Basic principles of analytical 2D radial magnetic bearing design

In this paper an analytical solution for the forces of a magnetic bearing is presented. The development of the equations is based on the magnetic scalar potential Φ. Since the consideration of a plain model (2D) is sufficient, we solve the Laplace equation in polar coordinates. The magnetic flux density B is derived from the scalar potential. The solution is a sum of a series of magnetic field waves with different orders. Only the radial and tangential components of the magnetic flux density (Br ,Bt) along a closed line surrounding the levitating object are relevant for the calculation of forces. In the next step the Maxwell stress tensor is calculated for each point on this line. This results in mechanical stress and the integration of the mechanical stress yields to the desired force. The same procedure applies for the torque in electrical machines. First, the equation for tangential forces (which forms the torque) are derived. It was found, that only magnetic field components with the same order generate a resulting torque. More interesting for a magnetic bearing are the force generating field components. There only field waves, whose orders differ by ±1, yield a resulting force. Finally, the equation of forces for a magnetic bearing can be represented in an analytical way as a series of magnetic field components. These kind of representation is useful for further design and optimization considerations.