# 1. Introduction

Dr Ir E. M. H. Kamerbeek is with Philips Research Laboratories, Eindhoven.

In 1725 Jacob Leupold wrote in his book Theatrum Machinarum Hydraulicarum: 'Whosoever therefore seeks to achieve with his Machines as much as the Theory teaches should do away as far as possible with all friction...'. Today the limitation of friction losses in bearings is still an engineering challenge. The magnetic bearing is distinguished from other types of bearing by an almost complete absence of friction. A difficulty, however, is that in practice magnetic levitation is always accompanied by an unstable equilibrium of forces, so that height control with feedback is necessary. This article shows how the load-carrying capacity and stiffness of magnetic bearings can be calculated and the problems of stability can be solved in certain cases.

Magnetic levitation has a number of advantages as a means of supporting a shaft in a bearing. There is virtually no friction and no wear, so that magnetic bearings can have a long life and run at high speeds. They are quiet, and are particularly suitable for applications in which there should be no contamination from lubricants, as in vacuum. In addition it is fairly easy to adjust the position of the axis of rotation via the bearing forces and suppress the effects of unbalance.

From Earnshaw's theorem formulated in 1842, it can be shown that in a static magnetic field the equilibrium of the forces acting on a body is not stable if the magnetic permeability of the materials in the field is greater than 1. A stable equilibrium in a field of forces (Fx, Fy, Fz) requires negative  as well as negative ; see fig. 1. A necessary - but not sufficient - condition for stability is thus

 (1)

It can be shown that when an object is levitated in a static magnetic field in which there are materials with  > 1 then

The equilibrium of forces is then unstable and a small displacement from the unstable equilibrium state causes a greater deflection of the object. In an electrodynamic system - characterised by the action of forces on current-carrying conductors - in air and with constant currents we have

This is also true for systems with permanent magnets only with  = 1 ('hard' permanent magnets). In both the last two cases the stability condition is thus not satisfied.

Stable equilibrium is however possible in a magnetic system if the magnetic permeability  is locally less than 1. This is the case in systems with diamagnetic materials and in systems with materials in the superconducting state. In superconducting materials the magnetic permeability is effectively zero. In the diamagnetic materials with the lowest permeability, graphite and bismuth, the value of  differs only slightly from unity. For this reason the load-carrying capacity that can be achieved with these materials in magnetic levitation is very small. Temperatures near absolute zero are necessary to bring materials into the superconducting state. Although the load-carrying capacity that can then be achieved is greater than with diamagnetic materials, 'superconductive' bearing systems have as yet little practical usefulness.

To design a magnetic bearing for practical applications it is therefore necessary to compensate for the instability of the equilibrium of forces. By the expedient of adding feedback control loops it is then possible to satisfy the condition of eq. (1).

Fig. 1. A stable equilibrium of forces. An object with the gravitational force Fg acting on it is situated at the position (x0,y0,z0) in a rectangular coordinate system (x,y,z). Also defined is a field of forces (Fx,Fy,Fz), whose magnitude is given for a small region around the point (x0,y0,z0) at the bottom of the figure. (Fx, Fy and Fz are taken to be positive in the x-, y- and z-directions; the torques acting on the object are not considered.) When the object is displaced to the point (x0,y0,z0 + dz) the resultant force Fg - Fz( z0 + dz) will repel the object back to the starting position. The equilibrium of the forces that act on the object in the z-direction is thus stable. Similar considerations apply to displacements in the x-and y-directions. For stability it is thus necessary in any case to satisfy the conditions  < 0,  < 0 and  < 0.

An object freely levitated in space has in general six degrees of freedom: three translations and three rotations. A shaft supported in a bearing has had five of its degrees of freedom removed. To obtain a stable equilibrium we must therefore provide a maximum of five control loops for the magnetic bearing. With an appropriate geometry only one control loop, relating to a translation, may be sufficient in some cases. The remaining two rotations and two translations are then characterised by a negative stiffness for the corresponding torques and forces.

The method of calculating the load-carrying capacity and stiffness of magnetic bearings will next be described. This will be followed by a discussion of some bearing designs we have investigated that have fewer than five control loops. After examining the dynamic behaviour of the control loops, the article concludes with a discussion of the instability of a rotor at very high speeds in a magnetic bearing that we have designed.