Force Measurements on Permanent Magnets and Demagnetization Effects of Assembling Halbach Arrays

TVE 14 041 juni Examensarbete 15 hp Juni 2014

Andreas Östman Max Ivedal Abstract Force Measurements on Permanent Magnets and Demagnetization Effects of Assembling Halbach Arrays Andreas Östman and Max Ivedal

Teknisk- naturvetenskaplig fakultet UTH-enheten This project has studied axial forces for the attractive and repulsive cases between paris of NdFeB permanent Besöksadress: magnets at different distances. Demagnitization when Ångströmlaboratoriet Lägerhyddsvägen 1 assembling Halbach arrays has also been studied. The Hus 4, Plan 0 practical measurements of the magnetic forces corresponded to the performed simulations with some Postadress: exceptions. Those exceptions were due to Box 536 751 21 Uppsala measurement errors. Permanent demagnitization was not noticed when assembling the Halbach arrays nor Telefon: when pushing two repulsive magnets together in the 018 – 471 30 03 force measurements.

Telefax: 018 – 471 30 00

Hemsida: http://www.teknat.uu.se/student

Handledare: J. José Perez Loya and Stefan Sjökvist Ämnesgranskare: Henrik Olsson Examinator: Martin Sjödin ISSN: 1401-5757, TVE 14 041 juni Contents

1 Introduction 2 1.1 Background ...... 2 1.2 Motivation ...... 2 1.3 Aim of the project ...... 2

2 Theory 3 2.1 Permanent magnets ...... 3 2.2 Halbach Arrays ...... 4 2.3 Hysteresis curves ...... 5 2.4 Calculations ...... 6 2.4.1 Calculations in MATLAB ...... 7

3 Simulations 8 3.1 Simulations using FEM ...... 8 3.2 2D-force simulations between two cylindrical magnets ...... 10

4 Experiment preparations 10 4.1 Holder design ...... 11

5 Method 15 5.1 Calibration of load cell ...... 15 5.2 Force measurements ...... 16 5.2.1 Setup ...... 16 5.2.2 Procedure ...... 17 5.3 Assembly of Halbach Arrays ...... 18

6 Results 19 6.1 Calibration of load cell ...... 19 6.2 Force Measurements ...... 20 6.3 Demagnetization in force measurements (repulsive case) ...... 23 6.4 Demagnetization when assembling Halbach arrays ...... 23 6.5 Error calculations ...... 24 6.5.1 Instrument errors ...... 24 6.5.2 Calibration errors ...... 24 6.5.3 Force measurement errors ...... 24

7 Discussion 25 7.1 Causes of errors ...... 25 7.2 Improvements ...... 25

8 Conclusions 25

1 1 Introduction

1.1 Background Permanent magnets have got many application areas in society, ranging from small fridge magnets to smart mobile phones. In order to develop new technologies involving magnets and to improve existing ones, it is necessary to thoroughly understand the physics behind them.

1.2 Motivation This project was performed at the Division of Electricity at Uppsala University and studied how forces between two permanent magnets vary with the distance between the magnets. An additional assessment was to investigate whether demagnetization occurs when two repulsive magnets approach one another. Demagnetization in diﬀerent permanent magnet applications can cause undesirable eﬀects. Using models for demagnetization can help engineers to predict problems with their design.1

1.3 Aim of the project The goal of this project was to measure the forces that arise when permanent magnets are approaching each other. Since a magnetic ﬁeld suﬃciently strong acting on a magnet will cause it to become partly demagnetized, it is an important aspect to know exactly how close the magnets can be pushed without causing changes to its magnetic abilities. Two types of setups were studied. These included Halbach arrays and pairs of cylindrical magnets. In the case of the Halbach array setups the main object to assess was whether demagnetization occurs when a Halbach array is assembled. All magnets used were of the type Neodymium Iron Bromide (NdFeB). The dimensions of the cylindrical and cube magnets can be seen in table 1 and 2 respectively

Table 1: Dimensions and types of cylindrical magnets studied.

Magnet type Diameter [mm] Height [mm] N45 4 3 N45 4 2 N42 20 10

Table 2: Dimensions and the type of cuboidal magnet studied.

Magnet type Side length [mm] N45 3

1Ruoho S. “Modeling demagnetization of sintered NdFeB magnet materials in time-discretized ﬁnite element analysis”. Helsinki: Aalto print. 2011

2 2 Theory

2.1 Permanent magnets Permanent magnets are objects that produces magnetic fields. The magnetic fields are not visible for the human eye but the impact of these fields can be clearly seen in many everyday situations. The source of magnetism is partly a result of unfilled electron shells of atoms as well as due to the rotation of the electrons around the nucleus. Unfilled atomic shells gives rise to a net magnetic field because of the electrons spin. A filled atomic shell would have an equal amount of electrons with spin up and down, resulting in no net magnetic field . An unfilled shell on the other hand, would result in the small magnetic contributions from the orbiting electrons not canceling out and thereby give rise to a net magnetic field of the atom.2 Looking at a larger scale, a material will align its atoms and chunks of atoms (domains) in a different manner, namely the one that requires the lowest energy. Since each atom with an unfilled shell gives rise to a small magnetic field, the atoms themselves could informally be considered as small bar magnets. When the small ”magnets” are aligned in an non-ordered fashion the resulting field will be very weak but when they are aligned in an ordered fashion the resulting field will be strong, see figure 1 and 2.

Figure 1: Several atoms (left) illustrated as small bar magnets. When all of these “magnets” are aligned in the same direction the material will produce a net magnetic ﬁeld and as indicated by the arrow, behave as a permanent magnet.

2Jindal UC. Material Science and Metallurgy. Pearson Education India; 2013

3 Figure 2: Magnetic domains in a material with their directions of magnetization aligned differently. When the domains are ordered in this fashion, no significant net magnetic field will be observed.

There is a way of achieving an alignment of the magnetic domains in certain materials. This is done by placing a sample in an external field, causing its domains to align in the direction of the applied field. When the external field later is removed the magnetic domains might stay in its new directions and if that is the case, the material has become magnetized. A material will have different properties depending on how the magnetic domains will align after the external field have been removed. If the domains revert to their original directions the material is called paramagnetic. If on the other hand the domains will remain in the directions caused by the external field the material is called ferromagnetic. Examples of ferromagnetic materials are iron and nickel. There are also materials which are anti-ferromagnetic and diamagnetic. Anti-ferromagnetic materials align the domains in a pattern with neighboring spin causing no net magnetic field while diamagnetic materials align the domains in the opposite direction of a field when it is present. The magnetization of a magnet is dependent on temperature and a certain material has a critical point where its intrinsic magnetic moment change direction called the Curie temperature. In experiments it is important to consider these effects, especially when using materials with a critical point close to temperatures occurring in the experiment. Two permanent magnets that are placed with their net magnetic fields in opposite direction to one another will exert repulsive forces, pushing the magnets away from one another. This phenomena is used in a few applications for creating low friction bearings. To be able to predict how much weight that can be added to a magnetic bearing one needs to know how close the repulsive magnets will be positioned with an applied weight.

2.2 Halbach Arrays Halbach arrays is an arrangement of permanent magnets that augment the field on one side and nearly cancel the field on the other side, see figure 3. One sided magnetic fluxes were first discovered by Mallinson in 19733 and later independently discovered by Klaus

3Mallinson J.C. “One-Sided Fluxes − A Magnetic Curiosity?” . IEEE Transactions On Magnetics. 1973; 9 (4) : 678-682

4 Halbach 4. At the time Mallinson described what he called “A magnetic curiosity”, not many possible applications were considered. When Halbach later made the discovery he saw applications in particle accelerators among other things. The eﬀect of producing a strong one sided magnetic ﬁeld makes Halbach arrays useful in some applications, for example magnetic levitating trains.

Figure 3: An example of a Halbach array with the net magnetic ﬁeld pointing upwards. The ﬁeld on the other side of the array nearly cancels out.

2.3 Hysteresis curves To visualize the behavior of permanent magnets when exerted to an external field so called hysteresis curves are used. These curves, often referred to as BH-curves, shows how the flux density B of a magnet varies with the applied external field H. There are two curves that are studied on the hysteresis plots. These are the normal curve and the intrinsic curve. The intrinsic curve shows the same data as the normal curve, plus an additional term of M, the magnetization of the magnet itself. The magnets always operate on the normal line, so this is the line to study for design purposes. The hysteresis curves of the magnets used in the experiments are shown in figure 4 and 5.

4Rennie G. Magnetically levitated train takes ﬂight [Internet]. [Place unknown] : [U.S Department Of Energy Research News]; [Date unknown] [2004; cited 2014-05-23]. Available from: http://www. eurekalert.org/features/doe/2004-11/ddoe-mlt111104.php

5 Figure 4: Hysteresis plot for the N42 magnets showing the normal curve

Figure 5: Hysteresis plot for the N45 magnets showing the normal curve.

2.4 Calculations When approximating the forces between magnets, one has to make some simplifications in order to proceed. Consider a cylindrical permanent magnet as shown in figure 6. The magnet has a certain magnetization denoted M. This magnetization is a result of many magnetic dipoles aligned in the material. The source of the magnetic dipole moment can be modeled as a current-loop. This model of a permanent magnet can be seen in figure 7. If the magnetization inside the magnet is assumed to be uniform, the dipoles and their surface currents will cancel out in the interior of the magnet leaving only a net current on the sides. This approximation is described by

~ ~ Jm = 5 × M (1)

Where Jm is the volume current density.

6 It can be shown that eq. 1 can be simpliﬁed to only account for currents on the surface of the magnet and the expression then becomes

~ ~ Jms = M × ~an (2)

where ~an is the normal vector directed outwards from the surface of the magnet as shown in ﬁgure 7. The model derived for the magnet is basically the same as the models for a thin solenoid with a determined height. Because of this, force expressions for solenoids can be used to calculate the axial forces between permanent magnets.

Figure 6: A sample of a magnet type N42 with 20 mm diameter and 10 mm height used in the experiments

Figure 7: The model of an ideal permanent magnet. The arrow pointing from the side of the magnet corresponds to the direction of the normal vector on that side. The circles inside the rectangular area shows the magnetic dipoles and the arrows on these circles show the direction of the current of each dipole

2.4.1 Calculations in MATLAB To compare with measured results, the model of a permanent magnet previously derived was used and forces were calculated with MATLAB using the expression for thin solenoids

7 derived by Iwaza 5. These calculations are only valid for the attractive forces between magnets.

3 Simulations

3.1 Simulations using FEM Simulations were performed in order to verify the measured results. This was done for both attractive and repulsive cases using Comsol. In order to simulate the setups in Comsol some assumptions had to be made. The complete geometry with the diﬀerent domains can be seen in ﬁgure 8

• Due to symmetry only a slice of the setup needed to be simulated in the case of the cylindrical magnets.

• The surroundings of the magnets were assumed to be air,

• The boundaries of the air-domain were assumed to be perfect magnetic conductors in the case of attractive magnets. This means that the field lines penetrate the boundary perpendicular and all of the field is preserved throughout the boundary. When simulating the repulsive case the boundaries were set as magnetic insulators. It means that no magnetic field can traverse the boundary and this corresponds well with two repulsive magnets approaching each other. See figure 9 and 10 for details.

• Outside the surrounding air-layer an infinite element domain was selected. This means COMSOL approximates an infinite surrounding, see figure 8.

Figure 8: The geometry when simulating forces between cylindrical magnets in Comsol. Domain 1 corresponds to the surrounding air, domain 2 is the magnet and domain 3 is the inﬁnite element domain

5Iwaza Y. Case Studies in Superconducting Magnets: Design and Operational Issues. Second edition. New York, USA: Springer Science+Business Media; 2009. p.88-89

8 Figure 9: The selected boundary (2) and (9) shows where the perfect magnetic conductor boundary condition was set

Figure 10: The selected boundaries shows were the magnetic insulation boundary conditions were set

The magnets in these simulations were set to have a relative permeability that varies depending on the ﬁeld they were exerted to. This data was obtained from the magnet manufacturers data sheets of the BH-curves and were included in the simulations.6 The BH-curves were included in the simulations by setting the relative permeability to vary

6Neodymium-Iron-Boron Magnets: Summary Listing. Arnold Magnetic Technologies Corp. Available from: http://www.arnoldmagnetics.com/Neodymium_Literature.aspx

9 as a function of the remanent ﬁeld and the external ﬁeld. That way the assumption of having an ideal magnet is not needed to be done.

3.2 2D-force simulations between two cylindrical magnets The force simulations on two cylindrical magnets were made for attractive and repulsive cases with a 2D axissymetrically configuration. This means there is an axis which serves as an axis of symmetry for the simulation. In that way cylindrically shaped geometries can be simulated. The desired output of the simulations were the forces two cylindrical magnets exerted on each other depending on the distance between them. To achieve this the distance was varied by using a parametric sweep in COMSOL. Setting the boundary conditions along with initial conditions giving a contribution in the axissymmetrical direction, an approximation of the forces were made. Simulations were made for the different dimensions of cylinders shown in table 1. An example of a simulation made for the larger cylinders can be seen in figure 11

Figure 11: Example of a repulsive simulation showing the magnetic ﬁeld for the cylinders with 20 mm diameter and 10 mm height made in Comsol. The white part of the plot shows singularities in the simulation that do not inﬂuence the result of the simulation.

4 Experiment preparations

In order to make practical experiments it was necessary to design a setup for the force measurements. One magnet was to be attached into a holder and the holder was connected

10 to the load cell through a screw. The conﬁguration could be moved vertically towards the second magnet, and for diﬀerent positions, the force it exerted on the other magnet was measured. The holders were designed in Solidworks and later printed with a 3D-printer.

4.1 Holder design It was necessary to design a holder that was able to keep one magnet fixed while exerted to a force from the opposite magnet. Since force measurements was to be done on magnets of different sizes it was necessary to make a design that accounted for this. The initial design consisted of four different holders, two magnet holders in which the magnets would be glued , one main holder in which this magnet holder would be attached and one lower holder in which the second magnet holder would be attached. The main holder was to be assembled from three parts; a larger frame to which two arms were attached. These two arms would then be connected by screws to one of the magnet holders. The lower holder was attached to a larger aluminium frame to ensure that there were no objects close enough to influence magnetic fields of the magnets during the measurement. In the configuration described the magnet holders would expose one of its circular surfaces to the opposing magnet. The initial design attempt can be seen in figures 12 - 14. After examining the first design another design was however chosen. The horizontal screws were changed for vertically placed ones and the frame holding the magnet holder was constructed as one solid piece instead of being assembled from several pieces. Non- magnetic screws with a diameter of 8 mm were used for assembling the final design. The final design can be seen in figure 15 and 16. Note that the pieces are not shown in the orientation of the final assembly to show the details of each holder. The design of the lower holder is the same is in the initial attempt. An additional holder for the hall probe was designed to measure the magnetic field. The final design had a more stable construction as the upper magnet holder was screwed onto the main holder in the vertical direction. A vertical setup would avoid torque and thereby ensure that emerging forces would not alter the horizontal position of the upper magnet holder. This design allowed more powerful screws that kept the components in place when exerted to magnetic forces.

11 Figure 12: One of the arms of the ﬁrst design that was to be attached to the main holder and the small holder

Figure 13: The main holder of the ﬁrst design

12 Figure 14: The small holder of the ﬁrst design attempt

Figure 15: The small magnet holder of the ﬁnal design

13 Figure 16: The lower holder of the initial and ﬁnal design

Figure 17: The main holder of the ﬁnal design

14 Figure 18: Hall-probe holder of the ﬁnal design for measuring the magnetic ﬁeld

5 Method

5.1 Calibration of load cell The force measurements were made using a load cell that can detect applied force and depending on the magnitude display an output voltage. The setup used consisted of a voltage source producing 18V that was applied to the load cell. Inside the load cell there is a circuit called a Wheatstone bridge. A tension and compressive-sensitive resistor in the Wheatstone bridge will vary the resistance depending on the applied force to the load cell, resulting in a varied output voltage.7 For measuring these voltage diﬀerences the load cell was connected to a digital multimeter. Weights were successively added to a screw that in turn was attached to the load cell, see ﬁgure 19. Since the applied force is proportional to the output voltage a proportionality constant of the voltage to weight-relationship was computed.

7Newnes Dictionary of Electronics. 4th ed. Elsevier Science & Technology: 1999. Wheatstone bridge; p. 350

15 Figure 19: The procedure of calibrating the load cell.

5.2 Force measurements 5.2.1 Setup The setup consisted of an upper main holder attached to the load cell, a lower holder and two smaller plastic holders which in turn were attached the lower and upper holders. The load cell was attached to a drill-device which measures the distance that the upper holder was moved from a given starting position. A hall probe was also included in the configuration to measure the magnetic field strength of the magnets. This probe was placed in a holder of its own that was attached to the lower holder and the field was measured underneath the lower magnet. The two magnets were glued to holes in the smaller holders to prevent movements during the experiment and the smaller holders themselves were attached with screws to the larger ones. The complete setup with the holders can be seen in figure 20

16 Figure 20: The experiment setup showing the two magnet holders attached to a lower and upper holder.

5.2.2 Procedure The measurements procedure started with determining the distance between the upper and lower holders with a digital ruler. This is important due to the load cell displaying an erroneous voltage as the load cell would detect a compressive force when the upper holder is pressed against the lower. An additional aspect that had to be considered before starting the measurements was to align the upper and lower holders so the voltage output only was a result of forces in the vertical direction of the system. Once the starting distance between the holders was determined, measurements were made by pushing the upper holder down towards the lower holder in steps ranging from tenths of millimeters to a few millimeters. For each step the output voltage and the distance between the upper and lower holder were observed. The upper holder was continuously pushed towards the lower holder until the distance between the holders was as small as possible without causing compression to the load cell. Once one of these measurement series had been made, the magnets in the holders were removed and measurements were taken for new samples. Since demagnetization could occur in the repulsive case, the procedure always started with measuring the attractive forces between the magnets. To detect any potential changes in the repulsive cases, the magnetic ﬁeld strength was measured before and after the procedure started.

17 5.3 Assembly of Halbach Arrays To investigate whether demagnetization occurs when constructing a Halbach array of cuboidal magnets, plastic holders were printed for the assembly. Two different types of holders were designed, one cylindrical type and one rectangular type. In the cylindrical type the magnets were pushed one by one into the holder, see figure 21 for the design. The direction of magnetization of each magnet inside the cylindrical Halbach holder can be seen in figure 22. In the rectangular design a Halbach was assembled by first gluing the side-magnets onto the holder and then pushing the middle magnet in between and keeping it in position for approximately one minute before removing it. The middle magnet was assembled with the direction of magnetization in two different ways, see figure 23 - 26. To keep track of the direction of the magnetic fields of the magnets during the construction of a Halbach array, one of the sides of the magnet was colored. When assembling a Halbach in the cylindrical holder one of the sides of the cylinder was also colored to keep track of which side was up and down and to produce a net magnetic field in the desired direction. In both experiments the magnetic field was measured before and after the assembly to detect potential changes.

Figure 21: Design of the cylindrical Halbach array holder

Figure 22: Direction of positive ﬁelds for the assembly in the cylindrical Halbach holder. The net ﬁeld is directed upwards.

18 Figure 23: Direction of positive ﬁelds for the assembly when the net ﬁeld is pointing downwards

Figure 24: Assembly of the middle magnet in the rectangular Halbach holder. The arrow below the middle magnet indicates the direction of movement.

Figure 25: Assembly of the middle magnet in the rectangular Halbach holder. The arrow below the middle magnet indicates the direction of movement.

Figure 26: Rectangular Halbach holder with three cuboidal magnets assembled.

6 Results

6.1 Calibration of load cell The result of the calibration of the load cell can be seen in ﬁgure 27. From the measured points a linear regression was made and the relationship between the output voltage of the load cell and the applied weight was derived. This relationship can be seen in the

19 equation shown in the plot. Since the behavior of the load cell was known to be linear the few measurement points that were taken serves as suﬃcient data for reliably determining the voltage-to-weight relationship.

Figure 27: Result of calibrating the load cell

6.2 Force Measurements All of the measured forces for the diﬀerent cylindrical magnets can be seen in ﬁgure 28 - 33. For the attractive cases the analytical calculations made in MATLAB is included in the plots.

Figure 28: Experiment, simulation and analytical comparison for the attractive case. The cylindrical magnets were of type N45 and had a diameter of 4 mm and height of 2 mm

20 Figure 29: Experiment and simulation comparison for the repulsive case. The cylindrical magnets were of type N45 and had a diameter of 4 mm and height of 2 mm

Figure 30: Experiment, simulation and analytical comparison for the attractive case. The cylindrical magnets were of type N45 and had a diameter of 4 mm and height of 3 mm

21 Figure 31: Experiment and simulation comparison for the repulsive case. The cylindrical magnets were of type N45 and had a diameter of 4 mm and height of 3 mm

Figure 32: Experiment, simulation and analytical comparison for the attractive case. The cylindrical magnets were of type N42 and had a diameter of 20 mm and height of 10 mm

22 Figure 33: Experiment and simulation comparison for the repulsive case. The cylindrical magnets were of type N42 and had a diameter of 20 mm and height of 10 mm

6.3 Demagnetization in force measurements (repulsive case) Demagnetization effects while performing the different force measurements were studied. The magnetic field strength of each magnet was measured before and after the experiment. There were no signs of permanent demagnetization in any of the experiments.

6.4 Demagnetization when assembling Halbach arrays The initial and final magnitude of the magnetic field for each of the magnets before being assembled in the cylindrical Halbach holder design is shown in table 3. The direction of the magnets were as shown in figure 22. The results of the assembly in the rectangular Halbach holder configuration with the resulting magnetic field upwards and downwards as shown in figure 24 and 25 can be seen in table 4 and 5 respectively. As can be seen there is no significant difference in the magnitude before and after the assemblies. The reason some of the fields measured after the assembly are larger than before is most likely due to errors and inaccuracies using the measurement equipment.

Table 3: Field measurements on cuboidal magnets before and after assembly in cylindrical Halbach holder

Initial ﬁeld strength Final ﬁeld strength Positive orientation of magnet 0.369 ± 0.05 T 0.375 ± 0.05 T ↑ 0.387 ± 0.05 T 0.389 ± 0.05 T ← 0.389 ± 0.05 T 0.383 ± 0.05 T ↓

23 Table 4: Field measurements on cuboidal magnets before and after assembly in the rectangular holder.

Initial ﬁeld strength Final ﬁeld strength Positive orientation of magnet 0.393 ± 0.05 T 0.385 ± 0.05 T ↑ 0.379 ± 0.05 T 0.387 ± 0.05 T ← 0.404 ± 0.05 T 0.382 ± 0.05 T ↓

Table 5: Field measurements on cuboidal magnets before and after assembly in the rectangular holder.

Initial ﬁeld strength Final ﬁeld strength Positive orientation of magnet 0.385 ± 0.05 T 0.389 ± 0.05 T ↑ 0.393 ± 0.05 T 0.385 ± 0.05 T → 0.404 ± 0.05 T 0.404 ± 0.05 T ↓

6.5 Error calculations 6.5.1 Instrument errors The resolutions of the instruments used in the measurements were as follows:

• Digital multimeter: 0.001 mV

• Gaussmeter: 0.1 T

• Drill device: 0.01 mm

6.5.2 Calibration errors When performing the calibration of the load cell, there were some sources of errors to consider. The resolution of the tools for measuring the output voltage and the weights give rise to deviations from the true value, see section 6.5.1 The calibration error consist of the error in the weight applied to the load cell and of the error in the digital multimeter. The resolution in the digital scale was 0.1 g, which means it has a measurement error of ±0.05 g. The digital multimeter had a resolution of 1 × 10−7 V with a measurement error of ±5 × 10−7 V which corresponds to ±0.333 g (converted with proportionality constant from the calibration). The calibration error is calculated by adding the two measurement errors. The sum of the two errors are ±(0.05 g +0.333 g ) = ±0.383 g.

6.5.3 Force measurement errors In the force measurements the error from the digital multimeter needs to be considered once again. An additional term accounting for the resolution is added to the calibration error to compute the force measurement error. The error is therefore ±(0.383 g +0.333 g) = ±0.716 g for each force measurement.

24 7 Discussion

7.1 Causes of errors The fact that the magnets did not become permanently demagnetized was unexpected. The reason for the constant magnitude of magnetization is probably due to the high co- ercivity of the NdFeB-magnets that were used in the experiment. The external magnetic field has to be larger in order to demagnetize the magnets used. Some of the measured values differ from the simulations and this is due to measurement errors. It was difficult to accurately determine the starting distance between the two magnets in the holders and if this relative distance is not correct, all of the measured points will be offset from the simulated values with a certain distance. In the experiments using the 20 mm diameter and 10 mm high cylindrical magnets the measured values are close to the simulated ones. The difference that can be seen could be due to an inaccurate BH-curve being used in the simulations. There are several causes of potential errors that could have arisen in the experiment. It was hard to determine the exact starting distance between the holders. This was an important aspect to know in order to prevent the upper holder being pushed too far, resulting in the load cell being compressed and displaying an incorrect value. Another error source was that the plastic holders themselves were not perfectly planar. This made it impossible to push the upper holder towards the lower one until the distance between them was close to zero millimeters. If they would have been pushed to a distance of zero millimeters while being inclined, the upper and lower holder would have touched and the output voltage would again be erroneous. As the holders were assembled with screws, a small air-gap did arise in some of the assemblies. Since the screw holes in the holders were not threaded, the holes were exerted to a certain shear stress and a minor displacement occurred. In experiments using the large magnets, this stress could also deform the holder itself. The 3D-printer used was only able to print with an accuracy of 0.14 mm or 0.19 mm depending on the settings used. This means that in order for the measurements of a printing to be exact it was necessary for the size of the model to be a multiple of the chosen accuracy.

7.2 Improvements To reduce the errors more measurements could have been performed. The holes in the experimental setup could have been threaded to make the assembly more stable. This could also indirectly have solved the issue of an erroneous distance being measured if the holders were assembled without gaps. More accurate BH-curves could have been used to improve the simulations and get a better correspondence between measured and simulated values.

8 Conclusions

The initial hypothesis was that a magnet would change its magnetization if exerted to a suﬃciently large external ﬁeld. No signs of permanent demagnetization was observed,

25 not during the force measurements nor when assembling the Halbach arrays. An interesting observation was that immediately after two repulsive magnets had been pushed together in the force measurements, signs of demagnetization was noticed. This effect was however only temporary and the field strength reverted to its original magnitude rapidly. The experimental results did for the most part of the measurements resemble the analytical results. Reasons for differences between analytical and experimental is most probably due to measurement errors.