Three-Dimensional Analytical Calculation of the Torque Between Permanent Magnets in Magnetic Bearings

In the high-precision industry, accurate vibration isolation and magnetic levitation are extremely important. As a result, high-performance vibration isolation and magnetic bearings based on permanent magnets are increasingly considered. This paper proposes improved analytical expressions for the torque on cuboidal permanent magnets applied to a magnetic bearing. These novel expressions are valid for any relative magnet position, especially when surfaces of the different magnets are in the same plane. Further, the torque can be obtained with respect to any reference point. Although these equations seem rather complicated, they enable an extremely fast and accurate calculation of the torque on a permanent magnet in the presence of a magnetic field of another permanent magnet. These properties enable a fast design and optimization process of such bearings using fully analytical expressions.