16041

Examensarbete 15 hpJuni 2016

Demagnetization Characterization of Ferrites Independent project in materials engineering

Erik Borg, Martin Brischetto,Joel Johansson Byberg, Anton Karlsson,Daniel Koivisto, Pierre Sandin,Victor Thorendal

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Demagnetization Characterization of Ferrites

Erik Borg, Martin Brischetto, Joel Johansson Byberg, Anton Karlsson, DanielKoivisto, Pierre Sandin and Victor Thorendal

Increasing prices of permanent magnets based on rare earth metals has driven anelectric motor manufacturer to develop an efficient new series of synchronousreluctance motor that uses ferrite magnets instead. Ferrite magnets may bedemagnetized if placed in a strong magnetic field. Their ability to resistdemagnetization is dependent on the temperature and this project aimed todetermine the demagnetization characteristics of a strontium ferrite. It was examinedby a vibrating sample magnetometer in the temperature span -40C to 200C. It wasfound that an analytic model could describe the temperature dependence of themaximum acceptable internal field for an amount of demagnetization. This modelrequired the temperature dependence of Br, Hci in addition to two curve fittingparameters C1 and C2. The model agreed very well with experimentally obtaineddemagnetization curves. Simulations using finite element method were carried out totest effects of different setups around the magnets and their dimensions. Thesimulations found that the air gap between the magnet and the electrical steel shouldbe minimized to yield the strongest magnetic flux density in the electrical steel. Amarket research was compiled to compare the ferrite studied with other grades withregards to BHmax, Br and Hci. With respect to Br vs. Hci the experimentallyexamined ferrite was found to resemble Y30H-2 of the Chinese standard to thegreatest extent. C12 and HF26/30 are the grades that was found to resemble theexamined ferrite the most from their respective standard but not as much as Y30H-2.

ISSN: 1401-5773, UPTEC Q16 041Examinator: Enrico BaraldiÄmnesgranskare: Klas Gunnarsson, Helena FornstedtHandledare: Peter Isberg, Freddy Gyllensten

DemagnetizationCharacterization of Ferrites

Independent Project in Materials Engineering

Erik Borg, Martin Brischetto, Joel Johansson Byberg, Anton Karlsson,Daniel Koivisto, Pierre Sandin and Victor Thorendal

12th June 2016

Abstract

Increasing prices of permanent magnets based on rare earth metalshas driven an electric motor manufacturer to develop an efficient newseries of synchronous reluctance motor that uses ferrite magnets in-stead. Ferrite magnets may be demagnetized if placed in a strongmagnetic field. Their ability to resist demagnetization is dependenton the temperature and this project aimed to determine the demag-netization characteristics of a strontium ferrite. It was examined bya vibrating sample magnetometer in the temperature span -40◦C to200◦C. It was found that an analytic model could describe the tem-perature dependence of the maximum acceptable internal field for anamount of demagnetization. This model required the temperature de-pendence of Br, Hci in addition to two curve fitting parameters C1and C2. The model agreed very well with experimentally obtained de-magnetization curves. Simulations using finite element method werecarried out to test effects of different setups around the magnets andtheir dimensions. The simulations found that the air gap betweenthe magnet and the electrical steel should be minimized to yield thestrongest magnetic flux density in the electrical steel. A market re-search was compiled to compare the ferrite studied with other gradeswith regards to BHmax, Br and Hci. With respect to Br vs. Hci theexperimentally examined ferrite was found to resemble Y30H-2 of theChinese standard to the greatest extent. C12 and HF26/30 are thegrades that was found to resemble the examined ferrite the most fromtheir respective standard but not as much as Y30H-2.

1

Contents1 Introduction 8

1.1 Background of the DemagnetizationCharacterization . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Literature study 11

3 Theory 123.1 Demagnetizing field . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Crystal anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Demagnetization curve . . . . . . . . . . . . . . . . . . . . . . 15

Operating point[30, p. 478] . . . . . . . . . . . . . . . . . . . 16Maximum energy product . . . . . . . . . . . . . . . . . . . . 17Irreversible demagnetization . . . . . . . . . . . . . . . . . . . 18

3.4 Model for approximating maximum acceptableinternal field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Simulating magnets in electrical steel . . . . . . . . . . . . . . 20

4 Method 224.1 Methodological approach . . . . . . . . . . . . . . . . . . . . . 224.2 Sample characteristics . . . . . . . . . . . . . . . . . . . . . . 224.3 Vibrating Sample Magnetometer . . . . . . . . . . . . . . . . . 234.4 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . 26

Locating easy axis within the VSM sample cup . . . . . . . . 264.5 Execution of measurement . . . . . . . . . . . . . . . . . . . . 274.6 Simulations in COMSOL Multiphysics® . . . . . . . . . . . . . 28

Cases 1-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.7 Market research of available ferrite gradesand their characteristics . . . . . . . . . . . . . . . . . . . . . 30

5 Results and discussion 325.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 32

2

Analytic model . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 40

Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Comparison of grades . . . . . . . . . . . . . . . . . . . . . . . 44

6 Conclusions 50

7 Acknowledgments 52

8 Appendix 58

A Datasheet of examined ferrite 58

B Typical data for the electrical steel SURA® M400-50A 59

C Composition of examined ferrite 60

D Lines fitted to data 61

List of Figures1 Demagnetizing factor N for different shapes . . . . . . . . . . 132 Demagnetizing factor N for a cuboid . . . . . . . . . . . . . . 143 Easy axis of a hard ferrite . . . . . . . . . . . . . . . . . . . . 144 The demagnetization curve . . . . . . . . . . . . . . . . . . . . 165 Irreversible demagnetization . . . . . . . . . . . . . . . . . . . 186 Schematic layout of the VSM . . . . . . . . . . . . . . . . . . 257 Samples for measurement . . . . . . . . . . . . . . . . . . . . . 268 Simulation setup for case 1-3 . . . . . . . . . . . . . . . . . . . 299 Simulation setup for case 4 . . . . . . . . . . . . . . . . . . . . 3010 Output data, M -Ha-curve at 27◦C . . . . . . . . . . . . . . . 3211 Experimental M -Hi-curves for different temperatures. . . . . . 3412 Experimental B-Hi-curves for different temperatures . . . . . 3413 Temperature dependence of Br, Hci and µr . . . . . . . . . . . 35

3

14 Model fitted at data at 27◦C . . . . . . . . . . . . . . . . . . . 3615 Temperature dependence of C1 and C2 . . . . . . . . . . . . . 3716 Example of analytic model for 20 % acceptable demagnetization 3817 Valid interval for the analytic model for different percentage

of demagnetization . . . . . . . . . . . . . . . . . . . . . . . . 3918 Comparison with data sheet . . . . . . . . . . . . . . . . . . . 4019 Distance dependence of B and Neff in case 1 . . . . . . . . . . 4120 Air gap thickness dependence of B and Neff in case 2 . . . . . 4221 Shape dependence of demagnetization in case 3 . . . . . . . . 4322 BHmax values for Chinese grades and the examined ferrite . . 4523 BHmax values for European grades and the examined ferrite . 4624 BHmax values for American grades and the examined ferrite . 4625 Br and Hci comparison for Chinese and European grades . . . 4826 Br and Hci comparison for Chinese and American grades . . . 4927 Model fitted at data at -40◦C and -10◦C . . . . . . . . . . . . 6128 Model fitted at data at 27◦C and 80◦C . . . . . . . . . . . . . 6129 Model fitted at data at 140◦C and 200◦C . . . . . . . . . . . . 62

List of Tables1 Experimental demagnetizing factor N for different temperatures 332 Linear temperature dependence of Br, C1, Hci and µr . . . . . 373 Br, C1, Hci at 20◦C . . . . . . . . . . . . . . . . . . . . . . . . 384 Comparison with data sheet . . . . . . . . . . . . . . . . . . . 405 Direction of magnetization dependence for Br and Neff in case 3 426 Demagnetizing factor N in case 5 . . . . . . . . . . . . . . . . 447 Excluded grades for Br vs. Hci comparison . . . . . . . . . . . 47

4

Nomenclature

Variables & Parameters

ABr Intercept of Br(T ) [T]

AC1 Intercept of C1(T ) [AT/m]

AHciIntercept of Hci(T ) [A/m]

B Magnetic flux density [T]

BHmax Maximum energy product -

BA Applied magnetic flux density [T]

Bd Magnetic flux density at the operating point [T]

Br Remanence [T]

Bnewr New remanence after demagnetization [T]

BBr Slope of Br(T ) [T/K]

BC1 Slope of C1(T ) [AT/Km]

BHciSlope of Hci(T ) [A/Km]

C1 Curve fitting parameter [AT/m]

C2 Curve fitting parameter -

EH Electric field produced by the Hall effect [V/m]

Ha Applied field [A/m]

Hc Coercivity [A/m]

Hd Demagnetizing field [A/m]

Hi Internal magnetic field [A/m]

Hk Maximum acceptable demagnetizing field [A/m]

Hci Intrinsic coercivity [A/m]

Hd,eff Effective demagnetizing field [A/m]

M Magnetization [A/m]

5

N Demagnetization factor -

Nx Demagnetization factor along x-axis -

Ny Demagnetization factor along y-axis -

Nz Demagnetization factor along z-axis -

Neff Effective demagnetization factor -

OP Operating line [T]

P Operating point -

R2 Coefficient of determination -

RH Hall coefficient [m3/C]

UH Hall voltage [V]

H̄d,eff Volume average of the effective demagnetizing field [A/m]

M̄ Volume average of the magnetization [A/m]

µ Magnetic moment [Nm/T]

µ0 Permeability in vacuum [Tm/A]

µr Relative permeability -

ρ Density of Ex.Mag [g/cm3]

ρCaO Density of CaO [g/cm3]

ρFe2O3 Density of Fe2O3 [g/cm3]

ρSrO Density of SrO [g/cm3]

b Distance over potential difference [m]

j Current density [A/m2]

k Maximum acceptable amount of demagnetization -

kmax Maximum defined k -

Abbreviations

AC Alternating Current

6

DIN-IEC Deutsches Institut für Normung - International ElectrotechnicalCommission

Ex.Mag Experimental examined Magnet

IEEE Institute of Electrical and Electronics Engineers

PM-REE Permanent Magnets Rare Earth Elements

SSVT Single State Variable Temperature

SynRM Synchronous Reluctance Motor

VSM Vibrating Sample Magnetometer

7

1 Introduction

The emission of greenhouse gases is considered one of the most critical andimportant problems of the world. Recent trends suggest that current emis-sions may lead to severe environmental damage[1][2]. Contemporary climatechange has affected physical and biological systems on all continents and inmost oceans[3][4][5]. And the greenhouse gases from anthropogenic emissionsare primarily responsible for global warming[1][6][7]. To deal with this prob-lem, combustion of energy sources that produce greenhouse gas emissions,such as petroleum and natural gas, has to decrease. REN21’s Renewable2015 Global Status Report states that renewable energy and energy efficiencyis critical for addressing climate change[8]. In 2013, transportation accoun-ted for 28% of the total energy consumption in the United States, of which93 % comes from petroleum[9]. Thus, replacing petroleum with renewableenergy, such as electricity from wind or sunlight, would help to decrease apart of the source of global warming. To do so, combustion motors wouldhave to be substituted with electrical motors. It should also be mentionedthat electric motor driven systems are estimated to be responsible for 29% ofoverall global electricity consumption according to the Industrial EfficiencyTechnology Database[10]. Considering this, a small gain in energy efficiencyof an electric motor system, would lead to a huge gain in total global energyconsumption. There are mainly three types of electrical motors of interest.The induction motor, permanent magnet motor and synchronous reluctancemotor. With the development of high performance permanent magnets basedon rare-earth elements (PM-RRE), it became possible to build a compact per-manent magnet motor with high efficiency and power output with respectto size compared to its competitors[11][12]. Thus the permanent magnetmotor expanded commercially and a lot of research was put into further de-veloping the high performing PM-RRE, neglecting conventional permanentmagnets such as ferrites[11]. The PM-RRE are widely applied in electricmotors today. However, the benefits of PM-REE may be clouded by itslimited availability and high costs of their constituent rare earth elements.China controls approximately 95% of the market of these elements, and thecosts were drastically increased due to decreased availability and high globaldemands since 2010[13][14][15]. The mining and refining process of thesehas shown to impact the native environment and ecosystem negatively[16].Exposure to rare earth elements through labor and environmental related

8

conditions have also shown to increase the risk of lowered liver function andlung related diseases[17][18]. To avoid the dependence of PM-REE in electricmotors, further research is required to understand and improve conventionalpermanent magnets such as, in this case, ferrites.

1.1 Background of the DemagnetizationCharacterization

This study was launched to amplify the knowledge of ferrites at differenttemperatures. While present research is insufficient to fully describe its be-havior, further research would enable manufacturers to decide if the efficiencyof their electric motor that utilize ferrites could be increased. The ferrite isa small part of the complex system in an electric motor, yet an importantpart.Synchronous reluctance-motor (SynRM) is an electric motor that doesnot necessary utilize permanent magnets. A leading electric motor manu-facturer developed a new series of SynRM that use ferrites to achieve aneven higher efficiency than electric motors with PM-REE[19]. A ferrite is aconventional and widely applied permanent magnet which is cost effectiveand easily available due to its natural abundance, resulting in an ecologicallyand economically sustainable source[11][20]. The efficiency can be increasedeven more by further understanding the behavior of ferrites in the SynRM.Since initial and operating temperature may vary vastly, the temperaturedependence of the magnetic properties must be known. Particularly the tem-perature dependence of irreversible demagnetization; a damaging and costlycase due to a decrease of the efficiency and the electromagnetic torque inthe machine[21]. Irreversible demagnetization is not specified as a magneticmaterial property. But it is sufficiently described by the magnets hyster-esis curve, also known as the B-H-curve or the M-H-curve. Additionally,the temperature dependence of the coercivity Hc and the remanence Br aretwo parameters that are useful to describe the temperature dependence ofirreversible demagnetization[22].

1.2 Aim

The aim of this project is to characterize the demagnetization risk of a mag-netically hard ferrite by investigating the behavior at different temperatures

9

and different surroundings with real and simulated experiments. It is fulfilledby answering the following questions:

• Can an analytic model determine the point of irreversible demagnetiz-ation for temperatures between -40◦C to 200◦C for one specific gradeof ferrite?

• How does the interaction of multiple samples of the examined ferritesaffect their overall performance and demagnetization?

• How does commercially available grades of ferrites compare with re-spect to magnetic properties?

1.3 Delimitations

This chapter describes the boundaries set in this project. Presenting thedemarcations chosen to achieve the aim in the time frame. Necessary delim-itations are presented below:

Experiment: Only one grade of ferrite is examined. It would yieldinformation to determine if the analytic model is ap-plicable on various grades, but due to the time re-quired for the experimental equipment to measure onesample, time is insufficient.

Experiment: It would also be interesting to examine one specificgrade of ferrite provided from different manufactur-ers, to see if the manufacturing process affected themagnetic properties. However, it would require moremeasurements and has to be dismissed.

Simulation: Due to no experience in the field of simulating by finiteelement method, simple setups are used in addition toapproximations of the magnetic properties.

Market research: All magnetic properties are not compared against.Excluding those that are not commonly used as figuresof merit to describe a magnetic materials performance.

10

1.4 Outline

The following section gives a brief description of the framework of this study.That is, a literature study that investigates similar studies conducted byothers scholars. Next, there is a theory section to explain the underlyingphysics of the results and its discussion. The method section is presented afterthe theory which explains the methodological approach and the performedexperiments. With the theory and method sections accounted for, the resultsare presented and discussed. Finally, the results are summed up in the lastsection; conclusions.

2 Literature study

The literature study investigates if similar studies had been done that couldbe of use for this project. Several other scholars studied ferrites[23, 24, 25,26, 27, 28, 29]. Three are considered interesting and further discussed be-low. The methodological approach for the literature study will be explainedin section 4.1. In the article Maximum energy product at elevated temperat-ures for hexagonal strontium ferrite (SrFe12O19) magnet, the authors JihoonPark, Yang-Ki Hong Seong-Gon Kim et al. calculated the temperature de-pendence of the saturation magnetization Ms by calculating the electronicstructure of strontium ferrite. By the use of density functional theory andgeneralized gradient approximation. They found that Ms decrease as tem-perature increase. The authors presents a theoretical way of calculating thetemperature dependence of Ms, which is an interesting material propertytowards the aim for this project. The article could be of use to predict theoutcome of Ms. But understanding and applying the methods presentedwould exceed the scope of this project. In addition, it would also not besufficient to obtain an analytic model to determine the point of irreversibledemagnetization since more magnetic properties has to be considered. In thearticle Relation between the alignment dependence of coercive force decreaseratio and the angular dependence of coercive force of ferrites, the authorsY. Matsuuraa, N. Kitaib, S. Hosokawab and J. Hoshijimab investigated thetemperature dependence of coercivity in addition to the angular dependenceof coercivity. The dependence was investigated at temperatures 300 K, 333K, 373 K, 413 K and 453 K. They strongly suggest that the coercive force is

11

determined by magnetic domain wall motion and discuss the physical reasonbehind the behavior of the coercive force at different temperatures. The coer-civity is an important material property to determine the point of irreversibledemagnetization. However, the article does not evaluate temperatures be-low 0◦C and they did not produce a practical model, which is of interestin this project. In the article Comparison of Demagnetization Models forFinite-Element Analysis of Permanent-Magnet Synchronous Machines, theauthors S. Ruoho, E. Dlala and A. Arkkio presents a few simple demagnet-ization models for finite element calculations. The Exponent Function Modelpresented in this article will be used as inspiration when constructing theanalytic model. The model is applicable due to the fact that the parameterswhich describes it will be provided by the experiment. In addition, the shapeof the model is similar to the curve that describes a magnetic compound.

To sum up, the first two articles mainly investigate the physical reason behindthe magnetic properties of ferrite. Utilizing methods beyond the scope ofthis project. No simple method based on experimental results to describethe demagnetization process of a ferrite were found. The third mentionedarticle will be used in this project by using the Exponent Function Model.It suits well since the parameters that describes it will be obtained from theexperiment. And that the general shape of the model is similar to the curvethat describes a ferrite.

3 Theory

This chapter presents a theoretical explanation of magnetically hard ferritesin sections 3.1 and 3.2, and how their magnetic properties characterize theshape of the hysteresis curve in section 3.3. The model for approximatingthe demagnetization curve and maximum acceptable internal field will beexplained in section 3.4. The section 3.5 will cover the theory needed tounderstand the results of the simulations.

3.1 Demagnetizing field

As soon as a magnetic material is being magnetized and open magnetic cir-cuit is obtained, fictive monopoles appears at the end surfaces in the direction

12

of magnetization. These poles will generate an internal demagnetizing field.It is demagnetizing since it is oriented in the opposite direction of the mag-netization. This demagnetizing field is defined as:

Hd = −NM (1)

Where M is the magnetization and N is the demagnetizing factor. N isdependent on direction and the sum of the x-,y-,z- components equals 1. Itincreases with decreasing distance between the end surfaces relative theirsize. In figure 1 the demagnetizing factor N is presented for different geo-metries and directions For the shapes given in figure 2, it implies that Nz

and Ny generate the strongest demagnetizing field and Nx the weakest. Nx

is therefore the axis in which the material wants to be magnetized along andhard to demagnetize from. Nx will therefore become the easy axis in termsof shape anisotropy.

Figure 1: The figure shows the demagnetizing factor N for three different shapes. Even though N cannot be calculated for the long rod nor the thin film, the distance between the end surfaces relative theirsize is so large that N goes to 0 asymptotically.

The demagnetizing factor can only be calculated theoretically for ellipsoids;the uniformity of Hd can only be achieved in an ellipsoid[30, p. 52-53].Bozorth calculated N for ellipsoids with different length ratios[32]. By prop-erly fitting any geometry to the ellipsoid, N can be approximated by usingthe same geometric ratio. This is exemplified in figure 2. The approximationin figure 2 will be compared to the values given from the simulation explainedin section 4.6, case 4. To obtain N from an experimental graph, see section3.3.

13

Figure 2: An ellipsoid and a cuboid. The demagnetizing factor N can be approximated for the cuboidby calculating N for an ellipsoid using the same ratio; the ratio of y or z and x.

3.2 Crystal anisotropy

Magnetically hard ferrites, like the one used in this project, possess a strongmagnetocrystalline anisotropy, an intrinsic property that tends to direct themagnetization along a particular crystallographic axis, the so-called easyaxis. The cell has a hexagonal structure and the direction which is easy tomagnetize and hard to demagnetize is along the ±[001] axis in figure 3, whichis called the crystal easy axis. The [100] and [010] directions are found equallyhard to magnetize[30, p. 236]. Due to the cells tendency to crystallize withthe [001] axis parallel against each other, the bulk possess an easy axis[30,p. 487]. In addition, an external magnetic field during the manufacturingprocess is usually applied to increase the tendency.

Figure 3: Crystal structure of a hard ferrite. Easy axis of magnetization is ±[001] while [100] and [010]is equally hard.Image retrieved from: http://tinyurl.com/hdl6cvp 2016-05-05

14

The magnetocrystalline anisotropy give rise to a field which tries to hold themagnetization along the easy axis. The magnitude of this field will always belarger than the demagnetizing field for a magnetically hard ferrite. Therefore,the resulting easy axis due to shape anisotropy and crystal anisotropy isalways determined by the crystals easy axis[30, p. 238].

3.3 Demagnetization curve

A magnet is often characterized by its demagnetization curve; the secondquadrant of its hysteresis curve. The curve describes how the materials mag-netic field strength vary with respect to an internal field. It is obtained byplotting equation 2:

B(M,Hi) = µ0(M +Hi) (2)Where B is the magnetic flux density, µ0 is the permeability in vacuum andM the magnetization of the material[30, p. 17]. The internal magnetic fieldstrength Hi is defined in equation 3.

Hi = Ha +Hd (3)

Ha is the measured applied field and Hd the demagnetizing field[30, p. 411]according to equation 1. The remanence Br, the coercivity Hc and the relat-ive permeability µr, which are three important magnetic material properties,can be extracted from the plot. The remanence is the remaining internal fluxdensity in the material, after an applied field aligned all the magnetic mo-ments and the applied field is removed. The coercivity measures the strengthof the applied field needed to reduce the internal flux to zero. Note that Hc

differ from Hci, the latter measures the strength of the applied field for mag-netization reversal and is always larger than Hc in magnitude. The relativepermeability µr can be extracted from the slope of the plot with

µr = 1µ0

∂B

∂Hi

∣∣∣∣∣Hi=0

. (4)

Hc and Br are shown relative to the demagnetization curve in figure 4. Aregion of the demagnetization curve is defined as the knee which is the es-sential part for the model described in section 3.4. The region can be seenin figure 4.

15

Figure 4: The demagnetization curve B-H and the M -H-curve(dashed) are shown in the figure. Theremanence Br can be found at the B-intercept of the B-H-curve and Hc at H-intercept. Hci can be foundat H-intercept of the M -H-curve. An operating point P is marked at B = Bd and H = Hd. BHmax is apoint on the curve and is exemplified with the point S and will be explained further in section 3.3. Theknee area is marked with a dashed box.

To obtain N from experimental results equations 1 and 3 are used,

Hi = Ha −NM. (5)

This equation is then differentiated with respect to M to

∂Hi

∂M= ∂Ha

∂M−N. (6)

At M = 0, ∂Hi

∂M≈ 0 for permanent magnets. This yields

N = ∂Ha

∂M

∣∣∣∣∣M=0

. (7)

Operating point[30, p. 478]

The function of a magnet is to provide an external field. To do so, it musthave free poles to achieve an open magnetic circuit. Otherwise no external

16

field would be produced. The free poles are what causes the demagnetizingfield Hd. This field causes the produced field to be lower than the remanenceBr. The actual produced field from the magnet without any applied fieldis given by the static operating point P which can be calculated given thedemagnetization curve or the dimensions of the magnet. Initially, Ha = 0which according with equations 1, 2 and 3 yields:

Bd = µ0(M +Hd) = µ0

(−Hd

N+Hd

)= −µ0

( 1N− 1

)Hd

⇒ − Bd

µ0Hd

= 1N− 1 (8)

The slope of the line OP (known as the load line) in figure 4 is given by 1N−1

which can be obtained by the result of equation 8. The static operating pointP is given by where OP intersects with the B-H curve. Since N dependspurely of the geometry, the static operating point can be chosen properly bychoosing the dimensions of the magnet. The static operating point also tellswhich path is to be taken in the B-H curve when the magnet is exposed toan applied field. In a sense, it sets the starting point in the B-H curve. Forexample, when a positive applied field is turned on, the operating point Pmoves upwards in figure 4. When a negative applied field is turned on Pmoves downwards.

Maximum energy product

The performance of a permanent magnet is often specified by its value of themaximum energy product (BHmax). It is a measure of the stored energy pervolume a magnetic material produces[31, p. 193]. Magnetically hard ferritehas considerably lower BHmax compared to PM-REE (∼ 30 kJm−3 and∼ 400kJm−3, respectively). BHmax is shown qualitatively as point S in figure 4.Highest operating efficiency (from the perspective of a magnet) is said tobe obtained when the operating point coincides with BHmax. Due to thefact that a given volume of magnetic material would produce the strongestmagnetic flux at that point[33]. For the grade of ferrite which is examinedin this experiment, with Hd = 166.3 kAm−1 and Bd = 0.1984 T[35], theoperating point coincides with BHmax if N = 0.513.

17

Irreversible demagnetization

An irreversible demagnetization is the result when Br drops in magnitude.After Br is obtained as described in section 3.3. Br only drops in magnitudeif one or several magnetic moments deviates from the direction of magnet-ization M . A deviation can be caused by an applied field. An applied fieldthat causes magnetic moments to divert from their direction is said to bereversible, if the magnetic moments are re-diverted to the direction of mag-netization after the field is switched off. This occurs if the magnet operatesat the linear region of its demagnetization curve. In the same way, it is saidto be irreversible if the magnetic moments are not re-diverted to the directionof magnetization after the field is switched off. This occurs if the magnetoperates between two regions with different slopes. Hence, operating where∆B/∆Hi changes. The point where the two regions are distinguished at isexemplified in figure 5 as point c[30, p. 483-484]. The figure also explainsthe process of irreversible demagnetization.

Figure 5: Note that the figure does not reflect a true demagnetization curve of a magnetically hard ferrite,it is constructed to describe irreversible demagnetization. The figure shows the process of irreversibledemagnetization. The arrows to the right represent the direction of magnetic moments. Firstly, Br isobtained. Secondly, an applied field Ha moves the operating point from q tom, passing the linear endpointc. When Ha is removed the operating point moves back with the same slope as before (the slope of cq)until it reaches point n. Which in turn moves Br to Bnew. Since point c was passed some magneticmoments are diverted as seen in 2. Thus decreasing Br with ∆Br. The demagnetization is irreversiblesince the previous remanence Br can only be obtained by saturating the magnet again to align all themagnetic moments.

18

An irreversible demagnetization is problematic due to the fact the the magnetproduces a weaker field than before. Which may not suit the application,forcing maintenance costs. It can be avoided by controlling the strength ofthe applied field, or choosing the dimensions such that the operating point ismoved upwards in the demagnetization curve (decreasing the demagnetizingfactor).

3.4 Model for approximating maximum acceptableinternal field

To find the location of the knee, an approximation of the demagnetizationcurve was made to find the parameters it depends on. As can be seen in figure4, the curves are linear until they become close to vertical at Hci. Based onthis and inspiration from a similar approximation authored by S. Ruoho et.al.[29] equation 9:

B = Br + µ0µrH −C1

C2Hci +H+ C1

C2Hci

(9)

Where Br is the remanent magnetic flux density, µ0 is the permeability invacuum, µr is the relative permeability, B and H are the magnetic fluxdensity and the internal magnetic field, respectively. C1 and C2 are twocurve fitting parameters; C1 sets the sharpness of the knee and C2 is a fewpercent larger than unity. This approximation is valid from H = 0 to H =Hci. The first two terms represent the linear behavior of the B-H-curvebefore permanent demagnetization. The third term could be interpreted asa demagnetization term, that is, the deviation from the linear behavior, seefigure 4. The fourth term is a correction term which ensures that B = Br atH = 0. When a field Hk has demagnetized the magnet to an amount k andthe internal field has returned to zero, the new remanence Bnew

r will be

Bnewr = Br −

C1

C2Hci +Hk

+ C1

C2Hci

. (10)

If Bnewr = Br(1− k) the magnetic field Hk is

Hk = C1

kBr + C1C2Hci

− C2Hci. (11)

19

Hk could be interpreted as the position of the knee and depends on Br, C1,C2 and Hci. The temperature dependence of these four variables will beexamined in the experiment. The first term of equation 11 represents thedistance between the knee point and Hci(or rather, a value slightly largerthan Hci) and the second term is the value slightly larger than Hci. Thismeans that the second term will have the most overall impact on the value ofHk. It is important to note, as mentioned earlier, that the approximation isonly valid for magnetic fields weaker thanHci. This means that the maximumamount of demagnetization kmax this approximation can be used to calculateat a certain temperature is obtained from equation 11 by setting Hk = −Hci;

kmax = C1

BrHci(C2 − 1) −C1

BrC2Hci

. (12)

The model described above should be valid for any hard ferrite. After atemperature dependence of all parameters are inserted the model can beused to predict the maximum acceptable internal field Hi for a maximumacceptable amount of demagnetization k.

3.5 Simulating magnets in electrical steel

The program chosen for the simulations handles most of the physics withoutuser input and is based on much of the same theory as the rest of this section.The user can set specific magnetic characteristics on different parts of themodel such as permeability and the remanence.

In the magnet relevant to this study the direction of the magnetization wouldbe normal to the largest surface of the magnet. As explained in section 3.1,this causes the demagnetizing field to be larger than otherwise. Since thefields produced by the magnets will be partly pass through their neighboringmagnets, they will influence the fields inside them. When placed in electricalsteel the demagnetizing field inside a magnet will be weaker than in airbecause the magnetic circuit is partly closed, and thus the operating point ismoved upward[30, p. 483-484]. The sum of the magnet’s own demagnetizingfield in a specific environment and the fields caused by their neighbors couldbe considered an effective demagnetizing field Hd,eff when applied to an non-isolated situation. This effective demagnetizing field can be used to calculate

20

the effective demagnetizing factor Neff of a magnet using equation 1 with

Neff = −H̄d,eff

M̄(13)

where H̄d,eff and M̄ are volume averages of the effective demagnetizing fieldand the magnetization. The effective demagnetization factor depends in parton the field produced by the neighboring magnets, and since the producedfield is temperature dependent, Neff is also temperature dependent.

21

4 Method

The section explains the methodological approach and characteristics of theexamined ferrite, followed by how the theory sections 3.2, 3.1, 3.3 and 3.3connects to the experiment. It also gives a detailed explanation of the ex-perimental equipment, the preparation of the sample and how the easy axiswas identified within the experimental equipment. The approach for simu-lating the various cases is presented. Finally, it is presented how data for themarket research was compiled.

4.1 Methodological approach

The measurement technique that was used for the experiment was developedby S.Foner[42] and further developed by himself and others[38, 39, 40, 41].It is a well proven technique and technology for accurate measurements ofmagnetic compounds. The use of the machine was led and supervised byan experienced user. Simulations were done by finite element method inthe software COMSOL Multiphysics®. There exists no uncertainty regard-ing the credibility of the software. The uncertainty lies at the amount ofexperience the user possess to correctly transmit a physical problem to thesoftware. In addition, it also depends on the amount of knowledge the userhas in the field of electromagnetism to decide the credibility of the simulatedcase. It was handled by simulating simple setups, so that the outcome ofthe simulation could be discussed by basic knowledge in the field of electro-magnetism. Information for comparing commercially available grades wasgathered by visiting websites of ferrite traders. The method is straight for-ward and trustworthiness was confirmed by plausibility checks among thetraders.

4.2 Sample characteristics

One dimension of a ferrite with composition 85 wt% Fe2O3, 13,5 wt% SrO,0,8 wt% CaO and 0,7 wt% [Al2O3, SiO2, other](see appendix C) was ex-amined. As explained later, the volume of the sample is needed to calculatethe magnetization. To find the volume, the density of the material ρ was

22

approximated with

ρ = 0.85 · ρFe2O3 + 0.135 · ρSrO + 0.008 · ρCaO, (14)

where ρX is the density of material X. The densities obtained from SI Chem-ical Data[44] were 5.2 g/cm3, 4.7 g/cm3 and 3.3 g/cm3 for Fe2O3, SrO, andCaO, respectively. Using these densities and the equation above, the totaldensity was calculated to ρ = 5.08 g/cm3.

Considering the theory presented in section 3.2, the direction of the easy axiswas given by the ferrite provider. The operating point coincides with BHmax

if the demagnetizing factor N equals 0.513, according to the theory presentedin sections 3.1, 3.3 and 3.3. Where the purpose is to gain a high operatingefficiency. A demagnetizing factor of N = 0.513 is hard to obtain accurately.However, the higher the ratio between y:x or z:x (see figure 2) the better theaccuracy. The experimental equipment limits the allowed maximum size ofthe sample to 2.65x2.65x9 mm. It would otherwise not fit the equipment. Ifthe cup is to be used the upper size limit of the sample is roughly 2x2x2 mm(explained in the following section).

4.3 Vibrating Sample Magnetometer

The following section explains how the experimental equipment works as wellas how the hysteresis curves are obtained. Vibrating sample magnetometer(VSM) is a measurement technique developed by Simon Foner[42]. The meas-urement technique is based upon that a vibrating magnetic sample createsflux change in coils which induces a voltage. This voltage is proportional tothe magnetic moment of the sample. A schematic sketch of the VSM is shownin figure 6. Initially a single point calibration with a Ni sphere with knownmagnetic moment and mass is done to get the correlation between appliedfield H and magnetic moment µ. The Hall probe located between the pick-upcoils is used to measure the applied field, Ha. The Hall probe contains a thinconducting plate which a current flows through. As the conducting plate isexposed to the magnetic field the following expression holds[43].

EH = µ0Ha

RHj= UH

b(15)

Where the electric field EH is produced by the current density j according tothe Hall effect. EH is parallel to the surface of the conducting plate over a

23

distance b, yielding the Hall voltage UH . The applied field Ha is orthogonalto the conducting plate surface and RH is known for the conducting plate.The voltage UH can be translated to applied magnetic flux density BA thusone can get the magnetic moment µ due to the single point calibration. TheHall coefficient RH is known for the conducting plate.

For small samples, a sample cup which is fastened by threading is used.For large samples, it will not fit the sample cup and is attached directly ona ceramic piece or quartz rod by glue, the ceramic piece is screwed onto asample rod while the quartz rod is complete by itself. The sample rod isattached to a membrane. The membrane is driven by an AC source with thefrequency of 82 Hz. A lock-in amplifier prevents registration of any unwantednoise. The applied magnetic field from the electromagnets can be adjustedin both strength and direction. Eliminating induced currents in the pick-upcoils from the applied field is done by winding the connected coils reversely.The winding does not affect the induced current since the vibrating samplecreates an in-homogeneous field relative to the pick-up coils. The hysteresiscurve is obtained with the software IDEAS-VSM which gives a µ vs. H curve.By using equation 16 one obtains the M vs. H curve. This curve can thenbe recalculated to B vs. H using equation 2 with the total magnetization asin equation 16.The mass m of the sample is presented in section 4.4 and thedensity ρ is presented in section 4.2.

M = µ · ρm⇒M, [emu/cm3] = 103[A/m] (16)

For measurements at different temperatures, a Dewar thermos, a heat-exchanger,an oven known as single state variable temperature (SSVT), and the gasesAr and N2 are used. For measurements below 27◦C, N2 at room temperatureis passed through the heat exchanger which is submerged in the Dewar ther-mos. The Dewar thermos contains liquid N2 at -196◦C[44]. The N2 gas isthen transported to the SSVT through an isolated vacuum tube where it isheated up to the desired temperature. Measurement at or above 27◦C workssimilarly. The heat-exchanger is taken out of the Dewar thermos and the N2gas is exchanged with Ar gas at room temperature. The temperatures canbe set and controlled with the software IDEAS-VSM which can then createhysteresis curves for the set temperature.

24

Figure 6: Schematic layout of the VSM and description of components. 1: Computer with softwareIDEAS-VSM version 4. 2: Equipment containing Lock-in Amplifier and AC registration source for pick-up coils. 3: Oscillating membrane driven by AC source. 4: Sample rod. 5: One of four pick-up coils. 6:Hall-probe measuring the applied field. 7: SSVT surrounding the sample. 8: Inflow of Ar/N2 to SSVT.9: Flux lines of applied magnetic field. 10: Electromagnets producing the applied field. 11: Sample rod,sample cup containing the magnetic sample. The sample cup is screwed to the sample rod locking themagnetic sample in place.

25

4.4 Sample preparation

A ferrite is hard to process due to its brittleness. Initially, the sampleswould be prepared by diamond blade cutting to yield precise dimensions andconsistent shapes. However, due to the properties of the material, this wasnot viable. Hence, a more simplistic method was initiated to the cost of thedimensional accuracy. The samples were prepared by fastening a ferrite ofsize 6x13x54 mm to a vice, followed by several hits to the small end surfacewith a hard narrow tool. By repeatedly hitting the surface, small chips wereobtained. The process was repeated until a sample with a high ratio betweenshort and long axis was obtained. The sample can be seen in figure 7. Thechosen sample was weighed at 8.57 mg with an analytical balance. The masswas used to get the total magnetization M of the sample. Seen in section4.3.

(a) Top side of sample. (b) Bottom side of sample.

Figure 7: Sample placed on grid of 100 µm squares. Height of sample is roughly 1 mm

Locating easy axis within the VSM sample cup

The surface of the original ferrite which the easy axis is normal to had a spe-cific pattern. The pattern could be identified on the samples that containedthat surface and thus the easy axis of the sample could be located visually.Afterwards, the sample was put into the VSM sample cup. It was put sothat the easy axis was perpendicular to the vibration axis. Since the sample

26

cup requires rotation to be fastened, the direction of the easy axis was lost,and calibration was required to align the easy axis with the applied field.The alignment was done by measuring the magnetization M as the samplewas subjected to a constant applied field, while simultaneously rotating thesample cup. According to the theory presented in section 3.2, strongest Mwas obtained when the easy axis is parallel with the applied field. When apeak in M was measured, the easy axis was located and the rotation shutoff.

4.5 Execution of measurement

The measurements were planned over three days where the following tem-peratures were to be examined: -40◦C, -10◦C, 27◦C, 80◦C, 140◦C and 200◦C.For every temperature a rough measurement with few data points were madeto identify the approximate region of the knee. Thereafter the program wasadjusted to increase the amount of data points around the knee. This gen-erated a hysteresis curve with focus on the shape of the knee. To begin themeasurements, the sample had to be attached to the VSM. There were sev-eral ways to attach the sample, each method with different beneficial factors.The first two methods failed at different states, whereas the third one wassuccessful.

1. The sample was attached to a porous ceramic rod with silver glue,supposed to be consistent up to 150◦C. This was successfully done forthe temperatures -40◦C, -10◦C and 27◦C. Although approaching thenext temperature (80◦C) the threads were partially pulverized, makingthe ceramic piece no longer tightly screwed in so the sample couldrotate and therefore making the result inconsistent.

2. Instead of the porous ceramic rod, a rod of quartz was used in combin-ation with the silver glue. This was unsuccessful and the glue loosenedafter which the sample detached. From this another sample was at-tached to the rod, although with a different glue, supposed to be con-sistent to 1100◦C. This generated values without significance as theglue was magnetically active in itself.

3. The third option was to place the sample in the sample cup which wasattached to the rod by threading. This was successful and generated

27

data appropriate for further analysis.

During the transition to lower temperature measurements, the VSM hada breakdown whereafter it had to be rebooted which meant resetting theangle of the sample. After the VSM was back up running, the final twotemperatures were measured and the experiment was completed.

4.6 Simulations in COMSOL Multiphysics®

Five different cases of interest were studied. Cases 1, 2, and 3 had the samesetup but with different parameters varied to show their effect on the overallperformance of the magnets. Case 4 and 5 were carried out using their ownsetups. Case 5 was added to show the demagnetizing factor, N , due to theadversity during the sample preparation which caused an inability to performmeasurements of different dimensions of magnets. The magnetization of themagnets was assumed to be 0 in the z- and x-axis because the y-axis issupposed to be the magnets’ easy axis. From the experimental result a curvecould be created of magnetization against magnetic flux density. This M-B-curve was set as the magnetization of the easy axis on the magnets. Theelectrical steel used in the simulation cases 1-3 was based on an electricalsteel called SURA® M400-50A. Its electrical properties are approximatedusing appendix B. Instead of using a full B-H-curve it was approximated asa linear permeability with a set limit at 1.1 T. The five cases were:

1. Distance variation between magnets

2. Thickness variation of air pocket

3. Reversed direction of magnetization

4. Magnet exposed to external field from a torus core with gap

5. Demagnetizing factor N of different dimensions of magnets.

Cases 1-3

The first three simulations used three ferrites with the dimensions in mm:6x13x54 which formed a row. Separated by a distance of 5 mm by electricalsteel, the electrical steel was also enclosing the magnets and was large enough

28

to be considered infinite. Surrounding each ferrite there was also a 0.5 mmthick air pocket. The ferrites were all magnetized in the same direction,parallel to the row they formed. One measurement point was located in themiddle of the space between two magnets to see the magnetic flux densityin the steel. Another measurement point was located between the same twomagnets but at the height of the longest axis of the magnets, see figure 8.The middle magnet was measured for effective demagnetizing factor Neff foreach case.

Figure 8: Default setup for simulations of case 1-3. Three magnets was placed in electrical steel. Theelectrical steel is not shown in the figure, but perfectly encloses the magnets and stretches far enough tobe considered infinite. Each magnet was enclosed in a 0.5 mm thick air pocket between the magnet andthe electrical steel. Two points between two magnets were introduced to show local magnetic flux density,seen as two small black dots.

Case 4

The fourth simulation used an external magnetic field with increasing fluxdensity to show where in the magnet the demagnetization occurred first. Twomagnets were studied in case 4, one with the same dimensions as in cases 1-3and one with the dimensions 25x25x54 mm. The magnet studied was placedin a gap of a torus core of iron. A coil was winded around the left part of

29

the torus (see figure 9) to produce a magnetic field within it. This results ina relatively homogeneous magnet field over the magnet. See figure 9.

Figure 9: Simulation setup for case 4. A large iron torus with a coil marked in red, to produce a magneticfield. The magnet of interest is located in a small gap in the torus opposite the coil.

Case 5

The last case used a very simple setup. It consisted of a magnet in a large1 meter radius sphere of air and was used to determine the demagnetizingfactor N along the shortest axis in the magnet as that was the one usedfor magnetization. Two cuboid magnets with the same two dimensions usedin cases 1-3 and 4 as well as one ellipsoid with the dimensions 25x25x54were studied in this case. The ellipsoid magnet was included to show anydifference from its cuboid counterpart.

4.7 Market research of available ferrite gradesand their characteristics

By looking at what different ferrite traders were offering a general over-view of different grades were obtained. Further reading on their websitesgave an understanding on how the grade names work with respect to mag-netic factors such as BHmax, Hc etc. The standards that were chosen todescribe and presented were the American/British, the European, and the

30

Chinese. The traders found with the most information and which data is usedto present the standards were Eclipsemagnetics®[36] and Haofeng MagneticTechnology[37]. Using the calculation software Google Sheets™and Matlab®

a minor comparison were made of the chosen standards with respect to gradename vs. BHmax and Br vs. Hci. For easier comparisons the mean valuewas taken for Br and Hci. Using Google Sheets™three graphs were madeshowing BHmax against the grade name. Values for the examined ferritewere extracted from the data sheet which can be seen in appendix A.

31

5 Results and discussion

The following chapter presents the experimental results and the analyticmodel that was obtained. Then the results of the simulation is presented.Lastly, the market research of commercially available grades is presented. Adiscussion of each section throughout the chapter is given.

5.1 Experimental results

Figure 10 shows an example of the raw data obtained from a measurementat room temperature. The data is expressed in magnetization M and ap-plied field Ha(M-Ha-curve). Note how the distribution of data points areconcentrated around the knee in the second quadrant. Note also the lowconcentration of data points in the second half of the curve. The reasons forthis are explained in section 4.5.

Figure 10: Data point distribution of raw data at 27◦C. The measurements were concentrated to theknee in the second quadrant and diminished in the fourth quadrant.

To obtain the M-Hi-curve and the B-Hi-curve, Hi and B needed to be cal-

32

culated. B was calculated using equation 2 and Hi was calculated usingequations 1 and 3. To obtain the demagnetization factor N used in equation1 the slope of the M-Ha-curve at M=0 was calculated. N equals the inverseof this slope according to equation 7. As seen in table 1, N varied between∼0.16 and ∼0.23 for reasons unknown. This is unexpected since the shape ofthe sample remained constant throughout the experiments. Because of this,the average of N=0.1910 was used for all curves.

Table 1: The calculated values of the demagnetization factor at each temperature. The variation of thevalues are larger than expected.

T, ◦C N-40 0.1991-10 0.202127 0.162880 0.1672140 0.1858200 0.2291Av. 0.1910

Figures 11 and 12 shows the M-Hi and B-Hi-curves respectively. Note in thefigures how the two curves at lower temperatures(dashed lines) break fromthe general trend of the other curves. It was expected that Br should increasewith temperature. Also that the curves for the lower temperatures shouldhave the same slopes as the rest. This is likely caused by a rotation of thesample between measurements or because of thermal contraction during thecooling. A minor movement of the SSVT when extracting the heat-exchangerfrom the Dewar thermos could have caused the sample to rotate a bit. Sincethere was only time to do one experiment the measurements could not beredone. Further research could be made by redoing the experiment to seeif the deviation of the curves for lower temperature are due to experimentalerrors. To find the knee point of the demagnetization curve, equation 11needs the temperature dependence of Br, Hci, C1 and C2. To find C1 andC2, the model of the demagnetization curve, equation 9, needs Br, Hci andµr. From figure 11, Hci was obtained from the intersection of the Hi-axis,from figure 12, Br could be obtained from the intersection of the B-axis. µrcould be obtained from the slope of the B-H-curve using equation 4. The Br

and µr values at the temperatures -40◦C and -10◦C deviated from expected

33

Figure 11: Knee area of the M-Hi-curves. The curves at lower temperatures(dashed) break from thegeneral trend of the rest, this is interpreted as experimental errors.

values and was excluded when determining their temperature dependence,see figure 13. As seen in the figure, all three had a clear linear temperaturedependence. They are be examined further in section 5.1.

Figure 12: Knee area of the B-Hi-curves. The curves at lower temperatures(dashed) break from thegeneral trend of the rest, this is interpreted as experimental errors.

34

(a) Remanence Br vs. Temperature

(b) Intrinsic Coercivity Hci vs. Temper-ature

(c) Relative Permeability µr vs. Tem-perature

Figure 13: Br(a), Hci(b) and µr(c) plotted against temperature. The lower temperatures gave inconsist-ent values of Br and µr and are excluded from the line fitting. All three have a clear linear temperaturedependence.

Analytic model

Figure 14 shows equation 9 fitted to the B-H-curve at room temperature, seeappendix D for the fitting at the other temperatures. As seen in the figure,the curve fitted the data very well. The same can be said for the fitting atthe other temperatures.

35

Figure 14: Equation 9 fitted to measured B-H-curves at room temperature

The obtained curves gave the fitting parameters C1 and C2 at each temper-ature. Their temperature dependence are shown in figure 15. C1 showed alinear temperature dependence. C2 showed no temperature dependence butremained around a few percent above unity. However, it should be notedthat, if the deviation in Br and µr (see figures 13a and 13c) were also presentfor C1 and C2, and that the data points at lower temperatures ought to havebeen higher or excluded all together. The temperature dependence for C1could be interpreted as exponential or quadratic, and C2 having a highervalue. The measurement plan could have been done better. Maybe by meas-uring the sample from the Tmin to Tmax instead, would have produced lessdeviation for certain parameters. This could be a topic for further examin-ation but in this project all data points were included and the temperaturedependence for C1 was considered linear. A good thing about the develop-ment of the analytic model was that, if the temperature dependence of theparameters were inaccurate or that one wants to use it for a different per-manent magnet, one does not have to start from scratch. That is, equation11 is general.

The figures 13 and 15 show the temperature dependence of Br, Hci, C1, andC2. Unlike the others, C2 had no temperature dependence and was consideredto be constant with the value C2 = 1.039(11). The other three parameters

36

had a linear temperature dependence and their general forms were

Br(T ) = ABr +BBr · T [T], (17)C1(T ) = AC1 +BC1 · T [AT/m], (18)Hci(T ) = AHci

+BHci· T [A/m]. (19)

µr(T ) = Aµr +Bµr · T -. (20)(21)

The parameters of the equations above is shown in table 2.

(a) C1 show a linear temperature de-pendence.

(b) C2 shows no temperature depend-ence but remains around a few percentabove unity.

Figure 15: Temperature dependence of the curve fitting parameteres C1(a) and C2(b) in equation 11.

Table 2: The table shows the parameters of the linear temperature dependence of Br, C1, Hci and µr.

Function Intercept, A Slope, B R2

Br(T ) 0.614 [T] −7.25 · 10−4 [T/K] 0.9989C1(T ) 43.0 [AT/m] 6.78 [AT/Km] 0.8924Hci(T ) 1.11 · 105 [A/m] 652 [A/Km] 0.9927µr(T ) −1.18 · 10−4 - 1.1039 [K−1] 0.9714

Inserting equations 17-19 into equation 11 yields the general formula

Hk(T ) = · C1(T )k · (Br(T )) + C1(T )

C2·Hci(T )

− C2 ·Hci(T ). (22)

37

For example, to obtain the maximum acceptable internal field at T = 20◦C ifthe maximum acceptable amount of demagnetization is k = 20% you insertthe values in the table below into the equation. The table below showsthe values of the parameters at 20◦C. This yields the maximum acceptable

Table 3: The table shows the values of the parameters Br, C1 and Hci at 20◦C.

Parameter ValueBr(20◦C) 0.4017 [T]C1(20◦C) 2.03 · 103 [AT/m]Hci(20◦C) 3.02 · 105 [A/m]

internal field H0.2(20◦C) = −2.90 ·105 if the maximum acceptable amount ofdemagnetization is 20%. Figure 16 illustrates this calculation.

Figure 16: k stands for the acceptable percentage of demagnetization. For a specific temperature thisgives a value for the acceptable internal field Hi that gives a demagnetization of a certain percentage. Inthis figure an acceptable demagnetization is set to 20% which gives the internal field an acceptable valueof ∼0.3 MA/m.

As previously mentioned, the approximation is limited to |Hk| ≤ Hci. Thisis illustrated in figure 17. The maximum acceptable internal field is plottedagainst temperature for different amounts of acceptable demagnetization k.The yellow area represents the forbidden area |Hk| ≥ Hci. Within this area,

38

the equation does not apply and the maximum acceptable internal field isnot defined. As seen in the figure, this is mostly relevant at high amounts ofdemagnetization and at low temperatures. This limit can also be calculatedusing equation 12.

Figure 17: The maximum acceptable internal field Hi plotted against temperature for different amountsof maximum acceptable demagnetization k. The yellow area represents the area within which the curvesare not valid, |Hk| ≥ Hci. This is mostly relevant at high values of k and low temperatures.

A comparison of the data sheet seen in appendix A, and experimental valueswas made. The comparison is presented with figure 18 and table 4. As onemay predict, they were not identical. This may be due to their developmentbackground. Firstly, the accuracy and the measurement technique used tocreate the data sheet is unknown which could cause their difference. Secondlyour experiment was not done at 20◦C which is the temperature the datasheet is valid for. The figure and table contain interpolated values of theexperimental data using equations 2, 9 and the experimental temperaturedependence in table 2.

39

(a) Data sheet (DS) (b) Experimental data (Exp.)

Figure 18: TheM -H-curve is represented in red, The B-H-curve is represented in green and the BHmax-curve is represented in blue

Table 4: The table shows a comparison between the parameters given in the delivery data sheet(DS) andthe experimental data(Exp.), both at 20◦C.

Parameter DS Exp. Exp/DS (%)Br 0.4119 T 0.4017 T 97.5Bd 0.1984 T 0.2033 T 102Hd 166.3 kA/m 143.6 kA/m 86.4Hc 304.3 kA/m 269.4 kA/m 88.5Hci 324.5 kA/m 308.0 kA/m 94.9BHmax 33 kJ/m3 29 kJ/m3 87.9

As can be seen in the table above Br, Bd differed with a few percent whilethe rest of the parameters differed with 5 to 15 %.

5.2 Simulation results

The results from the simulations include the effect that varying parametershave on the magnetic flux density in the electrical steel between magnetsand effective demagnetizing factor in the magnets. It also includes models oflocal demagnetization in magnets placed in a demagnetizing field as well asdemagnetizing factor for different shapes of magnets.

40

Case 1

Results from case 1 are shown in figure 19. As the distance was increased,the flux density at both points in the electrical steel decreased, as seen infigure 19a. As the magnets moved further from the points, the magneticfields at the points got weaker. The effective demagnetizing factor, seen infigure 19b, increased because the magnetization effect the magnets had ontheir neighbors got weaker and tapered off.

(a) Effect on magnetic flux density inelectrical steel. Measured at the twosmall black dots as seen in figure 8

(b) Effect on Neff of the centered mag-net as seen in figure 8

Figure 19: Effect from varying distance between magnets. The added distance is filled with electricalsteel so air pockets around magnets are unaltered.

Case 2

The results show that the magnetic flux density in the electrical steel de-creased in both points, as seen in figure 20a, while Neff increased as the sizeof the air pockets increased, as seen in figure 20b. Air has a much lowerpermeability compared to the electrical steel and acted like an insulator. Airpockets can therefore be used to partially direct magnetic fields in a rotor.To get a stronger magnetic field in the electrical steel it is important have aslittle air as possible between the magnet and the electrical steel.

41

(a) Effect on magnetic flux density inelectrical steel. Measured at the twosmall black dots as seen in figure 8

(b) Effect on Neff of the centered mag-net as seen in figure 8

Figure 20: Effect air pocket size has on properties of the central magnet.When air pockets increase theamount of electrical steel between the magnets decrease so the distance between magnets remain the same.

Case 3

Table 5 shows the results from the case 3 simulation. Reversing the magnet-ization on the central magnet in the setup decreased B at the central pointbut increased it at the border point. The value at the central point was veryclose to zero because it was in the middle between two magnets both facingit, essentially canceling each other out locally. It is interesting that B washigher at the border point when the magnet was reversed than when it wasin-line with the others, this suggests that if you want a stronger magneticfield out to the sides you want the magnets alternating in direction of mag-netization. Neff was increased slightly from reversing the direction due tothe adjacent magnets’ fields act as demagnetizing fields.

Table 5: Table showing the effect on magnetic flux in the electrical steel and Neff from reversing thecentral magnet in figure 8.

B, Center point [T] B, Border point [T] Neff

Original 0.298 0.117 0.1330Reversed 0.000 0.533 0.1332

42

Case 4

The results of case 4 with an applied magnetic field of 0.1 T on the twomagnets can be seen in figure 21. As seen by the color grading, 21a had ahigher overall magnetization than 21b and the arrows showing the directionand size of the magnetic flux density more closely followed the direction of they-axis, which is the direction of the magnetization. The simulation indicatesthat the central region on surfaces perpendicular to the easy axis had thelowest magnetization. The magnetization was only calculated from the y-component of the magnetic flux density and in the corners the field divergedoutward as can be seen from the arrows in figure 21. It was expected thatthe corners of the magnet would demagnetize first and it is unclear if thesimulation is correct. The approximation that magnetization only occurs inthe easy axis may be interfering, but to simulate in other directions newVSM measurements would have to be made in an axis perpendicular to theeasy axis. This could be made in further studies. In a motor the magnetswould certainly not be exposed to a uniform magnetic field perpendicularto the easy axis and the resulting demagnetization will likely not appear inthe same way. In addition, the experiment could be done with the magnetssurrounded by electric steel in lieu of air. This, to further approximate theenvironment in the motor.

(a) Magnet with cubic cross section of25 × 25 mm.

(b) Magnet with rectangular cross sec-tion of 6 × 13 mm.

Figure 21: Magnets in the case 4 setup seen in figure 9 with applied demagnetizing field of 0.1 T.Arrows show direction and relative size of magnetic flux density B. Color range indicates strength ofmagnetization in two slices of each sample, one in the middle of the longest axis and one 1 mm inside thelargest surface of the magnets.

43

Case 5

The resulting demagnetizing factors of the different shapes are presentedin table 6. The similar results in the ellipsoid and cuboid with the samedimensions gives credibility to the approximation of a cuboid’s demagnetizingfactor with an ellipsoid’s of the same size. The lower N of the magnet withthe square cross section would be the reason why the overall magnetizationof that sample was higher than the one with rectangular cross section in case4.

Table 6: Demagnetizing factor N along shortest axis for different shapes of magnets.

Shape Dimensions [mm] NCuboid 6x13x54 0.621Cuboid 25x25x54 0.440Ellipsoid 25x25x54 0.457

5.3 Comparison of grades

The American standard is one of the earliest used for ferrite magnets. Thegrade name starts with the letter C and a following even number. Theletter C stands for Ceramic e.g. C5. The European standard is called DINIEC and starts with the two letters HF which stands for Hard Ferrite.[34]The following number is a rational number e.g. HF26/18. The numeratorrepresent a mean value of BHmax in SI-units, kJ/m3. The denominator is atenth of the mean value of the Hci, also in SI-units, As/m[34]. Example forclarification in equation 23.

HF26/18 :BHmax = 26− 27 [kJ/m3]Hci = 160− 190 [kA/m] ≈

≈ 26 [kJ/m3]/180 [kA/m]⇒ 26/18 (23)

The Chinese standard is the most common. As the other standard mentionedit starts with a letter and a following number. The letter Y equals theEuropean letter combination HF. Following number is a mean value ofBHmax

in cgs-units, MGOe e.g. Y28. Conversion to SI-units is given by 1 MGOe= 4π·10−2 kJ/m3. For both the Chinese and European standard, BHmax

increased with respect to the increasing number in the grade name. This

44

is not the case for the American standard. C5 has e.g. a higher BHmax

than C7. The Chinese standard, the European standard, and the Americanstandard can be seen in figures 22, 23, and 24 respectivly.

Figure 22: BHmax is plotted against the grade names for the Chinese standard with the examinedferrite named Ex.Mag. In this graph Y8T is excluded due to its very low value for BHmax.

45

Figure 23: BHmax is plotted against the grade names for the European standard with the examinedferrite named Ex.Mag.

Figure 24: BHmax is plotted against the grade names for the American standard with the examinedferrite named Ex.Mag.

46

With regards to BHmax the examined ferrite Ex.Mag resembles Y32H-1 toY34 in figure 22, HF32/17 to HF32/25 in figure 23, and C8B, C11, and C12in figure 24.

The other comparison was made in Matlab® where all of the chosen standardswere mapped with respect to Br and Hci, as seen in figures 25 and 26. Thesegraphs were made to easily see which grades correlates between standards.Some grades from the Chinese standard are removed due to having similarvalues. These can be seen in table 7.

Table 7: Table for the selection of removed and replaced grades for the Chinese standard.

Excluded Identical with Much alike Replaced withY28 x Y30Y26H-1 x Y26HY32H-1 x Y32Y28H-1 x Y30H-1Y30BH x Y27H

47

Figure 25: The graph shows a comparison of the Chinese and European standard with respect to Brand Hci. Grades with Br lower than 0.3 T or Hci lower than 0.16 MA/m are not shown. The Chinesestandard are represented with green stars with the name centered above the star. The European standardare represented with red stars with the name centered below the star. The black star with the nameEx.mag represent the actual examined magnet.

48

Figure 26: The graph shows a comparison of the Chinese and American standard with respect to Brand Hci. Grades with Br lower than 0.3 T or Hci lower than 0.16 MA/m are not shown. The Chinesestandard are represented with green stars with the name centered above the star. The American standardare represented with blue stars with the name centered below the star. The black star with the nameEx.mag represent the actual examined ferrite.

Ferrite magnet traders often equates C5 to Y30 and HF26/18[36][37]. Theyare not completely equal as can be seen in figure 25 and 26. There one canalso see that the examined ferrite Ex.Mag lays close to Y30H-1 and C12.

49

6 Conclusions

In this section the reports three research questions are answered.

Can an analytic model determine the point of irreversible demag-netization for temperatures between -40◦C to 200◦C for one specificgrade of ferrite?

The project concluded that the equation below can be used to determine themaximum acceptable internal field Hk for a maximum acceptable amount ofdemagnetization k.

Hk = C1

kBr + C1C2Hci

− C2Hci. (11)

Br is the remanence, Hci is the intrinsic coercivity and C1 and C2 are con-stants derived from the demagnetization curve. The equation is limited toHk > −Hci, further work could be done to extend the model to larger valuesof |Hk|. However, the results showed that this limit is only relevant for veryhigh acceptable demagnetization. Further work could be done on the con-stants C1 and C2 as well, to make them more accurate. C2 was experimentallydetermined to equal 1.039(11) and Br, C1, Hci and µr were experimentallydetermined to have a linear temperature dependence. The parameters oftheir linear equations are shown in table 2 below.

Table 2: The table shows the parameters of the linear temperature dependence of Br, C1, Hci and µr.

Function Intercept, A Slope, B R2

Br(T ) 0.614 [T] −7.25 · 10−4 [T/K] 0.9989C1(T ) 43.0 [AT/m] 6.78 [AT/Km] 0.8924Hci(T ) 1.11 · 105 [A/m] 652 [A/Km] 0.9927µr(T ) −1.18 · 10−4 - 1.1039 [K−1] 0.9714

How does the interaction of multiple samples of the examined fer-rites affect their overall performance and demagnetization?

Simulations of magnets in electrical steel showed that the size of the air pock-ets around the magnets is the most influential factor for the demagnetizingfactor in the magnet and flux density in the steel, it is essential that they arekept thin. If the purpose is to direct the magnetic field out perpendicular

50

to the magnetization for a similar case, the direction of the magnetizationshould alternate. The simulations of magnets in air and in a demagnetizingfield showed that the middle of the surfaces perpendicular to the magnets’easy axis were most easily demagnetized, while the corners of the magnetretained their magnetization well.

How does commercially available grades of ferrites compare withrespect to magnetic properties?

With respect to Br vs. Hci the experimentally examined ferrite (Ex.Mag)resembled Y30H-2 of the Chinese standard the most. C12 and HF26/30were the grades which resembles the examined ferrite the most from theirrespective standard but not as much as Y30H-2. In the case of BHmax,Ex.Mag still resembled the Chinese Y30H-2 and the American C12 but notthe European HF26/30. More specifically Ex.mag resembles Y32H-1 to Y34,HF32/17 to HF32/25 and C8B, C11 and C12 with respect to BHmax.

51

7 AcknowledgmentsAs a last chapter of the study, there are several people who have been very present in theproject and truly deserve an official written acknowledgement.

A big "Thank you!" to Björgvin Hjörvarsson who have lent us a huge amount of his timeand attention. All to listen to our wants and needs in terms of this project, in fact oureducation in general. As of this he contacted his former colleague, Peter Isberg, and madethis project happen. It has been truly fun, tricky, and inspiring.

We would really like to thank Klas Gunnarsson for being a great technical consultant andmentor in ways of providing us with both navigation through the discipline of magneticmaterials as well as laughter, and permission to use the coffee machine. Really lookingforward to the next semester when we will be navigated further into the jungle by takingthe course functional materials I.

The whole project has been reviewed by our business consultant, Helena Fornstedt. Thankyou for really being involved and continuously during the project giving us constructivefeedback in ways to keep us on track and do our utmost to achieve a good result.

Thank you, Daniel Hedlund for all the continuous help and feedback we have got duringthe project. You have always been a great resource, whether it has been finding the rightpeople to talk to, us having trouble understanding a part of the theory, or fixing theVSM when behaving as an exception to all known logical systems. We have always beenencouraged to ask for help, and in times of darkness, reminded of the importance of havingfun. As a final note on the topic Daniel, thanks for the ice cream!

And of course we would also like to thank Freddy Gyllensten and Peter Isberg for engagingus in this project. As said above, it has been very interesting and most definitely a valuableexperience.

When we found ourselves in times of trouble, Peter Svedlindh came to us. Speaking wordsof wisdom, ”What’s the fuss?”. Thank you for contributing to our project by kindly lendingus your expertise in magnetism and being willing to discuss unexpected matters. It hasbeen very helpful!

Many ”thank you” to Joakim Andersson, for allowing us to be a disturbance in youreveryday by asking the impossible. We have been thinking of a possible spin-off from thisproject as co-writers for a potential sequel of Mission Impossible, known as the Ferriteedition. Sincere thanks for your help!

Since simulation has been an area of uncharted waters, it has been great to receive helpfrom experienced and helpful people at the institution of electricity. Furthermore, thethesis provided by, and written by Stefan Sjökvist was used as a template on "writing athesis". Thank you Stefan Sjökvist and Juan Santiago!

For illustrating the feasibility of the project with the time management lecture, we wouldsincerely like to thank Jorge Brischetto. Due to this, we realized what we were up against,

52

and could make up a schedule to take turns in the chamber of screams and the cryingcorner. Once this essential piece of work was done, the project was pretty much manage-able.

53

References[1] Brysse K, Oreskes N, O’Reilly J, Oppenheimer M. Climate change prediction: erring

on the side of least drama? Global Environ. Change. 2013;23(1):327-337.

[2] United States Environmental Protection Agency [Internet]. Frequently Asked Ques-tions about global warming and climate change: Back to Basics. April 2009. Avail-able from: https://www3.epa.gov/climatechange/Downloads/ghgemissions/Climate_Basics.pdf

[3] TA Abeku, van Oortmarssen GJ, Borsboom G, de Vlas SJ and Habbema JDF. Spatialand temporal variations of malaria epidemic risk in Ethiopia: factors involved andimplications. Acta Tropica, Acta Trop. 2003;87:331-340.

[4] Asab A, Peterson PM, Shetler SG and Orli SS. Earlier plant flowering in spring as a re-sponse to global warming in the Washington, DC, area. Biodiversity and Conservation.2001;10:597.

[5] ACIA, Arctic Climate Impact Assessment. Cambridge University Press, Cambridge.2005, 1042 pp.

[6] Davis SJ and Caldeira K. Consumption-based accounting of CO2 emissions. Proceed-ings of the National Academy of Sciences of the United States of America, PNAS.June 2009:107(12):5687-5692.

[7] Bradford A. livescience [Internet]. [Place unknown]: Live Science;December 17 2014.Available from: http://www.livescience.com/37057-global-warming-effects.html

[8] REN21. ren21 [Internet]. San Francisco;REN21;2015. Available from: http://www.ren21.net/status-of-renewables/global-status-report/

[9] United States. U.S. Department of Energy. Center for Transportation Analysis: Trans-portation Energy Data Book. 30th ed. Washington DC: U.S. Department of En-ergy;2011.

[10] Industrial Efficiency Technology Database:Motor Systems [Internet]. Available from:http://ietd.iipnetwork.org/content/motor-systems

[11] Kramer MJ, McCallum RW, Anderson IA and Constantinides S. Prospects for Non-Rare Earth Permanent Magnets for Traction Motors and Generators. The Journal ofThe Minerals, Metals & Materials Society, JOM. July 2012;64(7):752-763.

[12] Goll D and Kronmüller H. High-performance permanent magnets. Naturwis-senschaften. 2000;87:423-438.

[13] United States. U.S. Geological Survey. Rare Earth Elements:Critical Resources forHigh Technology. USGS Fact Sheet 087-02; 2002.

[14] Hurst C, China’s Rare Earth Elements Industry: What Can the West Learn?, Journalof Energy Security, IAGS. March 2010.

54

[15] Sjökvist S, Demagnetization Studies on Permanent Magnets - Comparing FEM Simu-lations with Experiments, Division of Electricity Department of Engineering Sciences.Uppsala, June 2014

[16] Packey DJ, Kingsnorth D. The impact of unregulated ionic clay rare earth mining inChina. Resources Policy. June 2016;48:112-116.

[17] Rim KT, Koo KH, Park JS. Toxicological Evaluations of Rare Earths and TheirHealth Impacts to Workers: A Literature Review. Safety and Health at Work. March2013;4(1)12:26

[18] Sabbioni E, Pietra R, Gaglione P, Vocaturo G, Colombo F, Zanoni M, Rodi F.Long-term occupational risk of rare earth pneumoconiosis, a case report as invest-igated by neutron activation analysis. Science of the Total Environment. December1982;26(1):19-32

[19] ABB [Internet], [place unknown], [Cited 2014 April 3]. Available from: http://www.abb.com/cawp/seitp202/ad26393b09a61275c1257caf00217a16.aspx

[20] Liu C, Zhu J, Wang Y, Lei G, Guo Y and Liu X. 17th International Con-ference on Electrical Machines and Systems (ICEMS);2014 Oct. 22-25;Hangzhou,China:IEEE;2014. 397-400 p.

[21] Moosavia SS, Djerdira A et al. Demagnetization fault diagnosis in permanent mag-net synchronous motors: A review of the state-of-the-art. Journal of Magnetism andMagnetic Materials. October 2015;391(1):203-212.

[22] Gu Q, Jiang X, Cai L and Chen S. Irreversible demagnetization loss for electricmachines with arc-shaped permanent magnets stabilized in air. Electrical Engineering.December 1994;78(1):19-28.

[23] Shirk BT and Buessem WR, Temperature Dependence of Ms and K 1 of BaFe12O19and SrFe12O19 Single Crystals, Journal of Applied Physics. 1969;40(1294). Publishedonline Nov 17 2003. Available from: http://scitation.aip.org/content/aip/journal/jap/40/3/10.1063/1.1657636

[24] Nishio H and Yamamoto H, Effect of Important Factors on Temperature Vari-ation of Coercivity in Sr-La-Co High-Performance Sintered Ferrite Magnets, IEEETRANSACTIONS ON MAGNETICS [Internet]. October 2009;47(10). Available from:http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6027754&tag=1

[25] Maurya JC, Janrao PS, Datar AS, Kanhe NS, Bhoraskar SV, Mathe VL. Evidenceof domain wall pinning in aluminum substituted cobalt ferrites. Journal of Magnetismand Magnetic Materials. August 2016;412:165-171.

[26] Zheng H, Han M, Zheng L, Deng J, Zheng P, Wu Q, Deng L, Qin H. Magnetic proper-ties of hexagonal barium ferrite films on Pt/MgO(111) substrates annealed at differenttemperatures. Journal of Magnetism and Magnetic Materials. September 2016;413:255-29.

55

[27] Park J, Hong Y, Kim S et al. Maximum energy product at elevated temperatures forhexagonal strontium ferrite (SrFe12O19) magnet. Journal of Magnetism and MagneticMaterials. April 2014;355:1-5.

[28] Matsuuraa Y, Kitaib N, Hosokawab S and Hoshijimab J. Relation between the align-ment dependence of coercive force decrease ratio and the angular dependence of coer-cive force of ferrite magnets. Journal of Magnetism and Magnetic Materials. August2016;411:1-6.

[29] Ruoho S, Dlala E and Arkkio A. Comparison of Demagnetization Models for Finite-Element Analysis of Permanent-Magnet Synchronous Machines. IEEE TRANSAC-TIONS ON MAGNETICS. November 2007;43(11):3964-3968.

[30] Cullity BD, Graham CD, Introduction to Magnetic Materials, Wiley & Sons 2nd ed.New Jersey: Wiley & Sons; 2009. p:52,487.

[31] McCurrie RA, Ferromagnetic Materials Structure and Properties. San Diego: Aca-demic press inc.; 1994

[32] Bozorth RM, Ferromagnetism, Van Nostrans (1952); reprinted IEEE (1993). p:846-847.

[33] Long GJ, Grandjean F, editors. Supermagnets, Hard Magnetic Materials. In: Funda-mental Concepts and Units in Magnetism. Netherlands:Springer Science & BusinessMedia;2012. p. 44-46.

[34] supermagnete.de - Supermagnete online [Internet]. [Place unknown]. [Cited 2016-05-02]. Available from:https://www.supermagnete.de/eng/faq/What-does-Y35-stand-for

[35] Gyllensten F. Composition of examined ferrite [Internet]. Message to: VictorThorendal. 2016-04 [Cited 2016-05-02].

[36] Ferrite Magnets/Ceramic Magnets Datasheet [Internet]. England: Eclipse Magnet-ics. [Cited 2016-05-02]. Available from: http://www.eclipsemagnetics.com/media/wysiwyg/datasheets/magnet_materials_and_assemblies/ferrite_magnets-ceramic_magnets_datasheet_v1.pdf

[37] Magnetic Properties - Ferrite Magnet [Internet]. China: Haofeng Magnetic Techno-logy - Ferrite Magnet. [Cited 2016-05-02]. Available from: http://www.ferrite-magnet.org/Magnetic-Properties-p2.html

[38] Foner S. Further improvements in vibrating sample magnetometer sensitivity. Amer-ican Institute of Physic: Review of Scientific Instruments. 1975:45;1425. Publishedonline September 2008. Available from: http://scitation.aip.org/content/aip/journal/rsi/46/10/10.1063/1.1134021

[39] Foner S. The vibrating sample magnetometer: Experiences of a volunteer. AmericanInstitute of Physic: Review of Scientific Instruments. 1996:79;4740.

[40] Foner S. Very Low Frequency Integrating Vibrating Sample Magnetometer(VLFVSM) with High Differential Sensitivity in High dc Fields. American Instituteof Physic: Review of Scientific Instruments. 1968:39;171. Published online November

56

2003. Available from: http://scitation.aip.org/content/aip/journal/rsi/39/2/10.1063/1.1683309

[41] Foner S, Guertin RP. Application of a vibrating sample magnetometer to magneticmeasurements under hydrostatic pressure. American Institute of Physic: Review ofScientific Instruments. 1974:45;863. Available from: http://scitation.aip.org/content/aip/journal/rsi/45/6/10.1063/1.1686758

[42] Foner S. Versatile and Sensitive Vibrating-Sample Magnetometer. May 1959. [Cited2016-05-09]. Available from: http://scitation.aip.org/content/aip/journal/rsi/30/7/10.1063/1.1716679

[43] Hofmann P. Solid State Physics an introduction. Wiley-VCH 2nd ed. Germany:Wiley-VCH; 2015. p:81-82

[44] Aylward G, Findlay T. SI Chemical Data. Wiley & Sons 6th ed. Australia: Wiley &Sons; 2008.

57

8 Appendix

A Datasheet of examined ferrite

58

B Typical data for the electrical steel SURA®

M400-50A

59

C Composition of examined ferrite

60

D Lines fitted to data

Figure 27: Equation 9 fitted to measured B-H-curves at -40◦C(left) and -10◦C(right).

Figure 28: Equation 9 fitted to measured B-H-curves at 27◦C(left) and 80◦C(right).

61

Figure 29: Equation 9 fitted to measured B-H-curves at 140◦C(left) and 200◦C(right).

62