tezĂ de doctorat - cern · efficiently if the magnetic hysteresis effects in the cores of the...

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Universitatea Transilvania din Brașov

Școala Doctorală Interdisciplinară

Departament: Inginerie Electrică și Fizică Aplicată

Ing. Valentin PRICOP

TEZĂ DE DOCTORAT

Conducător științific

Prof.univ. dr.ing. Gheorghe SCUTARU

BRAŞOV, 2016


Școala Doctorală Interdisciplinară

Departament: Inginerie Electrică și Fizică Aplicată

Ing. Valentin PRICOP

TEZĂ DE DOCTORAT

EFECTELE HISTEREZISULUI DIN MATERIALELE

FOLOSITE PENTRU CIRCUITELE MAGNETICE ALE

ACCELERATOARELOR DE PARTICULE

HYSTERESIS EFFECTS IN THE CORES OF PARTICLE

ACCELERATOR MAGNETS

Domeniul de doctorat: INGINERIE ELECTRICĂ

Comisia de analiză a tezei:

Conf.dr.ing. Carmen GERIGAN Președinte, Universitatea Transilvania din Brașov

Prof.dr.ing. Gheorghe SCUTARU Conducător științific,


Prof.dr.ing. Horia GAVRILA Referent oficial, Universitatea Politehnică din București

Prof.dr.ing. Gheorghe MANOLEA Referent oficial, Universitatea din Craiova

Dr.ing. Davide TOMMASINI Referent oficial, CERN, Geneva, Elveția

Prof.dr.ing. Elena HELEREA Referent oficial, Universitatea Transilvania din Brașov

Data susținerii: 26/02/2016

Contents

Introduction .............................................................................................................................. 1

1. Current status of research and development of particle accelerator magnets ............... 7

1.1. Particle accelerators..................................................................................................... 7

1.2. Materials used in the core of particle accelerator magnets ....................................... 10

1.2.1. Alloys of iron with silicon ................................................................................. 10

1.2.2. Alloys of iron with nickel .................................................................................. 12

1.2.3. Alloys of iron with cobalt .................................................................................. 13

1.3. The induction in the gap of the magnet ..................................................................... 13

1.3.1. Governing equations of particle accelerator magnets ........................................ 13

1.3.2. The ramping rate of the magnets in a synchrotron ............................................ 16

1.3.3. Magnet gap induction control methods.............................................................. 17

1.4. Conclusions ............................................................................................................... 18

2. Characterization of ferromagnetic materials used in the cores of particle accelerator

magnets ................................................................................................................................... 19

2.1. Magnetic testing methods.......................................................................................... 19

2.1.1. Magnetic measurement methodologies.............................................................. 20

2.1.2. Magnetic measurement tools ............................................................................. 22

2.1.3. Discussion .......................................................................................................... 24

2.2. New procedure for testing soft magnetic materials ................................................... 25

2.2.1. Measurement principle and procedure ............................................................... 25

2.2.2. Development of iterative measurement procedure ............................................ 31

2.2.3. Assessment of the measurement uncertainty ..................................................... 33

2.2.4. Critical analysis of different measurement procedures ...................................... 44

2.2.5. Development of new curve fitting method ........................................................ 52

2.3. Experimental characterization of Fe-Si alloys .......................................................... 61

2.3.1. The spread of the magnetic properties of Fe-Si alloys ...................................... 62

2.3.2. The anisotropy of Fe-Si alloys ........................................................................... 65

2.3.3. The effect of annealing Fe-Si alloys .................................................................. 76

2.3.4. Comparison of Fe-Si alloys with identical grading ........................................... 79

2.3.5. The influence of the chemical composition on the magnetic and electric

properties of electrical steels ............................................................................................ 81

2.4. Conclusions ............................................................................................................... 86

3. Modelling and simulation of the magnetic hysteresis ..................................................... 89

3.1. Magnetic hysteresis models ...................................................................................... 89

3.1.1. The Jiles-Atherton model of hysteresis.............................................................. 89

3.1.2. The Preisach model of hysteresis ....................................................................... 91

3.1.3. Conclusion ......................................................................................................... 94

iv Hysteresis effects in the cores of particle accelerator magnets

3.2. Identification of the Preisach model.......................................................................... 94

3.2.1. Methods to construct the Preisach weight function ........................................... 94

3.2.2. Development of FORC interpolation method .................................................... 98

3.2.3. Development of FORC level selection method ............................................... 102

3.3. Validation of the developed methods ...................................................................... 106

3.3.1. The samples and testing procedure .................................................................. 106

3.3.2. The experimental results .................................................................................. 109

3.4. Conclusions ............................................................................................................. 111

4. Assessment of hysteresis effects in magnetic circuits.................................................... 113

4.1. Hysteresis modelling of the gap induction of an experimental demonstrator magnet

113

4.1.1. Design of the magnetic circuit ......................................................................... 115

4.1.2. Structural considerations .................................................................................. 122

4.1.3. The model and the measurement procedure .................................................... 129

4.2. Hysteresis modelling of the gap induction of the U17 magnet ............................... 137

4.2.1. Description of the magnetic circuit of the U17 magnet ................................... 137

4.2.2. Identification of the mathematical model ........................................................ 139

4.2.3. Benchmarking of the model against experimental measurements ................... 141

5. Final conclusions .............................................................................................................. 145

5.1. Conclusion ............................................................................................................... 145

5.2. Personal contributions ............................................................................................. 149

5.3. Outlook .................................................................................................................... 150

Bibliography ......................................................................................................................... 151

Abstract ................................................................................................................................. 165

Curriculum Vitae ................................................................................................................. 167

Statement of copyright ........................................................................................................ 169

CD with annexes ................................................................................................................... 171

Annex 1. LabView code used to automate the magnetic measurement procedure .............. 17p.

Annex 2. Matlab code used for the curve fitting procedure .................................................. 8 p.

Annex 3. Matlab code used to process bh files with limited number of points ..................... 2 p.

Annex 4. Matlab code used to process the measured first order reversal curves ................ 13 p.

Annex 5. Matlab code used to generate the Preisach function .............................................. 7 p.

Annex 6. Matlab code used to model the field induction in the gap of a magnet ...................... 6 p.

INTRODUCTION

Particle accelerators are used for a large number of applications. These applications include

synchrotron radiation, high-energy physics experiments, medical applications, or ion

implantation. The numbers of particle accelerators currently in operation around the world is

in the order of tens of thousands. These devices require special technology like magnets,

vacuum, RF cavities, cryogenics, power converters, beam instrumentation, injection and

extraction related hardware, and geodesy and alignment. This work tackles one of the operating

challenges of a particle accelerator magnet, namely, the hysteretic characteristic of the field

induction.

Motivation

At the beginning of the previous century research into the structure of matter was advancing

rapidly and this work inspired the development of the first accelerators. In his experiments

Rutherford used alpha particles from radioactive disintegration to observe the pattern of

particles scattered by atoms [1]. Rutherford deduced that the nucleus was a tiny but massive

central element of the atom. The energy of the alpha particles used in this experiment are in the

order of 10 MeV. To improve on the observations particles of higher energies and in a steady

supply are required.

More powerful accelerators have been developed and other applications have been identified.

They have also been adopted for producing isotopes and for cancer treatment [2]. Many

facilities employ electron rings of a few GeV, typically in the order of 2.5 GeV [3], to generate

photons in the infrared to hard X-ray spectre for experiments which investigate the structure of

complex molecules. Proton accelerators of about 1 GeV produce beams of neutrons which are

used to study the structure of materials [4]. Also, a large number of lower energy accelerators

are used in industry for sterilisation and ion implantation in the fabrication of sophisticated

CPU chips [5].

In a synchrotron the beam is maintained on a circular path using magnetic fields and the

acceleration is provided by electric fields in RF cavities [6]. At the moment of injection, the

particles have a low energy and thus the steering magnetic field is also low. The magnetic field

is increased in proportion to the momentum of the particles as they are accelerated. The

magnetic and electric fields are operated independently and they have to be synchronised to

keep the beam stable [7]. The magnetic field is provided by a slender ring of individual

magnets.

The diameter of a synchrotron, its size and cost for a given energy are given by the bending

radius which depends on a magnetic rigidity. This rigidity increases with the momentum of the

particles and it imposes constraints to the bending field which for iron-dominated magnets

2 Hysteresis effects in the cores of particle accelerator magnets

saturates at approx. 1.7 T. Any improvements to the relevant quantities of the magnetic field

(quantity and quality) can affect the size and, implicitly, the cost of a synchrotron.

Proposed problem

Normal conducting magnets are electro-magnets in which the excitation field is generated by

coils made of aluminium or copper [6]. These magnets rely on a core made of ferromagnetic

material to guide and to concentrate the magnetic flux. The magnetic induction provided by

these magnets in their aperture rarely exceeds 1.7 T due to the saturation of the material in the

core [8]. The core provides a closure path for the magnetic flux with little use of the magneto-

motive force and the profile of the pole determines the path of the magnetic field in the gap.

Today’s practice for building the cores of particle accelerator magnets is to use cold-rolled non-

grain-oriented electrical steel laminations [9]. Although laminated yokes require extra labour

and tooling they offer a number of advantages: reproducible steel quality over a large

production, magnetic properties within tight tolerances, and the material is relatively cheap. A

source of optimization in magnet design is the reduction of hysteresis effects which is achieved

by using materials with a narrow hysteresis cycle. Nevertheless, these materials come with

increased price both for the raw material and for its processing. Therefore, research on magnet

field reproducibility with consideration to magnetic hysteresis is a topic of interest for the field

of particle accelerator physics. Additionally, existing infrastructure could be used more

efficiently if the magnetic hysteresis effects in the cores of the magnets can be accurately

modelled.

Objectives

The goal of this doctoral thesis is to develop a method to predict the hysteresis effects of the

field in the gap of a particle accelerator magnet with the purpose to increase the reproducibility

of this value.

To achieve this goal, the following specific objectives have been set:

1. The development of an advanced method for measuring the magnetic properties of

the soft magnetic materials used in the cores of particle accelerator magnets at low

frequencies and with sinusoidal polarization waveform control by means of

iterative augmentation of the magnetizing current.

2. Development of advanced methods to improve the modelling of the magnetic

hysteresis.

Doctoral thesis 3

3. Modelling and simulation of the magnetic hysteretic behaviour of an experimental

demonstrator magnet and of an iron dominated particle accelerator magnet to

approximate the gap induction with increased accuracy.

Research methodology

This work relies on recent works in the field of electrical engineering, books, articles, doctoral

thesis and software instruments.

In-depth studies of the magnetic circuit of particle accelerator magnets, of the magnetic

properties measurement methods, and of the modelling methods of magnetic hysteresis are

required to achieve the goal of the thesis. Advanced notions in the field of electromagnetism

have been used for the analysis of the magnetic circuits.

Starting from the notions found in literature a new method for measuring the magnetic

properties of soft-magnetic materials at low frequency and with sinusoidal magnetization

waveform has been developed. This method uses numerical methods notions and it has been

implemented using the LabViev and Matlab programming environments, which allowed access

to many readily available functions and tools for processing the involved signals.

Statistics notions have been used for the development of a new method which analyses analog

signals characterised by noise and for the analysis of the errors of the developed magnetic

measurement system. The analog signals analysis method has considerable value for this work

as it has been the main driver for increasing the resolution, and implicitly the accuracy, of the

Preisach model.

Magnetic measurements have been performed on various magnets in the CERN laboratories in

Switzerland. Experimental magnets have been designed by means of the finite element

methods, using the Opera, FEMM and COMSOL software. Mechanical design has been

performed for various components by means of computer aided design software like Inventor

and AutoCAD. Project management notions have been used during the development of the

experimental magnets and while performing the magnetic measurements.

Scientific contribution of the results

In this work are covered theoretical notions and practical applications which are connected to

magnetic hysteresis. To achieve the accuracy requirement of the application new methods and

procedures have been developed for magnetic measurements and hysteresis modelling.

A new magnetic measurement method which perform measurements of the magnetic properties

of electrical steel samples at low frequency (down to 0.01 Hz) and with sinusoidal


magnetization waveform has been developed. The main challenge of these measurements is

given by the difficulty in processing the analog signals which have very low amplitude and

very high signal-to-noise ratio.

Another novelty developed in this work is the method for analysis of the experimental results

which relies on linear regression analysis. This method allows the analysis of an experimental

signal and of its first and second derivative. The main challenge encountering in developing

this method has been the calculation of the systems of equations which give the solutions.

Two novel procedures have been developed which allow the identification of the specific

function of the Preisach model and its use with high accuracy. The first of these methods allow

the identification of the first order reversal curves for any resolution relying on limited input

data. The second method allows the determination of the optimum amplitude of the reversal

curves where the measurements required for the identification have to be performed. The main

challenges of encountered in implementing these methods has been handling the very high

volume of experimental data which had to be processed with consideration to the constraints

of the Preisach model of hysteresis.

Another novelty is the simulation of the hysteretic characteristic of the magnetic induction in

the gap of a magnet relying on an analytical model. These simulations allow the analysis of the

performance of a magnetic circuit with the consideration of the magnetic hysteresis

phenomenon. The main challenge in implementing this model consisted in configuring the

models to operate simultaneously: one model for the magnetic circuit and the second model for

the magnetic hysteresis of the material in the core.

The practical value of the work

The practical value of the work has more sources: the magnetic measurements procedure, the

presented experimental results, the tools developed for the analysis of the experimental results,

and the developed mathematical and analytical models. Therefore, several applications are

identified:

Development of magnetic measurement instrumentations at low magnetization

frequencies and with arbitrary waveform.

Optimization of particle accelerator magnet development.

Development of control systems which rely on advanced information of analog

waveforms.

Development of real-time control systems which rely on magnetic hysteresis modelling.

Doctoral thesis 5

Dissemination of the results

The works published during the research program comprise 6 peer-reviewed articles published

as main author in national and international conference proceeding and journals in the field of

electrical engineering.

Outline

The doctoral thesis covers theoretical and experimental topics in the field of electrical

engineering with regard to hysteresis effects found in the gap of particle accelerator magnets.

The doctoral thesis is structured in five chapters:

1. Current status of research and development of particle accelerator magnets, where an

analysis of the magnetic circuit of a particle accelerator magnet is performed. The

analysis highlights the influence of the magnetic material to the performance of a

magnet.

2. Characterization of ferromagnetic materials used in the cores of particle accelerator

magnets, where the current status of experimental characterization of magnetic materials

is presented and the development of a magnetic measurement procedure and tools is

described. Also, experimental measurements performed on electrical steels with

different silicon content and thickness are analysed.

3. Modelling and simulation of the magnetic hysteresis, where magnetic hysteresis models

proposed in literature are analysed and the development of advanced methods for the

identification of the Preisach model weight function are presented.

4. Assessment of hysteresis effects in magnetic circuits, where a demonstrator magnet is

designed, and the modelling of the magnetic field using the developed models is cross-

checked with experimental measurements for the demonstrator magnet and for a specific

magnet to validate the developed mathematical models.

5. Final conclusions, where the general conclusions of the work, the personal

contributions, and the outlook of the future work are described.

Acknowledgements

This doctoral thesis is the result of the scientific research performed during 2012-2015 in the

field of electrical engineering within Transivania University of Brasov and within the doctoral

student program at the European Organization for Nuclear Research (CERN).


I would like to thank Prof. univ. dr. ing. Gheorghe SCUTARU as scientific coordinator of the

work for the collaboration and his valuable support. I would also like to thank

Prof. univ. dr. ing. Elena HELEREA for the scientific support and for the passion she showed

during our numerous collaborations.

I would like to thank Dr. Davide TOMMASINI and Dr. Daniel SCHOERLING from CERN

for the support of this work and for their valuable technical and scientific contributions to this

work. Also, I would like to thank the members of the Magnets-Normal Conducting (MNC)

section, within the Magnets, Superconductors and Cryostats (MSC) group, the Technology

(TE) department, from CERN for the technical and scientific contribution to this work.

Last but not least, I would like to thank my family and friends for their enthusiastic support

during the entire research period.

1. CURRENT STATUS OF RESEARCH AND DEVELOPMENT OF

PARTICLE ACCELERATOR MAGNETS

The first particle accelerators have been inspired by the early experiments in nuclear physics.

In his 1924 PhD thesis De Broglie proposed the existence of an inverse relationship between

the momentum of a particle, and hence of the energy, and the wavelength of its representation

in quantum mechanics [10]. It was argued that higher energy particles having shorter

wavelengths could better reveal the structure of the atom that Rutherford has detected. Such

arguments led to the development of the first particle accelerators and have sustained the

development of accelerators with increasing energy. At first, the physicists used accelerators

to test the structure of the atom, later, with increasing energy levels the structure of the newly

discovered fundamental particles has been tested. Higher energy levels required the

development of larger accelerators. Also, it was discovered that the best way to have readily

available high energy particles was to keep them on a circular path whose radius is proportional

to the energy of the particle and to the magnetic flux density used to bend the trajectory of the

particles.

1.1. Particle accelerators

Particle accelerators are complex installations used in the field of high energy physics to

accelerate particles to high energies and to keep them on a given trajectory. Accelerator physics

is a vast and varied field due to the broad range of beam parameters and due to the diverse

technologies employed in accelerators. The accelerated particles range from electrons to heavy

ions and their energies range from a few electron volts (eV) to several TeV.

Based on the trajectory of the particles the accelerators can be divided into:

Linear accelerators (linacs) maintain particles on a straight trajectory [11]. These type

of accelerators have the advantage that the particles emit very low amounts of

synchrotron radiation. Small electron, proton or ion linacs are used for medical therapy

and diagnosis. Large proton linacs are injectors for large particle colliders or proton

drivers for neutron or neutrino production. Large electron linacs are often injectors at

GeV levels into storage rings which are used to produce synchrotron radiation, or as

𝑒−/𝑒+ colliders [12].

Cyclotrons use a fixed magnetic field and a radio-frequency (RF) cavity to accelerate

particles in orbits of increasing radius. The cyclotrons can produce continuous particle

beams. The first cyclotron had a diameter of 11 cm and was built at Beckley in

1931 [13].


Synchrotrons maintain the particles on a fixed orbit. The magnetic field used to steer the

particle beam is ramped in proportion to the energy of the particles. The magnets are

often divided into separate units to allow simplified construction. Therefore, the cross-

section of the magnet's gap is smaller and the magnets are less expensive. Synchrotrons

are used as storage rings and they can be cascaded for different energy levels. To this

day the largest synchrotron is the Large Hadron Collider at CERN which started

operation in 2010 and it is designed for energies of 7 TeV per beam [14].

In an accelerator the properties of the particles are changed by means of the Lorentz force:

𝐅 = 𝑞(𝐄 + 𝐯 × 𝐁) = 𝑞𝐄 + 𝑞(𝐯 × 𝐁) = 𝐅E + 𝐅B (1.1)

where 𝑞 is the charge of the particle (C), 𝐄 is the electric field vector (V/m), 𝐯 is the speed

vector of the particle (m/s), 𝐁 is the magnetic induction vector (T), 𝐅E is the electric field

component of the Lorentz force (N), 𝐅B is the magnetic field component of the Lorentz force

(N).

Due to technical limitations the voltage in an accelerator is limited to several tens of kV and,

therefore, the electric field component of the Lorentz force is limited. On the other hand, the

magnetic field component of the Lorentz force can easily have much larger values. For

instance, by assuming that 𝐯 ⊥ 𝐁, and that the particles travel at the speed of light

(𝑣 ≈ 3 ⋅ 108 m/s), by using a magnetic field of 1 T can be obtained a force acting on the

particle which would otherwise require an electric field of 3 ⋅ 108 V/m. Therefore, in high

energy particle accelerators the electric fields are used to increase the energy of the particles

while the magnetic fields are used to steer the particles on the desired trajectory.

The basic layout of a synchrotron is presented in Fig. 1.1.

Fig. 1.1: Basic layout of a synchrotron

Doctoral thesis 9

In a particle accelerator, the energy of the particles is increased by applying an electric field

oriented along the trajectory of the particle. In this process particles pass through cavities

excited by radio frequency (RF) generators. When the particles exit from these cavities they

will have gained an increment in energy from the electric field.

The energy associate to a particle of mass 𝑚 is given by Einstein’s equation:

𝐸 = 𝑚𝛾𝑐2 , (1.2)

where 𝑐 is the speed of light (299 792 458 m/s), and 𝛾 is the Lorentz factor which is described

by the equation:

𝛾 =1

√1 − 𝛽2 , (1.3)

with 𝛽 = 𝑣/𝑐, and 𝑣 is the speed of the particle (m/s).

As the energy of a particle increases with its velocity the total energy can be expressed as:

𝐸 = 𝐸0 + 𝐸K , (1.4)

where 𝐸0 is the energy of the particle at rest, and 𝐸K is the kinetic energy of the particle.

In particle physics the energy is expressed in eV (electron-volts), where 1 eV is the energy

acquired, or lost, by an electron when moving across an electric potential of 1 V. Therefore:

1 eV = 1.602 ⋅ 10−19 C ⋅ 1 V ≅ 1.602 ⋅ 10−19 J . (1.5)

The momentum of the particles increases with their energy, therefore, the value of the

centripetal force required to maintain a particle on a given trajectory has to be adjusted

accordingly. The source of the centripetal force in a particle accelerator is the magnetic field

component of the Lorentz force:

𝐹L =

𝑝𝑣

𝑟𝐹L = 𝐹B

, (1.6)

where 𝐹L is the amplitude of the centripetal force (N), 𝑝 is the momentum of the particle (eV/c),

and 𝑟 is the bending radius of the magnets (in Fig. 1.1).

Particle accelerators come in different sizes, depending on their application, and have the

purpose to supply high energy particles. In an accelerator particles travel through vacuum


chambers and the method to act upon their energy and phase state properties is through the

Lorentz force. Thus, an electric field is applied parallel and in the direction of the particles

speed vector to increase their energy, and a magnetic field is applied perpendicular to the speed

vector to change the trajectory of a particle.

With circular machines the same set of RF cavities is used for each turn when an increment of

energy is added. Once accelerated the particles may circulate indefinitely on the orbital

trajectory at their top energy. The synchrotrons are the best solution to obtain high energy

particles as the cyclotrons are limited by their diameter and magnetic flux density. Also,

relative to the cyclotrons the synchrotrons bring great economies in the cost per unit length of

the magnet system. A significant development in accelerator magnet technology has been to

use superconducting magnets which, due to the higher magnetic fields, reduce the

circumference of the machine by a factor between 3 and 5 [15].

The discussion in this thesis is focused on iron dominated particle accelerator magnets. The

magnetic properties of iron are characterized, among others, by non-linearity and hysteresis.

Therefore, the magnetic field in the gap of an iron dominated particle accelerator magnet will

also be influenced by these characteristics.

1.2. Materials used in the core of particle accelerator magnets

The yoke of a magnet has the purpose to guide and to concentrate the magnetic flux in the gap

of the magnet. The field in the gap of a magnet is characterized by hysteresis mainly due to the

hysteretic characteristic of the magnetization. The ramping rates of particle accelerators are

close to quasi-static and therefore an accurate characterization of the material used in the core

of particle accelerator magnets has to be performed also in quasi-static conditions. Current

magnet design and field control methods consider the properties of the material as a black-box

and disregards the hysteretic properties of the magnetization.

Magnetic materials are classified according to their alloying elements, metallurgical state and

physical properties [16]. Additionally, magnetic materials are classified according to their

coercive force in soft-magnetic materials (with a coercivity below 1000 A/m) and in hard-

magnetic materials (with a coercivity above 1000 A/m). In order to minimize the hysteresis

effects the coercivity of a material used in the core of an accelerator magnet is desired to be as

small as possible, usually materials with coercivity smaller than 100 A/m are used.

1.2.1. Alloys of iron with silicon

The alloys of iron with silicon destined for electro-technical applications are commonly called

electrical steels. In addition to silicon additional alloying elements exist. Combining iron with

silicon increases the electrical resistivity of the iron which presents advantages mainly for AC

applications. The silicon quantity in the alloy has the following effects [17, 18]:

Doctoral thesis 11

The relative permeability increases and the coercive force decreases, the exact values

being influenced by the chemical composition, grain size, manufacturing process and

crystallo-graphic orientation.

The saturation polarization decreases, from 2.15 T for pure iron to 1.3 T for 6.5 % Si

content.

The electrical resistivity increases, from 9.8 × 10−8 Ωm for pure iron to

70 × 10−8 Ωm for 6.5 % Si content.

The magnetostriction of the alloy decreases with the content of silicon, from

𝜆100 > 20 × 10−6 and 𝜆111 < −20 × 10

−6 for high purity Fe, becoming very small

for 6.5 % Si: 𝜆100 = 0.5 × 10−6 and 𝜆111 = 2 × 10

−6.

Following the manufacturing process, the properties of electrical steels are characterised by a

spread [19] which affects the magnetic identity between the magnets of a series [20].

Therefore, laminations are shuffled prior to being assembled into a core [21].

The Goss texture was one of the ground-breaking inventions in the history of electrical steels

improvement. It was patented by Norman Goss in 1934 [22] and described in 1935 [23]. With

Goss’s invention a grain texture is obtained by a suitable combination of annealing and cold

rolling. The grains which have the (001) direction oriented along the rolling direction and the

(110) plane along the lamination surface are privileged to grow. The steels obtained in this way

profit from the fact that the iron crystal has the best magnetic properties in (100) direction.

Therefore, with grain oriented (GO) electrical steel the main effort is made to obtain relatively

large grains ordered in one direction [24].

Excellent magnetic properties along rolling direction can be successfully exploited when the

excitation field is applied in this direction. On the other hand, when the excitation field is

applied in another direction than the rolling direction significant deterioration of the magnetic

performance is expected [24, 25].

For the applications where the magnetic flux is not aligned with the rolling direction, like in

rotating machines, employing non-grain oriented (NGO) steels is advised. NGO steel exhibits

lower magnetic performance compared to GO steel but are characterised by lower anisotropy.

They have lower Si content and simpler production process. Thus, if lower magnetic properties

are acceptable NO steels are a feasible alternative to GO steels.

Carbon and sulphur content in electrical steels is a major source of losses [26, 27]. The purpose

of the annealing process of electrical steel is to remove carbon and other unwanted impurities

from the bulk of the steel, to stimulate grain growth and relief the mechanical stress. Annealing

can lead to significant magnetic performance benefits by reducing core loss and increasing the

permeability [28, 29].


The European standards classify the electrical steels according to their maximum total specific

loss and according to their thickness. According to the IEC standard 60404-8-4 the steel name

of cold-rolled NGO steels strips and sheets in fully processed state comprises the

following [30]:

The letter M for electrical steel;

One hundred times the specified value of the maximum specific total loss, in watts per

kilogram, at 1.5 T and 50 Hz or 60 Hz, depending on the material;

One hundred times the nominal thickness of the material, in millimetres;

The characteristic letter A for cold-rolled non-oriented electrical strip or sheet in the

fully processed state;

One tenth of the frequency at which the maximum specific total loss is specified (5 or 6).

The steel names of grain-oriented electrical strip and sheets designated according to the IEC

standard 60404-8-7 are assigned similarly as for non-oriented steel strips. The difference is

with the testing levels of the specific total loss, at 1.7 T for grain-oriented steels, and the

characteristic letter is S for conventional grades and P for high permeability grades [31].

The chemical composition and the manufacturing process of electrical steels determine their

magnetic and electric properties [32]. The content of silicon in an iron alloy greatly diminishes

the eddy-currents by increasing the electrical resistivity [33]. The electrical resistivity and the

grain size are influenced by the alloying elements of the steel: aluminium increases the size of

the grain [34, 35, 36, 37, 38], but the element is oxygen-avid and alumina incursions create

domain wall pinning sites which increase the energy losses; manganese has the effect of

increasing the electrical resistivity [39]; sulphur created domain wall pinning sites and

decreases the grain size [27]; copper increases the grain size and slightly decreases the

permeability [40]. Also, the grain size is influenced by the annealing temperature of the

steel [41], and the losses and the coercivity are influenced by the quenching temperature [42].

1.2.2. Alloys of iron with nickel

Useful magnetic properties can be achieved by alloying iron with nickel, most notable is the

significant increase of the magnetic permeability. With increasing nickel content and after a

well-tailored annealing process, the Fe-Ni alloys present the following properties [17]:

The coercivity can decrease to 0.4 A/m for alloys containing 80 % Ni.

The permeability can increase up to 100.000 for alloys containing 80 % Ni.

The value of the electrical resistivity changes from 75 × 10−8 Ωm for 36 % Ni to

16 × 10−8 Ωm for 80 %Ni.

Doctoral thesis 13

The saturation polarization decreases, from 1.3 T for 36 % Ni to 0.7 T for 80 % Ni.

Although nickel is an expensive material its alloys with iron present magnetic properties useful

for distinct applications: magnetic shielding, magnetic cores which require high permeability

and low coercivity [43]. These alloys are highly sensitive to mechanical stress and, therefore,

their handling, mechanical processing and additional annealing operations create additional

costs.

1.2.3. Alloys of iron with cobalt

The alloys of iron with cobalt are less versatile and more expensive due to the price of cobalt

(27,100 US$/tonne for Co vs. 14,565 US$/tonne for Ni for 20 August 2013). By alloying iron

with cobalt the saturation and the Curie point are increased. An alloy 50 % iron and 50 %cobalt

can offer a saturation level up to 2.45 T, the highest for any bulk material at room temperature,

and the Curie temperature can reach 980 °C. Also, by adding vanadium to the alloy is increased

the machinability and the electrical resistivity is increased [18].

Although the Fe-Co alloys have some very attractive magnetic properties they have a

prohibitive price. Thus, these alloys are employed in applications which can fully exploit their

high saturation level.

1.3. The induction in the gap of the magnet

Usually around two thirds of the circumference of a particle accelerator is covered by dipole

magnets, therefore, in the following section the magnetic circuit of a dipole magnet will be

analysed. In Fig. 1.2 the simplified magnetic circuit of a C-shaped dipole magnet with iron core

is presented. The analytical calculations rely on Feynman’s model presented in his lectures on

physics [44].

1.3.1. Governing equations of particle accelerator magnets

The quantities shown in Fig. 1.2 are: the yoke of the magnet (in blue), the powering coil (in

red); surface 𝑆 is a sphere of infinite radius whose shell intersects the horizontal symmetry

plane of the magnet; Γ is a closed curve representing the average path of the magnetic field

strength in the circuit; 𝑙Fe is the length of curve Γ in the yoke; 𝑙g is the length of curve Γ in the

gap of the magnet; Φ is the magnetic flux which intersects surface 𝑆 at 𝑆Fe and 𝑆g; 𝑆Fe is the

cross-sectional area in the yoke through which the upward pointing flux passes; 𝑆g is the cross-

sectional area in the gap through which the down pointing flux will close; 𝐽 is the current

density which is given by the magnetomotive force 𝑁𝐼.


Fig. 1.2: Simplified circuit of a C-shaped dipole magnet

For this calculation it is assumed that the magnetic flux closes through a surface of constant

area (𝑆Fe = 𝑆g) and is perpendicular to this surface, therefore, the following relation is

established:

𝐵Fe𝑆Fe = 𝐵g𝑆g , (1.7)

where 𝐵Fe = Φ/𝑆Fe is the magnitude of the magnetic induction in the iron, and 𝐵g = Φ/𝑆g is

the magnitude of the magnetic induction in the gap.

Under the simplifying assumptions that the magnetic field strength vector is oriented along

curve Γ, then Ampere’s law along this curve gives the relation:

∮ 𝐻 d𝑙

Γ

= 𝑁𝐼 , (1.8)

Where 𝐻 is the magnitude of the magnetic field strength and 𝑁𝐼 = ∫ 𝐽 d𝑠𝑆Γ

is the

magnetomotive force (assuming that 𝐉 ∥ d𝐬).

Eq. (1.8) can be rewritten as:

𝐻Fe𝑙Fe + 𝐻g𝑙g = 𝑁𝐼 , (1.9)

where 𝐻Fe is the magnitude of the magnetic field strength in the core and 𝐻g is the magnitude

of the magnetic field strength in the gap.

Doctoral thesis 15

Considering the constitutive law 𝐵 = 𝜇0(𝐻 +𝑀) and Eq. (1.7), then Eq. (1.8) can be rewritten

in the following form:

𝑀Fe𝑙g + 𝐻Fe(𝑙g + 𝑙Fe) = 𝑁𝐼 . (1.10)

The operating point of a magnet is given by the simultaneous solution of the iron’s

magnetization functional relation 𝑀Fe = 𝑓(𝐻Fe) and Eq. (1.10):

𝑀Fe𝑙g + 𝐻Fe(𝑙g + 𝑙Fe) = 𝑁𝐼

𝑀Fe = 𝑓(𝐻Fe) . (1.11)

The operating point of a magnet can be identified by plotting a graph of Eq. (1.10) (the straight

interrupted line in Fig. 1.3) on the same graph with the functional relation

𝑀Fe = 𝑓(𝐻fe) (the solid line in Fig. 1.3). The solution is found at the intersection of the two

curves. Fig. 1.3 shows the evolution of the operating point of a magnet with varying current.

Fig. 1.3: The operating point of a magnet for 𝑙g = 10−5 m and 𝑙Fe = 1 m

For a given current 𝐼 the graph of Eq. (1.10) is a straight line, represented with interrupted line

in Fig. 1.3. Different current values will shift this line horizontally. From Fig. 1.3 it can be seen

that for a given current there are several solutions depending on the history of the

magnetization. Considering that in the initial state the material is demagnetized

(𝐻Fe = 0, 𝑀Fe = 0) when the current is increased from 0 to 𝐼1 the magnetization will follow

the path of the first magnetization curve and the operating point is 𝑎. After the current is

increased to a very high positive value and then is decreased back to the value 𝐼1 then the

operating point is 𝑏. After the current is decreased to a very high negative value and is then

brought back to 𝐼1 the operating point is 𝑐.

To value of the residual field is used to characterize the hysteresis effects induced by a material

when the geometrical parameters of the magnetic circuit are known (𝑙Fe and 𝑙g). The value of


this field gives the magnetic induction which is found in the gap of a magnet when the current

in the coils has been brought to zero (the 𝐼0 = 0 line in Fig. 1.3). Thus, this value is

approximated as:

𝐵rez = −𝜇0𝐻c𝑙Fe𝑙g , (1.12)

where 𝐻c is the coercivity of the hysteresis cycle (as presented in Fig. 1.3).

The transfer function of a magnet is given by the ratio 𝐵g/𝐼. This quantity is used to calculate

the powering requirements of the magnetic circuit. Considering the simplified magnetic circuit

the relation describing the transfer function is approximated starting from Eq. (1.9) and by

considering the constitutive law 𝐵 = 𝜇0𝜇r𝐻:

𝐵g

𝐼=

𝜇0𝑁

𝑙g +𝑙Fe𝜇r

, (1.13)

where 𝜇r is the relative permeability of the material in the core. From Eq. (1.13) it can be

observed that the permeability of the material has to be as high as possible to achieve the

highest efficiency of the circuit. For a given magnet gap height, the transfer function of a

magnet is limited to the value 𝜇0𝑁/𝑙g.

Due to the hysteretic characteristic of the iron’s magnetization the field in the gap of the magnet

is also characterised by hysteresis. In order to accurately reproduce the induction in the gap of

a magnet two models have to be used: one for the magnetic circuit given by Eq. (1.10) and one

for the functional relationship 𝑀Fe = 𝑓(𝐻Fe).

1.3.2. The ramping rate of the magnets in a synchrotron

The operation mode of a magnet in a synchrotron is given by its task. For instance, a kicker

magnet will be fast pulsed, a septa magnet will be operated in continuous mode [45], and the

main magnets of a synchrotron will be ramped in sync with the increase of the particles

energy [46]. An example of the magnetic induction over time in the gap of the SPS main

magnet at CERN is presented in Fig. 1.4.

Doctoral thesis 17

Fig. 1.4: The magnetic induction in the gap of the SPS main magnet

Several cycles can be identified in Fig. 1.4. Although the cycles have different peak levels they

are characterized by identical ramping rate, with d𝐵

d𝑡≈ 1.2 T/ 200 ms. The ramping rate of the

magnet is given by the type of the particle in the beam and by the characteristics of the RF

system. Therefore, for a given particle accelerator configuration, operating with a given type

of particle, the ramping rate of the dipoles is constant.

Considering a model of the magnet based on Eq. (1.11) the repeatable and accurate prediction

of the magnetic induction in the gap is linked to the accurate reproduction of the hysteretic

characteristic of the material in the yoke. By using standard magnetic measurement

methodologies the best estimate of the magnetic properties are achieved under quasi-static

testing [47]. Nevertheless, depending on the material's physical properties, final geometry, and

magnetization ramping rate the shape of the hysteresis cycle is altered [18]. The standard

measurement methodologies have no recommendations for testing materials with controlled

rate of change of the magnetization with values in the range of the ramping rates of accelerator

magnets [48]. Therefore, in order to obtain the best estimate of the magnetic properties of a

material used in the core of an accelerator magnet a measurement methodology which controls

the ramping rate of the magnetization during testing is required.

1.3.3. Magnet gap induction control methods

One key operational concern for particle accelerator magnets is field reproducibility. This

requires careful attention to powering history due to the hysteretic characteristic of the yoke's

magnetization. Therefore, magnet pre-cycling or meticulous cycle configurations are

employed. For a required field level in the gap of a magnet the challenge is to establish the

value of the powering current required to be supplied.

Two approaches are employed for controlling the current: feedback and feed-forward control:


The feedback approach requires the knowledge of the instantaneous field in the magnet.

A possible configuration is for the field to be measured in the gap of a reference magnet

and this value is used to calculate the error required as input in the feedback loop. Several

sources of uncertainty are identified for this method: temperature drift, iron hysteresis,

eddy current, ageing of the material [46].

The feed-forward approach requires a mathematical or numerical model of the magnet.

For the LHC a semi-empirical model (FiDeL [49]) has been developed. Look-up tables

were generated using a large database of test results. The particle momentum is used to

determine the required field which is then used as input in the look-up table to find the

value of the supply current.

The key objective of a magnet control system is to achieve a rapid and easy conversion between

a beam parameter, the field in the gap, and ultimately the supply current. For iron dominated

particle accelerator magnets an accurate control method is required which incorporates as many

characteristics of the magnetic behaviour of the material as possible.

1.4. Conclusions

Particle accelerators are devices which use electric and magnetic fields to increase the energy

of charged particles and to keep them on a well-defined trajectory. In a particle accelerator the

magnets have the purpose to generate the magnetic field required for the deflection of the

particles. The ferromagnetic core of a magnet significantly improves the 𝐵/𝐼 ratio of a magnet

system but come with the inherent drawbacks of non-linearity and hysteresis. Current field

control methods rely on either feedback systems which are expensive to operate or on feed-

forward systems which require vast amounts of input data and are not able to predict the output

for unknown disturbances. Therefore, a model driven control system which relies on few input

parameters would be a major contribution to the field of particle accelerator physics.

2. CHARACTERIZATION OF FERROMAGNETIC MATERIALS USED

IN THE CORES OF PARTICLE ACCELERATOR MAGNETS

The magnetic properties of a material are dependent on a series of factors like: chemical

composition, thermal and mechanical history, and dynamic effects. This chapter of the thesis

firstly describes a review of the available magnetic measurements techniques with

consideration to the application of particle accelerator (PA) magnet core. Secondly, the

development of a methodology for magnetic testing methodology, the evaluation of this

methods uncertainty, and the development of a method to analyse experimental data are done.

And thirdly, experimental measurements performed on various materials commonly used to

build cores for particle accelerator (PA) magnets, and the implications of the findings to the

operation of PA magnets are presented.

2.1. Magnetic testing methods

The magnetic properties of a ferromagnetic material can be described by the family of

concentric symmetric hysteresis cycles. The relevant information obtained from these cycles is

presented in Fig. 2.1.

Fig. 2.1: Determination of the magnetic properties of a material

The first magnetization curve can be approximated by connecting the locus points of the

symmetric hysteresis cycles [18]. Thus, the normal magnetization curve can be used to

approximate the curve starting from (0,0) and passing through point 𝑎 in Fig. 1.3. Also, the

value of the coercivity required to calculate the residual field (Eq. (1.12)) can be approximated

from these cycles.

The quantities usually required during magnet design are:


The coercivity curve (𝐻c(𝐵)) is used to approximate the residual field of the magnet.

The total amplitude permeability of the normal magnetization curve (𝜇r(𝐵)) is used to

approximate the transfer function of the magnet.

When a conductive material is subjected to an applied time-varying magnetic field, loops of

electric current (eddy-currents) will form in this material due to the Faraday’s law of induction.

The value of the magnetic field due to eddy-currents which oppose the applied magnetic field

in a thin lamination is estimated with the relation [18]:

𝐻eddy = 𝜎𝑑2

8 (2.1)

where: is the variation in time of the magnetic induction in the thin lamination [T/s] (for

ferro-magnetic materials ≈ 𝐽, and 𝐽 is the magnetic polarization: 𝐽 = 𝜇0𝑀), 𝜎 is the

conductivity of the material [S/m], and 𝑑 is the thickness of the lamination [m]. In the following

sections of the thesis the term 𝐽 is used to express the magnetic polarization.

The accurate computation of the field in the gap of a magnet is linked to the accurate

measurement of the magnetic hysteresis in the core, as shown by Eq. (1.11). The shape of the

hysteresis cycle of a material's magnetization depends inter alia on the ramping rate of the

magnetization during testing as highlighted by Eq. (2.1) and by work presented in

literature [50]. Therefore, a measurement methodology and installation is required to test

magnetic materials at the foreseen ramping rate of the magnet.

2.1.1. Magnetic measurement methodologies

The rate of change of the induction in the gap of a PA magnet is in the order of 10 T/s.

Therefore, quasi-static measurements provide the best estimate of a material magnetic

properties. The control methods available for quasi-static magnetic measurements are shown

in Fig. 2.2.

Fig. 2.2: The magnetic measurements control methods

The IEC standard 60404-4 describes two open-loop methodologies for measuring d.c. magnetic

characteristics [47]. These are the ballistic method and the continuous recording method. In the

Quasi-static magnetic test

method

Open-loop(standard d.c.)

Closed-loop(feedback)

Feed-forward

Doctoral thesis 21

ballistic method, the excitation field is switched, in a step-like fashion, between two symmetric

values. The cycling is performed several times to allow for the sample to stabilise on a cycle.

After each switching a period of holding time exists which allows the eddy-currents to decay.

The value of the magnetic induction is measured from a flux integrator.

In the continuous recording method, the excitation field is varied slowly between two

symmetric values in a time between 30 and 60 seconds. The reading of the excitation field is

connected to the 𝑥-axis of a plotter and the reading from the flux integrator is connected to the

𝑦-axis of the plotter. After several cycles the material stabilizes on a symmetric cycle and the

values are read with the plotter. However, the standard does not recommend a procedure to

control the magnetic induction during testing and this leads to dynamic effects during

measurements, as Fiorillo highlighted [18].

The lack of control of magnetization rate of change gives rise to additional rate-dependent

losses, affecting the shape and area of the hysteresis cycle if the material has a fast and non-

linear response. Another detrimental effect of uncontrolled magnetization rate is the peaked

shape of the signal induced in the coils of the testing hardware whose accurate measurement is

limited by the dynamic range of amplifiers and A/D converters. This effect is prominent for

extra-soft magnetic materials, which exhibit near-rectangular hysteresis loops. In order to

improve the quality of magnetic measurements the rate of change during magnetic testing has

to be controlled.

In order to control the waveform of the magnetization non-standard magnetic testing methods

have to be employed. The two basic ways to control the rate of change of the magnetic

induction during tests are:

real-time control of the sample induction by means of feedback, and

producing a suitable current waveform 𝑖(𝑡) through iterative augmentation of the input

by means of an inverse approach (feedforward).

Various feedback topologies can be found in the literature. In his paper, Lyke [51] presents a

setup which performs magnetic measurements at 60 Hz and uses a microcomputer to determine

the values of the variables required for the feed-back loop. In his work Fiorillo presents a

control method which exploits waveform control by feedback and digital programming [52].

These methods are flexible and can be implemented in various test systems but they are highly

dependent on the electronic components and are prone to oscillations when working with

materials characterized by strong non-linear responses. Additionally, at low frequency the

induced voltage is very low and the primary circuit is mostly resistive. This leads to a series of

draw-backs [18]: difficult control of the drift signal and the control system may follow the

measured noise.

For high permeability materials the iterative method, which is a feed-forward method, is more

versatile and effective. Here, a suitable waveform of the excitation field is programmed by

means of iterative augmentation using an inverse approach. At every iteration a new waveform


of the excitation cycle is applied to the material and the procedure is repeated until the

convergence criterion is achieved. Several implementations can be found in the literature with

various performances.

One implementation is presented by Stan Zurek et. al. in [53], where the working principle is

similar to a feedback controller: the difference between the reference waveform and the

acquired waveform at the previous iteration is computed; the difference is normalized with

respect to the reference waveform; the error waveform is obtained by multiplying the

normalized waveform with the excitation waveform from the previous iteration; the error

waveform is summed with the previous excitation waveform thus obtaining the excitation

waveform for the new iteration. The author reported that the algorithm achieved convergence

in 20 minutes for GO electrical steel at 𝐵peak = 1.9 T, and 8 minutes for NGO electrical steel

at 𝐵peak = 1.6 T.

Matsubara et. al. presents in [54] a technique to accelerate the above mentioned method: at the

first step an initial excitation signal of sinusoidal waveform is applied to the experimental

setup; at the second step the voltage to be applied to the setup is computed using the equivalent

circuit ( 𝑣o = 𝑅𝑖 + 𝐿d𝑖

d𝑡+ 𝑁𝑆

d𝐵

d𝑡); from the third step the conventional feedback method is

used. With this acceleration technique the author reports a reduction in the number of iterations

to about 1/6 of the previous method.

Anderson presents in [55] another technique of the iterative method. The algorithm has the

following steps: an excitation field is applied with the fundamental waveform of the desired

𝐵(𝑡) signal; the descending branch of the hysteresis cycle is isolated and shifted along the 𝐻

axis so that 𝐻 = 0 corresponds to 𝐵 = 0; the obtained curved is fitted to a 30th order

polynomial of the form: 𝐻(𝐵) = ∑ 𝑎𝑖𝐵𝑖30

𝑖=1 ; the shift which was previously removed from the

𝐻 axis is reintroduced in the equation; the 𝐵ref(𝑡) signal is used with the newly developed

𝐻(𝐵) equation to generate the 𝐻(𝑡) waveform required to obtain 𝐵ref(𝑡); the process is

repeated starting from the second step until 𝐵(𝑡) approaches the desired waveform. The author

reports convergence of the algorithm within 3 iterations for GO silicon steel and excitation

peak field of 2000 A/m.

2.1.2. Magnetic measurement tools

Electrical steel in the form of sheets is the material usually employed for manufacturing the

cores of PA magnets [8]. Standardised tools used for d.c. testing magnetic materials in the form

of sheets are: the ring core (Fig. 2.3 (a)), the single sheet tester or permeameter (Fig. 2.3 (b))

and the Epstein frame (Fig. 2.3 (c)).

Doctoral thesis 23

(a) The ring core (b) The single sheet tester (c) The Epstein frame

Fig. 2.3: Magnetic measurement tools

A. The ring core

The ring core is a straightforward topology whose assumptions allow Ampere's law and the

magnetic induction law to be easily applied. The magnetic field is assumed to be homogenous

along the magnetic path and the value of the applied magnetic field strength 𝐻 is determined

by measuring the magnetizing current. In [18] the limitations of the ring method are presented

in detail. A brief summary of these limitations include: preparation of the sample and manual

winding of the coils is tedious; for automated test setups which use two half coils the electrical

contacts is a technical problem which can lead to reliability issues; if strip-wound samples are

used bending stress will appear; in some cases it may be difficult to achieve saturation of the

material; non-homogeneous distribution of the magnetization along the sample cross-section

may appear for some sample configurations.

B. The Epstein frame

The Epstein frame was initially proposed in 1900 as a 50 cm square frame [56]. The smaller

version of 25 cm, proposed later by Burgwin in 1941 [57] is standardised and used [47]. The

Epstein frame was accepted as the standard measurement device due to the advantages of

relatively easy assembly of the samples in the magnetic circuit and good reproducibility. On

the other hand, a limitation of the Epstein frame is a systematic error which appears due to the

double overlapping corners [58, 59]. The double-overlapping corners form a significant

inhomogeneity of the circuit and the measurements are thus influenced by the permeability of

the material.

Some advantages of the Epstein frame are: it provides averaged measurements, which are

representative for a larger mass of material; the material properties can be measured at any

angle with respect to the rolling direction. Some of the drawbacks of the Epstein frame as

highlighted by several authors [18, 58] are: samples of high permeability materials require

stress relief annealing after being cut, which is a tedious operation; compared to the single sheet


tester, loading of samples in the Epstein frame is a tedious operation; some materials (magnetic

domain refined materials) require tedious sample manufacturing.

C. The single sheet tester

To counter some of the drawbacks of the Epstein frame, especially the tedious sample

preparation and loading, the single sheet tester was developed and standardized (first standard

was developed in 1982) and it was expected to replace the Epstein frame. Due to the strong

adherence to the Epstein frame, the first standard (the so-called SST(82)), required that the

measurements were calibrated by means of Epstein strips, initially measured in the Epstein

frame . The standard was revised in 1992 (SST(92)) with several studies for design parameter

alternatives being taken into account, including single/double yoke construction, lamination

modifications, corrections for the loss in the yoke, several methods for measuring the magnetic

field strength 𝐻 and power losses [60, 61, 62, 63].

Later studies [60, 64, 65] confirmed that the procedure specified in SST(82) showed

considerable scattering and poor reproducibility, and that SST(92) greatly improved on these

drawbacks. Still, some drawbacks remain: the measurements are recommended only for

applied magnetic fields with strength above 1000 A/m [47], and air flux compensation is

problematic for thin laminations and films [18].

2.1.3. Discussion

In order to obtain the best estimate of the magnetic properties of a material the rate of change

during magnetic testing has to be matched to the rate of change of the magnet. This can be

achieved by employing the correct measurement methodology.

The feedback methods do not require time consuming iterative procedures but have the

drawbacks of sensitivity to the quality of the electronic components, difficult handling of the

noise, and may require controller retuning for different samples. On the other hand, the iterative

methodologies require less electronic components, as they are software implemented and use

readily available software functions, and produce more reliable results. The major limitations

of these methodologies lie in the number of iterations required to achieve convergence and in

the computing power required to process the measured data. Depending on the algorithm and

on the teste material, the convergence of the iterative methodologies is achieved between three

and several tens of iterations. Also, the algorithms may use curve fitting which is computing

intensive and does not always produce accurate results.

Most magnetic materials are characterized by anisotropy and magnetic measurements for

different directions of the excitation field with respect to the rolling direction provide relevant

information on a material. For this reason, the ring core has not been selected for this study as

under these conditions it does not provide the most accurate measurements. The advantages of

the SST over the Epstein frame lie in the speed of assembly of the samples in the circuit.

Nevertheless, the SST is recommended for measurements above 1000 A/m, therefore,

magnetic properties cannot be measured in a very important operating region of the material.

Doctoral thesis 25

For the current research an Epstein frame has been used as it offers the best trade-off between

measurement speed and quality. Also, a new magnetic testing procedure which controls the

waveform of the polarization by means of iterative augmentation with a quickly converging

algorithm has been developed.

2.2. New procedure for testing soft magnetic materials

In order to estimate the magnetic properties of the materials used in the cores of PA magnets a

measurement procedure which maintains sinusoidal waveform of the magnetization has been

developed. The procedure has been implemented using recursive digital control because this

method requires less electronic components and allows for more data processing options. The

iterations required for convergence is usually 3 to 5. This performance of this method is

confirmed by the similar results obtained by Kuczmann [66]. The new procedure brings several

contributions to the measurement process: improvements to the scattering of the measured

cycles and to the processing of the measured curves.

2.2.1. Measurement principle and procedure

The measurement setup is comprised of a power supply (PS), a shunt resistor 𝑅s, a standard

Epstein frame (EF and AFCC), the samples (S), a data acquisition (ADC) and waveform

generating (DAC) device, and a PC which controls the process and stores measurement data.

The block diagram of the measurement setup is presented in Fig. 2.4.

DAC

ADCPC

Rsis(t)

>

us(t)

u2(t)

AFCC

EF

PS

S

N

Fig. 2.4: Block diagram of the developed measurement setup

The items presented in Fig. 2.4 are:

PS is a voltage controlled power supply (KEPCO bipolar BOP 6-36ML, max. current

6 A);


𝑅s is a shunt resistor of 1 Ω and 15 W;

EF is a 25 cm Epstein frame (in accordance with IEC 60404-2);

S is the test sample;

AFCC is the air flux compensation coil (integrated in the body of the Epstein frame);

ADC is the analog to digital converter (NI PCI-6154);

DAC is the digital to analog converter (NI PCI-6154);

PC is the personal computer with LabView software;

𝑢2 is the voltage induced in the secondary winding of the Epstein frame;

𝑢s is the voltage drop on the shunt resistor.

In the measurement setup the PC with the control software has the purpose to control the

measurement procedure by interacting with the ADC and DAC through the conventional PCI

local bus. At each iteration the PC calculates the required waveform of the excitation current

which is then scaled to the level required as input by the PS. The DAC converts the digital

values required for the control voltage to analog values. The output current of the PS follows

the waveform of the voltage generated by the DAC. The primary electrical circuit closes

through the PS, the 𝑅s, the AFCC and the EF. The value of the current passing through the

primary circuit is calculated using Ohm's law (𝑖s(𝑡) =𝑢s(𝑡)

𝑅s). The variation in time of the

magnetic polarization in the test sample induces a voltage in the secondary winding of the EF,

𝑢2(𝑡), whose value is acquired in sync with the value of 𝑢s(𝑡).

Measurement principle

Starting from Ampere's law the waveform of the magnetic field strength 𝐻(𝑡) acting upon the

sample S is:

𝐻 =𝑁𝑖s(𝑡)

𝑙 . (2.2)

The magnetic induction in the sample is determined using Faraday's law:

𝑢 = −dΦ

d𝑡 . (2.3)

A mutual inductance which links the primary and the secondary windings of the EF will exist

due to the air flux. The AFCC has the purpose to balance this mutual inductance and thus to

Doctoral thesis 27

cancel the voltage induced in the secondary windings due to the air flux. The voltage induced

in the secondary windings of the AFCC will subtract from the voltage induced in the secondary

windings of the EF. The number of windings in the secondary of the AFCC is adjusted such

that 𝑢2(𝑡) is zero when there is no sample in the EF. Therefore, voltage 𝑢2(𝑡) is due to the

magnetic polarization of the sample alone:

𝐽(𝑡) = 𝐽0 +1

𝑁𝑆∫𝑢2(𝜏) d𝜏

𝑡

0

, (2.4)

where: 𝐽0 is a constant value [T], 𝑁 is the number of windings in the secondary coils of the

Epstein frame, 𝑢2 is the voltage induced in the secondary coils of the Epstein frame [V], and 𝜏

is the integration time constant [s].

The measurement procedure was implemented in LabView on the PC. The PS is controlled by

the control voltage generated by the DAC. The current of the PS will follow the waveform of

the control voltage with a gain of -6/10. The magnetizing current 𝑖s(𝑡) is modulated by this

waveform. Due to the limitations of the PS the following limitations are imposed to the results:

the noise in the current limits the minimum amplitude of an excitation cycle to approx. 5 A/m;

the maximum available current is 6 A, therefore, the peak value of the magnetic field strength

is 4468 A/m.

A. Assessment of the measurement resolution

When measuring the family of symmetric hysteresis cycles the scattering of the amplitudes of

the measured cycles has to be properly adjusted in order to obtain high quality measurements.

A simple method to obtain the measurement levels is to increase the peak amplitude of the

excitation field on the 𝐻-axis with constant increment. Nevertheless, this method leads to poor

resolution of the curves in the linear region of the hysteresis characteristic and high point

resolution in the saturation region. On the other hand, by linearly spacing the points on the 𝐵-

axis would lead to high point density in the linear region of the hysteresis characteristic and

low point density in the saturation region.

The field levels where the measurement has to be densely scattered are in the regions where

the BH curve has the highest curvature. These are the reversible domain movement region

(close to the origin) and the knee region (before the material saturates). Optimal scattering has

been achieved with the curve presented in Fig. 2.5.


(a) Test levels of the magnetic polarization (b) The histogram of the test levels

Fig. 2.5: The scattering of the measurement points

The curve presented in Fig. 2.5 (a) has been obtained by shifting and normalizing the tangent

hyperbolic function between the values -1.2 and 2:

𝐽p,𝑖

𝐽max=tanh(𝑥𝑖) − tanh(−1.2)

tanh(2) − tanh(−1.2) (2.5)

The scattering of the points presented in Fig. 2.5 (b) has been obtained using Eq. (2.5). The

intense scattering in the low field region allows the observation of the reversible wall

movement phenomenon. The increase scattering of the measurements in the high field region

ensures that the saturation of curves with different shapes is well defined. Good resolution at

all field levels for the performed measurements has been obtained by using Eq. (2.5).

B. The waveform of the magnetic polarization

The signal measured on the secondary windings of the Epstein frame is proportional to the

derivative of the magnetic polarization in the sample. By using a smooth waveform of the

polarization, like the sine or the cosine, it is ensured that the signal induced in the secondary

winding of the Epstein frame is continuous and it is maintained in the dynamic range of most

A/D converters.

Between two iterations a short time period will exist when no voltage variation exists at the

output of the D/A converter and, therefore, the excitation field will be maintained at the last

value. Thus, during this period no voltage is induced in the secondary windings of the Epstein

frame and no voltage will exist on the A/D converter. When applying a cosine waveform in a

new iteration the initial value of the induced voltage is 0 V. On the other hand, by using a sine

waveform a rapid transition will exist from 0 V (no induced voltage between two iterations) to

the initial value of the voltage. Therefore, minimum loss of information has been ensured by

using a cosine waveform for the magnetization.

Doctoral thesis 29

C. Demagnetizing the sample

One of the assumptions of the developed method is that the measured cycles are symmetric.

Any remanent magnetization in the material will shift the measured cycles on the vertical axis.

Therefore, the sample's remanent magnetization has to be cancelled before the start of the

measurement procedure. The cancelling of the remanent magnetization is accomplished by

degaussing the material. The procedure for degaussing is achieved by applying a powering

signal with slowly decreasing amplitude to the sample. The waveform of the demagnetizing

signal has been obtained using the equation:

𝐼(𝑡)

𝐼max

= sin(2𝜋𝑓𝑡) ⋅ 𝑒−𝑘𝑡 . (2.6)

The optimum value for the coefficient 𝑘, of the frequency 𝑓, and for the length of the signal

have been obtained by empirical observations on a broad range of materials. The value for

coefficient 𝑘 has been set to 0.15, for the frequency 𝑓 the value has been set to 1 Hz, and length

of the signal has been set to 40 seconds. The waveform of the demagnetizing waveform is


Fig. 2.6: Waveform of degaussing signal

The amplitude of the sinusoidal cycles will decrease with time. Most of the cycles will be in

the low current range, which is an important feature that ensures the demagnetization of

materials with a narrow hysteresis cycle.

D. Convergence of the iterative algorithm

The shape of the hysteresis cycle is different for every material, therefore, the waveform of the

excitation field required to achieve a cosine magnetization waveform is determined iteratively

for each measurement point. Two criteria have been defined for the convergence of the iterative

algorithm: the difference between the coercivity of the cycles of two consecutive iterations is


below 1 A/m; and the form-factor of the polarization waveform is within 0.2 % of the value

1.1107 (the form-factor of a pure cosine waveform). The form-factor of a waveform is [47]:

𝐹𝐹 =𝑓RMS𝑓AVG

(2.7)

Empirical observations of the measurements of a broad range of materials have shown that

these two conditions are sufficient to obtain good resemblance of the polarization waveform to

the cosine waveform. The number of iterations has been limited to 20. If the measurement

procedure reaches this number of iterations without achieving convergence then the process is

stopped.

E. Setting the gain of the cycle

The iterative algorithm uses normalized waveforms during processing. Once a new excitation

waveform has been determined its amplitude is scaled to the value required to achieve the

desired peak value of the magnetic polarization. The peak value of the new excitation cycle is

difficult to estimate due to the non-linear characteristic of the magnetization, and it is, therefore,

approached with every iteration. The hysteresis cycles of two consecutive iterations are


Fig. 2.7: Determination of the gain for the new iteration

The hysteresis cycle in hashed line presented in Fig. 2.7 is the cycle measured at the previous

iteration and it has the following parameters: the peak value of the excitation waveform is

𝐻p,𝑖−1, and the peak value of the magnetic polarization is 𝐽p,𝑖−1. The desired value of the peak

polarization for the new iteration is 𝐽p,𝑖. The problem is to identify the peak value for the new

excitation cycle 𝐻p,𝑖. It is assumed that the branch which connects the tips of the two cycles is

a straight line with the slope equal to the differential permeability at point (𝐻p,𝑖−1, 𝐽p,𝑖−1). The

gain of the excitation cycle for the 𝑖th iteration is:

Doctoral thesis 31

𝐺 =𝐻p,𝑖

𝐻p,𝑖−1= 1 +

𝐽p,𝑖 − 𝐽p,𝑖−1

𝐻p,𝑖−1𝜇r,inc (2.8)

For quasi-static measurements a number of three iterations are usually required to reach the

desired peak polarization and convergence of the iterative algorithm. After the first iteration

the peak polarization value is approach and the polarization waveform is very distorted. After

the second iteration the desired peak polarization is reached and the waveform of the

polarization is less distorted. After the third iteration the polarization waveform has the desired

waveform and amplitude.

2.2.2. Development of iterative measurement procedure

The proposed procedure measures the family of symmetric hysteresis cycles of a sample using

the Epstein frame. The waveform of the magnetic polarization for every cycle is modulated as

a cosine. The logical diagram of the measuring procedure is presented in Fig. 2.8.

The measurement procedure consists in the following steps:

1. The input data are provided: the characteristics of the sample (length, width, mass

of the samples) and the testing conditions (magnetization frequency 𝑓, maximum

value of the polarization 𝐽max, and number of hysteresis cycles 𝑁h).

2. Calculation of the constants: cross-sectional area 𝐴 of the sample S according to

IEC60404-2, the reference waveform for the magnetic polarization, and the

degaussing waveform (Fig. 2.6).

3. The sample is demagnetized.

4. The first excitation cycle is applied to the sample. This cycle is a sinusoidal signal

of 3.25 periods with the amplitude of 5 A/m. This value offer the best trade-off

between measurement noise and complete sample information. The first

polarization measurement level, as in Fig. 2.5, is zero, therefore, this level will be

overwritten and the cycle will have the polarization amplitude associated to the

magnetic field strength of 5 A/m. For the first excitation cycle the first quarter of a

period will be discarded and only the following three cycles will be used for

processing. For the following iterations the cycles will be comprised of three

consecutive cycles. Three cycles are required in order to ensure that no information

is lost due to the phase shift between the maximum values of 𝐽(𝑡) and 𝐻(𝑡).


6. Was convergence

reached?

1. Input data

Start

2. Calculate variables

3. Degauss sample

4. Apply magnetization cycle and signals

acquisition

5. Process acquired signals

7. Determine waveform for new

powering cycle

NO

8. Scale amplitude of new cycle to

reach desired J level

11. Was Jmax reached?

10. Store cycle to memory

YES

9. Technical limitations reached?

NO

14. Create measurement

report files.

13. Extract quantities from

stored cycles

YES

YES

A

A

12. Determine next J level

NO

End

Fig. 2.8: Logical diagram of the measurement procedure

5. Signal processing is done in several steps. Firstly, the data series is down-sampled

to 5000 samples per cycle by averaging the extra samples. Thus, noise filtering of

the resulting waveforms is achieved. Secondly, the value of the excitation field

𝐻(𝑡) is calculated using Eq. (2.2) and the value of the polarization 𝐽(𝑡) is

calculated using Eq. (2.3) implemented with the trapeze method. Thirdly, the

second cycle is identified and selected in the 𝐽(𝑡) waveform for further processing.

Doctoral thesis 33

6. The convergence criteria are verified.

7. A new waveform for the excitation cycle is modulated if the convergence criteria

are not fulfilled at step 6. Firstly, the waveform of the polarization is normalized

and a lookup table is created using the descending branch of the hysteresis cycle.

Secondly, the values of the normalized reference waveform are found in the lookup

table and the associated values of the magnetic field strength are extracted. Thirdly,

a copy of the obtained waveform is negated and concatenated to the original

waveform, thus, closing the hysteresis cycle.

8. Determine the gain of the excitation cycle for the current iteration.

9. Verify if the excitation waveform for the current iteration reached the following

limitations: maximum current of the power supply, and the maximum number of

iterations. If these conditions are met then the waveforms from the current iteration

are discarded and the software proceeds to the report generation section.

10. The current cycle is stored to memory.

11. Verify if the maximum polarization level of 𝐽max has been reached. If so, then the

software proceeds to the report generation section.

12. The next 𝐽p,𝑖 level is selected from the list generated with Eq. (2.5) and presented

in Fig. 2.5 and the iterative loop is resumed from step 8.

13. Information is extracted from the stored cycles: the loci of the hysteresis cycles,

the coercivity, the remanence and the magnetic energy loss.

14. The measurement report files are generated.

The excitation cycle created following the iteration of the loop with steps 4, 5, 6, 7, 8, and 9

will generate a waveform for the magnetic polarization which will bear a close resemblance to

the reference waveform. The implementation of the measurement procedure in the LabView

programming environment is presented in Annex 1.

2.2.3. Assessment of the measurement uncertainty

The objective of the measurement operation is to estimate as close as possible the measured

value. However, no measurement can exist without error. Therefore, a value estimated through

a measurement procedure should be accompanied by another quantity, the uncertainty, which

represents the degree of dispersion of the results around the estimate. The International

Organization for Standardisation with the support of various international


organizations1 undertook the task of creating a Guide to the expression of uncertainty in

measurement [67], referred to as the guide in the following section. The definitions of several

general metrological terms relevant to the discussion in this section are presented in Table

2.1 [67].

Table 2.1: Definition of general metrological terms

Term Definition

value magnitude of a particular quantity generally expressed as a unit of

measurement multiplied by a number

measurement set of operations having the object of determining a value of a quantity

method of

measurement

logical sequence of operations, described generically, used in the

performance of measurements

measurement

procedure

set of operations, described specifically, used in the performance of

particular measurements according to a given method

measurand particular quantity subject to measurement

accuracy of

measurement

closeness of the agreement between the result of a measurement and a true

value of the measurand

repeatability

closeness of the agreement between the results of successive

measurements of the same measurand carried out under the same

conditions of measurement

reproducibility closeness of the agreement between the results of measurements of the

same measurand carried out under changed conditions of measurement

experimental

standard

deviation

the quantity characterizing the dispersion of the results for a series of

measurements of the same measurand

uncertainty

parameter, associated with the result of a measurement, that characterizes

the dispersion of the values that could reasonably be attributed to the

measurand

error (of

measurement) result of a measurement minus a true value of the measurand

relative error error of measurement divided by a true value of the measurand

A. Estimation of a measured value and of its uncertainty

Any measurement is affected by some kind of stochastic behaviour. By repeating a

measurement under identical conditions the measured values will be scattered. When the

measurements are ordered according to representation frequency the probability distribution

function of the measurements can be identified. In his 1809 monograph [68] Gauss introduces

1 BIPM (Bureau International des Poids et Mesures), IEC (International Electrotechnical Commission), IFCC

(International Federation of Clinical Chemistry), ISO (International Organisation for Standardization), IUPAC

(International Union of Pure and Applied Chemistry), IUPAP (International Union of Pure and Applied Physics),

OIML (International Organization of Legal Metrology)

Doctoral thesis 35

several important statistical concepts in order to interpret the astronomical observations, such

as the method of least squares, the method of maximum likelihood, and the normal distribution.

The measurements of a quantity can be considered to be affected by many small random

contributions generated by a large number of sources. By denoting with 𝑥(1), 𝑥(2), … , 𝑥(𝑛) a

series of 𝑛 observations of the measurable quantity 𝑥, the probability of finding the expected

value within a prescribed interval (𝑥, 𝑥 + d𝑥) is given by the normal distribution function:

𝑓(𝑥) =1

𝜎√2𝜋exp(−

(𝑥 − 𝜇)2

2𝜎2) , (2.9)

where: 𝜇 is the most probable value of the observed quantity and is estimated by the arithmetic

mean of the observations (); 𝜎 is called standard deviation and it provides a measure of the

dispersion of the observations around 𝜇.

The arithmetic mean of the observations is determined using:

=1

𝑛∑𝑥(𝑘)𝑛

𝑘=1

. (2.10)

The standard uncertainty provides a quantitative evaluation of the dispersion of the values and

is estimated by the standard deviation of the mean:

𝑢() ≡ 𝑠() =𝑠(𝑥(𝑘))

√𝑛= √

∑ (𝑥(𝑘) − )2𝑛𝑘=1

𝑛(𝑛 − 1) , (2.11)

where: 𝑢() is the standard uncertainty of the estimation of 𝑥, and 𝑠() is the standard

deviation of the mean.

The guide defines two methods for expressing the uncertainty of a measurement:

For type A method the probability densities are obtained from observed frequency

distributions,

For type B method the uncertainty is derived from systematic effects.

When applying the Type B method, the distribution function of the measured values has to be

assumed. If a calibration certificate is provided, the distribution is assumed to be of normal

type. On the other hand, there are situations when it is possible to estimate only the upper and

lower bounds 𝑥0− and 𝑥0+ for the values that 𝑥 can take. In this situation a reasonable

assumption is that 𝑥 is equally likely to belong anywhere in the interval (𝑥0−, 𝑥0+). The

variance and the uncertainty associated to the most probable value = (𝑥0− + 𝑥0+)/2 are


expressed by the rectangular distribution. Considering the half-width of the distribution

𝑎 = (𝑥0+ − 𝑥0−)/2, then the variance and the uncertainty of the estimate are defined as:

𝑢2() =𝑎2

3 , 𝑢() =

𝑎

√3 . (2.12)

B. Combined uncertainty

In the usual case a quantity is not determined directly but is related by a functional relationship

to a certain number of input quantities. Considering the output quantity 𝑦 and the input

quantities 𝑥1, 𝑥2, … , 𝑥N, then the functional relationship between the output and the input

quantities is:

𝑦 = 𝑔(𝑥1, 𝑥2, … , 𝑥N). (2.13)

The best estimate of the output quantity is:

= 𝑔(1, 2, … , N). (2.14)

The problem becomes one of determining the uncertainty of from knowledge of the

quantities 1, 2, … , N. For the following calculations two assumptions are made:

The measured quantities have normal distribution, and

The function 𝑔 and its derivative are continuous around the value .

A Taylor series, truncated to the first order, will give, for small intervals (𝑥𝑖 − 𝑖):

𝑦 − =∑(𝜕𝑔

𝜕𝑥𝑖) (𝑥𝑖 − 𝑖)

N

𝑖=1

. (2.15)

The square of Eq. (2.15) is:

(𝑦 − )2 =∑(𝜕𝑔

𝜕𝑥𝑖)2

(𝑥𝑖 − 𝑖)2

N

𝑖=1

+ 2∑ ∑𝜕𝑔

𝜕𝑥𝑖

𝜕𝑔

𝜕𝑥𝑗

N

𝑗=𝑖+1

(𝑥𝑖 − 𝑖)(𝑥𝑗 − 𝑗)

N−1

𝑖=1

. (2.16)

By interpreting the differences in Eq. (2.16) as experimental samples and taking the averages,

the variance of the output quantity can be expressed as a combination of the variances 𝑢2(𝑖)

and of the covariances 𝑢(𝑖, 𝑗) of the input estimates. The law of propagation of uncertainty

is determined:

Doctoral thesis 37

𝑢c2() = ∑(

𝜕𝑔

𝜕𝑥𝑖)2

𝑢2(𝑖) +

N

𝑖=1

2∑ ∑𝜕𝑔

𝜕𝑥𝑖

𝜕𝑔

𝜕𝑥𝑗

N

𝑗=𝑖+1

𝑢(𝑖, 𝑗)

N−1

𝑖=1

(2.17)

where: 𝑢c2() is the combined variance, and 𝑢c() is the combined standard uncertainty. The

partial derivatives in Eq. (2.17) are the sensitivity coefficients. When these coefficients are

denoted by 𝑐𝑖, then Eq. (2.17) becomes:

𝑢c2() =∑𝑐𝑖

2𝑢2(𝑖) +

N

𝑖=1

2∑ ∑ 𝑐𝑖𝑐𝑗

N

𝑗=𝑖+1

𝑢(𝑖 , 𝑗)

N−1

𝑖=1

. (2.18)

If an estimate is associated with both Type A and Type B uncertainties then the variance is

calculated as:

𝑢2(𝑖) = 𝑢A2(𝑖) + 𝑢B

2(𝑖) . (2.19)

If the input quantities are uncorrelated or they have a very low degree of correlation such that

the covariance 𝑢(𝑖 , 𝑗) ≈ 0, the combined uncertainty is expressed as:

𝑢c() = √∑𝑐𝑖2𝑢2(𝑖)

N

𝑖=1

. (2.20)

Considering an output quantity of the form 𝑦 = 𝑚 ⋅ 𝑥1𝑝1 ⋅ 𝑥2

𝑝2 ⋅ … ⋅ 𝑥N𝑝N, with 𝑚 a constant

coefficient, then the relative standard uncertainty, 𝑢c()/ is expressed based on the relative

variance 𝑢c2()/2:

𝑢c2()

2=∑𝑝𝑖

2𝑢2(𝑖)

𝑖2

N

𝑖=1

. (2.21)

C. Expanded uncertainty and confidence level

The information of the measurement can be collected into two parameters: the best estimate

and the combined uncertainty. A 𝑦 quantity described by a normal distribution function,

characterized by the best estimate and a standard deviation 𝜎 is assumed. A confidence level

𝑝 is achieved by integrating the distribution function over a confidence interval ±𝑘𝜎 around .

For the coverage factors 𝑘 = 1, 2, and 3, the confidence levels are 𝑝 = 68, 95.5 and 97.7 %,

respectively. The expanded uncertainty 𝑈 is the value which defines the interval


− 𝑈 ≤ 𝑦 ≤ + 𝑈, to which the true value of the measurand is expected to belong with a

confidence level 𝑝. The result of a measurement can be declared in the form:

𝑦 = ± 𝑈 (2.22)

According to the guide, the expanded uncertainty is defined as:

𝑈 = 𝑘𝑢c() . (2.23)

The coverage factor depends on the desired level of confidence and can be found if there is

extensive knowledge of the probability distribution of each input quantity. The value of the

coverage factor is calculated based on the effective degrees of freedom of the measurement

and the typical values are found in Table 2.2 [67].

Table 2.2: Values for the coverage factor

Degrees

of

freedom

𝒗𝐞𝐟𝐟

𝒑 [%]

68.27 90 95 95.45 99 99.73

1 1.84 6.31 12.71 13.97 63.66 235.80

2 1.32 2.92 4.30 4.53 9.92 19.21

3 1.20 2.35 3.18 3.31 5.84 9.22

4 1.14 2.13 2.78 2.87 4.60 6.62

5 1.11 2.02 2.57 2.65 4.03 5.51

6 1.09 1.94 2.45 2.52 3.71 4.90

7 1.08 1.89 2.36 2.43 3.50 4.53

8 1.07 1.86 2.31 2.37 3.36 4.28

9 1.06 1.83 2.26 2.32 3.25 4.09

10 1.05 1.81 2.23 2.28 3.17 3.96

15 1.03 1.75 2.13 2.18 2.95 3.59

20 1.03 1.72 2.09 2.13 2.85 3.42

25 1.02 1.71 2.06 2.11 2.79 3.33

30 1.02 1.70 2.04 2.09 2.75 3.27

35 1.01 1.70 2.03 2.07 2.72 3.23

40 1.01 1.68 2.02 2.06 2.70 3.20

45 1.01 1.68 2.01 2.06 2.69 3.18

50 1.01 1.68 2.01 2.05 2.68 3.16

100 1.005 1.660 1.984 2.025 2.626 3.077

∞ 1.000 1.645 1.960 2.000 2.576 3.000

Doctoral thesis 39

The value of the effective degrees of freedom is calculated according to the guide using the

Welch-Satterthwaite formula:

𝑣eff =𝑢c4(𝑦)

∑𝑢𝑖4(𝑦)𝑣𝑖

𝑁𝑖=1

, (2.24)

where 𝑣𝑖 is the degrees of freedom and is equal to: 𝑛 − 1 for a single quantity estimated by the

arithmetic mean of 𝑛 observation, 𝑛 − 2 if 𝑛 observations are used to determine the slope and

the intercept of a straight line by means of least squares, or 𝑛 −𝑚 if a least-squares fit of 𝑚

parameters to 𝑛 data points is performed. For the case where Type B uncertainty is used, the

uncertainty is considered to be completely defined, and, therefore 𝑣𝑖 → ∞.

D. Experimental determination of the uncertainty

Experimental measurements have been performed to determine the uncertainty of the normal

magnetization curve measured with the new method. In particular, the uncertainty of the

magnetic polarization 𝐽 at a given applied field 𝐻 has to be determined. Test strips of NGO

electrical steel have been used for the experimental measurements. The characteristics of the

sample are presented in Table 2.3.

Table 2.3: The sample used to determine the measurement uncertainty

Characteristic Value

Alloy Fe-Si

Quality isovac 250-35 A HP

Manufacturer voestalpine

Cutting technology Laser

Stress relief As cut

Direction of magnetizing field Parallel to rolling direction

Density [kg/m3] 7650

Mass [kg] 0.28999

Number of laminations 12

Lamination thickness [mm] 0.35

Strip size [mm] 30 × 300

Considering that in Eq. (2.4) the waveform of voltage 𝑢2 is sinusoidal, then the estimate of the

polarization on the magnetization curve for a given applied field 𝐻 is:

𝐽 = p1

𝑁𝑆+ 𝛿(𝐽)𝑢 + 𝛿(𝐽)𝑎 + 𝛿(𝐽)𝑇 + 𝛿(𝐽)𝑆 , (2.25)


where 𝑢p is the peak voltage measured for the respective 𝐽 value, 𝛿(𝐽)𝑢 is the bias correction

for voltage reading, 𝛿(𝐽)𝑎 is the bias correction for residual air-flux, 𝛿(𝐽)𝑇 is the bias correction

for temperature, and 𝛿(𝐽)𝑆 is the bias correction for the sample cross-sectional area.

The relative standard uncertainty of the polarization value is then expressed through the Type

A and Type B contributions (Eq. (2.21)) as:

𝑢c(𝐽)

𝐽 = √

𝑢A2(𝐽)

𝐽2+𝑢c2()

2+𝑢B2(𝐽)𝑎

𝐽2+𝑢B2(𝐽)𝑇

𝐽2+𝑢B2(𝑆)

𝑆2 , (2.26)

where: 𝑢A2(𝐽) is the Type A variance of the polarization readings, 𝑢c

2(𝑢) is the combined

variance of the voltage reading, 𝑢B2(𝐽)𝑎 is the Type B variance of the residual air-flux, 𝑢B

2(𝐽)𝑇

is the Type B variance of the temperature, and 𝑢B2(𝑆) is the Type B variance of the sample

cross-sectional area.

The sample cross-sectional area 𝑆 has been determined by the precise measurement of the total

mass 𝑚 and of the strip length 𝑙:

𝑆 =𝑚

4𝑙𝛿= 3.1589 × 10−5 m2 , (2.27)

where 𝛿 is the specific density of the material. The relative standard uncertainty of the sample

cross-sectional area is (Eq. (2.21)):

𝑢B(𝑆)

𝑆= √

𝑢B2(𝑚)

𝑚2+𝑢B2(𝑙)

𝑙2+𝑢B2(𝛿)

𝛿2 (2.28)

The combined variance 𝑢B2(𝑆) of quantity 𝑆 is obtained by combination of the variances of 𝑚,

𝑙 and 𝛿. For 𝛿 it is assumed a rectangular distribution of semi-amplitude

𝑎 = 25 kg/m3 [18]. The length 𝑙 has been measured using a digital calliper (Mitutoyo

500-505-10) of 10 𝜇m resolution, therefore, it is assumed a rectangular distribution of semi-

amplitude 𝑎 = 5 𝜇m. The mass 𝑚 has been measure using a compact balance (KERN PCB

1000-2) of 10 mg resolution, therefore, it is assumed a rectangular distribution of semi-

amplitude 𝑎 = 5 mg. The uncertainty budget for the determination of the sample cross-section

is presented in Table 2.4.

Doctoral thesis 41

Table 2.4: Uncertainty budget in the determination of the sample cross-sectional area

Source of

uncertainty

Distribution

function Divisor

Relative

uncertainty

Sensitivity

coefficient

Degrees of

freedom

Density Rectangular √𝟑 𝟏. 𝟗 × 𝟏𝟎−𝟑 1 ∞

Length reading Rectangular √𝟑 𝟗. 𝟔 × 𝟏𝟎−𝟔 1 ∞

Mass reading Rectangular √𝟑 𝟏𝟎−𝟓 1 ∞

Combined relative

standard uncertainty Rectangular - 𝟏. 𝟗 × 𝟏𝟎−𝟑 - ∞

The value of the uncertainty of the air-flux compensation is assumed to be

𝑢B(𝐽)𝑎/𝐽 = 5 × 10−4, as found in similar works [18]. The measurements have been performed

at room temperature and for the employed alloy any contribution to the uncertainty of the

temperature is considered negligible: 𝑢B(𝐽)𝑇/𝐽 ≈ 0. The voltage measurement has been

performed using the analog to digital converter of the NI PCI-6154. The combined standard

uncertainty of the voltage measurement 𝑢c(𝑢) has been determined using Eq. (2.28)(4.4) by

combining the Type A standard uncertainty of the readings and the Type B standard uncertainty

determined from the accuracy specifications of the manufacturer [69]. The standard Type B

uncertainty associated with the manufacturers’ specifications is evaluated by assuming that the

stated accuracy provides the half-width value 𝑎 of a rectangular distribution. The accuracy of

the voltage readings has been determined according to the specifications of the manufacturer

and the Type B uncertainty is presented in Fig. 2.9.

Fig. 2.9: Type B uncertainty of the voltage readings

The Type A uncertainty of the voltage measurements is presented in Fig. 2.10.


Fig. 2.10: Type A uncertainty of the voltage readings

The relative uncertainty of the voltage measurements 𝑢c(𝑢)

𝑢 is presented in Fig. 2.11.

Fig. 2.11: The relative uncertainty of the voltage readings

The Type A relative uncertainty 𝑢A(𝐽)

𝐽 of the polarization readings is presented in

Doctoral thesis 43

Fig. 2.12: The Type A relative uncertainty of the polarization observations

The combined relative standard uncertainty of the polarization 𝑢c(𝐽)

𝐽 is presented in Fig. 2.13.

Fig. 2.13: The combined relative uncertainty of the polarization measurement

The effective number of degrees of freedom have been calculated using Eq. (2.24) and the

values were found in the order of 106, therefore, the number of degrees of freedom has been

assumed to be ∞ and a coverage factor of 2 associated to a 95.45 % confidence level has been

selected. The expanded uncertainty for the measurement is presented in Fig. 2.14.


Fig. 2.14: The expanded uncertainty of the polarization measurement (𝑝 = 95%)

The uncertainty budget in the measurement of the magnetic polarization is presented in Table

2.5.

Table 2.5: Uncertainty budget in the measurement of the magnetic polarization

Source of

uncertainty

Distribution

function Divisor

Relative

uncertainty

Sensitivity

coefficient

Degrees of

freedom

Voltage reading Normal 1 Fig. 2.11 1 ∞

Polarization Normal 1 Fig. 2.12 1 14

Cross-sectional area Rectangular √3 1.9 × 10−3 1 ∞

Sample temperature Rectangular √3 0 1 ∞

Residual air-flux Rectangular √3 5 × 10−4 1 ∞

Combined relative

standard uncertainty Normal - Fig. 2.13 - ∞

Expanded uncertainty

(95 % confidence

level)

- - Fig. 2.14 - ∞

On average the expanded uncertainty of the measurement, with a 95.45 % confidence level,

has been measured to be 0.018 T. The experimental determination has revealed that the

expanded uncertainty of the polarization varies with the level of the polarization with a small

non-linear characteristic (Fig. 2.14). The two major sources of the uncertainty are the Type B

voltage measurement uncertainty and the uncertainty of the specific density of the material.

2.2.4. Critical analysis of different measurement procedures

A critical analysis is performed to analyse the performance of the different magnetic

measurement methods when testing a material used in the cores of particle accelerator magnets.

Doctoral thesis 45

The tested methodologies are the continuous recording method described by the IEC standard

60404-4 [30] and the polarization waveform control method described in this thesis [70]. The

critical analysis relies on experimental measurements performed with the experimental setup

described in the previous section.

A. Description of the sample and test procedure

The new measurement procedure has been cross-checked with the standard measurement

procedure by testing a sample of electrical steel with the two methods. The characteristics of

the sample are presented in Table 2.6.

Table 2.6: The sample used for the analysis of the new measurement procedure

Characteristic Value

Alloy Fe-Si

Quality M 270-50 A

Manufacturer C.D. Wälzholz

Cutting technology Laser

Stress relief As cut

Direction of magnetizing field Parallel to rolling direction


Electrical resistivity [𝜇Ω ⋅ cm] 55

Mass [kg] 1.0598

Number of laminations 28

Lamination thickness [mm] 0.5

Strip size [mm] 30 × 300

Two methods of measurements are applied: the continuous recording method (described by

standard IEC60404-4 and presented in section 2.1.1) with cycling periods for the excitation

field of 30 s and 60 s, and the flux waveform control method, with frequency of the polarization

of 1 Hz and 0.1 Hz. In order to cross-check the quantities measured with the different cycles

the peak polarization will be limited to the value of 1.5 T. The measurement cycles and their

symbols are presented in Table 2.7.

Table 2.7: Measurement cycles used for the measurement procedure analysis

No Symbol Description

1 M30 Standard procedure with cycle period of 30 s

2 M60 Standard procedure with cycle period of 60 s

3 MN1 New procedure with sinusoidal polarization at 1 Hz

3 MN10 New procedure with sinusoidal polarization at 0.1 Hz


The selected sample is made of a material likely to be used in the core of a particle accelerator

magnet and the selected measurement cycles highlight the differences of the measurement

methods. Therefore, the critical analysis highlights which is the optimal method to be used to

characterize a magnetic material intended to be used in the core of a particle accelerator magnet.

B. Experimental results

Fig. 2.15 show the waveform of the excitation field and Fig. 2.16 show the waveform of the

magnetic polarization for the M30 measurement cycle. The current is ramped linearly with a

cycling period of 30 s and the resulting waveform of the polarization follows the material

properties. The scale of the time axis has been set to 60 s for visual cross-checking with the

M60 measurement cycle.

Fig. 2.15: Polarization waveform for M30 cycle


Fig. 2.17 show the waveform of the excitation field and Fig. 2.18 show the waveform of the

magnetic polarization for M60 measurement cycle. The current is ramped linearly with a

cycling period of 60 s and the resulting waveform of the polarization follows the material

properties.

Doctoral thesis 47

Fig. 2.17: Excitation waveform for M60 cycle


Fig. 2.19 shows the waveform of the excitation field 𝐻(𝑡) and Fig. 2.20 shows the waveform

of the magnetic polarization 𝐽(𝑡) for MN1 measurement cycle. The waveform of the excitation

current is modulated such that the waveform of the polarization is sinusoidal at 1 Hz.


Fig. 2.19: Excitation waveform for MN1 cycle

Fig. 2.20: Polarization waveform for MN1 cycle

Fig. 2.21 show the waveform of the excitation field 𝐻(𝑡) and Fig. 2.22 show the waveform of

the magnetic polarization 𝐽(𝑡) for MN10 measurement cycle. The waveform of the excitation

current is modulated such that the waveform of the polarization is sinusoidal at 0.1 Hz.

Doctoral thesis 49

Fig. 2.21: Excitation waveform for MN10 cycle

Fig. 2.22: Polarization waveform for MN10 cycle

In Fig. 2.23 are presented the superimposed magnetic hysteresis cycles for the MN1 and MN10

measurements.

Fig. 2.23: Major hysteresis cycles for MN1 and MN10 measurements


The broadening of the hysteresis cycle visible in Fig. 2.23 is due to the dynamic effects which

increase with the frequency, as highlighted by Eq. (2.1).

The results of the measurements are presented in Table 2.8.

Table 2.8: Experimental results of the measurement procedure analysis

Quantity M30 M60 MN10 MN1

𝐽peak [T] 1.5 1.5 1.5 1.5

𝐻peak [A/m] 703 703 702 707

𝑊m [mJ/kg] 31.8 31.1 31.3 34.1

𝐻c [A/m] 48.9 47.9 48.7 52.4

𝐽r [T] 0.39 0.39 0.39 0.41

𝐽c [T/s] 1.05 0.53 0.94 9.42

𝐻eddy,c [A/m] 0.13 0.07 0.12 1.18

𝑇 [s] 30 60 10 1

The quantities presented in Table 2.8 are:

𝐻peak is the peak value of 𝐻 of the cycle;

𝐽peak is the peak value of 𝐽 of the cycle;

𝑊m is the magnetic energy loss per unit mass; it is obtained by dividing the area of the

hysteresis cycle to the density of the material;

𝐻c is the coercivity of the cycle;

𝐽r is the remanence of the cycle;

𝐽c is the rate of change of the magnetic polarization at the coercivity point;

𝐻eddy,c is the magnetic field due to eddy-currents which opposes the applied magnetic

field at the coercivity point in the measurement cycle;

𝑇 is the period of the cycle.

C. Critical analysis

The source of the differences observed in the measurement results originate mostly in the

dynamic effects which appear during the measurements and which are proportional to the rate

of change of the polarization, 𝐽. Measurements M30 and MN10 have very similar values for 𝐽

and, therefore, can be cross-checked. Some of the parameters of these two measurements have

similar values (𝑊m, 𝐻c, 𝐽r).

Doctoral thesis 51

For M30 and M60 measurement cycles the results are influenced by the magnetic properties of

the sample. This can be shown by expressing the rate of change of the polarization as:

𝐽 =d𝐽

d𝐻⋅d𝐻

d𝑡= 𝜇r,inc𝜇0

d𝐻

d𝑡 (2.29)

where: 𝜇r,inc is the incremental permeability of the material. Therefore, measurement cycles

M30 and M60, which use the standard measurement procedure, will be affected by dynamic

effects which are proportional to the incremental permeability of the material.

Measurement cycle MN1 has the ramping rate of the polarization close to the operating value

of a particle accelerator magnet. For this measurement cycle the values of the quantities which

characterise the hysteresis cycle increase with approx. 7 %. Therefore, by using the new

measurement procedure a material can be tested with the magnetization regime foreseen for

the magnet. Under these conditions the new method provides faster and more accurate

measurements.

With the standard recording method high quality electronic components are required to prevent

the drift of the signal and to ensure sufficient sampling rate. On the other hand, by controlling

the rate of change of the induction the uncertainty of the measurement is improved by

decreasing the strain on the dynamic range of the ADC and by decreasing the integrating time.

One of the limitations of using the new measurement procedure is that low error d.c.

measurements may require more time to complete despite the fact that a small number of

iterations are required to achieve convergence of the algorithm. Similar measurement

procedure found in literature use a mathematical model to track the hysteresis characteristic.

By using look-up tables instead of mathematical models the error for tracking the hysteresis

characteristic is decreased. Nevertheless, by cycling the material more than once between the

same levels ensures that the error of the measurement due to the assumption that material

operates on a closed cycle is minimized.

The procedure can be modified to create random waveforms of the magnetic polarization. Thus,

the new magnetic measurement procedure is the tool required to predict the behaviour of a

material with controlled magnetizing conditions.

By measuring the magnetic properties of a material with sinusoidal polarization waveform at

1 Hz a ramping rate of the polarization similar to the one found in the core of a particle

accelerator magnet has been achieved. At this ramping rate the measured quantities showed a

7 % increase compared to standardised d.c. magnetic measurements. Therefore, measurements

performed with controlled rate of change of the polarization are better suited to estimate the

magnetic properties of a material used in the yoke of a PA magnet.


2.2.5. Development of new curve fitting method

The analysis of materials with non-linear magnetic properties using finite element analysis

requires the modelling of the reluctivity 𝑣 as a function of 𝐵2 [71, 72]. This curve is required

to be continuous, monotonically varying and must have continuous first derivative to fully

exploit the quadratic convergence properties of the Newton process. Also, mathematical

models of physical systems require processing of information extracted from experimental data

for their identification [73, 74, 75]. Since experimental data is characterized by noise and

manual correction of data is unworkable, a computer based procedure is required to identify

the fundamental waveform of experimental data characterized by noise. The identification is

done by finding the parameters of a model equation.

A. Background

The first attempts to estimate the magnetization characteristic of ferromagnetic materials

started with the observation of Rayilegh [76] which compared the magnetization curve in the

low field area with a parabola. The Frölich equation is a classical equation used to approximate

the magnetization curve [77]:

𝐵 =𝐻

𝑐 + 𝑏|𝐻| , (2.30)

where 𝑐 and 𝑏 are the constants describing the curve parameters. This equation is preferred due

to its simplicity, but it yields very rough approximations. Later, many analyses were performed

for different algebraic functions [78, 79, 80].

In their work, Trutt et al [81] and Schenk et al [82] proposes the approximation of the

reluctivity by means of linear interpolation of data points by using numerical methods. The

computing time is reported to decrease and the magnetization characteristic is well

approximated. Further improvement of this procedure is presented by Chatterjee [83] which

proposes a mixed scheme which employs quadratic interpolation of the values intermediate the

provided points.

The rational fraction is a simple method of approximating the magnetization characteristic

which provides a compromise between accuracy and simplicity [84, 85, 86]. Due to the small

number of operations the solutions to these equations are very fast to compute. On the other

hand, in order to accurately approximate the magnetization curve for the full range the number

of parameters has to be increased considerably.

The estimation of the magnetization curves by means of sum of polynomials is a method to

store the information in a compact form [87]. By using a large number of parameters the

magnetization curve can be approximated very accurately. On the other hand, for increasing

number of parameters their identification becomes laborious and the result may give non-

monotone fits.

Doctoral thesis 53

The magnetization curve begins in the origin and rises to approach asymptotically a constant

value as the excitation field increases. This behaviour can be roughly estimated using one or a

series of exponential functions [88, 89, 90]. However, it has been found that an exponential or

any summation of exponential leads to a large error, especially around the origin and the knee

of the curve [89].

Another curve-fitting techniques employed for the representation of the magnetization curve

or for close hysteresis cycles is by using Fourier series [91, 92]. With this technique, a larger

number of coefficients leads to lower error in reproducing the curve at the expense of execution

time [91].

The tangent and cotangent hyperbolic functions are also employed where a simple and gross

approximation of the magnetization characteristic is required. They are often used with the

Jiles-Atherton model of hysteresis to approximate the bulk magnetization [93, 94, 95, 96, 97].

Just like all the other functions the hyperbolic functions can be used to accurately approximate

the magnetization curve only for some regions.

In the literature can be found additional methods to approximate the magnetization curve.

Hejda [98] uses the natural logarithmic function to approximate the magnetization

characteristic. Levi et al [99] uses a two parameter analytical function to approximate the

inverse magnetization characteristic.

In order to obtain a very smooth curve from experimental measurements the raw data has to be

fitted to a prototype function. The literature review has shown that a single-function

approximation of the magnetization curve is rarely satisfactory over the entire magnetization

range. Therefore, a curve fitting procedure which uses very simple relations to approximate

many small portions of the curves has been developed [100, 101]. The requirement for finite

element analysis is that the first derivative of the magnetization curve is continuous.

Nevertheless, the second derivative of this curve can provide valuable information, therefore,

the new procedure has been developed such that the second derivative of the curve is

continuous.

B. The procedure

The input data are decomposed in a series of segments and for each segment is determined the

coefficients of the cubic polynomials by means of linear regression. The coefficients of the

polynomials are constrained by the condition of continuity of the fitted curve and its first and

second derivatives, which will ensure the continuity and smoothness of the resulting curve and

its derivative. Due to the imposed constraints the polynomial for each segment can be rewritten

with the coefficients of the previous segment, and so on down to the first segment; thus with

increasing number of segments, the complexity of the 𝑛th polynomial will also increase. In the

following are presented the equations describing the polynomials, for any number of segments

used for regression, and the method to determine the coefficients of the polynomials by means

of linear regression.


When making magnetic measurements usually is obtained only one measurement of the

dependent quantity (𝐽, 𝐻c, 𝐽r) for a single independent quantity (𝐻). For this reason a certain

distribution of the measured values around the mean has to be assumed. For the developed

method the Gaussian distribution is assumed. A series of assumptions are made when fitting a

curve to a set of data points by means of regression analysis: the independent quantity is error

free, the response of the dependent quantity is a linear combination of the regression

coefficients and of the independent variables, all values are characterized by the same

deviation, the error of one point does not influence the error of another point [102, 103, 104,

105, 106, 107, 108].

In order to be able to study the derivative of the fitted curves cubic polynomial functions will

be used which will ensure that the derivative of the resulting curve is both continuous and

smooth. The polynomial function can be expressed as:

𝑦(𝑥) = ∑𝑎𝑘𝑥𝑘

𝑚

𝑘=0

, (2.31)

where: 𝑦 is the dependent variable estimated by the polynomial, 𝑎𝑘 are the coefficients of the

polynomial, and 𝑥 is the independent variable.

Assuming normal error distribution for the experimental data, the probability of obtaining the

observed set of measurements is [102]:

𝑃(𝑎0, … , 𝑎𝑚) =∏(1

𝜎√2𝜋) ⋅ exp(−

1

2∑

1

𝜎2(𝑦𝑖 −∑𝑎𝑘𝑥

𝑘

𝑚

𝑘=0

)

2

) . (2.32)

The goodness of fit parameter is defined from Eq. (2.32) as:

𝜒2 =∑(1

𝜎2(𝑦𝑖 −∑𝑎𝑘𝑥

𝑘

𝑚

𝑘=0

))

2

. (2.33)

The method of least squares requires to improve the goodness of the fit to the data, therefore,

𝜒2 is required to be minimized with respect to the parameters 𝑎𝑘. By assuming that the standard

deviation 𝜎𝑖 is constant for all input data this term can be neglected in the following steps. The

minimum is determined by setting to zero the partial derivatives of Eq. (2.33) with respect to

each parameter. A set of 𝑚 + 1 coupled linear equations are obtained.

The first step of the method is to break the input data into 𝑛 segments (Fig. 2.24), to align them

(Fig. 2.25), and to resample each segment to 𝑝 points using numerical methods. The curve

presented in Fig. 2.24 is the normal magnetization curve for AFK502R, a Fe-Co alloy.

Doctoral thesis 55

Fig. 2.24: Five level segmentation of the normal magnetization curve

Fig. 2.25: Segmented and aligned input data

The equations presented in this section take into account the assumptions implicit to least

square fit of a curve to a polynomial by means of regression analysis. The order of the

polynomial has to be higher than three in order to accommodate the constraints and to allow

for a degree of freedom. The constraints which impose the continuity and smoothness of the

resulting curves and their derivatives are defined as:

𝑓𝑛(Δ𝑥) = 𝑎3,𝑛Δ𝑥𝑛3 + 𝑎2,𝑛Δ𝑥𝑛

2 + 𝑎1,𝑛Δ𝑥𝑛 + 𝑎0,𝑛 ,

𝑓𝑛(0) = 𝑓𝑛−1(𝑋𝑛) ,

𝑓𝑛′(0) = 𝑓𝑛−1

′ (𝑋𝑛) ,

𝑓𝑛′′(0) = 𝑓𝑛−1

′′ (𝑋𝑛) ,

(2.34)


where: Δ𝑥𝑛 = 𝑥 − 𝑥(𝑙𝑛−1) is the value of the input data on each segment with respect to the

first value on the segment, 𝑋𝑛 = 𝑥(𝑙𝑛) − 𝑥(𝑙𝑛−1) is the end value of each segment, 𝑙𝑛 is the

level of the segment and 𝑛 is the order of the segment.

From the set of equations (2.34) it results:

𝑎0,𝑛 = 𝑎3,𝑛−13 𝑋𝑛

3 + 𝑎2,𝑛−12 𝑋𝑛

2 + 𝑎1,𝑛−1𝑋𝑛 + 𝑎0,𝑛−1 ,

𝑎1,𝑛 = 3𝑎3,𝑛−1𝑋𝑛2 + 2𝑎2,𝑛−1𝑋𝑛 + 𝑎1,𝑛−1 ,

𝑎2,𝑛 = 6𝑎3,𝑛𝑋𝑛 + 2𝑎2,𝑛−1

(2.35)

From relations (2.34) and (2.35) it can be seen that the coefficients of the polynomial from one

segment are combined with the coefficients from the previous segments. With increasing

number of intervals 𝑛 the complexity of 𝑓𝑛 also increases. In order to do the least squares

estimation it is required to obtain a general equation for each 𝑓𝑛. Considering the constraints

in (2.34) the equation for the polynomial at the 𝑛th segment is:

𝑓𝑛(Δ𝑥𝑛) = 𝑎0,1 + 𝑎1,1 (Δ𝑥𝑛 +∑𝑋𝑖

𝑛−1

𝑖=1

) + 𝑎2,1 (Δ𝑥𝑛 +∑𝑋𝑖

𝑛−1

𝑖=1

)

2

+

∑𝑎3,𝑠∑((Δ𝑥𝑖 +∑𝑋𝑡

𝑖−1

𝑡=𝑠

)

3

− (Δ𝑥𝑖 + ∑ 𝑋𝑡

𝑖−1

𝑡=𝑠+1

)

for 𝑖≠𝑠

3

)

𝑛

𝑖=𝑠

𝑛

𝑠=1

.

(2.36)

The system of equations which has to be solved to determine the coefficients of the polynomials

is determined as:

𝐷𝑛 =∑(𝑦𝑛,ℎ − 𝑓𝑛(Δ𝑥𝑛,ℎ))2

𝑝

ℎ=1

,

𝑆 =∑𝐷𝑖

𝑛

𝑖=1

,

𝜕𝑆

𝜕𝑎𝑧,1= 0; 𝑧 = 0,1,2

𝜕𝑆

𝜕𝑎3,𝑛= 0

,

(2.37)

Doctoral thesis 57

where 𝑝 is the number of points in each data sets. The system will have 𝑛 + 3 equations; which

is the number of parameters of the regression. The derivatives in the system of equations (2.37)

are determined:

𝜕𝑆

𝜕𝑎0,1= 0

⇒ 𝑎0,1 𝑛𝑝⏟𝐴1,1

+ 𝑎1,1∑∑(Δ𝑥𝑖,ℎ +∑𝑋𝑡

𝑖−1

𝑡=1

)

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴1,2

+ 𝑎2,1∑∑(Δ𝑥𝑖,ℎ +∑𝑋𝑡

𝑖−1

𝑡=1

)

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴1,3

2

+∑

(

𝑎3,𝑠∑∑((Δ𝑥𝑖,ℎ +∑𝑋𝑡

𝑖−1

𝑡=1

)

3

− (Δ𝑥𝑖,ℎ + ∑ 𝑋𝑡

𝑖−1

𝑡=𝑠+1

)

for 𝑖≠𝑠

3

)

𝑝

ℎ=1

𝑛

𝑖=𝑠⏟ 𝐴1,𝑠+3 )

𝑛

𝑠=1

=∑∑𝑦𝑖,ℎ

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝑌1

.

(2.38)

The following notations are made:

𝜆 = Δ𝑥𝑖,ℎ +∑𝑋𝑡

𝑖−1

𝑡=1

,

𝛿 =∑∑((Δ𝑥𝑖,ℎ +∑𝑋𝑡

𝑖−1

𝑡=1

)

3

− (Δ𝑥𝑖,ℎ + ∑ 𝑋𝑡

𝑖−1

𝑡=𝑠+1

)

for 𝑖≠𝑠

3

)

𝑝

ℎ=1

𝑛

𝑖=𝑠

.

(2.39)

Then:

𝜕𝑆

𝜕𝑎1,1= 0 ⇒ 𝑎0,1∑∑𝜆

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴2,1

+ 𝑎1,1∑∑𝜆2𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴2,2

+ 𝑎2,1∑∑𝜆3𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴2,3

+∑

(

𝑎3,𝑠 𝛿∑∑𝜆

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴2,𝑠+3 )

𝑛

𝑠=1

=∑∑𝑦𝑖,ℎ𝜆

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝑌2

,

(2.40)


𝜕𝑆

𝜕𝑎2,1= 0 ⇒ 𝑎0,1∑∑𝜆2

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴3,1

+ 𝑎1,1∑∑𝜆3𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴3,2

+ 𝑎2,1∑∑𝜆4𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴3,3

+∑

(

𝑎3,𝑠 𝛿∑∑𝜆2

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴3,𝑠+3 )

𝑛

𝑠=1

=∑∑𝑦𝑖,ℎ𝜆2

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝑌3

.

(2.41)

Considering 𝑣 = 1…𝑛, then:

𝜕𝑆

𝜕𝑎3,𝑣= 0 ⇒ 𝑎0,1 𝛿⏟

𝐴𝑣+3,1

+ 𝑎1,1 𝛿∑∑𝜆

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴𝑣+3,2

+ 𝑎2,1 𝛿∑∑𝜆2𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝐴𝑣+3,3

+∑(𝑎3,𝑠 𝛿2⏟𝐴𝑣+3,𝑠+3

)

𝑛

𝑠=1

=∑∑𝑦𝑖,ℎ𝛿

𝑝

ℎ=1

𝑛

𝑖=1⏟ 𝑌𝑣+3

.

(2.42)

The system of equations in (2.37) can now be written in matrix form:

(

𝐴1,1 ⋯ 𝐴1,𝑛+3⋮ ⋱ ⋮

𝐴𝑛+3.1 ⋯ 𝐴𝑛+3,𝑛+3

) ⋅ (

𝑎0,1⋮𝑎3,𝑛

) = (𝑌1⋮

𝑌𝑛+3

) (2.43)

The proposed algorithm has been implemented in Matlab and the solutions to the matrices have

been obtained using the built-in mldivide function. The Matlab code for the procedure is

presented in Annex 2.

C. Experimental results

The new curve fitting method has been used to process the normal magnetization curve of

AFK502R (Fe-Co alloy). The number of segments has been set to 10. The first derivative of

the curves has been investigated using the incremental relative permeability 𝜇r′ . The

experimental incremental relative permeability has been obtained using numerical methods

applied to experimental data:

𝜇r,e′ =

Δ𝐵

𝜇0Δ𝐻 . (2.44)

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The fitted incremental relative permeability has been obtained using the analytical expression

of the derivative of the fit (𝑣 = 1… . 𝑛):

𝜇r,f′ =

1

𝜇0𝑓𝑣′(Δ𝐻𝑣) =

1

𝜇0(3𝑎3,𝑣ΔH𝑣

2 + 2𝑎2,𝑣Δ𝐻𝑣 + 𝑎1,𝑣) . (2.45)

The experimental and the fitting result of the normal magnetization curve of AFK502R are


Fig. 2.26: Experimental and fitted normal magnetization curve of AFK502R

The experimental and the fitting result of the incremental permeability curve of AFK502R are


Fig. 2.27: Experimental and fitted incremental permeability of AFK502R

Fig. 2.26 and Fig. 2.27 show that the new curve fitting procedure generates a continuous and

smooth output for both the resulting curve and for its first derivative. As the fitting has been


performed on cubic polynomials the unique possibility to observe the curves second derivative

is possible (𝑣 = 1… . 𝑛):

𝜇′′ = 𝑓𝑣′′(Δ𝐻𝑣) = 6𝑎3,𝑣Δ𝐻𝑣 + 2𝑎2,𝑣 . (2.46)

In Fig. 2.28 is presented the second derivative of the normal magnetization curve of AFK502R.

Fig. 2.28: The second derivative of the normal magnetization curve of AFK502R

Fig. 2.28 show that the second derivative of the curve obtained analytically from the fit is

continuous and its values can be used for further processing. One possible application of the

second derivative is the identification of the curvature regions of a curve. In these regions the

absolute value of the second derivative will increase. This property has been used to increase

the scattering of the points in the curvature regions when storing information of the normal

magnetization curve with a limited number of points (max 50 points for Opera bh files).

In Fig. 2.29 are presented the points selected to be saved to the bh file with increased scattering

in the curvature regions.

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Fig. 2.29: Points selected with high scattering in the curvature regions

The script developed to create the material characteristic files for Opera, with increased

scattering in the curvature regions, is presented in Annex 3. The experimental data obtained

using the new procedure for magnetic testing is processed (at step 14 in Fig. 2.8) using the

developed curve fitting method.

2.3. Experimental characterization of Fe-Si alloys

Electrical steels offer a very good trade-off between price and performance and are the usual

choice for manufacturing the cores of particle accelerators. The measurements performed

during the experimental research are used to analyse these materials. The experimental

measurement campaign is aimed at investigating the following characteristics:

The variation of the magnetic properties of Fe-Si alloys.

The anisotropy of Fe-Si alloys.

The effect of annealing Fe-Si alloys.

The properties of Fe-Si alloys with identical grading.

The influence of the chemical composition on the magnetic and electric properties.

Samples of electrical steel have been tested using the newly developed procedure, presented in

section 2.2. The family of symmetric hysteresis cycles has been measured for each sample with

sinusoidal waveform of the magnetic polarization at 1 Hz. The curves analysed in the following

sections have been obtained by processing these measurements using the methods described in

section 2.2.


2.3.1. The spread of the magnetic properties of Fe-Si alloys

The magnetic properties of electrical steels from the same charge which were cut into different

coils have been investigated. The steel samples were produced from two grades of NGO

electrical steel manufactured by C.D. Wälzholz. The presented measurements are for the

samples cut in the rolling direction. The details of the samples are presented in Table 2.9.

Table 2.9: Details of the samples from different coils

Sample Grade Charge Coil no. Position

SM1.1 M 800-50 A 00970111 12968837 Middle

SM1.2 M 800-50 A 00970111 12968740 Middle

SM2.1 M 470-50 A 00456981 12988173 End

SM2.2 M 470-50 A 00456981 12979551 End

SM2.3 M 470-50 A 00456981 12988173 End

The normal magnetization curves of the samples SM1.1 and SM1.2 cut from grade

M 800-50 A are presented in Fig. 2.30.

Fig. 2.30: The normal magnetization curves for grade M 800-50 A

The coercivity curves of the samples SM1.1 and SM1.2 cut from grade M 800-50 A are


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Fig. 2.31: Coercivity curves for M 800-50 A

The significant measurement results of samples SM1.1 and SM1.2 are summarized in Table

2.10.

Table 2.10: Analysis of the normal magnetization curve and of the coercivity curve of M 800-50 A

𝑱 [𝐓] 𝑯 [𝐀/𝐦] 𝑯𝐜 [𝐀/𝐦]

SM1.1 SM1.2 SM1.1 SM1.2

0.1 41.6 46.7 10.7 11.5

0.25 60.1 67.7 31.4 35.4

0.5 70.2 78.7 51.4 57.8

0.75 78.7 86.6 59.3 66.4

1 90.2 96.5 64.7 72.8

1.25 127.8 133.3 70.4 78.9

1.5 467.5 461.6 78.6 88.2

1.7 3240 3016 80.4 90.6

The maximum difference between the two curves is approx. 0.25 T. This value is observed

when the material operates on a 75 A/m cycle.

The normal magnetization curves of the samples SM2.1, SM2.2 and SM2.3 cut from grade

M 470-50 A are presented in Fig. 2.32.


Fig. 2.32: The normal magnetization curves for M 470-50 A

The maximum difference between the curves is approx. 0.24 T. This value is observed when

the material operates on a 30 A/m cycle.

The coercivity curves of the samples SM2.1, SM2.2 and SM2.3 cut from grade M 470-50 A

are presented in Fig. 2.33.

Fig. 2.33: The coercivity curves for M 470-50 A

The significant measurement results of samples SM2.1, SM2.2 and SM2.3 are summarized in

Table 2.11.

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Table 2.11: Analysis of the normal magnetization curve and of the coercivity curve of M 470-50 A

𝑱 [𝐓] 𝑯 [𝐀/𝐦] 𝑯𝐜 [𝐀/𝐦]

SM2.1 SM2.2 SM2.3 SM2.1 SM2.2 SM2.3

0.1 19.3 15.9 20.3 5.7 4.8 5.9

0.25 27.8 23.3 29.2 15 12.3 15.8

0.5 34 29.6 35.6 23.5 19.7 24.4

0.75 41.9 38.2 43.5 28 23.7 29.3

1 62 59.5 62.8 31.8 27.5 33.2

1.25 135.3 136.4 129.7 35.5 30.9 37.2

1.5 896.1 951 788.4 37.9 32.4 39.8

1.65 3536 3657 3186 38.2 32.9 40.2

The coercivity of sample SM2.2 is lower compared to SM2.1 and SM2.3. The observed

difference is approx. 7.5 A/m for magnetization cycles above 1.5 T.

The magnetic characteristics of sheets of NGO electric steel produced from the same charge

had a variation in the range of 20 % for M 470-50 A and 12 % for M 800-50 A. The variation

of the magnetic properties in a charge is implicit and the magnet designer has to take this

variation into consideration when setting the tolerances. The variation of the magnetic

properties can be decreased, during steel manufacturing and processing, by paying close

attention to [19]: avoidance of stress in assembly, provision of appropriate annealing,

avoidance of burrs of laminations and avoidance of inter-lamination short-circuits.

2.3.2. The anisotropy of Fe-Si alloys

A comparative analysis of samples manufactured from two grades of conventional grain

oriented (CGO) steel and two grades of non-grain oriented (NGO) steel has been performed.

The samples have been laser cut at angles from 0 to 90 degrees in steps of 5 degrees with

respect to the rolling direction. The details of the samples are presented in Table 2.12.

Table 2.12: Details of the samples used for the anisotropy study

Sample Grain texture Grade Manufacturer

GO1 Grain oriented M 140-35 S Cogent

GO2 Grain oriented M 165-35 S Arcelor-Mittal

NGO1 Non-grain oriented isovac 250-35 A HP voestalpine

NGO2 Non-grain oriented M 235-35 A Cogent

Also, the direction where a material exhibits best performance is called easy magnetization

axis and the direction where the material exhibits worst performance is called hard

magnetization axis.


A. The anisotropy of NGO Fe-Si alloys

The normal magnetization curves for sample NGO1 at 0 and 90 degrees with respect to the

rolling direction are presented in Fig. 2.34.

Fig. 2.34: The normal magnetization curves of NGO1

The dependence of the excitation field on the direction of magnetization for 1 T and 1.5 T in

sample NGO1 is presented in the form of a polar plot in Fig. 2.35.

Fig. 2.35: Excitation vs. orientation for NGO1

In Fig. 2.34 and Fig. 2.35 one sees that the easy magnetization axis of sample NGO1 is directed

along the rolling direction and that the hard magnetization axis is oriented along the 55 degrees

direction. Although for samples NGO1 the hard magnetization axis appears to be oriented

along the 55 degrees angle, as for GO1 and GO2 samples, the performance of the sample at 90

degrees is very similar to the performance at 55 degrees. The anisotropy of the magnetization

for the cycles measured at 1 T was 25 %, while for the cycles measured at 1.5 T was 29.2 %.

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The coercivity curves for sample NGO1 at 0 and 90 degrees with respect to the rolling direction


Fig. 2.36: Coercivity curves of NGO1

The coercivity of sample NGO1 for cycles with peak magnetic polarization amplitude of 1 T

and 1.5 T versus the direction of the magnetic flux is presented in the form of a polar plot in

Fig. 2.37.

Fig. 2.37: Coercivity vs. orientation for NGO1

The experimental measurements show that the coercivity of sample NGO1 is characterized by

increased isotropy when compared to the isotropy of the excitation field. The lowest coercivity

values have been observed in the rolling direction and the highest values have been observed

transversal to rolling direction. The anisotropy of the coercivity for the cycles measured at 1 T

is 19.5 %, while for the cycles measured at 1.5 T is 17.6 %.

The normal magnetization curves for sample NGO2 at 0 and 90 degrees with respect to the



Fig. 2.38: The normal magnetization curves of NGO2


sample NGO2 is presented in the form of a polar plot in Fig. 2.39.

Fig. 2.39: Excitation vs. orientation for NGO2

In Fig. 2.38 and Fig. 2.39 one sees that for sample NGO2 the easy magnetization axis is

oriented along the rolling direction and that the hard magnetization axis is oriented along the

80 degrees direction. Similarly to sample NGO1 the difference between the performances of

sample NGO2 along the hard magnetization axis and the 90 degrees direction are very similar.

The anisotropy of the magnetization for the cycles measured at 1 T has been 24.2 %, while for

the cycles measured at 1.5 T has been 29.3 %.

The coercivity curves for sample NGO2 at 0 and 90 degrees with respect to the rolling direction


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Fig. 2.40: Coercivity curves of NGO2

The coercivity of sample NGO2 for cycles with peak magnetic polarization amplitude of 1 T

and 1.5 T versus the direction of the magnetic flux is presented in the form of a polar plot in

Fig. 2.41.

Fig. 2.41: Coercivity vs. orientation for NGO2

In Fig. 2.40 and Fig. 2.41 one sees that the coercivity of sample NGO2 follows a similar

characteristic as for sample NGO1. The lowest coercivity is found in the rolling direction while

the largest coercivity is found transversal to rolling direction. The anisotropy of the coercivity

for the cycles measured at 1 T was 20.8 % and for the cycles measured at 1.5 T was 17.4 %.

The measurements of the anisotropy of steel samples NGO1 and NGO2 are summarized in

Table 2.13.


Table 2.13: Experimental measurement results for NGO steels

No. Angle

[degrees]

𝑯(𝟏 𝐓) [𝐀/𝐦]

𝑯(𝟏. 𝟓 𝐓) [𝐀/𝐦]

𝑯𝐜(𝟏 𝐓) [𝐀/𝐦]

𝑯𝐜(𝟏. 𝟓 𝐓) [𝐀/𝐦]

NGO1 NGO2 NGO1 NGO2 NGO1 NGO2 NGO1 NGO2

1 0 192.7 169.1 1159 1538 40.1 29.3 45.3 32.7

2 5 188 195.3 1024 1554 42.4 32.9 48.3 36.1

3 10 194 170.6 1179 1530 41.4 30.3 46.6 33.7

4 15 200.8 170.4 1174 1519 43.4 31.7 49.3 34.6

5 20 210.5 193.4 1383 1508 43 33.7 48.1 37

6 25 209.2 200 1351 1567 44.6 35 50.5 37.7

7 30 230.3 203.7 1597 1560 45.5 36.5 50.6 39.8

8 35 237.2 206.9 1694 1643 46.7 37.2 52.3 39.6

9 40 238.9 213.8 1809 1740 48.7 37.3 54 40.3

10 45 252 222.2 1873 1841 50.6 37.7 55.9 40.1

11 50 291.8 232 2186 1881 49.7 37.7 54.3 40

12 55 285.7 240.4 2114 2045 52.6 38.4 57.3 40.8

13 60 288.6 253.2 2075 2220 24 38.3 59.6 40.8

14 65 333.9 254.2 2143 2222 55.6 38.9 60.2 41.4

15 70 313.7 260.5 1983 2291 58.7 39.8 63.8 42.1

16 75 364.3 262.6 2077 2568 58.1 40.8 63.3 43.4

17 80 304.3 277.4 1815 2810 55.9 42.4 61.3 45.1

18 85 336.6 257.5 1854 2553 58.6 42.5 64.2 44.8

19 90 321 277 1746 2391 59.5 44.7 64.6 46.5

For the tested NGO steels the easy magnetization axis appears to be oriented along the rolling

direction and the hard magnetization axis oriented at 55 degrees with respect to the rolling

direction for sample NGO1 and at 80 degrees with respect to the rolling direction for sample

NGO2. The coercivity has been observed to be minimum when measured along the rolling

direction and maximum when measured transversal to the rolling direction. Both NGO samples

exhibited similar anisotropy values: the magnetization anisotropy has been approx. 29 % at

1.5 T and approx. 25 % at 1 T, while the coercivity anisotropy has been approx. 17 % at 1.5 T

and approx. 20 % at 1 T.

B. The anisotropy of GO Fe-Si alloys

The normal magnetization curves for sample GO1 at 0, 55 and 90 degrees with respect to the


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Fig. 2.42: The normal magnetization curves of GO1


sample GO1 is presented in the form of a polar plot in Fig. 2.43.

Fig. 2.43: Excitation vs. orientation for GO1

In Fig. 2.43 one sees that for sample GO1 the orientation of the hard magnetization axis changes

its direction with the level of polarization. Thus, up to approx. 1.1 T the direction of the hard

magnetization axis is at 90 degrees with respect to the rolling direction and for larger

polarization values this direction changes to 55 degrees. For the samples cut at 55 degrees the

measurement setup was not able to generate an excitation field high enough to reach a

polarization level of 1.5 T. This behaviour is typical for the Goss texture [24, 25]. The

anisotropy of the magnetization for the cycles measured at 1 T was 48.2 %, while for the cycles

measured at 1.2 T was 68.5 %.

The coercivity curves for sample GO1 at 0, 55 and 90 degrees with respect to the rolling

direction are presented in Fig. 2.44.


Fig. 2.44: Coercivity curves of GO1

The coercivity of sample GO1 for cycles with peak magnetic polarization amplitude of 1 T and

1.2 T versus the direction of the magnetic flux is presented in the form of a polar plot in Fig.

2.45.

Fig. 2.45: Coercivity vs. orientation for GO1

In the measurements presented in Fig. 2.44 and Fig. 2.45 one sees that the coercivity of sample

GO1 is not influenced by the hard magnetization axis. The lowest coercivity is exhibited by

the samples cut in rolling direction and the largest coercivity is exhibited by the samples cut

transversal to rolling direction. Also, for angles from 25 to 40 degrees there appears to be a

flat-top region where the coercivity has constant value. The anisotropy of the coercivity for the

cycles measured at 1 T was 38.7 %, while for the cycles measured at 1.2 T was 37.8 %.

The normal magnetization curves for sample GO2 at 0, 55 and 90 degrees with respect to the


Doctoral thesis 73

Fig. 2.46: The normal magnetization curves of GO2


sample GO2 is presented in the form of a polar plot in Fig. 2.47.

Fig. 2.47: Excitation vs. orientation for GO2

In the measurements presented in Fig. 2.46 and Fig. 2.47 one sees that the hard magnetization

axis of sample GO2 follows the same pattern as sample GO1. Up to polarization values of

approx. 1.1 T the hard magnetization axis is oriented transversal to the rolling direction. For

higher polarization values the orientation of the hard magnetization axis changes its orientation

to 55 degrees with respect to the rolling direction. The anisotropy of the magnetization for the

cycles measured at 1 T was 60.1 %, while for the cycles measured at 1.2 T was 71.4 %.

The coercivity curves for sample GO2 at 0, 55 and 90 degrees with respect to the rolling

direction are presented in Fig. 2.48.


Fig. 2.48: Coercivity curves of GO2

The coercivity of sample GO2 for cycles with peak magnetic polarization amplitude of 1 T and

1.2 T versus the direction of the magnetic flux is presented in the form of a polar plot in Fig.

2.49.

Fig. 2.49: Coercivity vs. orientation for GO2

In Fig. 2.48 and Fig. 2.49 one sees that the coercivity of sample GO2 follows a similar patters

to that of sample GO1. The lowest coercivity is found with the samples cut in rolling direction

and the highest coercivity is found with the samples cut transversal to rolling direction. Also,

the flat-top region is visible for the range of angles from 25 to 40 degrees. The anisotropy of

the coercivity for the cycles measured at 1 T was 54.1 %, while for the cycles measured at

1.2 T was 54.8 %.

The measurements of the anisotropy of the GO steel samples are summarized in Table 2.14.

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Table 2.14: Experimental measurement results for GO steels

No. Angle

[degrees]

𝑯(𝟏 𝐓) [𝐀/𝐦]

𝑯(𝟏. 𝟐 𝐓) [𝐀/𝐦]

𝑯𝐜(𝟏 𝐓) [𝐀/𝐦]

𝑯𝐜(𝟏. 𝟐 𝐓) [𝐀/𝐦]

GO1 GO2 GO1 GO2 GO1 GO2 GO1 GO2

1 0 199.6 110.8 249 134.8 16.7 11.1 18 12

2 5 216.9 121.9 258.1 146.9 17.9 13 18.9 13.7

3 10 217.6 142.1 264.3 173.7 17.8 14.3 19.1 14.9

4 15 215.8 147.4 276.2 184.9 19.2 16.2 20.3 17

5 20 231.7 163 320.4 211.8 19.6 18.2 20.7 19.1

6 25 236.5 167.1 368.3 241.3 20.9 18.5 22.3 19.6

7 30 255.7 161 481.7 277.9 21.3 18 22.9 18.7

8 35 261.4 167.8 663.3 427.6 21.3 17.8 22.7 20.1

9 40 292.8 178.3 941.5 581.4 20.7 16.6 22 17.2

10 45 303.6 197.6 1216.5 708.8 21.4 17 22.6 18.5

11 50 335 227.7 1282.6 870.7 22.5 19.3 24.2 21.1

12 55 360 256.2 1331.4 806.6 25.2 22.7 26.7 25.1

13 60 407.3 304.1 1306.3 806.9 30.3 28.4 32.4 32.2

14 65 458.4 340.4 1153.5 753.5 34.3 32.7 36.6 34.9

15 70 497.9 368.8 1025.6 641.2 36.6 34.9 39 38.3

16 75 511.8 391.8 852.1 575.1 37.6 36.6 40.3 39.5

17 80 555.1 419 776.6 551.6 38.2 38.1 40.5 41.2

18 85 562.2 422.8 722.5 517.8 39 39.2 41.5 41.8

19 90 571.5 444.3 732.2 535.5 37.9 38.2 39.9 41.3

For the tested GO steels the hard magnetization axis appears to change orientation depending

on the polarization level. The easy magnetization axis has been found to be oriented along the

rolling direction and the hard magnetization axis changed orientation from 90 degrees to

55 degrees for polarization levels above 1.1 T. The coercivity has been found to be minimum

when measured along the rolling direction and maximum when measured transversal to rolling

direction. The anisotropy of the magnetization at the investigated polarization levels had a

maximum of 71.4 %. For higher polarization levels the anisotropy of the magnetization tends

to 100 % as the samples saturates at approx. 1.4 T along the hard magnetization axis. The

anisotropy of the coercivity had a relatively small variation at the investigated polarization

values: approx. 38 % for sample GO1, and approx. 54 % for sample GO2.

Table 2.15 shows a summary of key performance parameters of GO and NGO electrical steels.


Table 2.15: Summary of performance parameters of GO and NGO electrical steels

Parameter GO NGO

Easy magnetization axis Rolling direction Rolling direction

Hard magnetization axis Perpendicular to rolling

direction

55 deg. with respect to

rolling direction

Easy axis saturation Approx. 1.9 T Approx. 1.6 T

Hard axis saturation Approx. 1.25 T Approx. 1.6 T

Minimum coercivity 15-20 A/m

(in rolling direction)

30-45 A/m

(in rolling direction)

Maximum coercivity

Approx. 40 A/m

(perpendicular to rolling

direction)

45-60 A/m

(perpendicular to rolling

direction)

The main difference between GO and NGO electrical steels is the level of anisotropy. The GO

electrical steels exhibit very good performance along the easy magnetization axis (high

saturation and low coercivity), but along the hard magnetization axis they have low saturation

values, as presented in Table 2.15. NGO electrical steels are characterised by magnetic

properties with increased isotropy compared to GO steel, but at the expense of overall

performance (increased coercivity and lower saturation value). For building cores for particle

accelerator magnets GO electrical steels present very attractive magnetic properties but special

care has to be taken during design to prevent saturation along the hard magnetization axis.

2.3.3. The effect of annealing Fe-Si alloys

The magnetic performance of electrical steel before and after annealing has been investigated.

The characteristics of the tested samples are presented in Table 2.16.

Table 2.16: Description of the samples used for the study of annealing effects

Sample name Annealing Grade Manufacturer Orientation

M15-SRA Stress relief M15 (NGO) AK Steel Rolling direction

M15-AS As cut M15 (NGO) AK Steel Rolling direction

M47-SRA Stress relief M47 (NGO) AK Steel Rolling direction

M47-AS As cut M47 (NGO) AK Steel Rolling direction

The stress relief anneal is a low temperature anneal used to minimize adverse effects of

production and machining process. This anneal is expected to lead to small change to the grain

structure. The procedure for stress relief annealing of these samples was: heating of the samples

at a rate of 2-4 °C per minute; holding the temperature of 700 °C (± 20 °C) for at least 60

minutes; cooling of samples down to 550 °C at a rate of 2-4 °C per minute; maintaining the

sample in a protected atmosphere down to 300 °C.

Doctoral thesis 77

The normal magnetization curves of samples M15-SRA and M15-AS are presented in

Fig. 2.50.

Fig. 2.50: The normal magnetization curves of M15

The coercivity curves of samples M15-SRA and M15-AS are presented in Fig. 2.51.

Fig. 2.51: Coercivity curves o M15

The normal magnetization curves of samples M47-SRA and M47-AS are presented in Fig.

2.52.


Fig. 2.52: The normal magnetization curves of M47

The coercivity curves of samples M47-SRA and M47-AS are presented in Fig. 2.53.

Fig. 2.53: Coercivity curves of M47

All the measurements are summarized in Table 2.17.

The stress relief annealing procedure improved the magnetization performance of the M15

grade uniformly for all polarization levels. The annealing process led to a decrease of the

coercivity by approx. 2 A/m for polarization values above 0.5 T.

The stress relief annealing process led to great improvement of the magnetization performance

for grade M47 for polarization values up to 1.5 T. The coercivity decreased by approx. 10 A/m

for polarization values above 1 T.

The annealing process leads to stress and strain relief, additional grain growth and

decarburizations of the material. The tested samples showed an overall improvement of the

magnetic properties (improved magnetization response and decreased coercivity).

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Table 2.17: Experimental measurement results of annealing process

𝑱 [𝐓]

𝑯 [𝐀/𝐦] 𝑯𝐜 [𝐀/𝐦]

M15 M47 M15 M47

SRA AS SRA AS SRA AS SRA AS

0.1 10 11.8 13.2 18.9 2.7 3.3 3.9 5.3

0.25 15.7 17.7 19.9 29.2 8.3 9.4 10.2 15

0.5 20.6 23 24.6 39.3 13.6 15.8 17.1 24.3

0.75 28.3 31.5 29.1 53.1 17.7 19.5 21.2 29.4

1 44.8 51 39.4 76.6 20.5 22.6 24.1 33.6

1.25 96.2 121.8 70.8 136 23.6 25.5 27.2 36.9

1.5 753.3 1041.2 385.4 598.2 25.7 27.1 30.3 39.6

2.3.4. Comparison of Fe-Si alloys with identical grading

From the magnetic performance point of view, the naming of the electrical steels take into

consideration only the total losses. The value of the total losses at 1.5 T and line frequency

offer little insight to the magnetic behaviour of a material when operating in quasi-static

regime. The magnetic behaviour of two grades of non-oriented electrical steel provided by two

different manufacturers, and with the same total loss ratings has been investigated. The

properties of the tested samples are presented in Table 2.18.

Table 2.18: Description of the electrical steel samples with similar power loss rating

Sample name Grade Manufacturer Orientation

SG1 isovac 270-50 A voestalpine Rolling direction

SG2 M 270-50 A C.D. Wälzholz Rolling direction

Fig. 2.54: The normal magnetization curves of SG1 and SG2


The two samples have identical power loss ratings of 2.7 W/kg, the same thickness of 0.5 mm,

but are manufactured by different companies and, therefore, it is assumed that they have

different chemical composition and have been subjected to different manufacturing processes.

The normal magnetization curves of samples SG1 and SG2 are presented in Fig. 2.54.

The coercivity curves of samples SG1 and SG2 are presented in Fig. 2.55.

Fig. 2.55: Coercivity curves of SG1 and SG2

The measurements of samples SG1 and SG2 are summarized in Table 2.19.

Table 2.19: Experimental measurement results of samples SG1 and SG2

𝑱 [𝐓]

𝑯 [𝐀/𝐦] 𝑯𝐜 [𝐀/𝐦]

SG1 SG2 SG1 SG2

0.1 11.2 36.4 3.2 11.2

0.25 16.1 60.7 8.8 24.4

0.5 19.9 92.1 14.2 37.1

0.75 25.3 119.4 16.9 42.7

1 40.2 151.8 19.8 47

1.25 91.6 212.7 22.4 50.4

1.5 531.5 717.6 23.9 52

1.65 2496 3186 24.5 52.2

Above 1.5 T samples SG1 and SG2 will have a similar magnetic characteristic. On the other

hand, up to the polarization level of 1.5 T the two steels are characterized by very different

magnetization curves. For the whole magnetization range the coercivity of sample SG1 is

approx. two times higher than the coercivity of sample SG2.

The standard naming of electrical steel grades does not offer a complete picture of the material's

magnetic behaviour. The experimental measurements highlighted that the magnetic properties

Doctoral thesis 81

of steel grades with identical total loss ratings can have a very large variation. Therefore, in

order to correctly evaluate a material for building the core of a particle accelerator magnet,

magnetic measurements over the entire magnetization range are required.

2.3.5. The influence of the chemical composition on the magnetic and electric

properties of electrical steels

A total number of 11 grades of electrical steels have been analysed: two grades of GO electrical

steel and nine grades of NGO electrical steel. The samples are described in Table 2.20, where

the value 𝑑 represents the thickness of the sample.

Table 2.20: Details of the samples used for the chemical composition study

Sample

name Steel name Texture Manufacturer 𝒅 [𝐦𝐦]

S1 Rotor steel NGO AK Steel 1.5

S2 M 140-35 S GO Arcelor-Mittal 0.35

S3 M 165-35 S GO TATA Steel 0.35

S4 M 330-50 A HP NGO C.D. Wälzholz 0.5

S5 M 400-50 A NGO C.D. Wälzholz 0.5

S6 isovac 250-35 A HP NGO voestalpine 0.35




S10 M 1400-100 A NGO voestalpine 1

S11 Low carbon steel NGO 1.5

The chemical composition has been determined using spark optical spectroscopy with a readily

available system, the PMI-MASTER PRO manufactured by Oxford Instruments. The

specimens are sheets of material of 30 mm x 300 mm. The impurities at the surface of the

specimen (oxides and insulation layer) have been removed by grinding. The measurement

method is based on the development of high energy sparks between the specimen and an

electrode. The electric arc will melt, evaporate and excite the elements at the surface of the

specimen. When the atoms in the plasma relax they will emit light of characteristic wavelength.

The light emissions are compared to known standards to provide quantitative results. For each

sample a minimum of three measurements have been performed and the average of these

measurements is reported.

The electrical resistivity has been measured using the four points method recommended by the

IEC standard 60404-13 [109]. The surface insulation of the sample has been removed using

sand paper. A support frame and a contact holder have been built in support of these

measurements (Fig. 2.56).


Fig. 2.56: Sample and contact holder for the measurement of electrical resistivity

The sample and contact holder is connected as in the circuit presented in Fig. 2.57. The

measurement instrumentation is the same as the one used for the new magnetic measurement

procedure. In Fig. 2.57 is presented the block diagram of the electrical resistivity measurement

setup.

Fig. 2.57: Electrical connection diagram for the measurement of electrical resistivity

A current is applied to the sample through two brush-like contacts on the narrow side of the

sample and it is measured using the shunt resistor. The voltage drop on the surface of the sample

between a fixed distance is measured using two point-like contacts. The current is increased

from 1 A in steps of 0.5 A until the variation of two consecutive measurements falls below

1 %.

The magnetic parameters which have been investigated are the saturation polarization (𝐽sat),

the coercivity associated to the saturation induction (𝐻c,sat) and the energy losses associated

to the saturation induction (𝑊m,sat). For this study the saturation was considered to be achieved

when the relative permeability of the material's normal magnetization curves falls under 1000.

This allowed for a common reference for all the samples.

Steel samples S1 to S11 have been tested for the chemical composition, for the electrical

resistivity and for the magnetic properties. In Table 2.21 are presented the experimental results.

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Table 2.21: Experimental results on the chemical, magnetic and the electric properties

Sample

name

Si

[wt.%]

Al

[wt.%]

Mn

[wt.%]

𝑱𝐬𝐚𝐭 [𝐓]

𝑯𝐜,𝐬𝐚𝐭

[𝐀/𝐦]

𝑾𝐦,𝐬𝐚𝐭

[𝐦𝐉/𝐤𝐠]

𝝆

[𝝁𝛀𝐜𝐦]

S1 3.31 0.663 0.137 1.431 107.5 90.3 51

S2 3.27 0.0029 0.296 1.898 21.1 25.4 50

S3 3.11 <0.001 0.0078 1.919 10.2 19.5 48

S4 2.7 0.925 0.212 1.579 50.8 33.6 52

S5 2.22 0.418 0.209 1.534 61.8 45.8 44

S6 1.95 1.22 0.322 1.504 45.3 41.3 50

S7 1.37 0.327 0.289 1.545 60.2 46.1 35

S8 1.18 0.129 0.213 1.603 77.9 70.3 30

S9 1.13 0.153 0.191 1.594 76.3 66.6 31

S10 0.45 0.463 0.496 1.609 55.3 57.1 27

S11 <0.02 0.0152 0.272 1.561 95.7 85.4 11

The graphical representation of the chemical composition of the samples if presented in Fig.

2.58.

Fig. 2.58: The chemical composition of steel samples S1 to S11

The silicon content of grades S2 and S3 has similar values, the main difference being in the

content of manganese (0.296 wt. % for S2 and 0.0078 wt. % for S3). The difference in the

chemical composition lead to slightly different values of the electrical resistivity of the two

samples (50 𝜇Ω ⋅ cm for S2 and 48 𝜇Ω ⋅ cm for S3). The measurements show that sample S2

has a saturation level 20 mT lower than sample S3. Also, the coercivity of sample S2

(21.1 A/m) is greater than for samples S3 (10.2 A/m) and the energy losses of sample S2

(25.4 mJ/kg) are larger than the energy losses of samples S3 (19.5 mJ/kg).

The major hysteresis cycles of samples S2 and S3 are presented in Fig. 2.59.


Fig. 2.59: The hysteresis cycles of steel samples S2 and S3

The different alloying elements of steel grades S2 and S3 determine entirely different shapes

of the hysteresis cycles. The a.c. performance of sample S2 is enhanced due to the extra

manganese content which increases the electrical resistivity. On the other hand, the slope of

the branches of the hysteresis cycle, and, therefore, the permeability of the material, decrease.

Therefore, a particle accelerator magnet employing steel grade S3 in its core would operate

with improved performance compared to a magnet which employs steel grade S2.

The silicon and manganese content of samples S4, S5 and S6 are in the same range. On the

other hand, the aluminium content has a great variation (in the range 0.418…1.22 wt.%) and,

therefore, the effect of aluminium to the magnetic properties can be investigated. Steel sample

S5 (2.22 wt. % Si, 0.418 wt. % Al and 0.209 wt. % Mn) shows a coercivity of 61.8 A/m while

sample S6 (1.95 wt. % Si, 1.22 wt. % Al and 0.322 wt. % Mn) shows a coercivity of 45.3 A/m.

The results show that the extra aluminium and manganese content of sample S6, compared to

sample S5, increases the electrical resistivity at the expense of decreased saturation and

permeability, as seen in the hysteresis cycles presented in Fig. 2.60.

Fig. 2.60: The hysteresis cycles of steel samples S4, S5 and S6

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The measurements show that the branches of the hysteresis cycles of steel samples S4 and S5

are steeper compared to sample S6. The chemical composition of sample S6 optimises the

performance for a.c. applications but for the application of particle accelerator magnet steel

grades S4 and S5, which have fewer alloying elements, would be a better choice.

Steel samples S8 and S9 have higher silicon content than sample S10. On the other hand,

sample S10 has considerably higher Al and Mn content which which its electrical resistivity.

The measurements show that the electrical resistivity of samples S8, S9 and S10 have similar

values. On the other hand, sample S10 has lower coercivity (55.3 A/m) than sample S8

(77.9 A/m) and sample S9 (76.3 A/m). The hysteresis cycles of samples S8, S9 and S10 are



The measurements of samples S8, S9 and S10 show that the steepness of the branches of the

hysteresis cycles and, therefore, the permeability of the material decrease with the content of

silicon. The shape of the hysteresis cycles and the saturation level of these grades have similar

values, therefore, sample S10 would be a better choice for the core of a particle accelerator

magnet since its coercivity (55.3 A/m) is approx. 20 A/m lower than for grades S8 (77.9 A/m)

and S9 (76.3 A/m).

The measurements performed on samples S1, S7 and S11 show the influence of the silicon

content in a NGO steel to its saturation level and to the value of its electrical resistivity. Sample

S1 (3.31 wt. % Si) is characterised by the highest value of the electrical resistivity (51 µΩ∙cm)

and the lowest saturation (1.431 T), while samples S7 (1.37 wt. % Si) and S11 (<0.02 wt. % Si)

have decreasing values of the electrical resistivity (35 µΩ∙cm and 11 µΩ∙cm, respectively) and

increasing saturation level (1.545 T and 1.561 T, respectively). The hysteresis cycles of

samples S1, S7 and S11 are presented in Fig. 2.62.



The measurements have shown that the magnetic permeability is associated to the silicon

content in the material. Nevertheless, the complexity of the shape of the cycles do not allow to

establish a simple relations between the two quantities. Also, the measurements have shown

that sample S7 is characterised by the lowest coercivity level (60.2 A/m) compared to samples

S1 (107.5 A/m) and S11 (95.7 A/m), and, therefore, would be the better option to manufacture

the core of a particle accelerator magnet.

The measurements have shown that by adding silicon to the content of an NGO steel the

electrical resistivity increases and the coercivity decreases at the expense of the saturation level.

The addition of aluminium and manganese to the content of electrical steels raises the electrical

resistivity at the expense of magnetic permeability. The addition of manganese has a similar

effect as aluminium but with a significant toll on the permeability. Although some steels have

very good a.c. loss rating, like S2, for the application of particle accelerator magnets steels with

fewer alloying elements are a better choice, like S3.

2.4. Conclusions

The accurate computation of the field in the gap of a magnet is linked to the accurate

measurement of the magnetic hysteresis of the material in the yoke. In order to obtain the best

estimate of the magnetic properties of a material the rate of change of the magnetization during

magnetic testing has to be matched to the rate of change of the magnetic induction in the gap

of the magnet. This can be achieved by employing the correct measurement methodology. This

chapter of the thesis presented the development of such a measurement methodology and the

obtained experimental results.

From the control system point of view, the measurement methods described by the standard

are open loop systems and are not able to control the waveform of the magnetization. The

performance of feedback (closed-loop) systems depends on the quality of the electronic

Doctoral thesis 87

component used to build the control system, it is sensitive to the noise which is inherent to the

quasi-static magnetic testing, and the feedback loop may in fact follows this noise. The

feedforward systems adjust the control variable based on knowledge of the process in the form

of a mathematical model. On the other hand, the shape of the hysteresis cycle changes with the

peak level of the magnetization, and, therefore, the model of the feed-forward system has to be

updated. Feedforward systems use a pre-defined model and will continue to augment the model

by means of iterative algorithms until the output of the model fulfils some criteria. For the

current research a feedforward method has been developed, with a look-up table for the model,

because it is a purely software implementation and, therefore, allows for great flexibility and

the use of a wide range of data processing tools.

The newly developed and the d.c. standard magnetic testing methodologies have been tested

and cross-checked using experimental measurements. Using the new testing method steel

samples have been tested with sinusoidal polarization waveform at 1 Hz, thus, yielding a rate

of change of the polarization in the range of 10 T/s, a value similar to the one which is found

in PA magnets. The comparative analysis has revealed that the shape of the hysteresis cycle is

affected even for such low frequencies: the variation of the coercive force of the hysteresis

cycles measured with the standard method and with the new method has been approx. 5 %.

Therefore, magnetic test performed with sinusoidal polarization waveform at 1 Hz provide the

best trade-off between speed and measurement accuracy.

Usually, the magnets of the same type in a particle accelerator are connected in series and

identical field is expected in all the magnets. In order to have identical output the core of the

magnets are required to have identical magnetic properties. The magnetic characteristics of

electrical steels vary within the same charge. The experimental measurements have shown that

the coercivity has a variation of approx. 10 A/m for samples from different coils of the same

charge. Therefore, during manufacturing of the magnets the homogenization of the magnetic

properties of the steel has to be achieved, usually by shuffling the laminations.

The magnetic field in the yoke of a magnet closes in a loop which covers all the rotation angles

in the plane of the lamination. When the material in the core reaches saturation the transfer

function of the magnet decreases considerably. The experimental measurements have shown

that electrical steels can saturate along the hard magnetization axis at significantly lower values

than their rating. Therefore, in order to prevent saturating regions the cross-sectional area of

the yoke has to be increased along the hard magnetization axis. To prevent saturation in the

cores manufactured using NGO steels a 10 % increase of the cross-sectional area of the yoke

is required in the regions where the magnetic field is oriented along 90° w.r.t. the rolling

direction of the sheet. On the other hand when using GO steels a 30 % and a 23 % increase are

required for the regions oriented along 55° and 90°, respectively.

Another parameter which greatly influences the magnetic properties of electrical steels is the

mechanical history and the grain size. Mechanical processes can induce stress in a material and

decrease its performance. Also, increased grain size favours mobility of the domain walls, thus

enhancing the magnetic performance of a material. Heat treatment annealing operations are

designed to remove internal stresses and to stimulate grain growth. The experimental


measurements have shown that for some material significant improvements of the magnetic

properties can be obtained after heat treatment operations. Therefore, in order to maximize the

performance of a material its mechanical history should be monitored and magnetic annealing

processes should be applied.

The standard grading of electrical steels take into consideration the energy losses at line

frequency. The chemical composition of an alloy affects its electrical resistivity which in turn

has a significant contribution to reducing the energy losses. On the other hand, with increasing

content of alloying elements the slope of the hysteresis cycle is reduced. Experimental

measurements have shown that similar a.c. grading can be achieved for steels with different

chemical composition. Therefore, if dynamic effects in a magnet are negligible steels with

fewer alloying elements will operate with increased performance. Otherwise, steels

characterised by large electrical resistivity are recommended to be used.

In this chapter the current status of experimental characterization of magnetic materials has

been presented. Also, the development of a magnetic testing method, and experimental

measurement results obtained with this method have been presented. The developed method

improves the quality of the experimental measurements by tackling a series of limitations of

the standard method: the lack of control of magnetization rate of change during the

measurement, the resolution and distribution of the measured cycles and the demagnetization

of the samples. The presented experimental measurements analyse the magnetic properties of

the materials from the point of view of particle accelerator magnet core.

3. MODELLING AND SIMULATION OF THE MAGNETIC

HYSTERESIS

The modelling of magnetic hysteresis is very important for solving problems which involve

time varying electromagnetic fields in ferromagnetic materials [110, 111, 112, 50, 113, 114,

115, 116]. The magnetization of these materials is characterized by non-linearity and

hysteresis. The hysteresis characteristic leads to the dependence of the magnetization not only

on the current status of the excitation field but also on the history of magnetization.

This chapter describes the two magnetic hysteresis models which are the most widely used for

hysteresis modelling: the Jiles-Atherton model and the Preisach model. The numerical

implementation of the Preisach model is analysed in detail the methods required to improve its

performance are developed: an interpolation method for the input data and a method to

minimize the amount of input data. The performance of the model is cross-checked with

experimental measurements.

3.1. Magnetic hysteresis models

The magnetic hysteresis models can be categorised as physical and phenomenological (at the

lowest level of resolution). A physical model relies on the laws of physics and require in depth

information of the physical properties of a material. Phenomenological or semi-physical

models rely on understanding of the physical system to a limited extent and use black-box

modelling. Also, the models can be classified as numerical and analytical [73, 74].

3.1.1. The Jiles-Atherton model of hysteresis

The Jiles-Atherton model [93, 94] decomposes the magnetization process into two components

associated to the observed behaviour of the magnetization: a reversible (𝑀rev) and an

irreversible (𝑀irr) component. The output of the model is the sum of the two components:

𝑀 = 𝑀rev +𝑀irr . (3.1)

The model relies on the response of the magnetic material without hysteresis losses, which

originally has been estimated with a modified Langevin equation:

𝑀an(𝐻) = 𝑀s (coth (𝐻 + 𝛼𝑀

𝑎) −

𝑎

𝐻 + 𝛼𝑀) , (3.2)

where 𝛼 and 𝑎 are the parameters of the model, 𝑀s is the saturation magnetization, and 𝐻 is

the excitation field. The anhysteretic magnetization represents the effects of the magnetic


moment rotation but it does not take into consideration the losses due to domain wall

movement.

Under the assumption of rigid and planar domain walls, the energy losses associated to the

magnetic moment rotation expressed as a function of the irreversible changes of the

magnetization are defined as:

𝐸pin(𝑀irr) = 𝜇0𝑘 ∫ d 𝑀irr

𝑀irr

0

, (3.3)

where 𝑘 is the “pinning” constant [A/m].

The differential susceptibility of the irreversible magnetization is written as:

d𝑀irrd𝐻e

=𝑀irr −𝑀an

𝑘𝛿 , (3.4)

where 𝐻e is the effective magnetic field defined as:

𝐻e = 𝐻 + 𝛼𝑀 , (3.5)

and 𝛿 is a directional parameter which can take the values:

𝛿 = 1, for d𝐻 / d𝑡 > 0−1, for d𝐻 / d𝑡 < 0

. (3.6)

However, domain wall movement does not occur in a step-like fashion. The domain wall are

flexible and bend when are pinned. The bending of the domain walls is associated to the

reversible changes in the magnetization process. The reversible component of the

magnetization is defined as:

𝑀rev = 𝑐(𝑀an −𝑀irr) , (3.7)

where 𝑐 is the reversibility coefficient.

By integrating Eq. (3.7) into Eq. (3.1) the magnetization is rewritten as:

𝑀 = 𝑀irr + 𝑐(𝑀an −𝑀irr) . (3.8)

The total differential susceptibility of the system is obtained by differentiating Eq. (3.8) with

respect to 𝐻:

Doctoral thesis 91

d𝑀

d𝐻= (1 − 𝑐)

𝑀an −𝑀irr𝑘𝛿 − 𝛼(𝑀an −𝑀irr)

+ 𝑐d𝑀and𝐻

. (3.9)

Eq. (3.9) is the model differential equation which gives the value of the magnetization as a

function of the excitation field 𝐻. The model requires the identification of five parameters

(𝛼, 𝑎, 𝑘, 𝑐, 𝑀s) from experimental measurements. The physical meaning of the parameters are

presented in Table 3.1.

Table 3.1: Physical meaning of J-A model parameters

Parameter Property

𝛼 Linked to domain interaction

𝑎 Linked to the shape of 𝑀an

𝑘 Linked to hysteresis losses

𝑐 Reversibility coefficient

𝑀s Saturation magnetization

A procedure for identification of the model parameters from experimental measurements is

presented by Jiles, Thoelke and Devine in [117]. The parameters are identified from

experimental data after some developments of Eq. (3.2), (3.4) and (3.7). The identification

requires an iterative procedure, the method is numerically sensitive, and does not

systematically converge [118].

3.1.2. The Preisach model of hysteresis

The Preisach model of hysteresis has been first introduced in 1935 by Preisach [75]. Since then

it received many improvements and extensions [119, 120, 121, 122, 123, 124]. The model is a

mathematical tool which is able to describe the hysteresis phenomenon of any nature and at all

resolution levels. The success of the model comes from its ability to have different behaviours

depending on the direction of the excitation field (increasing or decreasing). The model is

capable of describing the branching of the hysteresis cycle and other complex magnetization

processes. Its limitations reside in some of which properties which cannot be found in magnetic

materials: the congruency and deletion properties. Nevertheless, it shows good performance

and, therefore, it is very often used.

The model introduced by Preisach uses a basic building block called a hysteron for the

description of a single hysteresis element, and a density function associated to each hysteron.

The operation of the hysteron is presented in Fig. 3.1.


Fig. 3.1: The elementary hysteresis operator

The equation which defines the model is [75]:

𝑓(𝑡) = ∬𝜇(𝛼, 𝛽)𝛼𝛽𝑢(𝑡)

𝛼>𝛽

d𝛼 d𝛽 , (3.10)

where 𝛾𝛼𝛽 is the hysteresis element associated to 𝛼 and 𝛽, which are commutation values of

the element, 𝜇(𝛼, 𝛽) is the density function associated to the 𝛼 and 𝛽 values, and 𝑢(𝑡) is the

input of the model. The Preisach function is defined on the 𝛼𝛽 domain, above the first bisector

defined by the 𝛼 = 𝛽 line. This triangle formed by the upper limits in the 𝛼𝛽 domain will be

further called 𝑇. Outside 𝑇 it is considered that 𝜇(𝛼, 𝛽) = 0. The evolution in time of the input

of the model is memorized by the hysterons, which will, thus, create a line which will separate

the triangle 𝑇 in two regions: the 𝑆+(𝑡) region has the 𝛾𝛼𝛽 operators in the +1 status, while the

𝑆−(𝑡) region has the 𝛾𝛼𝛽 operators in the −1 status. The triangle 𝑇 and its regions for random

history of 𝑢 is presented in Fig. 3.2.

Fig. 3.2: The Preisach triangle

Doctoral thesis 93

The tip of the separation line which forms between the two regions (the bolded line in Fig. 3.2)

has the coordinates 𝛼 = 𝛽. The last segment of this line is horizontal and moving upward if 𝑢

is increasing, or is vertical and moving from right to left if 𝑢 is decreasing. Therefore, Eq. (3.10)

can be rewritten as:

𝑓(𝑡) = ∬𝜇(𝛼, 𝛽) d𝛼 d𝛽

𝑆+

− ∬𝜇(𝛼, 𝛽) d𝛼 d𝛽

𝑆−

. (3.11)

The Preisach model can be easily implemented using numerical methods by considering the

hysterons and the weights as sets of parallel operators stored into two matrices. The output of

the model will be the sum of all the elements of the resulting product matrix. In Fig. 3.3 is

presented the logical diagram for the numerical implementation of the Preisach model for 𝑛

threshold levels.

Fig. 3.3: Block diagram of the numerical implementation of the Preisach model

One of the properties of the Preisach model is the symmetry of the saturating states: the area

of the triangle 𝑇 is constant and, therefore, when area 𝑆+ or 𝑆− fill the triangle the absolute

value of the output is the same. The elimination property is observed when the coordinate of

the tip of the separation line is greater the a previous output: if, for instance in Fig. 3.3,

𝛼2 > 𝛼1 then the information regarding the 𝛼1 level would have been lost. The congruency

property is observed when the model is cycled between two constant values: if, for instance in

Fig. 3.3, the input would have continued to oscillate between 𝛼2 and 𝛽2 the output of the model

would have been oscillated with a constant value given by the area of triangle 𝑇 between 𝛼2

and 𝛽2. The classical Preisach model is static by its nature: the value of the output does not

depend on the rate of change of the input.

Since the model cannot describe many of the details in the behaviour of the magnetization,

modifications based on physical reasoning have been developed. The DOK model [125] has


been developed to incorporate a magnetization-dependent reversible component. This model

is further improved by the CMH model [126] which ads a more complex state-dependent

reversible component. The congruency limitation has been tackled by means of an output

dependent model as the moving model [127] or the product model [128]. The deletion property

has been tackled by including aftereffect model [129] or accommodation model [130]. In its

original form the Preisach model is a scalar model, nevertheless, adaptations to vector models

have been developed: Mayergoyz model [122], Oti and Della Torre use pseudo-particles in

their model [131, 132], and the coupled-hysteron model developed by Vajda and Della

Torre [133].

3.1.3. Conclusion

The Preisach and the Jiles-Atherton models of hysteresis are the most widely used models in

the calculation of electromagnetic fields. The main advantages of the Jiles-Atherton model are

that its coefficients have a physical meaning and that it is simple to implement from the

computational aspect (it is favoured when studying complex electromagnetic systems,

especially when using finite element analysis). Although the Preisach model was initially based

on physical phenomena it is really a mathematical method for curve fitting model rather than a

truly physical model. Of the two models the Preisach model is reported to produce the most

satisfactory results for any materials [118, 134, 135]. Therefore, the Preisach model of

hysteresis has been used in this work to model the hysteretic behaviour of the magnetization

and some methods to improve its performance have been developed.

The identification of the Preisach model requires the determination of the weight function

𝜇(𝛼, 𝛽). The only limitation to the accuracy of this model lies in the accuracy of the

identification of the weight function. Therefore, this work will be further focused on the

development of two methods which improve its identification procedure.

3.2. Identification of the Preisach model

The output of the Preisach model depend on the variables of the model (Eq. (3.10)): the

weight/density function 𝜇(𝛼, 𝛽), the hysterons associated to each element 𝛾𝛼𝛽, and on the input

𝑢. The value of the hysterons change according to the history of the input, while, the weight

function contains the information regarding the behaviour of the system. This section presents

some of the currently employed methods used to construct the Preisach weight function and

the development of the methods required to increase the resolution of the memorised model.

3.2.1. Methods to construct the Preisach weight function

The experimental determination of the Preisach weight function is a tedious operation: it

requires time and accurate equipment, and the result is affected by the noise in the experimental

determinations. For this reason analytical functions, whose parameters are identified to best

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reproduce the behaviour of the system, are usually used to construct the weight function. This

section presents a review of the methods described in literature for the identification of the

Preisach weight function.

A. Identification by parametric methods

One way to determine the weight function is by using a distribution function. One of the

following four distributions are usually employed [136, 124]: exponential,

Factorized-Lorentzian, Gauss-Gauss, or Lognormal-Gauss distribution function.

The exponential distribution is given by the equation:

𝜇(𝛼, 𝛽) =

exp [−

(𝛼 − 𝛽 − 𝑐)2

10𝑎−(𝛼 + 𝛽 − 𝑑)2

10𝑏] , for 𝛼 + 𝛽 ≤ 0

exp [−(𝛼 − 𝛽 − 𝑐)2

10𝑎−(𝛼 + 𝛽 + 𝑑)2

10𝑏] , for 𝛼 + 𝛽 > 0

𝛼 ∈ [−1,1]𝛽 ∈ [−1,1]

, (3.12)

where 𝑎, 𝑏, 𝑐 and 𝑑 are the parameters of the model.

If the weight function is assumed as a Lorentzian distribution, the function is given by the

equation:

𝜇(𝛼, 𝛽) =2

𝜋𝜎c2𝐻02 (𝜋2 + arctan

1𝜎c)⋅

1

[1 + (𝛼 − 𝐻0𝜎𝐻0

)2

] [1 + (−𝛽 − 𝐻0𝜎𝐻0

)2

]

, (3.13)

where 𝜎c is the standard deviation and 𝐻0 gives the position of the peak.

The Gauss-Gauss distribution uses two Gauss distributions to estimate the weight function,

therefore the distribution is characterized by two standard deviations:

𝜇(𝛼, 𝛽) =1

𝜋𝜎c𝜎u𝐻02

2

erf(𝜎c√2) + 1 exp

[ −(𝛼 − 𝛽2 − 𝐻0)

2

2𝜎c2𝐻02

] ⋅ exp

[ −(𝛼 + 𝛽2 )

2

2𝜎u2𝐻02

] , (3.14)

where 𝜎c and 𝜎u are the standard deviations for each diagonal, 𝐻0 is the position of the peak,

and erf is the error function2.

2 The assumed normalization id erf(𝑥) =2

√𝜋∫ (−𝑢2) d𝑢𝑥

0


The Lognormal-Gauss distribution is described by two deviations similar to the Gauss-Gauss

distribution:

𝜇(𝛼, 𝛽) =1

𝜋𝜎c𝜎u𝐻02

𝐻0𝛼 − 𝛽

exp [−ln2 (

𝛼 − 𝛽2𝐻0

)

2𝜎c2] exp [−

(𝛼 + 𝛽)2

8𝜎u2𝐻02 ] , (3.15)

where 𝜎c and 𝜎u are the two deviations and 𝐻0 is the location of the peak.

The parametric methods have the form of two-dimensional probability distribution and create

the weight function based on few parameters. The motivation for using these methods is that

they can describe the shape of the weight function analytically and therefore, are able to store

the information on the behaviour of the material in a small number of variables. Nevertheless,

these methods are able to reproduced idealized shapes of the hysteresis cycles and are not able

to reproduce the exact shape of the hysteresis cycle.

B. Mayergoyz’ method

Mayergoyz presents a method for identifying the weight function from measurements of first

order reversal curves (FORCs) [122]. The FORCs are formed when a monotonic increase (or

decrease) along the limiting hysteresis branch is followed by a subsequent monotonic decrease

(or increase). The FORC is the curve which originates in the limiting branch following the

change of the input. The formation of a FROC is presented in Fig. 3.4.

Fig. 3.4: Description of the first order reversal curve

In Fig. 3.4 the limiting branches of the hysteresis cycle are depicted with solid line. A FORC

is presented with dotted line within the area of the hysteresis cycle. This curve has been

obtained following a monotonic increase from negative saturation to a random value 𝑢0,

followed by a monotonic decrease to negative saturation.

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The quantities used by the method in the identification of the weight function are presented in

Fig. 3.5.

Fig. 3.5: The output of the Preisach model and the 𝛼 − 𝛽 diagram

To apply this method, at the first instance the system is brought to negative saturation. In the

next step, the input is increased until the value 𝛼0 and the limiting ascending branch is followed.

At this instance the output of the system is 𝑓𝛼0. Next, a FORC is formed until the value 𝛽0. At

this instance, the output of the system is 𝑓𝛼0𝛽0. The 𝛼 − 𝛽 diagram in Fig. 3.5 show that

following this operation the triangle 𝑇(𝛼0, 𝛽0) is subtracted from region 𝑆+ and added to region

𝑆−.

The following function is defined:

𝐹(𝛼0, 𝛽0) =1

2 (𝑓𝛼0 − 𝑓𝛼0𝛽0) . (3.16)

Using Eq. (3.11) the following relation can be established:

𝑓𝛼0 − 𝑓𝛼0𝛽0 = −2 ∬ 𝜇(𝛼, 𝛽) d𝛼 d𝛽

𝑇(𝛼0,𝛽0)

. (3.17)

Equation (3.16) and (3.17) are combined:

𝐹(𝛼0, 𝛽0) = ∬ 𝜇(𝛼, 𝛽) d𝛼 d𝛽

𝑇(𝛼0,𝛽0)

, (3.18)

and this equation can be rewritten as:


𝐹(𝛼0, 𝛽0) = ∫ (∫ 𝜇(𝛼, 𝛽) d𝛼

𝛼0

𝛽0

)d𝛽

𝛼0

𝛽0

. (3.19)

By differentiating Eq. (3.19) we find:

𝜇(𝛼0, 𝛽0) = −𝜕2𝐹(𝛼0, 𝛽0)

𝜕𝛼0𝜕𝛽0 . (3.20)

Using FORCs for the identification of the weight function has some clear advantages: with

sufficiently small increments the finest detail of the hysteresis cycle can be memorized, and

secondly, measurement of these curves is a simple process as they start from a well-defined

state (positive or negative saturation). Nevertheless, obtaining the experimental measurements

for a large number of FORCs can be a time-consuming process which can make the Preisach

model prohibitive.

C. Conclusion

The parametric identification methods allow the description of the weight function by means

of analytical equations with a small number of parameters. On the other hand, with Mayergoyz’

method the weight function is stored in a matrix whose size can be adjusted to capture the

desired level of details. Classical limitations of this method is the necessity of differentiation

of experimental data, and the requirement of large amount of experimental data.

The first limitation of Mayergoyz’ method can easily be overcome by using the method

developed in section 2.2.5. Also, by developing the methods which are able to minimize the

amount of experimental data and to reconstruct FORCs of any resolution, Mayergoyz’ method

is very attractive for the identification of the Preisach function. Therefore, this method has been

used to identify the weight function.

3.2.2. Development of FORC interpolation method

The classical Preisach model of hysteresis evolved to meet the demands of realistic models. At

the same time, the interest in the FORC identification method diminished because of the

numerical derivation procedure which has to be applied to experimental data and due to large

amount of data required.

Considerable effort has been undertaken to interpret experimental FORCs. For instance,

Stoleriu and Stancu propose a simple and numerically efficient interpolation algorithm to

approximate the weight function in any point [137]. They use large increments to determine an

initial weight distribution and the rest of the points are determined using a linear interpolation

algorithm. Nevertheless, information on the shape of the curve is lost due to the large increment

used in the first step of the algorithm. Similarly, Shiffer and Ivanyi propose a wavelet

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interpolation method instead of linear interpolation [138]. In his work Fuzi proposes a linear

interpolation algorithm between adjacent FORCs to ensure the continuity of the output of the

model [136]. Nevertheless, his method relies also on large initial increments.

To capture all the details of the hysteresis cycle the size of the increments used during the

identification of the weight function has to be decreased. Empirical evidence has shown that

an adequate number of threshold levels has to be at least 1000. The method presented in this

section ensures the interpolation of the measured FORC to any number of threshold levels.

This method relies on the identification of the segments of shortest distance between two

adjacent FORCs. The starting point of the FORC is identified on the lower limiting branch of

the hysteresis cycle. The interpolated FORC is determined by connecting the points on the

shortest distance segments. The location of the points is identified by studying the proportions

of the starting points of the bounding FORCs and of the interpolated FORC.

Firstly, experimental measurements are performed to obtain the initial FORCs. The

measurements are performed according to the method presented in section 2.2.2. The limiting

branch is determined by calculating the average of the major branches of the measurements.

The FORCs are then corrected to the averaged value and their non-monotonic points are

removed. In Fig. 3.6 is presented an example of measured FORCs and the lower limiting cycle.

Fig. 3.6: The measured FORCs and the lower limiting cycle

Secondly, the noise in the experimental data has to be removed. The noise is averaged out using

the method presented in section 2.2.5. If required, an extrapolation is performed by using the

derivative at the extremity of the FORCs. Then the FORCs are trimmed to the extremity of the

limiting cycle and delimited to each other. The experimental data is resampled to 10.000 points

and is saved to a table file which contains the coordinated of the limiting cycle and of the

processed FORCs. The program developed to automate the experimental data processing

procedure is presented in Annex 4.

Thirdly, an array of 𝑛 + 1 linearly spaced elements between the maximum and minimum

values of the limiting cycle is generated. These elements define the threshold values on the


𝛼 − 𝛽 diagram (Fig. 3.1 and Fig. 3.2). For each of the 𝑛 central values a FORC originating on

the limiting cycle at point 𝐻𝑖=1…𝑛 will be interpolated.

Following, an example with 𝑛 = 1000 is shown. The interpolation of the FORC originating in

𝐻514 = 100 A/m is demonstrated. Firstly, the bounding FORCs are identified, as presented in

Fig. 3.7.

Fig. 3.7: FORC interpolation – identification of interpolating elements

Next, the FORCs are resampled to 𝑛 points and the distance between all the point in the upper

branch and all the points in the lower branch is calculated using the relation:

𝐷2 =

(

(

𝐻u,1 ⋯ 𝐻u,1⋮ ⋱ ⋮𝐻u,𝑛 ⋯ 𝐻u,𝑛

) − (

𝐻l,1 ⋯ 𝐻l,𝑛⋮ ⋱ ⋮𝐻l,1 ⋯ 𝐻l,𝑛

)

𝐻c

)

2

+

(

(

𝐽u,1 ⋯ 𝐽u,1⋮ ⋱ ⋮𝐽u,𝑛 ⋯ 𝐽u,𝑛

) − (

𝐽l,1 ⋯ 𝐽l,𝑛⋮ ⋱ ⋮𝐽l,1 ⋯ 𝐽l,𝑛

)

𝐽r

)

2

(3.21)

where 𝐻u and 𝐽u are the 𝐻 and 𝐽 values of the upper FORC, 𝐻l and 𝐽l are the H and J values of

the lower FORC, 𝐻c is the value of the coercivity of the limiting cycle, and 𝐽r is the absolute

value of the remanence of the limiting cycle. The normalization of the FORCs to the coercivity

and remanence is required due to the very high difference in the order of magnitude between 𝐽

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and 𝐻. Each line in matrix 𝐷 will contain the distances from the point in the upper FORC to

all the points in the lower FORC. The indexes of the points in the lower FORC associated to a

point in the upper FORC is identified and stored to an array:

𝐷m = (min(𝐷1)l

⋮min(𝐷𝑛)l

) (3.22)

The ratio of the polarization of the FORC 𝐽F and of the magnetic field strength of the FORC

𝐻F are identified based on the start coordinates:

𝑟𝐽 =𝐽u,𝑛 − 𝐽F,𝑛𝐽u,𝑛 − 𝐽l,𝑛

; 𝑟𝐻 =𝐻u,𝑛 − 𝐻F,𝑛𝐻u,𝑛 − 𝐻l,𝑛

, (3.23)

where 𝑟𝐽 is the ratio of the polarization, and 𝑟𝐻 is the ratio of the magnetic field strength. The

values of the interpolated FORC are identified using the relations:

𝐽F = 𝐽u𝑟𝐽 + 𝐽l(𝐷m) ⋅ (1 − 𝑟𝐽);

𝐻F = 𝐻u𝑟𝐻 + 𝐻l(𝐷m) ⋅ (1 − 𝑟𝐻) . (3.24)

The result of the interpolation procedure is presented in Fig. 3.8.

Fig. 3.8: The result of the FORC interpolation process

The weight function is identified using Mayergoyz’ procedure as described in section 3.2.1.B.

The Matlab program developed to automate the execution of the method is presented in

Annex 5.

The experimental measurements presented in the following sections show that the method

reproduces very well the hysteresis behavior of magnetic materials. The weight function is


characterized by a very complex shape, which cannot be approximated by any of the usually

employed analytical functions. The method has been proven to work successfully for a wide

range of 𝑛 values (𝑛 = 103…105). The required computation time (tens of minutes to hours)

gave the upper limit to 104.

3.2.3. Development of FORC level selection method

The newly developed FORC interpolation method provide best results as long as the segments

on the limiting branch has a linear characteristic between two consecutive reversal points. For

strong non-linearity of these segments, the starting point of the interpolated FORC will not

originate on the limiting branch. Therefore, the curvature regions of the limiting branch have

to be densely scattered with reversal points.

A method which is able to determine the second derivative of an experimental data curve is

presented in section 2.2.5. This method has been adapted to identify the curvature regions of

the limiting branch and to minimize the number of experimental FORCs required to identify

the weight function. This method is applied before the measurement process and it determines

the optimum values of the FORC reversal levels.

Firstly, the lower limiting branch of the hysteresis cycle is measured (Fig. 3.9)

Fig. 3.9: The shape of the experimentally determined limited branch

Secondly, the second derivative of the limiting branch is determined using the method

presented in section 2.2.5 (Fig. 3.10).

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Fig. 3.10: The second derivative of the limiting branch

Next, the following quantity is defined:

𝜒2 = |d2𝐽

d𝐻2| . (3.25)

𝜒2 is then normalized with respect to is average, 2:

𝜒2,n =𝜒22 . (3.26)

The peak amplitude of 𝜒2,n is then limited to the peak value 𝑘1 ⋅ 2,𝑛:

𝜒2,t = 𝜒2,n , for 𝜒2,n < 𝑘1 ⋅ 2,n

𝑘1 ⋅ 2,n , for 𝜒2,n > 𝑘1 ⋅ 2,n . (3.27)

To 𝜒2,t is added a constant value proportional to its average (𝑘2 ⋅ 2,t):

𝜒2,d = 𝑘2 ⋅ 2,t + 𝜒2,t . (3.28)

For 𝑘1 = 5 and 𝑘2 = 0.1 the shape of 𝜒2,d is presented in Fig. 3.11.


Fig. 3.11: The curvature regions are identified

The reversal points are determined by identifying the values on the horizontal axis which split

the area under the 𝜒2,d curve in 𝑁 exact portions. The area under the 𝜒2,d curve is calculated

with the relation:

𝐴 = ∫ 𝜒2,d d𝐻

𝐻

. (3.29)

The reversal points 𝐻F,𝑖 (𝑖 = 1…𝑁) are found:

∫ 𝜒2,d d𝐻

𝐻F,𝑖+1

𝐻F,𝑖

=𝐴

𝑁 . (3.30)

The location of the reversal points on the limiting branch for 𝑁 = 15 are presented in Fig. 3.12.

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Fig. 3.12: The reversal points identified for 𝑘1 = 5; 𝑘2 = 0.1; 𝑁 = 15

By adjusting the value of coefficient 𝑘1 the scattering of the reversal points is increased in the

curvature regions. By adjusting the value of 𝑘1 = 50, the reversal points are more densely

scattered in the curvature regions, as presented in Fig. 3.13


By adjusting the value of coefficient 𝑘2 the levels of the reversal points are adjusted to be

loosely scattered over the entire range of 𝐻. By adjusting 𝑘2 = 0.5, the scattering of the

reversal points occurs more evenly over the entire range of 𝐻, as presented in Fig. 3.14.



The developed method allows for the identification of the curvature regions of the limiting

branch of a hysteresis cycle. Also, the scattering of the reversal points can be adjusted by means

of two constants. By using the developed method the amount of experimental data required for

the identification of the Preisach function has been limited to 15 curves.

3.3. Validation of the developed methods

For the validation of the developed methods the weight function of the classical Preisach model

has been identified using Mayergoyz’s method together with the methods developed in the

framework of this work. Magnetic measurements of the FORCs have been performed using the

measurement procedure presented in section 2.2.1 and 2.2.2, and the FORC level selection

method presented in section 3.2.3. The experimental data has been processed using the curve

fitting method presented in section 2.2.5. The Preisach weight function has been identified

using the FORC interpolation method presented in section 3.2.2. A desired magnetic

polarization waveform has been supplied to the model to generate the required excitation

waveform. The obtained waveform has been supplied to the sample and the measured

polarization waveform has been compared to the desired one.

3.3.1. The samples and testing procedure

The samples selected for this test have been manufactured from a low carbon steel which has

been heat treated to improve its magnetic performance. The characteristics of the samples are

presented in Table 3.2.

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Table 3.2: The characteristics of the low carbon steel sample

Property Value

Length [mm] 280

Width [mm] 30

Mass [kg] 1.16426


Thickness [mm] 1.469

Number of samples 12

Chemical composition Mn(0.272 %), Cr(0.0235 %) Al(0.0153 %),

Cu(0.0466 %) Mo(0.0077 %), Fe(balance)

The samples have been loaded into the Epstein frame and demagnetized as described in section

2.2.1.C. The maximum negative excitation field has been applied to the sample followed by a

slowly increasing field to positive maximum (6 A peak → 4468 A/m peak). The measured

ascending branch of the hysteresis cycle (Fig. 3.9) has been used as input for the procedure

presented in 3.2.3. Thus, the reversal levels for the origin of the measured FORCs have been

obtained.

As the currently employed model cannot take into account the dynamic effects the maximum

ramping rate of the polarization during measurements has been limited to 0.1 T/s. The FORCs

have been measured by modulating the polarization to a sinusoidal waveform, by adapting the

procedure described in section 2.2.2, and with consideration to the maximum ramping rate. The

obtained polarization waveform is presented in Fig. 3.15.

Fig. 3.15: The waveform of the polarization during FORC measurement

The quantity 𝐽(𝐻𝑖) presented in Fig. 3.15 is the magnetic polarization associated to the reversal

levels 𝐻𝑖=1…𝑁.


The FORCs have been processed as described in section 3.2.2 and the weight distribution of

the Preisach model has been identified as described in section 3.2.1.B. The waveform of the

polarization desired to be reproduced is presented in Fig. 3.16.

Fig. 3.16: The waveform of the polarization fed to the Preisach model

It is assumed that the material is fully demagnetized after having been subjected to the

demagnetizing procedure. The polarization is brought to the maximum level following a

sinusoidal waveform. Next, using the model the polarization is modulated as a sinusoidal

waveform between the following levels: −𝐽max, 0 T, -1 T, 1 T, 0 T, 𝐽max. The resulting

waveform of the excitation field is presented in Fig. 3.17.

Fig. 3.17: The excitation waveform generated by the Preisach model

The steps followed for the determination of the excitation waveform are the following:

1. The hysterons matrix has been split along the secondary diagonal: the hysterons in the

upper region have been set to -1, while the hysterons in the lower regions have been set

to +1.

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2. The first magnetization curve has been obtained by calculating the incremental output

of the model from 0 to maximum 𝛽.

3. The first segment of the excitation waveform (from 0 T to −𝐽max) has been constructed

by identifying the values of the first segment of the polarization waveform on the first

magnetization curve.

4. The hysterons matrix has been reset to the value 𝛽 = 𝐻(−𝐽max). For segment (−𝐽max

to 0 T), the incremental output of the model has been calculated from the current 𝛽 level

to maximum 𝛼.

5. On the resulting 𝐽𝐻 characteristic the values of the excitation field 𝐻 associated to the

desired 𝐽 have been identified.

6. The final 𝐻 value (associated to 0 T) has been used to reset the hysterons matrix of the

model.

7. For the next segment (from 0 T to -1 T) the incremental output of the model has been

calculated from the current status (0 T) to −𝐽max.

8. The values of the excitation waveform for the second segment have been identified on

this second 𝐽𝐻 characteristic.

9. The values of the hysterons matrix has been reset to the new final value of 𝐻.

10. The process has been repeated for the remaining levels.

3.3.2. The experimental results

This section presents the experimental results of the magnetic measurements performed with

the excitation waveform generated using the Preisach model. The waveform of the excitation

current 𝑖s(𝑡) required to be applied to the windings of the Epstein frame has been determined

from the waveform of the excitation field (Fig. 3.17) using Eq. (2.2). The waveform of the

polarization has been obtained by integrating the voltage induced in the secondary windings of

the Epstein frame with time (Eq. (2.4)) and by centring the integrated waveform between its

peak values.

In Fig. 3.18 (a) are presented the simulated and the measured polarization waveforms. In

Fig. 3.18 (b) is presented the absolute error of the model, which has been obtained with:

𝜖a = 𝐽sim − 𝐽meas , (3.31)

where 𝐽sim and 𝐽meas are the simulated and the measured polarization waveforms, respectively.


(a) (b)

Fig. 3.18: The error of the Preisach model

The simulated and measured 𝐽𝐻 characteristics are presented in Fig. 3.19.

Fig. 3.19: The modeled and measured magnetic hysteresis

The values of the quantities in some of the key points of the experimental results are presented

in Table 3.3.

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Table 3.3: Experimental results of magnetic hysteresis modeling

No. 𝒕 [𝐬] 𝑱𝐬𝐢𝐦 [𝐓] 𝑱𝐦𝐞𝐚𝐬 [𝐓] 𝝐𝐚 [𝐓]

1 0.0 0.000 0.000 0.000

2 7.5 -0.841 -0.938 -0.097

3 15.0 -1.690 -1.654 0.036

4 22.7 -0.812 -0.810 0.002

5 30.3 -0.002 0.029 0.031

6 34.8 -0.524 -0.494 0.030

7 39.4 -1.000 -0.978 0.022

8 48.8 0.008 -0.012 -0.020

9 58.1 1.000 0.972 -0.028

10 62.8 0.495 0.484 -0.011

11 67.6 0.000 -0.018 -0.018

12 75.0 0.852 0.691 -0.161

13 82.5 1.690 1.612 -0.078

The experimental measurements have shown that the classical Preisach model together with

the developed identification methods can be successfully used to reproduce the magnetic

hysteresis of a material. The modelling error for the first magnetization curve has been in the

order of 100 mT. The model has shown similarly high errors, and larger (approx. 160 mT),

when the model operates on cycles which originate close to the descending limiting branch.

The very high errors of the simulated waveform are associated to the operation of the model

on curves which originate in locations whose past magnetization is not clearly defined: the

demagnetized state and the upper limiting branch (when the lower limiting branch has been

used in the identification). On the other hand, when the model operates on minor cycles which

originate close to the ascending limiting branch (which has been used as origin for the

identification FORCs) the error of the model decreases to approx. 30 mT and less.

3.4. Conclusions

The theoretical background of the Jiles-Atherton and of the Preisach models of hysteresis have

been analysed. The Preisach model has been selected for the task of modelling the magnetic

hysteresis due to its advantage of memorizing fine details of the hysteresis cycles. Nevertheless,

this feature of the Preisach model requires the advancement of the identification technique, in

three topics:

Smoothing of the experimental data;

Interpolation of first order reversal curves;

Minimize the required input data.


For the first topic a curve smoothing method has been developed. The method relies on fitting

segments of experimental data to low order polynomials by means of constrained linear

regression analysis. The second derivative of experimental data has been investigated by using

this method.

For the second topic a new curve interpolation method has been developed. This method

identifies the shortest distances between the points of two curves. The new FORC is identified

on these segments by relating the location of the reversal point to the reversal points of the

bounding FORCs. This method has been used to identify the weight function of the Preisach

model with high degree of detail (any number of threshold levels).

For the third topic has been developed a method which identifies the locations of the curvatures

of the limiting branch. The method is capable to increase the scattering of the reversal points,

where FORCs are measured, in these locations. This method has been used to limit the number

of acquired FORCs to 15 while capturing all the details of the hysteresis cycle.

The experimental results have shown that the classical Preisach model identified with the newly

developed identification method has among the lowest errors when it reproduces minor

asymmetric cycles which originate in the limiting ascending branch. This characteristic can be

exploited to model the behaviour of a material in the core of a magnet with improved

performance.

4. ASSESSMENT OF HYSTERESIS EFFECTS IN MAGNETIC

CIRCUITS

In this chapter the modelling procedure of the magnetic field induction in the gap of a magnet

is described. The model developed in section 1.3 has been adapted to consider fewer

assumptions and it has been used to model the magnetic circuit. The material in the core of the

magnet has been modelled using two models: a hysteresis model based on the Preisach model

with its weight function identified using the methods presented in chapter 3, and an anhysteretic

model based on the normal magnetization curve of the material obtained using the methods

presented in section 2.2.

The models have been applied to two magnets of different size and configuration of the

magnetic circuit. The first magnet is a small size H-dipole magnet which has been designed

and developed for this experiment. This magnet has a small gap which gives a small

demagnetizing coefficient. The second magnet is a confined function magnet which is used in

a test bench at CERN. This magnet has an aperture of 50 mm and the magnetic induction

generated in its gap has both a dipolar and quadrupolar component. The calculated quantity has

been the value of the magnetic induction in the centre of the gap, normal to the horizontal plane.

Experimental measurement results have been performed using an excitation cycle which

resembles a super-cycle used in a particle accelerator. The modelling results obtained using the

two models are benchmarked against the experimental measurement and the effectiveness of

the hysteresis model is highlighted.

4.1. Hysteresis modelling of the gap induction of an experimental demonstrator magnet

An experimental, small size H-dipole magnet has been designed and developed for the purpose

to model the hysteretic behaviour of the magnetic field induction in the gap of a magnet. The

design of the magnet aims to obtain a simple magnetic circuit comply as close as possible to

the assumptions of the idealised circuit. Therefore, the following two design constraints have

been defined: the value of the magnetic induction in the gap of the magnet has a very close

value to the one in the core, and the magnetic flux closes through a surface of constant area.

Table 4.1 presents the design constraints of the experimental magnet.


Table 4.1: Design constraints of the experimental magnet

Parameter Value Unit Remark

Gap induction (𝐵g) 2 T Value sufficient to saturate most

material

Gap height (𝑙g) 2 mm Height of air gap

Core field length (𝑙Fe) 600 mm The length of the magnetic field

path in the core (approximate value)

Maximum current (𝐼max) 6 A KEPCO BOP6-36ML

Maximum voltage (𝑉max) 36 V KEPCO BOP3-36ML

Maximum current density (𝐽max) 1 A/mm2 Air cooled coil

Saturation permeability (𝜇r) 200 Value of the relative permeability

used during calculation

The ampere turns required to achieve the required magnetic flux density is determined starting

from Ampere’s law and using the design constraints of the magnet as [139]:

𝑁𝐼 = ∫𝐵

𝜇 d𝑙 = ∫

𝐵

𝜇0 d𝑙

gap

+ ∫𝐵

𝜇0𝜇r d𝑙

iron

𝑁𝐼 =𝐵

𝜇0(𝑔 +

𝐿

𝜇r) =

2 T

𝜇0(2 mm + 3 mm) = 7957.7 A

(4.1)

Considering the maximum current, the required number of turns is:

𝑁𝐼max = 7957.7 A ⇒ 𝑁 = 1326 (4.2)

Considering the maximum current density allowed for air cooling of the coil the minimum

cross section of the conductor is determined:

𝑆min ≥𝐼max𝐽max

≥ 6 mm2 (4.3)

The available conductor is rectangular enamelled copper wire with cross-section of 1.6 mm by

3.75 mm for a total cross-section of 6 mm2. Considering the thickness of the insulation to be

0.075 mm the approximate total cross-sectional area of the winding is:

𝑆 = 𝑁 ⋅ (1.6 + 0.15) ⋅ (3.75 + 0.15) = 9050 mm2 (4.4)

In order to take into account the area losses in the surface due to air gaps and layer jumps, and

to provide sufficient tolerance for the mechanical design process, the cross-section used in the

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following design steps is 𝑆 = 9600 mm2. The resistance of the coil will be calculated after the

optimization of the coil and core design as the length of the conductor is not yet defined.

4.1.1. Design of the magnetic circuit

Modelling and simulations of the magnetic circuit have been performed using finite element

method (FEM) software (Vector Fields - Opera) with the purpose to achieve the design

constraints of the magnet. It is not the purpose of this work to cover the theoretical background

of the FEM, the topic being covered extensively in literature [140, 141]. The following design

steps have been performed:

Optimization of the coil aspect ratio, with the purpose to maximize the value of the

magnetic induction in the gap for a given current.

Optimization of the pole face, with the purpose to obtain a value of the gap induction as

close as possible to the value found in the core.

Optimization of the magnetic field distribution in the core, with the purpose to obtain

constant flux closing area along the entire magnetic circuit.

Optimization of the distribution of the magnetic induction in the gap.

A. Optimization of the coil aspect ratio

The optimization of the coil cross-section aspect ratio (the ratio between the cross-section width

and height) aimed at maximizing the value of the magnetic induction in the gap of the magnet

for a given current. Using Opera 2d the coils have been modelled as current carrying

rectangular regions with the total cross-sectional area previously determined. Between the

modelled coil region and the yoke region a 5 mm air gap has been reserved for the coil

insulation. In order to decrease the weight of the individual parts the coil has been separated in

two parts: one for the upper side of the core and one for the lower side. The distance between

the two parts of the coil, along the vertical axis, has been modelled to 6 mm. The symmetry of

the magnetic circuit allows that only one fourth of the magnet is simulated, thus saving on

computational time. The geometry of the model, which represents one quarter of the cross-

sectional area of the magnet in the vertical plane, is presented in Fig. 4.1.


Fig. 4.1: The initial 2d FEM model

In Fig. 4.1, with red is depicted the cross-section of the coil with the width 𝑤 and height ℎ.

With blue is depicted the magnetic core. In the core have been modelled the holes (∅ 6.3 mm)

which are required to hold the stack of laminations together. The material in the core has been

modelled using the normal magnetization curve of M 1300-100 A. The magnetic flux lines in

the circuit have been depicted with yellow lines.

For the optimization of the geometry of the coil a number of 41 simulations have been

performed where the width of the coil has been varied from 30 mm to 110 mm. The height of

the coil has been calculated to obtain the required coil cross-sectional area. For each simulation

the coil aspect ratio has been calculated and the value of the magnetic induction has been

measured in the centre of the gap. The results of the simulations are presented in Fig. 4.2.

Fig. 4.2: Simulation results for coil cross-sectional area aspect ratio optimization

The simulation results presented in Fig. 4.2 show that the peak gap induction is achieved for

an aspect ratio of 1.5. Therefore, the height ℎ of the coils has been set to 55 mm, and the width

𝑤 to 87.3 mm.

1.5

1.51

1.52

1.53

1.54

1.55

1.56

1.57

1.58

0 0.5 1 1.5 2 2.5 3

B (

T)

Coil aspect ratio (w/h)

𝑤

ℎ

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B. Optimization of the pole face

In the gap, the magnetic field will tend to spread outward in the “fringe field” and, thus, the

equivalent area through which the magnetic flux closes increases and the value of the gap

induction decreases. The shape of the pole face has been optimized to balance the level of the

gap induction to the level found in the core. The chamfer radius at the edge of the pole, see R

in Fig. 4.3, has been increased thus reducing the gap flux closing area.

Fig. 4.3: Pole-face edge chamfer

A number of 30 simulations have been performed where the value of the pole-face chamfer

radius has been varied from 0.2 mm to 7.2 mm. The value of the induction in the centre of the

gap (x=0 and y=0) and the value of the induction in the yoke at coordinates x=0 and y=50 mm

have been measured in the simulation. The results of the simulations are presented in Fig. 4.4.

Fig. 4.4: Results of the pole-face optimization

The simulations have revealed that for a pole-face chamfer radius of R=5.25 mm the value of

the induction in the gap is the same as the value of the induction found in the core of the magnet.

C. Optimization of the magnetic flux path

By analysing the magnetic flux path, as presented in Fig. 4.1, it appears that the flux closing

area is not constant along the circuit because of the holes designed to hold the laminations in

the core together. In the following section is described the optimization of the geometry of the

magnetic core with the purpose to comply with the requirement of constant flux closing area.

1.55

1.6

1.65

1.7

1.75

1.8

0 1 2 3 4 5 6 7

B (

T)

Pole radius R (mm)

B(0;0)

B(0;50)

R


The bottlenecks of the magnetic flux in the initial core design are presented in Fig. 4.5 (the

regions in red). The colour code represents the value of the magnetic induction: with dark-blue

are represented the regions with 1.5 T, while with magenta are represented the regions with

2.2 T.

Fig. 4.5: Highlight of magnetic flux bottlenecks within the circuit

In the bottleneck regions the magnetic flux density increases and as a result the operation point

of the material will shift. By removing the bottlenecks of the magnetic flux the core will operate

on the same point. As long as the surface normal to the magnetic flux vector is constant along

the path of the magnetic flux, bottlenecks of the magnetic flux can be avoided. In Fig. 4.6 is

presented the orientation of the magnetic flux around the holes and the additional material in

the core cross-section required to cancel the bottleneck.

Fig. 4.6: Magnetic flux lines in the region of the holes

Additional simulations have been performed where extra material has been added around the

holes, and the inner and outer corners of the core have been chamfered such that the cross-

sectional area normal to the magnetic flux path is constant. The location of the holes has been

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adjusted to maximize the homogeneity of the magnetic induction and extra material has been

added to simulate a clamping system. The simulation of the optimized magnetic circuit is


Fig. 4.7: Map of the magnetic induction in the optimized simulation

The homogeneity of the magnetic induction distribution, as presented in Fig. 4.8, has been

calculated with respect to the value of the induction measured in the centre of the gap using the

following relation:

Δ𝐵 =𝐵(𝑥, 𝑦, 𝑧) − 𝐵(0,0,0)

𝐵(0,0,0) , (4.5)

where: Δ𝐵 is the relative variation of the magnetic induction, 𝐵(𝑥, 𝑦, 𝑧) is the magnetic

induction measured at coordinates (𝑥, 𝑦), and 𝐵(0,0) is the value of the magnetic induction

measured in the centre of the gap. The 𝑧 axis is used for three-dimensional simulations and for

the current case it is considered that 𝑧 = 0 mm. The homogeneity of the magnetic induction

distribution is presented in Fig. 4.8. Here, two cases are presented: the material operates in a

high-permeability point when the simulated current density 𝐽 = 0.1 A/mm2, and the material

operates close to saturation when the simulated current density 𝐽 = 0.85 A/mm2.


(a) Homogeneity map for 𝐽 = 0.1 A/mm2 (b) Homogeneity map for 𝐽 = 0.85 A/mm2

Fig. 4.8: Magnetic induction homogeneity in the optimized model

From the simulation results presented in Fig. 4.8 one can see that for all magnetization levels

the magnetic flux closes through a path where the magnetic induction has a variation less than

5 %. Therefore, the design constraint that the flux closing area is constant has been achieved,

for the two-dimensional model, within a 5 % error.

D. Optimization of the magnetic induction distribution in the gap

The purpose of this optimization procedure has been to ensure small variation of the value of

the magnetic induction over the region where the Hall probe will be placed to perform the

measurements. This optimization process has been performed on the three-dimensional (3D)

model of the magnet. The general layout of the 3D model is presented in Fig. 4.9.

Fig. 4.9: The general layout of the 3D model of the demonstrator magnet

The possible optimization handles are the thickness of the core along the 𝑧 axis, and the

positioning along the 𝑦 axis of the central hole (in the previous figures it has been centred

around 𝑦 = 25 mm and 𝑥 = 0 mm). A number of 21 simulations have been performed where

the position of the central hole on the 𝑦 axis has been adjusted with linearly spaced values from

15 mm to 45 mm. In each simulation the value of the induction in the centre of the gap has

been measured and the variation of the gap induction (Δ𝐵) has been calculated using Eq. (4.5)

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for the coordinates 𝑥 = 10 mm, 𝑦 = 0 mm, and 𝑧 = 0 mm. The results of the simulations are


Fig. 4.10: Optimization of central hole position

The simulation results presented in Fig. 4.10 show that by adjusting the height of the central

hole the variation of the gap induction along the 𝑥 axis and the value of the induction in the

centre of the gap can be optimized. The screw which is placed in the central hole has the

purpose to hold together the laminations in the region of the pole face and it is desirable to

maintain the height of the hole as low as possible. Therefore, the height of 35 mm has been

selected as it offers the best trade-off between magnetic and mechanical performance.

The variation of the gap induction value along the 𝑧 axis can be optimized by adjusting the

value of the core length along this axis. A number of 21 simulations have been performed

where the length of the core has been varied from 40 mm to 80 mm in linearly spaced

increments. In each simulation the value of the induction in the centre of the gap has been

measured and the variation of the gap induction (Δ𝐵) has been calculated using Eq. (4.5) for

the coordinates 𝑥 = 0 mm, 𝑦 = 0 mm, and 𝑧 = 10 mm. The results of the simulations are

shown in Fig. 4.11.

Fig. 4.11: Optimization of the yoke thickness

0.12%

0.14%

0.16%

0.18%

0.20%

0.22%

0.24%

1.74

1.75

1.76

1.77

1.78

1.79

1.8

15 20 25 30 35 40 45

ΔB

B (

T)

Height of the central hole (mm)

B(0;0)

ΔBmax

0.00%

0.10%

0.20%

0.30%

0.40%

0.50%

0.60%

1.7

1.72

1.74

1.76

1.78

1.8

1.82

40 50 60 70 80

ΔB

B (

T)

Core length (mm)

B(0;0)

|Δbmin|


The simulation results presented in Fig. 4.11 show that by increasing the length of the core the

value of the gap induction can be increased and the variation of the gap induction along the 𝑧

axis can be decreased. The length of the core has been selected to be 72 mm, as any higher

value would lead to decrease of the gap induction variation below 0.15 %, the value of the

variation along the 𝑥 axis. Also, further increase of the core length, beyond 72 mm, would lead

to increase of the gap induction but at the expense of additional material and, therefore, the

core length of 72 mm has been considered the best trade-off between manufacturing price and

magnetic performance.

4.1.2. Structural considerations

This section presents the methods used to determine the manufacturing tolerance for key

dimensional values, the determination of the forces acting on key locations of the magnet, and

to present the thermal analysis of the coils.

A. Gap height tolerance

The tolerance for the distance between the pole faces can be determined analytically starting

from Eq. (4.1). For a given gap height value, the gap induction can be approximated to be:

𝐵1 =𝜇0𝑁𝐼

𝑔 +𝐿𝜇r

. (4.6)

Considering that the gap height has been manufactured with an error 𝜖, then the value of the

gap induction is:

𝐵2 =𝜇0𝑁𝐼

𝑔 + 𝜖 +𝐿𝜇r

. (4.7)

Then the relative variation of the gap induction due to the manufacturing error 𝜖 is:

Δ𝐵 =𝐵1 − 𝐵2𝐵1

= 1 −𝑔 +

𝐿𝜇r


=𝜖


. (4.8)

Then, the value of the manufacturing error as a function of the desired induction variation is:

𝜖 =Δ𝐵

1 − Δ𝐵⋅ (𝑔 +

𝐿

𝜇r) . (4.9)

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By assuming that 𝑔 ≫𝐿

𝜇r, and that Δ𝐵 ≪ 1, then the following simple relation of the gap

manufacturing error can be obtained:

𝜖 = Δ𝐵 ⋅ 𝑔 . (4.10)

The best trade-off between magnetic performance and manufacturing price can be obtained by

considering a value for the relative variation Δ𝐵 = 5 × 10−3, then the manufacturing tolerance

for the gap height is 𝜖 = 0.01 mm.

B. The force acting between the two pole faces

The force acting on the pole faces along the 𝑦 axis is determined in this section. This force is

determined starting from the magnetic pressure found in the gap of the magnet:

𝑃PF =𝐵𝑦2

2𝜇0=

22

2 ⋅ 4𝜋10−7= 1.59 ⋅ 106 Pa = 1.59 N/mm2 . (4.11)

Then, the force acting upon the pole faces in the gap is:

𝐹PF = 𝑃PF ⋅ 𝐴g = 1.59 ⋅ 50 ⋅ 72 = 5724 N . (4.12)

Between the pole faces a piece of non-magnetic material (brass) has been employed to maintain

the gap height and to provide a centring path for the Hall probe. The calculated force is

sufficiently low such that no deformation is considered to occur to the separating brass piece.

C. The force acting on the end-plate

The stack of laminations is bounded by a plate whose purpose is to ensure the mechanical

rigidity of the stack. This section presents the calculation method of the force which acts on

the end plates at the level of the pole face. The effect of this force is a spreading of the

laminations.

A conservative estimate of the pressure acting on the end-plate can be obtained using the value

obtained using Eq. (4.11). The surface over which the magnetic pressure acts is the hashed

region presented in Fig. 4.12. The area of the hashed surface is 1227.34 mm2 and, therefore,

the conservative estimate of force acting at the tip of the end-plate is:

𝐹EP = 𝑃PF ⋅ 𝐴EP = 1.59 ⋅ 1227.34 = 1951 N . (4.13)


Fig. 4.12: Detail of the end-plate subjected to the action of the force

The material selected for the manufacture of the end-plate is stainless steel, of quality 304L,

5 mm thick and with Young’s modulus of 200 GPa. The maximum deformation of the end-

plate is:

𝑓EP =𝐹EP𝑙

3

3𝐸𝐼=

𝐹EP𝑙3

3𝐸 ⋅𝑡𝑤3

12

=1951 ⋅ 31.853

3 ⋅ 200 ⋅ 103 ⋅5 ⋅ 403

12

= 1.5 ⋅ 10−3 mm , (4.14)

where: 𝑓 is the deflection of the end-plate (mm), 𝐹 is the force acting at the tip of the end-plate

(N), 𝑙 is the length of the end-plate from the rigid point to the tip (mm), 𝐸 is Young’s modulus

for the material of the end-plate (MPa), 𝐼 is the area moment (mm4), 𝑡 is the thickness of the

end-plate (mm), 𝑤 is the width of the end-plate (mm).

Although the magnetic pressure value used during the calculations has been a very conservative

value, the calculated end-plate deformation is sufficiently low to be considered negligible to

the operation of this magnet.

D. Coil thermal analysis

This section presents the thermal analysis of the coils of the magnet. The calculations have

been performed analytically starting from the optimized design of the magnetic circuit. The

view of the coils with the geometrical dimensions are presented in Fig. 4.13.

Doctoral thesis 125

(a) view of the coil in the 𝑥𝑧 plane (b) cross-setion view of the coil in the 𝑥𝑦 plane

Fig. 4.13: View of the coils

The number of windings per layer for both coils was determined to be 28 and the number of

layers to be 51. The approximate total length of the conductor in the coils has been calculated:

𝐿 = 28 ⋅∑(2 ⋅ 98 + 2 ⋅ 49.3 + 2 ⋅ 𝜋 ⋅ (7 + 0.875 + 1.75 ⋅ 𝑖))

50

𝑖=0

= 884 m (4.15)

Considering the electrical resistivity of copper of 16.78 nΩ ⋅ m and the cross-sectional area of

the conductor of 6 mm2, then the electrical resistance of the coil is 2.47 Ω. For a conductor of

similar cross-section, manufacturer catalogue data specifies the length per weight ratio of

18.1 m/kg. Therefore, for the current design the expected weight of the coils is 48.9 kg.

Starting from the first law of thermodynamics, the equation that governs the temperature

change with time is determined as follows:


g − out = st

g = 𝑅𝐼2

out = ℎ𝐴(𝑇 − 𝑇sur)

st =d

d𝑡(𝜌𝑉𝑐𝑇)

𝑅𝐼2 − ℎ𝐴(𝑇 − 𝑇sur) = 𝜌𝑉𝑐d𝑇

d𝑡

d𝑇

d𝑡=𝑅𝐼2 − ℎ𝐴(𝑇 − 𝑇sur)

𝜌𝑉𝑐

Steady − state:d𝑇

d𝑡= 0 ⇒ 𝑅𝐼2 = ℎ𝐴(𝑇 − 𝑇sur)

(4.16)

where: g is the thermal energy generation rate (W); out is the rate of energy outflow due to

convection and radiation (W); st is the change in energy storage due to temperature change

(W); 𝑅 is the electrical resistance of the conductor (Ω); 𝐼 is the current passing through the

conductor (A); ℎ is the convection heat transfer coefficient (W/m2K); 𝐴 is the area of the coils

block (m2); 𝑇 is the temperature of the coils block (K); 𝑇sur is the temperature of the

surrounding environment (K); 𝜌 is the density of the coils material (kg/m3); 𝑉 is the volume of

the coils (m3); 𝑐 is the specific heat of the coils material (kJ/kgK). In Eq. (4.16) the outflow

due to radiation has been considered negligible.

Heat transfer problems can be solved by numerical simulation or by using empirical correlation

based on experiments and numerical simulation. In the following, an approximate solution will

be determined by using the second method as detailed in reference [142]. Each of the surfaces

of the coil block will be characterised by a different heat transfer coefficient, therefore the term

out will become:

out = ℎt𝐴t(𝑇 − 𝑇sur) + ℎv𝐴out(𝑇 − 𝑇sur) + ℎv𝐴in(𝑇 − 𝑇sur)

+ ℎb𝐴t(𝑇 − 𝑇sur), (4.17)

and Eq. (4.16) becomes:

𝑅𝐼2 = [𝐴t(ℎt + ℎb) + ℎv(𝐴out + 𝐴in)](𝑇 − 𝑇sur) , (4.18)

Doctoral thesis 127

where: 𝐴t is the surface of the top side of the coils block as presented in Fig. 4.13-(a) (m2),

𝐴out is the surface of the coils block described by the outer perimeter in Fig. 4.13-(b) (m2), 𝐴in

is the surface of the coils (m2), ℎt is the convection heat transfer coefficient for a horizontal

surface when the lower surface is cooled (W/m2K), ℎb is the convection heat transfer

coefficient for a horizontal surface when the upper surface is cooled (W/m2K), ℎv is the

convection heat transfer coefficient for a vertical surface (W/m2K).

For each of the situations the heat transfer coefficient will be determined from the Nusselt

number [142]:

Nu =ℎ𝐿

𝜆⇒ ℎ =

Nu𝜆

𝐿, (4.19)

where: 𝐿 is the characteristic length and will be defined accordingly for each of the situations,

𝜆 is the thermal conductivity of air, given the present situation of air cooled coils. For vertical

surfaces, the Nusselt number is, as defined by [143]:

Nu = 0.825 + 0.387 ⋅ [Ra ⋅ 𝑓1(Pr)]162

, (4.20)

where: Ra is the Rayleigh number, Pr is the Prandtl number and the function 𝑓1, according

to [144], as long as 10−3 < Pr < ∞, is defined as:

𝑓1(Pr) = [1 + (0.492

Pr)

916]

−169

. (4.21)

For vertical surfaces, the characteristic length is equal to the height of the surface.

For horizontal surfaces, when the lower surface is cooled, the Nusselt number is, as proposed

by [145] and adapted by [142]:

Nu = Nu = 0.766 ⋅ [Ra ⋅ 𝑓2(Pr)]

15, when Ra ⋅ 𝑓2(Pr) ≤ 7 ⋅ 10

4

Nu = 0.15 ⋅ [Ra ⋅ 𝑓2(Pr)]13, when Ra ⋅ 𝑓2(Pr) ≥ 7 ⋅ 10

4, (4.22)

where the function 𝑓2, as long as 0 < Pr < ∞,is defined as:


𝑓2(Pr) = [1 + (0.322

Pr)

1120]

−2011

. (4.23)

For horizontal surfaces, when the upper surface is cooled, the Nusselt number, as presented

in [142], is:

Nu = 0.6 ⋅ [Ra ⋅ 𝑓1(Pr)]15 . (4.24)

For horizontal surfaces, the characteristic length is equal to the ratio between the surface and

its perimeter.

The Rayleigh number is defined as [142]:

Ra =𝛽𝑔(𝑇 − 𝑇sur)𝐿

3

𝑣𝜅 , (4.25)

where: 𝑔 is the acceleration of gravity (m/s2), Δ𝑇 is the temperature difference (K), 𝐿 is the

characteristic length (m), 𝑣 is the kinematic viscosity (m2/s), 𝜅 is thermal diffusivity (m2/s) and

𝛽 is the isobaric volume expansion (1/K) coefficient which can be approximated as:

β =2

(𝑇 + 𝑇sur) . (4.26)

The dependence of the convection heat transfer coefficient with temperature for the given coil

geometry is presented in Fig. 4.14.

Fig. 4.14: Dependence of the convection heat transfer coefficient with temperature

20 30 40 50 60 70 80 900

1

2

3

4

5

6

7

T (oC)

h (

W/(

m2K

))

Dependence of the convection heat transfer coefficient on temperature

h vertical

h top

h bottom

Doctoral thesis 129

The coefficient varies with temperature because of the dependence of the Rayleigh number and

because the properties of the air have been modelled as a function of temperature [146].

Assuming that the coils operate in an environment at room temperature, the dependence of the

coils temperature as a function of the applied current, determined according to Eq. (4.18), is

presented in Fig. 4.15

Fig. 4.15: Dependence of coil temperature with current in steady-state conditions

The calculations have revealed that under steady-state conditions and for maximum current,

the coils will operate at about 85 °C. This value does not represent any danger to the electrical

operation of the coils. Nevertheless, care shall be taken during the measurement process to

avoid lengthy measurements which can lead to the heating of the core.

4.1.3. The model and the measurement procedure

The objective of this section is to describe the method used to model the hysteretic behaviour

of the induction in the gap of the experimental demonstrator magnet. The hysteresis model has

been identified using the procedures presented in the previous chapter and it has been used to

obtain the hysteresis cycles of the magnetization as a function of the prescribed excitation

current. Additionally, the gap induction has been modelled using the normal magnetization

curve. The modelling results have been analysed and benchmarked against the experimental

measurements.

A. The experimental setup

This section describes the developed experimental magnet and the method used to perform the

measurements. The core has been manufactured using 101 laminations of electrical steel of

quality M 270-50 A. A Hall probe has been inserted in the centre of the gap to measure the gap

induction. A picture of the assembled experimental magnet is presented in Fig. 4.16.

0 1 2 3 4 5 620

30

40

50

60

70

80

90

I (A)

T (

oC

)

Dependence of temperature with current (steady-state)


Fig. 4.16: Picture of experimental magnet

In series with the coils of the magnet has been connected a four-point shunt resistor which has

been used to measure the value of the current through the circuit. The measurement has been

automated using a software developed in the LabView programming environment. The block

diagram of the measurement setup is presented in Fig. 4.17

Fig. 4.17: Block diagram of the measurement setup of the experimental magnet

The elements presented in Fig. 4.17 are:

PS is a voltage controlled power supply (KEPCO bipolar BOP 6-36ML, max. current

6 A);

𝑅s is a shunt resistor of 1 Ω and 15 W;

DAC

ADCPC

Rsis(t)

>

us(t)

PS H

Gaussmeter

RS232

PCI

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H is the test Hall probe;

ADC is the analog to digital converter (NI PCI-6154);

DAC is the digital to analog converter (NI PCI-6154);

PC is the personal computer with LabView software;

𝑢s is the voltage drop on the shunt resistor.

GAUSSMETER is the gauss-meter from F.W. BELL model 6010

During the measurement the waveform of the excitation current is sent through the PCI port to

the DAC which, in turn, will drive the PS. The value of the current is determined by measuring

the voltage drop 𝑢s on the shunt resistor. The value of the gap induction is measured with the

Gauss-meter through the RS232 port. The measurements of the two quantities are performed

synchronously, at a rate of 5 samples per second. The low measurement rate is given by the

communication speed of the Gauss-meter through the RS232 port.

B. The mathematical models

This section presents the mathematical models used to model the gap induction of the

experimental magnet. The modelling operation involved two mathematical models, as

presented in section 1.3: one for the magnetic circuit (which uses concentrated parameters and

assumes that the magnetic flux in the gap is homogenous) and one for the material in the gap.

The mathematical model of the magnetic circuit has been extended to take into account the

effects of the saturation of the material.

The mathematical model of the magnetic circuit is determined starting from the topology of

the circuit presented in Fig. 1.2. It is assumed that the area through which the magnetic flux

closes through the gap is dependent on the value of the magnetic induction found in the core

of the magnet, 𝑆g(𝐵Fe). It is also assumed that the length of the magnetic field path in the core

is dependent on the value of the induction in the core, 𝑙Fe(𝐵Fe). These two dependencies have

been identified using simulation performed on the 3D FEM model developed using Opera

software.

Starting from Ampere’s law, considering the previously mentioned assumptions, and

considering the magnetic field components are aligned to the magnetic field path, the

mathematical model of the gap induction is:

𝑀Fe𝑙g

𝑆Fe𝑆g(𝐵Fe)

+ 𝐻Fe (𝑙Fe(𝐵Fe) + 𝑙g𝑆Fe

𝑆g(𝐵Fe)) = 𝑁𝐼

𝑀Fe = 𝑓(𝐻Fe)

, (4.27)


where: 𝑀Fe is the magnetization of the material in the core (A/m), 𝑙g is the length of the

magnetic circuit in the gap (m), 𝑆Fe is the area through which the magnetic flux closes in the

core (m2), 𝑆g is the area through which the magnetic flux closes in the gap (m2), 𝐵Fe is the

value of the magnetic induction in the core (𝐵Fe = 𝜇0(𝐻Fe +𝑀Fe)) (T), 𝐻Fe is the value of the

magnetic field strength in the core (A/m), 𝑙Fe is the length of the magnetic circuit in the

core (m), 𝑁 is the number of windings in the coils of the magnet, and 𝐼 is the value of the

excitation current (A).

The 3d FEM model of the magnet has been used to identify the 𝑆g(𝐵Fe) and the 𝑙Fe(𝐵Fe)

characteristics. The material in the core has been modelled using the normal magnetization

curve, as presented in Fig. 4.18.

Fig. 4.18: The normal magnetization curve of the material in the core of the experimental magnet

A number of 15 simulations have been performed with the current density in the coils varying

from 0.01 A/mm2 to 0.85 A/mm2. The value of the magnetic flux in the circuit has been

determined by calculating the field integral on a patch which encompasses the cross-section of

the outer leg of the magnet. The value of the magnetic induction in the core has been determined

by dividing the value of the flux to the value of the area over which it has been calculated. The

value of the flux closing area in the gap has been determined by dividing the value of the flux

to the value of the gap induction. The 𝑆Fe/𝑆g(𝐵Fe) dependence is presented in Fig. 4.19.

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Fig. 4.19: The 𝑆Fe/𝑆g = 𝑓(𝐵Fe) dependence for the experimental magnet

It appears that for the largest range of the magnetic induction the ratio has a constant value of

approx. 0.85, and it increases exponentially as the material approaches saturation.

The length of the magnetic flux math in the core has been determined using the 2D FEM model

of the circuit. A number of 15 simulations have been performed with the current density in the

coils varying from 0.01 A/mm2 to 0.85 A/mm2. The value of the core induction has been

determined similarly to the 3D model method but without the 𝑧 axis. For each simulation the

coordinates of the point along the flux line originating in the 𝑥 = 10 mm and 𝑦 = 0 mm have

been determined, and the length of the line in the core has been calculated using numerical

methods. The results of the simulations are presented in Fig. 4.20.

Fig. 4.20: The 𝑙fe = 𝑓(𝐵Fe) dependence for the experimental magnet

Using the dependencies presented in Fig. 4.19 and Fig. 4.20 the mathematical model of the

magnetic circuit has been fully identified.


C. The modelling results

This section presents the experimental measurement results performed on the demonstrator

magnet and the benchmarking of the modelling results against the measurements. The

waveform of the excitation current has been modulated with consideration to the powering

cycles of a particle accelerator magnet, as presented in Fig. 1.4. The imposed excitation

waveform reflects the characteristics of these cycles with increased complexity. The core

magnetization, 𝑀Fe = 𝑓(𝐻Fe), has been modelled using two models of the material

M270-50A: the hysteretic model of the magnetization using the Preisach model identification

according to the procedure described in section 3.2, and the anhysteretic model of the

magnetization using the normal magnetization curve of the core material. The waveform of the

imposed excitation current is presented in Fig. 4.21.

Fig. 4.21: The waveform of the excitation current

The measurement operation has been preceded by three cycles of the power supply between

0 A and 3 A. The operating point of the magnet has been modelled by calculating the solutions

of the system of equations (4.27). For the anhysteretic model the normal magnetization curve

has been used. On the other hand, with the hysteretic model, after each reversal of the current

ramping direction (up or down) a new branch of the magnetization hysteresis cycle has been

calculated and used in system (4.27). The curves which characterize the two models are


Fig. 4.22: The magnetization models of the core material of the experimental magnet

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The benchmark of the modelling results against the experimental measurements is presented

in Fig. 4.23.

Fig. 4.23: Benchmarking of gap induction modelling against experimental measurements

The top section of Fig. 4.23 presents the superimposed values of the gap induction: the

measured waveform with blue, the waveform obtained using the hysteresis model with red, and

the waveform obtained using the anhysteretic model with yellow. The differences are too small

to be noticed without any further analysis.

The central section of Fig. 4.23 presents the superimposed values of the absolute modelling

errors of the two models using the following relation:

𝜖a = |𝐵g,meas − 𝐵g,model| , (4.28)

where: 𝐵g,meas is the value of the gap induction measured experimentally, and 𝐵g,model is the

value of the gap induction obtained with the models. As highlighted in Fig. 4.23, the absolute

error of the hysteresis model is significantly lower than the absolute error of the anhysteretic

model.

The lower section of Fig. 4.23 presents the superimposed values of the relative modelling errors

of the two models, referenced to the measured value. The value of the relative error has been

obtained using the following relation:


𝜖r =𝜖a

𝐵g,meas⋅ 100 . (4.29)

The benchmarking results show that the hysteresis model has approximated the gap induction

with an error of approx. 1 %. A selection of the values presented in Fig. 4.23 are presented in

Table 4.2.

Table 4.2: Modelling results of the experimental magnet

Time (s) 𝑰 (A) 𝑩𝐠 (T) 𝑩𝐠,𝐡𝐲𝐬𝐭

(T)

𝝐𝐚,𝐡𝐲𝐬𝐭

(mT)

𝝐𝐫,𝐡𝐲𝐬𝐭

(%)

𝑩𝐠,𝐚𝐧

(T)

𝝐𝐚,𝐚𝐧

(mT)

𝝐𝐫,𝐚𝐧

(%)

5 0.003 0.01 0.02 1.57 11.05 0.00 14.24 100.00

178 1.200 0.86 0.86 4.31 0.50 0.86 2.62 0.31

650 0.598 0.45 0.45 2.32 0.51 0.43 23.04 5.07

747 0.461 0.33 0.33 0.70 0.21 0.32 10.37 3.12

1115 1.773 1.22 1.21 14.69 1.20 1.20 16.58 1.36

1317 0.309 0.24 0.24 2.16 0.90 0.21 26.37 10.95

1473 1.345 0.96 0.96 6.68 0.70 0.96 4.67 0.49

1570 1.192 0.87 0.88 5.34 0.61 0.86 15.54 1.78

1720 0.384 0.29 0.28 2.85 1.00 0.27 17.49 6.12

1811 0.449 0.34 0.34 8.61 2.50 0.32 21.13 6.14

1908 0.010 0.02 0.02 1.29 6.64 0.00 19.46 100.00

2139 1.784 1.20 1.20 0.49 0.04 1.20 3.41 0.28

2485 0.675 0.51 0.51 2.40 0.47 0.48 23.07 4.54

2663 1.778 1.20 1.20 1.53 0.13 1.20 3.64 0.30

3009 0.675 0.51 0.51 2.37 0.47 0.48 23.28 4.58

3143 1.347 0.96 0.96 7.31 0.76 0.96 4.54 0.48

3239 1.192 0.87 0.88 5.55 0.64 0.86 15.69 1.79

3397 0.153 0.13 0.13 0.18 0.14 0.11 24.41 18.59

3495 0.302 0.22 0.23 1.11 0.49 0.21 9.70 4.32

3593 0.159 0.12 0.12 2.87 2.46 0.11 10.49 8.98

3722 0.332 0.65 0.65 1.20 0.19 0.65 1.58 0.24

3849 0.152 0.13 0.13 0.76 0.58 0.11 24.27 18.57

3947 0.309 0.22 0.23 1.96 0.87 0.21 9.75 4.34

4118 1.480 1.05 1.06 13.89 1.33 1.06 8.65 0.83

4290 0.299 0.24 0.24 0.15 0.06 0.21 26.29 10.91

The spiking error values of the hysteresis model are due to the very low referenced value. On

the other hand, in addition to the spiking error values due to the low referenced value, the

modelling results obtained using the anhysteretic model have additional high error regions

especially when the excitation current is ramped down. The source of these errors are

highlighted by analysing the shapes of the modelled magnetic hysteresis cycles modelled for

the material in the core, as presented in Fig. 4.22.

Doctoral thesis 137

The results presented in Fig. 4.22 show the source of the second set of spikes in the waveform

of the relative error. When the current is decreased, the core material follows a branch of the

hysteresis cycle which is poorly approximated by the normal magnetization curve.

Nevertheless, the normal magnetization curve offers a fair approximation of the ascending

branches of the hysteresis cycles, especially in the saturating region.

The gap induction of an experimental magnet has been modelled with a complicated waveform

of the excitation current powering history. The developed magnetic circuit model together with

the hysteretic magnetization model have successfully approximated the gap induction within

approx. 1 % error for the entire magnetization range. Additionally, the gap induction has been

modelled using an anhysteretic model which showed an error of approx. 1 % for the most

favourable case, and 100 % error when the current is brought to 0 A.

4.2. Hysteresis modelling of the gap induction of the U17 magnet

The U17 magnet is a test-bench which has the topology of the main magnets used in the Proton

Synchrotron (PS) accelerator at CERN [147]. This section presents the modelling results of the

gap induction of the U17 magnet. Firstly, the topology of the magnetic circuit is presented.

Secondly, the magnetic measurement and modelling procedures are described. And lastly, the

modelling results of the dipolar field component are presented.

4.2.1. Description of the magnetic circuit of the U17 magnet

The main PS magnet is a C-shaped combined function magnet which has both dipolar and

quadrupolar field components in its gap. Additionally, the magnet has two sections where the

quadrupolar field is either focusing or defocusing. The design of the magnet is presented in

reference [148]. A picture of the magnet is presented in Fig. 4.24.


Fig. 4.24: Picture of the U17 magnet at CERN

The magnet consists of 10 blocks, half of them having a focusing quadrupolar field and the

other half having a defocusing quadrupolar field. The cross-sectional view of a defocusing

block of the magnet is presented in Fig. 4.25. The difference between the focusing and the

defocusing blocks is the orientation of the slope of the pole face.

Fig. 4.25: Cross-sectional view of the U17 magnet through one of the defocusing blocks

The material in the core of the magnet (presented with grey in Fig. 4.25) is a low carbon steel

subjected to heat treatment operations to enhance its magnetic properties. The coils of this

magnet (presented with red in Fig. 4.25) have a complicated topology meant to correct the

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various harmonic components of the gap induction. For the current model only the main coils

(the ones with the larger cross-section) have been modelled to reduce the complexity of the

model. The number of windings in the coils is 10.

The blue line in Fig. 4.25 represents the magnetic field line which passes through the centre of

the gap. Using a finite element model and qualitative corrections while benchmarking the

model against the measurements, the length of this line in the gap has been calculated to be

50.2 mm. The amplitude of the quadrupolar component of the induction is linked to the dipolar

component through a constant. Therefore, only the dipolar component has to be modelled to

obtain a full description of the gap induction.

4.2.2. Identification of the mathematical model

This section presents the mathematical models used to model the gap induction of the U17

magnet. The same model as for the experimental magnet has been used, as described by

Eq. (4.27). Magnetic measurements have been performed on a material samples obtained from

the lamination of one of the PC main magnets to perform the identification of the Preisach

model, as described in section 3.2.

The 3D FEM model of the magnet developed using the Opera software has been used to

identify the 𝑆g(𝐵Fe) and the 𝑙Fe(𝐵Fe) characteristics. The material in the core has been

modelled using the normal magnetization curve, as presented in Fig. 4.26.

Fig. 4.26: The normal magnetization curve of the core material of the U17 magnet

A number of 12 simulations have been performed with varying current density in the main

coils. The value of the magnetic flux in the circuit has been determined by calculating the field

integral on a patch which encompasses the cross-section of the core in the median plane. The

value of the magnetic induction in the core has been determined by dividing the value of the

flux to the value of the area over which it has been calculated. The value of the flux closing

area in the gap has been determined by dividing the value of the flux to the value of the gap


induction. To the 𝑆Fe/𝑆g(𝐵Fe) dependence calculated using the simulations a qualitative

correction factor of 0.783 has been applied and the resulting curve is presented in Fig. 4.27.

Fig. 4.27: The 𝑆Fe/𝑆g = 𝑓(𝐵Fe) dependence for the U17 magnet

It appears that for the largest range of the magnetic induction the ratio has a constant value of

approx. 0.76 and, similarly to the characteristic of the experimental magnet, it increases

exponentially as the material approaches saturation.

The length of the magnetic flux math in the core has been determined using the 2D FEM model

of the circuit. A number of 14 simulations have been performed with varying current density.

The value of the core induction has been determined similarly to the 3D simulation method but

eliminating the 𝑧 axis. For each simulation the coordinates of the point along the flux line

originating in the centre of the gap has been determined, and the length of the line in the core

has been calculated using numerical methods. The results of the simulations are presented in

Fig. 4.28.

Fig. 4.28: The 𝑙fe = 𝑓(𝐵Fe) dependence for the U17 magnet

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The length of the magnetic field line in the U17 magnet has a significant variation, from 0.9 m

to 1.37 m, compared to the experimental magnet which has a relatively narrow magnetic

circuit.

Using the dependencies presented in Fig. 4.27 and Fig. 4.28 the mathematical model of the

magnetic circuit has been fully identified. The core magnetization, 𝑀Fe = 𝑓(𝐻Fe), has been

modelled using two models: the hysteretic model of the magnetization using the Preisach model

identification according to the procedure described in section 3.2, and the anhysteretic model

of the magnetization using the normal magnetization curve of the core material.

4.2.3. Benchmarking of the model against experimental measurements

This section presents the experimental measurement results performed on the demonstrator

magnet and the benchmarking of the modelling results against the measurements. The

waveform of the excitation current used for the experimental magnet, as presented in Fig. 4.21,

has been scaled to the maximum available current of the U17 magnet power supply of approx.

5350 A. The maximum ramping rate of the current has been limited to 45 A/s to minimize the

dynamic effects. The waveform of the imposed excitation current is presented in Fig. 4.29.

Fig. 4.29: The weveform of the excitation current applied to the U17 magnet

The bock diagram of the experimental setup is similar to the one for the experimental magnet,

as presented in Fig. 4.17, with the following particularities: the power supply is

NR71-002 MainPS MIEBACM PS1 6400-60V, the Gaussmeter is from Projekt Elektronik

model FM302 with hall probe AS-NTM-2, the data acquisition card is from National

Instruments model USB 6251, and the shunt resistor has been replaced by a high quality DCCT.

The waveform presented in Fig. 4.29 has been preceded by three cycles between 0 A and

5350 A.

The operating point of the magnet has been modelled by calculating the solutions of the system

of equations (4.27). For the anhysteretic model the normal magnetization curve has been used.

On the other hand, with the hysteretic model, after each reversal of the current ramping

direction (up or down) a new branch of the magnetization hysteresis cycle has been calculated

and used in system (4.27). The curves which describe the model of the material are presented

in Fig. 4.30.


Fig. 4.30: The modelled hysteresis cycles in the U17 magnet

The benchmark of the modelling results against the experimental measurements is presented

in Fig. 4.31. The matlab script used to generate the modelled values is in Annex 6.

Fig. 4.31: The results of field modelling in the U17 magnet

The top section of Fig. 4.31 presents the superimposed values of the gap induction: the

measured waveform with blue, the waveform obtained using the hysteresis model with red, and

the waveform obtained using the anhysteretic model with yellow. The differences are too small

to be noticed without any further analysis.

The central section of Fig. 4.31 presents the absolute errors of the two models, obtained using

Eq. (4.28): with blue for the hysteretic model and with red for the anhysteretic model. The

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results show that the absolute error of the hysteresis model is significantly lower than the

absolute error of the anhysteretic model.

The lower section of Fig. 4.31 presents the superimposed values of the relative errors of the

two models, referenced to the measured value, obtained using Eq. (4.29). The benchmarking

results show that the hysteresis model has approximated the gap induction with an error in the

order of 0.2 %. A selection of the values presented in Fig. 4.31 are presented in Table 4.3.

Table 4.3: Modelling results of the U17 magnet

Time (s) 𝑰 (A) 𝑩𝐠 (T) 𝑩𝐠,𝐡𝐲𝐬𝐭

(T)

𝝐𝐚,𝐡𝐲𝐬𝐭

(mT)

𝝐𝐫,𝐡𝐲𝐬𝐭

(%)

𝑩𝐠,𝐚𝐧

(T)

𝝐𝐚,𝐚𝐧

(mT)

𝝐𝐫,𝐚𝐧

(%)

5 11 0.00 0.00 0.06 1.38 0.00 0.19 4.32

130 3196 0.80 0.79 1.18 0.15 0.79 1.96 0.25

330 819 0.21 0.21 0.65 0.32 0.20 2.33 1.13

398 1087 0.27 0.27 0.11 0.04 0.27 1.07 0.39

590 2152 0.54 0.54 0.56 0.10 0.54 3.86 0.72

679 683 0.17 0.17 0.09 0.05 0.17 1.73 1.01

772 2539 0.63 0.63 0.29 0.05 0.63 1.54 0.24

861 1087 0.27 0.27 0.27 0.10 0.27 2.79 1.02

929 817 0.20 0.20 0.03 0.01 0.20 1.40 0.69

994 684 0.17 0.17 0.33 0.19 0.17 2.35 1.36

1061 146 0.04 0.04 0.13 0.35 0.04 1.18 3.20

1215 4816 1.17 1.18 6.68 0.57 1.17 3.43 0.29

1350 1086 0.27 0.27 0.22 0.08 0.27 2.81 1.03

1483 4816 1.17 1.18 7.03 0.60 1.17 3.39 0.29

1617 1086 0.27 0.27 0.34 0.13 0.27 2.85 1.04

1700 2540 0.63 0.63 0.17 0.03 0.63 1.74 0.28

1783 1354 0.34 0.34 0.28 0.08 0.34 3.13 0.92

1861 145 0.04 0.04 0.03 0.09 0.04 2.35 6.17

1931 281 0.07 0.07 0.21 0.30 0.07 1.80 2.52

1991 513 0.13 0.13 0.57 0.45 0.13 0.18 0.14

2077 1488 0.37 0.37 0.92 0.25 0.37 1.66 0.45

2157 146 0.04 0.04 0.15 0.39 0.04 2.01 5.28

2233 1087 0.27 0.27 0.28 0.10 0.27 0.58 0.21

2323 2540 0.63 0.64 1.00 0.16 0.63 2.65 0.42

2423 280 0.07 0.07 0.26 0.37 0.07 2.19 3.06

The spiking error values of the hysteresis model are due to the very low referenced value, as

observed for the experimental magnet. In addition to the spiking error values due to the low

referenced value, the modelling results obtained using the anhysteretic model have additional

high error regions especially when the excitation current is ramped down. The source of these

errors are highlighted by analysing the shapes of the modelled magnetic hysteresis cycles

modelled for the material in the core, as presented in Fig. 4.30.


The results presented in Fig. 4.30 show the source of the second set of spikes in the waveform

of the relative error. When the current is decreased, the core material follows a branch of the

hysteresis cycle which is poorly approximated by the normal magnetization curve. The normal

magnetization curve offers a fair approximation of the magnetization in the saturating region.

The developed magnetic circuit model together with the hysteretic magnetization model have

successfully approximated the gap induction within approx. 0.2 % error for the entire

magnetization range. Additionally, the gap induction has been modelled using an anhysteretic

model which showed an error of approx. 0.2 % for the most favourable case, and 100 % error

when the current is brought to 0 A.

5. FINAL CONCLUSIONS

The work presented in this document has shown that the goal of the thesis has been achieved.

An innovative method for the advanced characterization of the soft magnetic materials used in

the cores of particle accelerator magnets has been developed. The magnetic hysteresis has been

modelled using the Preisach model of hysteresis which has been identified using a newly

developed FORC interpolation techniques. The magnetic induction in the gap of two magnets

has been simulated using a hysteretic and an anhysteretic model, and the simulations have been

cross-checked with experimental measurements.

5.1. Conclusion

The theoretical and experimental characterization of soft magnetic materials through advanced

measurement and modelling methods represent a source of improvement to the performance of

particle accelerator magnets. The research performed in the framework of this doctoral thesis

allowed the accomplishment of the imposed specific objectives.

The first chapter of the thesis – Current status of research and development of particle

accelerator magnets – had the objective to analyse the magnetic circuit of a particle accelerator

magnet and to highlight the influence of the material to the performance of the magnet.

Particle accelerators come in different sizes, depending on their application, and have the

purpose to supply high energy particles. In an accelerator particles travel through vacuum

chambers and the most convenient method to act upon their energy and phase state properties

is through the Lorentz force. Thus, an electric field is applied parallel and in the direction of

the particles speed vector to increase their energy, and a magnetic field is applied perpendicular

to the speed vector to change the trajectory of a particle.

Normal conducting magnets are electro-magnets in which the excitation field is generated by

coils made of conducting elements (like aluminium or copper) which oppose an electrical

resistance to the current flow. These magnets rely on a core made of ferro-magnetic material

to guide and to concentrate the magnetic flux. The magnetic properties of iron are

characterized, among others, by non-linearity and hysteresis. Due to the hysteretic

characteristic of the magnetization the field in the gap of the magnet is also characterised by

hysteresis. In order to accurately reproduce the magnetic induction in the gap of a magnet two

models have to be used: one for the magnetic circuit and one for the functional relationship

between the magnetization and the magnetic field strength in the material.

The repeatable and accurate prediction of the magnetic induction in the gap is linked to the

accurate reproduction of the hysteretic characteristic of the material in the core. By using

standard magnetic measurement methodologies the best estimate of the magnetic properties are

achieved under quasi-static conditions. Nevertheless, depending on the material's physical

properties, final geometry, and magnetization ramping rate the shape of the hysteresis cycle is


altered. The standard measurement methodologies have no recommendations for testing

materials with controlled rate of change of the magnetization with values in the range of the

ramping rates of accelerator magnets. Therefore, in order to obtain the best estimate of the

magnetic properties of a material used in the core of an accelerator magnet a measurement

methodology which controls the ramping rate of the magnetization during testing is required.

Current field control methods rely on either feedback systems which require an additional

magnet to operate or on feed-forward systems which require vast amounts of input data and

are not able to predict the output for unknown disturbances. Therefore, a model driven control

system which relies on few input parameters would be a major contribution to the field of

particle accelerator physics.

The second chapter of the thesis – Characterization of ferromagnetic materials used in the

cores of particle accelerator magnets – had the objective to develop advanced magnetic

measurement methods and tolls for the measurement of the magnetic properties of soft

magnetic materials. Experimental measurements performed on electrical steels with different

silicon content and thicknesses have been presented.

In order to obtain the best estimate of the magnetic properties of a material the rate of change

during magnetic testing has to be matched to the rate of change of the magnet. This can be

achieved by employing the correct measurement methodology. The feedback methods are very

fast but have the drawbacks of sensitivity to the quality of the electronic components and

difficult handling of the noise. On the other hand, the iterative methodologies require less

electronic components and produce more reliable results. The major limitations of these

methodologies lie in the number of iterations required for convergence and in the computing

power required to process the measured data. The convergence is achieved between three and

several tens of iterations. These methods usually rely on curve fitting by means of regression

analysis, which is computing intensive and does not always produce accurate results.

Most magnetic materials are characterized by anisotropy and magnetic measurements for

different directions of the excitation field with respect to the rolling direction provide relevant

information on a material. For this reason, the ring core has not been selected for this study as

under these conditions it does not provide the most accurate measurements. The single sheet

tester (SST) has the advantage of fast assembly of the samples in the circuit. Nevertheless, the

SST is recommended for measurements above 1000 A/m, therefore, magnetic properties

cannot be measured in a very important operating region of the material. For the current

research an Epstein frame has been used and a new magnetic testing procedure which controls

the waveform of the polarization has been developed.

With the developed measurement method a number of three iterations are usually required to

reach the desired peak polarization and convergence of the iterative algorithm. After the first

iteration the peak polarization value is approached and the polarization waveform is distorted.

After the second iteration the desired peak polarization is reached and the waveform of the

polarization becomes less distorted. After the third iteration the polarization waveform has the

desired waveform and amplitude.

Doctoral thesis 147

With the developed measurement setup and method the two major sources of the uncertainty

are the Type B voltage measurement uncertainty and the uncertainty of the specific density of

the material. The experimental determination has revealed that the expanded uncertainty of the

polarization varies with the level of the polarization with a small non-linear characteristic. On

average the expanded uncertainty of the measurement, with a 95.45 % confidence level has

been measured to be 0.018 T.

By measuring the magnetic properties of a material with sinusoidal polarization waveform at

1 Hz, a ramping rate of the polarization similar to the one found in the core of a particle

accelerator magnet has been achieved. At this ramping rate the measured quantities showed a

7 % increase compared to standardised d.c. magnetic measurements. Therefore, measurements

performed with controlled rate of change of the polarization are better suited to estimate the

magnetic properties of a material used in the core of a particle accelerator magnet.

An analytical method for segmented and constrained fitting of nonlinear curves to cubic

polynomials by using the least squares method has been developed. The technique is a new

way of fitting experimental data, which provides accurate solutions in situations where the

definition of a prototype function, required for regression analysis, is very difficult. The

developed fitting method is a powerful tool which allows the analysis of the second derivative

of curves obtained from experimental data.

Usually, the magnets in a particle accelerator are connected in series and identical field is

expected in the magnets of the same type. In order to have identical output in a series of

identical magnets, the cores are required to have identical magnetic properties. But, the

magnetic characteristics of electrical steels vary even within the same charge and a variation

of the magnetic properties of the core material is expected. The experimental measurements

have shown that the coercivity has a variation of approx. 10 A/m for samples from different

coils but from the same charge. Therefore, during manufacturing of the magnets the

homogenization of the magnetic properties of the steel has to be achieved, usually by shuffling

the laminations.

The magnetic field in the core of a magnet closes in a loop which covers all the rotation angles

in the plane of the lamination. When the material in the core reaches saturation the transfer

function of the magnet decreases considerably. The experimental measurements have shown

that electrical steels can saturate along the hard magnetization axis at significantly lower values

than their rating. Therefore, in order to prevent saturating regions, the cross-sectional area of

the core has to be increased along the hard magnetization axis. Therefore, for the cores

manufactured using NGO steels a 10 % increase of the cross-sectional area of the core is

required in the regions where the magnetic field is oriented along 90° w.r.t. the rolling direction

of the sheet. On the other hand, when using GO steels a 30 % and a 23 % increase are required

for the regions oriented along 55° and 90°, respectively.

Another parameter which greatly influences the magnetic properties of electrical steels is the

mechanical history and the grain size. Mechanical processes can induce stress in a material and

decrease its performance. Also, increased grain size favours mobility of the domain walls, thus

enhancing the magnetic performance of a material. Heat treatment operations are designed to


remove internal stresses and to stimulate grain growth. The experimental measurements have

shown that for some material significant improvements of the magnetic properties can be

obtained after heat treatment operations. Therefore, in order to maximize the performance of a

material its mechanical history should be monitored and magnetic annealing processes should

be applied.

The standard grading of electrical steels is based on the energy losses at line frequency (50 or

60 Hz, depending on the region). The chemical composition of an alloy affects its electrical

resistivity which in turn has a significant contribution to reducing the energy losses. On the

other hand, the chemical composition influences the shape of the hysteresis cycle.

Experimental measurements have shown that similar a.c. grading can be achieved for steels

with different chemical composition. The analysis of the measurements has shown that if

dynamic effects in a magnet are negligible, steels with fewer alloying elements will operate

with increased performance. Otherwise, steels whose chemical composition increase the

electrical resistivity of the alloy should be used.

The third chapter of the thesis – Modelling and simulation of the magnetic hysteresis – had the

objective to analyse the magnetic models proposed in literature and the development of

advanced methods for the identification of the weight function of the Preisach model.

The theoretical background of the Jiles-Atherton and of the Preisach models of hysteresis have

been analysed. The Preisach model has been selected for the task of modelling the magnetic

hysteresis due to its ability to memorize the shapes of the hysteresis cycles with high degree of

accuracy. Nevertheless, this feature of the Preisach model requires the advancement of the

identification technique, in three topics: smoothing of the experimental data; interpolation of

first order reversal curves; minimize the required input data.

For the first topic a curve smoothing method has been developed. The method relies on fitting

segments of experimental data to low order polynomials by means of constrained linear

regression analysis. The second derivative of experimental data has been investigated by using

this method.

For the second topic a new curve interpolation method has been developed. This method relies

on numerical methods to identify the segments of shortest distance between the points of two

first order reversal curves (FORCs). The new FORC is identified on these segments by relating

the location of the reversal point to the reversal points of the bounding FORCs. This method

has been used to identify the weight function of the Preisach model with high degree of detail

(any number of threshold levels).

For the third topic a method has been developed which identifies the locations of the curvatures

of the limiting branch. The method is capable to increase the scattering of the reversal points,

where FORCs are measured, in these locations. This method has been used to limit the number

of acquired FORCs to 15 while capturing many of the fine details of the hysteresis cycle.

The experimental results have shown that the classical Preisach model identified with the newly

developed identification method has best performance when it models hysteresis branches

Doctoral thesis 149

which originate in the limiting ascending branch. This characteristic can be exploited to

improve the modelling performance of a material in the core of a magnet.

The fourth chapter of the thesis – Assessment of hysteresis effects in magnetic circuits – had

the objective to present the design and development of a demonstrator magnet, and to present

the cross-check between experimental measurements of the gap induction of a magnet and the

modelling of the magnetic induction using a hysteretic and an anhysteretic model.

The design of an air cooled, H dipole magnet is proposed. The purpose of this magnet is the

research of the effects of magnetic hysteresis in the cores of accelerator magnets by helping to

confirm the proposed mathematical models for magnetic hysteresis. The design maximizes the

value of the generated field and optimizes its homogeneity in the measurement region. The

experimental measurements have shown that the error to the hysteretic model has been in the

order of 1 %. On the other hand, the error of the anhysteretic model has been in the order of

1 % only when the model is operated in the saturating region. With this model, errors above

10 % have been observed when the model is operated at values <0.2 T.

Furthermore, the main PS magnet, a combined function magnet (dipole + quadrupole), has

been analysed. Three-dimensional finite element simulations have been used to identify the

parameters of the magnetic circuit model and experimental measurements of the laminations

in the two magnets have been used to identify the material models. For the PS magnet, the

relative error of the hysteresis model has been in the order of 0.2 % for all induction values.

On the other hand, for the anhysteretic model the relative error has been observed to have the

characteristic increased errors, with this magnet >1.5 %, when the model operated at

values <0.2 T. By analysing the hysteresis characteristic obtained using the hysteretic model it

is obvious that the accuracy of the anhysteretic model increased only when it operated in the

saturating region where the two models produced similar outputs.

The developed magnetization hysteresis modelling procedure together with the developed

magnetic circuit models can be used as a magnet field control method which require relatively

few input data and is very cheap to operate, as opposed to the currently employed strategies

which require either an additional magnet or vast amount of input data. Also, this work allows

the analysis of magnet performance with a coherent approach between beam-physics

requirements, magnet design and magnetic measurements.

5.2. Personal contributions

The contributions of the author within this work are summarized in the following list:

Development of a mathematical model of a magnet’s circuit which allowed the critical

reviewing the influence of the magnetic properties of the core material on the

performance of particle accelerator magnets.

Literature review which allowed critical reviewing of the d.c. magnetic measurement

methods for soft magnetic materials.


Development of a low frequency magnetic measurement method with sinusoidal

polarization waveform which is able to provide the measurements required to perform

the analysis of a material’s performance using the energy loss separation model.

Development of a magnetic measurement system for low frequency magnetic

measurement for soft magnetic materials and performing experimental measurements

on a wide range of Fe-Si alloys which allowed the analysis of a magnet’s performance

with consideration to the experimental measurement results.

Development of software for the automated acquisition and analysis of experimental

data, up to the second derivative of an experimental curve, which allowed performing

advanced analysis operations of the experimental data.

Development of a new method for the determination of the Preisach weight function

and of a new method for reducing the amount of required input data which allowed the

use of the Preisach model at very high resolution levels.

Design and manufacture of an experimental demonstrator magnet which allowed the

analysis of the performance of the developed models.

Development of benchmarking method and tools to compare the modelling results of

the hysteresis model and of an anhysteretic model against experimental measurements

performed on a demonstrator magnet and on the main PS magnet used at CERN. This

work allowed the analysis of the hysteresis effects in the cores of particle accelerator

magnets.

5.3. Outlook

The theoretical and experimental results obtained in this work describe the source of the

hysteresis characteristic of the gap induction of a particle accelerator magnet. These results

have been achieved by establishing a comprehensive mathematical model of the magnetic

circuit, by identifying the improvements which are required to obtain very high level of detail

from the Preisach hysteresis model, and by providing the input data required during magnet

manufacturing. From this work particle accelerator physicists, magnet designers and electrical

steel manufacturers can greatly benefit.

The following points may further improve the methods presented in this thesis:

Development of an identification procedure of the magnetic circuit model starting from

the geometry of the circuit alone;

Development of the identification procedure of the hysteresis model which includes

dynamic effects;

Integration of the developed models in an easy to use software which is capable to

evaluate the performance of a magnetic circuit with consideration to the magnetic

hysteresis;

Evaluate the feasibility of using the developed methods to control a magnet in real-

time.

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Doctoral thesis 165

REZUMAT ÎN LIMBA ROMÂNĂ

Prin intermediul acestei lucrări s-a efectuat un studiu privind efectele histerezisului din

materialele folosite pentru circuitele magnetice ale acceleratoarelor de particule. În acest studiu

s-au avut în vedere magneții de acceleratoare în conducție normală și cu miezuri fabricate din

materiale feromagnetice.

S-a realizat modelarea circuitelor magnetice prin intermediul modelelor dezvoltate: un model

pentru circuitul magnetic și unul pentru magnetizația materialului din miez. Parametrii

modelului circuitului magnetic au fost identificați cu ajutorul simulărilor ce folosesc metoda

elementelor finite (Opera 3D), iar parametrii modelului de histerezis al magnetizației au fost

identificați prin măsurători experimentale efectuate cu ajutorul unei metode dezvoltate în

cadrul acestui studiu. Pentru validarea rezultatelor modelărilor s-au efectuat măsurători

experimentale pe doi magneți: unul de mici dimensiuni proiectat și construit special pentru

această lucrare, și unul care este folosit în mod curent într-unul din acceleratoarele de particule

din complexul de la CERN.

Modelele dezvoltate în cadrul acestei lucrări au permis analiza formelor de undă a

magnetizației din timpul funcționării unui magnet și creșterea reproductibilității valorii

inducției din întrefierul magneților al căror circuit a fost modelat.

ABSTRACT

A study of the hysteresis effects in the cores of particle accelerator magnets has been performed

in the framework of the work presented in this thesis. This study has been focused on normal

conducting particle accelerator magnets whose cores are manufactured using ferromagnetic

materials.

The magnetic circuits have been modelled using the developed models: one model for the

magnetic circuit and one for the magnetization of the material in the core. The parameters of

the magnetic circuit model have been identified with the help of simulations which rely on the

finite element method (Opera 3D), while the parameters of the magnetic hysteresis model have

been identified through experimental measurements performed using a method developed in

the framework of this work. The modelling results have been validated by means of

experimental measurements performed on two magnets: one small size magnet which has been

specifically designed and manufactured, and one magnet which is currently used in a particle

accelerator within the CERN complex.

The models developed in the framework of this work allowed the analysis of the waveforms of

the magnetization during the operation of a magnet and the increase of the reproducibility of

the magnetic induction value in the gap of the modelled magnets.

Doctoral thesis 167

INFORMAȚII

PERSONALE

Nume

Adresă

Email

Data nașterii

PRICOP Valentin

Str. Fagurului, Nr. 21, 500484, Brașov, România

[email protected]

13.12.1982

Experiență profesională

Feb. 2013 – Ian. 2016 Doctorand în cadrul Universității “Transilvania” din Brașov –

Facultatea de Inginerie Electrică și Știința Calculatoarelor, și

în cadrul CERN din Geneva, Elveția.

Feb. 2012 – Iul. 2012 Stagiar în cadrul CERN din Geneva, Elveția.

Sept. 2010 – Ian. 2012 Inginer – S.C. PREMS TP S.A., Brașov, România

Mar. 2006 – Aug. 2010 Operator CNC – S.C. PREMS TP S.A., Brașov, România

Iul. 2005 – Ian. 2006 Electronist – S.C. KM Systems S.R.L., Brașov, România

Iul. 2002 – Mar. 2005 Electronist – S.C. GCS Electronics S.R.L., Brașov, România

Educație

Oct. 2012 – Feb. 2016 Studii de doctorat – Universitatea “Transilvania” din Brașov

– Facultatea de Inginerie Electrică și Știința Calculatoarelor

Oct. 2010 – Sept. 2012 Studii de master – Universitatea “Transilvania” din Brașov –

Facultatea de Inginerie Electrică și Știința Calculatoarelor

Oct. 2006 – Sept. 2010 Studii de licență – Universitatea “Transilvania” din Brașov –

Facultatea de Inginerie Electrică și Știința Calculatoarelor

Limbi străine Engleză (avansat), Franceză (mediu), Italiană (începător)

Aptitudini și competențe

tehnice

Redactare Crearea de articole, postere și prezentări pentru reviste și

conferințe din domeniul ingineriei electrice.

Comunicare Adaptabilitate la medii multiculturale dobândită în urma

experienței profesionale

Programare Cunoștințe avansate de informatică: limbaje de programare

(C, C++, C#, VBA, Matlab, LabView), achiziții de date,

proiectare 3D asistată (AutoCAD, Inventor, Opera,

COMSOL)

Management Abordare structurată și organizată a muncii, stabilirea de

priorități și sarcinile având în vedere rezultatele dorite

Permis de conducere Cat. B

Informații adiționale În timpul studiilor liceale am obținut locul III la faza

județeană a olimpiadei de electrotehnică (1998) și locul I la

faza județeană a concursului de matematică KANGOUROU

mailto:[email protected]


PERSONAL

INFORMATION

Name

Address

Email

Birthday

PRICOP Valentin

Str. Fagurului, Nr. 21, 500484, Brașov, România

[email protected]

13.12.1982

Professional experience

Feb. 2013 – Jan. 2016 Doctoral student within “Transilvania” University of Brașov

– Faculty of Electrical Engineering and Computer Science,

and within CERN in Geneva, Switzerland

Feb. 2012 – Jul. 2012 Technical student, internship – CERN, Geneva, Switzerland

Sept. 2010 – Jan. 2012 Engineer – S.C. PREMS TP S.A., Brașov

Mar. 2006 – Aug. 2010 CNC operator – S.C. PREMS TP S.A., Brașov

Jul. 2005 – Jan. 2006 Electronics technician – S.C. KM Systems S.R.L., Brașov

Jul. 2002 – Mar. 2005 Electronics technician – S.C. GCS Electronics S.R.L., Brașov

Education

Oct. 2012 – Feb. 2016 Doctoral studies – “Transilvania” University of Brașov –

Faculty of Electrical Engineering and Computer Science

Oct. 2010 – Sept. 2012 Masters studies – “Transilvania” University of Brașov –


Oct. 2006 – Sept. 2010 Bachelors studies – “Transilvania” University of Brașov –


Foreign languages English (advanced), French (medium), Italian (beginner)

Aptitudes and technical

competences

Authoring Authoring articles, posters and presentations for journals and

conferences in the electrical engineering field

Communication Adaptability to multi-cultural environments gain through my

working experience

Programming Advanced knowledge in the field of computer science:

programming (C, C++, C#, VBA, Matlab, LabView), data

acquisition, 3D aided design (AutoCAD, Inventor, Opera,

COMSOL).

Management Structured and organised approach towards work, able to set

priorities and plan tasks with results in mind

Drivers licence Cat. B

Additional information During the high-school studies at the county level I took the

3rd place at the electro-technics Olympiad (1998) and 1st place

for the KANGOUROU mathematics contest (2000)

mailto:[email protected]

Doctoral thesis 169

STATEMENT OF COPYRIGHT

DECLARAȚIE

Subsemnații: Ing. Pricop Valentin

în calitate de

student - doctorand al IOSUD: Universitatea “Transilvania” din Brașov

autor al tezei de doctorat cu titlul: EFECTELE HISTEREZISULUI DIN MATERIALELE

FOLOSITE PENTRU CIRCUITELE MAGNETICE ALE ACCELERATOARELOR DE

PARTICULE

și

Prof. Dr. Ing. Scutaru Gheorghe

în calitate de Conducător de doctorat al autorului tezei

la instituția Universitatea “Transilvania” din Brașov

(denumire instituție)

declarăm pe proprie răspundere că am luat la cunoștință de prevederile art.143 alin (4) si (5)*

şi art. 170** din Legea educației naționale nr.1/2011 și ale art. 65, alin.5 – 7***, art. 66, alin

(2)**** din Hotărârea Guvernului nr.681/2011 privind aprobarea Codului Studiilor

universitare de doctorat și ne asumăm consecințele nerespectării acestora.

Semnătură Semnătură

Student doctorand Conducător de doctorat


((4 )îndrumătorii lucrărilor de licență, de diplomă, de disertație, de doctorat răspund solidar cu autorii acestora

de asigurarea originalității conținutului acestora

(5) este interzisă comercializarea de lucrări științifice în vederea facilitării falsificării de către cumpărător a

calității de autor al unei lucrări de licență, de diplomă, de disertație sau de doctorat.

** (1)În cazul nerespectării standardelor de calitate sau de etică profesională, Ministerul Educației, Cercetării,

Tineretului și Sportului, pe baza unor rapoarte externe de evaluare, întocmite, după caz, de CNATDCU, de CNCS,

de Consiliul de etică și management universitar sau de Consiliul Național de Etică a Cercetării Științifice,

Dezvoltării Tehnologice și Inovării, poate lua următoarele măsuri, alternativ sau simultan:

a) retragerea calității de conducător de doctorat;

b) retragerea titlului de doctor;

c) retragerea acreditării școlii doctorale, ceea ce implică retragerea dreptului școlii doctorale de a organiza

concurs de admitere pentru selectarea de noi studenți-doctoranzi.

(2)Reacreditarea școlii doctorale se poate obține după cel puțin 5 ani de la pierderea acestei calități, numai în urma

reluării procesului de acreditare, conform art. 158.

(3)Redobândirea calității de conducător de doctorat se poate obține după cel puțin 5 ani de la pierderea acestei

calități, la propunerea IOSUD, pe baza unui raport de evaluare internă, ale cărui aprecieri sunt validate printr-o

evaluare externă efectuată de CNATDCU. Rezultatele pozitive ale acestor proceduri sunt condiții necesare pentru

aprobare din partea Ministerului Educației, Cercetării, Tineretului și Sportului.

(4)Conducătorii de doctorat sunt evaluați o dată la 5 ani. Procedurile de evaluare sunt stabilite de Ministerul

Educației, Cercetării, Tineretului și Sportului, la propunerea CNATDCU.

***(5) teza de doctorat este o lucrare originală, fiind obligatorie menționarea sursei pentru orice material

preluat.

(6) studentul - doctorand este autorul tezei de doctorat și își asumă corectitudinea datelor și informațiilor

prezentate în teză, precum și a opiniilor și demonstrațiilor exprimate în teză

(7) conducătorul de doctorat răspunde împreună cu autorul tezei de respectarea standardelor de calitate sau de

etica profesională, inclusiv de asigurarea originalității conținutului, conform art. 170 din Legea nr. 1/2011.

**** protecția drepturilor de proprietate intelectuală asupra tezei de doctorat se asigură în conformitate cu

prevederile legii.

ANNEX 1. LABVIEW CODE USED TO AUTOMATE THE MAGNETIC

MEASUREMENT PROCEDURE

The main interface of the program with the measurement parameters pane visible is presented

in Fig. A1.1.

Fig. A1.1: The main interface of the measurement software

The main interface of the program with the DAQ configuration pane visible is presented in

Fig. A1.2.


Fig. A1.2: The data acquisition card configuration pane of the main interface

The variables and the interface are initialised with the values used in the previous instance of

the software and which have been saved to .dat files. The different section of the program are

contained in frame blocks which ensures the control of the order of the program execution. In

the first section of the block diagram the variables are loaded from external files, as presented

in Fig. A1.3.

Doctoral thesis 173

Fig. A1.3: The block which load the variables saved in the previous instance of the program

All the blocks of the program are encompassed in a while loop whose completion is triggered

by the actuation of the Exit button (see Fig. A1.4).

Fig. A1.4: The Exit button

The parameters clusters are contained into a tab control block. The delay between two

executions of the while loop is 10 ms. The advancement to the next stage of the program is

triggered by the actuation of the Start button. In the first stage of the program, the access to the

clusters is disabled, the Start and Exit buttons are greyed and disabled, and the Stop button is

enabled. The block used modify the graphical user interface (GUI) during the measurement are

presented in Fig. A1.5.


Fig. A1.5: The blocks which modify the graphical user interface before the measurement

Next the waveform of the demagnetizing signal and the levels where the measurements will be

performed are generated, using the blocks presented in Fig. A1.6.

Fig. A1.6: The blocks used to generate the demagnetizing waveform and the testing levels

Additionally, the constants which define the proportionality between the measured shunt

resistor voltage (H_I), the proportionality between the required magnetic field strength and the

voltage supplied to the power converter (Ur_H), the cross-sectional area of the sample

(MatSurfaceArea), and the flux integration constant are calculated, as described in Fig. A1.7.

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Fig. A1.7: The blocks used to calculate the constants

Next, a Windows dialog is generated which prompts the user to select the destination folder of

the report files. If an error existed during this process or the user has canceled the folder

selection the value of the Start button is switched to false to prevent further execution of the

code and the GUI is modified to the default state. The block diagram used to perform these

operations is presented in Fig. A1.8.

Fig. A1.8: Block diagram used to prompt the user to select the destination folder and to revert the GUI to default

If nor error occurred during folder selection, then the variables are saved to external files. The

block diagram used to perform this operation is presented in Fig. A1.9.


Fig. A1.9: Block diagram used to save the variables to external files

The files containing the value of the variables are saved to the folder containing the application,

under the filenames: params.dat for the measurement parameters data cluster, and

DAQ_params.dat for the data acquisition data cluster. The block diagram used to save the

variables to files is presented in Fig. A1.10.

Fig. A1.10: Block diagram used to save the variables values to files

Next the program enters the main execution while loop whose execution is interrupted if the

Stop button is pressed or when measurements have been performed at all the specified

frequencies. The execution within the main while loop is controlled by a flat sequence

structure. In the first frames of the sequence the data acquisition card (DAQ), which is assumed

that has id Dev1, is reset, the text block indicating the current measurement is updated, a 500 ms

delay is imposed and then the power supply is turned on. The block diagram used to perform

these actions are presented in Fig. A1.11.

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Fig. A1.11: The blocks used to initialize the measurement process

The block diagram of the procedure used to turn on the power supply is presented in Fig. A1.12.

Fig. A1.12: The block diagram of the procedure used to control the relay which turns the power supply on and

off

Next, the frequency array is indexed with the count of the main while loop and the value is

stored to the measurement parameter cluster, in the Current Frequency variable. Also, the

measurement range of the DAQ is set to maximum. A mathscript block is used to calculate the

sampling frequency required to maximize the sampling rate as a multiple of 4000 samples per

cycle, as well as, generate the reference waveforms and constants used in later blocks. The

block diagram used to perform these actions is presented in Fig. A1.13.


Fig. A1.13: The blocks used to initialize the measurement for the current frequency

In the next frames of the main while loop a time delay of 500 ms is applied and the dc offset

of the DAQ is measured by applying a 0 V signal of half a measurement cycle to the DAQ.

The blocks used to perform these actions is presented in Fig. A1.14.

Fig. A1.14: The blocks used to measure the dc offset of the DAQ

In the next frame the waveform of the demagnetizing signal is sent to the DAQ. The data

acquisition and generation is handled by a separate procedure. This procedure is comprised of

three section: the analog output (AO) task configuration section (Fig. A1.15), the analog input

(AI) task configuration section (Fig. A1.16), and the task synchronous start and memory

flushing section (Fig. A1.17).

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Fig. A1.15: The blocks used to configure the AO task

Fig. A1.16: The blocks used to configure the AI tasks


Fig. A1.17: The blocks used to start the DAQ tasks and to clean the tasks from memory once they finish

Next, the measurement range of the DAQ is set to minimum, the initial excitation waveform

(sinusoidal with 5 A/m peak amplitude) is generated and applied to the Epstein frame, the

acquired data is processed and the waveforms of the magnetic field strength and of the

induction are generated. The last value in the voltage waveform is stored to the lastU variable

which will later be used to ensure a linear transition between the levels of two iterations. The

blocks which perform these steps are presented in Fig. A1.18.

Fig. A1.18: The blocks used to perform the initial measurement

The data processing procedure performs the average of every m samples, where m is the

variable defined using the blocks presented in Fig. A1.13. The block diagram of the wave

processing procedure is presented in Fig. A1.19.

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Fig. A1.19: The block used to perform the waveform processing

The polarization and excitation waveforms are calculated from the processed waveforms using

the constants stored in the DAQ_params cluster. The blocks used to perform this procedure are

presented in Fig. A1.20.

Fig. A1.20: The blocks used to calculate the waveforms of the polarization and of the excitation field

Next the program enters another while loop which has the purpose to perform measurements

at the levels defined using the blocks presented in Fig. A1.6 and stored to variable j. At the

beginning of each iteration the last peak polarization value is checked and the index of the j


variable is adjusted to point to the immediately larger value. The blocks used to perform these

operations are presented in Fig. A1.21.

Fig. A1.21: The blocks used at the beginning of each cycle of the while loop which parses the testing levels

Next, another while loop is used to modulate the waveform of the excitation cycle by means of

iterative augmentation, until the convergence criteria described in section 2.2 are achieved. In

the first part of this loop the second of the three measured cycles is identified. The blocks used

to perform this operation are presented in Fig. A1.22.

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Fig. A1.22: The blocks used to identify the second measurement cycle

The identified cycle is passed to the interpolation procedure. Further, this procedure isolates

the descending branch of the hysteresis cycle and modulates the waveform of the excitation

cycle which is required to obtain the waveform of the polarization reference cycle. The block

diagrams used to implement this procedure is presented in Fig. A1.23.


Fig. A1.23: The blocks used to perform the waveform interpolation procedure

Next, the obtained Up and Down waveforms are concatenated to obtain a three cycle excitation

waveform. The block diagram used to implement the concatenation procedure is presented in

Fig. A1.24.

Fig. A1.24: the blocks used to implement the waveform concatenation

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Next, a signal is generated with the purpose to ensure a smooth transition between the last value

of the excitation waveform from the previous iteration to the first value of the excitation

waveform in the current iteration. Also, if the peak amplitude of the generated AO signal is

larger than 10 V then the error flag is set to true. The block used to implement this procedure

is presented in Fig. A1.25.

Fig. A1.25: The block used to generate the leveling signal

The generated leveling waveform and the new excitation waveform are sent to the DAQ. The

measured data is processed, the waveforms of the polarization and of the magnetic field

strength are generated, and the data required to analyze the convergence criteria is calculated.

The block used to implement this functionality is presented in Fig. A1.26.

Fig. A1.26: The blocks used to perform the measurement and process data in the current iteration


Next, the measured hysteresis cycle is displayed to the user and the convergence criteria are

verified. The block diagrams used to perform these operations are presented in Fig. A1.27.

Fig. A1.27: The blocks used to display the measured hysteresis cycle and to evaluate the convergence criteria

If the convergence criteria have been achieved, then the iterative while loop is stopped and the

last measured cycles are used to extract the relevant quantities of the measurement. The blocks

used to implement this functionality are presented in

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Fig. A1.28: The blocks used to extract the useful information from the measured cycle

When measurements have been performed up to the last test level the control voltage of the

power supply is brought to 0. Additionally, the data extracted from each measured cycle is

bundled into a single cluster. The blocks used to implement this functionality are presented in

Fig. A1.29.

Fig. A1.29: The blocks used to bring the power supply control voltage to 0 and to bundle the measurement data


After all measurements have been completed (all the test levels for all the measurement

frequencies) a flat sequence structure is used to sequence the following processes (see

Fig. A1.30): the application of a demagnetization procedure, the turning off of the power

supply, The generation of the measurement report files, and resetting of the GUI to the default

values.

Fig. A1.30: The blocks used at the end of the measurement procedure

ANNEX 2. MATLAB CODE USED FOR THE CURVE FITTING

PROCEDURE

function [x,y,yp,ypp] = LS_3thO(H,B,N,NO,from_zero)

yi = B;

xi = H;

if from_zero

Bi = B;

Hi = H;

B = [0 ; Bi];

H = [0 ; Hi];

end

p = 10;

indexes = linspace(1,length(H),N*p+1);

H = interp1(H,indexes);

B = interp1(B,indexes);

Hinit = H;

Binit = B;

%% Define basic variables and the intervals

L = length(H);

n = round(linspace(1,L,N+1));

Hi = H(n);

H = zeros(N,p);

B = H;

for i = 1:N


H(i,:) =

interp1(n(i):n(i+1),Hinit(n(i):n(i+1)),linspace(n(i),n(i+1),p),'linear');

B(i,:) =

interp1(n(i):n(i+1),Binit(n(i):n(i+1)),linspace(n(i),n(i+1),p),'linear');

end

for i = 1:N

H(i,:) = H(i,:) - Hi(i);

end

Hi = zeros(N,1);

for i=1:N

Hi(i) = H(i,p);

end

%% Define the system of equations

A = zeros(N+3);

Y = zeros(N+3,1);

%fill the Y matrix

Y(1) = sum(sum(B));

for i=1:N

for h=1:p

Y(2) = Y(2) + B(i,h) * ( H(i,h) + sum( Hi(1:i-1) ) );

Y(3) = Y(3) + B(i,h) * ( H(i,h) + sum( Hi(1:i-1) ) )^2;

end

end

for s=4:N+3

v = s - 3;

for i = v:N

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tempS = zeros(1,p);

for h = 1:p

tempS(h) = ( H(i,h) + sum(Hi(v:i-1)) )^3;

if i~=v

tempS(h) = tempS(h) - ( H(i,h) + sum(Hi(v+1:i-1)) )^3;

end

end

Y(s) = Y(s) + sum(B(i,:) .* tempS);

end

end

%fill the A matrix for term d

A(1,1) = N * p;

for i=1:N

for h=1:p

A(2,1) = A(2,1) + ( H(i,h) + sum( Hi(1:i-1) ) );

A(3,1) = A(3,1) + ( H(i,h) + sum( Hi(1:i-1) ) )^2;

end

end

for v=1:N

for i = v:N

for h = 1:p

tempS = ( H(i,h) + sum(Hi(v:i-1)) )^3;

if i~=v

tempS = tempS - ( H(i,h) + sum(Hi(v+1:i-1)) )^3;

end

A(v+3,1) = A(v+3,1) + tempS;

end


end

end

%fill the A matrix for term c

A(1,2) = A(2,1);

A(2,2) = A(3,1);

for i=1:N

for h=1:p

A(3,2) = A(3,2) + ( H(i,h) + sum(Hi(1:i-1)) )^3;

end

end

for v=1:N

for i = v:N

for h = 1:p

tempS1 = ( H(i,h) + sum(Hi(1:i-1)) );

tempS2 = ( H(i,h) + sum(Hi(v:i-1)) )^3;

if i~=v

tempS2 = tempS2 - ( H(i,h) + sum(Hi(v+1:i-1)) )^3;

end

A(v+3,2) = A(v+3,2) + tempS1 * tempS2;

end

end

end

% fill the A matrix for term b

A(1,3) = A(3,1);

A(2,3) = A(3,2);

for i=1:N

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for h=1:p

A(3,3) = A(3,3) + ( H(i,h) + sum(Hi(1:i-1)) )^4;

end

end

for v=1:N

for i = v:N

for h = 1:p

tempS1 = ( H(i,h) + sum(Hi(1:i-1)) )^2;

tempS2 = ( H(i,h) + sum(Hi(v:i-1)) )^3;

if i~=v

tempS2 = tempS2 - ( H(i,h) + sum(Hi(v+1:i-1)) )^3;

end

A(v+3,3) = A(v+3,3) + tempS1 * tempS2;

end

end

end

%fill the A matrix for term a

for s=1:N

for i = s:N

tempS1 = zeros(1,p);



for h = 1:p

tempS1(h) = (H(i,h) + sum( Hi(1:i-1) ) );

tempS2(h) = (H(i,h) + sum( Hi(1:i-1) ) )^2;

tempS3(h) = (H(i,h) + sum( Hi(s:i-1) ) )^3;

if i~=s


tempS3(h) = tempS3(h) - ( H(i,h) + sum(Hi(s+1:i-1)) )^3;

end

end

A(1,s+3) = A(1,s+3) + sum(tempS3);

A(2,s+3) = A(2,s+3) + sum(tempS3 .* tempS1);

A(3,s+3) = A(3,s+3) + sum(tempS3 .* tempS2);

end

end

for v = 1:N

for i = 1:N

tempP = 0;

for s = max(v,i):N

tempS1 = ( H(s,:) + sum(Hi(i:s-1)) ).^3;

if i~=s

tempS1 = tempS1 - (H(s,:) + sum(Hi(i+1:s-1))).^3;

end

tempS2 = ( H(s,:) + sum(Hi(v:s-1)) ).^3;

if v~=s

tempS2 = tempS2 - (H(s,:) + sum(Hi(v+1:s-1)) ).^3;

end

tempP = tempP + tempS1 .* tempS2;

end

A(v+3,i+3) = sum(tempP);

end

end

%% solve the system

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if from_zero

A(1,:) = 0;

A(1,1) = 1;

Y(1) = 0;

end

Ans = mldivide(A,Y);

%% plot the data

f = @(x,a,b,c,d) a.*x.^3 + b.*x.^2 + c.*x + d;

fp = @(x,a,b,c) 3.*a.*x.^2 + 2.*b.*x + c;

fpp = @(x,a,b) 6.*a.*x + 2.*b;

b = zeros(N,1);

c = zeros(N,1);

d = zeros(N,1);

d(1) = Ans(1);

c(1) = Ans(2);

b(1) = Ans(3);

a = Ans(4:end);

for i = 2:N

d(i) = f(Hi(i-1),a(i-1),b(i-1),c(i-1),d(i-1));

c(i) = fp(Hi(i-1),a(i-1),b(i-1),c(i-1));

b(i) = 1/2*fpp(Hi(i-1),a(i-1),b(i-1));

end

x = linspace(Hinit(1),Hinit(end),NO)';

y = zeros(1,length(x))';

yp = y;

ypp = y;

for i = 1:length(x)


for k = 1:N

if x(i)>=Hinit(n(k)) && x(i)<=Hinit(n(k+1))

y(i) = f(x(i)-Hinit(n(k)),a(k),b(k),c(k),d(k));

yp(i) = fp(x(i)-Hinit(n(k)),a(k),b(k),c(k));

ypp(i) = fpp(x(i)-Hinit(n(k)),a(k),b(k));

end

end

end

end

ANNEX 3. MATLAB CODE USED TO PROCESS BH FILES WITH

LIMITED NUMBER OF POINTS

clear;

clc;

filename = 'C:\Hysteresis models\Cycle1.csv';

fname = 'E:\Hysteresis measurement\PS_Laminations\PS_mix_BH.bh';

file_content = csvread(filename,1,0);

B = file_content(:,2); Bo = B;

H = file_content(:,1); Ho = H;

L = length(H);

[H,B,yp,murD2] = LS_3thO(H,B,10,2e4,0);

if length(B)>2

H_div = 0;

B_div = 0;

murD2(murD2>mean(murD2)*5) = mean(murD2)*5;

murD2 = mean(murD2)/8 + murD2;

murD2_area = sum(murD2);

A = 0;

for i=1:length(murD2)

A = A + murD2(i);

if A > murD2_area/3000

H_div = [H_div ; H(i)];

B_div = [B_div ; B(i)];


A = 0;

end

end

H_div = abs([H_div ; H(end)]);

B_div = abs([B_div ; B(end)]);

L = length(H_div);

H_div = interp1(1:L, H_div, linspace(1,L,50)');

B_div = interp1(1:L, B_div, linspace(1,L,50)');

N = length(B_div);

fileID = fopen(fname,'w');

fwrite(fileID,sprintf('\t%d\t1\t1\r\n',N));

for i=1:N

fwrite(fileID,sprintf('\t%1.5f\t%1.5f\r\n',B_div(i),H_div(i)));

end

fclose(fileID);

end

ANNEX 4. MATLAB CODE USED TO PROCESS THE MEASURED

FIRST ORDER REVERSAL CURVES

% Script for processing the measurements required for the Preisach model

% The files have to be in csv format

% The naming convention of the files is: FORC*[%05d(H reversal)].csv

% The waveform of the polarization has to be sinusoidal

% FORCs_folder is the folder containing the measurement files

% Out_file is the name of the output file containing the results

% Nout is the length of the output arrays

function [FORC] =

Preisach_process_measurements_v03(FORCs_folder,Out_file,Nout)

% Inspect the folder and retrieve the list of measurement files

FORC_list = dir([FORCs_folder,'\FORC*.csv']);

% The number of measurement files

Nf = length(FORC_list);

Nf=Nf;

% Determine the sorted indexes of the list of FORCs based on H reversal

sort_list = 1:Nf;

for i = sort_list

sort_list(i) = str2num(FORC_list(i).name(end-8:end-4));

end

[~,sort_in] = sort(sort_list);


% Pre-allocate the arrays for the FORC's raw measurement

H_FORC = zeros(Nout , Nf , 2);

B_FORC = zeros(Nout , Nf , 2);

% Pre-allocate the arrays for the averaged FORCs

H_FA = zeros(Nout , Nf);

B_FA = zeros(Nout , Nf);

% Pre-allocate the arrays for the major branches raw measurement

H_major = zeros(Nout , Nf , 2);

B_major = zeros(Nout , Nf , 2);

% Pre-allocate the arrays for the averaged major branches

H_MA = zeros(Nout , Nf);

B_MA = zeros(Nout , Nf);

% Read the FORCs and the major branches from the measurement files

for i=1:Nf

% Read the file

filecontent =

csvread([FORCs_folder,'\',FORC_list(sort_in(i)).name]);

% The number of inputs in the file

Ni = length(filecontent);

% Find the reversal points in the measurement considering that

there are 1000 points per major cycle

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P = zeros(1,5);

P(1) = (Ni - 1000) / 4;

P(2) = (Ni - 1000) - ( (Ni - 1000) / 2 );

P(3) = Ni / 2+1;

P(4) = Ni / 2+1 + P(1);

P(5) = Ni / 2+1 + P(2);

% Resample and assign the FORCs and the major cycles

H_FORC(:,i,1) = interp1((1:(P(2)-P(1)+1))' ,

filecontent(P(1):P(2),1) , linspace(1,P(2)-P(1),Nout)' );

H_FORC(:,i,2) = interp1((1:(P(5)-P(4)+1))' ,


H_FA(:,i) = mean([H_FORC(:,i,1) , -H_FORC(:,i,2)],2);

B_FORC(:,i,1) = interp1((1:(P(2)-P(1)+1))' ,


B_FORC(:,i,2) = interp1((1:(P(5)-P(4)+1))' ,


B_FA(:,i) = mean([B_FORC(:,i,1) , -B_FORC(:,i,2)],2);

H_major(:,i,1) = interp1((1:(P(3)-P(2)+1))' ,


H_major(:,i,2) = interp1((1:(Ni-P(5)+1))' , filecontent(P(5):Ni,1)

, linspace(1,Ni-P(5),Nout)' );

H_MA(:,i) = mean([H_major(:,i,1) , -H_major(:,i,2)],2);

B_major(:,i,1) = interp1((1:(P(3)-P(2)+1))' ,


B_major(:,i,2) = interp1((1:(Ni-P(5)+1))' , filecontent(P(5):Ni,2)

, linspace(1,Ni-P(5),Nout)' );

B_MA(:,i) = mean([B_major(:,i,1) , -B_major(:,i,2)],2);

end

% Determine the average major descending branch

HM = mean([H_major(:,:,1) , -H_major(:,:,2)] ,2);


BM = mean([B_major(:,:,1) , -B_major(:,:,2)] ,2);

% Centre and smooth the major branch

HM = HM + ( max(HM) + min(HM) ) / 2;

BM = BM + ( max(BM) + min(BM) ) / 2;

[BM_S, HM_S, ~, ~, ~] = LS_3thO(-BM, -HM, 100, Nout, false);

HM_S = HM_S - ( max(HM_S) + min(HM_S) ) / 2;

BM_S = BM_S - ( max(BM_S) + min(BM_S) ) / 2;

% Determine the maximum B value

Bsat = max(BM);

% Pre-allocate the array for the monotonic major branches

H_MM = H_MA;

B_MM = B_MA;

% Pre-allocate the array for the minimum indexes

HM_mi = zeros(Nf,1);

BM_mi = zeros(Nf,1);

FM_mi = zeros(Nf,1);

% Enforce the monotonicity of the major branches

for i = 1:Nf

% Find the indexes of the minimum of H and B of the FORC

[~,HM_mi(i)] = max(H_MA(:,i));

[~,BM_mi(i)] = max(B_MA(:,i));

FM_mi(i) = max([HM_mi(i) , BM_mi(i)]);

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% Assign NaN values to all array locations before minimum index

if FM_mi(i)>1

H_MM(1:FM_mi(i)-1 , i) = NaN;

B_MM(1:FM_mi(i)-1 , i) = NaN;

end

% Initialize the index of the last monotonic value

last_m = FM_mi(i); i1=0; i2 = 0;

% Assign Nan values to all non-monotonic values

for j = FM_mi(i)+1:Nout

if H_MM(j,i)>=H_MM(last_m,i) || B_MM(j,i)>=B_MM(last_m,i)

H_MM(j,i) = NaN; i1= i1+1;

B_MM(j,i) = NaN; i2= i2+1;

else

last_m = j;

end

end

% Assign the non-NaN values in the arrays to temporary arrays

non_nans = find( ~isnan(H_MM(:,i)) & ~isnan(B_MM(:,i)) );

temp_HM = H_MM(non_nans,i);

temp_BM = B_MM(non_nans,i);

% Resample the temporary arrays and into the original arrays

no_nan = length(temp_HM);


H_MM(:,i) = interp1q( (1:no_nan)' , temp_HM ,

linspace(1,no_nan,Nout)');

B_MM(:,i) = interp1q( (1:no_nan)' , temp_BM ,


end

% Pre-allocate the arrays for the averaged and corrected FORCs

H_FC = H_FA;

B_FC = B_FA;

% Remove the demagnetizing characteristic and scale to Bsat

for i = 1:Nf

% Remove from the FORC the same amount that was removed from the

% major branch due to averaging at each H value

B_FC(:,i) = B_FA(:,i) + interp1(-HM_S,-BM_S,H_FA(:,i)) -

interp1(H_MM(:,i),B_MM(:,i),H_FC(:,i));

end

% Pre-allocate the array for the minimum indexes

HF_mi = zeros(Nf,1);

BF_mi = zeros(Nf,1);

F_mi = zeros(Nf,1);

% Pre-allocate the arrays for the monotonic arrays

B_FM = B_FC;

H_FM = H_FC;

% Enforce the monotonicity of the FORCs

for i = 1:Nf

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% Find the indexes of the minimum of H and B of the FORC

[~,HF_mi(i)] = min(H_FC(:,i));

[~,BF_mi(i)] = min(B_FC(:,i));

F_mi(i) = max([HF_mi(i) , BF_mi(i)]);

% Assign NaN values to all array locations before minimum index

if F_mi(i)>1

H_FM(1:F_mi(i)-1 , i) = NaN;

B_FM(1:F_mi(i)-1 , i) = NaN;

end

% Initialize the index of the last monotonic value

last_m = F_mi(i); i1=0; i2 = 0;

% Assign Nan values to all non-monotonic values

for j = F_mi(i)+1:Nout

if H_FM(j,i)<=H_FM(last_m,i) || B_FM(j,i)<=B_FM(last_m,i)

H_FM(j,i) = NaN; i1= i1+1;

B_FM(j,i) = NaN; i2= i2+1;

else

last_m = j;

end

end

% Assign the non-NaN values in the arrays to temporary arrays

non_nans = find( ~isnan(H_FM(:,i)) & ~isnan(B_FM(:,i)) );

temp_HF = H_FM(non_nans,i);

temp_BF = B_FM(non_nans,i);


% Resample the temporary arrays and into the original arrays

no_nan = length(temp_HF);

H_FM(:,i) = interp1q( (1:no_nan)' , temp_HF ,


B_FM(:,i) = interp1q( (1:no_nan)' , temp_BF ,


end

% Pre-allocate arrays for smoothed FORCs and their derivatives

H_FS = zeros(Nout,Nf);

B_FS = zeros(Nout,Nf);

d1_FS = zeros(Nout,Nf);

d2_FS = zeros(Nout,Nf);

% Smooth the FORCs using regression analysis

for i = 1:Nf

% Calculate the number of segments for the regression analysis

n_seg = ceil( (Bsat - B_FS(1,i)) / (8 * Bsat) * 100 );

% Perform the regression analysis smoothing

[B_FS(:,i), H_FS(:,i), d1_FS(:,i), d2_FS(:,i),~] =

LS_3thO(B_FM(:,i), H_FM(:,i), n_seg, Nout, false);

end

% Pre-allocate arrays for the extended FORCs

H_FE = H_FS;

B_FE = B_FS;

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% Extend the FORCs to intersect the major branch

for i = 1:Nf

% Create an array half way to -Bsat with the FORCs increment

tempB_F = (B_FS(1,i) : (B_FS(1,i)-B_FS(2,i))/10 : (-

Bsat+B_FS(1,i))/2)';

tempH_F = H_FS(1,i) + (tempB_F - B_FS(1,i)) * abs(d1_FS(1,i));

% Find the index where the FORC surpasses the major branch

trim_idx = find(tempB_F < interp1(-HM_S, -BM_S, tempH_F), 1,

'last');

% Trim and append the extension to the FORC

tempB_FA = [tempB_F(trim_idx:-1:2) ; B_FS(:,i)];

tempH_FA = [tempH_F(trim_idx:-1:2) ; H_FS(:,i)];

% Resample and store the temporary arrays

n_p = length(tempH_FA);

H_FE(:,i) = interp1( (1:n_p)' , tempH_FA , linspace(1,n_p,Nout)' );

B_FE(:,i) = interp1( (1:n_p)' , tempB_FA , linspace(1,n_p,Nout)' );

end

% Pre-allocate arrays for the extended FORCs

H_FEH = zeros(Nout,Nf);

B_FEH = zeros(Nout,Nf);

% Extend FORCs to maximum H or B of the major branch

for i = 1:Nf

% Create an array extending beyond saturation


tempH_FH = (H_FE(end,i) : (H_FE(2,i)-H_FE(1,i)) : HM_S(end) +

100*abs(HM_S(end)-H_FE(end,i)) )';

tempB_FH = B_FE(end,i) + (tempH_FH - H_FE(end,i)) / d1_FS(end,i);

% Append the extension to the FORC

tempB_FA = [B_FE(:,i) ; tempB_FH];

tempH_FA = [H_FE(:,i) ; tempH_FH];

% Resample and store the temporary arrays

n_p = length(tempH_FA);

H_FEH(:,i) = interp1( (1:n_p)' , tempH_FA , linspace(1,n_p,Nout)'

);

B_FEH(:,i) = interp1( (1:n_p)' , tempB_FA , linspace(1,n_p,Nout)'

);

end

% Pre-allocate the arrays of the lower constrained FROCs

H_FLC = H_FEH;

B_FLC = B_FEH;

% Constrain the FORCs to their previous FORC

for i = 1:Nf

% Determine which is the previous FORC

if i == 1

H_FP = HM_S;

B_FP = BM_S;

else

H_FP = H_FLC(:,i-1);

B_FP = B_FLC(:,i-1);

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end

% Find the start index in the FORC where constraining is required

con_idxH = find(B_FEH(:,i) < interp1(H_FP , B_FP , H_FEH(:,i)) ,1);

% Assign values to the indexes where the FORCs are not constrained

if ~isempty(con_idxH) && length(con_idxH)>1

con_idxL = find(B_FP > B_FEH(con_idxH-1,i) ,1);

tempH_FLC = [H_FEH(1:con_idxH-1,i) ; H_FP(con_idxL:end)];

tempB_FLC = [B_FEH(1:con_idxH-1,i) ; B_FP(con_idxL:end)];

n_p = length(tempH_FLC);

H_FLC(:,i) = interp1( (1:n_p)' , tempH_FLC ,

linspace(1,n_p,Nout)' );

B_FLC(:,i) = interp1( (1:n_p)' , tempB_FLC ,


end

end

% Pre-allocate the arrays of the upper constrained FROCs

H_FUC = H_FLC;

B_FUC = B_FLC;

% Constrain the FORCs to their previous FORC

for i = Nf:-1:1

% Determine which is the previous FORC and add a dummy point

if i == Nf

H_FP = -HM_S(end:-1:1);

B_FP = -BM_S(end:-1:1);


else

H_FP = H_FUC(:,i+1);

B_FP = B_FUC(:,i+1);

end

% Find the start index in the FORC where constraining is required

con_idxL = find(B_FLC(:,i) > interp1(H_FP , B_FP , H_FLC(:,i)) ,1);

% Assign values to the indexes where the FORCs are not constrained

if ~isempty(con_idxL) && length(con_idxL)>1

con_idxH = find(B_FP > B_FLC(con_idxL-1,i) ,1);

tempH_FUC = [H_FLC(1:con_idxL-1,i) ; H_FP(con_idxH:end)];

tempB_FUC = [B_FLC(1:con_idxL-1,i) ; B_FP(con_idxH:end)];

n_p = length(tempH_FUC);

H_FUC(:,i) = interp1( (1:n_p)' , tempH_FUC ,


B_FUC(:,i) = interp1( (1:n_p)' , tempB_FUC ,


end

end

% Pre-allocate the array according to the file format

FORC = zeros(Nout,Nf+2);

% The first column in the file contains the values for B of the major

% ascending branch with cosine distribution

FORC(:,1) = linspace(min(HM_S) , max(HM_S) , Nout)';

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% The second column in the file contains the values for H of the major

% branch corresponding to the values of B in the first column

FORC(:,2) = interp1(HM_S , BM_S , FORC(:,1));

% The next columns in the file contain the values of H for each FORC

% corresponding to the values of B in the first column

for i = 1:Nf

FORC(:,i+2) = interp1(-H_FUC(:,i) , -B_FUC(:,i) , FORC(:,1) );

% Values under the reversal point are coerced to 0

FORC(isnan(FORC(:,i+2)),i+2) = 0;

end

% Write the data file with sufficient resolution

dlmwrite(Out_file,FORC,'delimiter',',','precision',8);

end

ANNEX 5. MATLAB CODE USED TO GENERATE THE PREISACH

FUNCTION

% Identify the Preisach triangle

% FORCs_file - the file containing the measured FORCs

% Nout - the number of output points for the major branch

% PT - is a structure containing the matrix of the triangle and additional

% variables

function [PT] = Preisach_gen_triangle_v12(FORCs_file,Nout)

% multiplier - a variable used to resample the processed curves to at

% least 1000 samples

if Nout < 1000

multiplier = ceil(1000/Nout);

else

multiplier = 1;

end

% The number of points used during processing

Nproc = Nout * multiplier;

% Read the file containing the FORCs

filecontent = csvread(FORCs_file,0,0);

% The branch of the major hysteresis cycle

Hm = filecontent(:,1);

Bm = filecontent(:,2);


% The FORCs

Bf = filecontent(:,2:end);

% Determine how many FORCs are available

[np , nf] = size(Bf);

% Find the reversal indexes of all the FORCs

F_idx = ones(nf,1); F_idx(end) = np;

for i=1:nf

F_idx(i) = find(Bf(:,1+nf-i)~=0, 1, 'last' );

end

% Create the matrix of the FORCs

HM = interp1((1:np)' , Hm , linspace(1,np,Nproc)');

BM = interp1((1:np)' , Bm , linspace(1,np,Nproc)');

HF = zeros(Nproc,nf); HF(:,end) = -HM(end:-1:1);

BF = zeros(Nproc,nf); BF(:,end) = -BM(end:-1:1);

% Determine the value of the coercivity of the major branch

Hs = 1.5 * abs( interp1(BM , HM , 0) );

% Determine the value of the remanence of the major branch

Bs = 0.75 * abs( interp1(HM , BM , 0) );

% Pass through every FORC and resample to Nproc points

for i=1:nf-1

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tempB = Bf(1:F_idx(i),nf-i+1);

tempH = Hm(1:F_idx(i));

HF(:,i) = interp1((1:F_idx(i))' , tempH ,

linspace(1,F_idx(i),Nproc)');

BF(:,i) = interp1((1:F_idx(i))' , tempB ,

linspace(1,F_idx(i),Nproc)');

end

% Create the threshold levels of the triangle and the FORC levels

Hlevel = linspace(min(HM) , max(HM) , Nout+1)';

% Generate the empty matrices of the FORCs

H = zeros(Nproc,Nout);

B = zeros(Nproc,Nout);

% The last FORC is the branch of the major hysteresis cycle

H(:,end) = -HM(end:-1:1);

B(:,end) = -BM(end:-1:1);

% Create a FORC for each threshold level

for i=1:Nout-1

% Find the index of the lower available FORC

il = floor(interp1(HF(end,:),1:nf,Hlevel(i)));

% Find the index of the upper available FORC

iu = ceil(interp1(HF(end,:),1:nf,Hlevel(i)));

% Assign the upper FORC to temporary variables


if ~isnan(iu)

tempHu = HF(:,iu);

tempBu = BF(:,iu);

else

% If the reversal point is under the first FORC

tempHu = HF(:,1);

tempBu = BF(:,1);

end

% Assign the lower FORC to temporary variables

if ~isnan(il)

tempHl = HF(:,il);

tempBl = BF(:,il);

else

% If the reversal point is under the first FORC

if i == 1

tempHl = linspace(Hlevel(i),Hlevel(i),Nproc)';

tempBl = interp1(HM,BM,tempHl);

else

tempHl = Hlevel(1:i);

tempHl = interp1((1:i)' , tempHl , linspace(1,i,Nproc)');

tempBl = interp1(HM,BM,tempHl);

end

end

% Create square matrices from the arrays which define the branches

Hu_mat = repmat(tempHu,1,length(tempHu));

Bu_mat = repmat(tempBu,1,length(tempBu));

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Hl_mat = repmat(tempHl',length(tempHl),1);

Bl_mat = repmat(tempBl',length(tempBl),1);

% Calculate the distance of each point in the upper branch to each

% point in the lower branch

distances = sqrt( ((Hu_mat-Hl_mat)./Hs).^2 + ((Bu_mat-

Bl_mat)./Bs).^2 );

% Find the indexes where the distances are minimum

[~,idx_min] = min(distances,[],2);

% Determine the B of the interpolated FORC on the minimum distance

segments

B_start = interp1(HM,BM,Hlevel(i));

rl = (tempBu(end)-B_start)/(tempBu(end)-tempBl(end));

ru = 1-rl;

B(:,i) = tempBu.*ru + tempBl(idx_min).*rl;

% Determine the H of the interpolated FORC on the minimum distance

segments

rl = (tempHu(end)-Hlevel(i))/(tempHu(end)-tempHl(end));

ru = 1-rl;

H(:,i) = tempHu.*ru + tempHl(idx_min).*rl;

end

% Create the empty matrix of the weights

w = zeros(Nout);


% Pass through each FORC level and fill the Preisach triangle

% Variable i represents the index on the beta axis

for i=2:Nout

% Variable j represents the index on the alpha axis

j = i:-1:2;

% The incremental variation of B on the beta axis for the ith FORC

X = interp1q(H(:,i),B(:,i),Hlevel(j)) -

interp1q(H(:,i),B(:,i),Hlevel(j-1));

% The sum of the values in the cells on the alpha axis

Y = sum(w(Nout+1-i:end,j))';

% Correct interpolation errors

X(isnan(X)) = 0;

Y(isnan(Y)) = 0;

% Assign the values to the matrix of the Preisach triangle

w(Nout+1-i,j) = X-Y;

end

% Create the matrix of the hysterons spited diagonally

fi = ones(Nout) .* rot90(tril(ones(Nout).*-2)+ones(Nout),2);

% Build the output structure

PT = struct('w' , w ,...

'fi' , fi ,...

'HM' , HM ,...

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'BM' , BM ,...

'Hlevel' , Hlevel ,...

'prevH' , 0 ,...

'dH' , max(BM) / sum(sum(w)) );

end

ANNEX 6. MATLAB CODE USED TO MODEL THE FIELD

INDUCTION IN THE GAP OF A MAGNET

% Analytical model of the U17 magnet clear; clc;

%% Define the constants of the models % The permeability of free space mu0 = 4 * pi * 1e-7; % The number of windings in the magnet's coil N = 10; % The average induction in the iron for determining the area in the gap BFeA =

[0;0.00710168000000000;0.00812199800000000;0.0412744740000000;0.08309866600

00000;0.209042706000000;0.419061472000000;0.628766861000000;0.8379032730000

00;1.04575570400000;1.24981586300000;1.60607267100000;1.68256338600000;1.71

622002600000;1.78001399400000]; % The area over which the flux has been calculated AFe = 2 * 2.13 * 0.36; % The ratio of the area through which the flux closes in the gap Ag =

0.783*AFe*[1.68100000000000;1.68114187800000;1.68131103000000;1.68098003700

000;1.68034898800000;1.68022406400000;1.67979148400000;1.67914446000000;1.6

7832345400000;1.67670722900000;1.67219721700000;1.65666501800000;1.63559180

100000;1.60803587400000;1.56662206400000]; % The average induction in the iron for determining the length of the path BFel =

[0;0.010012682;0.0113959920000000;0.0287735580000000;0.0579577250000000;0.1

16542422000000;0.292618082000000;0.585967779000000;0.878362970000000;1.1670

0467400000;1.27592133800000;1.32705216500000;1.37536761900000;1.42040022600

000;1.45974678400000;2]; % The length of the field line in the iron lFe = 1e-

3*[1080;1048.36456800000;1024.02934500000;939.066191800000;914.318994800000

;937.653195800000;1013.16259200000;1141.59900800000;1239.55051100000;1314.0

2053700000;1337.41701000000;1347.16621800000;1355.41832600000;1361.09989200

000;1365.00807500000;1400]; % The length of the field line in the gap lg = 0.9845 * 1e-3 * mean(

[50.7147905187458;51.0363905077541;50.9032087099209;50.9032087099209;50.903

1794370866;51.2248087099209;50.9031794370866;50.9032087099209;51.0363905187

458;50.8479137379353;51.5278279860396;50.9052744925649;50.6595833872465;50.

9052850374636]);

Im = -[0, 5350, 0, 5350, 0, 5350, 0]; % The precycling values

% The initial Preisach triangle load('PS_PT10k'); PT = PT10k;

% The number of reversal points Nr = length(Im);

% The number of points on each segment NP_seg = 10;

% Preallocate the arrays I_seg = zeros(NP_seg,Nr-1);


Bg_seg = zeros(NP_seg,Nr-1); Hi_seg = zeros(NP_seg,Nr-1); Bi_seg = zeros(NP_seg,Nr-1); HF_seg = linspace(PT.Hlevel(1) , PT.Hlevel(end) , 1e6)'; % BF_seg = zeros(NP_seg,1);

% Construct the BH curve in the material if Im(2) > Im(1) [H,B,PT] = Preisach_evaluate_v01(PT,-.1,sum(sum(PT.w .* PT.fi)) *

PT.dH); else [H,B,PT] = Preisach_evaluate_v01(PT,0.1,sum(sum(PT.w .* PT.fi)) *

PT.dH); end

% The maximum number of iterations and value of the iterating error err_M = 0.1; icM = 100;

% Imitialize the previous values of H and B in the iron HF = 0; BF = 0;

% Go through each segment for i = 1:Nr-1 % Construct the waveform of the current I_seg(:,i) = linspace(Im(i) , Im(i+1), NP_seg)';

% Go through each point of the segment for j = 1:NP_seg % Initiallize the value of the iterating error and of the iterating

counter; err = 100; ic = 0;

% Find B in the yoke through itterative process while (err > err_M && ic < icM && ic < 3) % The area of the flux in the gap Ag_ij = interp1(BFeA , Ag , abs(BF) );

% The length of the magnetic circuit in the yoke lFe_ij = interp1(BFel , lFe , abs(BF) );

% The valid solutions to the equation of the magnetic circuit BF_seg = mu0 * (N * I_seg(j,i) - HF_seg .* lFe_ij) ./ (lg * AFe

/ Ag_ij);

% The index on the model curve where the solution is found [~,BF_i] = min( abs(B - interp1(HF_seg , BF_seg , H)) );

% The relative error between two consecutive iterations err = abs(BF - B(BF_i)) / abs(B(BF_i)) * 100;

% Assign the value of B in the iron for the current iteration BF = B(BF_i);

% Increment the iteration counter ic = ic + 1; end

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% The value of the induction in the gap Bg_seg(j,i) = -BF * AFe / Ag_ij;

% The value of the magnetic field strength in the yoke HF = PT.Hlevel(

interp1(PT.Hlevel,(1:length(PT.Hlevel))',H(BF_i),'nearest') ); Bi_seg(j,i) = -BF;

plot(HF_seg,BF_seg,H,B,HF,BF,'o'); grid on;

end

% Calculate the new BH curves [H,B,PT] = Preisach_evaluate_v01(PT,HF,BF); end

load('workspace_U17_BHstudy_fullanalysis.mat'); selcycle=3; idx = [250, 1000, 1700, 3100, 4800, 6500, 9400, 12300, 14000, 15600,... 16500, 17100, 17800, 18400, 19100, 19900, 21600, 23300, 25500,

27850,... 29500, 31250, 32050, 32700, 33350, 33950, 34650, 35450, 37150,

37950,... 38600, 39250, 39850, 40550, 41350, 43050, 43800, 44500, 45150,

45800,... 46450, 47100, 47700, 48350, 49000, 49700, 50350, 51150, 51750,

52400,... 53050, 53700, 55100, 56800, 58500, 60750, 63100, 64800, 66150,

66850,... 67500, 68100, 68750, 70150, 71850, 74150, 76450, 78150, 79550,

80200,... 80850, 81500, 82150, 83550, 84350, 85000, 85600, 86300, 86950,

87750,... 89150, 89850, 90500, 91550, 92350, 93050, 93700, 94350, 94950,

95750,... 96550, 97250, 97850, 98550, 99150, 99550, 101050, 101850, 102550,... 103150, 103850, 104050, 105250, 106400, 107150, 107850, 108450,... 109150, 109800, 110550, 111650, 113350, 114150, 114850, 115450,... 116150, 116750, 117550, 119250, 120350, 121150, 121800, 122500]; Nr = length(idx); idx_M = zeros(101,Nr); for i = 1:101 idx_M(i,:) = idx - 51 + i; end I_m = -mean( dataAnalysed.(datafieldsselcycle).current(idx_M) )'; I_full = dataAnalysed.(datafieldsselcycle).current; t_m = dataAnalysed.(datafieldsselcycle).time(idx); t_full = dataAnalysed.(datafieldsselcycle).time; Bg_mo = -mean( dataAnalysed.(datafieldsselcycle).fieldHall(idx_M) )'; B_full = dataAnalysed.(datafieldsselcycle).fieldHall; Bg_m = zeros(NP_seg,Nr-1); Hi_m = zeros(NP_seg,Nr-1); Bi_m = zeros(NP_seg,Nr-1); I_m_seg = zeros(NP_seg,Nr-1);

for i = 1:Nr-1 % Construct the waveform of the current I_m_seg(:,i) = linspace(I_m(i) , I_m(i+1), NP_seg)';


% Calculate the new BH curves if i>1 && ( (I_m(i+1) > I_m(i) && I_m(i)<I_m(i-1)) || (I_m(i+1) <

I_m(i) && I_m(i)>I_m(i-1)) ) [H,B,PT] = Preisach_evaluate_v01(PT,HF,BF); end



% Find B in the yoke through itterative process while (err > err_M && ic < icM) % The area of the flux in the gap Ag_ij = interp1(BFeA , Ag , abs(BF) );


% The valid solutions to the equation of the magnetic circuit BF_seg = mu0 * (N * I_m_seg(j,i) - HF_seg .* lFe_ij) ./ (lg *

AFe / Ag_ij);




% Increment the iteration counter ic = ic + 1; end % The value of the induction in the gap Bi_m(j,i) = -BF; Bg_m(j,i) = -BF * AFe / Ag_ij;

% The value of the magnetic field strength in the yoke HF = H(BF_i); Hi_m(j,i) = -HF;

end

end

B_model = [Bg_m(1,1);Bg_m(end,:)']; Bg_mo = -Bg_mo; I_m = -I_m; I_m_seg = -I_m_seg;

I_W = zeros(NP_seg * (Nr-1) , 1); H_w = I_W; B_w = I_W; for i = 1:Nr-1 I_W(1+NP_seg*(i-1) :NP_seg*i) = I_m_seg(:,i);

Doctoral thesis 225

H_W(1+NP_seg*(i-1) :NP_seg*i) = Hi_m(:,i); B_W(1+NP_seg*(i-1) :NP_seg*i) = Bg_m(:,i); end

abs_error = abs(Bg_mo - B_model); rel_error = abs_error ./ abs(Bg_mo) * 100;

% Tho model using the Normal Magnetization Curve Bg_NMC = zeros(NP_seg,Nr-1); Bi_NMC = zeros(NP_seg,Nr-1); Hi_NMC = zeros(NP_seg,Nr-1); filecontent = csvread('PS_NMC.csv'); H_NMC = [0 ; interp1((1:length(filecontent))' , filecontent(:,1) ,

linspace(1,length(filecontent),2e5)')]; B_NMC = [0 ; interp1((1:length(filecontent))' , filecontent(:,2) ,

linspace(1,length(filecontent),2e5)')];

for i = 1:Nr-1 % Construct the waveform of the current I_m_seg(:,i) = linspace(I_m(i) , I_m(i+1), NP_seg)';

H = H_NMC; B = B_NMC;



% Find B in the yoke through itterative process while (err > err_M && ic < icM) % The area of the flux in the gap Ag_ij = interp1(BFeA , Ag , abs(BF) );


% The valid solutions to the equation of the magnetic circuit BF_seg = mu0 * (N * I_m_seg(j,i) - HF_seg .* lFe_ij) ./ (lg *

AFe / Ag_ij);




% Increment the iteration counter ic = ic + 1; end % The value of the induction in the gap Bi_NMC(j,i) = -BF; Bg_NMC(j,i) = -BF * AFe / Ag_ij;


% The value of the magnetic field strength in the yoke HF = H(BF_i); Hi_NMC(j,i) = -HF;

end

end B_model_NMC = -[Bg_NMC(1,1);Bg_NMC(end,:)']; abs_eNMC = abs(Bg_mo - B_model_NMC); rel_eNMC = abs_eNMC ./ abs(Bg_mo) * 100;

tezĂ de doctorat - cern · efficiently if the magnetic hysteresis effects in the cores of the...

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