3. Active magnetic bearings

In active magnetic bearings a stable equilibrium is achieved by means of one or more control loops. In general, five degrees of freedom of the rotor have to be stabilised; see fig. 5. For each degree of freedom a sensor is then necessary to measure the displacement of the rotor. The difference between the measured position z and the desired position z0 is converted in the control loop into a change of the current through the coils that generate the load-carrying capacity.


Fig. 5. General configuration of a magnetic bearing for a rotor with five control loops. Each of the pairs of electromagnets 1-1, 2-2, 3-3, 4-4 and 5-5 has its own control loop (shown here simplified for 1-1) with a displacement sensor. The output signal of the sensor Se, which is a measure of the position z of the rotor shaft, is compared with a reference signal for the desired position z0. The difference signal is converted into a current  via a controller R and an amplifier Am. This signal is added to a direct current i in the one coil and subtracted from the same current in the other coil of the pair of magnets. This produces a resultant force on the shaft, which drives it back to the desired position. The magnet pairs 1-1 and 3-3 control two degrees of freedom of the rotor: a rotation and a translation. The same applies to the magnet pairs 2-2 and 4-4. The magnet pair 5-5 only controls a translation.

The five-fold control system required for the magnetic configuration shown in fig. 5 has the advantage that the parameters associated with each degree of freedom can be chosen more or less independently of each other. (The control loop for one pair of magnets is shown in a highly simplified form.) The associated electronics, however, makes the bearing rather complicated and expensive. For this reason magnetic bearings have been designed that have one or more non-controlled degrees of freedom. In these bearings the stiffness in the associated directions is usually smaller than that for the controlled degrees of freedom; but there is a 'natural' stable equilibrium of forces in these directions. We have built three such magnetic bearings, partly based on data from the literature and have tested them. Let us now take a closer look at the types of bearings we investigated, which were designed primarily for use in vacuum.

3.1 The magnetic configurations

Fig. 6 shows a configuration in which a control loop is only necessary for the axial direction. In the radial directions the magnetic forces and the external forces are in stable equilibrium. The stiffness and load-carrying capacity in these directions are increased by concentric grooves in the pole pieces. (Approximate equations for calculating the radial stiffness as a function of the groove geometry are given in the next subsection.) The coils are energised by a constant current on which a control current is superimposed. The magnitude of the control current is determined by a control loop with feedback of the axial rotor position, measured by the inductive displacement sensor Se.


Fig. 6. Magnetic bearing with only one control loop, for the axial direction (see fig. 12). The stiffness in the radial direction is in-creased by means of concentric grooves in the pole pieces (see inset). In spite of this the stiffness in the radial direction is substantially less than that in the - controlled - axial direction. Se displacement sensor. CC coils energised by a direct current with a super-imposed control current; see fig. 5. The magnitude of the control current is determined by the control loop. The schematised magnetic lines of force are indicated in blue.

The configuration in fig. 6 is simple but has the disadvantage that heat is constantly dissipated in the coils. The magnetic bearing in fig. 7 does not have this disadvantage because it has annular permanent magnets. In coil CC the only heat dissipated is the small amount generated by the control current. In the sensor coil SeC a voltage is induced when an axial displacement of the rotor changes the linked flux. The output voltage of the sensor coil is a measure of the speed of the rotor, so that this voltage can provide speed information for feedback in the control loop. As in fig. 6, a displacement sensor can also be used. The permanent magnets in the right-hand part serve not only for generating voltage in the sensor coil but also, like the permanent magnets in the left-hand part, for generating radial load-carrying capacity.


Fig. 7. Magnetic bearing with permanent magnets PM. Compared with the bearing in fig. 6, less heat is dissipated, since the coil CC is energised only by the control current from the control loop. SeC speed sensor. Instead of the axial displacement of the rotor the axial velocity here is the feedback quantity in the control loop (see fig. 13). The pole pieces have concentric grooves as in fig. 6. The schematised magnetic lines of force due to the permanent magnets are shown in red, the lines of force due to the coil CC are shown in blue. From the directions of these lines it appears that a current in CC increases the flux density in one air gap and reduces it in the other. The current  from the control loop can thus be supplied directly to coil CC (see fig. 5).

As we have seen in the foregoing, the load-carrying capacity of magnetic bearings is proportional to the square of the flux density in the air gap. It is therefore important to make this flux density large, i.e. to ensure that magnetic circuits have a low reluctance. This means that the air gaps must be small: practical considerations dictate a gap of between 0.1 and 0.2 mm. A disadvantage of the magnetic configurations in figs 6 and 7 is that the effect of the rotor temperature on the size of the air gaps is not always negligible. Furthermore, the considerable distance between the outer pole pieces imposes critical manufacturing tolerances on the rotor and the other components.

These disadvantages do not apply to the bearing in fig. 8, in which the axial distance between the pole pieces is much smaller. This bearing has three control coils at 120° spacing, three corresponding displacement sensors and one annular permanent magnet. The annular magnet gives a constant magnetic flux in the annular air gaps. There are three control loops. In the radial directions the equilibrium of forces is stable, and the stiffness is again increased by concentric grooves in the pole pieces.


Fig. 8. Magnetic bearing in which the magnitude of the air gaps is less dependent on the rotor temperature than for the bearings in figs 6 and 7. There is one annular permanent magnet PM and there are three control coils CC spaced at 120° around the circumference. There are also three displacement sensors Se and three control loops, each with its own coil CC. The pole pieces have concentric grooves. The paths of the schematised lines of force (the red ones relate to PM, the blue ones to CC) indicate that a current in CC increases the flux density locally in one outer air gap and reduces it in the other outer air gap. The three loops control three degrees of freedom: a translation (in the axial direction) and two rotations (about axes perpendicular to the axis of rotation). The three control loops are coupled in a special way for this purpose.

3.2 Calculation of the radial load-carrying capacity

Let us first consider fig. 9a, which gives a diagram of an iron armature drawn into the air gap of an electromagnet energised by direct current. The force F acting on the armature will be calculated, with the assumption that the air gaps have the same height g so that the vertical forces are in unstable equilibrium. We shall also assume that the gaps between armature and magnet are much smaller than the height of the armature, i.e. , and that in the iron  is constant and very large. We can calculate F by using equations (4) and (5) for the force densities at a surface enclosing the armature. Fig. 9b gives a schematic representation of the lines of force inside and outside the air gap. (The magnetic lines of force are always perpendicular to the surface of a material for which , since the tangential component of the magnetic field-strength at this surface must be zero.) The red line in the figure is the line of intersection of a 'convenient' enclosing surface with the plane of the figure. In the part of the surface indicated by dashed lines in fig. 9c the force densities can be neglected, since the flux density there is small compared with that in the air gap (). The force densities p1 and p3 on opposite sides of the surface cancel one another out, since the tangential. component Bt of the flux density is zero in the associated parts of the surface. Contributions to the force F come only from the force densities p2. In the associated parts of the surface the normal component Bn of the flux density is equal to zero. The tangential component Bt delivers a negative normal force density, which therefore has the opposite sense to the unit vector n. From (4) we then have
 
(9)

where B is the magnitude of the flux density in the air gap- Using (9) we can calculate the approximate magnitude of the force F on the armature:
 
(10)

where l is the length of the air gap.
 

a
b
c
Fig. 9. Magnetic model for calculating the radial load-carrying capacity of the bearings in figs 6, 7 and 8.
a) The magnetic circuit, in which various dimensional quantities are indicated that are used for calculating the force F on the armature An. The displacement x of the armature is positive in the direction indicated. The flux density results from a current in the coil Cl. The armature is situated in the centre of the gap of height ha.
b) Enlarged cross-section perpendicular to the armature with schematised lines of force. The red line is the line of intersection with a 'convenient' surface S' (see also fig. 2a), which completely encloses the armature.
c) The force densities at the surface S'. Owing to the low flux density in the region indicated by the dashed lines () the force densities here are negligible. The force densities p1 and p3 oppose one another at opposite sides and make no contribution to F. The force density p2 does make a contribution, however. The unit vector n is perpendicular to S' and positive outwards.

Next we provide the pole pieces on both sides of the air gaps with a large number of grooves perpendicular to the x-direction, as shown in fig. 10a. Now the force does depend on the displacement x, and is in addition proportional to the number of grooves Nt

(11)

a
b
c
Fig. 10. Modification of the model in fig. 9 with grooves in the pole pieces.
a) Various dimensional quantities used in the calculation. (The grooves are perpendicular to the plane of the drawing.)
b) The geometrical factor f as a function of the armature displacement x for t = s and h = 10g (from [13]). For this geometry f = f(x) can be approximated by a sine function.
c) Application of the model in (a) for annular pole pieces. Integration of the calculated value of F (see fig. 9) over the whole annular surface gives Frad, the radial load-carrying capacity. Fax axial load-carrying capacity. Frad cannot be greater than about 0.08 Fax.

The geometrical factor f depends on x and also takes account of the groove geometry, determined by the ratios of the dimensions w, t and h to the air gap g. (B in (11) is the flux density at the narrowest part of the air gap.) Fig. 10b shows the variation of f for t = s and h/g = 10 [13]. It can be seen that f is approximately linear with x for values of x up to about w/10.

The mean magnitude of the 'suction force' F on the grooved armature can be calculated from equation (11):
 
(12)

where

The factor  in equation (12) is equal to the value of the force density pn perpendicular to the pole pieces, given by (4). Field calculations show that fm reaches a maximum value of 0.0196 when  and h >> g [11].

The above expressions can be used for estimating the forces between annular grooved pole pieces of magnetic bearings; sec fig. 10c. If the function f = f(x) in fig. 10b is approximated by a sine function, integration of the force calculated from (12) over the annular surface gives a ratio of the radial to the axial bearing force of

(13)

Substituting the maximum value for fm and the corresponding ratio , we find that in this case the radial force cannot become greater than about 8% of the axial force.

3.3 Calculation of the length width ratio of the rotor

In the magnetic bearing of fig. 6 stability in the axial direction is obtained through control of the magnitude of the bearing gap. Instability can still occur, however, during an angular rotation of the rotor about a line perpendicular to the centre-line. This instability only occurs when the ratio of the outside diameter to the distance between the pole pieces exceeds a critical value.


Fig. 11. The forces and torques acting on the rotor in fig. 6 when lit is rotated through an angle  about an axis perpendicular to the axis of rotation. 2l length of the rotor. krad radial stiffness of a half-bearing; krad is positive for the direction chosen here for the vectors kradl torque acting on one half-bearing. Mt resultant torque. Rm mean radius of the annular pole pieces.

Fig. 11 shows the torques that act on the rotor if it is given a small angular rotation  as described above. The total torque Mt acting on the shaft is then:

(14)

where  is the torque produced at each pole piece in the tilted position, 2krad is the total radial stiffness of the bearing and 2l is the distance between the pole pieces.  can be calculated from the force densities in the axial direction. For an annular pole-piece surface:

(15)

where kax. is the axial stiffness  for one half of the bearing, see fig. 10c, and Rm is the mean radius of the annulus. The condition for stability is that the stiffness  should be negative. This leads to the condition:

(16)

The ratio 2kdrad/kax is almost independent of the radius Rm. In the bearing of fig. 6 that we designed this ratio is about 0.26. To ensure a sufficiently high value for G we made .

The condition (16) does not apply for a bearing like the one in fig. 8. The three independent control loops here do not only control the axial displacement; they also control the angular rotation about a line perpendicular to the centre-line.