3d analytical field calculation of pyramidal and other polyhedral shaped permanent magnets for a gravity compensator

3d analytical field calculation of pyramidal and other

polyhedral shaped permanent magnets for a gravity

compensator

Citation for published version (APA):

Janssen, J. L. G., Paulides, J. J. H., & Lomonova, E. A. (2009). 3d analytical field calculation of pyramidal and

other polyhedral shaped permanent magnets for a gravity compensator. In Proceedings IEEE International

Magnetics Conference, 4-8 May 2009, Sacramento, California (pp. AU-01). Institute of Electrical and Electronics

Engineers.

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Published: 01/01/2009

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INTERMAG 2009

AU-01

3d analytical field calculation of pyramidal and other polyhedral shaped permanent mag-nets for a gravity compensator.

J. Janssen, J. Paulides, E. Lomonova

Electrical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands

Permanent Magnets (PMs) are ever more used in high-performance applications such as electrical machines, linear servo actuators, contactless magnetic bearings, etc. This paper focuses on PM modeling for a contactless lithographic vibration isolation device with an integrated PM-based gravity compensator. This device needs to position a high mass (tons) with micrometer accuracy, with permanent magnets providing zero-power gravity compensation, and active devices ensuring stability and accurate positioning. To minimize the energy consumption of the active devices due to modeling errors it is essential that the PM modeling accuracy is maximized. Further, low calcu-lation time is required for quick and efficient topology optimization.

The 3D Finite Element Method (FEM), although accurate, consumes significant amounts of time, especially for models with small stroke compared to overall device dimensions. The 3D analytical charge model, on the other hand, provides field expressions which are exact for ironless structures, i.e. they are not approximations and, in contrary to FEM, remain very time inexpensive. Further-more, these analytical field solutions exhibit significantly lower noise (150dB) compared to FEM solutions, and consequently provide an increased accuracy, especially at large magnetic field gra-dients (e.g. the magnet edges). Analytical charge modeling has been commonly used in literature to analytically describe the 3D magnetic fields induced by PMs, e.g. in [1,2] to analyze the field of and the interaction between cuboidal permanent magnets.

It has been illustrated in [3] that the PM topology of the gravity compensator is of significant influ-ence on the interaction force. More specifically, topologies which focus the magnetic flux to a small volume, such as the Halbach topology, provide the highest force density. In [3], solely cuboidal PMs were used to investigate the achievable force density for bidirectional arrays and Halbach arrays. Differently shaped PMs can contribute to increased flux focusing properties, hence, increase force density.

This paper specifically presents novel analytical field expressions of the pyramidal frustum, or 3D trapezoidal PM (Fig. 1a), for the gravity compensator. It is defined such, that the bottom and top surface are rectangular, parallel, and share the vertical symmetry axis. By reducing the horizontal dimensions of the top surface towards zero, a pyramid is easily obtained. Fourteen rectangular and triangular shaped surfaces are distinguished on the contours of the PM. A charge is applied to each surface, and the respective magnetic fields are obtained analytically as described in [1] and [4], respectively. A summation of these fields results in the exact analytical expressions for the mag-netic field exhibited by the 3D trapezoidal PM. The derivation of all surfaces, rotations, field expressions and the summations which result in the total field are elaborated on in the paper. The paper discusses anomalies in the mathematical field description, and methods to overcome those, which results in a smooth field solution. Comparison with 3D FEM solutions (Fig. 2b) demon-strates the high accuracy of the analytical solution (Fig. 2a) for the pyramidal frustum.

Besides the pyramidal frustum, additional PM shapes, such as the hexagonal or octagonal prism (Fig. 1b), may contribute to an increased force density. The analytical model of such PMs is obtained by superposition of cuboidal PMs and triangular PMs, described in [1] and [4], respec-tively. In the final paper, a comparison with FEM shows the high accuracy for such PM shapes.

Using the analytical field expressions obtained for exotic-shaped PMs, novel array topologies are investigated. Due to space considerations, this paper presents force density for a selection of pos-sible topologies to be incorporated in the gravity compensator.

[1] J.P. Yonnet and G. Akoun, “3D analytical calculation of the forces exerted between two cuboidal magnets,” IEEE Trans. Magn., vol. 20, no. 5, pp. 1962-1964, Sept. 1984.

[2] F. Bancel, “Magnetic Nodes”, J. App. Phys., vol. 32, pp. 2155-2161, 1999.

[3] J.L.G. Janssen, J.J.H. Paulides and E. Lomonova, “Passive Magnetic Suspension Limitations for Gravity Compensation.” Proc. 11th international Symposium on Magnetic Bearings. Nara, Japan, pp. 83-90, 2008. [4] J.L.G. Janssen, J.J.H. Paulides and E. Lomonova, “Skewed linear PM Actuator: 3D analytical fields using triangular magnet segments”, to be published.

(a) pyramidal frustum and (b) top view of hexagonal and octagonal prism showing the cuboidal and triangular shapes

(a): analytically obtained absolute value of the magnetic flux density, parallel to one of the side surfaces (projected in the figure). (b): difference with results obtained by FEM.