We present analytical formulations, based on a coulombian approach, of the magnetic field created by permanent-magnet rings. For axially magnetized magnets, we establish the expressions for the three components. We also give the analytical 3-D formulation of the created magnetic field for radially magnetized rings. We compare the results determined by a 2-D analytical approximation to those for the 3-D analytical formulation, in order to determine the range of validity of the 2-D approximation.

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This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is axial. Such a configuration corresponds to a passive magnetic bearing. All the calculations are determined by using the Coulombian model. This paper also discusses the optimal ring dimensions in order to have a great force or a great stiffness between the rings. Such properties are commonly searched in passive magnetic bearings and we propose a three-dimensional method allowing to optimize these parameters. Furthermore, an important result is established in this paper: the exact relative position of the rings for which the force is the strongest depends on the air gap dimension. As the expressions presented in this paper give this exact relative position, manufacturers can easily optimize their passive magnetic bearings. It is noted that this result is new because the curvature effect is taken into account in this paper. Furthermore, such semi-analytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite element method. In addition, semi-analytical expressions have a low computational cost whereas the finite element method has a high one. Thereby, as shown in this paper, such calculations allow an easy optimization of quadripolar lenses or devices using permanent magnets. Index TermsMagnetic forces, analytical calculation, ring permanent magnet, magnetic bearing

This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is axial. Such a configuration corresponds to a passive magnetic bearing. All the calculations are determined by using the Coulombian model. This paper also discusses the optimal ring dimensions in order to have a great force or a great stiffness between the rings. Such properties are commonly searched in passive magnetic bearings and we propose a three-dimensional method allowing to optimize these parameters. Furthermore, an important result is established in this paper: the exact relative position of the rings for which the force is the strongest depends on the air gap dimension. As the expressions presented in this paper give this exact relative position, manufacturers can easily optimize their passive magnetic bearings. It is noted that this result is new because the curvature effect is taken into account in this paper. Furthermore, such semi-analytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite element method. In addition, semi-analytical expressions have a low computational cost whereas the finite element method has a high one. Thereby, as shown in this paper, such calculations allow an easy optimization of quadripolar lenses or devices using permanent magnets. Index TermsMagnetic forces, analytical calculation, ring permanent magnet, magnetic bearing