tezĂ de doctorat · folosite pentru circuitele magnetice ale acceleratoarelor de particule...

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CERN-THESIS-2016-009 26/02/2016 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

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  • CER

    N-T

    HES

    IS-2

    016-

    009

    26/0

    2/20

    16

    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

  • Universitatea Transilvania din Brașov

    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,

    Universitatea Transilvania din Brașov

    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

  • 4 Hysteresis effects in the cores of particle accelerator magnets

    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).

  • 6 Hysteresis effects in the cores of particle accelerator magnets

    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].

  • 8 Hysteresis effects in the cores of particle accelerator magnets

    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

  • 10 Hysteresis effects in the cores of particle accelerator magnets

    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].

  • 12 Hysteresis effects in the cores of particle accelerator magnets

    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.

  • 14 Hysteresis effects in the cores of particle accelerator magnets

    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

  • 16 Hysteresis effects in the cores of particle accelerator magnets

    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:

  • 18 Hysteresis effects in the cores of particle accelerator magnets

    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:

  • 20 Hysteresis effects in the cores of particle accelerator magnets

    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

  • 22 Hysteresis effects in the cores of particle accelerator magnets

    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

  • 24 Hysteresis effects in the cores of particle accelerator magnets

    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);

  • 26 Hysteresis effects in the cores of particle accelerator magnets

    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.

  • 28 Hysteresis effects in the cores of particle accelerator magnets

    (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

    presented in Fig. 2.6.

    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

  • 30 Hysteresis effects in the cores of particle accelerator magnets

    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

    presented in Fig. 2.7.

    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).

  • 32 Hysteresis effects in the cores of particle accelerator magnets

    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 cl