Demagnetization Characterization of Ferrites Independent Project in Materials Engineering

16041 Examensarbete 15 hp Juni 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 Abstract Demagnetization Characterization of Ferrites

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

Teknisk- naturvetenskaplig fakultet UTH-enheten Increasing prices of permanent magnets based on rare earth metals has driven an electric motor manufacturer to develop an efficient new series of synchronous Besöksadress: reluctance motor that uses ferrite magnets instead. Ferrite magnets may be Ångströmlaboratoriet Lägerhyddsvägen 1 demagnetized if placed in a strong magnetic field. Their ability to resist Hus 4, Plan 0 demagnetization is dependent on the temperature and this project aimed to determine the demagnetization characteristics of a strontium ferrite. It was examined Postadress: by a vibrating sample magnetometer in the temperature span -40C to 200C. It was Box 536 751 21 Uppsala found that an analytic model could describe the temperature dependence of the maximum acceptable internal field for an amount of demagnetization. This model Telefon: required the temperature dependence of Br, Hci in addition to two curve fitting 018 – 471 30 03 parameters C1 and C2. The model agreed very well with experimentally obtained

Telefax: demagnetization curves. Simulations using finite element method were carried out to 018 – 471 30 00 test effects of different setups around the magnets and their dimensions. The simulations found that the air gap between the magnet and the electrical steel should Hemsida: be minimized to yield the strongest magnetic flux density in the electrical steel. A http://www.teknat.uu.se/student market research was compiled to compare the ferrite studied with other grades with regards to BHmax, Br and Hci. With respect to Br vs. Hci the experimentally examined ferrite was found to resemble Y30H-2 of the Chinese standard to the greatest extent. C12 and HF26/30 are the grades that was found to resemble the examined ferrite the most from their respective standard but not as much as Y30H-2.

Handledare: Peter Isberg, Freddy Gyllensten Ämnesgranskare: Klas Gunnarsson, Helena Fornstedt Examinator: Enrico Baraldi ISSN: 1401-5773, UPTEC Q16 041 Demagnetization Characterization 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 metals has driven an electric motor manufacturer to develop an efficient new series of synchronous reluctance motor that uses ferrite magnets instead. Ferrite magnets may be demagnetized if placed in a strong magnetic field. Their ability to resist demagnetization is dependent on the temperature and this project aimed to determine the demagnetization characteristics of a strontium ferrite. It was examined by a vibrating sample magnetometer in the temperature span -40◦C to 200◦C. It was found that an analytic model could describe the temperature dependence of the maximum acceptable internal field for an amount of demagnetization. This model required the temperature dependence of Br, Hci in addition to two curve fitting parameters C1 and C2. The model agreed very well with experimentally obtained demagnetization curves. Simulations using finite element method were carried out to test effects of different setups around the magnets and their dimensions. The simulations found that the air gap between the magnet and the electrical steel should be minimized to yield the strongest magnetic flux density in the electrical steel. A market research was compiled to compare the ferrite studied with other grades with regards to BHmax, Br and Hci. With respect to Br vs. Hci the experimentally examined ferrite was found to resemble Y30H-2 of the Chinese standard to the greatest extent. C12 and HF26/30 are the grades that was found to resemble the examined ferrite the most from their respective standard but not as much as Y30H-2.

1 Contents

1 Introduction8 1.1 Background of the Demagnetization Characterization...... 9 1.2 Aim...... 9 1.3 Delimitations...... 10 1.4 Outline...... 11

2 Literature study 11

3 Theory 12 3.1 Demagnetizing ﬁeld...... 12 3.2 Crystal anisotropy...... 14 3.3 Demagnetization curve...... 15 Operating point[30, p. 478]...... 16 Maximum energy product...... 17 Irreversible demagnetization...... 18 3.4 Model for approximating maximum acceptable internal ﬁeld...... 19 3.5 Simulating magnets in electrical steel...... 20

4 Method 22 4.1 Methodological approach...... 22 4.2 Sample characteristics...... 22 4.3 Vibrating Sample Magnetometer...... 23 4.4 Sample preparation...... 26 Locating easy axis within the VSM sample cup...... 26 4.5 Execution of measurement...... 27 4.6 Simulations in COMSOL Multiphysics® ...... 28 Cases 1-3...... 28 Case 4...... 29 Case 5...... 30 4.7 Market research of available ferrite grades and their characteristics...... 30

5 Results and discussion 32 5.1 Experimental results...... 32

2 Analytic model...... 35 5.2 Simulation results...... 40 Case1...... 41 Case2...... 41 Case3...... 42 Case4...... 43 Case5...... 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 Figures

1 Demagnetizing factor N for different shapes...... 13 2 Demagnetizing factor N for a cuboid...... 14 3 Easy axis of a hard ferrite...... 14 4 The demagnetization curve...... 16 5 Irreversible demagnetization...... 18 6 Schematic layout of the VSM...... 25 7 Samples for measurement...... 26 8 Simulation setup for case 1-3...... 29 9 Simulation setup for case 4...... 30 ◦ 10 Output data, M-Ha-curve at 27 C...... 32 11 Experimental M-Hi-curves for different temperatures...... 34 12 Experimental B-Hi-curves for different temperatures..... 34 13 Temperature dependence of Br, Hci and µr ...... 35

3 14 Model fitted at data at 27◦C...... 36 15 Temperature dependence of C1 and C2 ...... 37 16 Example of analytic model for 20 % acceptable demagnetization 38 17 Valid interval for the analytic model for different percentage of demagnetization...... 39 18 Comparison with data sheet...... 40 19 Distance dependence of B and Neff in case 1...... 41 20 Air gap thickness dependence of B and Neff in case 2..... 42 21 Shape dependence of demagnetization in case 3...... 43 22 BHmax values for Chinese grades and the examined ferrite.. 45 23 BHmax values for European grades and the examined ferrite. 46 24 BHmax values for American grades and the examined ferrite. 46 25 Br and Hci comparison for Chinese and European grades... 48 26 Br and Hci comparison for Chinese and American grades... 49 27 Model fitted at data at -40◦C and -10◦C...... 61 28 Model fitted at data at 27◦C and 80◦C...... 61 29 Model fitted at data at 140◦C and 200◦C...... 62

List of Tables

1 Experimental demagnetizing factor N for diﬀerent temperatures 33 2 Linear temperature dependence of Br, C1, Hci and µr ..... 37 ◦ 3 Br, C1, Hci at 20 C...... 38 4 Comparison with data sheet...... 40 5 Direction of magnetization dependence for Br and Neff in case 3 42 6 Demagnetizing factor N in case 5...... 44 7 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]

AHci Intercept 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] new Br New remanence after demagnetization [T]

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

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

BHci Slope 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 - 3 RH Hall coefficient [m /C]

UH Hall voltage [V]

H¯d,eff Volume average of the eﬀective demagnetizing ﬁeld [A/m] M¯ 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] 3 ρCaO Density of CaO [g/cm ] 3 ρF e2O3 Density of Fe2O3 [g/cm ] 3 ρSrO Density of SrO [g/cm ] 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 Electrotechnical Commission 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 and important problems of the world. Recent trends suggest that current emissions may lead to severe environmental damage[1][2]. Contemporary climate change has affected physical and biological systems on all continents and in most oceans[3][4][5]. And the greenhouse gases from anthropogenic emissions are primarily responsible for global warming[1][6][7]. To deal with this problem, combustion of energy sources that produce greenhouse gas emissions, such as petroleum and natural gas, has to decrease. REN21’s Renewable 2015 Global Status Report states that renewable energy and energy efficiency is critical for addressing climate change[8]. In 2013, transportation accounted for 28% of the total energy consumption in the United States, of which 93 % comes from petroleum[9]. Thus, replacing petroleum with renewable energy, such as electricity from wind or sunlight, would help to decrease a part of the source of global warming. To do so, combustion motors would have to be substituted with electrical motors. It should also be mentioned that electric motor driven systems are estimated to be responsible for 29% of overall global electricity consumption according to the Industrial Efficiency Technology Database[10]. Considering this, a small gain in energy efficiency of an electric motor system, would lead to a huge gain in total global energy consumption. There are mainly three types of electrical motors of interest. The induction motor, permanent magnet motor and synchronous reluctance motor. With the development of high performance permanent magnets based on rare-earth elements (PM-RRE), it became possible to build a compact permanent magnet motor with high efficiency and power output with respect to size compared to its competitors[11][12]. Thus the permanent magnet motor expanded commercially and a lot of research was put into further de- veloping the high performing PM-RRE, neglecting conventional permanent magnets such as ferrites[11]. The PM-RRE are widely applied in electric motors today. However, the benefits of PM-REE may be clouded by its limited availability and high costs of their constituent rare earth elements. China controls approximately 95% of the market of these elements, and the costs were drastically increased due to decreased availability and high global demands since 2010[13][14][15]. The mining and refining process of these has 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 and lung related diseases[17][18]. To avoid the dependence of PM-REE in electric motors, further research is required to understand and improve conventional permanent magnets such as, in this case, ferrites.

1.1 Background of the Demagnetization Characterization

This study was launched to amplify the knowledge of ferrites at different temperatures. While present research is insufficient to fully describe its behavior, further research would enable manufacturers to decide if the efficiency of their electric motor that utilize ferrites could be increased. The ferrite is a small part of the complex system in an electric motor, yet an important part.Synchronous reluctance-motor (SynRM) is an electric motor that does not necessary utilize permanent magnets. A leading electric motor manufacturer developed a new series of SynRM that use ferrites to achieve an even higher efficiency than electric motors with PM-REE[19]. A ferrite is a conventional and widely applied permanent magnet which is cost effective and easily available due to its natural abundance, resulting in an ecologically and economically sustainable source[11][20]. The efficiency can be increased even more by further understanding the behavior of ferrites in the SynRM. Since initial and operating temperature may vary vastly, the temperature dependence of the magnetic properties must be known. Particularly the temperature dependence of irreversible demagnetization; a damaging and costly case due to a decrease of the efficiency and the electromagnetic torque in the machine[21]. Irreversible demagnetization is not specified as a magnetic material property. But it is sufficiently described by the magnets hysteresis 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 are two parameters that are useful to describe the temperature dependence of irreversible demagnetization[22].

1.2 Aim

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

9 and different surroundings with real and simulated experiments. It is fulfilled by answering the following questions: • Can an analytic model determine the point of irreversible demagnetization for temperatures between -40◦C to 200◦C for one specific grade of ferrite? • How does the interaction of multiple samples of the examined ferrites affect their overall performance and demagnetization? • How does commercially available grades of ferrites compare with respect to magnetic properties?

1.3 Delimitations

This chapter describes the boundaries set in this project. Presenting the demarcations chosen to achieve the aim in the time frame. Necessary delimitations are presented below: Experiment: Only one grade of ferrite is examined. It would yield information to determine if the analytic model is applicable on various grades, but due to the time required for the experimental equipment to measure one sample, time is insufficient. Experiment: It would also be interesting to examine one specific grade of ferrite provided from different manufacturers, to see if the manufacturing process affected the magnetic properties. However, it would require more measurements and has to be dismissed. Simulation: Due to no experience in the field of simulating by finite element method, simple setups are used in addition to approximations of the magnetic properties. Market research: All magnetic properties are not compared against. Excluding those that are not commonly used as figures of 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 by others scholars. Next, there is a theory section to explain the underlying physics of the results and its discussion. The method section is presented after the theory which explains the methodological approach and the performed experiments. With the theory and method sections accounted for, the results are presented and discussed. Finally, the results are summed up in the last section; conclusions.

2 Literature study

The literature study investigates if similar studies had been done that could be of use for this project. Several other scholars studied ferrites[23, 24, 25, 26, 27, 28, 29]. Three are considered interesting and further discussed below. The methodological approach for the literature study will be explained in section 4.1. In the article Maximum energy product at elevated temperatures for hexagonal strontium ferrite (SrF e12O19) magnet, the authors Jihoon Park, Yang-Ki Hong Seong-Gon Kim et al. calculated the temperature dependence of the saturation magnetization Ms by calculating the electronic structure of strontium ferrite. By the use of density functional theory and generalized gradient approximation. They found that Ms decrease as temperature increase. The authors presents a theoretical way of calculating the temperature dependence of Ms, which is an interesting material property towards the aim for this project. The article could be of use to predict the outcome of Ms. But understanding and applying the methods presented would exceed the scope of this project. In addition, it would also not be suﬃcient to obtain an analytic model to determine the point of irreversible demagnetization since more magnetic properties has to be considered. In the article Relation between the alignment dependence of coercive force decrease ratio and the angular dependence of coercive force of ferrites, the authors Y. Matsuuraa, N. Kitaib, S. Hosokawab and J. Hoshijimab investigated the temperature dependence of coercivity in addition to the angular dependence of coercivity. The dependence was investigated at temperatures 300 K, 333 K, 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 reason behind the behavior of the coercive force at different temperatures. The coercivity is an important material property to determine the point of irreversible demagnetization. However, the article does not evaluate temperatures below 0◦C and they did not produce a practical model, which is of interest in this project. In the article Comparison of Demagnetization Models for Finite-Element Analysis of Permanent-Magnet Synchronous Machines, the authors S. Ruoho, E. Dlala and A. Arkkio presents a few simple demagnetization models for finite element calculations. The Exponent Function Model presented in this article will be used as inspiration when constructing the analytic model. The model is applicable due to the fact that the parameters which describes it will be provided by the experiment. In addition, the shape of the model is similar to the curve that describes a magnetic compound. To sum up, the first two articles mainly investigate the physical reason behind the magnetic properties of ferrite. Utilizing methods beyond the scope of this project. No simple method based on experimental results to describe the demagnetization process of a ferrite were found. The third mentioned article 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 the experiment. And that the general shape of the model is similar to the curve that describes a ferrite.

3 Theory

This chapter presents a theoretical explanation of magnetically hard ferrites in sections 3.1 and 3.2, and how their magnetic properties characterize the shape of the hysteresis curve in section 3.3. The model for approximating the demagnetization curve and maximum acceptable internal ﬁeld will be explained in section 3.4. The section 3.5 will cover the theory needed to understand the results of the simulations.

3.1 Demagnetizing ﬁeld

As soon as a magnetic material is being magnetized and open magnetic circuit 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 magnetization. This demagnetizing field is defined as:

Hd = −NM (1)

Where M is the magnetization and N is the demagnetizing factor. N is dependent on direction and the sum of the x-,y-,z- components equals 1. It increases with decreasing distance between the end surfaces relative their size. In figure1 the demagnetizing factor N is presented for different geo- metries and directions For the shapes given in figure2, 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 and hard to demagnetize from. Nx will therefore become the easy axis in terms of shape anisotropy.

Figure 1: The figure shows the demagnetizing factor N for three different shapes. Even though N can not be calculated for the long rod nor the thin film, the distance between the end surfaces relative their size 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 properly fitting any geometry to the ellipsoid, N can be approximated by using the same geometric ratio. This is exemplified in figure2. The approximation in figure2 will be compared to the values given from the simulation explained in section 4.6, case4. To obtain N from an experimental graph, see section 3.3.

13 Figure 2: An ellipsoid and a cuboid. The demagnetizing factor N can be approximated for the cuboid by 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 strong magnetocrystalline anisotropy, an intrinsic property that tends to direct the magnetization along a particular crystallographic axis, the so-called easy axis. The cell has a hexagonal structure and the direction which is easy to magnetize and hard to demagnetize is along the ±[001] axis in ﬁgure3, which is called the crystal easy axis. The [100] and [010] directions are found equally hard to magnetize[30, p. 236]. Due to the cells tendency to crystallize with the [001] axis parallel against each other, the bulk possess an easy axis[30, p. 487]. In addition, an external magnetic ﬁeld during the manufacturing process 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 the magnetization along the easy axis. The magnitude of this field will always be larger than the demagnetizing field for a magnetically hard ferrite. Therefore, the resulting easy axis due to shape anisotropy and crystal anisotropy is always determined by the crystals easy axis[30, p. 238].

3.3 Demagnetization curve

A magnet is often characterized by its demagnetization curve; the second quadrant of its hysteresis curve. The curve describes how the materials magnetic ﬁeld strength vary with respect to an internal ﬁeld. It is obtained by plotting equation2: B(M,Hi) = µ0(M + Hi) (2)

Where B is the magnetic flux density, µ0 is the permeability in vacuum and M the magnetization of the material[30, p. 17]. The internal magnetic field strength Hi is defined in equation3.

Hi = Ha + Hd (3)

Ha is the measured applied field and Hd the demagnetizing field[30, p. 411] according to equation1. The remanence Br, the coercivity Hc and the relative permeability µr, which are three important magnetic material properties, can be extracted from the plot. The remanence is the remaining internal flux density in the material, after an applied field aligned all the magnetic moments and the applied field is removed. The coercivity measures the strength of 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 magnetization reversal and is always larger than Hc in magnitude. The relative permeability µr can be extracted from the slope of the plot with

∂B 1 µr = . (4) µ0 ∂Hi Hi=0

Hc and Br are shown relative to the demagnetization curve in figure4.A region of the demagnetization curve is defined as the knee which is the essential part for the model described in section 3.4. The region can be seen in figure4.

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

To obtain N from experimental results equations1 and3 are used,

Hi = Ha − NM. (5)

This equation is then diﬀerentiated with respect to M to

∂Hi ∂Ha = − N. (6) ∂M ∂M

∂Hi At M = 0, ∂M ≈ 0 for permanent magnets. This yields

∂H a N = . (7) ∂M M=0

Operating point[30, p. 478]

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

16 field would be produced. The free poles are what causes the demagnetizing field Hd. This field causes the produced field to be lower than the remanence Br. The actual produced field from the magnet without any applied field is given by the static operating point P which can be calculated given the demagnetization curve or the dimensions of the magnet. Initially, Ha = 0 which according with equations1,2 and3 yields:

Hd 1 Bd = µ0(M + Hd) = µ0 − + Hd = −µ0 − 1 Hd N N Bd 1 ⇒ − = − 1 (8) µ0Hd N

1 The slope of the line OP (known as the load line) in figure4 is given by N −1 which can be obtained by the result of equation8. The static operating point P is given by where OP intersects with the B-H curve. Since N depends purely of the geometry, the static operating point can be chosen properly by choosing the dimensions of the magnet. The static operating point also tells which path is to be taken in the B-H curve when the magnet is exposed to an applied field. In a sense, it sets the starting point in the B-H curve. For example, when a positive applied field is turned on, the operating point P moves upwards in figure4. When a negative applied field is turned on P moves downwards.

Maximum energy product

The performance of a permanent magnet is often specified by its value of the maximum energy product (BHmax). It is a measure of the stored energy per volume a magnetic material produces[31, p. 193]. Magnetically hard ferrite −3 has considerably lower BHmax compared to PM-REE (∼ 30 kJm and ∼ 400 −3 kJm , respectively). BHmax is shown qualitatively as point S in figure4. Highest operating efficiency (from the perspective of a magnet) is said to be obtained when the operating point coincides with BHmax. Due to the fact that a given volume of magnetic material would produce the strongest magnetic flux at that point[33]. For the grade of ferrite which is examined −1 in this experiment, with Hd = 166.3 kAm and Bd = 0.1984 T[35], the operating 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 magnitude if one or several magnetic moments deviates from the direction of magnetization M. A deviation can be caused by an applied field. An applied field that causes magnetic moments to divert from their direction is said to be reversible, if the magnetic moments are re-diverted to the direction of magnetization after the field is switched off. This occurs if the magnet operates at the linear region of its demagnetization curve. In the same way, it is said to be irreversible if the magnetic moments are not re-diverted to the direction of magnetization after the field is switched off. This occurs if the magnet operates between two regions with different slopes. Hence, operating where ∆B/∆Hi changes. The point where the two regions are distinguished at is exemplified in figure5 as point c[30, p. 483-484]. The figure also explains the 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 irreversible demagnetization. The arrows to the right represent the direction of magnetic moments. Firstly, Br is obtained. Secondly, an applied field Ha moves the operating point from q to m, passing the linear endpoint c. 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 magnetic moments are diverted as seen in 2. Thus decreasing Br with ∆Br. The demagnetization is irreversible since the previous remanence Br can only be obtained by saturating the magnet again to align all the magnetic moments.

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

3.4 Model for approximating maximum acceptable internal ﬁeld

To find the location of the knee, an approximation of the demagnetization curve was made to find the parameters it depends on. As can be seen in figure 4, the curves are linear until they become close to vertical at Hci. Based on this and inspiration from a similar approximation authored by S. Ruoho et. al.[29] equation9:

C1 C1 B = Br + µ0µrH − + (9) C2Hci + H C2Hci

Where Br is the remanent magnetic flux density, µ0 is the permeability in vacuum, µr is the relative permeability, B and H are the magnetic flux density and the internal magnetic field, respectively. C1 and C2 are two curve fitting parameters; C1 sets the sharpness of the knee and C2 is a few percent 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-curve before permanent demagnetization. The third term could be interpreted as a demagnetization term, that is, the deviation from the linear behavior, see figure4. The fourth term is a correction term which ensures that B = Br at H = 0. When a field Hk has demagnetized the magnet to an amount k and new the internal field has returned to zero, the new remanence Br will be

new C1 C1 Br = Br − + . (10) C2Hci + Hk C2Hci

new If Br = Br(1 − k) the magnetic ﬁeld Hk is C H 1 − C H . k = C1 2 ci (11) kBr + C2Hci

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 be examined in the experiment. The ﬁrst term of equation 11 represents the distance between the knee point and Hci(or rather, a value slightly larger than Hci) and the second term is the value slightly larger than Hci. This means that the second term will have the most overall impact on the value of Hk. It is important to note, as mentioned earlier, that the approximation is only valid for magnetic ﬁelds weaker than Hci. This means that the maximum amount of demagnetization kmax this approximation can be used to calculate at a certain temperature is obtained from equation 11 by setting Hk = −Hci;

C1 C1 kmax = − . (12) BrHci(C2 − 1) BrC2Hci The model described above should be valid for any hard ferrite. After a temperature dependence of all parameters are inserted the model can be used to predict the maximum acceptable internal ﬁeld Hi for a maximum acceptable amount of demagnetization k.

3.5 Simulating magnets in electrical steel

The program chosen for the simulations handles most of the physics without user 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 the model such as permeability and the remanence. In the magnet relevant to this study the direction of the magnetization would be 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 the fields produced by the magnets will be partly pass through their neighboring magnets, they will influence the fields inside them. When placed in electrical steel the demagnetizing field inside a magnet will be weaker than in air because the magnetic circuit is partly closed, and thus the operating point is moved upward[30, p. 483-484]. The sum of the magnet’s own demagnetizing field in a specific environment and the fields caused by their neighbors could be 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 equation1 with

H¯d,eff Neff = − (13) M¯ where H¯d,eff and M¯ are volume averages of the effective demagnetizing field and the magnetization. The effective demagnetization factor depends in part on the field produced by the neighboring magnets, and since the produced field is temperature dependent, Neff is also temperature dependent.

21 4 Method

The section explains the methodological approach and characteristics of the examined ferrite, followed by how the theory sections 3.2, 3.1, 3.3 and 3.3 connects to the experiment. It also gives a detailed explanation of the experimental equipment, the preparation of the sample and how the easy axis was identiﬁed within the experimental equipment. The approach for simulating the various cases is presented. Finally, it is presented how data for the market research was compiled. 4.1 Methodological approach

The measurement technique that was used for the experiment was developed by 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 of magnetic compounds. The use of the machine was led and supervised by an experienced user. Simulations were done by finite element method in the software COMSOL Multiphysics®. There exists no uncertainty regard- ing the credibility of the software. The uncertainty lies at the amount of experience the user possess to correctly transmit a physical problem to the software. In addition, it also depends on the amount of knowledge the user has in the field of electromagnetism to decide the credibility of the simulated case. It was handled by simulating simple setups, so that the outcome of the simulation could be discussed by basic knowledge in the field of electromagnetism. Information for comparing commercially available grades was gathered by visiting websites of ferrite traders. The method is straight forward and trustworthiness was confirmed by plausibility checks among the traders.

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 appendixC) was examined. As explained later, the volume of the sample is needed to calculate the magnetization. To ﬁnd the volume, the density of the material ρ was

22 approximated with

ρ = 0.85 · ρF e2O3 + 0.135 · ρSrO + 0.008 · ρCaO, (14) where ρX is the density of material X. The densities obtained from SI Chem- 3 3 3 ical Data[44] were 5.2 g/cm , 4.7 g/cm and 3.3 g/cm for Fe2O3, SrO, and CaO, respectively. Using these densities and the equation above, the total density was calculated to ρ = 5.08 g/cm3. Considering the theory presented in section 3.2, the direction of the easy axis was given by the ferrite provider. The operating point coincides with BHmax if the demagnetizing factor N equals 0.513, according to the theory presented in sections 3.1, 3.3 and 3.3. Where the purpose is to gain a high operating efficiency. A demagnetizing factor of N = 0.513 is hard to obtain accurately. However, the higher the ratio between y:x or z:x (see figure2) the better the accuracy. The experimental equipment limits the allowed maximum size of the sample to 2.65x2.65x9 mm. It would otherwise not fit the equipment. If the 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 well as how the hysteresis curves are obtained. Vibrating sample magnetometer (VSM) is a measurement technique developed by Simon Foner[42]. The measurement technique is based upon that a vibrating magnetic sample creates flux change in coils which induces a voltage. This voltage is proportional to the magnetic moment of the sample. A schematic sketch of the VSM is shown in figure6. Initially a single point calibration with a Ni sphere with known magnetic moment and mass is done to get the correlation between applied field H and magnetic moment µ. The Hall probe located between the pick-up coils is used to measure the applied field, Ha. The Hall probe contains a thin conducting plate which a current flows through. As the conducting plate is exposed to the magnetic field the following expression holds[43].

µ0Ha UH EH = = (15) RH j b

Where the electric ﬁeld EH is produced by the current density j according to the 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 orthogonal to the conducting plate surface and RH is known for the conducting plate. The voltage UH can be translated to applied magnetic flux density BA thus one can get the magnetic moment µ due to the single point calibration. The Hall 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 on a ceramic piece or quartz rod by glue, the ceramic piece is screwed onto a sample rod while the quartz rod is complete by itself. The sample rod is attached to a membrane. The membrane is driven by an AC source with the frequency of 82 Hz. A lock-in amplifier prevents registration of any unwanted noise. The applied magnetic field from the electromagnets can be adjusted in both strength and direction. Eliminating induced currents in the pick-up coils from the applied field is done by winding the connected coils reversely. The winding does not affect the induced current since the vibrating sample creates an in-homogeneous field relative to the pick-up coils. The hysteresis curve 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 then be recalculated to B vs. H using equation2 with the total magnetization as in equation 16.The mass m of the sample is presented in section 4.4 and the density ρ is presented in section 4.2. µ · ρ M = ⇒ M, [emu/cm3] = 103[A/m] (16) m For measurements at different temperatures, a Dewar thermos, a heat-exchanger, an oven known as single state variable temperature (SSVT), and the gases ◦ Ar and N2 are used. For measurements below 27 C, N2 at room temperature is 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 is then transported to the SSVT through an isolated vacuum tube where it is heated up to the desired temperature. Measurement at or above 27◦C works similarly. The heat-exchanger is taken out of the Dewar thermos and the N2 gas is exchanged with Ar gas at room temperature. The temperatures can be set and controlled with the software IDEAS-VSM which can then create hysteresis curves for the set temperature.

24 Figure 6: Schematic layout of the VSM and description of components. 1: Computer with software IDEAS-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 the magnetic sample in place.

25 4.4 Sample preparation

A ferrite is hard to process due to its brittleness. Initially, the samples would be prepared by diamond blade cutting to yield precise dimensions and consistent shapes. However, due to the properties of the material, this was not viable. Hence, a more simplistic method was initiated to the cost of the dimensional accuracy. The samples were prepared by fastening a ferrite of size 6x13x54 mm to a vice, followed by several hits to the small end surface with a hard narrow tool. By repeatedly hitting the surface, small chips were obtained. The process was repeated until a sample with a high ratio between short and long axis was obtained. The sample can be seen in ﬁgure7. The chosen sample was weighed at 8.57 mg with an analytical balance. The mass was used to get the total magnetization M of the sample. Seen in section 4.3.

(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 contained that 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 so that 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 sample was subjected to a constant applied field, while simultaneously rotating the sample cup. According to the theory presented in section 3.2, strongest M was obtained when the easy axis is parallel with the applied field. When a peak in M was measured, the easy axis was located and the rotation shut off.

4.5 Execution of measurement

The measurements were planned over three days where the following temperatures 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 made to identify the approximate region of the knee. Thereafter the program was adjusted to increase the amount of data points around the knee. This generated a hysteresis curve with focus on the shape of the knee. To begin the measurements, the sample had to be attached to the VSM. There were several ways to attach the sample, each method with different beneficial factors. The first two methods failed at different states, whereas the third one was successful. 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 for the temperatures -40◦C, -10◦C and 27◦C. Although approaching the next temperature (80◦C) the threads were partially pulverized, making the ceramic piece no longer tightly screwed in so the sample could rotate and therefore making the result inconsistent. 2. Instead of the porous ceramic rod, a rod of quartz was used in combination with the silver glue. This was unsuccessful and the glue loosened after which the sample detached. From this another sample was attached to the rod, although with a different glue, supposed to be consistent to 1100◦C. This generated values without significance as the glue was magnetically active in itself. 3. The third option was to place the sample in the sample cup which was attached 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 had a breakdown whereafter it had to be rebooted which meant resetting the angle of the sample. After the VSM was back up running, the ﬁnal two temperatures were measured and the experiment was completed.

4.6 Simulations in COMSOL Multiphysics®

Five different cases of interest were studied. Cases1,2, and3 had the same setup but with different parameters varied to show their effect on the overall performance of the magnets. Case4 and5 were carried out using their own setups. Case 5 was added to show the demagnetizing factor, N, due to the adversity during the sample preparation which caused an inability to perform measurements of different dimensions of magnets. The magnetization of the magnets was assumed to be 0 in the z- and x-axis because the y-axis is supposed to be the magnets’ easy axis. From the experimental result a curve could be created of magnetization against magnetic flux density. This M- B-curve was set as the magnetization of the easy axis on the magnets. The electrical steel used in the simulation cases 1-3 was based on an electrical steel called SURA® M400-50A. Its electrical properties are approximated using appendixB. Instead of using a full B-H-curve it was approximated as a 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 electrical steel, 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 mm thick air pocket. The ferrites were all magnetized in the same direction, parallel to the row they formed. One measurement point was located in the middle of the space between two magnets to see the magnetic flux density in the steel. Another measurement point was located between the same two magnets but at the height of the longest axis of the magnets, see figure8. The middle magnet was measured for effective demagnetizing factor Neff for each case.

Figure 8: Default setup for simulations of case 1-3. Three magnets was placed in electrical steel. The electrical steel is not shown in the figure, but perfectly encloses the magnets and stretches far enough to be considered infinite. Each magnet was enclosed in a 0.5 mm thick air pocket between the magnet and the 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 flux density to show where in the magnet the demagnetization occurred first. Two magnets were studied in case 4, one with the same dimensions as in cases 1-3 and one with the dimensions 25x25x54 mm. The magnet studied was placed in a gap of a torus core of iron. A coil was winded around the left part of

29 the torus (see figure9) to produce a magnetic field within it. This results in a relatively homogeneous magnet field over the magnet. See figure9.

Figure 9: Simulation setup for case 4. A large iron torus with a coil marked in red, to produce a magnetic ﬁeld. 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 large 1 meter radius sphere of air and was used to determine the demagnetizing factor N along the shortest axis in the magnet as that was the one used for magnetization. Two cuboid magnets with the same two dimensions used in cases 1-3 and 4 as well as one ellipsoid with the dimensions 25x25x54 were studied in this case. The ellipsoid magnet was included to show any diﬀerence from its cuboid counterpart.

4.7 Market research of available ferrite grades and their characteristics

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

30 Chinese. The traders found with the most information and which data is used to present the standards were Eclipsemagnetics®[36] and Haofeng Magnetic Technology[37]. Using the calculation software Google Sheets™and Matlab® a minor comparison were made of the chosen standards with respect to grade name vs. BHmax and Br vs. Hci. For easier comparisons the mean value was taken for Br and Hci. Using Google Sheets™three graphs were made showing BHmax against the grade name. Values for the examined ferrite were extracted from the data sheet which can be seen in appendixA.

31 5 Results and discussion

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

5.1 Experimental results

Figure 10 shows an example of the raw data obtained from a measurement at room temperature. The data is expressed in magnetization M and applied ﬁeld Ha(M-Ha-curve). Note how the distribution of data points are concentrated around the knee in the second quadrant. Note also the low concentration of data points in the second half of the curve. The reasons for this are explained in section 4.5.

Figure 10: Data point distribution of raw data at 27◦C. The measurements were concentrated to the knee 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 equation2 and Hi was calculated using equations1 and3. To obtain the demagnetization factor N used in equation 1 the slope of the M-Ha-curve at M=0 was calculated. N equals the inverse of this slope according to equation7. As seen in table1, N varied between ∼0.16 and ∼0.23 for reasons unknown. This is unexpected since the shape of the 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 the values are larger than expected.

T, ◦C N -40 0.1991 -10 0.2021 27 0.1628 80 0.1672 140 0.1858 200 0.2291 Av. 0.1910

Figures 11 and 12 shows the M-Hi and B-Hi-curves respectively. Note in the figures how the two curves at lower temperatures(dashed lines) break from the general trend of the other curves. It was expected that Br should increase with temperature. Also that the curves for the lower temperatures should have the same slopes as the rest. This is likely caused by a rotation of the sample between measurements or because of thermal contraction during the cooling. A minor movement of the SSVT when extracting the heat-exchanger from the Dewar thermos could have caused the sample to rotate a bit. Since there was only time to do one experiment the measurements could not be redone. Further research could be made by redoing the experiment to see if the deviation of the curves for lower temperature are due to experimental errors. To find the knee point of the demagnetization curve, equation 11 needs the temperature dependence of Br, Hci, C1 and C2. To find C1 and C2, the model of the demagnetization curve, equation9, needs Br, Hci and µr. 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. µr could be obtained from the slope of the B-H-curve using equation4. 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 the general 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 temperature dependence. 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 the general trend of the rest, this is interpreted as experimental errors.

34 (a) Remanence Br vs. Temperature

(b) Intrinsic Coercivity Hci vs. Temper- (c) Relative Permeability µr vs. Tem- ature perature

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

Analytic model

Figure 14 shows equation9 fitted to the B-H-curve at room temperature, see appendixD 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 at the other temperatures.

35 Figure 14: Equation9 ﬁtted to measured B-H-curves at room temperature

The obtained curves gave the fitting parameters C1 and C2 at each temperature. Their temperature dependence are shown in figure 15. C1 showed a linear temperature dependence. C2 showed no temperature dependence but remained around a few percent above unity. However, it should be noted that, if the deviation in Br and µr (see figures 13a and 13c) were also present for C1 and C2, and that the data points at lower temperatures ought to have been higher or excluded all together. The temperature dependence for C1 could be interpreted as exponential or quadratic, and C2 having a higher value. The measurement plan could have been done better. Maybe by measuring the sample from the Tmin to Tmax instead, would have produced less deviation for certain parameters. This could be a topic for further examin- ation but in this project all data points were included and the temperature dependence for C1 was considered linear. A good thing about the development of the analytic model was that, if the temperature dependence of the parameters were inaccurate or that one wants to use it for a different permanent magnet, one does not have to start from scratch. That is, equation 11 is general.

The ﬁgures 13 and 15 show the temperature dependence of Br, Hci, C1, and C2. Unlike the others, C2 had no temperature dependence and was considered to 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 table2.

(a) C1 show a linear temperature de- (b) C2 shows no temperature depend- pendence. ence but remains around a few percent above 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 −4 Br(T ) 0.614 [T] −7.25 · 10 [T/K] 0.9989 C1(T ) 43.0 [AT/m] 6.78 [AT/Km] 0.8924 5 Hci(T ) 1.11 · 10 [A/m] 652 [A/Km] 0.9927 −4 −1 µr(T ) −1.18 · 10 - 1.1039 [K ] 0.9714

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

C1(T ) Hk(T ) = · − C2 · Hci(T ). (22) C1(T ) k · (Br(T )) + C2·Hci(T )

37 For example, to obtain the maximum acceptable internal ﬁeld at T = 20◦C if the maximum acceptable amount of demagnetization is k = 20% you insert the values in the table below into the equation. The table below shows the 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 Value ◦ Br(20 C) 0.4017 [T] ◦ 3 C1(20 C) 2.03 · 10 [AT/m] ◦ 5 Hci(20 C) 3.02 · 10 [A/m]

◦ 5 internal ﬁeld H0.2(20 C) = −2.90 · 10 if the maximum acceptable amount of demagnetization is 20%. Figure 16 illustrates this calculation.

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

As previously mentioned, the approximation is limited to |Hk| ≤ Hci. This is illustrated in figure 17. The maximum acceptable internal field is plotted against 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 is not defined. As seen in the figure, this is mostly relevant at high amounts of demagnetization and at low temperatures. This limit can also be calculated using equation 12.

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

A comparison of the data sheet seen in appendixA, and experimental values was made. The comparison is presented with figure 18 and table4. As one may predict, they were not identical. This may be due to their development background. Firstly, the accuracy and the measurement technique used to create the data sheet is unknown which could cause their difference. Secondly our experiment was not done at 20◦C which is the temperature the data sheet is valid for. The figure and table contain interpolated values of the experimental data using equations2,9 and the experimental temperature dependence in table2.

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

Figure 18: The M-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) and the experimental data(Exp.), both at 20◦C.

Parameter DS Exp. Exp/DS (%) Br 0.4119 T 0.4017 T 97.5 Bd 0.1984 T 0.2033 T 102 Hd 166.3 kA/m 143.6 kA/m 86.4 Hc 304.3 kA/m 269.4 kA/m 88.5 Hci 324.5 kA/m 308.0 kA/m 94.9 3 3 BHmax 33 kJ/m 29 kJ/m 87.9

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

5.2 Simulation results

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

40 Case1

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 in figure 19a. As the magnets moved further from the points, the magnetic fields at the points got weaker. The effective demagnetizing factor, seen in figure 19b, increased because the magnetization effect the magnets had on their neighbors got weaker and tapered off.

(a) Effect on magnetic flux density in (b) Effect on Neff of the centered mag- electrical steel. Measured at the two net as seen in figure8 small black dots as seen in figure8

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

Case2

The results show that the magnetic flux density in the electrical steel decreased in both points, as seen in figure 20a, while Neff increased as the size of the air pockets increased, as seen in figure 20b. Air has a much lower permeability compared to the electrical steel and acted like an insulator. Air pockets 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 as little air as possible between the magnet and the electrical steel.

41 (a) Effect on magnetic flux density in (b) Effect on Neff of the centered mag- electrical steel. Measured at the two net as seen in figure8 small black dots as seen in figure8

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

Case3

Table5 shows the results from the case 3 simulation. Reversing the magnetization on the central magnet in the setup decreased B at the central point but increased it at the border point. The value at the central point was very close to zero because it was in the middle between two magnets both facing it, essentially canceling each other out locally. It is interesting that B was higher at the border point when the magnet was reversed than when it was in-line with the others, this suggests that if you want a stronger magnetic field out to the sides you want the magnets alternating in direction of magnetization. Neff was increased slightly from reversing the direction due to the adjacent magnets’ fields act as demagnetizing fields.

Table 5: Table showing the effect on magnetic flux in the electrical steel and Neff from reversing the central magnet in figure8.

B, Center point [T] B, Border point [T] Neff Original 0.298 0.117 0.1330 Reversed 0.000 0.533 0.1332

42 Case4

The results of case4 with an applied magnetic field of 0.1 T on the two magnets can be seen in figure 21. As seen by the color grading, 21a had a higher overall magnetization than 21b and the arrows showing the direction and size of the magnetic flux density more closely followed the direction of the y-axis, which is the direction of the magnetization. The simulation indicates that the central region on surfaces perpendicular to the easy axis had the lowest magnetization. The magnetization was only calculated from the y- component of the magnetic flux density and in the corners the field diverged outward as can be seen from the arrows in figure 21. It was expected that the corners of the magnet would demagnetize first and it is unclear if the simulation is correct. The approximation that magnetization only occurs in the easy axis may be interfering, but to simulate in other directions new VSM measurements would have to be made in an axis perpendicular to the easy axis. This could be made in further studies. In a motor the magnets would certainly not be exposed to a uniform magnetic field perpendicular to the easy axis and the resulting demagnetization will likely not appear in the same way. In addition, the experiment could be done with the magnets surrounded by electric steel in lieu of air. This, to further approximate the environment in the motor.

(a) Magnet with cubic cross section of (b) Magnet with rectangular cross sec- 25 × 25 mm. tion of 6 × 13 mm.

Figure 21: Magnets in the case 4 setup seen in figure9 with applied demagnetizing field of 0.1 T. Arrows show direction and relative size of magnetic flux density B. Color range indicates strength of magnetization in two slices of each sample, one in the middle of the longest axis and one 1 mm inside the largest surface of the magnets.

43 Case5

The resulting demagnetizing factors of the diﬀerent shapes are presented in table6. The similar results in the ellipsoid and cuboid with the same dimensions gives credibility to the approximation of a cuboid’s demagnetizing factor with an ellipsoid’s of the same size. The lower N of the magnet with the square cross section would be the reason why the overall magnetization of that sample was higher than the one with rectangular cross section in case 4.

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

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

5.3 Comparison of grades

The American standard is one of the earliest used for ferrite magnets. The grade name starts with the letter C and a following even number. The letter C stands for Ceramic e.g. C5. The European standard is called DIN IEC 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 numerator 3 represent a mean value of BHmax in SI-units, kJ/m . The denominator is a tenth of the mean value of the Hci, also in SI-units, As/m[34]. Example for clariﬁcation in equation 23.

3 BHmax = 26 − 27 [kJ/m ] HF 26/18 : ≈ 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 mentioned it starts with a letter and a following number. The letter Y equals the European letter combination HF. Following number is a mean value of BHmax in cgs-units, MGOe e.g. Y28. Conversion to SI-units is given by 1 MGOe −2 3 = 4π·10 kJ/m . 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 American standard 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 examined ferrite 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 examined ferrite named Ex.Mag.

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

46 With regards to BHmax the examined ferrite Ex.Mag resembles Y32H-1 to Y34 in figure 22, HF32/17 to HF32/25 in figure 23, and C8B, C11, and C12 in figure 24. The other comparison was made in Matlab® where all of the chosen standards were mapped with respect to Br and Hci, as seen in figures 25 and 26. These graphs were made to easily see which grades correlates between standards. Some grades from the Chinese standard are removed due to having similar values. These can be seen in table7.

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

Excluded Identical with Much alike Replaced with Y28 x Y30 Y26H-1 x Y26H Y32H-1 x Y32 Y28H-1 x Y30H-1 Y30BH x Y27H

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

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

Ferrite magnet traders often equates C5 to Y30 and HF26/18[36][37]. They are not completely equal as can be seen in ﬁgure 25 and 26. There one can also 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 demagnetization for temperatures between -40◦C to 200◦C for one speciﬁc grade of ferrite? The project concluded that the equation below can be used to determine the maximum acceptable internal ﬁeld Hk for a maximum acceptable amount of demagnetization k. C H 1 − C H . k = C1 2 ci (11) kBr + C2Hci

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 to Hk > −Hci, further work could be done to extend the model to larger values of |Hk|. However, the results showed that this limit is only relevant for very high acceptable demagnetization. Further work could be done on the con- stants C1 and C2 as well, to make them more accurate. C2 was experimentally determined to equal 1.039(11) and Br, C1, Hci and µr were experimentally determined to have a linear temperature dependence. The parameters of their linear equations are shown in table2 below.

Table2: The table shows the parameters of the linear temperature dependence of Br, C1, Hci and µr.

How does the interaction of multiple samples of the examined ferrites affect their overall performance and demagnetization? Simulations of magnets in electrical steel showed that the size of the air pockets around the magnets is the most influential factor for the demagnetizing factor in the magnet and flux density in the steel, it is essential that they are kept thin. If the purpose is to direct the magnetic field out perpendicular

50 to the magnetization for a similar case, the direction of the magnetization should alternate. The simulations of magnets in air and in a demagnetizing ﬁeld showed that the middle of the surfaces perpendicular to the magnets’ easy axis were most easily demagnetized, while the corners of the magnet retained their magnetization well. How does commercially available grades of ferrites compare with respect 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 HF26/30 were the grades which resembles the examined ferrite the most from their respective 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 not the 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 Acknowledgments

As a last chapter of the study, there are several people who have been very present in the project and truly deserve an official written acknowledgement. A big "Thank you!" to Björgvin Hjörvarsson who have lent us a huge amount of his time and attention. All to listen to our wants and needs in terms of this project, in fact our education in general. As of this he contacted his former colleague, Peter Isberg, and made this project happen. It has been truly fun, tricky, and inspiring. We would really like to thank Klas Gunnarsson for being a great technical consultant and mentor in ways of providing us with both navigation through the discipline of magnetic materials as well as laughter, and permission to use the coffee machine. Really looking forward to the next semester when we will be navigated further into the jungle by taking the course functional materials I. The whole project has been reviewed by our business consultant, Helena Fornstedt. Thank you for really being involved and continuously during the project giving us constructive feedback 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 during the project. You have always been a great resource, whether it has been finding the right people to talk to, us having trouble understanding a part of the theory, or fixing the VSM when behaving as an exception to all known logical systems. We have always been encouraged to ask for help, and in times of darkness, reminded of the importance of having fun. 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 engaging us in this project. As said above, it has been very interesting and most definitely a valuable experience. When we found ourselves in times of trouble, Peter Svedlindh came to us. Speaking words of wisdom, ”What’s the fuss?”. Thank you for contributing to our project by kindly lending us your expertise in magnetism and being willing to discuss unexpected matters. It has been very helpful! Many ”thank you” to Joakim Andersson, for allowing us to be a disturbance in your everyday by asking the impossible. We have been thinking of a possible spin-off from this project as co-writers for a potential sequel of Mission Impossible, known as the Ferrite edition. Sincere thanks for your help! Since simulation has been an area of uncharted waters, it has been great to receive help from experienced and helpful people at the institution of electricity. Furthermore, the thesis provided by, and written by Stefan Sjökvist was used as a template on "writing a thesis". Thank you Stefan Sjökvist and Juan Santiago! For illustrating the feasibility of the project with the time management lecture, we would sincerely 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 crying corner. Once this essential piece of work was done, the project was pretty much manage- able.

53 References

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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: Equation9 ﬁtted to measured B-H-curves at -40 ◦C(left) and -10◦C(right).

Figure 28: Equation9 ﬁtted to measured B-H-curves at 27 ◦C(left) and 80◦C(right).

61 Figure 29: Equation9 ﬁtted to measured B-H-curves at 140 ◦C(left) and 200◦C(right).