1.
non-oriented Fe-Si steels, about 16% are grain-oriented Fe-Si steels, and
only about 4% of all soft magnetic materials are other materials.
Although electrical steels only are the subject of this study, the
dependencies, presented in this book, have a more general outcome
towards other magnetic materials [Turgut03]. One may note that the
value of this 4% other magnetic materials is about 20% of the total costs
of soft magnetic materials, produced in the world [Schoppa00].
1.3. Electrical steels
The history of the development of electrical steels starts from using cast
iron and wrought iron due to its availability in the earliest times.
However, the permeability of cast iron is not very good, see Fig. 1.6.
Fig. 1.6. Magnetic curves of Fe materials at moderate applied fields [Beckley02].
Due to large eddy currents induced in solid cores as magnetization
changes, modern electromagnetic cores are produced as laminations of
thin steel sheets. During the twentieth century, electrical steels have
been constantly improved to better permeability and smaller energy
losses. Among the parameters to improve, there are electrical resistivity,
thickness of sheet, grain size, texture, etc.
Introduction
1.3.1.
21
Laminations.
Iron is a good electrical conductor, so that if solid iron is being used as a
magnetic core, the surface of the iron could be considered as a lowresistance short-circuit enclosing the core. It leads to a considerable
amount of energy, dissipated in the solid core.
The method to reduce the eddy currents is to reduce the
thickness of the sheets. This leads to an increase of the production costs,
moreover, very thin lamination reduces the percentage of the space
occupancy by the steel, when stacking the laminations. Also, the surface
acts as a primary source of domain pinning sites, which increase some of
the energy loss components. Despite those drawbacks, the majority of
electromagnetic devices uses electrical steels with thickness of 0.2 to 1.0
mm [Beckley02].
1.3.2.
Resistivity and various alloys.
As seen from Fig. 1.7, the maximum resistivity can be obtained when
silicon is added to the alloy of iron.
Indeed, at 3% silicon the resistivity of iron alloy raises 5 times,
and at 6% almost 8 times in comparison with the case of absence of
silicon. Aluminum also raises resistivity very well, but there is a
negative side effect of high affinity of aluminum for oxygen
[RosYanez03]. Carbon cannot be used due to the formation of nonmagnetic carbides at 100 °C, which reduce the mobility of the domain
walls [Devine96]. Other elements were tested, but silicon remains the
preferable alloying element to increase resistivity.
When the silicon content increases up to 6.5%, magnetostriction
and related problems of noise and vibration diminish to a negligible
level. Unfortunately, high silicon steels are difficult to produce costeffectively. The diffusion annealing after hot dipping is a feasible
solution [RosYanez03]. The hot dipping production route can result in
an increased silicon content, either constant over thickness of the steel
sheet, or varying over thickness with a lower content in the center and a
higher content at the surface of the lamination.
Hot dipping by diffusion annealing is a relatively new and costeffective process that increases the Si and Al content to obtain better
magnetic properties of electrical steel [RosYanez03].
22
Chapter 1.
Fig. 1.7. Resistivity of additions of various elements to iron [Bozorth51].
1.3.3.
Grain size and orientation of grains.
Grain boundaries are prime pinning sites for domain wall movement.
Impurities such as sulfur or carbon also pin domain walls. The larger the
grains, the smaller the number of grain boundaries per unit volume, the
smaller the overall amount of obstruction of wall movement. Large
grains can be obtained by annealing at a temperature of around 850 °C.
The iron lattice has favorable directions for magnetization. As
illustrated in Fig. 1.8, those directions can be exploited for uniaxial
magnetic flux applications, such as transformers. In 1930, Goss
developed grain orientation processing, i.e. suitably applied cold rolling
and heat treatment regimes, resulting in a selective growth of grains,
having their easy directions in the strip rolling direction.
It was believed for a long time that the larger the grains, the
better. However, with the advent of high-frequency power supplies,
using pulse width modulation, the size of grains has to be optimized to
avoid extra losses. Those extra energy losses occur in large grains mainly
due to large domain wall spacing, associated with micro eddy currents
and, consequently, extra losses [Beckley02]. Thus, the optimal grain size
depends on the electromagnetic application of the material.
Introduction
23
Fig. 1.8. a) cubic lattice, b) magnetization curves for a single crystal. The [100]
direction is the easy cube edge direction, [110] is the hard cube face diagonal
direction, and [111] is the hardest cube body diagonal direction [Beckley02].
Electrical steel is a polycrystalline material, having a large amount of
grains. The statistical distribution of the orientation of the grains along
various directions in the material is called texture. In grain-oriented steel
the direction of easy magnetization, or [100] direction in Fig. 1.8, lays in
the plane of the sheet, parallel to the RD. The worse direction is the one
that makes a 58-degree angle with the RD as it corresponds to the hard
magnetic axis of the iron crystal [Bozorth51].
Due to the fact that grain-oriented steel is often used for uniaxial
magnetization in the RD, the engineering interest for a 2D study for
these materials is limited. Nevertheless, there remains a scientific
interest in 2D behavior of grain-oriented steels.
In contrast, nonoriented steels are widely used under complex
2D magnetic conditions. Generally, non-oriented steels are assumed to
24
Chapter 1.
be completely isotropic in the plane of the sheet. However, this is not the
case. There is always a difference in magnetic behaviour for various
directions, such as the RD and the TD.
1.3.4.
Production of electrical steels.
The production of electrical steels has always been the interplay between
magnetic properties, physical properties, manufacturing methods and
production costs. Larger grains, more silicon content, high purification
of steel, low carbon content, and other “better quality” parameters
depend on workability of technology, as well as how feasible it is to
apply one or another costly method to achieve certain goals. Some
processes are crucial to one type of electrical steel, but less important to
another one.
The art of electrical steel production has been gradually
developed to minimize the cost and to increase the quality. When a new
piece of knowledge about the application of electrical steels appears at
the horizon, a change of production process might be a result, in order to
improve the output quality of electrical steel. However, it takes sufficient
effort to implement new changes to an existing technology. Before
making the decision to change the existing or to implement a new
technology in electrical steel production, one should take a look at
limitations of the technology-in-use.
When talking about production, electrical steel is always
mechanically rolled. The main objective for rolling is to obtain a given
thickness of the sheet in order to decrease energy loss due to eddy
currents. Another objective of rolling is to obtain a certain texture, or
sheet-plane anisotropy. In a majority of production methods, the steel is
rolled in one direction, called then the rolling direction. Consequently,
the transverse direction is the direction perpendicular to the RD in the
plane of the sheet.
It is known that the magnetic properties of non-oriented steel are
different in the RD and the TD. Hence, non-oriented electrical steels can
be named isotropic only to some extent. The future progress for the soft
magnetic sheets lays in the improvement of their texture, and
particularly in obtaining large [100] components [Mekhiche96].
Novel techniques such as cross rolling could bring the desired
result. The magnetic performance of cross-rolled sheets turns out to be
better compared to conventional non-oriented sheets [Mekhiche96].
Introduction
25
Rolling procedures introduce mechanical stresses in the plane of
the sheet, which strongly influence the mechanical and especially
magnetic properties of electrical steel. In order to reduce the damage
from stresses and deformations, the stress relief annealing is often
applied to improve magnetic properties of electrical steel sheets.
Carbon and sulphur have also very negative effects on the
magnetic properties of electrical steel as they act as pinning sites for
domain wall movement. When steel is rolled to thin final thickness, it is
then often decarburized and desulphurized. To achieve low carbon
content the steel is exposed to a decarburizing atmosphere such as wet
hydrogen at 800 °C. Carbon diffuses to the surface and reacts with the
furnace atmosphere gas. To reduce the sulphur content, the traditional
technique of removing liquid slag is applied together with various
chemical methods. The stress relief annealing is applied at the final
stages of steelmaking. The control parameters are temperature, time,
furnace atmosphere, and cooling period. The optimum annealing
temperature for electrical steel ranges from 720 °C to 840 °C. The
annealing time usually varies between 1.5 to 2.5 hours depending on
materials or type of furnace. The cost of these treatments is considerable,
but very low sulphur metal, low carbon content as well as reduced
stresses, gives excellent magnetic performance [Beckley02].
1.4. Stress effects in electrical steels.
The question of quality control for electrical steels arises each time a new
technological process is applied to steel. The quality control of electrical
steels particularly implies the control of the magnetic properties, such as
energy loss and permeability.
The lifetime of electrical steel starts with the steel manufacturer,
producing required quality steel. Then, the consumer of electrical steel
applies some technological procedures first to make the desired shape of
the steel products and then to build the electromagnetic core. The
magnetic properties of electrical steel usually deteriorate due to the
influence of these procedures on the microstructure.
Before electrical steel is finally used in an electrical machine, it is
exposed to various mechanical stresses. The influence of stress on
magnetic properties of electrical steels often results in a decreased
magnetic performance and extra energy losses. Various stress effects are
the subject of numerous researches all over the world. Here, the
26
Chapter 1.
summary of most of this research is presented as a preface to this work.
1.4.1.
Production stress in electrical steels.
Rolling with plastic deformation is an inevitable procedure during the
production of electrical steels. However, plastic deformation leads to an
increased number of defects in the microstructure. Therefore, many
thermal treatments are applied as described in section 1.3.4. Stress relief
annealing often minimizes the internal stresses from those procedures.
To the interest of this study, the magnetic properties of electrical
steel just after it is produced are assumed as the “primary” or the
standard properties of steel. Those properties, mechanical and magnetic,
can be found in various steel catalogues [Nippon92], [Beckley02].
The completing stage of the production process is the coating
procedure. The main purpose of coating is to prevent eddy currents
between neighbouring steel sheets in a stacked magnetic circuit. Other
purposes of coating are to make punching and welding easier and less
damaging to the steel sheet, and to prevent corrosion [Lindenmo00].
Coating also decreases the damage to the cutting tools. Sometimes, a
special coating is applied to introduce a small tensile force in the
preferable direction [Beckley02], [Grimm78]. The recent tendency to
apply composite coatings leads to some reduction of coating thickness
and improvement of its performance in a lamination stack.
It is often the case that the magnetic properties of one roll are
different from another, produced e.g. a day later. Moreover, the steel
batch can suffer from various stresses due to transportation. When the
steel batch comes to the machine factory, there may be a series of
magnetic measurements and testing. Those measurements could take
into account the actual standard properties for electrical steel grades
before the use.
1.4.2.
The effects of punching.
Electrical steel production is finalized in rolls. In order to use it in
electromagnetic devices, the electrical steel lamination should have a
certain shape, for example, a tooth-tip shape for rotating machines.
The required shape is obtained by cutting techniques, such as
punching, guillotine, laser cutting, photocorrosion [Emura03]. All these
techniques result in various cut edges.
Introduction
27
When the steel sheet is cut, the material properties next to the
cutting edge deteriorate drastically. In 1971 Carlberg first suggested that
a 1 mm wide degraded area could be expected adjacent to a sheared
edge of an electrical steel strip [Moses00]. Schmidt identified a cut-edge
hardening region 0.35 mm wide, showing an increase of loss of 30 to
40% and a corresponding induction drop of 70% for the same applied
magnetic field strength [Schmidt75]. Nakata stated that the degradation
of magnetic properties of non-oriented silicon steel sheet due to cutting
is present as far as 10 mm from the cut edge, and that the deterioration is
particularly pronounced up to 5 mm from the edge.
Despite the differences in early opinions, the engineering society
has agreed upon the average affected cut edge, having a width equal or
larger than the thickness of the lamination [Beckley02]. There are many
cutting parameters that affect mechanical and magnetic properties, such
as steel hardness, ductility, ultimate tensile strength, yield strength,
coating favorable to burr minimization, grain size, clearance, etc.
Burr and clearance have a strong impact on interlayer shortcuts
as well as on the cut edge properties [Baudouin03]. A mechanical strain
caused by shearing stresses from cutting electrical steel sheets results in
a deterioration of the magnetic characteristics of the sheet. One result is
that, when magnetized parallel to the cut edge of a sheet the magnetic
flux density measured in the cut edge region is less than the flux density
in the center of the sample [Moses00]. The steel grades with larger grains
ought to be more affected by cutting than the steels with smaller grain or
than the steels with less silicon [Schoppa99], [Schoppa00].
There are few methods to improve the magnetic properties after
cutting. The magnetic properties may be restored by heat treatment,
such as stress relief annealing after punching, as shown in Fig. 1.9. The
quality of the punching can be improved as well by an appropriate
choice of a cutting technique, which results in less deterioration of the
cutting edge. A lot of research has been devoted to improve the quality
of cutting and to introduce new cutting techniques. For example, an
abrasive waterjet cutting under high pressure could be applied
[Schoppa03]. The method has a minor effect on magnetic properties of
electrical steel due to lower deformation of the cut edge and a cooling
effect of water. Schoppa believes that the abrasive waterjet cutting is the
best currently known cutting method for the production of laminations
for magnetic cores used in electrical machines.
28
1.4.3.
Chapter 1.
The effects of core building.
After the electrical steel is cut into the required shape and before the
copper windings are applied in the production process for electrical
machines, the stack of laminations is usually built into the stator or the
rotor, and then mounted into the frame of the electrical machine.
Stress relief annealing can be applied either before the stack of
laminations is in the frame, or after it is built into the frame. The effect of
heat treatment is very useful to improve the magnetic properties of steel.
However, when the lamination is stacked into the frame, new sources of
mechanical stress appear.
There are different fixing methods, such as automatic stacking,
bolting, riveting, gluing, interlocking, laser or electrical bead welding,
and die casting of rotors [Beckley02], [Schoppa02]. When considering
which technique of core assembly to apply for lamination sheets, there is
a compromise between easiness and low cost of the technology and the
effect of various stresses and deformations due to assembly on electrical
machine parameters. The significance of the stress effects has often been
appreciated due to a negative effect on energy losses.
The first side effect of the core building is an introduction of
interlaminar short circuits, leading to increased eddy currents and
energy losses. This is the question of production quality and quality
control. Engineers often make a choice of slower or costly technology in
order to keep the electrical steel undamaged. Easy welding leads to a
short circuit between the laminations. Compared with welding, the
influence of interlaminar short circuits by burs on the magnetizing
behavior of electrical steels can be practically neglected [Schoppa02].
The second side effect of core building is the production of
mechanical stress, mainly compressive stresses. The effect of
compressive stress due to assembling is considered further in section 2.1.
Generally, the effect of the cutting and the influence of the cut edge is
much higher than the effect of core building due to the large flux density
coming through the cut edges in the teeth [Schoppa02].
Introduction
29
Fig. 1.9. Cut edges and pictures of grains in grain-oriented steel "as cut" and
"after stress relief annealing" [Beckley02].
1.5. Engineering choice of electrical steels.
There are two polar approaches among engineers about which type of
nonoriented steels to use in electromagnetic devices. The effect of stress
is different in each case.
One group of engineers choose semi-processed steels (SP),
having poor magnetic properties but at lower cost. The idea of further
processes is to improve the magnetic properties. During production and
assembling of the electrical machine, SP electrical steel is first cut to the
required shape of the lamination and then annealed. The drawback of
that approach is the need for costly annealing systems after punching.
30
Chapter 1.
That approach, favorite in the USA, results in a better quality of the
electrical steel in the electrical machine, although obtained from initially
poor-quality material.
Another approach is to use a material having very good
magnetic properties and high costs. Those fully-processed (FP) electrical
steels are then cut to produce the required shape of sheets. The cutting
process, of course, leads to a deterioration of the magnetic properties.
That approach, favorite in particular in Europe and Japan, diminishes
the quality of initially good electrical steel.
It is worth mentioning that the above choice does not exist in
grain-oriented steels. Usually, there is no need for annealing due to a
high-quality punching and a "step-lap" technique of core assembling that
allows magnetic flux passing without cutting edges.
This study aims to bring a clear understanding of the effect of
mechanical stress on the magnetic properties of electrical steels, applied
in both approaches. However, before investigating the stress
dependence, the area of working conditions for electrical steel as well as
the methods for magnetic measurements are considered.
Working conditions of electrical steels
31
To believe with certainty we must begin with doubting.
Stanislaus Lescynski
CHAPTER 2. WORKING CONDITIONS OF ELECTRICAL STEELS.
The choice of the electrical steel to be used in electromagnetic devices is
often made based on steel characteristics, obtained by means of standard
methods [IEC 404-2], [IEC 404-3]. These standards are useful to make a
primary comparison between various steel grades. However, actual
working conditions for electrical steel in electromagnetic devices often
differ considerably from the standard ones. This chapter aims to
investigate the actual 2D magnetic and mechanical conditions in
electrical machines. The main purpose for that investigation is to define
the practical area of further research on the stress effect.
The measured iron loss in electrical machines usually exceeds the
value obtained by a straight multiplication of the weight of electrical
steel in the machine with the standard loss density. This difference in
losses is generally taken into account by the so-called building factor
[Nakata84]. The building factor is the ratio of the measured iron losses in
the machine and the losses obtained under standard conditions.
The reasons for a building factor greater than 1 are production
reasons and design reasons. Production reasons are short-circuits
between neighbouring laminations, quality of technology, influence of
mechanical and thermal treatment of the material during the
construction of the machine, etc. Design reasons are non-uniform flux
distribution due to local saturation, harmonics in flux patterns due to the
tooth structure or the non-sinusoidal electrical supply, local rotational
magnetization and changes of the magnetic characteristics due to
various thermal and mechanical stresses present in electrical machines.
For many years, engineers have applied various coefficients and
techniques in order to predict the performance of electrical steel in
electrical machines. These coefficients were developed by decades of
experience in electrical machine design. These techniques had many
advantages, such as simplicity in obtaining good engineering results,
understanding basic relationships, making decisions about design and
performance compromises. Disadvantages were a very low accuracy and
sometimes unpredictable results or bad machine performance. In the
past, as long as heuristic design rules resulted in a machine that worked
satisfactory, the design was considered acceptable. Later however,
32
Chapter 2.
design costs and operating costs, i.e. efficiency, got more attention.
When advanced calculation methods became possible, many numerical
techniques have been developed to predict the design factors mentioned
above more accurately. For example, the analysis of local flux patterns
and local iron losses has become possible for further investigations of the
parameters of electrical machines [Gyselinck00]. The development has
usually reached the highest level. Development departments of machine
producers say that the cost improvement of one cent results in millions
of profit and a competitive advantage on the market. But where to find
that one-cent improvement? One of the ways is to take a good look back.
Calculation of actual working conditions is still a question of
approach. In many cases, rather conventional methods and coefficients
are used. For example, a design technique in Russia employs two
different magnetization curves for the same electrical steel, one is in the
tooth region and the other is in the yoke. Basically, that technique takes
into account the change of magnetic properties in the tooth region of
electrical machine due to punching. Indeed, new studies worldwide
have been investigating the modification of the magnetic properties due
to cutting [Nakata92], [Ossart00], [Schoppa99], [Schoppa00],
[Fujimura04]. Baudouin and Pulnikov have recently studied in details
the effect of punching in electrical steels [Baudouin02], [Pulnikov04].
2.1 Standard characteristics of electrical steel.
As it was mentioned in section 1.4, the primary properties of electrical
steels are the result of the steel production process. Basically, the
primary state of steel is what a steel manufacturer provides to a steel
customer. Output parameters of steel production are:
- chemical composition,
- thickness and its deviation,
- type of coating,
- state of heat treatment,
- hardness.
Chemical composition is sometimes hidden behind other parameters.
The final thickness after few stages of rolling usually ranges from 0.2 to
1.0 mm. Coating could be organic or non-organic. The state of heat
treatment means either fully processed or semi-processed.
The main characteristics of electrical steel grades can be found in
the certificate for a steel grade, catalogues of steel manufacturers and
Working conditions of electrical steels
33
other open sources [Beckley02].
Steel manufacturers provide typical physical and mechanical
properties as well as typical magnetic properties. Typical physical
properties include:
- density,
- silicon content,
- resistivity.
The density of electrical steel ranges from 7600 to 7850 kg/m3
[Bozorth51]. The silicon content varies from 1% for low silicon steel to
6.5% for high silicon steel. The resistivity ranges from 10 to 60·10-8 Ω·m.
The common silicon composition for electrical steels is about 3%-wt
content. It results in an increase of resistivity, see Section 1.3.2.
Typical mechanical properties are:
- tensile strength, or ultimate tensile strength,
- yield strength, or yield point,
- elongation, or plastic strain,
- hardness, etc.
From a practical point of view, the stress-strain characteristic is the true
basis for any analysis of mechanical conditions of electrical steels under
externally applied uniaxial mechanical load [Iordache03]. A typical
stress-strain characteristic is shown in Fig. 2.1.
Fig. 2.1. A typical true tensile stress – true strain characteristic for nonoriented steel at two stages of hardening rate [Iordache03].
34
Chapter 2.
Generally, the mechanical properties of a steel grade depend on
the type of electrical steel produced, see Section 1.5. For example, when
the steel sheet is annealed, then the number of internal defects of the
lattice, which is directly related to the dislocation density of the material,
is reduced. Consequently, fully processed electrical steel typically has
mechanical properties of a low-deformed material. On the contrary, high
dislocation density of semi-processed steel leads to a reduction of tensile
strength and a deterioration of other mechanical, and especially,
magnetic properties, see Chapter 4.
Typical properties for various electrical steels can be found in
[Beckley02]. In fully processed non-oriented electrical steels, the ultimate
tensile strength usually varies from 300 to 550 MPa. The yield strength is
about 15 to 30% less than the tensile strength. On a stress-strain
characteristic, the yield point corresponds to the beginning of plastic
deformation, or to the 0.2% proof strength. In non-oriented steels, the
difference between most mechanical properties in the RD and the TD is
very small and usually less than 5%. The elongation on 80 mm gauge
length can usually vary from 15 to 40%. Other mechanical properties are
less important to the present study.
In grain-oriented electrical steels, most physical and mechanical
properties are within a similar range as in non-oriented steels. The
typical thickness of the sheet varies from 0.23 to 0.5 mm. The typical
density is about 7650 kg/m3. The silicon content is usually 2 to 3.4%. The
typical resistivity is about 35 to 55·10-8 Ω·m [Bozorth51].
The ultimate tensile stress is different for the RD and the TD,
ranging from 300 to 400 MPa. The difference between the RD and TD
can be as much as 20%. The yield strength is smaller than the ultimate
tensile stress by 8% in the RD to 22% in the TD. The elongation on 80
mm gauge length is 5 to 15% in the RD and 30 to 40% in the TD.
The magnetic properties of electrical steels are the most
important data in steel catalogues for electrical machine manufacturers.
Typical magnetic properties include:
- specific core loss at power frequency 50-60 Hz,
- DC magnetization, DC hysteresis, DC coercive field,
- incremental, or relative permeability,
- core loss and permeability at frequencies up to 10 kHz, etc.
The core loss is presented either as a few critical values at different
induction levels, or as a complete graph. The typical specific core loss,
measured in W/kg, is usually taken at an induction of 1.5 T and for the
Working conditions of electrical steels
35
power frequency of 50-60 Hz. From an engineering point of view, the
unit W/kg is very useful, when multiplying the core weight to the core
loss at different magnetic inductions. Furthermore, a magnetization
curve (peak induction versus Ampere-turns) ranging from 0.5 to 1.8 T is
easy to use.
As a starting point of this discussion, the standard characteristics
are considered. The magnetic properties of electrical steels are presented
as the results obtained by an Epstein apparatus according to [IEC 404-2]
or by a single sheet tester according to [IEC 404-3]. According to [IEC
404-2], half of the samples of the non-oriented steel under study is
sheared parallel and half of the samples is sheared transverse to the RD.
This practice leads to an averaging of typical magnetic properties in 2D,
which is acceptable for the design of electrical machines.
Despite a general belief of isotropic non-oriented electrical steel,
the anisotropy is present in non-oriented steels. It results in different
properties along various directions in the plane of the sheet. The
difference is observed not only in mechanical properties as was stated
above, but most importantly also in magnetic properties. Thus, any
change of the magnetic anisotropy is important to be taken into account
for the prediction of the actual performance of the electrical steels.
Externally applied stress introduces a considerable change of the
magnetic anisotropy, and hence, the stress effect is very significant at
actual working conditions of electrical steels.
Some extra information and recommendations for final annealing
to reduce the degree of internal stress is also provided in catalogues. For
example, the following precautions are issued for non-oriented steel by
Nippon Steel Corp. [Nippon92]. "During stress-relief annealing, carbon
contamination should be prevented, fast and uniform heating of laminations
should be pursued, excessive oxidation from the annealing atmosphere should be
avoided." Those precautions ought to maintain a high quality of the steel.
Modern control techniques, applied by steel manufacturers,
result in a better control of the material parameters and characteristics.
In fact, fully processed non-oriented and grain-oriented steel, coming
from manufacturers, might be considered today as top quality materials,
which do not require any further annealing.
Although the control of properties is of major interest to electrical
steel factories, actual properties of a steel grade could still be different
from the typical properties, provided by the manufacturer. There is a
strong belief among engineers that the catalogue data only give a
36
Chapter 2.
general idea about the quality of steel grade. Good practice is the actual
magnetic investigations of the materials before use.
In general, standard parameters are the only information a steel
manufacturer provides about its electrical steel grade. It is often the case
that the amount of information an engineer requires in order to make a
decision about a new design of an electrical machine, is much larger
than the data available either from a catalogue or from classical magnetic
measurements. For example, a new design requires the application of a
compressive stress due to the preferred machine design. The data about
the stress effect on the magnetic properties of electrical steel either do
not exist, or are hard to find and difficult and costly to obtain.
What an engineer should do in that case is first to take into
account a general tendency of stress effect on magnetic properties of
various steels. Up to now the general rule is that only a small tensile
stress could be useful in the direction of the applied magnetic field
[Beckley02]. All other in-plane stresses lead to a deterioration of the
magnetic properties of electrical steels. This study is to extend the
knowledge of the stress effect. Here, the main question arises: what are
the target working conditions, mechanical and magnetic, that are of interest for
various electrical steels used in different electromagnetic devices?
2.2. Target research of working conditions in electrical steels.
Two types of electrical machines, and consequently, two types of
electrical steels are considered: non-oriented steels in rotating machines
and grain-oriented steels in transformers. In each case, theoretical and
practical approaches are applied both from a steel producer and a device
constructor point of view. Why use different approaches?
The first reason to use these approaches is just to take a different
look at the same problem. The second reason is to have a more complete
view on possible working conditions of various electrical steels. The
third reason is the possibility to make a choice of these working
conditions, which are either of most importance, or easier to simulate in
a novel measurement setup. Finally, a combined approach can bring a
new area of research.
2.2.1. Target research in grain-oriented steels.
The first type of considered electrical steels is grain-oriented steel. Grain-
Working conditions of electrical steels
37
oriented steels exhibit excellent magnetic properties when excited along
the RD. A typical application of grain-oriented steel is in power
transformers.
In medium and large transformers the steel sheets are cut in such
a way that the magnetic field in the steel laminations should be mainly
uniaxial, parallel to the easy axis, i.e. [100] direction of the lamination.
Indeed, one of the main objectives in a transformer design is to decrease
core losses, and consequently, total energy losses during the energy
transformation.
The modern construction practice of large transformers requires
cutting laminations in a sophisticated way in order to achieve the
required core properties and at the same time being economically sound.
The lamination of a large transformer core consists of hundreds of
sheets, having different dimensions. The cutting line itself is a large
automated piece of machinery. The reason to use such an expensive
equipment for production of the transformer core is the reduction of
core loss.
In three phase transformer cores, there are regions where the
magnetic field is not directed along the easy axis of the material, such as
the T-joints regions and corners. In these regions, magnetic field exhibits
an angle with the RD. Moreover, it is known that energy losses are much
larger when a nonzero angle between the RD and the magnetic field is
present [Dupre00]. The effect of the T-joints regions can be reduced by
cutting and stacking techniques, applying larger number of sheets per
layer. Further, a special tensile coating is applied to grain-oriented steels
to reduce energy loss even more.
From a practical point of view, the interest in stress dependence
in grain-oriented steel is limited. Generally, the mechanical stress in
transformers can be considered in two aspects. The first point is the
presence of 2D stresses, e.g. due to the weight of the material, in the
plane of the sheet, which are normally avoided by the present
technology. Due to mostly uniaxial magnetic flux in the RD of steel, the
case of alternating flux in the RD under different phase of in-plane
mechanical stress is of great interest. The second point is the presence of
the stress orthogonal to the sheet, e.g. due to the clamping of the
lamination package. This stress is not considered in the present study.
The effect of the orthogonal compression is likely to be similar to the inplane tension [Liorzou99].
The in-plane mechanical stress is usually controlled by the
38
Chapter 2.
construction methods. To control an external stress to the steel core,
various constructions are applied to keep the mechanical stress in the
core as low as possible (rarely exceeding 2 to 10 MPa). For example, the
upper yoke is connected to the lower yoke by external construction bolts
and frames to reduce the effect of weight of the upper yoke to vertical
parts of the transformer core. If this is not done, the weight of the upper
yoke might be sufficient enough to produce compressive stress in the
transformer core. The latter should be avoided.
The practical interest of further research might be in the
investigation of the following combination of working conditions. The
alternating magnetic flux is applied in the RD of grain-oriented steel,
while the stress is applied in any direction in the plane of the sheet.
However, a 1D investigation could be sufficient in grain-oriented steels
from an engineering point of view. A further extension of the 1D
investigation to an alternating magnetic field in any direction of grainoriented steel can be also considered.
From material science point of view, the 2D investigation of the
stress dependence of magnetic properties should be of most interest due
to the controlled orientation of the grains in that type of soft magnetic
material. In other words, it is quite possible to predict the stress
dependence in such fine soft magnetic materials as grain-oriented steel.
If the prediction, based on physical understanding, could be confirmed
experimentally, the stress dependence can be confirmed.
The alternating 2D investigation (an unidirectional magnetic
excitation for different directions in the plane of the sheet) has been
approached by many researchers [Moses81], [Dupre00]. One conclusion
is common, i.e. a possibility to predict 2D behavior of grain-oriented
steel by limited information about alternating properties in the RD and
the TD. The domain theory plays a major role in a prediction of soft
magnetic material behavior.
A mechanical stress could be applied in the RD as it corresponds
mainly with the [100] direction, or the easy cube edge direction. Other
directions might also be of interest, for example, the hard axis direction.
The magnetic field, on the other hand, could have a complex 2D pattern,
ranging from alternating in any directions to elliptical and circular
rotational magnetic flux. Many studies have been performed on the 2D
behavior of magnetic properties of grain-oriented steels. However, very
few studies were about the stress effect on 2D magnetic properties of
grain-oriented steels [Moses80], [Moses89].
Working conditions of electrical steels
39
2.2.2. Target research in non-oriented steels.
Non-oriented electrical steel is the most common soft magnetic material
used in various electrical machines. The main reason to use non-oriented
steel is its quasi-isotropic magnetic properties in different directions in
the plane of the sheet. The difference between the RD and the TD, or
more general, the anisotropy of magnetic properties, is very small and
often not taken into account. Therefore, non-oriented steels are usually
applied when complex 2D magnetic conditions are present in the
electromagnetic device, such as in an induction machine.
From a practical point of view, the effect of the 2D mechanical
conditions has always been of great interest. Mechanical conditions
present in the machine are at least of three different natures:
construction, design and operations.
The first cause of stress is the effect of various construction
technologies. For example, cutting and punching techniques usually
lead to a deterioration to a major extent of mechanical and magnetic
properties of steel sheets. Next type of technologies is core building,
which often results in extra compressive stresses in magnetic cores of the
stator and the rotor of an induction machine [Pulnikov04]. For example,
rotor laminations can be built on the rotor shaft by means of a thermal
expansion technique. The pre-cooled rotor is expanding when reaches
ambient temperature. It results in various compressive stresses in the
rotor laminations. The same is true for the stator, when the stator core is
pressed into the preheated frame. The frame cools down, resulting in a
compressive stress up to 50 MPa [Pulnikov04], [DeWulf04]. Other
technological reasons are tensile coating, thermal or residual stresses.
To predict the consequences of construction technologies, one
should carefully assess these technologies and their effects on steel.
Many studies have been done about existing technologies and their
effects on magnetic properties of electrical steels [Landgraf00],
[Emura03].
The second cause is the effect of the design. For example, if a new
design is based on using a compressive stress to be applied to the
electromagnetic core, the result of such a design would most certainly be
worse than expected. If a design engineer would not modify the design
properties of the tooth regions, then the performance of the electrical
machine could be worse than expected. In other words, a careful
combination of technology and design could result in a high quality
40
Chapter 2.
cost-effective machine, having lower energy loss and better
performance.
The third cause are the various operating conditions in electrical
machines. The operating conditions may introduce various effects. One
is the ageing effect, when the energy loss increases in time due to
migrating carbon and nitrogen atoms in the steel.
From a practical point of view, the 2D investigation of the stress
dependence of magnetic properties under 1D mechanical stress should
be of most interest to different electrical steels. From a material science
point of view, the complete 2D investigation of the stress dependence of
magnetic properties should be of most interest under both alternating
and rotational 2D magnetic conditions. Application of biaxial stresses is
a very interesting subject [Alves04]. It could result in a statistical
database of knowledge on how energy loss of various non-oriented
steels change under mechanical stress.
Lately, there has been a tremendous research activity on that
matter. Many research centers have been studying the effect of tensile or
compressive stresses on energy losses and magnetic properties of
various non-oriented electrical steels. Before considering an actual 2D
study of electrical steels under stress, the working conditions of steel in
the machine should be defined.
2.3. 2D magnetic working conditions in induction machines.
To have a deeper insight on 2D magnetic working conditions in
induction machines, an extra study was carried out, both from
theoretical and experimental point of view.
Type
IP
D100LB
54
No
D846
076
kW
3.0
Table 2.1. Parameters of the induction machine.
Hz Rpm Y- V Y- A phase T. °C
50
1420
380
6.7
3
80
A 3kW-induction machine was chosen as the first object for the
2D flux study. It is a four-pole induction machine with a closed
unskewed squirrel-cage rotor. The main features of this machine is
shown in Table 2.1.
This induction machine was equipped with search coils, which
were built inside the machine, fitted around each stator tooth close to the
Working conditions of electrical steels
41
air-gap. The used data-acquisition of the voltage signals from the search
coils was transferring the measurement data to a PC with commercial
software. The measurement results were stored in binary format and
then integrated by means of software developed in EELAB. Thus, local
magnetic fluxes are measured in each tooth of the stator near the air gap.
For numerical evaluation of the magnetic working conditions of
electrical steels, a 2D finite element dynamic calculation software was
used [Gyselinck00]. This software allows the evaluation of local
magnetic flux patterns in the same places where the search coils are
placed in the machine. The local magnetic flux waveforms obtained from
the numerical calculations and the measurements are compared in order
to verify the measurement and calculation techniques. This results in
information about the magnetic conditions of the electrical steel in
different parts of the magnetic core.
The stator of the 3kW-induction machine has 36 slots and the
rotor has 32 slots. This type of induction machine has three full-pitch
slots per phase. Therefore, two teeth lay in one phase and the third tooth
is between two phases. Hence, three teeth in a row are sufficient to
analyze both measurements and calculations, see Fig. 2.2.
Fig. 2.2. Location of the search coils (black dots), shown in three teeth in a row.
42
Chapter 2.
Fig. 2.3. 3kW induction machine equipped with search coils in EELAB.
To obtain the magnetic fluxes as they are present in this
induction machine during operational regimes such as no-load and
different loads, an electromechanical setup was constructed. The setup
for no-load test consists of the 3kW-induction machine without any
additional mass on the shaft, an autotransformer, control devices, such
as a voltmeter, an amperemeter and a stroboscope, and the dataacquisition system, see also Fig. 2.3.
By integrating the voltages induced in the search coils, local
magnetic flux waveforms at no-load were obtained, see Fig. 2.4. Flux
waveforms at rated load are shown in Fig. 2.5.
The set-up for different load regimes consists of two electrical
machines, the shafts of which are mechanically connected to each other:
the 3kW-induction machine and a shunt generator as a load to this
machine. There is again, like in the no-load condition, an
autotransformer, a rheostat, excitation shunt circuit with rheostat and
amperemeter, control devices, such as voltmeter, amperemeter and
stroboscope, and the data-acquisition system.
Similar results were obtained for the following series of different
loads: 0.25 of rated load, 0.5, 0.625, 0.75, 0.875, 1.0, 1.125, 1.25, 1.375 of
rated load. The control of the load was done by measuring rotating
speed, according to Table 2.2.
Working conditions of electrical steels
Teeth 1
43
Teeth 2
Teeth 3
1.00
0.80
0.60
Flux (mWb)
0.40
0.20
0.00
0
5
10
15
20
25
30
35
40
-0.20
-0.40
-0.60
-0.80
-1.00
Time (ms)
Fig. 2.4. Measured local flux waveforms at no-load in 3 subsequent teeth.
Teeth 1
Teeth 2
Teeth 3
1.0
0.8
0.6
Flux (mWb)
0.4
0.2
0.0
0
5
10
15
20
25
30
35
40
-0.2
-0.4
-0.6
-0.8
-1.0
Time (ms)
Fig. 2.5. Measured local flux waveforms at rated load in 3 subsequent teeth.
Table 2.2. Load regimes and speed control.
Speed, rpm
Slip, %
Load ratio
1480
1460
1450
1440
1430
1420
1410
1400
1385
1.333
2.667
3.333
4.0
4.667
5.333
6.0
6.667
7.667
0.25
0.5
0.625
0.75
0.875
1.0
1.125
1.25
1.375
44
Chapter 2.
In Fig. 2.4 and Fig. 2.5, "Tooth 1" is located between two phases while
"Tooth 2" and "Tooth 3" are located between coils of the same phase.
Therefore, the local magnetic flux waveform of Tooth 1 differs from
those of Tooth 2 and Tooth 3. In order to analyze more accurately the
local magnetic flux waveforms a Fourier analysis was performed for
each curve for each working regime.
A Fourier analysis results in the separation of the most
significant frequencies in the flux waveform, as well as a quantification
of their amplitudes and phases. A complete analysis was performed for
all working regimes. With varying load, the Fourier coefficients vary as
well. However, the set of the most significant frequencies turns out to
remain approximately constant. This set consists of 3, 5, 15, 17, 31, 33.
These frequencies can be divided into three groups: 3 and 5, 15 and 17,
31 and 33. The first group is due to saturation, the second and the third
groups are due to tooth frequencies.
The tooth frequencies can be determined by [Vandevelde94]:
(2.1)
f stator = f 0 ⋅ (1 + k ⋅ (1 − s) ⋅ S rotor / N P )
f rotor = f 0 ⋅ ( s + k ⋅ (1 − s) ⋅ S stator / N P )
(2.2)
Here, f0 is the power frequency, fstator and frotor are the tooth frequencies, s
is the slip, Srotor and Sstator are the numbers of teeth of rotor or stator, NP is
the number of pole pairs of the machine, and k = …-2, -1, 0, 1, 2…
Considering the induction machine having 32 teeth of the rotor
and 36 teeth of the stator with 4 poles, eq. (2.1) and (2.2) give:
(2.3)
f stator = f 0 ⋅ (1 + 16 ⋅ k ⋅ (1 − s))
f rotor = f 0 ⋅ ( s + 18 ⋅ k ⋅ (1 − s))
(2.4)
Taking into account that the no-load regime gives the slip s → 0, the
most significant frequencies in the stator are approximately 15, 17, 31
and 33. This set of theoretical frequencies is confirmed by the results of
the actual measurements of by the pick-up coils, presented above.
Of course, measuring the flux at the tooth tips is not sufficient to
estimate local flux patterns, e.g. for different height along a tooth.
Although it might be possible to provide a stator tooth with several
search coils, divided along its height, this is far from easy to implement.
Since the subject of this study is not the extensive experimental
evaluation of the magnetic flux patterns in each place of the induction
machine, the experimental part is limited to the search coils, placed at
the tip of each tooth of the stator. Therefore, the use of the FE software is
crucial to predict various magnetic conditions of electrical steel in the
Working conditions of electrical steels
45
considered induction machine.
To justify the software for further research of 2D magnetic
conditions of electrical steel, a comparison is made between measured
and calculated magnetic flux patterns. The latter were calculated in the
same places where search coils were measuring the local magnetic
fluxes, as shown on the finite element mesh in Fig. 2.6.
The compared calculations and measurements were performed
for the same operating conditions, according to Table 2.2. For both
experimental and computational results, the relevant frequencies are the
same: 3 and 5, 15 and 17, 31 and 33. Comparison of the measurements
and the computations shows that the deviation of the amplitudes and
the phases of the relevant frequencies was found to be of the order of
25%. This deviation may originate from measurement and calculation
errors and from different assumptions in the numerical model.
Despite this large quantitative deviation for the frequencies, it is
assumed that the software could be used for further analysis of 2D
magnetic working conditions of electrical steel in the considered
induction machine. The FE software output is the magnetic vector
potential in the 2D plane. According to the physical interpretation of the
magnetic vector potential in 2D field computations with in-plane B and
H vectors, the magnetic flux per unit length passing through an arbitrary
curve between two points is the difference of the vector potential values
in these two points.
Thus, local magnetic flux patterns were obtained as a difference
of magnetic vector potential values, calculated in two nodes of the mesh
as a function of time.
The advanced study of 2D magnetic flux patterns in various
places of the induction machine, as in Fig. 2.7, is done by means of the
software. Local magnetic fluxes are considered for rated load only, as it
is the most important working condition.
In order to analyze more accurately local magnetic flux
waveforms, the Fourier analysis was performed for each waveform. The
ratio between the amplitude of each frequency and the fundamental or
power frequency has been calculated for the 9 cross sections, shown in
Fig. 2.7. The results are given in Table 2.3.
46
Chapter 2.
Fig. 2.6. Finite elements mesh, including the positions of the measuring coils
(black dots) around the 3 teeth in a row.
Fig. 2.7. Local magnetic fluxes of the most target interest: Flux 0 to Flux 9.
Working conditions of electrical steels
47
Table 2.3. Highest percentages of the set of frequencies.
Results
% at # 3
% at # 5
% at # 15
% at # 17
% at # 31
% at # 33
1
1
# of tooth
1
2
3
1
2
3
1
2
3
1
2
Flux 0
Flux 1
Flux 2
Flux 3
Flux 4
Flux 5
Flux 6
Flux 7
Flux 8
Flux 9
2.5
2.4
2.2
2.0
1.9
1.5
1.5
0.9
0.3
0.4
1.5
1.4
1.8
2.0
2.0
1.4
3.4
0.3
1.0
0.3
0.4
0.4
0.8
0.9
0.9
2.3
1.3
0.5
0.4
0.3
0.8
0.8
1.1
1.2
1.2
1.4
0.3
0.4
0.4
0.5
2.3
2.4
2.7
2.8
2.8
2.6
1.9
1.0
0.3
0.4
1.9
1.8
1.7
1.7
1.7
1.2
1.8
0.6
0.2
0.3
3.3
2.0
0.5
0.3
0.3
0.8
0.9
1.2
1.1
1.0
4.1
3.3
3.0
3.0
2.9
2.3
2.2
1.2
1.3
0.3
4.5
3.7
3.3
3.3
3.3
1.7
3.2
0.5
0.4
1.2
0.3
1.2
2.1
2.3
2.4
1.8
1.7
0.5
0.7
0.5
3.5
2.4
2.0
1.9
1.9
1.4
1.0
0.6
0.3
0.2
3
2
3
2
3
3.6 0.7 0.9 0.8 0.3 0.4 0.3
2.4 0.4 0.7 0.6 0.2 0.3 0.1
2.0 0.2 0.5 0.6 0.3 0.3 0.2
1.8 1.8 0.9 1.5 0.3 0.5 0.4 -
The general conclusion from the data presented in Table 2.3 is as
follows. The distortion in magnetic flux patterns appears mainly in a
tooth region of the magnetic core of the induction machine. A set of high
frequencies induced by by saturation as well as by teeth frequencies are
of major importance to the distortion of the magnetic flux in the
machine. Therefore, magnetic flux patterns cannot be considered
sinusoidal in the tooth region.
Fig. 2.8. Rotational behavior of local flux patterns in points 1 to 15 of stator.
48
Chapter 2.
Table 2.4. Axis ratios in the points of Fig. 2.8.
Point #
1
2
3
4
5
6
7
8
9
Tooth1
Tooth2
Tooth3
0.1
0.1
0.1
0.1
0
0.1
0.1
0.1
0.1
0
0
0
0
0
0.1
0
0
0
0
0
0
0
0
0
0.1
0.1
0.1
11
12
13
14
15
0.4 0.3
0.4 0.2
0.4 0.3
10
0.2
0.2
0.2
0.5
0.3
0.3
0.2
0.1
0.1
0
0
0
The FE software is also an important numerical tool for the investigation
of the rotational properties of the local magnetic flux patterns. As a first
approximation, the local B loci can be described as an ellipse quantified
by the maximum axis and by an axis ratio a. The axis ratios a for all
points considered in Fig. 2.8 for three subsequent teeth in the stator of
the induction machine are collected in Table 2.4. Up to point # 9, the
behavior of the magnetic flux is almost uniaxial.
Fig. 2.9. Uniaxial magnetic fluxes in point # 7 for three subsequent teeth.
Fig. 2.10. Elliptical magnetic fluxes in point # 13 for three subsequent teeth.
Working conditions of electrical steels
49
Little rotational effects were found in the tip of the three considered
teeth in a row. However, the magnetic flux in points # 10 to 14 exhibits
an elliptical behaviour with axis ratios up to a = 0.5. Point #15 has an
uniaxial flux. Typical uniaxial and elliptical magnetic fluxes are
observed in the middle of the tooth (point # 7), and in the yoke (point #
13), respectively, see Fig. 2.9 and Fig. 2.10.
The electrical steel used in that induction machine was semiprocessed steel of 0.5 mm thickness. Steel sheets were first punched to
form the stator and rotor laminations, and were then stress relief
annealed to reduce the effect of punching. Then the laminations were
assembled into a magnetic core of the induction machine. For the
calculations described above, the magnetic properties of the laminations
were assumed the same for both tooth and yoke regions.
Similar studies have been done by other researchers worldwide. For
example, IEN Galileo Ferraris in Turin, Italy, have been recently
studying rotational fluxes in stator cores [Botta02].
A three-phase stator (Fig. 2.11), taken from a four-pole induction
machine, has been equipped with pick-up coils. A barless rotor makes
the no-load test better as electrical losses in the rotor bars induced by
higher harmonic magnetic fields are not present. A test device, driven by
a synchronous motor, has been employed for a thorough investigation of
the magnetic behavior of the stator core during the no-load test.
Fig. 2.11. Pick-up coils on the stator (A-L) and rotor (M-O) [Botta02].
50
Chapter 2.
Fig. 2.12. Measured and computed flux density loci at the tooth root [Botta02].
The analysis has been developed under sinusoidal, six-step and twelvestep voltage supplies. Experimental results have been compared with
the results provided by the 2D FE model. A good agreement has been
found in the prediction of the flux density loci.
To study magnetic flux patterns and distortion, several pick-up
coils were placed in different parts of the stator, denoted as A, D for the
yoke, B, C, F, E for the tooth (Fig. 2.11). The rotor was also equipped
with coils to check if there are higher harmonics induced in the rotor
core. To study the rotational behavior, two sets of orthogonal coils,
denoted as I-L and G-H, were placed on the stator in the region between
tooth and yoke. Measured and computed magnetic flux density loci are
shown in Fig. 2.12. Here, BL is the tangential component and BI is the
radial component of the flux density. The axis ratio a is around 0.3.
The last example of the experimental investigation of rotational
flux patterns in the stator shows how difficult it is to equip actual
electromagnetic devices with pick-up coils in the right places. On the
other hand, it shows how the simulation of actual working conditions in
a test device could open new areas and extend the experimental results.
Finally, a numerical analysis is a very powerful tool to predict the
properties of local magnetic flux patterns in an induction machine
[Gyselinck00], [Botta02], [Pulnikov04].
Based on the present studies, a measurement setup for the
investigation of electrical steels under stress should be capable of
creating the following magnetic conditions. First, it should create
sinusoidal and distorted magnetic flux patterns in electrical steel.
Working conditions of electrical steels
51
Second, it should create alternating and rotational magnetic fluxes at
least up to a = 0.5.
2.4. 2D mechanical working conditions in induction machine.
The mechanical conditions under which electrical steel are used in
rotating machines strongly depend on the power and the type of the
induction machine.
Magnetic cores of large machines usually consist of segmented
pieces of the stator. The segments are fixed by a trapezoidal or circleshaped fixing element on the outer diameter. These elements introduce
negligibly small mechanical stresses on the outer diameter of the stator
segment. Nevertheless, there is another source for mechanical stress,
present in large induction machines, i.e. an orthogonal compression by
means of bolts and grips. The effect of over-compression or undercompression could be significant, but this source of the orthogonal
mechanical stress is out of the framework of the current research.
Fig. 2.13. Stator core, combined from a number of electrical steel laminations.
a
a)
b)
stresses
Fig. 2.14. Assembling compression of stator (a) and rotor (b) [Pulnikov04].
52
Chapter 2.
Medium power induction machines are usually produced from one
sheet, so there are no segments. Despite that, the fixation mechanism is
very similar to large machines. It does not introduce any mechanical
stresses in the plane of the lamination.
Low power induction machines usually have a simple circle
shape for the outer diameter. The attention of the current research is
focused particularly on this case. The magnetic core of the stator or rotor
of small induction machine consists of laminations, produced from semiprocessed or fully-processed non-oriented electrical steels, see Fig. 2.13.
When a magnetic core of the stator is assembled into a frame of
induction machine, various mechanical stresses could be introduced by
the assembling procedure, see Fig. 2.14a. Similar stresses could be
observed in the rotor core, see Fig. 2.14b.
In low power induction machines, the stator core is usually
compressed into the heated frame. This is a very simple technology for
clamping the stator core, see Fig. 2.14a. The friction between the frame
and the stator core is sufficient in order to prevent any movement of the
stator. If there was no friction, the stator would rotate under the
electromagnetic force. The rotor laminations are usually shrunk on the
rotor shaft, which introduces the required friction, see Fig. 2.14b.
Few studies have been done to simulate such external
compression and its effect on energy losses in induction machine. The
effects of radial stress to the flux distribution and power loss were
extensively studied in a test device, simulating an induction machine
[Moses89]. To generate radial stress, a series of 30 pistons provided
pressure on the outer surface of the stator stack of 165 mm diameter.
Stress up to 4 MPa was applied by hydraulic pressure built up in a large
cylinder connected to the pistons.
Two cores used for stator laminations were compared, each
produced from the same non-oriented low silicon steel with 0.65 mm
thickness. However, one core was annealed and the other one was not
annealed. The laminations were magnetized as in an induction motor
stator core. The flux density and the local power loss were measured at
various flux densities using arrays of search coils. The radial component
of the flux in the teeth was checked with search coils looped around
every second tooth. Orthogonal search coils were wound through 0.30
mm diameter holes 5 mm apart at predetermined locations in the teeth
and the yoke regions of the stator core. Those coils were used to
determine the magnetic flux in the stator back under stress, see Fig. 2.15.
Working conditions of electrical steels
53
Fig. 2.15. Flux density at stress free (solid) and 4 MPa (dotted) [Moses89].
The ellipses in Fig. 2.15 signify the presence of rotational flux patterns of
varying degree. The largest axis ratio of a = 0.4 was found behind the
teeth.
The applied stress makes the flux more rotational behind the
teeth and less rotational behind the slots. The axis ratio due to the radial
stress of 4 MPa was around 0.5 instead of 0.4 at stress-free conditions.
The applied stress tends to make the magnetization in radial directions
more difficult so the flux is forced further outwards behind the slots and
drops behind the teeth [Moses89]. Similar changes were found in the
unannealed core at magnetic conditions of 1.0 and 1.3 T at 50 Hz.
The stator geometry was divided into 5 regions. Region 1 denotes
the teeth and the other 4 are regions of equal width in the core back, as
shown in Fig. 2.16. The power loss was measured locally using 0.5 mm
thick, 0.5 mm long microchip thermistors.
The method based on the rate of change of the temperature was
used to estimate the local loss [Thomas80]. The local power loss
increases due to stress by 10% to 40%. Initially, the losses are lower
behind the slots than behind the teeth because the rotational flux is
lower in that region.
An application of stress increases the loss more behind the teeth
than behind the slots because the stress makes the rotational flux higher
as seen in Fig. 2.15.
54
Chapter 2.
Fig. 2.16. Local loss in annealed (left) and unannealed (right) cores [Moses89].
At all regions of the stator yoke a rapid increase occurs at a stress of
around 2 MPa, see Fig. 2.16. The loss increases rapidly in regions 2 and
3, i.e. the inner region of the core yoke, where most rotational flux
occurs. The stator core assembled from annealed electrical steel shows a
higher dependence on stress, see Fig. 2.16 on the left. Although the core
made from unannealed steel shows a smaller dependence on stress, the
magnitude of the loss is much higher.
The described approach of Moses has one large weakness. The
applied radial stress is not a proper way to build a stress dependence of
power loss in such a complex geometry as the stator core. In order to
understand the distribution of actual mechanical stresses in the stator or
rotor core, mechanical modeling of radial deformation on the stator
should be used to simulate the effect of that deformation into localized
mechanical stresses.
A FE calculation of mechanical stresses due to the assembling
process of the stator core into a heated frame is based on the approach
described in [Pulnikov04] or [DeWulf04]. Assuming the frame of the
stator has a constant diameter, then a displacement, i.e. the difference
between the inner diameter of the house and the outer diameter of the
Working conditions of electrical steels
55
stator core, is numerically applied to the stator core. After cooling down
of the frame, the stator core is subjected to a uniform radial compression,
which is similar to the above approach of [Moses89]. If a small
displacement equal to 20 µm is applied towards the center of the sheet, it
results in a mechanical stress distribution as shown in Fig. 2.17. For
example, a further displacement of 40 µm leads to twice compressive
stresses in certain places of the magnetic core. In fact, the maximum
stress was found in the inner region of the core back, where most
rotational flux occurs, according to previous case [Moses89]. Moreover,
the stress distribution in the yoke consists of both radial and
circumferential stresses [Pulnikov04]. Assuming there is no change of
the inner diameter of the stator due to the displacement, the maximum
stresses are concentrated in the inner region of the core back. The tooth
body remains unaffected by these stresses produced by core assembling,
see Fig. 2.17. However, the electrical steel at the edge of the tooth is
strongly affected by punching and stress relief annealing, see Fig. 1.9. If
not annealed, the tooth is subjected to residual stresses as well as
deformation at the edge.
70 MPa
90 MPa
Fig. 2.17. Possible compression in stator and rotor cores after assembling.
56
Chapter 2.
The effect of rotor assembling is also important. When a rotor core is
shrunk onto a rotor shaft, the compression could reach as much as 90
MPa according to the same approach proposed in [Pulnikov04]. Again,
the tooth itself remains unaffected by those stresses, but only affected by
the effect of punching, see Fig.2.17.
A similar study of the stress due to core assembling has been presented
recently by OCAS, Arcelor Group in cooperation with EELAB
[DeWulf04]. The press fitting procedure, applied to assemble the stator
of the induction machine into the housing, leads to mechanical stresses
inside the stator core. The FE computation of radial and tangential
stresses has been performed for a thick elastic cylinder loaded by
uniform internal and external pressures.
According to these computations, a radial displacement of 20
µm, applied externally to the stator core of the induction machine,
results in an internal stress of 45 MPa. The radial displacement of 100
µm leads to the stress of 230 MPa. These values of internal stresses that
are introduced by the fitting of the stator into the housing of the
induction machine can affect the behaviour of electrical steel and the
performance of the induction machine.
2.5. Conclusions.
Based on theoretical and experimental research regarding the actual
working conditions, the following conclusions can be drawn.
Firstly, standard magnetic characteristics of electrical steels are
very limited. In fact, they give only basic information about the idealistic
properties. When it comes to the actual needs of the customers of
electrical steels, more information is required in order to design and
produce an efficient electromagnetic device.
Secondly, each type of electrical steel may be used in different
types of electrical machines, and therefore, exhibits different working
conditions.
Grain-oriented steel used in transformers works mainly under
uniaxial magnetic conditions. T-joint regions of transformers are the
main exceptions. Here, the magnetic flux patterns are rotational. From a
scientific point of view, 2D stress dependence under complex 2D
magnetic conditions is of most interest, since grain-oriented steel has a
large anisotropy between different directions of the lamination.
Working conditions of electrical steels
57
Non-oriented steels, the most used magnetic material today, are
normally working under complex 2D magnetic and mechanical
conditions. The study of 2D stress effect on 2D magnetic properties of
annealed and unannealed non-oriented steels should be very interesting
both from experimental and theoretical point of view.
Thirdly, actual magnetic working conditions of electrical steels
are usually non-sinusoidal and non-uniaxial. Non-sinusoidal magnetic
flux patterns were observed mainly in the tooth regions of the induction
machine. Harmonics lead to an extra power loss and should be taken
into account.
Rotational behavior of the magnetic flux above the teeth may
lead to an extra power loss. From an experimental point of view, an
elliptical flux with axis ratio a = 0.5 is the largest rotational flux found
during studies presented in this Chapter. However, the axis ratio a = 1.0,
i.e. circular rotational behavior of magnetic flux is the most extreme
theoretical condition for 2D rotational magnetization. Therefore, it is
preferable to apply magnetic conditions starting from sinusoidal and
distorted uniaxial excitations to elliptical and even circular.
Fourthly, mechanical working conditions of electrical steels show
complex 3D compressive and tensile elastic stresses, as well as plastic
deformations due to punching of the laminations. By eliminating the
third dimension from the framework of the present study, complex 2D
mechanical working conditions could be simplified to the following
options. In grain-oriented steels, the uniaxial stress could be applied in
different directions in comparison with the rolling direction of the steel.
In non-oriented steels, the uniaxial stress could be applied virtually to
any direction, since non-oriented steels could be considered quasiisotropic. Therefore, it is preferable to use uniaxial mechanical stress,
applied either as compression or tension up to plastic deformation.
Finally, engineering society agreed upon the fact that the actual
2D magnetic measurements are required in order to take into account
the effect of rotational magnetic conditions on energy loss in electrical
steels [Beckley02]. Advanced experiments have been performed during
the last decades, starting from testing machines, simulating 2D
magnetomechanical conditions, to advanced 2D measurement systems.
The latter give the opportunity to simulate different effects or
separate conditions and investigate the magnetic properties of electrical
steel samples. Using a sample is more convenient than carrying out
measurements inside the induction machine or the power transformer.
58
Chapter 2.
Therefore, the way to study the stress effects in electrical steels is to use
advanced measurement setups, suitable for the target study.
The next chapter will study conventional and unconventional
measurement setups and measurement techniques. The main question
will be: which measurement techniques are suitable to simulate the above
mentioned working conditions in different kinds of electrical steels?
Magnetic measurements under stress
59
Knowledge is power.
Sir Francis Bacon, 1597
CHAPTER 3. MAGNETIC MEASUREMENTS UNDER STRESS
"Measurements mean knowledge" [Fiorillo04]. Indeed, measurements are
indispensable to science, industry, trade and economy. They are the
prerequisite for any conceivable development in production and trade.
Actual measurements are the basis for statistical observation, which any
science or research is based upon. For electrical steels, magnetic
measurements are the primary source of information about the magnetic
properties. Measurements are required in all stages of a lifecycle of
electrical steels, starting from the production, followed by inspection
and trading, as well as in the actual use of electrical steels by electrical
machine designers and constructors.
Measurements require a good judgment and this is only possible
if the underlying scientific issues and their applications are understood.
Whatever the specific aim of measurements, there is no shortcut to
physically sound experimental methods [Fiorillo04]. The accuracy and
reproducibility are the most important key values to evaluate the
experimental methods.
Measurements are widely used when the measurement
standards are fixed. Basic magnetic measurements in electrical steels
have been collected in a group of standards IEC 60404. The International
Electrotechnical Committee has agreed upon characterization methods
for electrical steels, known as [IEC 404-2] and [IEC 404-3]. These
standards are applicable to grain-oriented and non-oriented "sheets and
strips" for DC and AC measurements at frequencies up to 400 Hz.
The engineering society is constantly working on the perfection
of existing standards and on the development of new ones. National
Metrological Institutes (NMIs) and international metrological and
standardization organizations play a key role through the mechanism of
an international programme for "intercomparison" by NMIs, such as
PTB, IEN, NPL and others [Sievert96].
What happens when there isn't any standard on specific measurements
available but required both for science and industry?
One approach is to use the standard measurement methods and
try to modify them in order to include a new measurement task. For
example, a standard system could be extended by one or another
60
Chapter 3.
mechanism or measurement instrument. When the new measurement
system is complete, it is calibrated to the standard system. Calibration to
the standards gives a great advantage of comparison between new
results and proven measurements performed by a standard system.
Unfortunately, not every new system can be calibrated to comply with a
standard measurement system. Moreover, the method of
intercomparison for non-standard techniques gives sometimes rather
different measurement results [Sievert96]. The difference, if not
explained, could become the obstacle for approval of a new standard by
the international standardization institutes.
Another approach is more flexible but less reliable than the
applied method of the intercomparison of the results. If a new
measurement system is so specific that its results are difficult to calibrate
to the results of a standard method, then the new system could produce
results on a relative basis. It simply means that all measurements are
performed in a single system, under the same conditions, by the same
instruments and methods. Two requirements the new system must
comply with are the accuracy and the reproducibility. If the accuracy of
the instruments and parts of a new system is good enough, then the
results obtained by the new system could be comparable to each other,
i.e. on relative basis only. The reproducibility of the results should lay in
a given area of the acceptable error, which might depend on all parts of
the system. To start with a new measurement system, first, the
conventional methods must be studied thoroughly.
3.1. Standard magnetic measurements in electrical steels.
The standards have been mentioned many times. Herewith a brief
description of the standard methods is made.
3.1.1. Epstein frame.
One of the most used standard is the standard IEC 60404-2 "Magnetic
Materials. Part 2: Methods of measurement of magnetic, electrical and
physical properties of magnetic sheet and strip." It is also called the
"Epstein frame" standard [IEC 404-2].
The Epstein frame is actually an unloaded transformer. It
comprises a primary winding, a secondary winding and the specimen to
be tested as a magnetic core. The use of the Epstein frame is applicable to
Magnetic measurements under stress
61
flat strip specimens obtained from magnetic sheets and strips of any
quality. The frame itself is shown in Fig. 3.1 and it is a closed circuit. The
closed circuit is typical for standard characterization of soft magnetic
materials and it is applied in several standards IEC 60404-2-3-4-6-10.
The magnetic circuit is made of a core constructed with the strips
to be tested, assembled in a square, having double-lapped joints to form
four branches of equal length and equal cross-sectional area. The steel
samples for Epstein frame measurements must be cut burr-free, having a
width of 30 ± 0.2 mm and a length of 280 ± 0.5 mm. The average
magnetic path length is agreed by the NMIs to be equal to 0.94 m. The
number of steel samples to be used in the tests is at least 12 or a
quadruple. Furthermore, it is permissible to apply a tangential force of
about 1 N to each corner.
To measure the polarization µ0M instead of the induction B, an
air flux compensation completes the test setup. A mutual inductor for air
flux compensation is connected in series with the primary and
secondary windings of the Epstein frame. The voltage induced in the
secondary winding of the mutual inductor by the primary current does
compensate the voltage induced in the secondary winding of the empty
Epstein frame by the flux attributed to the primary current. Last but not
least, the waveform of the secondary induced voltage shall be controlled
sinusoidal, having a form factor of 1.111 within ± 1%.
Fig. 3.1. The Epstein frame according to IEC 60404-2 [IEC 404-2].
62
Chapter 3.
The standard IEC 60404-2 defines several methods to identify the
various properties of the test specimens in the Epstein frame. The
following methods are described:
- determination of specific total losses by the wattmeter method,
- determination of magnetic characteristics by the bridge methods,
- determination of the magnetic field strength, the excitation
current and the specific apparent power,
- determination of the magnetic flux density in a DC field,
- determination of the density of the magnetic sheet,
- determination of the resistivity of the magnetic sheet and strip,
- determination of the coefficient of the surface insulation
resistance,
- determination of the stacking factor,
- appendices on air flux compensation, complex permeability and
density.
The purpose of these methods is to define terms and specify techniques
for the measurement of magnetic, electrical and physical properties of a
magnetic sheet.
3.1.2. Single sheet tester.
The second mostly used standard is IEC 60404-3 "Magnetic materials.
Part 3: Methods of measurement of the magnetic properties of magnetic
sheet and strip by means of a single sheet tester." It is also called the
"SST" standard [IEC 404-3].
Here, the test specimen comprises a sample of the magnetic sheet
and is placed inside two windings, an exterior primary winding (the
magnetizing winding in Fig. 3.2.) and an interior secondary winding. To
minimize the effect of the weight and pressure on the test specimen, the
upper yoke shall be provided with a means of suspension which allows
part of its weight to be counterbalanced as shown in Fig. 3.2.
In order to reduce the effect of eddy currents and provide a more
homogeneous distribution of the flux , the yokes are a pair of U-cores or
a glued stack of laminations. Each yoke is in the U-form, made up of
insulated sheets of grain-oriented silicon steel or nickel iron alloy. The
yoke pole faces have a width of 25 ± 1 mm. The air gap between the
opposite pole faces of the yokes shall not exceed 0.005 mm in order to
form a suitable closed circuit.
Magnetic measurements under stress
63
180 – 300
mm
25 mm
magnetizing winding
secondary winding
suspension
test specimen
500 mm
Fig. 3.2. The single sheet tester with counterbalances [Fiorillo04].
The test specimen for the SST shall have a length of at least 500 mm and
a width as large as possible but less or equal to the width of the yokes.
The standard defines the width of the yokes equal to 500 mm,
however, it is recognized that other yoke dimensions can be used. The
main condition to modify the dimensions is very simple. The yoke
dimensions can be modified provided that the compatibility of the
results can be demonstrated. In other words, a researcher could use a
smaller SST but the results must be comparable to the results by the
standard SST. The sample used in the standard SST may be square,
which allows rotating the sample 90 degrees. Thus, only one sample of
electrical steel can be used to define the magnetic properties both in the
rolling direction and in the transverse direction.
The air flux compensation is done in a similar way as in an
Epstein frame. The primary winding of the mutual inductor is connected
in series with the primary winding of the test apparatus, while the
secondary winding of the mutual inductor is connected in series with
the secondary winding of the test apparatus. The adjustment of the
value of the mutual inductance is done when passing an alternating
current through the primary windings in the absence of the specimen in
the apparatus, the voltage measured between the non-common
terminals of the secondary windings shall be less than 0.1% of the
voltage appearing across the secondary winding alone.
Thus, the average value of the rectified voltage induced in the
64
Chapter 3.
combined secondary windings is proportional to the peak value of
magnetic induction in the test specimen. Furthermore, the waveform of
the secondary induced voltage shall be controlled sinusoidal, having a
form factor of 1.111 within ± 1%. The standard is applicable at power
frequencies to grain-oriented and non-oriented magnetic sheets for:
- specific total loss, specific apparent power, r.m.s. value of the
magnetic field strength at magnetic induction 1.0 to 1.8 T in
grain-oriented steels,
- specific total loss, specific apparent power, r.m.s. value of the
magnetic field strength at magnetic induction 0.8 to 1.5 T in nonoriented electrical steels,
- peak value of the magnetic induction, peak value of the magnetic
field strength at magnetic field strength 10 kA/m in grainoriented steels,
- peak value of the magnetic induction, peak value of the magnetic
field strength at magnetic field strength 1 kA/m in non-oriented
steels.
Annex B of IEC 60404-3 proposes the non-obligatory calibration of the
SST "for those who wish to obtain the correlation between measurements taken
by this method and the Epstein frame method" [IEC 404-3]. The calibration
consists of the determination of the effective length of its magnetic
circuit from the measurement of the specific total loss in the Epstein
frame. This calibration should be performed for each grade of material
and each magnetic flux density.
3.1.3. DC magnetic measurements of iron and steel.
The third standard of interest is the standard IEC 60404-4 "Magnetic
materials. Part 4: Methods of measurement of d.c. magnetic properties of
iron and steel." This part of a series IEC 60404 specifies the techniques of
measuring the DC magnetic properties of steel in a closed magnetic
circuit using either the ring or the permeameter method.
The ring method is used to obtain the magnetization curve and
the hysteresis loop for a magnetic field strength up to 10 kA/m. A higher
magnetic field strength could cause an overheating of the test specimen.
The latter is a homogeneous unwelded ring with a rectangular or
circular cross-section, which ranges from 100 to 500 mm2.
A temperature sensor is attached to the test specimen. A
secondary winding of insulated copper wire is wound uniformly around
Magnetic measurements under stress
65
the core. A magnetizing winding capable of carrying the maximum
magnetizing current and of a sufficient number of turns to produce the
maximum required magnetic field strength is wound in one or more
layers on the core. Various cooling methods can be applied as well. The
magnetizing field strength H is proportional to the magnetizing current,
the number of turns of the magnetizing winding, and inversely
proportional to the path length.
3.1.4. Magnetic measurement techniques used in standard methods.
The principles behind the standard methods are defined in the already
mentioned Maxwell equations (1.1) – (1.5). Moreover, the standard
methods usually employ a closed circuit. What are the advantages and
disadvantages of that approach?
It all starts from the Ampere's law (1.2). According to this law,
the line integral of the magnetic field H along a closed path lm equals the
total electrical current through a surface spanned over the path lm, and is
given by
∫ H ⋅ dl = I
total
= N ⋅I
(3.1)
Here, N is the number of turns, each carrying a current I. By assuming
that H is constant along the closed path lm, the magnetic field strength is
given by
H=
N ⋅I
lm
(3.2)
The expression (3.2) is used in IEC 60404-4 as a method for measuring
the magnetic field strength by the ring method. Here, H is the magnetic
field strength in A/m, N is the number of turns of magnetizing winding
of the ring, lm is the mean magnetic path length of the ring in meters, and
I is the magnetizing current in A.
This is the first principle for the magnetic measurements: the
field is proportional to the current in the magnetizing winding.
The next method is based on Faraday's law, see eq. (1.1).
According to this law, the voltage induced in an electrical circuit is
proportional to the rate of change of the magnetic flux, linking the
circuit. Lenz's law indicates that the induced voltage is in a direction
which opposes the flux change producing it.
When the physical magnetic flux Φ is passing through a coil of N
66
Chapter 3.
turns, and dΦ /dt is the rate of change of the magnetic flux, then with the
average induction B and the cross section S, the induced emf E [in Volt]
is given by
E = −N
dΦ
dB
= − NS
dt
dt
(3.3)
This is the second principle for the magnetic measurements. The
induced voltage is proportional to the time derivative of the magnetic
flux density, or dB/dt.
The cross section S of the secondary windings in the Epstein
frame or the SST is larger than the magnetic cross section Sm of the test
specimen. Hence, the air flux in the secondary winding contributes to
the voltage induced in that winding. The magnetic flux density B'
integrated from the measured secondary voltage (3.3) differs from the
magnetic flux density B in the test specimen:
B' = B + µ 0 H
S − Sm
S
(3.4)
A series mutual inductor approximately compensates the error from the
second term on the right of (3.4) [IEC 404-2].
When the industrial society had realised that using a SST is much
easier than using the Epstein frame, in particular when considering the
preparation of the test specimens, the research centers developed novel
measurement techniques to measure the magnetic field H, because of
possible errors arising from the estimation of the magnetic path length lm
of the SST or the Epstein frame.
Therefore, other methods for measuring the magnetic field
strength were required. A practical method is the tangential magnetic
field measurement by an H coil [Pfutzner91]. The results obtained by the
H coils were compared with the results by the previously used current
method. The H coil method is considered a physically sound
measurement technique, based on the following principle. If a search
coil is placed in an alternating magnetic field in air, the emf U induced in
this coil is proportional to the time derivative of the magnetic field,
similar to (3.3):
U = − NS ⋅ µ 0
dH
dt
(3.5)
Here, the H coil measures the magnetic field strength in the air, passing
through the cross section NS. Once the H coil is calibrated to determine
the cross section NS, it can be used as a physically sound sensor. Indeed,
Magnetic measurements under stress
67
when a H coil is placed near to the surface of the sample with its axis
parallel to the surface, it can be used to measure H in the sample parallel
to the axis of the H coil, as the tangential component of H is continuous
at the surface of the specimen.
The calibration procedure for the H coil uses the Helmholtz coils,
for example, the ones shown in Fig. 3.3. The Helmholtz coils system
produces an uniform magnetic field in a central cylinder parallel to the
Helmholtz coil axis. This uniform cylinder has a diameter of 50 mm and
a length of 50 mm. The search coil is placed on top of the magnetically
and electrically non-conducting supporting blocks in the centre of the
uniform cylinder.
The water-cooled Helmholtz coils system of Fig. 3.3 used in
EELAB is capable to create a magnetic induction in the air up to 0.02 T,
which corresponds to a magnetic field of 16 kA/m. Rated current in the
Helmholtz coils is 35 A at the power frequency of 50 Hz.
The NS value of calibrated search coil is proportional to the
induced voltage in the search coil, and inversely proportional to the
excitation current of the Helmholtz coils system. Due to linearity, the
results are independent of the excitation current.
Fig. 3.3. The Helmholtz coils system by Walker Scientific Inc. EELAB.
68
Chapter 3.
3.2. State of the art in 2D magnetic measurements.
Starting from the standard methods, the art of advanced magnetic
measurements has been recently evolved into a new level of high
quality, accuracy and reproducibility. New digital methods have
replaced conventional analogue ones in various measurement systems.
Today, a modern software package can easily do a job of analogue
devices, for example, a job of integrators, when dB/dt is converted into B
as from (3.3). Modern measurement systems become more accurate and
advanced than ever before.
The question of accuracy rises from any kind of measurements,
including magnetic ones. Despite the agreement on the standard
methods, the engineering society is well aware of the existing systematic
errors in the standards. The main cause of these errors is nonhomogeneous magnetic field distribution along the path length. It
results from an assumption of the mean magnetic path length lm, equal to
0.94 m in the Epstein frame or 0.45 m in the SST. The errors can be taken
into account by extra coefficients. Otherwise, the deviation can be as
much as 10%. More about systematic errors can be found in [Sievert84],
[Pfutzner91], [Girgis98], [Sievert99].
Why users of standard methods need new measurement systems?
One of the reasons is that the dimensions of the test specimens in the
standard methods are fixed. When a smaller specimen is under
investigation, then a customized SST can be applied. A typical practical
example for a smaller SST application is the identification of the
properties of a stator lamination of an induction machine. A variety of
measurement systems, based on the standard ones, is produced today,
as shown in Fig. 3.4.
Another application of small SSTs is the characterization of
specimens obtained by production methods on laboratory scale
[DeWulf03], [RosYanez03].
In order to obtain reliable and accurate data about electrical steel
properties using specimens of small dimensions, usually a small SST is
calibrated to the standard one. Here, the calibration means the
procedure for the determination of the magnetic path length lm for
different yoke constructions as in Fig. 3.5. The procedure is based on
using the same material in different measurement systems, including the
standard such as the Epstein frame or the SST. The power loss, obtained
by the Epstein frame or the SST, is considered as the reference for
Magnetic measurements under stress
69
calibration of the miniature SSTs. An accuracy within 5% deviation is
quite feasible to achieve [DeWulf03].
Fig. 3.4. Various systems for punched parts and strips. Source: Brockhaus.net.
Fig. 3.5. Dimensions of various miniature single sheet testers [DeWulf03].
70
Chapter 3.
Small dimensions are fairly good reasons for using miniature
measurement systems, based on standard methods or novel
measurement techniques. However, the main reason to choose a new
measurement system is the limit of both magnetic and mechanical
working conditions of standard magnetic measurements.
Indeed, standard measurements by the Epstein frame or the SST
are performed at a controlled sinusoidal voltage waveform of the
secondary windings. The magnetic flux in the standard test specimens is
assumed as sinusoidal. Why a sinusoidal magnetic flux or a sinusoidal
induction B are employed in magnetic measurements?
The major reason for using a sinusoidal controlled magnetic flux
density is that in this way the experimental results of different
measurements can be easily compared.
As shown in Chapter 2, the actual working conditions in a
machine or a transformer are far from the standard 1D conditions. The
magnetic flux has a number of higher harmonics, which affect the
magnetic properties and the energy loss in electrical steels. Moreover,
due to clearly rotational behavior of magnetic flux in electrical steels, 2D
magnetic measurements are required. The mechanical conditions of
electrical steels can be included as well. Hence, novel measurement
setups are required for an advanced magnetomechanical study. The art
of advanced multidimensional magnetomechanical measurements has
started from modifying conventional systems by introducing a second
and even a third dimension into the measurement system [Zhu02].
A crucial choice in magnetic measurements is the choice of the
dimensions of the sample. The Epstein frame uses 12 (or more) steel
strips, having dimensions 30 mm x 300 mm. Its sample preparation is
more complicated than the preparation for the SST. The latter uses the
steel sample, having dimensions 500 mm x 500 mm, or smaller up to 30
mm for a miniature SSTs, as shown in Fig. 3.5.
Despite so large difference, a majority of 2D measurement setups
uses square or circle samples, having dimensions 60 to 100 mm
[Sievert96]. In fact, these dimensions of the samples are considered the
best in terms of measurement techniques and field strengths. Sometimes,
different measurement techniques and different sensors could be
successfully applied in a single measurement setup.
According to [Sievert96], many European research centers, such
as PTB, IWE, LEG, IEN, EBG, WCM, have developed various 2D
measurement setups to investigate the rotational behavior of magnetic
Magnetic measurements under stress
71
flux in electrical steels. They all had an objective to develop a reliable
and fairly accurate system of 2D magnetic measurements in electrical
steels. However, each center has used different measurement techniques.
Hereby, a brief study of 2D measurement setups, used in the above
European research centers, as well as some centers in Japan, is
presented.
3.2.1. 2D single sheet testers.
For 2D magnetic measurements, two sets of B and H probes placed in
the same area are required. The sets measure B and H in orthogonal
directions. If it is possible to create uniform magnetic conditions in the
area of the sample where the probes are, it can be assumed that the 2D
magnetic measurements are relatively accurate.
The first idea implemented in the 70s was an idea to use U-shape
yokes, similar to the ones in the standard SST. It is a typical benchmark
approach to use the proven SST technology and then calibrate it to the
SST standards.
Here, it is assumed that the two axes of the magnetic system are
independent from each other. In this case, each axis can be considered
separately. Thus, uniaxial results for each axis of magnetic excitation
should correspond to the results obtained by equivalent SST
measurements.
The double yokes of the setup in Fig. 3.6 are used for each of the
two axes, having a side of 300 mm or more. The yokes are laminated in
U-shape from grain-oriented steel. The measurement area is defined as a
result of a numerical analysis of the field distribution, and it corresponds
to an 80 mm central square [Nencib96]. The sensors are positioned in the
geometrical center of the system. The B coils are wound up through
holes, drilled in the sample. The H coils are placed exactly above the area
covered by the B coils.
The double H coils method is used. The H coils are respectively 3
and 8 mm above the sheet surface. The magnetic field H at the surface of
the specimen is obtained by linear extrapolating the values of H by the 3
mm and the 8 mm sensor [Nencib96]. The X and Y coils (XY plane =
plane of the specimen) are separately wound crosswise on two 80 x 80 x
2 mm nonmagnetic formers, which are rigidly linked together.
The system of sensors is then fixed to the yoke support in order
to ensure a precise adjustment and good repeatability. The test specimen
72
Chapter 3.
can be placed easily without moving the H coils.
The performance of the double yoke SST in Fig. 3.6 has been
evaluated by making a comparison under an unidirectional field with a
standard SST. Identical square samples of grain-oriented and nonoriented silicon steels were characterized with the new rotational single
sheet tester (RSST) and the conventional SST. The experiments were
carried out at 50 Hz along the rolling and transverse directions from 0.5
to 1.5 T. The maximum discrepancy between the two systems was
within 5% in losses and 15% in field strength. This can be attributed to
the fact that the SST field is measured with the magnetizing current
method while the RSST uses the double H coils.
An important part of the setup is the power and control system.
The 300 mm RSST was supplied by a two-phase generator via two
power amplifiers. A negative feedback technique was used for the two
axes in order to control the rotating flux in the sheet. It was implemented
taking into account different conditions, such as non- and grain-oriented
electrical steels, and easy and hard direction excitations. Corrector
components were added in order to avoid instabilities.
The six signals (four H signals and two B signals) were measured
using a Plug-In PC data acquisition card which offers an adaptable preamplification for each channel.
The symmetry of the device was verified by rotating the sample
90 degrees. Besides rotational property measurements, the RSST
performs an uniaxial excitation in any arbitrary direction.
The upper and lower yokes of the RSSTs on the right in Fig. 3.6
have also the U shapes, similar to the construction of the standard SST.
The laminated auxiliary yokes help to reduce the leakage fluxes
[Nakata93]. The position of the exciting windings can be as near as
possible to the test specimen in order to avoid leakage fluxes, as shown
in Fig. 3.7. In this setup, a large square sample is used, having a size
equal to 150 mm x 150 mm.
Magnetic measurements under stress
73
Fig. 3.6. The large double yoke RSSTs: left -[Nencib96], right - [Nakata93] .
Fig. 3.7. An improvement of the double yoke RSST [Nakano99].
74
Chapter 3.
Fig. 3.8. Construction of the two-axis cross-wound H coils [Nakata93].
In order to determine the area for the B and H sensors, the uniformity of
the magnetic field distribution in the specimen is measured and
calculated. As a result of the analysis, the distribution of the magnetic
field was found nearly uniform within a 40 mm square. The two H coils
method is used to measure the magnetic field in the rolling and in the
transverse direction. The H coils are wound on a single epoxy glass plate
to eliminate the angle error between two H coils in the X and Y
directions, see Fig. 3.8.
The thickness of the epoxy glass plate is equal to 1 mm. This is
considered as the best value for the H coils design [Pfutzner91].
3.2.2. Shift to a horizontal rotational single sheet tester.
The measurement setup of Fig. 3.7 was among the first in the world to
use an advanced yoke design with the auxiliary yokes. A further shift
from the SST-like U-shape design to an unconventionally flat design of a
RSST seemed to be logical, see Fig. 3.9.
Magnetising
coils
Core
Sample
Fig. 3.9. Shift from the SST-like (left) to a flat design (right) of the RSST.
Magnetic measurements under stress
75
Indeed, if the sample size is about 100 mm, then it is possible to
construct a RSST for the sample in such a way that the whole design
becomes flat and horizontal. With a sufficiently large cross section of the
yokes, the magnetic fluxes from both X and Y excitations can coexist in a
single core. Energy losses in the yokes can be neglected.
The idea of a flat design for a RSST became so popular, that
research centers created different types of these RSST. The one in EELAB
is shown in Fig.3.10.
When considering a two-phase system, the field homogeneity in
the samples for different shapes and sizes should be studied in 3D. Due
to the complexity of a 3D numerical model, 2D models were used for the
magnetic field analysis of the setup in Fig. 3.10: a XY model and a XZ
model. The XY model describes the field patterns in the sample at low
and high inductions, see Fig. 3.11 and Fig. 3.12. The XZ model describes
the flux lines passing from the yoke to the sample, see Fig. 3.13. The
study reveals that, firstly, the square samples yield the most uniform
field, and secondly, an increase of the air gap leads to a more uniform
field distribution in the sample [Makaveev99].
y
measurement
coils
sample
yoke
x
magnetizing
coils
air gap
z
x
Fig. 3.10. Rotational single sheet tester with air gaps [Makaveev00].
76
Chapter 3.
Fig. 3.11. XY models of square and circle samples [Makaveev03].
Fig. 3.12. XY models of square sample at high induction [Makaveev03].
field lines
z
yoke
sample
x
Fig. 3.13. XZ model for single sample and square yoke [Makaveev03].
Unfortunately, the magnetic field distribution in the XZ plane in Fig.
Magnetic measurements under stress
77
3.13 reveals some doubt about the accuracy of the magnetic field
measurements by the H coil technique.
The XZ model in Fig. 3.14 shows a substantial influence of stray
fluxes on the field distribution above the sample, even in the central
area, where the H coils usually perform the measurements.
In order to measure a homogeneous magnetic field strength, a
shielding method was applied, as shown in Fig. 3.14. Here, the samples
have an 80 mm size square shape.
When the H coil is shielded, i.e. laying between two similar
samples, there are no stray fluxes in the XZ plane between the samples.
When no shielding is applied as on the top of Fig. 3.14, the
inhomogeneous field distribution leads to an incorrect interpretation of
the signals of the H coils, resulting in different BH loop (Fig. 3.15).
The method for a stray flux shielding was proven to be very
effective in obtaining an accurate field strength measurements in the
RSST [Makaveev00]. Thus, on the one hand, the shielding leads to the
desired 2D homogeneous field pattern in almost the whole area of the
samples. On the other hand, a single H coil method can be successfully
applied instead of the double H coil method. Indeed, when the field is
homogeneous between two samples, there is no need for two H coils to
measure the magnetic field with sufficient accuracy.
Fig. 3.14. XZ model with and without shielding [Makaveev00].
Generally, the measurement of the magnetic field strength by the
78
Chapter 3.
H coil can only lead to a correct estimation of the field strength on the
surface of the studied material, if the air-flux lines are parallel to the
plane of the sample and thus do not penetrate the sample or the H coil in
the Z direction. Thus, the double H coil method has a doubtful physical
basis without shielding, as it does not remedy the 3D stray fluxes.
To ensure that the magnetization in the shield and in the sample
is approximately the same, the laminations should be made from the
same material, which guarantees an uniform magnetic field strength H
for the sample. Furthermore, the shielding laminations should be
mounted with their RD parallel to the RD of the test specimen.
Fig. 3.15. BH loop with and without shielding [Makaveev00].
exciting coils
air gap 0.1
B-coils
Y
X
unit : mm
closer yoke
specimen
21
Y
H-coils
X
0.1
80
260
Fig. 3.16. A 3-phase RSST [Pfutzner96] and novel setup by [Sugimoto04].
In other words, it is crucial for the measurements to keep these two
Magnetic measurements under stress
79
samples under the same magnetic, thermal or mechanical conditions.
There are some other designs of the RSST, each having its
advantages and disadvantages. Fig. 3.16 depicts two other types of
advanced RSSTs. These two have a higher limit for the magnetizing field
and the induction in the sample. The one shown on the left in Fig. 3.16
obtains the power excitation from a conventional three-phase power
supply [Pfutzner96]. The setup uses a hexagonal shape for the test
specimen.
Another setup (on the right in Fig. 3.16) has a square sample but
a very complicated yoke design and a small air gap of 0.1 mm
[Sugimoto04].
3.2.3. Typical operational algorithms for magnetic measurements.
The majority of modern 2D measurement setups use some kind of
operational algorithm in order to control the magnetic flux density in the
sample. This allows the intercomparison of the experimental results.
Each measurement setup consists of two major parts: an
excitation system enforcing a flux pattern to the test specimen and a set
of measurement instruments. By changing the waveform of the
excitation voltage it is possible to achieve the desired waveform of the
magnetic flux density in the sample.
Various control algorithms have been developed throughout the
history of magnetic measurements. An up-to-date scheme of the
measurement control and operations, as applied in the majority of
modern measurement setups, is shown in Fig. 3.17.
The upper part of the scheme in Fig. 3.17 represents the
excitation, generated by the software and amplified by a power
amplifier.
Output
Software
Waveform
Control
Input
Data
Acquisition
Card
Power
Amplifier
Windings
Signal
Amplifier
B and H
probes
Fig. 3.17. Scheme of operational algorithm for magnetic measurements.
Generally, that kind of excitation can supply multi-phase
80
Chapter 3.
windings of the measurement setup. The lower part of the scheme
represents the measurement signals from the B and H probes, amplified
by signal amplifiers and read by the data acquisition card. The software
is responsible for all operations as well as for the waveform control to
obtain the true waveform of B according to the desired waveform, e.g. a
sinusoidal or distorted B. The data acquisition card converts analogue
signals of the sensors into digital information for the software, and vice
versa, from digital to analogue signals to the excitation windings.
A typical algorithm generates an output to the excitation
windings, and reads the B and H signals. The waveform control updates
the output generation in a cycle until the magnetic induction waveform
reaches the desired one.
The accuracy of the complete scheme depends on the design of
the setup and the choice of the measurement technique, as well as on the
accuracy of the amplifiers, the parameters of the data acquisition, the
features of the waveform control. Although the whole system looks
complicated, a high accuracy can be achieved if every part of the system
is designed and constructed at a high level.
The waveform control plays a crucial role in the magnetic
measurements. Generally, the waveform control depends on the
physical property to be controlled: B or H, the design and physical
parameters of the excitation windings, the design of the magnetic parts
of the setup, the presence and value of the air gap in the magnetic path
of the excitation flux, and so on. In case the flux density B is controlled
[Makaveev03], the voltage V for the excitation windings reads for the X
direction (analogously for the Y direction) :
V X = a X B X (t ) + b X
dB X (t )
dH X (t )
+ c X H X (t ) + d X
dt
dt
(3.7)
Here, aX, bX, cX, dX are coefficients depending on the physical parameters
of the setup.
For 2D magnetic measurements, the variables BX(t) and HX(t) are
projections of the corresponding vectors to the X axis. An analogous
equation for the Y direction can be used, similar to eq. (3.7) with all the
parameters and coefficients corresponding to the Y axis.
In case B and H coils are used, the signals from the B and H
sensors (search coils) are proportional to dB(t)/dt and dH(t)/dt,
respectively, see Section 3.1.4.
Magnetic measurements under stress
81
3.3. Magnetic measurements under stress.
In order to perform magnetic measurements with the sample subjected
to a mechanical load, it seems logical to start with the simplest case. A
few research centers did the same job, taking a standard SST or its
miniature copy and upgrading it with a mechanical system to apply
tensile or compressive mechanical load. The first and most logical way
to apply a tensile load is to use weight as shown in Fig. 3.18. Here, the
specimen is gripped by two parts of the vertical system. The
measurements are done by the SST.
The system of Fig. 3.18 provides an ideal mechanical load, which
is constant and uniaxial. Indeed, the uniaxial stress is proportional to the
weight and inversely proportional to the cross section of the test
specimen. The system is durable and simple, however, it has its limits.
Only tensile load can be applied. Moreover, the load has a maximum
value, limited not only by ultimate tensile strength of the material, but
also by the safety norms. For example, it takes 2000 kg to create a
sufficient load to start plastic deformation in non-oriented electrical steel
with 0.5 mm thickness and having a width of 80 mm. That kind of load
is difficult to provide in non-industrial working space.
Sample
B-coil
H-coil
primary
winding
SST
yokes (U-shape)
Weight
Fig. 3.18. Vertical SST under uniaxial stress due to weight [Hubert04].
82
Chapter 3.
In addition to mechanical limits, there are some limits related to
magnetic measurements. To use the U-shape yokes of the SST, it takes an
extra frame to hold these yokes vertically. The accuracy of the magnetic
measurements will depend on the air gap between the SST yokes and
the sample, which must be controlled by an extra supporting frame.
Here, the dimensions are very important. The standard SST of 500 mm is
not as convenient as a small SST, calibrated to the standard SST.
Fig. 3.19. Horizontal small SST under uniaxial stress ±50 MPa [DeWulf02].
Fig. 3.20. Horizontal measurement setup with uniaxial stress [Delage97].
Magnetic measurements under stress
83
Using a miniature SST, it is also possible to introduce uniaxial stress to a
small sample, as shown in Fig. 3.19. The mechanical load is applied
horizontally by a manual limb (on the right) and measured by a load cell
installed on the shaft (on the left). The sample dimensions are 30 mm
width and 90 mm length. The SST has a length of 60 mm.
When a weight system is used, as in Fig. 3.18, a compressive
stress is impossible to apply to the steel sample. A miniature system in
Fig. 3.19 allows applying uniaxial stresses, both tensile and compressive,
up to 50 MPa. A higher tensile stress is also feasible to achieve.
Another example of how to apply an external mechanical load to
a larger sheet of electrical steel is shown in Fig. 3.20. Translating the
mass along the lever arm creates a moment, amplified by the toothed
wheel work around the axis A. This creates a tangential force to the
cylinder C, normal to the axis A, transmitted to the specimen via a steel
strip and the pivot linking. The maximal applied force is 15 kN
[Delage97]. Typical dimensions for the sample used in this setup are a
width of 100 mm and a length of 400 mm. When considering a steel
sheet of 0.35 mm thickness, the mechanical system is capable of
externally applied tensile stress up to 425 MPa, which generally
corresponds to the elastic limits of fully processed electrical steels.
Higher thickness, however, would lead to lower externally applied
tensile stress. Plastic deformation of steel sheet is possible but unlikely.
An external compression is not possible.
The magnetic part of that 2D measurement setup consists of two
independent axes. Despite the first impression that the magnetic system
in Fig. 3.20 is similar to the SST in Fig. 3.2, it is not, actually. There are
two yokes in the considered system, a longitudinal and a transversal.
Each yoke in Fig. 3.20 is equipped with an excitation winding,
responsible for the magnetic excitation in the X or the Y directions. There
is no symmetry present in the system in Fig. 3.20 compared to the
standard SST in Fig. 3.2. The absence of a symmetrical yoke above the
longitudinal one or below the transversal one leads to a drastic error, if
using the same method as in the SST. Assuming the field being
proportional to the current in the excitation winding, the single yoke
system can give a systematic error up to 20% due to additional dynamic
losses in the sample [Pfutzner91]. A miniature air gap between the steel
sheet and a yoke results in drastic changes of the measured permeability
as in Fig. 3.15, causing the method to be unsuitable [Nakata86].
Therefore, other techniques than the SST method must be used.
84
Chapter 3.
Both B and H magnetic measurements can be performed locally
in the geometrical center of the system, as in Fig. 3.20. It seems that two
reliable measurement techniques could give very accurate results.
However, two questions remain: how to measure and what to measure.
The question how is mainly about the accuracy of local magnetic
measurements. However, the main question is what to measure, i.e. a
question of uniformity of the field in the steel sheet.
According to the analysis of the magnetic field distribution
[Delage97], the magnetic field is uniform only in a central area in Fig.
3.20. Moreover, the air gap between the steel sheet and the yokes has a
major effect on an uniform area. At constant exciting level, an increase of
the air gap extends the size of the area with the uniform field. With
increasing exciting level, the size of the area with the uniform field
gradually decreases. Despite the nonlinearity of the sample, a 3 mm
optimal air gap leads to a 4 cm2 central area, where the magnetic field is
uniform. A set of two crossed H coils is used close to the sample in order
to determine the magnetic field components, one for each direction.
The H-coil measurements are useful when the H-coil and B-coil
are covering exactly the same area with the uniform field [Pfutzner91].
Normally, in order to measure B, it is possible to drill small holes in a
sample, wound the B-coil locally through the holes and apply the
principle of eq. (3.3). However, for an externally applied mechanical
load, it is impossible to make holes in the sheet without introducing a
disturbing stress field. Therefore, nondestructive B sensors are required
to measure the components of the flux density. Thus, two pairs of special
needles are applied perpendicularly to the sample, see Fig. 3.21.
Fig. 3.21. Magnetic sensors: H coils and B needles [Delage97].
Magnetic measurements under stress
85
Fig. 3.22. Needle probe methods with 2 needles at points 1 and 2.
Needle probes have proved to be very effective to measure the magnetic
flux density B. In fact, the needle probe method was first proposed by
Werner in 1949. Since then the method was critically assessed a few
times by Japanese and European scientists. The needle probe method
makes it possible to measure the local flux, as shown in Fig. 3.22.
Generally, the needles can be used as shown in Fig. 3.22.
Faraday's law (1.1) for the loop 1-2-3-4-1 in the XY plane gives
dB
⋅ 1Z dS
dt
S/2
∫ E ⋅ dl = ∫
1− 2 −3− 4 −1
(3.8)
In general case of the path 1-2-3-4-1, eq. (3.8) is given by
dB
⋅ 1Z dS
dt
S/2
∫ E ⋅ dl + ∫ E ⋅ dl + ∫ E ⋅ dl + ∫ E ⋅ dl = ∫
12
23
34
41
(3.9)
Here, S/2 is the cross section 1-2-3-4. If the induced eddy currents are
parallel to the surface of the sheet in the area 1-2-3-4, then the path 3-4
gives zero as well as the two paths 2-3 and 4-1. Indeed, the E23 ┴ dl and
E41 ┴ dl which gives zero when integrating (3.9).
Thus, the path 1-2 is the only path remains non-zero in the left
part of eq. (3.9). If the needle probe method is used at a distance from the
edge, more or equal to three times the thickness of the steel sheet then
the assumption above is acceptable [Loisos01], and consequently, the
two-needle probe method can be easily used for the measurements.
The induced Vsensor in the needle probe circuit is equal to
Vsensor =
dB
dH
⋅ 1Z dS + ∫ µ 0
⋅ 1Z dS Air
dt
dt
S/2
S Air
∫
(3.10)
Here, the second term of the induced Vsensor in eq. (3.11) corresponds with
the loop formed by the surface of the specimen and the lead wires
86
Chapter 3.
connecting probes to the measuring instruments. This error can be large
in case of large distance between the sheet surface and the closing wire.
Therefore, to ensure a high quality measurements, the wire
between the needle probes must be placed as close as possible to the
surface of the studied sample. In addition, the technique to reduce the
second term is the twisting of the wires to reduce the area SAir.
It is essential to use the needle probes in the middle of the sample
to avoid arising errors due to using two needles at the edge [Loisos01]. If
the magnetic measurements are performed at the edge of the steel sheet,
then the general 4 needle probe method should be used to reduce a
possible error [Loisos01], [DeWulf02], [Pulnikov03].
An experimental study of using different distances between
needles in different electrical steels shows that for the measurements
over distances greater than 10 mm two needle probes technique can give
fairly accurate results [Loisos01].
At distances greater than 25 mm the two needle probe method
can give accurate results both for non-oriented and grain-oriented steels,
without any damage due to drilling holes for a search coils in the
sample. The latter is particularly important in case of applying tensile or
compressive stress to the sample of electrical steel.
3.4. 2D magnetic measurement system under stress.
Based on the presented overview of different magnetic measurement
systems, the following conclusions can be drawn.
The target of a novel measurement setup is to carry out various
2D magnetic measurements in a steel sample under externally applied
mechanical stress.
Because of the interest in both compressive and tensile stresses, a
measurement setup should be capable of creating either compressive or
tensile mechanical load.
An extra opportunity to apply a mechanical load above the
elastic limit is preferable in order to study the effect of plastic
deformation, or a plastic strain, on the 2D magnetic properties of various
electrical steels.
The 2D magnetic measurement system should resemble the
proved design of the RSST. The horizontal flat design of the symmetrical
RSST is preferable. It allows a waveform control of various 2D
conditions of alternating and rotational magnetic fluxes.
Magnetic measurements under stress
87
The measurement technique can be chosen freely from standard
methods, due to the absence of a standard on 2D magnetic
measurements.
The accuracy of every single measurement technique is crucial to
ensure the overall accuracy of the novel measurement system.
Calibration of each axis to the standard results is advisory but
not an obligation due to the different magnetic circuits and
measurement methods applied.
A local magnetic measurement technique by means of B needles
and H coils is preferable due to its high accuracy and sound physical
ground. If proper designed and then placed in the same area with
uniform conditions, they can provide a high degree of accuracy and
repeatability.
In case of B needles, the non-destructive method of two needle
probes allows avoiding drilling holes in the samples.
A shielding applied to achieve uniform magnetic flux
distribution is considered the best technique to use the H coils for the X
and the Y axis. A double H coil technique is unnecessary in that case.
The area of magnetic measurements and the sample dimensions
should be chosen wisely.
One of the features of electrical steels is the grain size, see
Chapter 1. In order to obtain macroscopic properties of electrical steel
under applied stress, the area of magnetic measurements should be large
enough to measure statistical data based on a large number of grains.
Furthermore, taking into account the minimal distances to use B
and H probes, described in Section 3.3, the measurement area covered by
the probes should be at least a square with a width of 25 mm, situated in
the center of the sample.
The dimensions of the sample can be chosen as usual to the range
of the RSST, e.g. an 80 mm square sample. A further 2D or 3D analysis of
the magnetic flux distribution is required to ensure that the central area
has uniform magnetic conditions.
The main question remains: how to combine all of the above into a
single measurement system?
Focusing on stress dependence of the magnetic properties, the
choice of the mechanical system is the main question.
The mechanical setup should be able to apply a tensile or a
compressive mechanical load to a square sample of 80 mm size up to 1
mm thickness.
88
Chapter 3.
A vertical mechanical system is unable to create compressive
stress. Furthermore, it requires an extensive weight capacity for the
building.
Instead, a horizontal mechanical system can be used, comprising
enough strength, a possibility to create either compressive or tensile load
as well as a high accuracy of the design, which should ensure that the
mechanical load is applied in a single direction only.
One of the drawbacks of the horizontal mechanical system is a
slight instability of the mechanical load due to a horizontal mechanical
load. The shaft of such a system has some degree of an overall elastic
capacity. In other words, once the load is applied by the manual winch,
the load does not remain constant but slightly decreases in time.
Another drawback is the need to hold the sample and to pull or
to compress it. That requires nonmagnetic grips with abrasive layers to
hold the sample. Thus, the shaft should also be a part of the magnetic
system, i.e. the part of the magnetic core of the RSST.
The following solution was created in EELAB to comply with the
above choices.
A horizontal mechanical subsystem consists of a solid frame to
carry a load, a manually driven shaft, and a combined system of grips,
parts of the magnetic core, placed on brass rolls, see Fig. 3.23. In fact,
only sample and core parts are magnetic. At one side of the shaft a
transducer is installed to measure the applied mechanical load.
The magnetic subsystem is horizontal but asymmetrical in
comparison with other RSSTs. The non-symmetry comes from the
mechanical subsystem, having the core parts as a part of the shaft, see
Fig. 3.23. Those parts are movable, and furthermore, they create a
magnetic flux in the longitudinal axis of the setup.
The rest of the RSST consists of two similar parts, having an Eshape, to provide the magnetic flux in the transverse axis by means of
the middle leg of the E-shape design, as shown in Fig. 3.24.
Magnetic measurements under stress
89
manual winch
grips
abrasive
samples
core parts
Fig. 3.23. Horizontal mechanical subsystem to apply uniaxial load to the
samples of steels up to 2 mm thick.
Fig. 3.24. Horizontal non-symmetrical magnetic subsystem, an analogue of flat
RSST with an extension to apply mechanical load.
90
Chapter 3.
3.4.1. The mechanical subsystem of the 2D setup.
The scheme of the mechanical subsystem is shown in Fig. 3.23 and Fig.
3.25. Its frame is 1.2 m long. The sample is held by means of brass grips
in order to apply uniaxial mechanical load. To increase the traction force
of the grips, special abrasive pieces have been used to hold the sample
under very high loads. The chosen design has been proved to be very
effective for a wide range of materials up to very high uniaxial loads.
Due to the high strength of the grips required to hold the sample,
an extra space is used to place the bolts, which contract two grips
together, as shown in Fig. 3.26. Therefore, a sufficiently large space of
the sample is used for holding the sample by means of grips, abrasive
and bolts under compression up to 40 kN. Moreover, a large air gap is
used between the sample and the core parts of the magnetic subsystem,
which are simultaneously the parts of the mechanical subsystem. The
following dimensions are used:
- 10 mm on each side is the air gap due to bolts,
- 20 mm on each side is the grips for clamping a sample,
- at least 40 mm is enough to use the B and H sensors.
Taking these dimensions into account, a minimal length of the sample
can be 100 mm.
Fig. 3.25. The mechanical subsystem: a) top view, b) front views.
Magnetic measurements under stress
91
Fig. 3.26. Holding a sample in a non-symmetrical magnetic subsystem.
The mechanical subsystem allows to create either a compression or a
tension. In order to achieve a compression, the left part of the
mechanical subsystem in Fig. 3.25 is fixed to the frame from the inside,
as shown in Fig. 3.27. The part with the winch moves towards the center
of the setup by means of non-fixed aluminum bars, as shown in Fig. 3.28.
The value of the applied compression is limited by the possible buckling
of the sample of the laminated steel. Thus, the limit of the compressive
stress is up to 100 MPa for a sample of 1 mm thickness.
Tension can be applied by means of fixing the part of Fig. 3.27
from outside the frame, while the part of Fig. 3.28 is movable outwards
the center of the setup by means of the manual winch. The tensile stress
can be applied manually in steps up to the elastic limit, and even further
up to destruction. In our experiments the sample is always destructed in
the middle between the grips, as shown in Fig. 3.29.
transducer
to fix from
the outside
to fix from
the inside
Fig. 3.27. The part, which can be fixed either from inside or outside.
92
Chapter 3.
manual winch
removable
aluminum
bars (used for
compression)
Fig. 3.28. The part, which is movable either towards or outwards the center.
Fig. 3.29. Plastically destroyed samples of different electrical steels.
3.4.2. The magnetic subsystem of the 2D setup.
To create a magnetic core of the flat horizontal RSST type taking into
account the non-symmetry, a three phase transformer core was used
with suitable E-shape laminations, having a width of 80 mm. The same
80 mm strips of the same steel were used to create two movable parts,
which are the components of the mechanical subsystem.
Four excitation windings provide the excitation in the
longitudinal direction, parallel to the direction of externally applied
mechanical load, according to the scheme of Fig. 3.24. Only two
excitation windings provide the excitation in the transverse direction,
perpendicular to the direction of the applied mechanical load, as shown
in Fig. 3.24 and 3.30.
Magnetic measurements under stress
93
Fig. 3.30. The magnetic subsystem with two switchboards of excitation
windings, having each 5 sections for interconnections.
A significant question for all RSSTs is the overall value of the air gap,
and consequently, the field distribution in the sample. Due to the nonsymmetry of the setup, and consequently, different field distributions
along the two excitation axes, the air gap and the dimensions of the
sample are the parameters to choose in order to achieve an uniform field
distribution in the center of the sample, where the B and H probes are
placed. Basically, the choice of the dimensions is one of the crucial points
to ensure that the magnetic measurements are accurate and repeatable.
The air gap in the longitudinal direction of the setup is equal to
20 mm as stated above. Its value is constant, however the length of the
sample may vary from 100 to 200 mm. In fact, the variation of the
sample length would result in the change of the magnetic field
distribution. The air gap in the transverse direction is equal to 10 mm,
i.e. 5 mm on each side of the sample. However, the width of the sample
can be reduced to enlarge the air gap in the transverse direction if
necessary. The width of the sample equals 80 mm or less, whereas the
length of the sample equals 100 mm or more.
Thus, the working dimensions of the sample were chosen as 80
mm x 120 mm. These dimensions allow the largest width of the sample
and the minimal length, taking into account the parts of the sample
under grips and the required space of the sensors.
To define the area for the magnetic measurements, many
calculations have been done. The field distribution was first analyzed in
2D: in the XY plane and in the XZ plane. An extra 2D analysis was
94
Chapter 3.
performed for the application of shielding, which is used to reduce the
stray fluxes, as was shown before in Fig. 3.14 and 3.15. Some numerical
results of the 2D model of the setup considering the XY and XZ planes
are shown in Fig. 3.31 – 3.32.
a)
b)
Fig. 3.31. XY models of the magnetic flux in different directions: a) in the X
axis direction, b) in the 60 degree direction.
Fig. 3.32. XZ model of the magnetic flux in the air with a single sample.
Magnetic measurements under stress
95
The doubt about the presence of a proper magnetic flux
distribution arises from the XZ model. An extra study was performed to
compare the magnetic flux distribution in the case that a single sample is
used with the case of two samples, as shown in Fig. 3.33. The shielding
should lead to an uniform magnetic flux distribution between two
samples, as shown in Section 3.2.2. An extra study of magnetic flux
patterns in the XZ plane has been performed in the central area of the
sample, where the H coils will measure the magnetic field, as shown in
Fig. 3.34 and Fig. 3.35.
When the single H coil is placed on top of a single sample, the
stray fluxes are measured by the H coil, as shown in Fig. 3.34. This leads
to the above described error. When the single H coil is placed between
two samples, the stray fluxes are shielded by the samples. The single H
coil measures the horizontal flux only, as shown in Fig. 3.35.
Fig. 3.33. Principle of application of shielding for accurate H measurements.
Fig. 3.34. XZ model of the flux in the H coil, placed on top of a single sample.
96
Chapter 3.
Fig. 3.35. XZ model of the flux in the H coil, placed between two samples.
Fig. 3.36. The flux density in ¼ of sample for the Y excitation [Pulnikov03].
Fig. 3.37. The flux density in ¼ of sample for the X excitation [Pulnikov03].
Magnetic measurements under stress
97
The flux density distribution under the excitation in the longitudinal
direction Y is uniform in almost the whole sample area, see Fig. 3.36.
However, in the transverse direction X the picture is completely
different. The flux density distribution under the excitation in the X
direction is highly non-uniform in the majority of the sample. Only a
central area can be considered as having uniform flux density
distribution, see Fig. 3.37. Thus, the magnetic measurements uniaxial
with the applied stress can be performed with a high degree of accuracy
considering the uniform area of the flux density in the sample.
As for the second axis, perpendicular to the applied stress, some
extra means are required to ensure uniform magnetic flux distribution in
the central area of the sample. This can be achieved, for example, by
adding removable magnetic pole shoes installed in the X direction.
Basically, the application of the pole shoes reduces the non-symmetry of
the magnetic subsystem and improves the flux density distribution as
shown in Fig. 3.37.
Taking into account the 2D analysis of the magnetic field
distribution with the application of the shielding, the following
conclusions can be drawn.
To achieve accurate magnetic measurements, it is recommended
to use two samples and to place the H coils between the two samples in
the geometrical center of the sample, see Fig. 3.31. When two samples
are used, both of them should be placed under the same externally
applied mechanical load to provide compressive or tensile stress
simultaneously to both samples. Similar mechanical conditions applied
to both samples ensure accurate magnetic measurements in the center by
the H coils.
A single sample is possible to use under uniaxial stress and
magnetization in the longitudinal direction only, because of a large
uniform area and a large air gap.
To avoid drilling holes, a non-destructive method of two needle
probes can be used to measure the flux density in the sample. For a
shielding solution, it is impossible to use B coils or B needles between
two samples. Therefore, the only way to measure B in the two sample
solution is to place B needles on top of the upper sample in the center of
the samples. Moreover, due to the movable shaft, the samples should
always be placed in the geometrical center of the magnetic subsystem.
Based on the above recommendations, the magnetomechanical
setup was created and tested in EELAB, as shown in Fig. 3.38.
98
Chapter 3.
The setup was first presented orally at the 15th Soft Magnetic
Materials Conference in Bilbao, Spain in 2001 by V. Permiakov.
Fig. 3.38. Novel 2D magnetic measurement setup with 1D mechanical stress.
3.4.3. Magnetic measurement technique.
The choice of the magnetic measurement technique has already been
discussed in the preceding sections. However, there are a few aspects to
be discussed separately.
When it comes to the accuracy of the measurements, the design
Magnetic measurements under stress
99
and calibration of the sensors are crucial. In the current study, a pair of
H coils was constructed manually. According to recommendations in
[Pfutzner91], the thickness of the epoxy layer in the middle of the H coil
is taken equal to 1 mm. The number of turns for each H coil is about 1200
turns. A 0.1 mm thick isolated copper wire was used to wind the H coils.
Thus, the dimensions of the H coils are: inside thickness of 1 mm,
outside thickness of 2 mm, width of 10 mm, and length of 15 mm. Each
H coil was calibrated by means of the Helmholtz coils shown in Fig. 3.3.
As a result of the calibration, accurate values of the cross sections NS
were obtained.
The design of the B needle probes is also important.
Commercially available needles GKS079, produced by INGUN and
shown in Fig. 3.39, were applied. The needle is a special multi-alloy
sensor with golden plating, having a spring force of 1.3 N and a stroke of
1.2 mm. This construction ensures an excellent electrical contact, even if
there is a strong coating on a steel sheet. A pair of needles is sufficient to
measure B in one direction accurately, if the distance between the
needles is sufficiently large and the needles are not too close to the edge
of the sheet, see Section 3.3.
Fig. 3.39. The holder (left) and the needle (right) geometry. Source: Ingun.com.
X and Y
pairs of
B needles
Fig. 3.40. The 4 needles in the holder to measure B in X and Y directions.
100
Chapter 3.
The distance between the needles, placed in the holder shown in
Fig. 3.40, is 42 and 30 mm for the X and the Y directions, respectively.
According to [Loisos01], these distances are sufficiently large to
measure the magnetic flux density accurately both in grain-oriented and
in non-oriented electrical steels. Thus, the total set of measurement
instruments consists of two H coils and two pairs of B needles. The
central area of 40 mm x 40 mm can be assumed to be sufficiently
uniform to perform accurate local magnetic measurements.
The signals from the B and H probes are amplified by signal
amplifiers as shown in Fig. 3.41. The accuracy of the signal amplifiers
affects the total accuracy of the magnetic measurements [Pfutzner91].
To reduce a possible error, an already proven design of signal
amplifiers [Makaveev03], equipped with operational amplifiers at
variable gain (1 to 10000), are used.
The excitation windings shown in Fig. 3.38 are supplied by
bipolar power amplifiers. The excitation voltage is controlled by the
software developed in EELAB. A stack of KEPCO bipolar power
amplifiers, each having 400 W power, is shown in Fig. 3.42. As many as
four KEPCO amplifiers have been used in the present study, connected
in series to increase the total gain factor from 3.6 to as much as 10.
DAC
contacts
B and H
signal
amplifiers
in X and Y
directions
Fig. 3.41. The 4 signal amplifiers and the board of data acquisition card (DAC).
Magnetic measurements under stress
101
Fig. 3.42. A stack of KEPCO power amplifiers for the excitation windings.
3.4.4. Waveform control.
As was mentioned in Section 3.1, it is crucial to ensure a sinusoidal
waveform of the magnetic flux density B in order to permit comparison
of different measurements.
Generally, there are two target approaches for the waveform
control: to control B or to control H. Consequently, the results, i.e. the
energy loss and the permeability, will depend on that choice. When a
closed magnetic circuit is used, for example, in a SST, the current
method is applied to obtain the magnetic field H. Of course, in that case
it is logical to operate with H. Although H is controlled in many SSTs,
the main criterion for the control remains a sinusoidal B, as was
discussed before.
When an air gap is present in the magnetic circuit, or when B and
H are measured locally by the test probes, then it is better to control the
magnetic flux density B in the sample. Indeed, by controlling B almost
any kind of flux density waveform can be created. The two axes of the
102
Chapter 3.
measurement setup are considered independent and so is the waveform
control. The ability to create any kind of magnetic flux density in each
axis of the sample is an advantage as virtually any mode of alternating
or rotational magnetization can be studied.
Various techniques for flux density waveform control have been
used by different researchers. To control B in the present study, the
technique of [Makaveev01] was applied. It works very well in the
measurement system having an air gap. High induction is difficult to
achieve due to high power requirements.
The control is based on an iteration procedure. The output
voltage to the excitation windings are created from the signals of the B
and H probes. The control modifies the next output voltage to achieve B
sinusoidal. A new voltage is generally calculated according to eq. (3.7).
Parameters of the equation (3.7) depend on the following:
- resistance and inductance of the excitation winding,
- number of turns and connections of the excitation windings,
- the overall air gap, or the equivalent air gap [Makaveev01],
- the number of the highest harmonic used in the algorithm, etc.
As a result of the waveform control, the magnetic flux density becomes
sinusoidal with an error as low as 1%. At the same time, the output
voltage V to the excitation windings changes gradually from the primary
sinusoidal to the distorted one as shown in Fig. 3.43.
0.2
Before a
waveform
control
After a
waveform
control
0.15
0.1
Voltage, V
0.05
0
0
50
100
150
200
-0.05
-0.1
-0.15
-0.2
time, msec
Fig. 3.43. Output voltage V to the excitation windings before and after control
in order to obtain a sinusoidal B, measured locally by the B needles.
Magnetic measurements under stress
103
3.5. Magnetic measurement procedure. Operational software.
Except the application of the mechanical load, which is done manually,
everything else is controlled by means of the operational software,
developed for this study. The software consists of several modules, each
responsible for one specific operation.
The following main modules are used in the software:
- a Global Setup Database module that includes all parameters for
measurements (Fig. 3.44),
- a user defined module, i.e. a choice of the automatic range of
measurements,
- an initialization module, an excitation algorithm based on the
primary data,
- an output generation module, the control of the two output
channels of the DAC,
- a data acquisition module, the main control of the measurements
(Fig. 3.45),
- an update module for the waveform control iterations (Fig. 3.46).
The Global Setup Database allows all data to be collected at one place,
and to be achieved easily from any module of the developed software.
The main clusters of data collected in the Global Setup Database
shown in Fig. 3.44 are:
- setup files, including data about samples, sensors, acquisition,
amplifiers,
- sensors, including data about NS values of B and H probes for X
and Y axes,
- signal and power amplifiers, including gains of B and H for X
and Y axes,
- acquisition, including input and output channels, scan/update
rates, buffer size,
- control, including signal type, frequency, amplitude, phase, axis
ratio, etc.
Other modules in the package are responsible for partial operations:
- a module for DC offset compensation that eliminates the DC
offset in the signals,
- a demagnetization module that demagnetize a sample with a
predefined waveform,
- a storage module that is responsible for storage of all data
obtained during the measurements.
104
Chapter 3.
Fig. 3.44. Global Setup Database of the developed software package.
Fig. 3.45. Modules of the data acquisition and experimental results.
Magnetic measurements under stress
105
Fig. 3.48. Update module, responsible for iterations of the waveform control.
3.6. Sample preparation.
It is crucial to prepare the samples for the study in a proper way. The
steel sheet has to be clean and flat, without any damages of the sheet
surface. Any damage or loss of quality of the sheet can affect the results
of the magnetic measurements.
The typical sample is cut along the rolling or any given arbitrary
direction, having 120 mm length and 80 mm width. When two samples
are used simultaneously, these should be cut from the same sheet with
the same direction along the longest size of the steel sample.
3.7. Conclusions.
Chapter 1 has introduced to the reader the subject of this study. Chapter
2 has described 2D magnetic and mechanical working conditions of
various electrical steels used in different electrical machines, both from
theoretical and experimental point of view.
This chapter has described various measurement methods,
applied worldwide for characterization of electrical steels. The
measurement setup along with a suitable measurement technique has
been chosen to meet the objectives of this study. The setup was specially
designed, constructed, tested and upgraded in EELAB.
106
Chapter 3.
The following features for the magnetomechanical measurements can be
used:
- manually applied uniaxial compressive load,
- manually controlled uniaxial tensile load up to elastic strength,
- various tensile plastic deformations up to destruction,
- alternating magnetization in arbitrary directions in the plane of
the sample,
- rotational magnetization up to circular rotation of the magnetic
flux density vector,
- sinusoidal (or distorted) controlled magnetic flux density,
- high accuracy due to the chosen measurement technique,
- fully automatic operations of the magnetic measurements.
The combination of the above features is quite unique. It allows to create
novel magnetomechanical conditions. The following chapters present
actual 1D and 2D magnetic measurements in various electrical steels
under uniaxial mechanical stress.
1D magnetic measurements under stress
107
An expert is a person who has made all the mistakes
that can be made in a very narrow field.
Niels Bohr (1885 - 1962)
CHAPTER 4. 1D MAGNETIC MEASUREMENTS UNDER STRESS.
The main instrument of the present research is the magnetic
measurement system for 2D magnetomechanical conditions. Hence, the
study is based on experimental observation, followed by a physically
sound explanation. As with any research instrument, the novel 2D
measurement system must first be extensively tested up to its limits.
Having a little experience with magnetic measurements, it is crucial to
gain the measurement skills, i.e. to make "all the mistakes that can be
made in a very narrow field".
To narrow the field of the measurements, a primary investigation
is done for the simplest case of uniaxial magnetization under uniaxial
applied stress. This allows us to make a comparison between our results
and the already known stress dependences of magnetic properties for
electrical steels. A comparison of the experimental results with the
known tendency from literature can give an idea about how the novel
measurement setup is operating under uniaxial conditions.
Thanks to uniform magnetic conditions for uniaxial magnetic
measurements in the longitudinal direction of the setup (see Chapter 3),
a number of uniaxial magnetic measurements under stress can be
carried out for a single sample of electrical steel with a high accuracy.
The externally applied mechanical load can be either compressive or
tensile, applied in the longitudinal axis of the setup, see Fig. 4.1. As
shown in Chapter 2, the uniaxial case is of interest to both grain-oriented
and non-oriented electrical steels. In grain-oriented steels, an uniaxial
compression and tension occurs in the transformer cores due to
assembling. In non-oriented steels, an uniaxial magnetization under
stress occurs in the tooth region of the laminations. Hence, an uniaxial
magnetization under stress is useful to study from both experimental
and theoretical point of view. Different measurement techniques,
described in the previous chapters, are put into practice in the 1D case.
To explain the experimental results, different theoretical approaches are
presented. Based on the study of Chapter 2, the actual magnetic working
conditions are included as well. A number of electrical steels have been
studied under complex uniaxial conditions.
108
Chapter 4.
Fig. 4.1. The magnetomechanical setup to apply uniaxial stresses.
4.1. Uniaxial measurements in electrical steels.
Based on the analysis presented in Chapters 2 and 3, a single sample of
electrical steel is used for uniaxial (1D) magnetomechanical
measurements.
The procedure for the 1D magnetic measurements includes all
the limits of the components of the measurement system, developed in
EELAB for 1D and 2D magnetic measurements under mechanical stress,
including hardware and software modules. According to Fig. 3.17, the
detailed procedure within the limits is as follows:
- a digital generation of an output voltage V to supply the
excitation windings,
- a data-acquisition of the voltage V in the range of ±10 V,
- an amplification of the voltage by a gain factor 3.6 or 5,
- a measurement of B and H signals by local probes,
- an analog amplification of the signals by a factor 1 to 1065,
- a data-acquisition of the signals in the range of ±10 V,
- a digital integration of the signals to obtain B and H waveforms,
- a construction of the BH loop, a calculation of energy loss.
In order to achieve the sinusoidal (or distorted) flux density in a sample,
the B waveform is compared to the desired sinusoidal (or distorted)
waveform. The primary mechanism is to update the amplitude of the
voltage V to increase or decrease the amplitude of B. This mechanism
has proved to be very fast and effective up to high inductions, but the
1D magnetic measurements under stress
109
form correction is also required. Hence, the second mechanism is a form
update. This was described in section 3.4.4.
An "elementary cycle" of strictly controlled magnetic
measurements consists of the above procedure, followed by a
construction of the update to ensure the desired sinusoidal B.
The elementary cycle is terminated when the required condition
is met, in this case, when the error between the desired and the obtained
B waveform in the sample is less than 1%. The elementary cycle was
developed to be fully automatic, controlled by a user-defined
operational algorithm.
When the externally applied stress is considered, the elementary
cycle has to be repeated each time a new stress is applied as follows:
- a new mechanical load is applied manually, shown by a digital
indicator,
- the parameters for the magnetic measurements are defined,
- the elementary cycle is applied automatically, controlled by the
software,
- the experimental results are saved automatically into a specific
data file,
- the demagnetization module of the software is applied after each
regime of magnetic measurements.
The following range of working parameters for the uniaxial magnetic
measurements can be obtained:
- the amplitude of the induction B is between 0.2 and 1.5 T,
- the frequency of magnetization ranges from 2 to 200 Hz.
If the magnetic measurements are performed under a range of
inductions or a range of frequencies, the operational algorithm for
magnetic measurements is the one as shown in Fig. 4.2.
Fig. 4.2. The algorithm of uniaxial magnetic measurements under stress.
110
Chapter 4.
The typical operations for the magnetic measurements consist of the
following steps:
- making a choice of the electrical steel and the preparation of the
sample,
- making a choice of the parameters for measurements according
to Fig. 4.2,
- conducting magnetic measurements under stress for the chosen
parameters,
- analyzing the experimental results, such as BH loops and energy
losses.
When the experimental results are analyzed, the next step is a
comparison with the knowledge, obtained by other measurements
setups which are limited either to small stresses, or to uniaxial
magnetization under tension only, or to magnetization at plastic
deformation only. Eventually, the present study should either confirm
the previous results, or bring a certain novelty into the subject of the
target research.
This is the first evaluation of the EELAB measurement setup for
the 1D case, followed by the 2D magnetic measurements under
mechanical stress.
4.1.1. Uniaxial magnetization in grain-oriented steels.
Grain-oriented steels are known for their large grain sizes and the
orientation of an easy axis of the crystals close to the rolling direction of
the steel sheet. In the present study for grain-oriented materials, the
mechanical stress is applied in the rolling direction.
A 0.30 mm thick grain-oriented steel sample has been cut along
the rolling direction, having a length of 120 mm and a width of 80 mm.
The measurement procedure described above has been applied for a
controlled sinusoidal B, within a range of peak inductions of 0.25, 0.5,
0.75, 1.0, 1.25 T at the power frequency. A higher induction level was not
possible to achieve due to the limitations of the excitation power because
of the large air gaps of the measurement setup.
As a result of the magnetic measurements, BH loops are obtained
for every peak induction level. Fig. 4.3 depicts the experimental BH
loops at peak induction of 0.5 T under a range of applied stresses from
compression of 10 MPa to elastic tension of 100 MPa. The BH loops are
the basic data about the magnetic properties of steel. Many other
1D magnetic measurements under stress
111
magnetic properties can be obtained from these loops. For example, the
energy loss is equal to the area of the BH loop as in (1.7) [Sievert84]. Fig.
4.4 depicts the experimental energy losses under stresses.
Fig. 4.3. The BH loops under a range of uniaxial stresses at 50 Hz.
Fig. 4.4. The energy losses under a range of uniaxial stresses at 0.25 to 1.25 T.
112
Chapter 4.
The application of a tensile stress in the rolling direction of a grainoriented steel leads to a considerable improvement of the magnetic
properties in this direction.
Indeed, the area of BH loops under tension is reduced in
comparison with the stress-free condition. In fact, a further increase of
the tensile stress up to 100 MPa results in a very little change of the BH
loops, see Fig. 4.3. On the contrary, a compressive stress of 10 MPa leads
to a drastic deterioration of the magnetic properties of grain-oriented
steel. The area of the BH loop at compression of 10 MPa is much larger
than the area at the stress-free condition. A similar tendency was
observed for other peak induction levels up to 1.25 T.
In fact, the improvement of energy losses under uniaxial stress in
the rolling direction of grain-oriented steels has been well-documented
before. Fig. 4.5 depicts the experimental data on the effect of applied
tensile stress on energy loss at 1.5 T at 50 Hz in different grain-oriented
steels having different average grain diameter [Shilling74].
As shown in Fig. 4.5, the largest loss reduction with an increasing
tensile stress occurs in a single crystal. A conventional grain-oriented
electrical steel with an average grain size of around 10 mm exhibits a
weaker improvement under tensile stress.
Below a certain optimum grain size (about 7 mm, according to
[Shilling74]), the energy loss increases with decreasing grain size after a
certain tensile stress, as shown in Fig. 4.5. The cause for the increase of
energy loss under large tensile stress is an increasing amount of domain
wall pinning due to a larger magnetostatic energy at the grain
boundaries [Shilling74].
Thus, in grain-oriented steels the tensile stress applied in the
rolling direction can reduce the energy loss. The effect of tensile stress
has been successfully exploited by the use of stress inducing coatings for
grain-oriented steels. The effect of different surface coatings on domain
structure and energy loss was studied e.g. in [Fukuda81].
1D magnetic measurements under stress
113
Fig. 4.5. The effect of applied tensile stress on energy loss [Shilling74].
Fig. 4.6. The effect of compressive (negative) and tensile (positive) stress
on energy loss in various grain-oriented steels [Moses80].
114
Chapter 4.
In another experimental study [Moses80], several grades of grainoriented steels have been tested at 1.5 T and 50 Hz. Fig. 4.6 depicts an
experimental comparison between the mean measured losses in several
steel grades. In fact, the right part of Fig. 4.6, i.e. the loss at tensile stress,
has a good correspondence with Fig. 4.5 by [Shilling74] and the
experimental results of the present study, as shown in Fig. 4.4. The left
part of Fig. 4.6, i.e. the energy loss at compressive stress, has a very good
correspondence with the experimental results of the present study at 10
MPa, as shown in Fig. 4.4.
Unfortunately, it was difficult to obtain high compressive stress
in the present setup due to buckling of the very thin sample of the
considered grain-oriented steel. It is worth mentioning here that a
compressive stress higher than 10 MPa is not likely to happen in actual
devices such as power transformers, as shown in Chapter 2. A similar
sharp increase of the energy loss under a small compressive stress up to
10 MPa can be observed in the past experimental results [Moses80].
4.1.2. Uniaxial magnetization in non-oriented steels.
Due to the orientation of the grains, grain-oriented steel is presented
first. As a matter of fact, the present research of 1D magnetization under
uniaxial stress had started from the non-oriented steels, and this was for
at least two reasons. Non-oriented electrical steels are the most common
magnetic materials, so is the interest in the properties. Furthermore, to
gain the skills in special magnetic measurements, it is much easier to
deal with a "simple" high-loss material such as non-oriented steel than
with a "fine" low-loss grain-oriented steel. Hence, the trial measurements
were performed on non-oriented electrical steels.
The thickness of non-oriented electrical steels ranges from 0.35
mm up to 1.0 mm, which results in much larger eddy current loss in the
laminations. Moreover, due to the higher thickness, magnetic
measurements are feasible at larger compressive stress without bending
of a steel sample, as occurs in grain-oriented steels.
As shown in Chapter 2, there can be very large compressive
stresses (above 50 MPa) in the stator and rotor cores of rotating
machines. Furthermore, the difference in magnetic properties between
arbitrary directions of non-oriented steels is much smaller than in the
case of grain-oriented steels. The mechanical stress can be applied in any
arbitrary direction of the steel sheet with a little difference in
1D magnetic measurements under stress
115
magnetomechanical properties between directions. By default, the
mechanical load is applied in the rolling direction (RD) of the steel
samples of the considered non-oriented electrical steels.
Samples of several fully-processed non-oriented electrical steels
with thickness of 0.50 mm and 0.65 mm have been used during the
present research. The above measurement procedure has been applied
for a controlled sinusoidal B, having a peak induction between 0.2 and
1.4 T. Only a selection of the experimental results is presented here.
One of the parameters of the operational control shown in Fig.
4.2 is stress. Three areas of applied mechanical stress are considered:
small elastic stresses including compression and tension, all elastic
stresses up to the elastic limit, and plastic strains. The typical range of
small compressive and tensile stresses is ±50 MPa. This is also the range
of stresses considered in most electrical steel research. A further increase
of tensile stress can be achieved up to the elastic limit, followed by
plastic deformation up to the destruction of the sample.
As a result of a series of magnetic measurements on 0.65 mm
steel, BH loops can be drawn for every considered peak induction.
Fig.4.7 depicts the experimental BH loops at peak induction of 1.0 T
under applied stresses ranging from compression under 60 MPa to
tension up to 150 MPa. Fig. 4.8 depicts the experimental BH loops at
peak induction of 1.4 T under a similar range of applied elastic stresses.
The application of tensile stress in the rolling direction of the
non-oriented steel leads to some improvement of the magnetic
properties in this direction. Indeed, the area enclosed by the BH loops
under small tension is reduced in comparison with the stress-free
condition. On the contrary, the application of compressive stresses leads
to a drastic deterioration of the magnetic properties of the considered
non-oriented steel. The area of the BH loop at compression of 60 MPa is
much larger than the area under the stress-free condition. A similar
tendency was observed for peak inductions up to 1.4 T.
Fig. 4.9 depicts the experimental energy losses under a range of
compressive (20, 40, 60 MPa) and tensile (30, 70, 100, 150 MPa) stresses at
several peak inductions up to 1.4 T.
Fig. 4.10 depicts the relative permeability changes under the
same range of applied stresses at peak inductions of 1.0 T and 1.4 T. The
relative permeability was determined in correspondence with the
maximum flux density for every BH loop.
116
Chapter 4.
Fig. 4.7. The BH loops under compression and tension at 1.0 T and 50 Hz.
Fig. 4.8. The BH loops under compression and tension at 1.4 T and 50 Hz.
1D magnetic measurements under stress
117
Fig. 4.9. The energy losses under a range of uniaxial stresses at 0.7 to 1.4 T.
Fig. 4.10. The relative permeability under a range of stresses at 1.0 and 1.4 T.
118
Chapter 4.
Considering magnetic properties versus stress, it is possible to
establish the value of a tensile stress corresponding to the minimum
energy loss (see Fig. 4.9) or to the maximum permeability (see Fig. 4.10).
This stress is called here a "critical stress", where the applied uniaxial
tension results in the best magnetic properties in the same direction. The
range where the critical stress is located can be observed at 20 to 60 MPa,
depending on the material. The critical stress phenomenon was found in
various grain-oriented and non-oriented steels in the present study.
In fact, the improvement of energy losses under uniaxial stress in
non-oriented steels has been experimentally studied very recently. Fig.
4.11 depicts the energy loss per cycle W under a range of uniaxial
stresses σ from compression of 50 MPa to tension of 75 MPa [LoBue00].
Here, the open symbols correspond to non-oriented steel with an
average grain size of 50 µm and the closed symbols correspond to nonoriented steel with an average grain size of 270 µm, see Fig. 4.11. Both
kinds of materials show a drastic increase of the energy loss under
compression and a moderate change under tension. The critical stress
can be observed at about 30 MPa for the material with a small grain size.
The non-oriented steel with larger grains shows a lower sensitivity to the
applied tension but a higher sensitivity to the applied uniaxial
compression.
Fig. 4.11. The energy loss per cycle under uniaxial stresses at 0.5, 1.0 and 1.5 T.
Grain size: open symbols - 50µm, closed symbols - 270µm [LoBue00].
1D magnetic measurements under stress
119
4.1.3. Uniaxial magnetization at tensile plastic deformation.
The condition of tensile plastic deformation requires a separate
attention.
According to Chapter 2, the production process for rotating
machines consists of the inevitable procedure of punching or cutting of
the sheet to produce the tooth-and-slot structure of the laminations. A
complex plastic deformation occurs due to cutting.
The simplest way to study the plastic deformation is to apply
uniaxial strain to the sample, using the same measurement setup as for
applying uniaxial compressive and tensile elastic stresses. Indeed, the
setup created in EELAB allows to apply tensile plastic deformation up to
the destruction of the sample, see Chapter 3.
The experimental procedure in case of plastic deformation is
different from the one used for elastic stress. When elastic stress is
studied, a mechanical load is applied and the measurements are
performed according to Fig. 4.2. Then the sample is unloaded to the
stress-free conditions. The fact that the physical and magnetic properties
of the unloaded sample are the same as before the applied mechanical
load ensures that the stress was indeed elastic.
When the tensile plastic deformation starts after the yield stress
σy (see Fig. 4.12), the application of a larger stress σf leads to a plastic
strain εf, depicted by the point A in Fig. 4.12. When the load is released
after point A, the material will not return to the unloaded "zero" strain,
but instead a residual strain εr occurs.
The experimental procedure to study the effect of tensile plastic
deformation on the magnetic properties of non-oriented electrical steels
consists of the following steps, as shown in Fig. 4.12.
Fig. 4.12. Applied experimental procedure 1-2-3-4 at a strain curve [Sablik04].
120
Chapter 4.
Magnetisation, T
Step 1 as shown by the arrow 1 in Fig. 4.12 corresponds to a
tensile strain εf2 , i.e. from the point A (strain εf) to the point B (strain εf2).
At the point B, a series of magnetic measurements, called here "under
stress", is performed according to Fig. 4.2.
Step 2 as shown by the arrow 2 in Fig. 4.12 corresponds to a
residual strain εr2 , i.e. from the point B (a loaded condition, or under
stress σf2) to the point C (an unloaded condition, or after the stress
release). At the point C, a series of magnetic measurements, called here
"after release", is performed according to Fig. 4.2.
Step 3 as shown by the arrow 3 in Fig. 4.12 corresponds to the
area of elastic stresses applied to the plastically deformed material
having a stress-free condition in the point C, i.e. at a plastic strain εr2 .
Step 3 can consists of a number of applied tensile elastic stresses from
the point C to the point B in Fig. 4.12.
Step 4 repeats Step 1, resulting in a new level of tensile plastic
deformation.
Fig. 4.13 depicts the evolution of BH loops under tensile elastic
stresses and tensile plastic deformation. In fact, the BH loops shown in
Fig. 4.13 were the first published experimental results obtained by
means of the novel magnetomechanical setup at EELAB.
The yield stress for this non-oriented steel with a thickness of
0.65 mm is equal to 430 MPa. So, the tensile plastic deformation is
presented here by the two largest BH loops, depicted by the dotted lines.
1.2
1
0.8
0.6
0.4
0.2
0
-250
0
-0.2
250
500
750
1000
1250
1500
1750
Magnetic Field, A/m
-0.4
-0.6
-0.8
No tensile stress, 20, 40, 60, 80, 100, 130, 165, 200, 250, 300, 350, 400 Mpa (-----)
and 450, 500 Mpa ( - - - )
-1
-1.2
Fig. 4.13. The BH loops at tensile stresses up to 400 MPa and two plastic
strains, depicted as two curves at 450 and 500 MPa.
1D magnetic measurements under stress
121
Fig. 4.13 includes another interesting information about the BH
loops behaviour. All BH loops at elastic stresses have two cross points in
the second and the fourth quadrants, very close to the remanence points.
The two largest BH loops (see Fig. 4.13) of plastic deformations are not in
the common cross point of all the others BH loops for tensile elastic
stresses.
Fig. 4.14. The BH loops at two plastic strains under stress and after release.
Fig. 4.15. The energy losses at plastic strains under stress and after release.
122
Chapter 4.
Fig. 4.16. The relative permeability changes at plastic strains.
Fig. 4.14 depicts the BH loops obtained according to the experimental
procedure as shown in Fig. 4.12. Fig. 4.15 depicts the energy loss "under
stress" and "after release" according to the experimental procedure.
The application of tensile plastic deformation leads to a very fast
deterioration of the magnetic properties at plastic strains less than 10%
and a steady deterioration at higher plastic strains up to destruction,
which occurs at 35% for the considered steel. The energy loss under
stress is considerably lower than the loss after release. The material
"after release" can be considered as a newly deformed material. When a
tensile elastic stress is applied "after release" (see arrow 3 in Fig. 4.12),
the magnetic properties can be improved. A physical interpretation of
this observation will be given later on. Fig. 4.16 depicts the change of the
relative permeability at the same plastic strains as in Fig. 4.15.
4.1.4. Uniaxial magnetization in semi-processed and annealed steels.
Semi-processed non-oriented electrical steels are widely used in the USA
and Japan. The idea to use semi-processed steels in rotating machine
laminations is simple. The cutting of steel is performed to a low-quality
material, which is annealed later to achieve the best magnetic properties,
comparable with the ones of the available fully-processed steels.
1D magnetic measurements under stress
123
To study the effect of uniaxial stress on semi-processed steels, a 0.65 mm
thick semi-processed steel was used to produce two kinds of samples: an
ordinary semi-processed sample and the one annealed according to
EN10126. The magnetic measurements under stress were performed and
the stress effect was compared with a 0.65 mm fully-processed steel. Fig.
4.17 depicts the stress-free BH loops for samples of fully-processed (FP),
semi-processed (SP) and annealed semi-processed (SPA).
Fig. 4.17. The stress-free BH loops of FP, SP, and SPA steels.
FP
SP
SPA
Fig. 4.18. The energy losses of FP, SP, and SPA steels under elastic stresses.
124
Chapter 4.
Fig. 4.19. The energy loss of FP (triangle), SP (circle), and SPA (square) steels
at plastic strains. Open symbols: under stress. Closed symbols: after release.
FP
SP
SPA
Fig. 4.20. The relative permeability of FP, SP, and SPA at a range of stresses.
1D magnetic measurements under stress
FP
125
SP
SPA
Fig. 4.21. The coercive force of FP, SP, and SPA steels at a range of stresses.
FP
SP
SPA
Fig. 4.22. The remanence of FP, SP, and SPA steels at a range of stresses.
126
Chapter 4.
Fig. 4.18 to Fig. 4.22 depict various magnetic properties of the three
compared steels under uniaxial stress and plastic deformation. This
work was presented by V. Permiakov at 2DM'2002.
Although a similar tendency can be observed for all materials in
comparison with the previous experimental results presented above, the
sensitivity of semi-processed steel to the applied stress and the tensile
plastic deformation is very low in comparison with the behaviour of the
same material after the annealing and with the one of the fullyprocessed steel. The annealed steel shows the best magnetic properties
and the largest dependence of the applied compressive and tensile
stress. A critical stress can be observed at uniaxial tension of 20 to 60
MPa, similar to the one observed above.
4.1.5. Uniaxial distorted magnetization under stress.
According to Chapter 2, the actual magnetic flux waveforms in machines
are distorted (non-sinusoidal) in most cases.
Although all magnetic measurements are usually performed at
sinusoidal flux density in the sample, a study of distorted magnetization
under stress can be easily performed in the novel magnetomechanical
system. The waveform control of the system allows to achieve virtually
any form of magnetic flux density in the material.
A study of the stress effect on the magnetic properties under
actual distorted magnetization has been performed for the experimental
flux patterns at rated load shown in Fig. 2.5, see Chapter 2.
Fig. 4.23 depicts the BH loops under a distorted flux versus a
sinusoidal flux of the same peak induction level at compressive stresses
[Permiakov03].
Fig. 4.24 depicts the BH loops under a distorted flux versus a
sinusoidal flux under plastic deformation [Permiakov03].
Fig. 4.25 depicts the distorted BH loops under a complete range
of applied stresses. Fig. 4.26 depicts the energy loss under distorted and
sinusoidal fluxes and "after release" [Permiakov03].
1D magnetic measurements under stress
127
Fig. 4.23. The BH loops at compressive stresses: sinusoidal vs. distorted.
Fig. 4.24. The BH loops at plastic deformations: sinusoidal vs. distorted.
128
Chapter 4.
Fig. 4.25. The BH loops at a range of applied stresses under distorted flux.
Fig. 4.26. The energy loss at a range of stresses and plastic deformations. Closed
symbols: under stress. Open symbols: after release. S: sinusoidal. D: distorted.
1D magnetic measurements under stress
4.1.6. Experimental
tendencies.
results
of
1D
129
measurements:
preliminary
Based on a large number of experiments for different electrical steels, of
which only a part was presented above, the following preliminary
conclusions can be drawn for grain-oriented and non-oriented electrical
steels under sinusoidal and distorted uniaxial magnetization:
- a compressive stress leads to a drastic deterioration of the
magnetic properties,
- grain-oriented steel is strongly sensitive to even a small
compressive stress,
- a tensile stress improves the magnetic properties in the direction
of the stress,
- a "cross point" can be observed for BH loops at tension in two
quadrants,
- the "critical tensile stress" of 20-60 MPa corresponds to the best
magnetic properties,
- grain-oriented steel is barely sensitive to tensile stress above the
critical tensile stress,
- semi-processed steel is less sensitive to stress than fullyprocessed steel,
- annealing of semi-processed steel leads to a higher sensitivity to
the applied stress,
- a tensile plastic deformation deteriorates the magnetic properties
of all steels,
- a tensile plastic elongation of less than 10% leads to a very fast
tendency for the deterioration,
- an elongation larger than 10% leads to a steady deterioration up
to destruction,
- a deformed steel has poorer properties "after release" than "under
stress",
- a distorted magnetization exhibits a similar stress effect as a
sinusoidal one,
- a plastically deformed material is less sensitive to a distorted
magnetization,
- uniaxial compression and plastic deformation are the worst
conditions, resulting in a large increase of energy loss and a large
decrease of permeability.
130
Chapter 4.
4.2. The magnetomechanical effect: theoretical approach.
The previous section presents a broad range of experimental results
obtained by means of the novel magnetomechanical system under
uniaxial stress and magnetization. For every aspect of the above
tendencies there must be a physically sound explanation.
To explain the experimental results, it is better to start with grainoriented steel. One of the features of grain-oriented steel is the
orientation of the [001] crystallographic direction of the (110) crystal
plane along the rolling direction of the material. Furthermore, due to
large grains, it is easy to obtain an experimental observation of the
domain structure for grain-oriented steels.
4.2.1. Energy theory and domain structures.
Domain theory is indeed a suitable tool for a physically sound
explanation of the stress effect in grain-oriented steels. Fig. 4.27 depicts
the domain structure at tension in the rolling direction (RD) of 0.30 mm
3%-Si steel without coating [Fukuda81].
A clear GOSS texture can be observed in Fig. 4.27, when the [001]
crystallographic direction of the grains is close to the RD of the sheet.
For this grain-oriented steel, the (110) crystallographic plane is parallel
to the RD of the sheet.
Several grains can be observed as well in Fig. 4.27. A stress-free
condition can be described by a small number of fairly large domains
oriented inside each grain close to the rolling direction. The application
of tensile stress parallel to the RD of grain-oriented steel leads to an
increase of the number of domains in every grain with a clearly
observed decrease of the width of each domain. The effect is described
as if a stress-free domain is split into a few longitudinal domains.
Moreover, in Fig. 4.27 the material is not magnetized.
1D magnetic measurements under stress
131
Fig. 4.27. Domain structures due to tensile stress in the RD [Fukuda81].
132
Chapter 4.
Fig. 4.28. (a) ideal domain structure, (b) effect of compression [Moses80].
Indeed, the application of tensile stress refines the wall spacing of "main
domains" and removes "supplementary structure", resulting in almost an
ideally oriented domain structure [Shilling74]. The stress effect can then
be easily understood based on the fact that the supplementary structure
consists mainly of transverse domains, oriented perpendicular to main
domains in the plane of the sheet. In other words, when a tensile stress is
applied in the RD of grain-oriented steel, there is at least a partial
elimination of transverse domains and a structural refinement, or a
"splitting", of main domains that are oriented parallel to the applied
tensile stress, see Fig. 4.27.
Indeed, due to the orientation of grains and the domains in the
direction parallel to the applied mechanical load, tensile stress seems to
improve the domain structure. Compression on the other hand leads to
completely different results in a demagnetized state. Fig. 4.28 depicts the
effect of compression on (110) [001] oriented electrical steel in the RD.
Consequently, when a magnetic field is applied in the rolling
direction of grain-oriented steel under tension or compression, the stress
effect is completely different. An uniaxial tension is favourable for the
uniaxial magnetization, resulting in smaller BH loops, see Fig. 4.3, and
smaller energy loss, see Fig. 4.4. On the contrary, an uniaxial
compression is unfavourable for the uniaxial magnetization, resulting in
much larger BH loops and a higher energy loss, see Fig. 4.3 and Fig. 4.4.
When magnetization is applied in the rolling direction parallel to
the direction of applied stress, the domain theory should be considered.
An introduction into the domain theory has been given in section 1.2.2.
Here, a more detailed explanation is required.
The ferromagnetic domains are magnetically ordered regions
1D magnetic measurements under stress
133
within which the magnetization is equal to the saturation magnetization.
The overall magnetization of the material is the vector sum of the
magnetization within all domains. When a magnetic field is applied to a
magnetic material, changes occur in the domain structure which
produce changes in the overall magnetization. Changes in the domain
structure can occur by two means. The primary mechanism is the
domain wall movement, i.e. when the boundary between two domains
moves. The second mechanism is the domain rotation, i.e. when the
magnetization within a domain rotates to another direction, e.g. parallel
to the applied field.
Fig. 4.29. Demagnetized domain structures in: a) isotropic material, b) single
crystal, c) grain-oriented steel, d) grain-oriented steel at tension [Shilling74].
In electrical steels, domain rearrangement occurs to a major extent by
wall movement between neighbouring domains. Domain rotation occurs
only at large applied magnetic fields. Due to the complex domain
structures in electrical steels, a prediction of the domain structure by
means of a minimization of crystal energy is a large scientific challenge.
So far, nobody succeeded in doing this. Instead, an adaptation of domain
structures and an approximation of the total energy are usually used. In
Fe-Si alloys such as electrical steels, the major energy components are:
- the magnetostatic energy,
- the magnetocrystalline anisotropy energy,
- the magnetoelastic energy,
- the exchange energy,
- the energy of the domains in the presence of an applied field.
The magnetostatic energy is the energy of a sample in its own field. In
order to minimize this energy, a convenient multidomain structure is
usually introduced, as shown in Fig. 1.5b. The magnetostatic energy
without a domain structure (se Fig. 4.29a) is very large. For a domain
structure shown in Fig. 4.29b, i.e. a "two-two" structure of two main and
two transverse domains, the magnetostatic energy in Fe-Si steels
134
Chapter 4.
corresponds to a minimum energy state in a demagnetized material.
The magnetic moments of the atoms have preferable directions in
the crystal structure. The magnetocrystalline anisotropy describes a
minimum energy for domains located parallel to certain crystallographic
directions, called "easy directions of magnetization."
The same atomic moment interactions giving rise to magnetism
and magnetic anisotropy produce forces between atoms, which tend to
strain the material lattice anisotropically. The magnetic energy
associated with these lattice strains is called the magnetoelastic energy,
usually characterized by two parameters λ100 and λ111. These parameters
are defined experimentally as the strains in the [100] and [111]
directions, when the magnetization is in the [100] and [111] directions,
respectively. Because of the dependence of magnetoelastic energy on
lattice strain, a strong interaction exists between the orientation of
domains and applied stress.
For example, if an uniform tensile stress is applied parallel to the
[100] easy direction of grain-oriented steel, the magnetoelastic energy
proportional to the applied stress reaches its minimum state for the
domains which lie parallel to the externally applied tensile stress.
However, if tensile stress is applied at some angle with respect to the
cube edges, domains can be expected to lie along the closest easy
direction to the tensile stress, but not parallel to it. Residual stresses can
also introduce a magnetoelastic energy component into the total energy.
There are two wall types in Fe-Si electrical steels: the 180-degree
walls and the 90-degree walls. A 180-degree wall reverses the flux by
180º and a 90-degree wall reverses the flux by 90º, as shown in Fig. 4.29b.
How can the total energy be minimized?
The magnetostatic energy can be reduced to zero by using four
90-degree walls and one 180-degree wall, see Fig. 4.29b. The anisotropy
energy is equal to zero for the domains oriented in easy directions. The
wall energy (exchange, anisotropy, magnetoelastic energy within the
wall region) can be neglected. Thus, the magnetoelastic energy in the
domains is the largest component existing in the considered domain
structure. In fact, the magnetoelastic energy is proportional to the
volume of the closure domains [Shilling74]. Therefore, by reducing the
volume of the supplementary structure, the magnetoelastic energy can
be reduced to some optimal value. When tension is applied, this optimal
value can be achieved, as shown in Fig. 4.29d.
From theory to practice, in order to explain the experimentally
1D magnetic measurements under stress
135
observed domain structures under tension shown in Fig. 4.27, the energy
behaviour is crucial. The magnetoelastic energy of the main domains
decreases if tensile stress is applied parallel to the rolling direction (i.e.
an easy direction) or increases if compressive stress is applied instead.
The magnetoelastic energy of transverse domains does the opposite.
Hence, based only on magnetoelastic energy the assumption can
be made that the supplementary structure of closure domains tends to
disappear under tension. However, the removal of the supplementary
structure by tensile stress increases the magnetostatic energy. Hence, a
simultaneous refinement of the wall spacing of main domains occurs
(see Fig. 4.29c,d) to reduce the magnetostatic energy [Shilling74].
When a magnetic field is applied to the material, domain wall
movement occurs, see Fig. 1.5.
In Fe-Si steels, magnetization occurs by the motion of 180º and
90º domain walls until the net force of all walls is zero. During
magnetization under tensile stress, the supplementary structure of
transverse domains that disappears under stress in demagnetized state
reappears again to support the magnetization process [Shilling74]. This
occurs when movement of the 180° walls separating main domains has
increased the magnetostatic energy to the point where the sum of the
magnetostatic energy and magnetoelastic energy is reduced by the
formation of a supplementary structure.
Thus, there is in fact an "energy balance" of domain structures,
depending on many parameters, such as grain size, impurities and other
pinning centers, residual stresses, and of course, the externally applied
stress and the magnetic field.
4.2.2. Applied stress and domain theory.
As shown in Fig. 4.3, the application of tensile stress at medium
magnetization has an useful effect on macroscopic energy loss due to the
reduction of the magnetoelastic energy of the domains. A further
increase of the tensile stress seems to have a moderate effect on energy
loss of the considered grain-oriented steel as shown in Fig. 4.4.
Applied tensile stress does not indefinitely refine the wall
spacing due to the reappearance of supplementary structure during
magnetization. The more stress is applied reducing the magnetoelastic
energy, the more supplementary structure is required for uniaxial
magnetization. Furthermore, pinning of domain walls by grain
136
Chapter 4.
boundaries can occur, although high-quality grain-oriented steel has
large grains and a limited number of pinning centers. In fact, in grainoriented materials with a smaller grain size, the stress dependence of
energy loss is different, see Fig. 4.5. Here, the increase of energy loss
after a certain tensile stress can be explained by an increasing number of
pinning centers at grain boundaries, resulted in the build-up of a larger
magnetostatic energy at higher stress.
At compressive stress in the direction of the magnetization, the
domain structure makes a transition from an ideal one shown in Fig.
4.28a to the compressed structure shown in Fig. 4.28b.
Here, in order to magnetize the domain structure modified by
compression, a considerably large domain rotation must occur
[Moses80]. The domain walls must move further and faster than in the
stress-free state. The amount of domain wall movement can be very
large due to a large volume of the closure domains, which are
unfavourable to the uniaxial magnetization in the rolling direction of
grain-oriented steel. A similar dependence can be observed in nonoriented steels under compression.
Fig. 4.30. Schematic of spin rotation in a 180-degree domain wall [Bulte02].
1D magnetic measurements under stress
137
A further increase of compressive stress, however, leads to a smaller rate
of deterioration of the magnetic properties, as shown in Fig. 4.6. Once all
the grains in the material have the stress pattern, further increase of
stress reduces the domain spacing. The difference in energy loss at
higher compressive stress becomes smaller, although more domains
must rotate at a higher stress. The amount of domain wall movement is
not greatly different since the volume of the closure domains is very
small at a high stress. The domains in different grains switch at different
stresses so the energy loss increases up to very high stresses. The
tendency for grain-oriented steels in Fig. 4.6 can be observed at large
compression in non-oriented steels, as shown in Fig. 4.9 and 4.11.
4.2.3. Levels of understanding: microscopic, domain, macroscopic.
A general difficulty to plot a reliable theoretical explanation of the stress
effect is derived from the fact that electrical steels are complex
polycrystalline soft magnetic materials. Different levels of physically
sound explanations can be applied:
- a macroscopic level, where the sample dimensions are
considered,
- a domain structure level, where domains in the grains are
considered,
- a microscopic level, where the atomic spins and the
micromagnetic energies are considered.
Generally, there is a range of different types and orientations of grains in
non-oriented electrical steels. Therefore, only a statistical approach on a
macroscopic level can be considered as a reliable method for research.
For example, if the average grain size is equal to 20 µm and the distance
between two needle probes is 40 mm, the magnetic behaviour of a
couple of thousand of grains is considered in an average way during
each magnetic measurement.
Although the grains in non-oriented steel have random
orientation, resulting in a little anisotropy in the plane of the sheet, the
magnetization processes are basically the same as in grain-oriented
steels. These include both the domain structure changes under stress as
well as the domain wall movement and the domain rotation.
Apparently, a physically sound explanation of the stress effect in
non-oriented steels can be approached from a microscopic level, and in
particular, by considering domain walls and spin rotations in detail, as
138
Chapter 4.
shown in Fig. 4.30 [Bulte02].
In a 90-degree wall, none of the spins within the wall will lie in
one of the easy directions. In a 180-degree wall, the center of the wall
will lie in an easy direction but the rest of the wall spins will all be in
non-easy directions, as shown in Fig. 4.30 [Bulte02].
The interactions on a microscopic level in the domain walls are
very complex. The key relationships are the spin-spin interaction, which
defines the exchange energy, and the interaction between the spin and
the magnetocrystalline preferable directions. In fact, the total energy per
unit wall area is a sum of the exchange energy and the anisotropy
energy. In a wall of thickness δ (see Fig. 4.30), the angle between
neighboring spins is proportional to the distance between the
neighboring atoms and inversely proportional to the thickness δ. Hence,
the exchange energy per unit wall area is inversely proportional to the
wall thickness and the distance between atoms [Bozorth51]. Moreover,
the anisotropy energy is proportional to the thickness of the wall. Hence,
the thickness of the wall is defined by the above energy balance.
When the anisotropy is modified by an externally applied strain,
the corresponding magnetoelastic energy is added to the anisotropy
energy. The magnetoelastic energy is proportional to the applied stress.
Thus, the thickness of the domain wall is modified according to a
new minimum for the wall energy. The latter is now the sum of the
exchange energy, the magnetocrystalline anisotropy energy and the
stress-induced anisotropy energy. In a single crystal, the externally
applied stress distorts the crystal lattice, and consequently, the cubic
directions will no longer be perpendicular to each other [Bozorth51].
Depending on the orientation of the lattice with regards to the
applied stress, the angles between the easy directions will increase or
decrease. Any spins which are not aligned with one of the easy
directions will be affected by these lattice distortions. When the easy
axes and the atomic spins move relative to each other, the induced
changes in the anisotropy energy and the exchange energy will modify
the total energy, which is required to keep the moments pointing in any
given direction. A new "energy balance" must be found to minimize the
total energy of the wall.
As domain walls are groups of magnetic moments, not aligned
with any easy direction, the walls will be affected by externally applied
stresses. A wall movement will occur for 90-degree walls as a result of
the applied stress. The 90-degree wall will experience an "effective
1D magnetic measurements under stress
139
pressure" as a response to the stress [Bulte02]. For example, if the tension
is applied along the [100] easy axis, and the orientation of the spins of a
90-degree starts from the [100] direction and ends at the [010] direction,
the magnetic moments of the wall will tend to align with the [100]
direction in order to decrease the magnetoelastic energy of the 90-degree
domain wall. This results in 90-degree wall movement "out" of the main
domain that is oriented parallel to the tension. A 180-degree wall may be
regarded as composed of two 90-degree walls. Therefore, the pressure
on 180-degree walls will be on both sides of the wall but in opposite
directions. Hence, the wall thickness will change slightly, but the wall
position will not be affected.
In the demagnetized state, due to this process alone the net
magnetization will remain zero. When a magnetic field is applied,
uniaxial stress will affect the magnetization as it will reduce
(compression) or increase (tension) the size of favourable domains by
means of the movement of 90-degree domain walls.
Thus, making a step up from a microscopic level to a domain
level, the growth of the main domains under tension occurs due to an
"effective pressure" on 90-degree walls. A single 180-degree wall in a
"two-two" minimum energy domain structure remains stress unaffected.
The four 90-degree walls are "pressed out" of the main domains and the
area of these walls shrinks, so does the volume of transverse domains.
At compression, the four 90-degree walls are "pressed into" the main
domains, forcing the apparent domain rotation to the direction
transverse to the applied compression.
The magnetomechanical effect is virtually the same for all Fe-Si
steel, although its contribution is different. Thus, to simplify the uniaxial
magnetomechanical effect, an external stress "presses" the 90-degree
walls "in" or "out" of the favourable domains under externally applied
uniaxial mechanical load. Other effects, which have been observed
experimentally, can be explained as well. The presented domain theory
and the domain wall movement approach can be used to explain a
"critical" stress, a sensitivity, an effect at plastic deformation, and other
conclusions drawn in Section 4.1.6.
4.2.4. Explanation of various effects: the cross points in two quadrants.
The behaviour of the BH loops, shown in Fig. 4.3, 4.7, 4.8, and 4.13, can
be explained as follows, starting from Fig. 4.13.
140
Chapter 4.
As the magnetization of the material moves along the loop from
the tip of the loop in the first quadrant to the remanence and further to
the second quadrant, the magnetic field is first reduced to zero, which
corresponds to the remanence. The remanence is a result of positive
internal field sufficient to maintain a reversible rotation of magnetic
moments. When a negative field of a larger amplitude is applied, the
point is reached where the field is insufficient to maintain reversible
rotation. The overall magnetization of the material is still positive due to
the fact that it is a sum of the internal and the applied fields. However,
no magnetic moments are experiencing a sufficient field strength to
overcome the stress-induced anisotropy. Thus, all moments are aligned
in the easy directions which are closest to the direction of the original
field. Consequently, no domain walls will exist within the grains.
At this point, there are no more non-easy-aligned spins. Thus, the
magnetization can be independent of the applied tensile stress in the
"cross points" in the second and the forth quadrants, see Fig. 4.13.
The magnetization remains dependent on the history, the applied
field and the temperature. Hence, all BH loops of that sample with the
same history, temperature and field strength will coincide at that cross
point, no matter which tensile stresses are applied to the sample. The
stress applies an effective pressure on 90-degree domain walls. If a
domain wall is considered as a series of non-easy-aligned magnetic
moments, then it is assumed that applied stress can only affect non-easyaligned spins. Hence, no 90-degree walls are present at the cross points.
At compression or plastic deformation, the cross point slips away
either because a domain rotation take place, or respectively, a plastically
deformed material with a higher dislocation density is considered in
comparison with the one at elastic tension. A further study is advised.
4.2.5. Explanation of various effects: the critical tensile stress.
Another visible effect from the experimental results is the "critical"
tensile stress, which corresponds to the best magnetic properties at
uniaxial case of stress and magnetization.
Why do the losses increase after the critical stress? From the above
theoretical explanations, the critical tensile stress corresponds to the
macroscopic "energy balance", where all energies are reduced to its
minimized states. The "effective pressure" on 90-degree domain walls
leads to partial elimination of the transverse domains, which "helps" to
1D magnetic measurements under stress
141
magnetize the material in the direction parallel to the applied tension.
This positive effect of uniaxial tension is limited by microstructure of
electrical steels.
A larger tension results in an increase of the magnetostatic
energy at a larger number of pinning centers at the grain boundaries. All
considered non-oriented steels seem to have the critical stress around 20
to 60 MPa. For the considered grain-oriented steel, a large tensile stress
does not lead to a loss increase (see Fig. 4.4) in comparison with the nonoriented steels. However, other researchers have found that tension
above the critical stress does increase the loss in a grain-oriented steel
with smaller grains, as shown in Fig. 4.5 [Moses80].
4.2.6. Explanation of various effects: the plastic deformation.
The energy losses are generally associated with the various
rearrangements of main and supplementary domain structures.
Considering the rearrangement of the main structure, the 180-degree
domain wall movement is the major mechanism.
The domain wall movement can be characterized by the velocity
and the distance of this movement. For a 180-degree domain wall
movement, not influenced by neighbouring walls, the energy loss will be
proportional to the square of the wall velocity. The wall velocity is
proportional to the spacing 2L between neighbouring 180-degree
domain walls. When a system of finely spaced walls is considered, the
eddy currents from the neighbouring walls interact.
Generally, the impurities, grain size, grain orientation and stress
might have a similar impact to the domain wall movement. If the motion
of an individual wall is interrupted, by pinning for example, the wall
will move slower than the average wall velocity when it is pinned and
faster than the average wall velocity for a short period of time after it
becomes unpinned. This process will result in an increase of losses
compared to uniform wall motion due to a square dependence of losses
on the wall velocity. Therefore, the presence of pinning centers is indeed
undesirable. The pinning centers can be impurities or grain boundaries
as well as any kind of dislocations in the lattice.
At elastic stress, the dislocation density remains unchanged.
However, the grain boundaries can be very effective pinning centers.
Two variables mainly affect the wall spacing: applied tensile stress and
grain size.
142
Chapter 4.
The wall velocity should decrease if the wall spacing 2L is
reduced, followed by a reduction of losses. When tensile stress is
applied, the 180-degree walls become more closely spaced. The decrease
of loss is observed up to the "critical" stress, when the domain wall
movement becomes harder, taking into account smaller wall spacing
and the interaction between domain walls.
To explain the behaviour of the magnetic properties under plastic
deformation, an increase of the dislocation density must be considered
due to the fact that dislocations of crystal lattice are the major pinning
centers for domain wall movement. Indeed, the main cause for a
hindered domain wall movement at plastic strain is a linear increase of
dislocation density and a formation of blocks of dislocations. A hindered
domain wall movement results in a large increase of energy loss at
plastic strains, as shown in Fig. 4.15 and 4.26. Amazingly, the increase of
loss at smaller strains is rapid in comparison with a steady increase at
larger strains, see Fig. 4.15. A possible reason for the non-linearity is the
fact that the deformation is non-uniform, especially at small strains,
when different parts of the material are deformed at different rate. At
larger strains, when necking of the sample occurs, a more uniform
deformation in the center of the sample might be a reason for a gradual
increase of energy loss at high strains.
In fact, dislocations or irregularities in the crystal lattice exist in
vast amounts in semi-processed steel. Therefore, the dependence of
energy loss in semi-processed steel is less "sensitive" to applied stress.
Here, the sensitivity to applied stress is somewhat similar to the
sensitivity at a highly strained material, see Fig. 4.16.
When a semi-processed steel is annealed, the dislocations
disappear, which results in a larger sensitivity of the material properties
to an applied mechanical stress, see Fig. 4.20 and 4.21.
The fact that the energy loss "after release" is larger than "under
stress" is simply because of the presence of residual stresses in the
deformed material. After the dislocation density, the residual stress is
the second major reason for worse properties. A separate study of the
residual stresses was performed in [Pulnikov04], see Chapter 5. Briefly,
when a tension is removed at pre-strained material, the residual stresses
caused by tensile strain must be transverse to the applied tension. The
amount of residual stresses can be evaluated indirectly. Generally, the
application of tensile elastic stress to the material with residual stresses
leads to a biaxial redistribution of internal stresses.
1D magnetic measurements under stress
143
Another effect is generally known as a frequency dependence.
Indeed, the main domain spacing varies with the excitation frequency.
As was mentioned above, the domain wall movement is not smooth and
continuous in electrical steels. One of the views on this subject assumes
that domain wall movement in Fe-Si steels is limited due to eddy
currents [Shilling74]. When the excitation frequency increases, the wall
cannot move fast enough to keep the local fields ahead of the wall below
the level required to nucleate a new domain. If the velocity of a 180degree domain wall is increased by increasing the magnetizing
frequency, the applied field must also increase in order to move the wall
to reach the same peak induction level. Thus, if the frequency of
magnetization is increased, the magnetic field must increase, which at
some point can lead to nucleation of new domains. This happens when
the activation energy for nucleation is exceeded and the pinned wall
becomes unpinned and starts to move.
Furthermore, skin-effect can be observed at higher frequencies.
More about these effects can be found in [Shilling74] and [Bertotti98].
To conclude the theoretical approach, different physical levels, from
a microscopic to domain to macroscopic, can be considered to explain
the behaviour of magnetic properties in electrical steels under stress.
4.3. Statistical theory of energy losses.
A common practice for the energy loss evaluation in electrical steels has
been the separation of losses [Bertotti98].
According to this theory, the average energy loss per unit
volume P of soft magnetic material consists of the sum of both a
hysteresis and a dynamic contribution, given by (4.1) [Bertotti88]:
P = P ( hyst ) + P ( dyn )
(4.1)
Both hysteresis and dynamic losses are investigated in a separate way.
The physical reason for such a decomposition is that P(hyst)
originates from the discontinuous character of the magnetization process
at a microscopic level (Barkhausen jumps), whereas P(dyn) is associated
with the macroscopic dynamic behaviour of the domain structure. It is
generally assumed that the effects of these two levels do not overlap
[Bertotti98].
The interpretation of the dynamic loss P(dyn) is usually based on
the induced electrical currents around the moving domain walls,
144
Chapter 4.
without taking into account microscopic behaviour of the domain wall.
The classical dynamic loss model does not take into account the domains
at all, but assumes that a magnetization process is perfectly
homogeneous in space. Neglecting skin effects at the range of low
magnetizing frequencies, the classical dynamic loss model for a
lamination having a thickness d predicts [Bertotti88]:
P ( dyn ) ≡ P ( class ) = π 2 ⋅ σ ⋅ d 2 ⋅ Bm2 ⋅ f 2 / 6
(4.2)
Here, σ is the electrical conductivity of the material, Bm is the amplitude
of the sinusoidal magnetic flux density, and f is the magnetizing
frequency.
The simplification of the dynamic loss to only a classical loss
component gives a known error. As a consequence of domain effects, the
dynamic loss P(dyn) is generally found to be larger than the classical loss
P(class). The difference between them is called the excess loss P(exc), which
can be much larger than the classical loss, for example, in case of grainoriented materials. Thus, the total energy loss is given by:
P = P ( hyst ) + P ( class ) + P ( exc )
(4.3)
Each component of total energy loss is discussed further in details.
4.3.1. Classical loss.
The classical loss can be considered as a side effect of magnetization.
This is in fact the dynamic eddy current loss obtained from the
Maxwell's equations (see Chapter 1), assuming that there are no
magnetic domains present. A general expression for the classical loss in
conductive laminations is given by [Bertotti98]:
P ( class ) = (σ ⋅ d 2 / 12)(dB / dt ) 2
(4.4)
Under a sinusoidal magnetic flux, when B(t) = Bmsin(2πft), the classical
loss is given by (4.2). Thus, the classical loss generally depends on:
- the electrical conductivity σ of the considered electrical steel,
- the square of the thickness d of the lamination,
- the square of the amplitude Bm of the magnetic flux density,
- the square of the magnetizing frequency f.
For every considered frequency and peak induction level, an electrical
steel with a smaller thickness (the square dependence!) or a smaller
resistivity (i.e. a higher Si content) has a smaller classical loss.
1D magnetic measurements under stress
145
4.3.2. Hysteresis loss.
The hysteresis loss P(hyst) can be obtained from the area of the quasi-static
hysteresis BH loop times the power frequency f :
P ( hyst ) = Wh ( Bm ) ⋅ f
(4.5)
Here, Wh(Bm) is a hysteresis energy related to the microstructure of the
material and defined by the quasi-static BH loop. Thus, the hysteresis
loss depends on:
- the magnetization frequency f,
- the material parameter Wh depending on the peak magnetic flux
density Bm.
The hysteresis loss can be obtained experimentally at a sufficiently low
frequency, such as below 0.1 Hz, when the dynamic effects are
negligible. Other methods can be applied to define the hysteresis energy
as well, see section 4.3.4.
4.3.3. Excess loss.
The excess loss P(exc) is given by the difference of the total loss and the
two other loss components (classical loss and hysteresis loss) derived
from eq. (4.3).
The understanding of the origin and the properties of the excess
loss P(exc) has been a challenge for many years. The primary idea was that
excess losses are the consequence of the existence of magnetic domains.
The guideline in the interpretation of excess losses has been, for a long
time, the model proposed by Pry and Bean [Bertotti88]. In this model,
only 180-degree domain walls perpendicular to the plane of the sheet are
considered. In the demagnetized state of the material, the width of each
domain is equal to 2L. The dynamic loss P(dyn) of an infinite lamination of
thickness d and containing a periodic array of domains is calculated
from Maxwell's equations. This model indicates that the main parameter
controlling excess losses is the ratio 2L/d between the domain size and
the lamination thickness, and predicts the following:
when
P ( exc ) << P ( class ) ←⎯
⎯⎯ 2 L / d << 1
(4.6)
when
P ( exc ) ≅ (1.63 ⋅ 2 L / d − 1) ⋅ P ( class ) ←⎯
⎯⎯ 2 L / d >> 1
(4.7)
146
Chapter 4.
Unfortunately, this model of (4.6) and (4.7) is very idealized. On
the one hand, excess losses have been found non-negligible to classical
loss at 2L/d <<1 as in (4.6). On the other hand, P(exc) / f shows a nonlinear
dependence on frequency f, which cannot be explained by (4.6) and (4.7)
[Bertotti88].
Based on many experiments for different magnetizing
frequencies, induction levels and a large number of materials, as well as
on physical statistical interpretation of domain wall movement, the
following conclusions have been drawn [Bertotti88]:
- the dependence of P(exc) / f on frequency f is nonlinear in general,
- P(exc) / f is linearly dependent on the square root of the frequency,
- for non-oriented and grain-oriented electrical steels the excess
loss per cycle can be expressed by:
P ( exc) / f = 2 ⋅ Bm ( (n0 ⋅ V0 ) 2 + 16 ⋅ σ ⋅ G ⋅ S ⋅ V0 ⋅ Bm ⋅ f − n0 ⋅ V0 )
(4.8)
Here, σ is the electrical conductivity of the material, G = 0.1356 is a
dimensionless coefficient, S is the cross section of the sample, n0 and V0
are the parameters related to microstructure.
The parameter V0 having the dimension of a magnetic field may
show a flux density dependence, but is frequency independent. In nonoriented steels, the parameter n0 often does not play any significant role
and can be neglected. Thus, the eq. (4.8) for the excess loss is given by:
P ( exc ) = 8 ⋅ σ ⋅ G ⋅ S ⋅V0 ( Bm ⋅ f ) 3 / 2
(4.9)
4.3.4. Loss separation technique.
The frequency dependence of the total losses in electrical steel can be
used for a successful loss separation and further analysis of the loss
behaviour. To perform the loss separation, energy terms are used
instead of power. The total energy is given by:
Wtot = Whyst + Wclas + Wexc
(4.10)
The loss in [W/kg] can be transformed into the energy in [J/m3] as
follows:
P[W / kg ] ⋅ ρ [kg / m3] / f [ s ] = W [ J / m 3 ]
(4.11)
The classical energy term in eq. (4.10) is simply equal to the classical loss
1D magnetic measurements under stress
147
per cycle, and for a sinusoidal induction is given by:
Wclas = P ( class ) / f = π 2 ⋅ σ ⋅ d 2 ⋅ Bm2 ⋅ f / 6
(4.12)
The hysteresis energy in eq. (4.10) is a material parameter Wh(Bm), given
in (4.5). The excess energy in eq. (4.10) can be derived from (4.9):
Wexc = P ( exc ) / f = 8 ⋅ σ ⋅ G ⋅ S ⋅V0 ⋅ Bm3 / 2 ⋅ f
(4.13)
For applying the loss separation technique, the classical energy term is
subtracted from the total energy due to its different nature.
Hence, the sum of the hysteresis and the excess energy can be
derived from (4.10)-(4.13) as follows:
Wtot − Wclas = Whyst + 8 ⋅ σ ⋅ G ⋅ S ⋅ V0 ⋅ Bm3 / 2 ⋅
f
(4.14)
Eq. (4.14) is nothing else than a straight line, i.e. y = a + bx, where
y = Wtot – Wclas and x = f1/2. To determine the parameters a and b, at least
two frequency points are required. The eq. (4.14) is used to identify the
microstructure dependent parameters Wh and V0.
A typical method for loss separation is described in [Bertotti88].
The method is based on the experimental frequency dependence of the
total losses at various peak inductions, which can be obtained by means
of the operational scheme in Fig. 4.2. The method consists of the
following steps:
- a series of magnetic measurements at a range of frequencies from
1 to 200 Hz and different induction levels,
- the calculation of the classical energy loss for each induction and
frequency according to (4.12),
- the simple subtraction of the classical energy term from the total
energy,
- the plotting of the result as a function of the square root of the
frequency as in (4.14),
- the extrapolation of the plot to zero frequency to define the
parameter Wh,
- the determination of the other significant material parameter V0.
In theory, two frequency points can be sufficient to perform the above
algorithm of the loss separation.
In practice, several experimental points from the lowest
frequency up to 200 Hz are usually required to increase the accuracy of
the extrapolation of the Wh.
The material parameter can be derived from (4.9), given by:
148
Chapter 4.
V0 = ( P ( exc ) ) 2 /(64 ⋅ σ ⋅ G ⋅ S ⋅ ( Bm ⋅ f ) 3 / 2 )
(4.15)
Fig. 4.31 depicts hysteresis and dynamic loops for a typical non-oriented
steel, assuming that the experimental loop at 5Hz can be considered as
the hysteresis loop. In fact, this assumption is very reliable for many
non-oriented steels due to the small change in BH loops below 20 Hz.
For grain-oriented steels this assumption is less reliable due to the much
larger ratio of the dynamic losses to the quasi-static losses.
1.3
Hysteresis loop
Dynamic loop
Hyst
Clas
Exc
B, T
Dyn
0
-1.3
-100
0
H, A/m
100
Fig. 4.31. Hysteresis and dynamic loops with classical and excess terms.
4.4. Separation of losses under applied stress.
The experimental results of Section 4.1 and the comparison with other
studies presented up to now in this Chapter have all been obtained at
power frequency 50 or 60 Hz.
To perform the loss separation, a range of frequencies has been
used at several peak induction levels and under different applied
mechanical stresses.
In fact, the first international publication on the study of loss
separation under tensile stress was presented at the 46th MMM
Conference in Seattle [Permiakov02].
1D magnetic measurements under stress
149
Fig. 4.32 depicts the frequency dependence of the BH loops at
peak induction of 1.0 T for low stress conditions: a stress-free condition
and a tensile stress of 240 MPa. The latter corresponds to the start of
plastic deformation, i.e. about 5% elongation.
The frequencies used in the loss separation were 5, 10, 15, 25, 50,
80, 100 Hz. The BH loops at only three frequencies are shown in Fig.
4.32. Although the frequency of 15 Hz cannot be considered as a target
quasi-static frequency, it was observed that BH loops at frequencies
below 25 Hz were the same for the considered material.
Fig. 4.33 and 4.34 depict the results of the loss separation, i.e.
hysteresis and excess loss versus applied stress. The application of
tensile stress below the critical tensile stress leads to a large reduction of
the hysteresis loss. Again, the critical stress can be observed around 20 to
40 MPa. After the critical stress, the hysteresis loss increases gradually
until the elastic limit.
Fig. 4.32. Frequency dependence at stress-free and 240 MPa [Permiakov02].
150
Chapter 4.
Elastic
Limit =
220 MPa
Fig. 4.33. Hysteresis loss at tensile stresses and strains [Permiakov02].
Elastic
Limit =
220 MPa
Fig. 4.34. Excess loss at a range of tensile stresses and strains [Permiakov02].
1D magnetic measurements under stress
151
Fig. 4.35. Loss separation at a whole range of stresses [Permiakov04].
At plastic deformation, the hysteresis loss exhibits a drastic
increase. The excess loss exhibits an almost similar stress dependence as
the hysteresis loss. The same critical tensile stress can be observed for the
excess loss, followed by a larger increase at inductions larger than 1 T in
comparison with a gradual increase observed for the hysteresis loss.
It was observed that at plastic deformation the excess loss
exhibits a weak increase in comparison with the drastic increase of the
hysteresis loss. It is likely that an increase of the dislocation density has a
primary effect on hysteresis loss rather than on excess loss, see Fig. 4.34.
A more recent study has supported this explanation by a more
accurate loss separation. Fig. 4.35 depicts the results of the loss
separation for another non-oriented steel with a large grain size of 240
µm (in comparison with 20 µm before).
A complete range of applied stresses are presented from
compression of 60 MPa to elastic tension with an elastic limit of 440
MPa. Additionally, three plastic strains are shown in the same figure.
The behaviour of the hysteresis and excess loss under elastic
stresses seems to be identical. Due to the choice of a non-oriented steel
with 10 times larger grains, the excess loss at 50 Hz is about 50% of the
hysteresis loss [Permiakov04]. A compressive stress leads to a drastic
152
Chapter 4.
increase of both hysteresis and excess loss, while a tensile stress results
in a reduction of both loss components below the critical stress. The
critical tensile stress was observed at 30 MPa for both hysteresis and
excess loss. The V0 parameter is shown in Fig. 4.36.
It is a remarkable fact that the V0 parameter at the critical tensile
stress is very small, in fact, it is almost zero. When the V0 parameter is
defined by eq. (4.15), a square dependence on the excess loss leads to, for
example, a decrease of the V0 parameter by a factor of 9 at the critical
tensile stress, whereas the excess loss is decreased only 3 times. At
compression, the doubling of the excess loss results in a 4-time increase
of the V0 parameter. Thus, the material parameter V0 is much more
sensitive to the applied stress than the energy loss.
Fig. 4.36. The V0 parameter at a whole range of stresses [Permiakov04].
The two material parameters, namely Wh and V0 , are usually considered
to depend only on the induction level. Furthermore, the V0 parameter is
often considered independent on the induction, except for the case of
high induction levels [Bertotti98]. Here, we found that the two material
parameters depend not only on the induction level, but also on the
applied mechanical stress or the plastic deformation. Thus, a new
dependence has been discovered, as described in [Permiakov04].
1D magnetic measurements under stress
153
4.5. Conclusions.
Taking into account the presented theoretical approaches, the
preliminary tendencies drawn from the experimental results (see Section
4.1.6) can be explained as follows.
a) A compressive stress leads to a drastic deterioration of the magnetic
properties.
A domain rotation occurs due to an effective pressure on 90-degree
domain walls in a way that more energy is required to magnetize a
sample in the direction of compression.
b) A grain-oriented steel is extremely sensitive to even a small compressive
stress.
Because of the [100] orientation of grains and the relevant domains along
the direction of applied stress, an uniaxial compression is the least
desirable regime for uniaxial magnetization.
c) A tensile stress improves the magnetic properties in the direction of
stress.
Tension leads to an effective pressure on 90-degree walls, followed by an
elimination of the transverse domains, which improves uniaxial
magnetization to some extent.
d) A "critical" stress corresponds to a condition of the best magnetic
properties.
The "energy balance" can be achieved at the "critical" tension. A higher
stress leads to an increase of energy at grain boundaries, the pinning
sites for domain wall movement.
e) A grain-oriented steel is less sensitive to tensile stress above the critical
stress.
Due to large grains, the increase of loss after the critical stress is rare in
oriented steels.
f) A semi-processed steel is less sensitive to stress than a fully-processed
steel.
Due to a high dislocation density, there is a natural limit for either a
positive or a negative effect of stress on domain structure and domain
wall movement.
g) An annealed semi-processed steel is more sensitive to stress as an
unannealed steel.
An annealing process leads to a reduction of the dislocation density,
resulting in an easier domain wall movement and a larger stress effect.
154
Chapter 4.
h) A tensile plastic deformation leads to a drastic deterioration for all
steels.
A higher dislocation density hinders domain wall movement to a major
extent.
i) A tensile plastic elongation of less than 10% leads to a very fast
deterioration.
A growth of dislocation density leads to a devastating effect on domain
wall movement.
j) An elongation larger than 10% leads to a steady deterioration up to
destruction.
A further increase of dislocation density at higher strains has a moderate
effect.
k) A deformed steel has the worse properties "after release" than "under
stress".
Residual stresses in a deformed material can be compensated by elastic
tension.
l) A distorted magnetization exhibits a similar stress effect as a sinusoidal
one.
Each harmonic exhibits the same stress effect as the fundamental one of
50 Hz.
m) A plastically deformed material is less sensitive to a distorted
magnetization.
Excess loss seems to be less affected by plastic deformation than
hysteresis loss.
n) An uniaxial compression and a plastic deformation are the worst
conditions.
The first is the worst condition for elastic stress, the second corresponds
to already modified material properties due to plastic deformation.
A new stress dependence has been discovered for the material
parameters, which can be useful to study further the stress effect under
uniaxial magnetization.
2D magnetic measurements under stress
155
All truths are easy to understand once they are discovered;
the point is to discover them.
Galileo Galilei (1564 - 1642)
CHAPTER 5. 2D MAGNETIC MEASUREMENTS UNDER STRESS.
The 1D magnetic measurements presented in the previous chapter
provide some remarkable characterization of various electrical steels
under applied compressive and tensile stress and plastic deformation.
Using the 1D case offers a few opportunities:
- to improve measurement skills and to gain measurement
experience,
- to create a range of 1D magnetomechanical conditions in a single
setup,
- to examine typical domain structures, for example, in grainoriented steels,
- to provide an explanation of the magnetic behaviour in terms of
domain structures,
- to use other theoretical approaches, from microscopic to
macroscopic,
- to observe and to explain various experimental results and stress
effects.
As a matter of fact the application of an uniaxial mechanical load to a
sample of electrical steel cannot be considered as the 1D case at all.
When the energy loss in the rolling direction changes under
applied stress, the macroscopic anisotropic behaviour of the magnetic
properties of the lamination also changes. Taking into account a primary
considered case of uniaxial magnetization, the stress effect must be
studied as a 2D effect, investigating a re-distribution of the 2D magnetic
properties in electrical steels.
In comparison with the 1D case, when the target was to confirm
and to explore some new information about the stress effects, a complete
2D investigation under 1D stress is a poorly studied subject due to its
complexity. Some studies have been done recently about the stress
effects under alternating 2D magnetic conditions [Pulnikov04] or various
rotational conditions without stress [Makaveev03].
As a result of the present 2D study, new stress effects in electrical
steels will be added to the understanding of the behaviour of electrical
steels under uniaxial stress.
156
Chapter 5.
5.1. Vector magnetization under uniaxial mechanical load.
When describing the magnetization processes corresponding with an
applied field in the 1D case, a vector magnetization must be taken into
account instead of an uniaxial scalar magnetization.
Indeed, the 1D magnetic measurements are the most simple case
of using one pair of sensors that measure B and H in the direction
parallel to the applied stress. A general case of 2D magnetic
measurements is presented in Fig. 5.1.
Theoretically, there is an angle between the vector of applied
field H and the vector of magnetic flux density B. This angle can be
nonzero, which is caused by the fact that electrical steels are
polycrystalline materials. In other words, the sum of the local
magnetization vectors of all domains can have an angle with the applied
field, as shown in Fig. 5.1. When uniaxial magnetic measurements are
performed in the 1D case, the actual B and H sensors measure the
projections of B and H vectors on the corresponding sensor axis (see Fig.
5.1), resulting in a simplification of eq. (1.6) to:
B* = µ0 (H*+M*)
(5.1)
Here, B* and H* correspond to the actually measured values of B(t) and
H(t) in the 1D case. The B and H sensors are assumed to be positioned
parallel to the direction of applied mechanical load, i.e. the 0-degree
direction. If the sensors have an angle with the 0-degree direction, a
measurement error occurs.
The above described simplification should be taken into account.
The difference between B and B* can result in an error as shown in Fig.
5.1. In order to avoid the error due to this simplification, it takes two
pairs of sensors for accurate measurements of the magnetization in the
sample. The latter will be explained below.
Fig. 5.1. The vector magnetization and the simplification at alternating field.
2D magnetic measurements under stress
157
Generally, if the angle between B and B* is small, the
simplification error is very small. This happens, for example, when the
field is parallel to the rolling direction of grain-oriented steel. Since the
easy axis in grain-oriented steels may be assumed parallel to the rolling
direction, the error of the simplification is acceptable, and therefore,
often neglected. In non-oriented steels, this error is rarely observed due
to the much smaller anisotropy in these materials. When the field is not
parallel to the rolling direction, or when the field vector is rotating, a
more accurate consideration of the vectors must be applied.
Two projections of the magnetization vectors to the X and Y axis,
obtained from the sensors, are sufficient to reconstruct the vectors in the
plane of the sheet:
⎡ B x (t ) ⎤
⎡ H x (t ) ⎤
B(t) = ⎢
, H(t) = ⎢
⎥
⎥
⎣ B y (t )⎦
⎣ H y (t )⎦
(5.2)
The measurement technique of the present study is based on the two
sets of sensors in the rolling (0-degree) and transverse (90-degree)
directions, see Chapter 3. Thus, in order to build a vector for any
considered 2D magnetic conditions, the two pairs of signals Bx(t), By(t)
and Hx(t), Hy(t) should be considered together according to eq. (5.2),
resulting in the B and H vectors, respectively.
In the 1D case, only the signals corresponding to the Bx(t) and
Hx(t) are considered, hence, the projections of the vectors on the Y axis in
eq. (5.2) are assumed zero. The error of the 1D case can be verified by
means of 2D magnetic measurements with four signals for the same
grain-oriented steel as in Chapter 4, see section 5.2.1.
5.2. 2D magnetic measurements in electrical steels.
Before continuing with the actual 2D measurements, it is critical to
determine the experimental procedures and the operational algorithms
in a similar way as in section 4.1.
Two principally different 2D magnetic conditions can be created:
an alternating magnetization in any arbitrary direction of the 2D plane
and a rotational magnetization in 2D. The first condition will be referred
to as "2D alternating measurements", and the second condition will be
referred to as "2D rotational measurements".
A particular uniaxial case of 2D alternating magnetization has
158
Chapter 5.
been studied in the previous chapter, i.e. when field and mechanical
load are applied in the rolling direction of electrical steels. An extensive
experimental study has been performed at various magnetic conditions.
Here, the research area includes the previous 1D case as well as other
alternating magnetic conditions, when the angle between the applied
mechanical load and the applied magnetic field varies.
According to Chapter 3, the 2D magnetic measurements are
performed with the four signals, corresponding to Bx(t), By(t) and Hx(t),
Hy(t), in comparison with only two signals used in the 1D case,
corresponding to Bx(t) and Hx(t).
Furthermore, in order to carry out 2D magnetic measurements,
two samples of the same material are used as described in Chapter 3.
The use of two samples ensures that both samples are in the same
magnetomechanical conditions.
The operational procedure resembles the one used in the 1D case
to some extent. In the hardware part, two output channels of the dataacquisition card are used instead of one, as well as two sets of power
amplifiers, corresponding to the X and Y axes, respectively. From an
operational point of view, the major difference is one more operational
variable that is added to the procedure presented in Fig. 4.2.
The new parameter is the angle between the direction of applied
field and the direction of applied mechanical load, ranging from 0 to 90
degrees. Hereafter, the direction of applied mechanical load is called the
"0-degree direction", whereas the perpendicular direction, or the
transverse direction is called the "90-degree direction".
The operational algorithm of the 2D alternating measurements is
shown in Fig. 5.2. The angle can vary with a fixed step, for example, 15,
30 or 45 degrees, so as to create an alternating magnetic field in a range
of angles between field and stress. The elementary cycle is basically the
same as in the 1D case, except for the waveform control procedure that
works for the B waveforms in the two axes instead of one.
The procedure for 2D rotational magnetization is even more
complex than the one for 2D alternating magnetization. Fig. 5.3 depicts
the operational algorithm for 2D rotational magnetic measurements
under stress. In fact, the procedures in Fig. 5.2 and Fig. 5.3 can be
combined in order to obtain virtually any 1D and 2D magnetic
condition, including any axis ratios and arbitrary angles between field
and stress. Hereafter, the axis ratio is the ratio between the minor and
the major axes of ellipse.
2D magnetic measurements under stress
159
The 2D rotational case is the most complex condition, which may
include features of 1D and 2D alternating conditions, see Fig. 5.3. If the
axis ratio is zero and the angle between field and stress is zero, then we
have the 1D case as in Chapter 4. If the axis ratio is zero but the angle is
nonzero, then we have the 2D alternating case. An elliptical field is
created when both the axis ratio and the angle between the major
elliptical axis (MEA) and stress are nonzero. If the axis ratio is equal to 1,
elliptical magnetization becomes circular rotational magnetization.
Fig. 5.2. The algorithm of 2D alternating magnetic measurements under stress.
Fig. 5.3. The algorithm of 2D rotational magnetic measurements under stress.
160
Chapter 5.
The more complex the operational procedure, the more time
consuming the magnetic measurements. To remind the reader, each
mechanical load is applied manually, whereas a set of strictly controlled
magnetic measurements can be done automatically. For 20 different
stresses it might take about 5 frequencies, 2 rotational directions, 5 axis
ratios, 7 angles and 5 induction levels. Thus, 35,000 magnetic
measurements would be required for a complete characterization of a
given electrical steel under the range of stresses and plastic strains.
However, it would take a large amount of time to perform these 1D and
2D magnetic measurements for just one material.
Does the target study require that large number of magnetic
measurements?
Instead of using the complete range of possibilities of the
developed measurement system and, of course, a range of features of the
created operational software, the target of the studied 2D magnetic
measurements can be carefully re-defined.
What conditions are of greater interest to this study?
Based on the conclusions of Chapter 2, the following 2D working
conditions are of practical or theoretical interest under applied stress.
For grain-oriented electrical steels, the interest lays in 2D
rotational conditions, elliptical or circular, for example, present in T-joint
regions of transformers. For non-oriented steels, both the 2D alternating
case and the 2D rotational case at the axis ratios up to 0.5 are present in
electrical machines. Based on the above, the following conditions are the
chosen targets of the present study:
- an alternating magnetization in the 0-degree and the 90-degree
directions,
- a circular rotational magnetization.
The first target condition can result in a complete characterization of the
2D behaviour of the magnetic properties of non-oriented steels at
alternating magnetization under applied uniaxial stress. Grain-oriented
steels can be skipped due to rare application of the 2D alternating
conditions, since the magnetic field is usually applied along the rolling
direction of grain-oriented steel.
For the second target, any elliptical condition can be considered
as a less extreme condition in comparison with a circular one that can be
considered as the ultimate 2D case. This is of great interest for both nonoriented and grain-oriented steels.
2D magnetic measurements under stress
161
5.2.1. Justification of the 1D measurements in Chapter 4.
Taking vector magnetization into consideration, the B and H loci can be
obtained for the same grain-oriented steel as in Chapter 4, see Fig. 4.3
and 4.4.
Fig. 5.4a depicts half (the first and the fourth quadrants) of the H
loci under compression of 10 MPa, stress-free state and tension of 30
MPa. Fig. 5.4b depicts half of the B loci under these conditions. Clearly,
the previous experimental results were obtained with an error due to the
assumption described in section 5.1, i.e. when the magnetic field is
applied parallel to the rolling direction, the flux density vector has an
angle with the applied field. Even in a stress-free case, the angle αfree
between the field and the induction is about 4 degrees, which results in
an error between the lengths of the vectors that is less than 0.3%.
At a compressive stress of 10 MPa, the angle αcompres between the
field and the magnetization is about 18 degrees, which results in about
5% error. The 5% error means that the controlled induction was around
1.1% higher than the required one. The 1.1% higher induction leads to
approximately 2% higher energy loss. Thus, the energy loss under
compression would be lower than the one depicted in Fig. 4.4.
Of course, these errors are not the only ones that can occur
during the magnetic measurements. There are other sources of errors,
i.e. errors due to sensors and their positions, amplifiers, data-acquisition
card, etc. Here, the simplification error can be avoided by a proper
measurement technique, i.e. by means of two sets of sensors.
A couple of interesting observations can be made from the H loci
at a tensile stress of 30 MPa in the grain-oriented steel. Firstly, the field
required to maintain the peak induction under uniaxial tension is
smaller than the field in a stress-free condition, as shown in Fig. 5.4a. On
the contrary, the applied compression requires a much larger field in the
direction of applied stress in order to maintain an alternating
magnetization in the RD. This leads to a lower permeability at
compression.
Secondly, the B and H vectors become parallel to each other
under uniaxial tension. In other words, uniaxial tension in the RD of
grain-oriented steel leads to aligning of the magnetization vector M in
the RD, see Fig. 5.1. In other words, if the vector M becomes parallel to
the direction of tensile stress that is applied in the rolling direction of
grain-oriented steel, the B and H vectors become parallel as well.
162
Chapter 5.
a)
b)
Fig. 5.4. The first and fourth quadrants of loci: (a) the H loci, (b) the B loci.
2D magnetic measurements under stress
163
Thus, uniaxial magnetic measurements in grain-oriented steels can be
justified for the case of uniaxial tension only. An uniaxial compression as
well as a stress-free condition requires 2D magnetic measurements with
two sets of sensors.
5.2.2. Alternating magnetization at 0 and 90 degrees with uniaxial
stress.
Being justified in the previous section, the 1D magnetic measurements
uniaxial with the applied stress can be easily expanded by means of 2D
alternating magnetic measurements at arbitrary angle with stress.
When measurements are performed in two axes in accordance
with eq. (5.2), the alternating magnetic field can be created in an
arbitrary direction in the plane of the sheet. Any angle between the
alternating magnetic field and the applied mechanical load can be easily
created and studied in the magnetomechanical setup of EELAB. There
have been previous attempts to study the multidirectional behaviour of
the magnetic properties of non-oriented steel under uniaxial stress
[Pulnikov04]. The local SST was used to create the magnetic field in a
few directions, such as 0, 30, 60, 90 degrees with the stress. The present
study applies the principle used in the rotational SST [Makaveev03] as
discussed in Chapter 3.
Based on the theoretical approaches from Chapter 4, it is possible
to predict the behaviour of energy loss in arbitrary directions of nonoriented steel. Since an uniaxial tension leads to a better magnetic
condition, a magnetization in the transverse direction, i.e. at 90 degree
with the stress, would result in worse magnetic properties. There must
be some kind of symmetry between applied stress in one direction and
applied compression in a perpendicular direction [Pulnikov04].
Does this prediction actually take place in non-oriented steels?
To answer this question, it is sufficient to carry out the magnetic
measurements in two directions with respect to the stress, the 0-degree
direction and the 90-degree direction.
A set of magnetic measurements was carried out at peak
inductions of 0.6, 0.8, 1.0, 1.2 T and for a range of elastic stresses from
compression of 50 MPa to tension of 200 MPa, as shown in Fig. 5.5. Only
a selection of the experimental results is presented here.
164
Chapter 5.
Fig. 5.5. The power losses at 0 and 90 degrees with elastic stresses.
Indeed, qualitatively the above prediction does take place at small elastic
stresses. The stress is applied only in the rolling direction of the steel
sample, i.e. the 0-degree direction for the alternating magnetic field. The
energy loss in the 0-degree direction under compression increases. A
similar increase can be observed for the loss in the 90-degree direction
under tension, see Fig. 5.5. A slight decrease of energy loss can be
observed both in the 0-degree direction under a small tension and in the
90-degree direction under a small compression. Although a first look
gives the impression of symmetry, there is no exact symmetry after all.
Based on the theoretical explanations presented in Chapter 4, the
following reasoning argues against the symmetry of the stress
behaviour. The mechanisms behind the stress effects are the "effective
pressure" on the 90-degree domain wall movement at tension [Bulte02]
and the mechanism of domain rotation at compression [Moses80], as
described in Chapter 4. Hence, when the tension is applied in the RD,
the magnetic properties under arbitrary alternating magnetization can
be explained by the effect on 90-degree domain wall movement.
2D magnetic measurements under stress
165
At tension, the uniaxial magnetization is preferable due to a
combined effect of both stress and field towards the growth of the
favourable domains and the shrinking of the 90-degree domain walls.
When the magnetization is in the transverse direction to the stress, the
stress and the field act in opposite directions. The stress is unfavourable
to the alternating magnetization in the 90-degree direction, resulting in a
higher field required for the growth of transverse domains and the
magnetization in the 90-degree direction.
At compression, the applied stress leads to domain rotation to
the transverse direction, which results in the decrease of energy loss for
the 90-degree magnetization. However, this decrease is smaller in
comparison with the decrease at uniaxial tension. Furthermore, the
energy loss for the 90-degree magnetization increases after the critical
compressive stress of 30 MPa. All this can be partly explained by the fact
that the considered non-oriented steel is not isotropic. The non-oriented
steel has a texture, which makes the RD more favourable for
magnetization.
Fig. 5.6. The power losses at 0 and 90 degrees up to the elastic limit.
166
Chapter 5.
Taking into account the whole range of elastic stresses, as shown in Fig.
5.6, the stress dependence once again has some new features.
Below the critical tensile stress of 50 MPa, the energy loss in the
0-degree direction decreases and the one in the 90-degree direction
increases. Above the critical tensile stress, the loss behaviour is similar at
0 and 90-degree magnetizations, i.e. the energy loss increases for both
directions. Due to the absence of a hard direction in non-oriented steels,
the loss at 2D alternating magnetization between 0 and 90 degree will be
higher than a 0-degree loss and lower than a 90-degree loss. This has
been experimentally verified, considering a few measurement points at
different angles of 15, 30, 45, 60, 75 degrees with stress.
The loss increases gradually from the critical tensile stress up to
the elastic limit. Based on the approach in Chapter 4, the loss increase
above the critical stress is likely to be caused by the grain boundaries
and impurities acting as pinning centers.
One interesting fact is that the energy loss for the 90-degree
magnetization has a reduced rate of increase above the critical tensile
stress. The rapid increase of the power loss at tensile stress less than the
critical stress was due to the "effective pressure" on the 90-degree
domain walls (see Chapter 4), resulting in an elimination of these walls
from the domain structure.
The mechanism of a smaller slope of increase after the critical
stress is less obvious. It seems that at the critical tensile stress there can
be a certain level of "critical energy" required in order to create some
necessary amount of 90-degree domain walls. Hence, a further increase
of uniaxial loss in the 0-degree direction can be due to magnetostatic
energy build-ups at grain boundaries.
An increase of energy loss at 90-degree magnetization with a
smaller slope of increase can be also explained as the sum of the critical
energy plus the effect of grain boundaries as well as impurities acting as
pinning centers for domain wall motion.
5.2.3. Circular rotational magnetization.
The 2D alternating magnetization seems to extend the knowledge about
the stress effect on magnetic anisotropy in electrical steels. The 2D
rotational magnetic measurements brings us to the next level. In fact, the
actual local working conditions in electrical machines and transformers
do not always correspond to the alternating magnetization.
2D magnetic measurements under stress
167
(e)
Fig. 5.7. a) B loci, b) H loci, c) BH at X, d) BH at Y, e) angle [Makaveev03].
To study the effect of rotational magnetization on the magnetic
properties of electrical steels, the B and H loci are usually obtained and
analyzed. According to section 5.1, the B and H loci are the result of the
time elimination from Bx(t), By(t) and Hx(t), Hy(t).
In fact, when there is a rotation of the B and H vectors, the
acquired signals of Bx(t), By(t) and Hx(t), Hy(t) have no physical basis. To
explain that, Fig. 5.7 depicts an example of circular rotational
magnetization at 50 Hz [Makaveev03]. The B vector is rotating with a
constant peak induction level of 1.3 T, see Fig. 5.7a. In comparison with
the circular B loci, the H loci is not at all a perfect circle, as shown in Fig.
5.7b. This is a good example of the fact that there is a texture in nonoriented steels. The interesting feature is that the BH loops shown in Fig.
5.7c,d are very different from typical BH loops. They are only the
projections of the B and H vectors to the X and Y axes, and therefore,
have no physical meaning as magnetization loops.
Therefore, the main tools to make an observation of rotational
magnetization are the B and H loci as well as an angle shift Θ between
the B and H vectors, as shown in Fig. 5.7e. Since the induction is
controlled during the magnetic measurements, the objective is to create
and control a rotating magnetic flux density vector. The output will be
the field loci, obtained at different magnetomechanical conditions.
168
Chapter 5.
Fig. 5.8. The H loci for grain-oriented steel at elastic stresses.
At small elastic stresses, the H loci at circular magnetization are
presented in Fig. 5.8 and Fig. 5.9 for the above considered grain-oriented
and non-oriented steels, respectively. The similarity of the results for
different electrical steels shows the overall tendency of the stress effect
under a circular magnetization at small inductions.
At compression, a larger field is required in the direction of
applied stress to maintain the circular rotation of the B vector. At
tension, a larger field is required in the transverse direction, and a
smaller field is required in the direction of tensile stress.
A larger tensile stress leads to a drastic increase of the field in the
90-degree direction in electrical steels, see Fig. 5.10 and Fig. 5.11. Fig.
5.11 also depicts the two cases of tensile plastic strains of 4.5% and 15%,
which correspond to the two largest H loci.
Fig. 5.12 depicts the H loci at an induction of 1.25 T. The nonsymmetry can be explained by measurement errors such as the limits of
the data-acquisition card. The interesting observation can be made that
there is a stretching of the H loci in the direction of the applied
compression or tension. When a compression of 10 MPa is applied, the
shape of the H loci is compressed along the 90-degree direction (TD) and
widen along the 0-degree direction (RD). At a tension of 10 MPa, the
2D magnetic measurements under stress
169
shape of the H loci is stretched along the 90-degree direction and shrunk
along the 0-degree direction.
Fig. 5.9. The H loci for non-oriented steel at elastic stresses.
Fig. 5.10. The H loci for grain-oriented steel at tensile stresses.
170
Chapter 5.
Fig. 5.11. The H loci for non-oriented steel at tensile stresses and plastic strains.
Fig. 5.12. The H loci at high induction level for grain-oriented steel.
2D magnetic measurements under stress
171
5.2.4. Energy loss at 2D magnetization under stress.
It is generally agreed that the rotational loss is smaller than or equal to
the sum of the alternating losses in the RD and the TD in non-oriented
electrical steels [Makaveev03]. In fact, at low inductions the rotational
loss is equal to this sum. Thus, the rotational loss can be twice the
alternating losses, assuming isotropic properties. At higher induction
levels, the rotational loss turns out to be smaller than the sum.
Fig. 5.13 depicts the dependence of circular rotational loss on the
induction levels in comparison with the alternating loss in the RD and
the TD as well as the sum of the alternating losses. Indeed, at an
induction level of 1.0 and 1.2 T the circular rotational loss is smaller than
the sum of the alternating losses. This general tendency is valid in the
stress-free condition as well as under a compression of 40 MPa and a
tension of 40 MPa, as shown in Fig. 5.13.
The stress dependence of the circular rotational loss is depicted
in Fig. 5.14, together with the energy loss in alternating magnetizations,
i.e. in the RD (0-degree) and in the TD (90-degree), similar to Fig. 5.6.
The same set of magnetic measurements was carried out for peak
induction levels of 0.6, 0.8, 1.0, 1.2 T and for a range of elastic stresses
from a compression of 50 MPa to a tension of 200 MPa.
Fig. 5.13. The energy losses vs. induction. SUM: RD + TD, CIR: circular loss.
172
Chapter 5.
Fig. 5.14. The 2D energy loss at the range of tensile stresses.
The circular rotational loss at small compressive and tensile stresses
shows a small dependence on the stress. In fact, they almost resembles
the sum of the alternating losses in 0-degree and 90-degree directions.
Since the alternating losses for the considered non-oriented steel
are not symmetrical with respect to the stress-free condition, the sum is
also asymmetrical. The circular rotational loss is not symmetrical with
respect to the stress-free condition as well.
At higher compression, the circular loss exhibits a huge increase.
In fact, this increase starts at the critical compressive stress of 30 MPa,
depicted at 2D alternating characteristics in the 90-degree direction.
After this critical stress, the circular loss increases drastically. The sum of
the 2D alternating losses increases as well.
At small tension below the critical tensile stress of 50 MPa, the
circular rotational loss exhibits a slight reduction. Apparently, the sum
of the alternating losses is increasing at these stresses. It is likely that a
small tension leads to the most preferable condition not only for the
uniaxial alternating magnetization but also for rotational magnetization.
2D magnetic measurements under stress
173
Here, a moderate reduction of the circular rotational loss at small
tension has been observed for steel with an average grain size of 240 µm.
It is the same material as the one used in Chapter 4, see Fig. 4.35.
Other non-oriented steels can exhibit a slight increase of the
circular rotational loss at small tension. Thus, a slight decrease or
increase of the circular rotational loss between the critical compressive
and the critical tensile stresses depend on the material and in particular
on material properties such as the average grain size and other
parameters of the microstructure.
A most remarkable observation was made at a stress above the
critical tensile stress. An increase of the circular rotational loss at the
critical stress can be observed for all considered peak induction levels up
to 1.2 T. The circular rotational loss at high elastic stresses was found to
be larger than the sum of the alternating losses in the 0 and 90-degree
directions, see Fig. 5.14.
To explain this behaviour, the previously presented discussions
on the critical tensile stress can be evaluated. As it was stated above, the
critical tensile stress corresponds to the point of the "energy balance" in
the 1D case, see Chapter 4.
When the 2D alternating case was considered, the critical tensile
stress was connected to the "critical energy" that is required for
alternating magnetization in the direction perpendicular to the stress. At
circular rotational magnetization, it may take a similar kind of "critical
energy" to create new 90-degree domain walls in order to maintain the
rotational magnetization under the applied mechanical stress.
Moreover, the energy required to maintain the circular rotational
magnetization is much larger than the one required in the 90-degree
alternating magnetization. This can be explained by the fact that a much
larger 90-degree domain wall movement is required under rotational
magnetization then under alternating magnetization.
Thus, to compensate the effect of high elastic stresses, additional
energy is required to maintain the circular rotation of the B vector.
Above the critical tensile stress, the circular rotational loss
increases gradually, similar to the increase of the 0 and the 90-degree
alternating losses, and probably, due to the same mechanisms, i.e. an
increase of magnetostatic energy at grain boundaries as well as the
presence of impurities in the microstructure of electrical steels.
174
Chapter 5.
5.2.5. 2D magnetization under tensile plastic deformation.
The tensile plastic deformation at 2D magnetization should be
considered separately as in Chapter 4. A similar procedure can be
applied as described in section 4.1.3.
Fig. 5.15 depicts the circular rotational loss as well as the
alternating loss in the 0-degree (RD) and the 90-degree (TD) directions
under a range of tensile plastic strains up to destruction of samples.
Taking a look back at Fig. 5.11, it is obvious that the required field in the
RD is constant at elastic stresses and increases at 4.5% and 15% strains.
In comparison with the alternating loss, the circular rotational
loss at plastic deformation exhibits a drastic increase, as shown in Fig.
5.15. For example, at 1 T the increase of the circular losses from the
stress-free condition to the 10% elongation is almost by a factor of 4. The
sum of the alternating loss is at least 50 % lower than the circular
rotational loss under stress.
Similar to the procedure in section 4.1.3, the four steps are
applied here as well. A similar tendency is present, i.e. a fast increase of
loss at small strains, followed by a moderate increase of loss at higher
strains. However, if in the 1D case the loss "after release" was larger than
the one "under stress", the circular loss "after release" is smaller than the
one "under stress", as shown in Fig. 5.16. This can be explained by a
much stronger effect of the increased dislocation density on the circular
rotational magnetization than on the alternating magnetization.
Generally, the degradation of the magnetic properties with
plastic strain results from two effects: an increase of dislocation density
and the residual stresses. The first effect leads to the pinning of domain
walls against metallurgical defects, or imperfections. The second effect
leads to anisotropic internal stresses.
When stress is applied after release, the alternating loss decreases
since the applied stress compensates the internal residual stresses. The
increase of dislocation density has a large effect on the level of loss as it
is increased. However, once it is increased, the residual stresses are the
ones responsible for the stress effect observed at 1D alternating
magnetization. In similar conditions, the circular rotational loss increases
due to a compensation of the residual stresses only in the 0-degree
direction, whereas the rotating magnetic flux is affected by both residual
stresses and increased pinning.
2D magnetic measurements under stress
175
Fig. 5.15. The alternating and circular losses at plastic strains.
Fig. 5.16. The power losses "under stress" and "after release"(see 4.1.3).
176
Chapter 5.
5.3. 2D magnetomechanical effect in literature.
Generally, the 2D case covers all aspects of the magnetomechanical
behavior of electrical steels. Similar theoretical approaches can be used
in the 2D case: a microscopic, a domain level, a macroscopic. Many
aspects of 2D magnetomechanical effects have been mentioned above:
- the magnetomechanical effect explained in the 1D case,
- the critical compressive and tensile stresses at 2D magnetizations,
- circular rotational and 1D alternating magnetizations in grainoriented steels,
- circular rotational and 2D alternating magnetizations in nonoriented steels,
- the tensile plastic deformation in non-oriented steels,
- some special effects of alternating and rotational magnetizations.
Here, an extra domain observation found in literature might be of great
help. Fig. 5.17 depicts the variation of the domain pattern with applied
tensile stress and applied DC field in the transverse direction, i.e. the 90degree direction [Phillips74].
The application of a DC field in the 0-degree direction of grainoriented steel only accelerates or retards the domain wall motion and
does not change the observed domain patterns. However, the
application of a transverse DC field has the effect opposite to the
application of a longitudinal tensile stress (i.e. in the 0-degree direction)
on the domain structure, see Fig. 5.17.
The domain pattern under 0-degree AC magnetization at 0.5 T is
shown in Fig. 5.17a.
Fig. 5.17. Variation of domain pattern with applied stress and transverse DC
field at AC flux density of 0.5 T in the 0-degree direction: (a) no DC field, no
stress, (b) 0.5 T transverse field, no stress, (c) 0.75 T transverse field, no stress,
(d) 0.75 T transverse field, tensile stress 6 MPa, (e) 0.75 T transverse field,
tensile stress 8 MPa [Phillips74]. (The vertical line separates the domains.)
2D magnetic measurements under stress
177
The addition of a transverse DC magnetization of 0.5 T breaks up
the pattern along the vertical line shown in Fig. 5.17b. The new domain
pattern observed over part of the grain is similar to the compressive
stress [Phillips74]. A further increase of the DC transverse flux density to
0.75 T causes the complete break up of the original domain pattern, as
shown in Fig. 5.17c.
The application of the tensile stress of 6 MPa in the 0-degree
direction causes the original domain pattern gradually to reoccur, as
shown in Fig. 5.17d. The increase of the tensile stress to 8 MPa completes
the return to the original configuration, as shown in Fig. 5.17e. Thus,
based on the domain observation it can be concluded that the
application of tensile stress in the 0-degree direction has an opposite
effect on the domain configuration compared to the application of a DC
field in the 90-degree direction.
The above presented domain observation by [Phillips74] has
confirmed that the applied field and the applied stress have a similar
action towards the domain structure.
Hence, the following conclusions can be drawn for the 2D
alternating magnetizations under stress applied in the 0-degree direction
in electrical steels:
- if the stress and the field are coaxial, the tension has a positive
effect on the alternating 0-degree magnetization, while the
compression has a negative effect on the alternating 0-degree
magnetization,
- if the stress and the field have a 90 degree angle with respect to
each other, the tension has a negative effect on the alternating 90degree magnetization, while the compression has a positive
effect on the alternating 90-degree magnetization.
These conclusions confirm the stress dependence under 2D alternating
magnetization in non-oriented steel, presented in Fig. 5.5.
Apparently, the same study of [Phillips74] includes the effect of
small stresses on power loss. Fig. 5.18 depicts the variation of power loss
per cycle with applied stress under various conditions at 1.5 T in grainoriented steel. Based on the results shown in Fig. 5.18, tensile stress has a
little effect on the power loss, while compressive stress increases the
power loss [Phillips74]. The percentage increase in the power loss is
much greater under 0-degree alternating magnetization than under 90degree alternating magnetization or under rotational magnetization. The
application of an additional DC field always increases the power loss but
178
Chapter 5.
does not change the overall shape of the characteristics, see the broken
lines in Fig. 5.18.
Fig. 5.19 depicts the loss in grain-oriented steel magnetized at
different directions under compression and tension less than 8 MPa
[Moses81]. As the angle of the magnetization is increased steadily from 0
degree to 90 degrees, not only does the loss under 2D alternating
magnetization increase, reaching the peak at about 60 degrees, but the
shape of the stress characteristic shows a gradual change from
compression to tension. This dependence is somewhat similar to the one
obtained for non-oriented steel in the present study, see Fig. 5.5.
Fig. 5.18. Variation of loss per cycle with strain (60·10-6 ≈ 12 MPa) under
different fluxes at 1.5 T: a) 0-degree, c) 90-degree, e) circular [Phillips74].
Fig. 5.19. Variation of power loss with stress in the 0-degree direction, when
magnetic flux in: (a) 0, (b) 15, (c) 60, (d) 90-degree [Moses81].
2D magnetic measurements under stress
179
5.4. Ratios between rotational and alternating losses.
An interesting parameter for characterization of energy loss in electrical
steels has been introduced recently in [Appino97], [Dupre00]. The
parameter defines the ratio between the rotational energy loss at various
axis ratios to the alternating energy loss.
Apparently, the ratio between rotational and alternating energy
losses depends on the induction level. For non-oriented steel, the ratio is
descending with higher induction levels [Appino97]. Furthermore, at
induction levels above 1.5 T the ratio can be below 1 in some elliptical
cases and especially in the circular rotational case. For grain-oriented
steel, the ratio can increase with induction level for the case of
alternating magnetization in the 90-degree direction [Dupre00]. The
ratios between circular loss and alternating loss in the 45, 30, 15, 0degree directions are decreasing with induction level similar to the ratios
in non-oriented steels.
When stress is applied in non-oriented steel, it changes the
magnetic anisotropy of the material. It can be presumed that the stress
dependence of the ratio between circular and alternating loss in different
directions will depend on the stress as well as on the induction level.
Similar to the V0 parameter that was extended by a stress dependence in
Chapter 4, the stress dependence of the ratio between rotational and
alternating loss can be determined as well.
For alternating magnetizations in the 0-degree and 90-degree
directions, the ratios of circular loss to alternating loss are given by:
R0 = Wc / Wa0 ,
R90 = Wc / Wa90
(5.1)
Here, Wc or Wa is the energy loss at circular or alternating
magnetizations, respectively.
Fig. 5.20 depicts the ratios R0 or R90 under small compression and
tension in the non-oriented steel, considered in the present chapter. The
known tendency of induction dependence of the ratio is a descending
tendency at stress-free condition, as shown by the ellipse in Fig. 5.20.
The ratio R0 between the circular loss and the alternating loss in
the 0-degree direction is slightly increasing from applied compression to
tension below 40 MPa. On the contrary, the ratio R90 between the circular
loss and the alternating loss in the 90-degree direction is decreasing from
applied compression to tension. Furthermore, the ratio R90 becomes less
than 1 at a tensile stress of 25 MPa and an induction of 1.2 T. In other
180
Chapter 5.
words, at certain tensile stress and induction levels the circular
rotational loss becomes smaller than the alternating loss in the direction
perpendicular to the applied tensile stress in non-oriented steel.
Thus, when tensile stress is applied in the considered nonoriented steel, a lower induction than the one in the stress-free state is
required to achieve the condition, when the circular rotational loss is
lower than the alternating loss in the 90-degree direction.
known effect
of descent
Fig. 5.20. The ratios R0 and R90 under small stresses in non-oriented steel.
5.5. Conclusions.
The presented 2D magnetomechanical measurements complete the
picture of the stress effect on 2D magnetic properties of electrical steels.
We recall that the stress is always applied in the 0-degree direction.
The following information has been confirmed and/or discovered
by means of 2D alternating magnetic measurements:
- applied stress and applied field have a similar effect on the
domain structure of electrical steel, the actions of field or stress
are either in the same or in the contrary sense,
2D magnetic measurements under stress
181
applied compression is qualitatively similar to an applied field in
the 90-degree direction; applied tension is qualitatively similar to
an applied field in the 0-degree direction; the actual values of loss
under stress depend on the material and can be non-symmetrical
with respect to the stress-free condition,
- high compression leads to worse magnetic properties at any
arbitrary direction,
- a critical compressive stress can be observed as well as a critical
tensile stress, the first one occurs at the 90-degree magnetization,
the second one occurs at the 0-degree magnetization, both are
useful for only specific magnetizations,
- high elastic tension leads to an increase of loss due to increased
magnetostatic energy at the grain boundaries and the impurities
such as carbon, energy loss in any arbitrary direction increases
after the critical tensile stress,
- tensile plastic deformation is characterized by a higher
dislocation density and residual stresses, which are partly
compensated by the applied tension in the 0-degree direction, so
that the energy loss in the 90-degree directions is higher than the
energy loss in the 0-degree direction.
The following information has been added based on 2D rotational
magnetic measurements:
- H loci at rotational magnetic flux describe the stress behaviour; at
small stresses, tension leads to a higher field required in the 90degree direction, while compression leads to a higher field
required in the 0-degree direction; at high tensile elastic stresses
more stress means more field is required, at plastic deformation
both fields in the 0-degree and 90-degree directions are
increased,
- rotational loss at small compression and tension is less
dependent on stress in comparison with the alternating loss in
the 0 or 90-degree directions, at compression the loss is observed
to be higher than at tension,
- rotational loss above the critical compressive stress (depicted by
2D alternating magnetization in the 90-degree direction)
increases drastically, below the critical tensile stress (depicted by
2D alternating magnetization in the 0-degree direction) reduces
at inductions below 1.2 T,
- rotational loss at the critical tensile stress exhibits an increase at
-
182
Chapter 5.
all inductions, and at high tensile elastic stress exhibits a
moderate increase similar to the one observed for alternating
losses in the 0-degree and 90-degree directions,
- rotational loss at plastic deformation increases drastically in
comparison with the alternating loss; a similar tendency of the
fast increase of loss at small plastic strains is observed for
alternating and rotational losses; rotational loss after release is
lower than under elastic stress in contrast to a higher loss after
release under the 0-degree and 90-degree alternating
magnetizations,
- rotational loss at plastic deformation can be more than twice the
alternating loss at plastic deformation, and more than 4 times the
standard alternating loss of stress-free material.
The parameter characterizing the 2D behaviour of the material under
alternating and rotational conditions, called the ratio between rotational
and alternating losses, usually depends on the induction level.
It has been discovered that this parameter also depends on the
applied stress. In the considered non-oriented steel, the ratio between
the circular rotational loss and the alternating loss in the 0-degree
direction slightly decreases or increases at applied compression or
tension below 40 MPa. However, the ratio between the circular
rotational loss and the alternating loss in the 90-degree direction
decreases from compression to tension so that it becomes less than 1 at a
tensile stress of 25 MPa and an induction of 1.2 T.
Stress effects in electrical steels.
183
Science may set limits to knowledge,
but should not set limits to imagination.
Bertrand Russell (1872 - 1970)
CHAPTER 6. STRESS EFFECTS IN ELECTRICAL STEELS.
The main question of the present study is: How an externally applied
mechanical stress affects the behaviour of different types of electrical steels?
Electrical steels are the most common soft magnetic material
used in various electromagnetic devices. From the practical point of
view, energy transformation, energy savings, energy efficiency – those are the
subjects being targeted by the present study.
A brief historical introduction into the subject of this research has
been made in Chapter 1. The practical approach to the subject has been
evaluated in Chapter 2 with an extra study of working conditions of
electrical steels in induction machines. In fact, the choice of further
subjects was made based on the conclusions of Chapters 1 and 2. Then,
the art of magnetic measurements has been presented in Chapter 3,
followed by the choice of the measurement techniques.
Thus, the present research has been performed by means of the
2D magnetomechanical system with uniaxial mechanical load, specially
created in EELAB (UGent). The system is a unique combination of the
past expertise in the 2D magnetic measurements and newly developed
possibilities to apply uniaxial compressive or tensile mechanical load.
A synergy of magnetic and mechanical features, enforced by the
advanced operational software developed for the target study, has
resulted in a broad range of possible 2D magnetomechanical conditions
being created in electrical steels. Any 2D magnetic condition can be
strictly controlled by the developed system.
The 1D magnetic measurements presented in Chapter 4 provide
a broad characterization of grain-oriented, semi- and fully-processed
non-oriented electrical steels under applied compressive and tensile
stress and tensile plastic deformations.
Many tendencies of stress effects discovered in the past have
been confirmed. Some new stress effects and dependencies of the
material parameters have been discovered.
In fact, the 1D magnetomechanical characterization can be seen
as an extension of the standard magnetic measurements, performed by
means of the Epstein frame or the single-sheet tester (SST).
184
Chapter 6.
As an extension of standard methods, the present 1D magnetic
measurements provide some valuable information about the behaviour
of electrical steels under uniaxial stress. Indeed, the standard magnetic
measurements are unable to evaluate the stress dependence that is of
great interest because of the actual mechanical conditions of electrical
steels in electromagnetic devices.
The need for studying stress effects under uniaxial magnetization
has been successfully fulfilled in the present study. Although other
research in the past has been dealing with a similar problem of uniaxial
magnetomechanical conditions, the present study has many distinct
features. For example, a complete range of measurements at tensile and
compressive stresses can be performed in a single system. None of the
past systems has provided such a wide range of magnetomechanical
conditions. Another unique feature is the automatic operational
algorithm that allows strictly controlled magnetic measurements in
electrical steels under sinusoidal or distorted excitations.
The analysis of the 1D stress dependences has been naturally
built based on the domain theory. The chosen way "from an observation
to an explanation" has resulted in different levels of understanding of
the stress effects in electrical steels. A valuable source of information
was the domain observation in grain-oriented steel under applied stress
in a demagnetized state. Eventually, a physically sound explanation has
been found for many stress effects, observed in different electrical steels.
A further analysis of the discovered stress effects has lead to the
conclusion that the 1D case is not sufficient to explain the behaviour of
electrical steel under stress. A vector magnetization under stress,
discussed in Chapter 5, has shown that the stress effects must be studied
as a 2D case. The error that occurs due to the described simplification of
the vector magnetization in the 1D case can be large, considering the
alternating magnetization in the rolling direction of grain-oriented steels
under applied compressive stress.
Creating 2D magnetic conditions in this unconventional
magnetomechanical system has been a challenge. The waveform control
and the 2D magnetic excitation are facing a non-symmetry of the
magnetic system as well as various limitations of the hardware, such as
limits of the data-acquisition card, the power amplifiers and the signal
amplifiers. Despite all these challenges and limitations of the
magnetomechanical system, a broad range of 2D magnetomechanical
conditions was presented in Chapter 5.
Stress effects in electrical steels.
185
The 2D magnetization can be separated into the 2D alternating
case and the 2D rotational case. The first one is the case of an alternating
magnetization that has an angle with the direction of applied mechanical
load. The 1D case presented in Chapter 4 is in fact a particular case of the
2D alternating magnetization, when stress and field are in parallel.
In comparison with the 1D case, the 2D alternating case presents
the stress effect in the in-plane anisotropy in electrical steels. Generally, a
small tension, preferable in the 0-degree direction of non-oriented steel,
leads to a large deterioration of magnetic properties in the 90-degree
direction. On the contrary, a compression in the 0-degree direction leads
to slightly better magnetic properties in the 90-degree direction.
Although the stress seems to have a symmetrical effect on the
alternating magnetization in the 0 and 90-degree directions, it is only
qualitatively. The presence of texture in non-oriented electrical steels
leads to a difference in the stress effects by compression and tension.
The second case of 2D magnetic conditions is the 2D rotational
magnetization. A rotational magnetization under stress opens new
horizons on the effect of uniaxial stress in electrical steels. First of all, the
BH loops under rotational magnetization have no physical meaning.
However, the B and H loci can be used to analyze the behaviour of
electrical steels under stress and rotational magnetization. When rotation
of a vector B with a constant amplitude is controlled, the H loci
represents the field required for circular rotational magnetization. When
compressive stress is applied in the 0-degree direction, a higher field is
required in the 0-degree direction. When a tensile stress is applied in the
0-degree direction, a higher field is required in the 90-degree direction.
Various stress effects has been discovered during the present
study, such as the critical tensile stress, the critical compressive stress,
the cross points, the stress dependence of the parameters Wh and V0 and
the ratios R0 and R90.
In the following, some valuable conclusions of the present study
are collected. Some extensions of the present study are mentioned as
well, in order to define other related research subjects.
Finally, some recommendations are presented to help various
professionals to better understand the stress effects in electrical steels
under complex 1D and 2D magnetic conditions, as they are present in
actual electromagnetic devices. Some conclusions of the present study
can be helpful for further academic research in other soft magnetic
materials as well [Turgut03].
186
Chapter 6.
6.1. Conclusions of the present study.
The general conclusions of the present study can be separated into the
following topics:
- the analysis of the working conditions in electrical steels,
- the art of 2D magnetomechanical measurements,
- the 1D magnetization under uniaxial applied stress,
- the 2D alternating magnetization under uniaxial applied stress,
- the 2D rotational magnetization under uniaxial applied stress,
- the material parameters related to the microstructure,
- the general tendencies of stress effects in electrical steels.
6.1.1. The art of 2D magnetomechanical measurements.
The present research has been using the novel measurement setup,
created on the basis of international experience in advanced magnetic
measurements. The following conclusions can be drawn for the 2D
magnetomechanical measurements presented in this study.
- The novel magnetomechanical system and the operational
software developed in EELAB allows a broad range of complex
2D magnetomechanical conditions under applied uniaxial
mechanical load.
- The operational algorithm allows to carry out strictly controlled
magnetic measurements under manually applied load.
- The operational algorithm is capable to vary the following
parameters: the direction of rotation, the magnetizing frequency,
the axis ratio, the angle between the direction of alternating
magnetization (or the major axis of the ellipse) and the direction
of applied stress, the peak induction level.
- For each set of the above defined parameters, the elementary
cycle of the operational algorithm is repeated until a predefined
error between the measured flux waveform and the desired
sinusoidal (distorted) one is achieved, in practice, from 1% to 5%.
- The elementary cycle consists of a digital generation of the
output voltage, an amplification of the voltage by a power
amplifier, the magnetic measurements by the set of sensors, an
analogue amplification of the signals from the sensors, a dataacquisition of the amplified signals, a digital integration of the
signals, a construction of the BH loops and the update, etc.
Stress effects in electrical steels.
-
-
187
The demagnetization of the sample is performed after each
elementary cycle.
The overall error of the measurements depends on many factors,
such as the error of positioning of the sensors, the error of
calibration of the sensors, the error of signal transportation and
signal amplification, the error of data-acquisition, the error of
digital integration, the stability of the mechanical subsystem, the
error of the power amplifiers, the discretization error of the
operational algorithm, etc. Many efforts have been made to
reduce the overall error of the measurements.
Finally, statistical information on a number of data has been used
for further analysis of experimental results.
6.1.2. The 1D magnetomechanical measurements.
The 1D case of alternating magnetization under uniaxial stress can be
considered as an extension of the standard magnetic measurements
methods, such as the Epstein frame or the SST.
Here, one more variable is added into the consideration, i.e. an
uniaxial mechanical load applied to the sample of electrical steel.
The following conclusions can be drawn from the 1D case:
- Compressive stress leads to a drastic deterioration of the
magnetic properties.
- Grain-oriented steel is extremely sensitive to even a small
compressive stress.
- Tensile stress below the critical tensile stress improves the
magnetic properties in the direction of stress.
- Semi-processed steel is less sensitive to stress than a fullyprocessed steel.
- Annealed semi-processed steel is more sensitive to stress than an
unannealed steel.
- Tensile plastic deformation leads to a drastic deterioration for all
steels.
- The magnetic properties at plastic strains can be improved by
tensile elastic stress.
- A distorted magnetization exhibits a similar stress effect as a
sinusoidal one.
- An uniaxial compression and a plastic deformation are the worst
stress conditions for uniaxial alternating magnetization.
188
Chapter 6.
6.1.3. The 2D alternating magnetomechanical measurements.
A 2D alternating magnetization means the alternating magnetization in
a direction different from the direction of the applied stress. In fact, the
previously considered 1D case is a particular case of the 2D alternating
magnetization when field and stress are in parallel.
The following conclusions can be drawn from the 2D alternating
magnetization in the 0 and 90-degree direction with uniaxial stress:
- An applied stress can act in the same sense as the applied
magnetic field (i.e. helpful at uniaxial tension) or not.
- An applied compression is qualitatively similar to an applied
field in the transverse direction; an applied tension is
qualitatively similar to an applied field in the longitudinal
direction; the actual values of energy loss under stress depend on
the material and may not be symmetrical with respect to the
stress-free condition.
- A high compression leads to the worst magnetic properties at
any arbitrary direction under elastic stress.
- A critical tensile stress and a critical compressive stress are
observed at the 0 and 90-degree magnetization. The first one
occurs at alternating magnetization in the longitudinal direction,
the second one occurs at alternating magnetization in the
transverse direction. Both critical stresses indicate the best
magnetic conditions under uniaxial stress for the considered
electrical steel at the 0 and 90-degree alternating magnetization.
- A high elastic tension leads to an increase of energy loss due to
the presence of grain boundaries and impurities.
- The increased dislocation density at tensile plastic deformation
leads to a deterioration of magnetic properties in any arbitrary
direction in the plane of the sheet.
6.1.4. The 2D rotational magnetomechanical measurements.
The 2D rotational magnetization is the most general case of 2D magnetic
measurements, which includes any elliptical magnetization as well as
any alternating magnetization at an axis ratio equal to zero.
The circular rotation of the magnetic flux density vector is the
ultimate case of rotational magnetization. The following conclusions can
be drawn from the 2D circular rotational magnetic measurements:
Stress effects in electrical steels.
-
-
-
-
189
H loci at circular rotational flux describe the stress behaviour and
the change of anisotropy of the material. At small stress, tension
leads to a higher field required in the 90-degree direction, while
compression leads to a higher field required in the 0-degree
direction. At high tensile elastic stresses more stress means more
field is required. At plastic deformation both fields in the 0degree and 90-degree directions are increased.
The circular rotational loss at small compression and tension is
less dependent on stress in comparison with the alternating loss
in the 0 and 90-degree directions.
The circular rotational loss increases drastically above the critical
compressive stress. The circular rotational loss at the critical
tensile stress exhibits an increase at all inductions, and at high
tensile elastic stress exhibits a moderate increase similar to the
one observed for alternating losses in the 0-degree and 90-degree
directions.
The circular rotational loss at plastic deformation increases
drastically in comparison with the alternating loss. The circular
rotational loss at plastic deformation can be more than twice the
alternating loss, and more than 4 times the standard loss of
stress-free material, obtained by the Epstein frame.
6.1.5. The parameters and the stress effects.
In the present study, the stress dependences of the following parameters
have been discovered.
The first parameter is the hysteresis energy Wh, which strongly
depends on induction level but also on applied stress or plastic strain.
The tensile plastic deformation leads to a large increase of the hysteresis
energy due to the increased dislocation density in the material.
The second parameter is the V0 field, which is constant at low and
medium inductions in the stress-free condition. The V0 parameter that is
related to the microstructure has a very strong dependence on the
applied stress. In fact, it is the most stress sensitive parameter in
comparison with other parameters considered in the present study.
The third parameter is the ratio between the circular loss and the
alternating loss in the 0 and 90-degree directions. Some tensile stress
may lead to a ratio between the circular loss and the alternating loss
lower than 1 at a much lower induction than in the stress-free condition.
190
Chapter 6.
The stress effects depend on many features of the microstructure
such as the grain size, the texture, the impurities such as carbon and
nitrogen, the lattice defects or dislocations, the residual stresses, etc. The
following stress effects have been observed in electrical steels:
- An alternating magnetization in the 0-degree direction shows the
energy loss minima at the critical tensile stress of 40-60 MPa.
- An alternating magnetization in the 90-degree directions shows
the loss minima at the critical compressive stress of 20-40 MPa.
- The permeability reaches its maximum at the critical stress. At
the critical tensile stress, uniaxial magnetization leads to a
maximum permeability. At the critical compressive stress, the 90degree magnetization leads to a maximum permeability.
- The BH loops at alternating uniaxial magnetization under tensile
elastic stresses have two cross points in the second and the fourth
quadrants close to the remanence induction. In these points the
alternating magnetization is stress independent.
- The energy loss exhibits a fast increase at plastic strains below
10%, followed by a moderate increase at higher strains.
6.2. Extensions of the present study.
The following related subjects have been left out of the present study.
6.2.1. Residual stresses.
A side effect of the plastic deformation in electrical steels is the residual
stress. A change of magnetic anisotropy after application of tensile strain
can only be explained by the presence of the compressive residual
stresses in the material, which are much more dominant than other
metallurgical features.
A method has been found to estimate indirectly the value of the
residual stresses by means of magnetic measurements [Pulnikov04]. The
idea of the method is the following. Since the plastic deformation is
tensile, the residual stresses should be compressive. Thus, all it takes is
to define the additional tensile stress that compensate the residual
compressive stress. The validity of this method was confirmed by the Xray measurements that confirmed the presence of the compressive
residual stresses, acting in the direction perpendicular to the direction of
applied mechanical load.
Stress effects in electrical steels.
191
6.2.2. NDE: non-destructive evaluation.
The indirect estimation of the residual stresses is an example of nondestructive evaluation.
Based on the well-known stress effect confirmed in the present
study, the reduction of magnetostriction occurs in the plastically
deformed electrical steel. In fact, the same happens in any carbon steel,
used in construction, suspension bridges, steel reinforced concrete, etc.
A local NDE by means of a simple closed circuit sensor can detect the
deformed areas of steel wires and other steel constructions.
Another area of the indirect characterization of electrical steel by
means of magnetic measurements is the study of a cyclic load and the
corresponding fatigue. The change of the magnetic properties under a
cyclic mechanical load can bring information about the change of the
material properties in time [Vandenbossche04]. This work can be critical
for such areas as nuclear reactors, where the deterioration of the
materials in time may lead to a devastating result.
6.2.3. Magnetostriction.
The physical nature of the magnetomechanical effect in electrical steels
has been related to the "effective pressure" on the 90-degree domain
walls. The same 90-degree domain walls are well-known to be the ones
responsible for the magnetostriction of electrical steels.
Magnetostriction is a change of the dimensions of the sample due
to an applied magnetic field. In electrical steels, it is known that
magnetostriction can be either positive or negative, depending on the
induction level and the metallurgical features. If magnetized, electrical
steel expands at positive magnetostriction, whereas at negative
magnetostriction it contracts. In high Si steels the magnetostriction is
negligible [RozYanez03].
Apparently, the magnetostriction is affected by stress. At the
critical tensile stress the magnetostriction in the direction of applied
mechanical load can be reduced to zero [Pulnikov04].
6.2.4. Biaxial stresses.
Electrical steels used in electromagnetic devices can be subjected to
biaxial stresses. This subject is even more challenging than the present
192
Chapter 6.
study of the 2D magnetization under uniaxial stress. The measurement
system to create and control the biaxial stress is twice as complicated as
the present system. Despite the challenges, a very recent research has
been done in this area by LMT-Cachan, France [Hubert04].
6.3. Applications of this research.
When writing this text, one of the ideas was to make the study as
interesting as possible in terms of practical applications, such as the use
of electrical steels in induction machines and transformers.
6.3.1. Magnetomechanical measurements in soft magnetic materials.
The presented art of magnetomechanical measurements can be further
used for various specific tasks. Some stress effects such as the cross
points can be studied further in details.
Future research can be carried out for any of the above
extensions or may have as aim the application of higher magnetic flux
density which was limited by the large air gap in the present study.
6.3.2. Recommendations to manufacturers of electrical steel.
The following recommendations to steel manufacturers can be drawn.
The smaller the amount of impurities and various defects of
microstructure, the better the magnetic properties of electrical steels.
The tensile coating in grain-oriented steels is indeed very helpful
to improve the magnetic properties in the rolling direction.
The larger the grains, the larger the positive effect of tensile stress
on the magnetic properties in the direction of applied tension.
Annealing of semi-processed steels is a valuable method to
improve the magnetic properties of non-oriented steel.
Stress relief annealing might be useful after any plastic
deformation applied to a sheet of non-oriented steel.
6.3.3. Recommendations to manufacturers of power transformers.
The following recommendations to transformer producers can be drawn
as well.
The use of grain-oriented steel under in-plane compressive stress
Stress effects in electrical steels.
193
is not recommended under any circumstances. Even a small compressive
stress can lead to a drastic deterioration of magnetic properties of grainoriented steel. For example, the weight of the upper yoke of transformer
should be compensated. That is normally done by an extra frame
supporting the upper yoke.
The orthogonal stress applied to the stack of transformer
laminations can be considered as having similar effect as the tension
applied in the rolling directions.
Thus, the compressive stress in the rolling direction is the worst
condition for a transformer core. Any means necessary should be
applied to avoid these compressive stresses.
6.3.4. Recommendations to manufacturers of rotating machines.
The following recommendations to induction machine designers and
producers can be drawn.
The actual magnetic measurements should be performed not
only at standard conditions, but also at various 2D magnetomechanical
conditions if possible.
The effect of punching or cutting should be carefully taken into
account, since the material properties in the teeth area of the laminations
change after punching. Plastic deformation generally deteriorates the
originally good magnetic properties of electrical steels. Due to plastically
deformed edges of the laminations, the effective tooth width to be used
in computations can be considered smaller than the actual size of the
laminations. Furthermore, the effective air gap can be considered larger
than the actual air gap, due to the deformed edges of the laminations.
The introduction of a compressive stress leads to a deterioration
of magnetic properties of steel and to higher energy loss in the machine.
Therefore, it is recommended to avoid if possible any clamping or other
means of compression of the stator and the rotor core.
6.4. Which engineering choice of section 1.5 is better?
In section 1.5, the two common choices were described: "from worse to
better" and "from better to worse".
The first choice means the use of semi-processed steel in the
punching and the core building processes, followed by a multi-stage
annealing process to improve the magnetic properties of semi-processed
194
Chapter 6.
electrical steel. The second choice means the use of fully processed steel
in the punching and the core building processes. This leads to a
deterioration of the magnetic properties from the best ones.
Based on the presented stress effects, it can be concluded that the
choice for the improvement of the magnetic properties of plastically
deformed semi-processed steel is a better approach than the choice for
the deterioration of fully processed steel by punching. However, it all
depends on the actual procedures and the quality of the punching and
the core building.
The latest improvements in cutting technologies may result in
less harmful production processes for electrical machine manufacture
even if fully processed electrical steel is used.
6.5. Conclusions.
The present study has brought a general overview on the effect of
uniaxial stress on 1D and 2D magnetic behaviour of electrical steels.
Some known tendencies and stress effects have been confirmed
and extended. Novel stress effects have been observed, explained and
introduced to the scientific society.
The author hopes that further research will be carried out to
extend this study to other specific areas.
List of international publications
195
INTERNATIONAL PUBLICATIONS OF V. PERMIAKOV
SCI - journal papers
1
2
3
4
5
6
7
8
Permiakov V, Dupré L, Makaveev D, Melkebeek J
Dependence of power losses on tensile stress for Fe-Si
nonoriented steel up to destruction
J APPL PHYS 91 (10): 7854-7856 Part 3 MAY 15 2002
Poulnikov A, Permiakov V, De Wulf M, Dupré L, Melkebeek J
Investigation of residual effects of tensile stress on magnetic
properties of nonoriented electrical steel
IEEE T MAGN 38 (5): 3204-3206 Part 1 SEP 2002
Pulnikov A, Permiakov V, De Wulf M, Dupré L, Melkebeek J
Measuring setup for the investigation of the influence of
mechanical stresses on magnetic properties of electrical steel
J MAGN MAGN MATER 254: 47-49 Sp. Iss. SI JAN 2003
De Wulf M, Makaveev D, Dupré L, Permiakov V, Melkebeek J
Comparison of methods for the determination of dc-magnetic
properties of laminated SiFe alloys
J APPL PHYS 93 (10): 8543-8545 Part 3 MAY 15 2003
Permiakov V, Pulnikov A, Dupré L, De Wulf M, Melkebeek J
Magnetic properties of Fe-Si steel depending on compressive and
tensile stresses under sinusoidal and distorted excitations
J APPL PHYS 93 (10): 6689-6691 Part 2 MAY 15 2003
Dupré L, De Wulf M, Makaveev D, Permiakov V, Melkebeek J
Preisach modeling of magnetization and magnetostriction
processes in laminated SiFe alloys
J APPL PHYS 93 (10): 6629-6631 Part 2 MAY 15 2003
Dupre L, De Wulf M, Makaveev D, Permiakov V, Pulnikov A,
Melkebeek J
Modelling of electromagnetic losses in asynchronous machines
COMPEL, Vol. 22, No. 4, 2003.
Pulnikov A, Permiakov V, Petrov R, Gyselinck J, Langelaan G,
Wisselink H, Dupré L, Houbaert Y, Melkebeek J
Investigation of residual stresses by means of local magnetic
measurement
J MAGN MAGN MATER 272-76 (Part 3): 2303-2304, MAY 2004
196
9
10
11
12
List of publications
Permiakov V, Dupré L, Punikov A, Melkebeer J
Loss separation and parameters for hysteresis modelling under
compressive and tensile stresses
J MAGN MAGN MATER 272-276 (Sup. 1): E553-E554, MAY 2004
Permiakov V, Dupré L, Pulnikov A, Melkebeek J
Rotational magnetization in nonoriented Fe-Si steel under
uniaxial compressive and tensile stresses
IEEE T MAGN 40 (4): 2760-2762, JUL 2004
Permiakov V, Pulnikov A, Dupré L, Melkebeek J
2D magnetization of grain-oriented 3%-Si steel under uniaxial
stress
J MAGN MAGN MATER 290-291: 1495-1498, 2005
Pulnikov A, Decocker R, Permiakov V, Dupre´ V, Vandevelde L,
Petrov R, Melkebeek J, Houbaert Y, Gyselinck J, Wisselink H
The relation between the magnetostriction and the hysteresis
losses in the non-oriented electrical steels
J MAGN MAGN MATER 290-291: 1454–1456, 2005
Proceedings
1
2
Permiakov V, Pulnikov A, Makaveev D, De Wulf M, Dupre L,
Melkebeek J
Magnetic measurements under compressive and tensile stresses
for nonoriented electrical steel
Proceedings of 2DM'2002 (ISBN 3-89701-992-2): 15-21, June 2003
Permiakov V, Pulnikov A, Dupré L, Melkebeek J
2D Magnetic Measurements under 1D stress
Proceedings of 2DM'2004 (PL ISSN 0033-2097): 68-72, May 2005
List of Attended Conferences and Awards
INTERNATIONAL
CONFERENCES
ATTENDED BY V. PERMIAKOV:
197
AND
WORKSHOPS
15th Soft Magnetic Materials Conference
SMM'15
Bilbao, Spain, September 5-7, 2001
46th Annual Conference on Magnetism and Magnetic Materials
MMM'2001
Seattle, Washington, November 12-16, 2001
7th International Workshop on 1&2 Dimensional Magnetic Measurement
and Testing
2DM'2002
Ludenscheid, Germany, September 16-17, 2002
47th Annual Conference on Magnetism and Magnetic Materials
MMM'2002
Tampa, Florida, November 11-16, 2002
International Conference on Magnetism
ICM'2003
Rome, Italy, July 27-August 1, 2003
9th Joint MMM-Intermag Conference
MMM'2004
Anaheim, California, January 5-9, 2004
8th International Workshop on 1&2 Dimensional Magnetic Measurement
and Testing
2DM'2004
Ghent, Belgium, September 27-28, 2004
198
AWARDS OF V. PERMIAKOV:
Laureat Presentatieprijs
2nd UGent – FTW – PhD Symposium, 12 December 2001
Laureat Presentatieprijs
3rd UGent – FTW – PhD Symposium, 11 December 2002
List of Awards
Bibliography
199
BIBLIOGRAPHY
[Alves04]
F. Alves, K. Azoum, A. Bensalah, M. Besbes, F. Bouillault, M. Chaabane,
L. Daniel, O. Hubert, V. Maurel, " Magneto-mechanical behavior of FeCo alloys submitted to biaxial stresses," 2DM Workshop, Ghent, 2004.
[Appino97]
C. Appino, F. Fiorillo, A. M. Rietto, "The energy loss components under
alternating, elliptical and circular flux in non-oriented alloys,"
Proceedings of the 2DM'1997, September 1997, Grenoble, pp. 55-61.
[Baudouin02]
P. Baudouin, Effect of laser and mechanical cutting on the magnetic
properties of non-oriented electrical steels, PhD, Ghent University, 2002.
[Baudouin03]
P. Baudouin, M. De Wulf, L. Kestens, Y. Houbaert, "The effect of the
guillotine clearance on the magnetic properties of electrical steels,"
Journal of Magnetism and Magnetic Materials, Vol. 256, pp. 32-40, 2003.
[Beckley02]
P. Beckley, Electrical Steels for Rotating Machines, IEE, London, 2002.
[Bertotti88]
G. Bertotti, "general properties of power losses in soft ferromagnetic
materials," IEEE Transactions on Magnetics, Vol. 24, pp. 621-630, 1988.
[Bertotti98]
G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, 1998.
[Botta02]
O. Bottauscio, M. Chiampi, M. Zucca, "Experimental and numerical
investigations on rotational fluxes in stator cores of three-phase motors,"
IEEE Transactions on Magnetics, Vol. 38, pp. 3294-3296, 2002.
[Bozorth51]
R. M. Bozorth, Ferromagnetism, D. Van Nostrand Company, 1951,
reissued by IEEE Press, New York, 1993.
200
Bibliography
[Bulte02]
D.P. Bulte, R.A. Langman, " Origins of the magnetomechanical effect,"
Journal of Magnetism and Magnetic Materials, 251, pp. 229–243, 2002.
[Delage97]
M. Delage, Cl. Ramiarinjaona, J. B. Desmoulins, J. F. Rialland,
"Measurement of magnetic characteristics of ferromagnetic materials
along arbitrary directions under mechanical stress," IEEE Transactions
on Magnetics, Vol. 33, pp. 4038-4040, 1997.
[Devine96]
M. K. Devine, D. C. Jiles, "Composition dependence of the
magnetomechanical effect and magnetostriction", IEEE Transactions on
Magnetics, Vol. 32, pp. 4740-4742, 1996.
[DeWulf02]
M. De Wulf, Karakterisering en energieverliezen onder unidirectionele
magnetisatie in relatie tot de microstructuur van zacht magnetische
materialen, PhD, Ghent University, 2002.
[DeWulf03]
M. De Wulf, D. Makaveev, Y. Houbaert, J. Melkebeek, "Design and
calibration aspects of small size single sheet testers," Journal of
Magnetism and Magnetic Materials, Vol. 254-255, pp. 70-72, 2003.
[DeWulf04]
M. De Wulf, E. Hoferlin, L. Dupre, S. Jacobs, "Induction machine
modeling taking into account manufacturing processes," 2DM
Workshop, Ghent, 2004.
[Dupre00]
L. R. Dupré, F. Fiorillo, J. Melkebeek, A. M. Rietto, C. Appino, "Loss
versus cutting angle in grain-oriented Fe-Si laminations", Journal of
Magnetism and Magnetic Materials, Vol. 215-216, pp. 112-114, 2000.
[Dupre03]
L. Dupré, M. De Wulf, D. Makaveev, V. Permiakov, A. Pulnikov, J.
Melkebeek, "Modelling of electromagnetic losses in asynchronous
machines," COMPEL, Vol. 22, pp.1051-1065, 2003.
[Emura03]
M. Emura, F.J.G. Landgraf, W. Ross, J.R. Barreta, "The influence of
Bibliography
201
cutting technique on the magnetic properties of electrical steels", Journal
of Magnetism and Magnetic Materials, Vol. 254-255, pp. 358-360, 2003.
[Fiorillo04]
F. Fiorillo, Measurement and Characterization of Magnetic Materials,
Elsevier Academic Press, Amsterdam, 2004.
[Fujimura04]
H. Fujimura, H. Yashiki, H. Kojima, T. Nakayama, "Effect of stress due
to stamping and interlocking on magnetic properties of non-oriented
electrical steel sheet", SMM'16 Conference Proceeding, 2004.
[Fukuda81]
B. Fukuda, K. Satoh, T. Ichida, Y. Itoh, H. Shimanaka, "Effect of surface
coating on domain structure in grain oriented 3% Si-Fe", IEEE
Transactions on Magnetics, Vol. 17, pp. 2878-2880, 1981.
[Girgis98]
R. S. Girgis, K. Gramm, J. Sievert, M. G. Wickramasekara, " The single
sheet tester. Its acceptance, reproducibility, and application issues on
grain-oriented steel," Journal De Physique IV, Vol.8 (P2), pp. 729-732,
1998.
[Glew98]
C. N. Glew, Stray load losses in induction motors: a challenge to
academia, Power Engineering Journal, February 1998.
[Grimm78]
W. Grimm, W. Jillek, A. Hubert, "Measurement and consequences of the
stresses induced by coating layers onto oriented transformer steel",
Journal of Magnetism and Magnetic Materials, Vol. 9, pp. 225-228, 1978.
[Gyselinck00]
J. Gyselinck, Twee-dimensionale dynamische eindige-elementen
modellering van statische en roterende elektromagnetische
energieomzetters, PhD thesis, Ghent University, 2000.
[Hubert04]
O. Hubert, M. Chaabane, J. Jumel, V. Maurel, F. Alves, A. D. Bensalah,
202
Bibliography
M. Besbes, K. Azoum, L. Daniel, F. Bouillault, "Magneto-mechanical
behavior of Fe-Co alloys submitted to biaxial stresses," 2DM'2004, Ghent.
[IEC 404-2]
IEC 60404-2, Magnetic materials – Part 2: Methods of measurement of
the magnetic properties of electrical steel sheet and strip by means of an
Epstein frame, International Electrotechnical Commission, 1978.
[IEC 404-3]
IEC 60404-3, Magnetic materials – Part 2: Methods of measurement of
the magnetic properties of magnetic sheet and strip by means of a single
sheet tester, International Electrotechnical Commission, 1992.
[Iordache03]
V. E. Iordache, E. Hug, N. Buiron, "Magnetic behaviour versus tensile
deformation mechanisms in a non-oriented Fe-(3 wt.%)Si steel",
Materials Science and Engineering A, 359 (1-2): 62-74 OCT 25 2003
[Jiles91]
D. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman &
Hall, New York, 1991.
[Kochmann96]
T. Kochmann, "Relationship between rotational and alternating losses in
electrical steel sheets," Journal of Magnetism and Magnetic Materials,
Vol. 160, pp. 145-146, 1996.
[Landgraf00]
F. J. G. Landgraf, M. Emura, J. C. Teixeira, M. F. de Campos, "Effect of
grain size, deformation, aging and anisotropy on hysteresis loss of
electrical steels", Journal of Magnetism and Magnetic Materials, Vol. 215216, pp. 97-99, 2000.
[Lindenmo00]
M. Lindenmo, A. Coombs, D. Snell, "Advantages, properties and types
of coatings on non-oriented electrical steels", Journal of Magnetism and
Magnetic Materials, Vol. 215-216, pp. 79-82, 2000.
[Liorzou99]
F. Liorzou, D. L. Atherton, "Effects of compressive stress on a steel cube
Bibliography
203
using tensor magnetization and magnetostriction analysis," Journal of
Magnetism and Magnetic Materials, Vol. 195, pp. 174-181, 1999.
[LoBue00]
M. LoBue, C. Sasso, V. Basso, F. Fiorillo, G. Bertotti, "Power losses and
magnetization process in Fe}Si non-oriented steels under tensile and
compressive stress," Journal of Magnetism and Magnetic Materials, Vol.
215-216, pp. 124-126, 2000.
[Loisos01]
G. Loisos, A. J. Moses, "Critical evaluation and limitations of localized
flux density measurements in electrical steels," IEEE Transactions on
Magnetics, Vol. 37, pp. 2755-2757, 2001.
[Makaveev99]
D. Makaveev, M. De Wulf, J. Melkebeek, "Field homogeneity in a twophase rotational single sheet tester with square samples," Journal of
Magnetism and Magnetic Materials, Vol. 196-197, pp. 937-939, 1999.
[Makaveev00]
D. Makaveev, M. von Rauch, M. De Wulf, J. Melkebeek, "Accurate field
strength measurement in rotational single sheet testers," Journal of
Magnetism and Magnetic Materials, Vol. 215-216, pp. 673-676, 2000.
[Makaveev01]
D. Makaveev, J. Maes, J. Melkebeek, "Controlled circular magnetization
of electrical steel in rotational single sheet tester," IEEE Transactions on
Magnetics, Vol. 37, pp. 2740-2742, 2001.
[Makaveev03]
D. Makaveev, Karakteriseren en neural-netwerkgebaseerd modelleren
van rotationele magnetisatie in niet-georienteerd elektrisch staal, PhD
thesis, Ghent University, 2003.
[Mekhiche96]
M. Mekhiche, Th. Waeckerle, B. Cornut, "A metallurgical and magnetic
study of {100} textured soft magnetic sheets," Journal of Magnetism and
Magnetic Materials, Vol. 160, pp. 125-126, 1996.
[Moses80]
A. J. Moses, D. Davis, "Influence of compressive stress on magnetic
properties of grain-oriented silicon iron magnetized in various
204
Bibliography
directions," IEEE Transactions on Magnetics, Vol. 16, pp. 454-460, 1980.
[Moses81]
A. J. Moses, "Effect of stress on the magnetic properties of grain-oriented
silicon-iron magnetized in various directions," IEEE Transactions on
Magnetics, Vol. 17, pp. 2872-2874, 1981.
[Moses89]
A. J. Moses, H. Rahmatizadeh, "Effect of stress on iron loss and flux
distribution of an induction motor stator core," IEEE Transactions on
Magnetics, Vol. 25, pp. 4003-4005, 1989.
[Moses00]
A. J. Moses, N. Derebasi, G. Loisos, A. Schoppa, "Aspects of the cut-edge
effect stress on the power loss and flux density distribution in electrical
steel sheets," Journal of Magnetism and Magnetic Materials, Vol. 215216, pp. 690-692, 2000.
[Nakano99]
M. Nakano, H. Nishimoto, K. Fujiwara, N. Takahashi, "Improvement of
single sheet testers for measurement of 2-D magnetic properties up to
high flux density," IEEE Transaction on Magnetics, Vol. 35, pp. 39653967, 1999.
[Nakata84]
T. Nakata, "Numerical analysis of flux and loss distributions in electrical
machinery", IEEE Transaction on Magnetics, Vol. 20, pp. 1750-1755, 1984.
[Nakata86]
T. Nakata, N. Takahashi, Y. Kawase, M. Nakano, M. Miura, J. D. Sievert,
"Numerical analysis and experimental study of the error of magnetic
field strength measurements with single sheet testers," IEEE Transaction
on Magnetics, Vol. 22, pp. 400-402, 1986.
[Nakata92]
T. Nakata, M. Nakano, K. Kawahara, "Effects of stress due to cutting on
magnetic characteristics of silicon steel", IEEE Transactions on Magnetics
in Japan, Vol. 7, pp. 453-457, 1992.
[Nakata93]
T. Nakata, N. Takahashi, K. Fujiwara, M. Nakano, "Measurement of
magnetic characteristics along arbitrary directions of grain-oriented
silicon steel up to high flux densities," IEEE Transaction on Magnetics,
Vol. 29, pp. 3544-3546, 1993.
Bibliography
205
[Nencib96]
N. Nencib, A. Kedous-Lebouc, B. Cornut, "Performance evaluation of a
large single sheet tester," Journal of Magnetism and Magnetic Materials,
Vol. 160, pp. 174-176, 1996.
[Nippon92]
Non-oriented electrical steel sheets, Nippon Steel Corporation, 1992.
[Ossart00]
F. Ossart, E.Hug, O. Hubert, C. Buvat, R. Billardon, "Effect of punching
on electrical steels: experimental and numerical coupled analysis", IEEE
Transactions on Magnetics, Vol. 36, pp. 3137-3140, 2000.
[Permiakov02]
V. Permiakov, L. Dupre, D. Makaveev, J. Melkebeek, "Dependence of
power losses on tensile stress for Fe-Si nonoriented steel up to
destruction," Journal of Applied Physics, Vol. 91, pp. 7854-7856, 2002.
[Permiakov03]
V. Permiakov, A. Pulnikov, M. DeWulf, L. Dupre, J. Melkebeek,
"Magnetic properties of Fe-Si steel depending on compressive and
tensile stresses under sinusoidal and distorted excitations," Journal of
Applied Physics, Vol. 93, No. 10, pp. 6689-6691, May 2003.
[Permiakov04]
V. Permiakov, L. Dupre, A. Pulnikov, J. Melkebeek, "Loss separation and
parameters for hysteresis modelling under compressive and tensile
stresses," Journal of Magnetism and Magnetic Materials, Vol. 272-276,
pp. e553-e554, 2004.
[Pfutzner84]
H. Pfutzner, C. Bengtsson, M. Zehetbauer, "Effect of torsion on domains
of coated Si-Fe sheets", IEEE Transactions on Magnetics, Vol. 20, pp.
1554-1556, 1984.
[Pfutzner91]
H. Pfutzner, P. Schonhuber, "On the problem of the field detection for
single sheet testers," IEEE Transactions on Magnetics, Vol. 27, pp. 778785, 1991.
[Pfutzner94]
H. Pfutzner, "Rotational magnetization and rotational losses of grain
oriented silicon steel sheets – fundamental aspects and theory", IEEE
206
Bibliography
Transaction of Magnetics, Vol. 30, pp. 2802-2807, September 1994.
[Pfutzner00]
H. Pfutzner, "Different designs of rotational single sheet testers –
suggestions for procedures of calibration", Proceedings of 2DM'2000,
Bad Gastein, Austria, September 2000.
[Pfutzner04]
H. Pfutzner, G. Krismanic, "The needle method for induction tests:
sources of error," IEEE Transactions on Magnetics, Vol. 40, pp. 16101616, May 2004.
[Phillips74]
R. Phillips, K. J. Overshott, "Domain configuration under rotational flux
and applied stress conditions in silicon-iron", IEEE Transaction on
Magnetics, Vol. 10, pp. 163-165, June 1974.
[Pulnikov02]
A. Poulnikov, V. Permiakov, M. DeWulf, L. Dupre, J. Melkebeek,
"Investigation of residual effects of tensile stress on magnetic properties
of nonoriented electrical steel," IEEE Transactions on Magnetics, Vol. 38,
pp. 3204-3206, September 2002.
[Pulnikov03]
A. Pulnikov, V. Permiakov, M. DeWulf, L. Dupre, J. Melkebeek,
"Measuring setup for the investigation of the influence of mechanical
stresses on magnetic properties of electrical steel," Journal of Magnetism
and Magnetic Materials, Vol. 254, pp. 47-49, 2003.
[Pulnikov04]
A. Pulnikov, Modification of magnetic properties of non oriented
electrical steels by the production of electromagnetic devices, PhD thesis,
Ghent University, 2004.
[RosYanez03]
T. Ros Yanez, Innovative production methods for high silicon-steel, Ph.D
thesis, Ghent University, 2003.
[Sablik04]
M. J. Sablik, T. Yonamine, F. J. G. Landgraf, "Modeling plastic
deformation effects in steel on hysteresis loops with the same maximum
flux density," IEEE Transaction of Magnetics, Vol. 40, pp. 3219-3226,
September 2004.
Bibliography
207
[Schmidt75]
K. H. Schmidt, "Influence of punching on the magnetic properties of
electric steel with 1% silicon," Journal of Magnetism and Magnetic
Materials, Vol. 2, pp. 136-150, 1975.
[Schoppa99]
A. Schoppa, J. Schneider, J.-O. Roth, " Influence of the cutting process on
the magnetic properties of non-oriented electrical steels," Journal of
Magnetism and Magnetic Materials, Vol. 215-216, pp. 100-102, 2000.
[Schoppa00]
A. Schoppa, J. Schneider, C. Wuppermann, " Influence of the
manufacturing process on the magnetic properties of non-oriented
electrical steels," Journal of Magnetism and Magnetic Materials, Vol. 215216, pp. 74-78, 2000.
[Schoppa02]
A. Schoppa, J. Schneider, C. D. Wuppermann, T. Bakon, "Influence of
welding and sticking of laminations on the magnetic properties of nonoriented electrical steels," Journal of Magnetism and Magnetic Materials,
Vol. 254-255, pp. 367-369, 2003.
[Schoppa03]
A. Schoppa, H. Louis, F. Pude, Ch. von Rad, "Influence of abrasive
waterjet cutting on the magnetic properties of non-oriented electrical
steels," Journal of Magnetism and Magnetic Materials, Vol. 254-255, pp.
370-372, 2003.
[Shilling74]
J. W. Shilling, G. L. Houze Jr., "Magnetic properties and domain
structure in grain-oriented 3% Si-Fe," IEEE Transactions on Magnetics,
Vol. 10, pp. 195-223, 1974.
[Sievert84]
J. D. Sievert, "Determination of AC magnetic power loss of electrical
steel sheet: present status and trends," IEEE Transaction of Magnetics,
Vol. 20, pp. 1702-1706, 1984.
[Sievert96]
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", Journal of Magnetism and Magnetic Materials, Vol.
208
Bibliography
160, pp. 115-118, 1996.
[Sievert99]
J. Sievert, "The measurement of magnetic properties of electrical sheet
steel – survey on methods and situation of standards," Journal of
Magnetism and Magnetic Materials, Vol. 215-216, pp. 647-651, 1999.
[Sugimoto04]
S. Sugimoto, S. Urata, A. Ikariga, H. Shimoji, T. Todaka and M.
Enokizono, "A New Measurement Device for Two-Dimensional VectorMagnetic-Property in High Magnetic Flux Density Range," 2DM
Workshop, Ghent, 2004.
[Thomas80]
B. Thomas, "A quantitative assessment of the change in magnetic
properties of CoFeV alloy when subjected to flux and stress conditions
similar to those encountered in an aircraft generator," IEEE Transactions
on Magnetics, Vol. 16, pp. 1299-1309, 1980.
[Turgut03]
Z. Turgut, J. C. Horwath, M. Q. Huang, R. T. Fingers, "Magnetic
properties of FeCo laminates subjected to axial loading," Journal of
Applied Physics, Vol. 93, pp. 6692-6694, 2003.
[Vandenbossche04]
L. Vandenbossche, L. Dupre, J. Melkebeek, " Magnetic evaluation of
cyclic mechanical loading and fatigue damage progression using the
Preisach formalism," 2DM Workshop, Ghent, 2004.
[Vandevelde94]
L. Vandevelde, J.A.A. Melkebeek, "Theoretical and experimental study
of radial forces in relation to magnetic noise of induction machines,"
Proceedings of ICEM 1994, Vol. 3, pp. 419-424, 1994.
[Zhu02]
J. G. Zhu, J. J. Zhong, Z. W. Lin, J. Sievert, "3D magnetic property tester –
design, construction, and calibration, " Proceedings of the 7th
International Workshop on 1&2-Dimensional Magnetic Measurement
and Testing, September 2002, Ludenscheid, pp. 97-102.
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