**Physik | Technik**

## Michal Oskedra, 2001 | Bottmingen, BL

**Classical Halbach Arrays (CHAs) are a type of magnet configuration used in the construction of magnetic bearings. The internal reluctance forces between the magnets of a CHA make them mechanically unstable, reducing their practically achievable size. In linear Stabilized Halbach Arrays (SHAs) those forces are significantly reduced. This work investigates whether radial SHAs could be applied in the construction of large-size magnetic bearings. A computational method was developed to calculatee the scale factors of stiffness and internal reluctance forces between SHAs and CHAs through FEM simulations. By applying the scale factors to the theory of CHAs, the stiffness and forces on SHAs were determined. Finally, a design for scalable-size SHAs was presented**

#### Introduction

Classical Halbach Arrays (CHAs) in linear configurations are patterns of alternatively polarized magnets, which concentrate the magnetic field one side of the row, while cancelling it out on the other. In radial configurations those arrays form a ring directing the magnetic field either inwards (IN arrays) or outwards (OUT arrays). Magnetic bearings are restricted by Earnshaw’s Theorem, which forbids the levitation by stationary currents or charges. Radial CHAs allow escaping this restriction by generating additional stiffness through dynamic induction. The CHAs can only be constructed in small sizes due to internal reluctance forces between the magnets. Those forces need to be counteracted to ensure structural integrity of the array, so in larger magnets, such arrays need a large amount of support material. In linear Stabilized Halbach Arrays (SHAs), magnets of one polarization are displaced by a certain offset into equilibrium. This significantly reduces stress, at the cost of a weaker magnetic field. The aim of this work is to investigate whether radial SHAs could be used for large-size magnetic bearings.

#### Methods

First, I derived an equation defining the stiffness of a magnetic bearing for a CHA. Then I defined variations of a SHA with a test rig, in which the magnets of one polarization could be radially displaced. I used the Finite Element Method (FEM) to simulate the magnetic fields and the internal reluctance forces of those arrays, from which I determined the scale factors of those properties. Scale factors allowed me to define the magnetic field of a SHA in relation to the CHA, which is already described in the literature. Finally, I determined the scale factors for two SHA variations by simulating them at different sizes. For this, I wrote the FindEq script to calculate the equilibrium points of a SHA and CompBField script to determine their scale factors. By inserting the scale factors into the previously derived equation, the stiffness of a magnetic bearing with a SHA was determined.

#### Results

Two variants of SHAs could be determined: The “normal” SHAs, in which the magnets of one polarization are radially displaced outwards, and “super” SHAs, in which the magnets of one polarization are displaced inwards. Simulations showed that for SHAs, the internal reluctance forces are directed inwards, in contrast to CHAs, where those forces are pointing both inwards and outwards. The shape of the magnetic field of SHAs and same-sized CHAs is identical. The third order SHA Super OUT provides 26% stiffness of a same-size CHA while being subject to 67% internal reluctance forces. The third order SHA Normal lN provides 35% of stiffness and is subject to 18% internal reluctance forces. These scale factors are constant for any size of magnets.

#### Discussion

The size of SHA Normal IN is limited due to constant scale factors: It can be constructed with magnets twice the size of the original CHA before being subject to same internal reluctance forces. This is not a significant improvement; nevertheless, such arrays allow for the reduction of the amount of support material. The scalable design of SHA Super OUT uses the fact that the internal reluctance forces are directed only inwards. Those forces contribute to the structural integrity, turning the disadvantage of CHAs into an advantage.

#### Conclusions

The comparative approach allowed the calculation of relevant properties of SHA for bearing design, such as internal reluctance forces and stiffness. The downside of this method was a lack of generality: Only two specific cases could be analyzed, leaving room for optimization of other variations of SHAs. The accuracy of the scale factors could be improved by applying better interpolation techniques and a mesh tailored for magnetic fields in the FindEq and CompBField scripts.

#### Würdigung durch den Experten

Dr. Christian Stamm

Mit grosser Ausdauer untersuchte Michal Oskedra die Anordnung von Permanentmagneten in sogenannten Halbach Arrays, die ihr Magnetfeld auf eine Seite konzentrieren. Motiviert wurde seine Arbeit durch mögliche Anwendung als Magnetlager in schnell rotierenden Schwungrädern zur Energiespeicherung. In seiner Arbeit kombiniert Michal Stabilitätsüberlegungen mit Experimenten und systematischen Simulationsrechnungen, und schlägt als Ergebnis eine neuartige Anordnung mit besonders hoher Stabilität vor. Wird seine Entdeckung bald eine Anwendung in hocheffizienten Energiespeichern finden?

#### Prädikat:

sehr gut

Sonderpreis Metrohm – Expo Sciences Europe (ESE)

**Gymnasium Oberwil
**Lehrer: Nils Detlefsen