Abstract-This paper is the third part of a series dealing with permanent magnet passive magnetic bearings. It presents analytical expressions of the axial force and stiffness in radial passive magnetic bearings made of ring permanent magnets with perpendicular polarizations: the inner ring polarization is perpendicular to the outer ring one. The main goal of this paper is to present a simple analytical model which can be easily implemented in Matlab or Mathematica so as to carry out parametric studies. This paper first compares the axial force and stiffness in bearings with axial, radial and perpendicular polarizations. Then, bearings made of stacked ring magnets with alternate polarizations are studied for the three kinds of polarizations, axial, radial and perpendicular. The latter correspond to Halbach structures. These calculations are useful for identifying the structures required for having great axial forces and the ones allowing to get great axial stiffnesses.

Abstract-This paper presents a general analytical formulation for calculating the three-dimensional magnetic field distribution produced by Halbach structures with radial or axial polarization directions. Our model allows us to study tile permanent magnets of various magnetization directions and dimensions. The three magnetic field components are expressed in terms of analytical and semi-analytical parts using only one numerical integration.Consequently, the computational cost of our model is lower than 1 s for calculating the magnetic field in any point of space. All our expressions have been checked with previous analytical models published in the literature. Then, we present two optimized permanent magnet structures generating sinusoidal radial fields.