According to Earnshaw's theorem, the ratio between axial and radial stiffness is always -2 for pure permanent magnetic configurations with rotational symmetry. Using highly permeable material increases the force and stiffness of permanent magnetic bearings. However, the stiffness in the unstable direction increases more than the stiffness in the stable direction. This paper presents an analytical approach to calculating the axial force and the axial and radial stiffnesses of attractive passive magnetic bearings (PMBs) with back iron. The investigations are based on the method of image charges and show in which magnet geometries lead to reasonable axial to radial stiffness ratios. Furthermore, the magnet dimensions achieving maximum force and stiffness per magnet volume are outlined. Finally, the calculation method was applied to the PMB of a magnetically levitated fan, and the analytical results were compared with a finite element analysis.
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Jungmayr et al. (2014) studied this question.