2. Theoretical background

2.1 Calculating the load-carrying capacity

There are various ways of determining the force acting on an object in a magnetic field in air or in vacuum. A very straightforward and general method is one in which Maxwell's equations in vector notation are used to calculate a fictitious force density  at a surface that completely encloses the object:

(2)

where the vector  is the magnetic flux density,  the outward-directed unit vector perpendicular to the enclosing surface and  the magnetic permeability of free space; see fig. 2a. The total vector force  acting on the object is then found from

(3)

where S' is the total area of the enclosing surface with surface element ds. The enclosing surface S' may coincide with the surface S of the object.

After some manipulation, expressions for the component  of the force density perpendicular to the surface ('tensile or compressive stress') and the component  along the surface ('shear stress') can be derived from (2):
 

	(4)
	(5)

where  is the normal component and  the tangential component of . The relation between the two components of  and the angle  that  makes with the surface is shown in fig. 2b: for  = 0 and for  the component  and  is at a maximum, and for  the component  and  is at a maximum.

In a magnetic bearing we have a cylindrical object that is levitated in a magnetic field. For the cylindrical surface of the shaft it is then desirable that p_n should be large and p_t small, since p_n produces the load-carrying capacity and p_t usually causes an opposing torque ('friction'). This requirement can be satisfied if  = 0 or if . For  = 0 the lines of force are tangential to the surface; see fig. 2c. It can easily be seen that in this case the force F_z, the force density integrated over the area of the surface, becomes smaller if the shaft is displaced from the region with the largest flux density, i.e. . The equilibrium of F_z with the gravitational force F_g is therefore stable. For  the lines of force cut the surface perpendicularly; see fig. 2d. We now have , so that the equilibrium between the magnetic force and the gravitational force is unstable.

The question is now whether the situation shown in fig. 2c is in conflict with Earnshaw's theorern as stated above. We shall find that this is not the case.

Fig. 2. a) Definition of quantities for calculating the force acting on an object in a magnetic field.  magnetic flux density; B_t is the tangential component, B_n the normal component. S' surface completely enclosing the object. S surface of the object.  force density; p_t is the tangential component, p_n the normal component.  unit vector perpendicular to S' (positive outward).  resultant force on the object, where ds is a surface element of S'.  angle of  to the surface S'.
b) The components p_t and p_n of  as a function of the angle , from equations (4) and (5).
c) Maximum  obtained at p_t, = 0 and  = 0°. In this way a stable equilibrium is established between the load-carrying capacity F_z and the gravitational force F_g, since .
d) A similar situation at . The equilibrium is now unstable, however, since . Fig. 3 shows how (c) and (d) are attained in practice.

2.2 Obtaining the required direction for the lines of force

A situation as shown in fig. 2c can be obtained by making use of the effect of superconductivity. If niobium or lead ('superconductors of the first kind') are in the superconducting state, any change in the magnetic flux density causes permanent eddy currents at the surface of the material, their magnitude and direction being such as to cancel the effect of this change. No magnetic flux can therefore be generated in these superconductors. This situation corresponds to that in a hypothetical material whose magnetic permeability is equal to zero. The lines of force representing the magnetic field cannot penetrate into the material; see fig. 2c. In fig. 3a this situation is shown again for the case where the centre-line of the shaft coincides with the plane of the diagram and for a different direction of the flux density. The maximum force density that can be achieved is , from (4). The flux density at the boundary surface of superconductors of the first kind must remain limited to between 0.1 and 0.3 T (tesla or Wb/m²), since otherwise the superconductivity would vanish. The corresponding force density is at the most 3 10⁴ N/m² or 0.3 bar. The load-carrying capacity that can be achieved is thus fairly small. Also, there is the great disadvantage that part of the rotor has to be cooled to a temperature of 10 K or lower. As explained above, the magnetic bearing with superconducting shaft ends does yield a stable equilibrium of forces. This is not in conflict with Earnshaw's theorem, since this theorem only applies to cases where  > 1.

Fig. 3b shows the situation for the case where the shaft ends are made of diamagnetic material. The angle that the lines of force make with the surface now has a value between 0 and . In the material the lines of force diverge, because the flux density here is less than in air. Since  differs very little from unity, the force density is limited. The load-carrying capacity is even less than in the case of fig. 3a. It should be noted that the situation shown in fig. 3b can also be obtained by means of an alternating electromagnetic field, because eddy currents opposing the alternating magnetic field are then induced into the shaft ends. The eddy currents cause losses, however. A rough calculation shows that the loss on levitating a copper plate, irrespective of its thickness, is about 2.8 kW per m² of bearing surface. The loss with an aluminium plate is about 1.3 kW/m²

Fig. 3. Three methods of 'levitating' a shaft in a magnetic field (see fig. 2).
a) With  = 0°. The shaft must be cooled to a temperature at which the material becomes superconducting. The magnetic permeability  is then effectively equal to zero.
b) With 0 <  < . The shaft must be made of diamagnetic material, for which 0 < , < 1. In both these cases the equilibrium of forces in the vertical direction is stable and the shaft 'floats' on the magnetic field.
c) With . The shaft now bas to be made of ferromagnetic material, for which  >> 1. Because of the reversal of the force density, the shaft is suspended in the magnetic field. As Earnshaw's theorem states, however, the equilibrium of forces is not stable.

Fig. 3c shows a situation where the direction of the flux is perpendicular to the bearing surface. As in fig. 3a, the force density is equal to . We can work here with materials of high magnetic permeability and high saturation flux density. In cobalt iron, for example, the flux density can amount to as much as 2.4 T. The maximum force density is then 23 10⁵ N/m² or 23 bars. The equilibrium of forces is not stable, however, since Earnshaw's theorem now applies. To obtain an effective bearing the size of the bearing gap must be controlled. The method of control will be discussed later on in this article.

2.3 Calculating the load-carrying capacity for a sinusoidal surface-current density

The mean force density  for the three cases of fig. 3 can be expressed in mathematical terms. We assume that the magnetic field along the surface of a medium of magnetic permeability  is generated by means of a sinusoidal surface-current density of peak value  (in A/m); see fig. 4a. Using equation (4) we then find for 
 

(6)

where  is the wavelength of the surface-current density.

Equation (6) can be derived as follows. We assume that the surface-current density  is located in a rectangular co-ordinate system (x,y,z) in the plane z = 0, with the currents perpendicular to the plane x = 0 and the surface-current density (current sheet) extending to infinity in the positive and negative y-directions. For  the permeability  is not equal to unity and is constant, and for z < a we have  = 1. Let the space be divided into three parts. In part 1, for , the magnetic potential function has the value ; in part 2, for , it has the value ; in part 3, for z < 0, it has the value . In each of these parts Laplace's equation à  = 0 applies, with i = 1, 2 and 3. We are considering a two-dimensional problem, so that the derivatives with respect to x are equal to 0 and . For the magnetic field-strengths in the three parts we can have:

With the aid of the above differential equations we can calculate the field-strengths along the plane z = a. The following boundary conditions then apply:

H_1y = H_2y and H_1z = H_2z, for z = a;
H_1y = H_1z = 0, for z = ;
H_3y - H_2y = A, and H_2z = H_3z, for z = 0;
H_3y = H_3z = 0, for z = 

The solutions for the potential functions must then satisfy:

in which the boundary conditions for z =  and z =  have already been included. The remaining boundary conditions give four equations, which we can solve for A₁, A₂, A₃ and B₂. We thus obtain equations for the field-strengths at the boundary plane z = a:

from this and (4) we obtain an equation for the force density p_n perpendicular to the boundary plane:

Integrating p_n with respect to y from  to  and dividing the result by , we find equation (6) for the mean force density perpendicular to the boundary plane.

Fig. 4. a) Definition of quantities for calculating the density of forces p_n at the interface of two media with  and  in the presence of a sinusoidal surface-current density of magnitude . The interface coincides with the plane z = a in a rectangular coordinate system (x,y,z). The surface-current density (current sheet) is situated in the (x,y)-plane and extends to infinity in the positive and negative y-directions. ,  and  magnetic potential functions in the spaces z a, 0  z < a and z < 0.
b) The factor  from equation (6) for the mean force density, plotted as a function of . Here  is the value of  for superconducting material,  the corresponding value for diamagnetic material and  is this value for ferromagnetic material. The diagram shows that higher force densities can be obtained with superconducting and ferromagnetic materials than with diamagnetic materials.

Since  is very small for diamagnetic materials, the force density obtained with these materials is not very large, as indicated earlier. For bismuth, the diamagnetic material with the lowest permeability, we have . The force density is thus about  times smaller than for materials in the superconducting state with = 0 or for ferromagnetic materials with  >> 1; see fig. 4b.

The mean force density at z = a in the z-direction is . After differentiating , with respect to a we then find the mean vertical stiffness per unit area in the z-direction:
 

(7)

We see from this that for diamagnetic and superconducting materials the mean vertical stiffness is negative. Thus, with these materials, as we have already seen, a stable equilibrium of forces is possible for levitation. With other, 'ordinary' materials this stiffness is positive; the equilibrium of forces is then unstable, as would be expected from Earnshaw's theorem. It also follows from (6) that the conductors of the surface-current density must lie as close as possible to the boundary plane z = a to produce the maximum force density. For a = 0 the mean absolute force density  for ferromagnetic and superconducting materials is given by:
 

(8)

In practice the force densities for superconducting materials and ferromagnetic materials are unequal, since, as we noted earlier, in superconducting materials the maximum flux density at the boundary plane must remain limited.