# Modelling and Experimental Testing of Passive Magnetic Bearings for Power Loss Reduction
## 2 PMB Model  
A test rig of a fully passive vertical rotor equipped with two PMB supports is developed to analyse a configuration of the bearings as both radial and thrust supports. The analysis of the PMB axial and radial forces as functions of the magnetic rings' relative position is necessary for identifying the conditions where the bearings operate as thrust-radial supports providing upward axial force, which partially compensates for the rotor weight and stabilises the system in the radial direction. The numerical evaluation of the magnetic forces is performed with the Coulomb surface charge density method used in [19], where magnets are modelled as equivalent surface distributions of fictitious charges. According to [20], it is possible to use the charge density method considering the boundary conditions of Equations (1) and (2) at the material interfaces:  
\[\
abla\times\mathbf{H}=0, \tag{1}\]  
\[\
abla\mathbf{\cdot}\mathbf{B}=0. \tag{2}\]  
Equation (1) implies that the tangential component of the magnetic field intensity \(\mathbf{H}\) is continuous at each material interface. Equation (2) implies that the normal component of the magnetic flux density \(\mathbf{B}\) is continuous at each material interface. Equation (3) defines the surface charge density \(\sigma_{m}\) of the equivalent fictitious charges located in the plane surfaces of the magnetic rings. \[\sigma_{m}=\mathbf{M}\cdot\mathbf{\hat{n}}, \tag{3}\]  
where \(\mathbf{M}\) is the ring magnetisation and \(\mathbf{\hat{n}}\) is the outward unit normal to the ring surfaces. According to [21], under the hypotheses of linear isotopic homogeneous magnetic material, rare earth material with relative permeability \(\mu_{r}\approx 1\) and uniform magnetisation, the modulus of \(\mathbf{M}\) can be expressed by Equation (4). \[\left|\mathbf{M}\right|=\frac{B_{r}}{\mu_{0}}, \tag{4}\]  
where \(\mathbf{M}\) is the ring magnetisation and \(\mathbf{\hat{n}}\) is the outward unit normal to the ring surfaces. Equation (5) expresses the fictitious infinitesimal charge \(q\), located on an infinitesimal plane surface \(A\) of a magnet, in terms of surface charge density. \[q=\sigma_{m}A. \tag{5}\]  
Thus, the force \(\mathbf{F}_{ij}\) exchanged between two fictitious infinitesimal charges \(q_{i}\) and \(q_{j}\) placed in two  
Figure 2 shows an example of the axial force \(F_{z}\), evaluated with Equation (8) in the case of PMB rings with axial magnetisation with the same direction. The values of \(F_{z}\) are evaluated at different configurations of the PMB, changing the position of the rotor magnet while keeping the stator magnet fixed. The rotor magnet position is changed in radial and axial directions, thus varying the PMB ring radial and axial offsets \(\Delta r\) and \(\Delta z\). The resulting force \(F_{z}\) is normalised with respect to its maximum value and plotted versus the radial and axial offsets \(\Delta r\) and \(\Delta z\), normalised considering the PMB radial airgap \(\delta\) and axial thickness \(h\), respectively. In the case shown in Figure 2, the axial force strongly depends on the axial offset \(\Delta z\), while it remains almost constant with respect to the radial offset \(\Delta r\), because the air gap between the stator and rotor PMB rings is small. The axial force can be directed either upwards or downwards according to the position and the magnetisation directions of the rotor and stator rings. Only the axial offset range where the axial force is directed upwards and the PMB functions as a thrust bearing is considered to achieve a partial compensation of the rotor weight. The radial stiffness \(k_{r}\), as the axial force, assumes both positive and negative values [22; 23], corresponding to stable and unstable behaviour in the radial direction, respectively, depending on the PMB axial offset. Therefore, the range of a second axial offset where radial stiffness \(k_{r}\) is positive and the PMB works as a radial bearing must be identified.