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 z_{0} is converted in the control loop into a change of the current through the coils that generate the load-carrying capacity.
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.
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.
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.
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 p_{1} and p_{3} on
opposite sides of the surface cancel one another out, since the tangential.
component B_{t} of the flux density is zero in the
associated parts of the surface. Contributions to the force
F come only from the force densities
p_{2}. In the associated parts of the surface the normal
component B_{n} of the flux density is equal to zero.
The tangential component B_{t} 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.
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 N_{t}
(11) |
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 p_{n} perpendicular to the pole pieces, given by (4). Field calculations show that f_{m} 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 f_{m} and the corresponding ratio , we find that in this case the radial force cannot become greater than about 8% of the axial force.
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 shows the torques that act on the rotor if it is given a small angular rotation as described above. The total torque M_{t} acting on the shaft is then:
(14) |
where is the torque produced at each pole piece in the tilted position, 2k_{rad} 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 k_{ax}. is the axial stiffness for one half of the bearing, see fig. 10c, and R_{m} 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 2kd_{rad}/k_{ax} is almost independent of the radius R_{m}. 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.