TVE 12 025 juni

Examensarbete 15 hpJune 2012

Review of Magnetic Materials Along With a Study of the Magnetic Stability and Solidity of Y40

Joakim KarlssonOla Söderström

Sammanfattning på svenskaPermanentmagneter används idag i större utsträckning än tidigare som källa till det magnetiska flödet i generatorer och elektriska motorer. En risk som finns med denna typ av konstruktion, till skillnad från när elektromagneter används, är att permanentmagneter vid påverkan från omgivningen kan helt eller delvis förlora sin magnetiska förmåga och därmed sänka effekten på motorn/generatorn. De faktorer som framförallt kan orsaka avmagnetisering är motriktade magnetfält och ökad temperatur. De vågkraftverk som utvecklas vid Uppsala universitet är ett exempel där permanentmagneter används i en generator. Ett sådant vågkraftverk utnyttjar den potentiella energin som finns i havsvågor för att alstra en ström i en generator på havsbotten. Generatorn är byggd enligt principen för en linjärgenerator och består av en rörlig translator samt en fast stator. Via en vajer fäst vid en boj på havsytan rör sig translatorn upp och ner i takt med vågorna och inducerar en ström i statorns lindningar. Till kommande vågkraftverk är en ny typ av permanentmagneter tänkta att användas och syftet med detta arbete har varit att undersöka hur dessa magneter kommer att klara av de påfrestningar som kan tänkas uppstå inuti kraftverket och om det finns någon risk för avmagnetisering. Arbetet ger även en relativt omfattande överblick över magnetiska material i stort, där visst fokus har lagts på magnetisk stabilitet. För att få en förståelse för hur den magnetiska förmågan kan komma att påverkas har mestadels litteraturstudier genomförts. Magneternas mekaniska egenskaper, närmare bestämt hållfastheten vid ett pålagt tryck, har undersökts på experimentell väg. Resultatet visade att det finns väldigt liten risk för avmagnetisering men att magneterna med stor sannolikhet kan komma att spricka på grund av sin utformning och det tryck de kommer utsättas för. Huruvida denna sprickbildning kommer att påverka generatorns egenskaper är däremot osäkert.

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

Review of Magnetic Materials Along With a Study ofthe Magnetic Stability and Solidity of Y40

Joakim Karlsson, Ola Söderström

Wave energy converters (WECs) are relatively new power sources under rapiddevelopment. WECs utilize permanent magnets to generate power and theperformance of these magnets have a great impact on the produced effects in theWECs. This paper is primarily constructed to investigate the magnetic and mechanicalproperties of a specific kind of permanent magnets, referred to as Y40. The paperalso gives a comprehensive review of magnetic materials in general, slightly focusingon magnetic stability. Literature studies has been made to get an understanding ofhow the Y40 magnets will perform under external influences such as reversed field,temperature change and mechanical stress. Further, a compression test has beenmade to examine the Y40s solidity. From the results of the tests and from theinformation provided in literature it is considered to be little to no risk for the Y40sto lose magnetization due to external influences. However, because of theirassembled structure, the Y40 magnets are very likely to break in their joints duringpressure lower than what is expected in the WECs.

ISSN: 1401-5757, UPTEC F** ***Examinator: Martin SjödinÄmnesgranskare: Teresa Zardan Gomez de la TorreHandledare: Boel Ekergård

Symbols and ConversionsA - Area [m2]a – Acceleration [m/s2]B – Magnetic flux density [T]Br – Residual induction [T]d – Distance [m]E – Anisotropy energy [J/m3]e – Electron charge [C]H – Magnetic field strength [A/m]Ha – Applied magnetic field [A/m]Hc – Coercivity [A/m]Hci – Intrinsic coercivity [A/m]Hd – Demagnetization field [A/m]I – Electrical current [A]J – Total angular momentum [Js]j – Current density [A/m2]K – Anisotropy constant [J/m3]L – Orbital angular momentum [Js]l – Length [m]M – Magnetization [A/m]m – Magnetic dipole moment [Am2]m – Mass [kg]me – Electron mass [kg]

Mr – Magnetic remenance [A/m]Ms – Magnetic saturation [A/m]N – Demagnetization factor P – Power [W]Pc – Permeance coefficientPci – Intrinsic permeance coefficientR – Electrical resistance [Ω]Rm – Magnetic reluctance [A/Wb]r – Radius [m]S – Spin angular momentum [Js]TC – Curie Temperature [K]TW – Working temperature [K]V – Volume [m3]v – Velocity [m/s]λs – Magnetostriction constant μ – Magnetic permeability [Vs/Am]μ0 – Permeability in free space [Vs/Am]μr – Relative permeabilityμrec – Recoil permeability [Vs/Am]σ – Tension/ Stress [N/m2]χ – Magnetic susceptibilityω – Angular frequency [rad/s]

All calculations will in this paper be performed in SI units, though some of the plots will require CGS units. The most relevant conversions are given in Table 1.

Table 1 Conversions between the SI unit system and the CGS unit system. Quantity CGS Unit SI Unit Conversion

Magnetic flux density (B) Gauss [G] Tesla [T] 1 G = 10-4 T

Magnetic field strength (H) Oersted [Oe] Ampere per meter [A/m] 1 Oe = 103/4π A/m

Magnetization (M) Gauss [G] or emu/cm3 Ampere per meter [A/m] 1 G = 103 A/m

Susceptibility (χ) - - 1 (CGS) = 4π (SI)

Permeability in free space (μ0 ) - Volt second per ampere meter [Vs/Am]

1 (CGS) = 4π∙10-7 Vs/Am

Table of Contents

1. Introduction 11.1 Permanent Magnets and Demagnetization 11.2 Connection to the Lysekil Project 11.3 Project Aim 2

2. Theory 32.1 Origin of Magnetism 32.2 Types of Magnetism 4

2.2.1 Diamagnetism and Paramagnetism 42.2.2 Ferromagnetism, Antiferromagnetism and Ferrimagnetism 5

2.3 Magnetostatics 62.4 Domain Theory and the Hysteresis Loop 8

2.4.1 Domain Theory 82.4.2 Magnetic Hysteresis 9

2.5 Hard Magnetic Materials 112.6 Magnetic Anisotropy and the Effect of Stress 12

2.6.1 Magnetocrystalline Anisotropy 122.6.2 Effect of Stress 13

2.7 Thermal Properties 142.8 Demagnetization due to Reversed Fields 152.9 Other Causes for Demagnetization 17

3. Method and Implementation 183.1 Literature Studies 183.2 Compression Test 18

4. Results 194.1 Demagnetization 19

4.1.1 Temperature 194.1.2 Reversed Magnetic Fields 20

4.2 Solidity 22

5. Discussion 245.1 Demagnetization 24

5.1.1 Temperature 245.1.2 Reversed Magnetic Field 245.1.3 Effect of Mechanical Shocks 255.1.4 Stress Induced Anisotropy 25

5.2 Solidity 26

6. Conclusions 25

7. References 25

1 Introduction

1.1 Permanent Magnets and DemagnetizationPermanent Magnets (PM) can be used as the source of magnetic fields both in electric motors and in generators. This is the case in the synchronous linear generators developed by Uppsala University used in the ocean wave power project outside Lysekil [1]. Since the absorbed energy of these machines is closely related to the delivered magnetic flux, it is useful to study how the magnetic properties are affected when the PM are subjected to various external influences (increased temperature, reversed field, mechanical stress etc). External influences can cause demagnetization i.e. losses in the PM magnetization. These losses can be of both reversible and irreversible character and in worst case, the PM can be completely demagnetized.

1.2 Connection to the Lysekil ProjectThe wave energy converters (WECs) at Uppsala University’s experimental facility outside Lysekil utilize the potential energy of waves created on the ocean to generate electrical energy. A buoy floating on the surface is attached via a cable to the movable part of a generator located on the seabed. The wave motion will lift the buoy up and down, resulting in a movement in the generator. The generator is designed according to the principle of a linear generator and comprises a fixed stator and a moveable translator. The translator is constructed from a large number of PM (Fig. 1-2) all which radiate a magnetic flux. When the translator moves, the magnetic flux relative to the stator varies which gives rise to a current induced in the stator windings.

The PM used in previously WECs are so called neodymium magnets (Nd-Fe-B). These PM have a strong magnetic flux density (Br ~ 1.3 T) and are well suited in generators since the amount of energy produced depends on the magnitude of the varying magnetic flux, described by Faraday's law of induction. Nd-Fe-B have, since their introduction in the 80th, been the most abundant PM type in applications where the PM strength is of importance. A disadvantage of these PM is their high cost. On financial grounds, there are

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Fig. 1 A simplified picture of a WEC showing a few vital parts. [2].

Fig. 2 The PM placement in the translator. 4 out of about 97 layers are shown with pole shoes between the thicker PM.

good reasons to change to so-called ferrites (often called ceramic magnets), which have lower magnetic properties but are more favorably in price.

The ferrites meant to be used in the next WEC (L11) have the trade name Y40 (Chinese nomenclature) and will throughout this paper be referred to as this name. These ferrites have a magnetic flux remanence of 0.45 T, which is about a factor three weaker than Nd-Fe-B. To be able to switch to these ferrites without making the generator less efficient, a new design which includes more PM will be used. The new design also means that the PM will be buried between pole shoes instead of surface mounted as previously. The difference between buried and surface mounted PM can be seen in Fig. 3. Since ferrites have not been used in large scale in any previous project, it is important to examine their magnetic properties so that they do not lose their magnetization while operating inside the generator.

To hold the PM in place and to protect them from cyclic stress, the translator is clamped together by a force of 350 kN. This force and the translators weight will result in a pressure of about 3 MPa on each of the PM mounted at the bottom of the translator. The solidity of the Y40 magnets are therefor also of great interest.

Fig. 3 To the left, PM mounted between pole shoes and to the right surface mounted PM. The arrows indicates the direction of the magnetic flux. [3].

1.3 Project AimThe purpose of this study is to give a comprehensive review of the basic theory of magnetic materials with a slight focus on how PM can lose their magnetic properties. Further, an investigation will be made on whether ferrites are suited for use in the WECs and if there is any risk of breakage and/or demagnetization.

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2 Theory

2.1 Origin of MagnetismAll matter consists of atoms and the behavior of the matter is tightly related to the structure of the atoms and their reciprocal interactions. In the atom, negatively charged electrons can be said to orbit the nucleus (The Bohr model Fig. 4) with an orbital angular momentum L. From the laws of classical mechanics it is given that

L=r×me v (1)

where in this case r, me and v is the electron's orbital radius, mass and velocity respectively. It can be shown that the magnetic dipole moment m for a small current loop can be written as

m= I A uN= I r2 π uN (2)

where A is the area inside the loop and uN the unit vector normal to A. Further, based on the fact that an electric current I is defined as charged particles in motion, the current can be written as

I =eω= e v2π r (3)

where e is the electron charge and ω is the angular frequency of the electron. This is illustrated in Fig. 5. Combining Eq. 1, 2 and 3 gives

m= −e2me

L (4)

where the factor -e/2me is called the gyromagnetic ratio (or as in [4], the magnetomechanical ratio).

Because of this, all atoms produce a magnetic dipole moment because of their moving electrons and all matter are therefore somewhat magnetic (or at least responds to an applied magnetic field). But the orbital angular momentum L of the electrons is not the only factor that plays a part in producing m. In quantum mechanics there exists something called intrinsic (spin) angular momentum S. Basically all elementary particles, including the electron, have spin and it can to

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Fig. 4 The Bohr Model of an electron orbiting a nucleus.

Fig. 5. Illustration of the magnetic dipole moment m.

some extent be seen as the particles angular momentum around its own axis. (Spin is a completely quantum mechanical phenomenon, and has no equivalent in classical mechanics). Since spin can be seen as an angular momentum, it can be understood that it too must give rise to a magnetic dipole moment. The total angular momentum J is therefore defined as

J=L+S . (5)

As stated by Mattis [5] and Gould [6], S plays a much bigger part in giving a material its magnetic properties than L. In fact, L is often considered negligible.

2.2 Types of MagnetismSince all matter somehow responds to a magnetic field, it is useful to introduce a definition of the concept magnetic materials. Magnetic materials can be classified by how their magnetic dipoles are oriented relative to each other; parallel, anti-parallel or not at all. If the individual dipole moments are randomly oriented so that Σ m = 0, the material is said to be paramagnetic or diamagnetic. In a non-physically context, these materials are often referred to as non-magnetic. However, paramagnetic and diamagnetic materials can be partially oriented if an external magnetic field is applied. If the dipole moments are not randomly oriented although no external magnetic field is applied, the material is said to be ordered [7]. Ordered magnetic materials can be divided into ferromagnets, antiferromagnets and ferrimagnets. Table 2 shows some examples different magnetic materials.

2.2.1 Diamagnetism and Paramagnetism

In diamagnetic and paramagnetic materials, the individual dipole moments m do not interact with each other and are therefore randomly oriented, resulting in a zero net magnetic moment. This is illustrated in Fig. 6. However, the dipole moments can become partially oriented if an external magnetic field is applied. If the applied magnetic field is removed, these materials will not retain any magnetization. The reason for this is that their intrinsic thermal energy is large relative to the potential energy that orders the dipoles and thermal agitation will randomize the orientations of m. The big difference between these two types of materials is the direction of the generated magnetic field; in paramagnets the generated magnetic field will be in the same direction as the applied field while in diamagnets it will be in the opposite direction. Diamagnetism occurs because of the electrons motion around its nucleus, i.e because of L. Lenz's Law implies that when an external field is applied, the electrons will change their speed in order to cancel the change in magnetic flux. Since diamagnetism is derived from the electrons motion around its nucleus it is present in all matter, though the effect is very small. In many materials diamagnetism is obscured by paramagnetism or ferromagnetism. Paramagnetism is difficult to describe properly without quantum physics. The

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Fig. 6 The dipole moments do not interact with each other and are therefore randomly oriented.

atoms in paramagnets have a nonzero electron spin S, which will align in the direction of the external field when it is applied and thus creating a field in the same direction as the external field.

2.2.2 Ferromagnetism, Antiferromagnetism and Ferrimagnetism

Ordered magnetic materials have an atomic/molecular structure which makes their dipole moment spontaneously align in certain directions relative to each other. The thermal agitation in these materials is not enough to randomize the dipole moments, and so they stay in fixed arrangements. In a ferromagnet, all dipole moments couples in parallel directions, see Fig. 7, and it is easy to see why these materials are magnetic since all dipole moments point the same way. Antiferromagnets and ferrimagnets have their dipole moments coupled in antiparallel arrangements, making them cancel each other out. The difference between antiferromagnets and ferrimagnets is the magnitude of this cancellation. In antiferromagnets the dipole moments in the sublattices are of the same magnitude and thus cancel each other out entirely, see Fig. 8, while the dipole moments in ferrimagnets are of different magnitude as illustrated in Fig. 9, and thus have a net magnetic moment separated from zero. On a macroscopic scale, this means that antiferromagnets behave like non magnetic materials, while ferrimagnets behave like ferromagnetic materials.

Table 2 Different types of magnetic materials. Compiled from [7] and [8].Magnetic order Example of Materials

Diamagnetic Copper (Cu), Carbon (C), Hydrogen (H2), Natrium chloride (NaCl)

Paramagnetic Aluminum (Al), Oxygen (O2)

Ferromagnetic Iron (Fe), Nickel (Ni), Ferronickel (FeNi)

Antiferromagnetic Chromium (Cr), Iron oxide (FeO), Nickel oxide (NiO)

Ferrimagnetic Strontium ferrite (SrFe12O19), Barium ferrite (BaFe12O19)

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Fig. 7 The alignments of the dipole moments in a ferromagnetic material.

Fig. 9 The alignments of the dipole moments in a ferrimagnetic material.

Fig. 8 The alignments of the dipole moments in an antiferromagnetic material.

2.3 MagnetostaticsA magnetic dipole moment m is said to point from south to north. A magnet can be compared to the sum of many dipoles and the magnetic field of a magnet is thus the net magnetic field of all its dipoles. There are three vector fields necessary to fully describe the state of magnet; the B-field, H-field, and the M-field

The B-field, or the magnetic flux density, is measured in Tesla [T]. It is generated by an electrical current and can be calculated from Ampere's Law (without Maxwell's correction):

∇×B=μ 0 j (6)

where μ0 = 4π∙10-7 Vs/Am is the permeability in free space and j is the current density. However, when a B-field passes through a magnetic material there can be some ambiguity regarding which part of the field that origins from the external field and which part that origins from the material itself. It is therefor motivated to introduce the magnetic field strength H and the magnetization M as

B=μ0 (H +M ) (7)

where both H and M are measured in ampere per meter [A/m]. B and H inside and around a PM is shown in Fig. 10.

Outside a magnetic material B and H are almost identical, both fields point in the same direction but differ in size by the factor μ = μ0μr where μ is the permeability for the specific medium and μr is the relative permeability,

B=μ 0 μ r H =μ H. (8)

Inside a magnetic material however, the magnetization M also have to be regarded. The magnetization corresponds to how strongly a region in a material is magnetized and is defined as

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Fig. 10 To the left, a PM where M goes from south to north. The two figures on the right represents the upper right half of the PM and the area just outside with the B-field and H-field in this region.

M =nmV (9)

where n is the number of magnetic dipoles and V is the volume of the region. The magnetization does not exist outside a magnetic material.

Eq. 7 can be expanded as

B=μ 0(H +M )=μ 0(1+χ ) H =μ H (10)

where χ is the magnetic susceptibility. χ is a material dependent constant which indicates the degree of magnetization M of a material in response to an applied H-field,

M =χ H . (11)

In vacuum χ = 0 while χ is a negative constant for diamagnetic materials (as explained in Section 2.2.1, a diamagnteic material will create a field in the opposite direction if an external H-field is applied) and a positive constant for paramagnetic materials. Ferromagnets and ferrimagnets on the other hand have a non linear dependency of χ giving rise to the hysteresis phenomenon described in Section 2.4.2. The factor (1+χ) in Eq 10 is the the relative permeability (μr) of the material and can be written as

μ r=(1+χ )=μμ0

. (12)

From Eq. 12 it can be seen that μ is the magnetic permeability of the specific material in question and that it is possible to express the same equations with both χ and μ. However, when studying ferrimagnetic and ferromagnetic materials, μ is usually the parameter of interest. The permeability can be used to calculate a materials reluctance Rm, that is a materials capability to conduct magnetism. It is given by the equation

Rm= lμ A (13)

where l is the length of the conducting material and A its cross section area. The reluctance can be seen as the magnetic equivalent to electrical resistance and is the reason why the magnetic fields preferably travel through iron cores (low reluctance) rather than through air (high reluctance). Ferrimagnetic and ferromagnetic can only conduct a certain amount magnetism, and when this maximum value is reached the material is said to be saturated. A magnetically saturated material has μ = μ0 and therefore the same Rm as air or vacuum.

Inside a PM, the magnetization M gives rise to a reversed magnetic field Hd . This field is called the demagnetization-field (or the stray-field) and can be calculated as

H d=−Ν M (14)

where Ν is the demagnetization-factor which has a value between 0 and 1 depending on the shape of the sample. For the special case of an ellipsoid Ν is a constant (e.g. for a sphere Ν = 1/3) but in general it is a tensor function of the samples shape. However, it can in often be approximated as a constant and in a uniform thin film

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magnetized in the plane Ν = 0 while the same film magnetized normal to the plane has Ν = 1. The stray-field is the H-field generated outside the sample while the demagnetization-field is the H-field inside the sample [9].

2.4 Domain Theory and the Hysteresis Loop2.4.1 Domain Theory

It was stated in Section 2.2.2 that in ferromagnets or ferrimagnets, the atomic moments are aligned in the same directions even if no external field is applied. But this does not explain why two pieces of iron (which are ferromagnetic) do not attract or repel each other. To understand this, magnetic domain theory is needed. A magnetic domain is a spontaneously magnetized region (typically a cluster of 1017-1021 atoms [10]) separated by domain walls. The domains in a material is arranged in such a way that their vector sum is zero, thus closed flux paths are created within the material. This alignment of domains leads to no observable net magnetization. The domain walls are either turned 180 degrees relative to each other or 90 degrees in the flux closure domains, Fig. 11a. This type of domain alignment occur spontaneously in all ferromagnetic material in order to reduce the magnetostatic energy associated with the leakage of magnetic flux to the surrounding medium [11].

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Fig. 11 A simplified demonstration of the domains in a ferromagnet. The blue arrows are the net dipole moments in the domains in the sample. a) The sample at the moment an external magnetic field is applied. b) The domain with vector fields in the same direction as the applied field has grown in size at the expense of the other domains (wall movement). c) All dipole moments in the sample have rotated to align with the applied field (wall rotation).

When an external magnetic field is applied to a ferromagnetic material, the domain walls moves so that those domains having vector-fields in a similar direction to the applied magnetic field will grow in size at the expense of the domains pointing in the opposite direction, Fig. 11b. This is often an effortless process and a relatively weak magnetic field can therefore cause a thorough wall movement. But to completely magnetize the sample, wall movements are not enough, the domains with magnetization vectors pointing in the direction of the easy-axis have to be rotated in order to align with the applied field. This process requires a substantially stronger magnetic field. The changes in domain structure are illustrated in Fig. 11. Fig. 12 shows the relation of an applied H-field and the magnetization of a sample.Magnetic Hysteresis

2.4.2 Magnetic Hysteresis

If a ferromagnetic sample has been completely magnetized through the process described in Section 2.4.1 and the external magnetic field is removed, the magnetization will not follow the original magnetization curve back to zero but instead stay at a higher level than expected. To regain zero magnetization an external field in the opposite direction must be applied. This phenomenon is called magnetic hysteresis and can be illustrated with the M-H curve in Fig. 13. The M-H curve has to be determined experimentally and is in fact a multivalued loop. It describes how a sample is magnetized or demagnetized when the strength and direction of an applied magnetic field is varied. Another curve, the B-H loop, can be derived from the relation in Eq. 7. This curve only differs slightly in appearance and contains similar information. Which one to use depends on the context; scientists and physicists tend to prefer the M-H loop while engineers usually focus on the B-H loop [10] [8].

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Fig. 12 When magnetizing a ferromegnetic material the wall movements of the domains is a relative effortless process. Wall rotation requires a substantially stronger magnetic field which can be seen as the curve flattens out.

It is important to understand that the H-field in both the M-H loop and the B-H loop is the sum of both an applied magnetic field Ha and the demagnetization field Hd

described in Section 2.3. The true field becomes

H =H a+H d . (15)

A consequence of this is that when a PM is used as a source to magnetism and no external magnetic field is applied, H will be in the opposite direction of B and M. This causes the operating point of a PM to be somewhere in the second quadrant of the hysteresis loop. From an engineering perspective, it is sufficient to study only the second quadrant since it provides all the information needed for design purpose. Therefore, the second quadrant has been given its own name, the demagnetization curve. Fig. 14 illustrates a typical demagnetization curve where the line represents the load line or the permeance coefficient (Pc). The point where this line intersects the normal curve is the operating point of the PM. Since Hd is dependent of the shape of the magnet (Eq. 14), so is also the load line.

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Fig. 13 An example of what the hysteresis curve of a PM might look like. The blue narrower loop is the B-H loop, while the red broader loop represents the M-H loop. Important locations on the curves are noted. The dashed curves are the virgin curves which a sample follows when it is magnetized for the first time.

When all magnetic dipoles in a sample is oriented in the same direction, the sample has reached magnetic saturation Ms. The remaining magnetization in the sample if the external H-field is removed is called the remanence Mr. The corresponding value for the magnetic flux density is called the residual induction (or the flux remanence) Br. These two values are measurements on how “strong” PM are.

The intrinsic coercivity Hci is the point where the M-H loop intersects the H-axis. This value of H corresponds to the strength of a reverse magnetic field required to force the magnetization back to zero, i.e. to completely demagnetize the PM. The reversed field required to restore the flux density B to zero is called the coercivity Hc and is the point where the B-H loop intersects the H-axis. The intrinsic coercivity and the coercivity are measurements on how easy or difficult it is to magnetize or demagnetize a certain material. If Hc and Hci have low values resulting in a narrow hysteresis loop the magnetized sample has properties of a soft magnetic material while if the hysteresis loop have a wider appearance the sample is said to be a hard magnetic material. All PM consists of hard magnetic material.

The product BH has the unit of Joules per cubic meter [J/m3]. The point in the second quadrant where the area under the B-H curve is maximized is called (BH)MAX, see Fig. 13. This value corresponds to the operating point where the PM can supply the most energy to an air gap. When manufacturing a PM, it is preferable to make this point the operating point. The value of (BH)MAX is of importance in many engineering applications [10].

2.5 Hard Magnetic MaterialsIn Section 2.4.2 it was stated that a high coercivity results in a broad hysteresis loop. Materials with this property are hard to demagnetize and are therefor referred to as hard or permanent. Although there are many compounds with hard magnetic properties, these can be divided into three groups; rare earth magnets, metal alloys

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Fig. 14 The demagnetization curve. The intrinsic curve is part of the M-H loop while the normal curve is part of the B-H loop.

and ferrites. Rare earth magnets (e.g. Nd-Fe-B and Sm-Co) is considered to have the best magnetic properties of all. These magnets have both high remenance and high coercivity and is therefor often preferred in applications where performance is of importance. The disadvantage of these PM are their relative low working temperature, high cost and low corrosion resistance. Metal alloys, for example Alnico magnets, have almost as high remenance as rear earth magnetss but have low coercivity (but still high enough to be counted as a hard magnetic material). Alnico magnets have very high working and Curie temperature (see Section 2.7) and can therefor be used in applications where the temperature is expected to be high. Ferrites (or ceramic magnets) usually refers to barium or strontium ferrites (Ba/SrFe12O19). The specific composition of ferrites is often not specified but more strontium usually means better properties and higher price [8]. Ferrites have lower remenance then both rare earth magnets and metal alloys but their coercivity typically lie somewhere in between these two. A great advantage of ferrites is their low price and inability to rust. Table 3 presents some typical material properties for different hard magnets.

Table 3 Typical values for the most important properties of some permanent magnets. “M” in the ferrite row denotes barium, strontium or lead. Summarized from [7] and [12].

Material Category Br [T] Hc [kA/m] Hci [kA/m] (BH)MAX

[kJ/m3]Tw [°C] Tc [°C] Br change

per °C [%]

MO∙6(Fe2O3) Ferrite 0.2-0.4 150-290 200-380 8-30 250 450 -0.2

Al-Ni-Co Metal Alloy 0.7-1.3 40-150 40-170 10-70 450-550 810-860 -0.02

SmCo5 Rare Earth 0.85-0.95 600-720 1300-2400 130-180 250-300 685-700 -0.04

Nd2Fe14B Rare Earth 1.0-1.4 760-1030 880-3300 190-400 125-150 310 -0.1

2.6 Magnetic Anisotropy and the Effect of StressThe physical phenomena of being directionally dependent is called anisotropy. The fact that the properties of a magnetic material varies depending on the direction they are measured in is called magnetic anisotropy. Magnetic anisotropy origins from crystal structure, sample shape, stress or atomic pair ordering. It can also be induced by factors such as magnetic annealing, stress annealing, plastic deformation and irradiation. Though most of these causes to magnetic anisotropy can be of importance in practice [8], only anisotropy from magnetocrystalline structure and stress will here be closer described.

2.6.1 Magnetocrystalline Anisotropy

All ferromagnetic materials have one or more crystal direction in which their magnetization will prefer to be oriented, this direction is called the easy-axis. To rotate the magnetization away from this direction it is necessary to overcome a certain energy level called the magetocrystalline anisotropy energy. The crystal structure of a magnetic material can be ether cubic (e.g. iron) or hexagonal (e.g. cobalt). In a hexagonal structures there is only one easy axis (usually the c axis) and all other axes are equally hard. The anisotropy energy E can therefor be described as

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a function of the angle θ between the magnetization and the c axis. The relevant equation becomes

E=K 1sin 2θ +K 2sin4θ (16)

where K1 and K2 are the anisotropy constants dependent on material an temperature. K1 and K2 can be either positive or negative. In a cubic structure this uniaxial symmetry does not exist and the equation to describe the anisotropy energy becomes more complicated. In PM, a high magnetocrystalline anisotropy is desirable. Comparing iron and cobalt is a good way of showing the principle of magnetocrystalline anisotropy and the difference between cubic and hexagonal structures, this is done in Fig. 15.

Fig. 15 Comparing iron and cobalt is a good way of showing the principle of magnetocrystalline anisotropy and the difference between cubic and hexagonal structures. Cobalt has its easy-axis along the c-axis and as seen in corresponding diagram, this is the direction in which it takes lowest H to reach saturation.

2.6.2 Effect of Stress

The dimensions of a sample can be changed when subjected to a magnetic field, this effect is called magnetostriction. The magnetostriction effect is usually very small, the relative length changes is typically of order 10-5. The reversed phenomena, i.e. an applied stress can alter the favored magnetization direction, is called stress induced anisotropy (or sometimes Villari effect or piezomagnetism). A simplified way to describe the energy from this anisotropy is due

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E=32

λ sσ sin2θ (17)

where λs is the magnetostriction constant, σ is the applied stress and θ the angle between the direction of Ms and σ. This equations assumes that the material is isotropic, i.e. the material has equal properties in all directions. λs can have either positive or negative sign which is also the case of σ. Tensile stress means positive σ and compressive stress negative σ. If the product λsσ is positive, lowest energy occurs when the stress and magnetization is parallel and if λsσ is negative, lowest energy is when the stress and magnetization are perpendicular.

2.7 Thermal PropertiesAll magnetic materials have temperature dependence, making them change their magnetic properties if exposed to increased or decreased temperatures. As explained in Section 2.2.2 ferromagnetic and ferrimagnetic materials obtain there magnetic properties due to alignment of their magnetic dipoles. If a magnetic sample is exposed to increased temperature, the intrinsic thermal energy might increase sufficiently to overcome the potential energy keeping the dipoles in place. This will lead to randomization of the magnetic dipoles and as a consequence, changed magnetic properties. In most magnetic materials, an increased temperature will lead to reduced remenance, coercivity and maximum energy product. The exception here is ferrites, in which the coersivity increases with higher temperature. All magnetic materials have two important temperatures which are of great importance when studying the thermal properties of the material; the working temperature TW and the Curie temperature TC. Up to TW the material can recover all of its former magnetization if cooled and these losses are therefore reversible losses. After TW the material will permanently lose some of the magnetization, see Fig. 16, these permanent losses are irreversible losses. TC marks the temperature where all of the magnetization in the material is lost, even if cooled afterward. It is possible to recover the irreversible losses if the material is cooled and exposed to an applied external H-field, making the domains align once more.

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Fig. 16 A general plot of the Br

dependence of temperature. Above TW

irreversible losses begins.

2.8 Demagnetization due to Reversed FieldsAs mentioned in Section 2.4.2 the operating point of a PM can be determined by the load line (or permeance coefficient, Pc-line). If an external field is applied so that the true field changes from Hd to H1 in Fig. 17, the operating point will move along the blue line to the right, following the permeability recoil line (μrec). μrec is a straight line with the same slope as the normal curve as it intersects the B-axis. If the applied field is removed, the operating point will more or less retrace this line back to its original value. If the applied field instead changes the true field from Hd to H2, the operating point will move along the red line, following the normal curve to the left. Removing this field will not make the operating point retrace the normal curve but instead follow μrec

to a new value causing a permanent loss in the PM strength.

However, this effect is hard to observe in rare earth magnets and in most ferrites since μrec in these materials coincides with the normal curve. Fig. 18 depict a typical demagnetization curve for a Nd-Fe-B magnet. A temporary change in the true field will not permanently lower the Br-value unless the operating point drops under the “knee” on the normal curve. When designing PM-applications it is usually enough to make sure that the operating point never drops bellows this knee. A thorough method on how to calculate the irreversible loss is given in [13] and brief summery of the method will be given bellow.

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Fig. 17 The original operating point is where the load line intersects the normal curve. If an external field is temporarily applied, it might lower the operating point.

Fig. 18 Typical appearance of demagnetization curves of a Nd-Fe-B magnet for four different temperatures. The upper left curves with the adjacent temperatures are the intrinsic curves, those on the lower right are the normal curves. The line originating from origo is the load line (Pc-line), irreversibly losses will occur if it intersects the normal curve bellow a knee.

Instead of studying the “normal” operating point it is also possible to use the intrinsic normal point. This is the point where the intrinsic permeance coefficient Pci intersects the intrinsic curve, see Fig. 19. Pci is calculated as

Pci=Pc+1. (18)

Note that both Pc and Pci are considered to be positive even though they mathematically should have a negative sign. When no external field is applied, the Pc-line intersects the normal curve at the same H-value as the Pci-line intersects the intrinsic curve. If a reversed field is applied this is not true and the Pc-line is no longer of interest. To calculate the loss the intrinsic recoil permeability is also needed. This value is obtained by simply subtracting 1 from the “normal” value for the recoil permeability. Since the recoil permeability often is just above 1 (1.05 for many Nd-Fe-B magnets) the intrinsic recoil permeability almost becomes a flat line. A new value for the residual induction Br' can be calculated as

B r '=Bdi+H d ' (μ rec−1) (19)

where the Hd' and Bdi are show in Fig 20, once again the minus sign should be dropped on Hd'.

In Section 2.7 it was explained that increased temperature can cause losses in the residual induction of a PM. Not surprisingly, a combination of temperature and a reversed field can also cause a loss. To calculate the loss of these two combined factors the same method described above can be applied. Simply use the curve appropriate for the specific temperature. Fig. 21 clarifies this effect.

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Fig. 19 When no external field is applied, the Pc-line intersects the normal curve at the same H-value as the Pci-line intersects the intrinsic curve.

Fig. 20 Shows a situation where the external field is high enough to cause irreversible losses in the PM strength.

2.9 Other Causes for DemagnetizationThere are a few other causes which can be proved to have an effect on magnetic properties than those already stated. These causes are in particular mechanical shocks, time, radio activity and chemical factors such as rust. Mechanical shocks are not considered a problem for modern PM unless they are physically damaged [14], and today’s PM are more likely to break before any demagnetization can be seen. Earlier PM were a lot more sensitive, and could lose a few percent of magnetization if repeatably dropped on the floor [15]. Time dependence of the intensity of magnetization is the result of thermal activation of metastable domain processes [16] and the changes are often so small they can be considered negligible in most applications. If old PM gets demagnetized it is more often the cause of shocks rather than the effects of time only. Exposing PM to radiation may cause changes in the electrons spin and thus change the magnetic moments of the atoms resulting in a demagnetization. These effects are only of consideration if the PM are used in areas where strong radiation might occur and can therefore for the most part be excluded. Rust is something that can severely damage a PM. When oxygen binds to the iron in the magnets their composition changes which might result in a change of the magnetic domains. Ceramic magnets on the other hand already consists of iron-oxide, making them immune to rust.

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Fig. 21 The Pci lines with and without an applied H-field for two different temperatures. When the temperature increases the PM gets more susceptible to external fields. The exception is ferrites where the coercivity increases with higher temperatures.

3 Method and Implementation

3.1 Literature StudiesThe examination of the demagnetization of Y40 were done almost exclusively with literature studies on what is already known about stability of ferrites. Some magnetic properties were provided from the seller and thus assumptions could be made about how the Y40 magnets would behave in the generator.

3.2 Compression TestThe compression tests were made on the Y40s with an oil-powered universal fatigue test machine (INSTRON 8516, Fig. 22) which could apply an increasing uniform pressure up to 100 kN and plot the compressive force over the specimens length contraction. The Y40 magnets meant to be used in the L11 consists of 8 smaller pieces (Fig. 23), which leads to joints where magnets are more likely to break. Limitations of the test machine made it impossible to perform tests on pieces larger than approximately 120x120 mm and so, to examine the solidity at the influence of applied pressure, the magnets were split in half to acquire the right dimensions. To even out the pressure and prevent the pieces from cracking due to unevenness on the surface, a 1 mm thick rubber cover were placed between the machine and the specimen (the same cover is supposed to protect the Y40s in the real generator). When the machine started it was allowed to go on until the Y40s broke or the maximum force of 100 kN had been applied. 5 tests were made on parts of the size approximately 110x100 mm.

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Fig. 23 The jointed Y40 supposed to be in the L11. It is made of 8 smaller pieces.

Fig. 22 INSTRON 8516. The equipment was used in the compression test.

4 Results

4.1 Demagnetization4.4.1 Temperature

Table 4 was provided from the seller and gives important information about the temperature dependence of Y40. Table 4 specifies TW = 250°C and TC = 450°C which agrees well with many other retailers or informers such as [17], [18] and [19]. The percentage change in Br per °C rated -0.2 does also conform with [17] and [19]. This means that when Y40 is heated it will temporarily lose approximately 0.2% of its Br

value per °C as shown in Fig. 24. After 100°C the linearity ends and Br will drop faster towards TW according to [17].

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Table 4 A table provided by the seller of the Y40s showing properties of different ferrites. Y40 is marked with red.

4.1.2 Reversed Magnetic Fields

To calculate the reversed field needed to cause irreversible demagnetization of the Y40 the method presented in Section 2.8 can be used. Since no demagnetization curve was provided from the seller the one used was taken from another retailer, see Fig. 25. The values of Hc and Hci in this curve does not match exactly with the corresponding values in Table 4 but the curve is still useful. If the load line is assumed to be 0.8 it would require a reversed H-field of about 3200 Oe ≈ 250 kA/m or equally a B-field of about 0.34 T before irreversible demagnetization occur.

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Fig. 25 Demagnetization curve of Y40. If the load line is 0.8 it intersects the knee of the normal curve at a reversed field of about 3200 Oe ≈ 250 kA/m. [20].

Fig. 24 The Br temperature dependence of Y40 between 20°C and 100°C. Note that the losses are reversible.

The reversed field arising from the stator windings is dependent of the electric current running through them. A simplified model of this would be to approximate the windings as one long, straight conductor (which we can since the air gap is 3 mm and the conductor about 200 mm) and then use Biot-Savarts formula (for a long straight conductor)

B=μ0 μ r

4π2 Id

, (20)

to calculate the magnitude of the B-field induced by the windings. I is the current in the windings and d the distance from the windings to the point of interest. Since the pole shoe is protecting the magnet from an external field, the B-field needed to permanently demagnetize Y40 would first have to saturate the pole shoe. The design of the pole shoes makes them reach saturation at about 1.5 T and the resulting B-field needed for irreversible demagnetization can then be calculated to 1.5 T + 0.34 T ≈ 1.8 T. Creating a worst case scenario with the magnets positioned closer to the windings, Fig. 27, we can use Eq. 20 to estimate the current required to create a B-field of 1.8 T at d = 3 mm. It turns out this would be I ≈ 26 kA.

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Fig. 27 Same as Fig. 19 But the PM are moved toward the stator windings to create a worst case scenario. [3].

Fig. 26 An approximate picture of the magnetic circuit for a generator with buried magnets. The magnetic flux is marked by closed curves and arrows indicate the direction of the flux. (Leakage flux through the aluminum plate is marked with a dashed line).[3].

4.2 Solidity5 compression tests were preformed and the results are plotted in Fig. 28-32 corresponding to test 1, 2, 3, 4 and 5 respectively. In test 1, 3, 4, and 5 the pieces broke at the joints when the pressure was applied, while the specimen in test 2 did not. In Fig. 30 and 32 it is relatively easy to determine the point at which the specimens in test 3 and 5 broke to about 10.5 kN and 3.5 kN or 0.95 MPa and 0.32 MPa respectively. In test 1 and 4 it is more difficult to determine the breaking force, especially for test 4 which plot has a very similar appearance as test 2. The notch in Fig. 29 and 31 are considered either small movements of the specimens or measurement errors, and it is therefore hard to tell when test 4 broke. The results are summarized in Table 5.

Table 5 The results of the 5 compression tests.Test number Result

1 Broke at unknown pressure < 9.1 MPa

2 Did not break

3 Broke at 0.9 MPa

4 Broke at unknown pressure < 9.1 MPa

5 Broke at 0.3 MPa

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Fig. 28 Test 1. The piece broke when the pressure was applied. It is hard to tell at which pressure the piece broke.

Fig. 29 Test 2. The piece held for forces up to 100kN. The notch is considered a measurement error or a displacement of the test piece in the machine.

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Fig. 30 Test 3. The piece broke when pressure was applied, most probably at 10.5 kN.

Fig. 31 Test 4. The piece broke when the pressure was applied. It is hard to tell at which pressure the piece broke. The notch is considered a measurement error or a displacement of the test piece in the machine.

Fig. 32 Test 5. The piece broke when pressure was applied, most probably at 3.5 kN.

5 Discussion

5.1 Demagnetization5.1.1 Temperature

A temperature study of a linear generator on the seafloor has been made in [21] without worrying results. However, this study might not give a very accurate picture on how the temperature will behave in L11. The positions of the temperature sensors in [21] can be argued not the best for our study and the time of the measurements might be too short for a proper analysis of the maximum possible temperature. Further, the increased temperature is mainly dependent of the resistive losses in the windings which are proportional to

P=I 2 R (21)

where P is the power, I the induced current and R the resistance in the windings. The L11 is calculated to induce a current 10 times as high as in [21], making the losses 100 times as high. However, based on the fact that the water outside the generator is about 4°C and acts as a very good passive cooling medium, the temperature is still not likely to increase all the way up to 250°C, which is needed for irreversible losses. It is also worth noting that ferrites have much higher working temperature than Nd-Fe-B which have been used in previously WEC. Since the temperature has not been a problem for the Nd-Fe-B magnets, it is unlikely to be a problem for the ferrites.

5.1.2 Reversed Magnetic Fields

Considering the risk of demagnetization due to a reversed field from the stator windings, the change from surface mounted Nd-Fe-B to buried ferrites entails one advantage and one disadvantage. The disadvantage is that ferrites have considerably lower coercivity and can therefor not restrain a reversed field in the same manner as Nd-Fe-B. The advantage is that the pole shoes will work as a protection making machines with buried PM more resistance against demagnetization due reversed fields according to [22] and [23]. This is since the induced field will rather go through the pole shoe steel than the magnets unless the steel is fully saturated. A simplified construction with PM buried between pole shoes is shown in Fig. 26. Since ferrites have the property of increasing Hc and Hci for higher temperatures, there is no higher risk for demagnetization if the temperature increases.

The current of 26 kA obtained in Section 4.1.2 is a much higher value than what we can expect the translator to induce in the windings. To show this we can use a simple set of equations:

F=ma (22)

P=Fv=RI 2 (23)

where m is the mass of the translator, a is the acceleration of the translator, F is the force acting on the translator, P the power created, v the velocity of the translator and R the resistivity in the windings. The largest possible I induced would implicate a =

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9,8 m/s2 (free fall). With a stator mass of m ≈ 10.000 kg, the force acting on the stator is F = 98 kN according to Eq. 22. If the stator falls from the top of the generator its velocity v would not have time to reach above 2 m/s and with R ≈ 1.5 Ω in the stator windings Eq. 23 can be be used to calculate I to about 360 A. In the calculations made in Section 4.4.2, the stator steel around the windings was neglected. In reality, this steel has even lower permeability than the pole shoe, making it work as another protection. This increases the current needed to demagnetize the magnets even more.

The calculations shows that the Y40 magnets will not be irreversible demagnetized due to external fields when the WEC is running properly. However, the effects of a short circuit currents or other fault condition we do not know. To examine this kind of situations a more thorough calculation should be made, perhaps with a FEM-analysis.

5.1.3 Effect of Mechanical Shocks

The Y40s are not likely to lose their magnetization due to shocks. Hadfield [15] gives a summery of impact tests on PM made by himself and others with the conclusion that PM can lose their magnetization due to shocks, but that it is very difficult to predict the effect on different PM because of their various compositions. These test were all made on alnico magnets and the results shows that a demagnetization limit of a few percent were reached after some 100 impacts. Since a demagnetization limit seems to be reached rather quickly, Hadfield [15] suggests a quick practical test to establish the facts in any specific case. In [24] similar test were performed with inconsistent results due to large spurious effects. [24] points out the difficulty of making good reliable impact tests on PM.

Very few impact tests have been made on ferrites, and none of practical relevance have been found to use in this paper. The reason for the lack of comforting results on ferrites is probably because of their ceramic properties. Ferrites like Y40 are easily cracked, and a larger impact will probably be needed to demagnetize the PM than it would take to break them. As many retailers like [17] and [19] points out, ferrites are ceramic magnets , making them very hard and brittle and will most probably break when exposed to mechanical shock before any demagnetization can be seen.

5.1.4 Stress Induced Anisotropy

Because of the way the PM are mounted in the translator, where the bottom PM will be subjected to a pressure of 3 MPa, it can be interesting to see if this pressure will effect the PM magnetic properties. The method here is to compare the induced anisotropy from the compressive stress with the intrinsic magnetocrystalline anisotropy and to see if it will lead to any change in the magnetization direction. Since ferrites have hexagonal structure, this can be done by assuming that the PM are isotropic and then directly compare Eq. 16 and Eq. 17. Ignoring the K2 term and dropping the sinus terms gives

σ =23

K 1

λ s. (24)

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The anisotropy constant for barium ferrite is K1 ≈ 330 kJ/m3 (more strontium gives a higher value) and as a value for the magnetostriction constant λs = 20∙10-6 can be used though this is a typical value for many materials [8]. These values indicates that a stress of 11,000 MPa is needed before the induced stress anisotropy equals with the intrinsic magnetocrystalline anisotropy. This is approximately 10 times the compression strength of ferrite magnets [17] and therefore not possible to reach. It is also approximately 3,500 times the pressure the PM are subjected to in the WEC. The calculation above is not very accurate though it ignores anisotropy from other sources such as shape. It also assumes that the PM is isotropic which is not true in our case.

5.2 SolidityIt should be said that the compression test was not designed to evaluate a mean value and standard deviation of the critical compression strength of the actual material, but merely to examine if the magnets had a tendency to break or not. 5 tests are to few to estimate a mean value but enough to clarify that there is a risk for the Y40 magnets to break. The compressive strength for ferrites is well known and is specified to around 895 MPa according to [17] and [18]

Since 4 out of 5 tested magnets broke in the joints and at least 2 of them did so at pressure lower than 3 MPa we can conclude that there is a risk for the Y40 magnets to break when mounted in the WEC L11. However, the question remains whether this would make any difference for the performance of the magnets. The Y40 magnets will be constantly clamped between the pole shoes in the translator and also magnetically attracted to the pole shoe which makes it quite difficult for the them to move even if they break in their joints. If the specimens are not moving the cracks that arise will probably have little to no effect on the magnetic performance. If the Y40s were designed without the joints, their compressive strength would probably rise to around 895 MPa, which would be a great increase in solidity.

Ferrites are very hard and brittle and therefore it is probably the rubber cover that has done almost all the length contractions where the specimens did not move or break. The fact that the specimens are hard and brittle might also be the reason of why it is difficult to determine the breaking force on some of the specimens; they might have broken without moving and therefore not leaving any trace in the plot. An important note is that the specimens did only break in the joints, which agrees with the compressive strength of 895 MPa.

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6 ConclusionsNon of the external influences; reversed external field, temperature change or exposure to mechanical shocks or pressure will be enough to severely affect the Y40s magnetic properties. Nor should the combined effect of a reversed field and increased temperature be be able to cause any irreversible losses. The only scenario where the irreversible losses might occur is in case of short circuit currents in the stator windings, for which we do not know the outcome. The compression tests indicates a risk for the Y40 to break in their joints when exposed forces of magnitudes possible to occur in the WECs. However, this is not considered a major problem since the Y40 will be clamped in place in the translator and therefore no change in the magnetization is likely to be seen. Furthermore, we suspect that there might be some problems mounting the magnets since they are very brittle and will easily crack when exposed to shocks.

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[13] Gulick D. 2007. “Calculating Losses due to Reverse Applied Fields”. Arnold Magnetics TECHNotes, TNO301.

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[15] Hadfield D. 1962. “Permanent Magnets and Magnetism”. New York: John Wiley & Sons, Inc.

[16] Folks L. 1994. “Analysis and interpretation of time dependent magnetic phenomena”. Journal of Applied Physics 76: 6391-6395. DOI: 10.1063/1.358276

[17] e-Magnets UK. Available at: www.ferrite-info.com (16-05-12)[18] MagnaWorks Technology, Inc. Available at: www.magnaworkstechnology.com

(16-05-12)[19] XinChang Magnet Ind.Co.,Ltd. Available at: www.ndfeb-magnet.cn (16-05-12)[20] DirectIndustry. Available at: http://pdf.directindustry.com/pdf/aic-

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