As stated above, one control loop is required for the magnetic bearing in fig. 6 or fig. 7 and three control loops are required for the bearing in fig. 8. Let us first consider the behaviour of a control loop with feedback of the displacement, with a 'PID controller'. A PID controller contains elements with proportional, integrating and differentiating action.
The movement of the rotor in the axial direction is described by the differential equation
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(17) |
where z is the displacement of the rotor from the centre position, M is the mass of the rotor, kPM is the (positive) stiffness of the rotor in the field of the permanent magnets in figs 7 and 8 (or in the field due to a constant current in coil CC in fig. 6), F is the force produced by the control current and Fs is a force that perturbs the equilibrium of the rotor. The minus sign for the last term indicates that the equilibrium of the rotor in the magnetic field is unstable. Fig. 12a shows the complete block diagram of the control circuit with displacement feedback. The figure also shows the separate transfer functions of the sensor, the controller, the control-coil current amplifier, the control coil and the mass of the shaft. The s in the various transfer functions represents the complex Laplace variable in the usual convention.
The transfer function H(s) of the open control loop is equal to the product of the transfer functions of the individual blocks. We have:
 |
(18) |
(The significance of the symbols is explained in the caption to fig. 12.)
The poles of H(s) are defined as the values of s
for which the denominator is equal to zero. There is therefore one positive
pole, 
,
where 
.
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where v is the output voltage of PID. Kc gain for
the proportional
action. 
and 
time
constants of the integrating and differentiating
actions. 
factor limiting the differentiating action for high signal frequencies.
s complex Laplace variable. Amp signal amplifier with transfer
function i = Av. i output current. A gain.
CC control coil with idealised transfer function F = ci,
neglecting eddy-current effects. F axial force.
Fs axial perturbing force. MR rotor with transfer
function z = (F +
Fs)/(Ms2 -
kPM). M rnass. kPM axial stiffness
due to the field of the permanent magnets. z axial rotor displacement.
Se displacement sensor with transfer function
vs, = Kz. vs output voltage.
K gain. 
of the transfer function of the open loop for sinusoidal signals of
frequency, 
is plotted from the origin
for 
.
Here
is the product of the transfer functions of all the blocks, with the imaginary
quantity
substituted for s. The distance from the origin O to the curve
is equal
to 
,
the angle between H and the positive real axis represents the
phase angle. The length of the vector a = 1 -
H, apart from a factor
containing ,
is a measure of the stiffness of the bearing to perturbing forces
Fs. 
to perturbing forces.
The behaviour of the control loop for sinusoidal signals of angular
frequency
can be studied by substituting the imaginary
quantity 
for s. We then obtain the following expression for the complex transfer
function 
:
|
(19) |
The stability of the control loop can be investigated by mapping the variation
of for
increasing values
of in
the complex plane, thus producing the Nyquist diagram as shown in fig.
12b.
For 
we
have 
and the phase angle
is 
(
);
for 
we
have H = 0 and the phase angle
is 
.
(Re and Im represent the real and imaginary parts
of .)
For Nyquist's stability criterion to be satisfied, when the curve is traced
out from 
to , the
vector from the point (-1, 0) to a point on the curve must rotate anticlockwise
through an
angle 
,
where n is the number of positive poles of the transfer function.
In our case with one positive pole s
= 
this
vector must therefore rotate through an
angle .
For the
point of the vector lies at infinity on the negative Re-axis. It is
true that
for a
phase
angle
was calculated
at 
, but
if the
term 
in the denominator of (19) is replaced by a
term
the phase
angle is
for
and 
.
Nyquist's stability criterion is thus satisfied and the control is stable.
In reality the behaviour of the control is less satisfactory than indicated in fig. l2b because eddy currents are induced in the pole pieces. This means that additional damping terms are included in the transfer function. By laminating the pole pieces or making them of a material of high electrical resistivity such as cobalt iron, this undesirable effect can to some extent be reduced.
The stiffness of the rotor to a perturbing force Fs can also be indicated in fig. 12b. Let us consider the simplified control loop shown in fig. 12c. It can be shown that
and from this the stiffness is
 |
(20) |
The term 
is equal to the length of the vector a from (-1, 0) to a point
on the curve of the Nyquist diagram. From equation (20) and the length of
a as a function
of it
follows that the stiffness to perturbing forces is very considerable
when
and 
.
At low frequencies the high stiffness is due to the integrating action of
the controller, at high frequencies it is due to the inertia of the rotor
mass. We see that for finite frequencies maximum stiffness is achieved when
the curve in the Nyquist diagram remains as far away as possible from the
point (-1, 0). This condition also gives the least amplification of sinusoidal
perturbations, i.e. the greatest possible stability.
The perturbing force Fs thus causes a change in the current in the coil CC in figs 6, 7 and 8, which acts to oppose the initial displacement due to Fs. A disturbance of the equilibrium therefore brings about extra heat dissipation in the coils. We shall see that with velocity feedback this dissipation is greatly reduced.
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. 
time constant.
The transfer function of the velocity sensor Se is equal to sK.
See also the caption to fig. 12a. 
the
end describes
a circle with centre
(
, 0) and
radius 
,
with 
and 
.
The continuous curve applies
for
and 
;
the dashed curve applies
for 
and 
.
Fs,.
In 3 FCC = 0, so that no heat is dissipated in the
control coils.
Fig. 13a shows a block diagram of the control loop with velocity feedback.
Instead of a PID controller a special controller is used here whose transfer
function
is ,
where
is a time constant and Kc a constant that determines the
proportional action [9].
Fig. 13b shows the controller network used, with the operational
amplifiers to give the required transfer function. Fig. 13c shows
the equivalent block diagram, in which it is assumed that the operational
amplifiers have an infinitely high negative gain. The transfer function of
the controller has a positive
pole 
.
The action of the controller alone is thus not stable, but when it is combined
with the other elements of the control loop a stable system is obtained.
The transfer function of the open loop of fig. 13a for sinusoidal
signals of angular
frequency
is:
With
the real part
of 
becomes:
and the imaginary
part becomes:
Fig. 13d gives the corresponding Nyquist diagram with curves
for , and
for , . The real axis
is intersected
for 
at
the
point 
.
The curves start
for 
at
the origin and also end there
for ,
in both cases tangential to the imaginary axis at the origin. The transfer
function H(s) has two positive poles, s
= 
and
s
= 
. For
Nyquist's stability criterion to be satisfied the vector from (-1, 0) to
the curve must therefore rotate anticlockwise
from
to through
an angle 
.
For 
the curve is traced out in such a way that the point (-1, 0) is enclosed.
Nyquist's stability criterion is then satisfied. It can be shown that
for the
curve becomes a circle with its centre at
(,0). The circle
shown in fig. 13d
with
and gives
the value 1
for 
at
all frequencies. The stiffness to perturbing forces of angular
frequency
is then, from
(20), 
.
As mentioned above, the system with velocity feedback takes less energy than the system with displacement feedback; see fig. 13e. If the rotor is in the central position, the forces of the permanent magnets oppose one another and a stabilised equilibrium of forces prevails. This is represented in the figure as position 1 of a 'ball on a hill'. If the stabilised equilibrium is perturbed by a force Fs, the ball rolls down the hill with increasing velocity. The velocity produces an opposing force, which is generated by a current in the control coils: position 2. The ball 'shoots over the hill' and finds a stabilised equilibrium at the other side: position 3. The current in the control coils is then virtually zero.
The position of the rotor with velocity feedback is therefore not always
the same but depends on the magnitude of the perturbing force. This is no
great problem, because the control behaviour described above takes place
in a small region - 10 to
20 
-of
the bearing gap. A disadvantage is that the control has to be 'started',
because the rotor is pulled to one side by one of the permanent magnets when
the bearing is uncontrolled. This problem is solved by energising the coil
briefly with a current surge from the appropriate direction, thus disengaging
the rotor and giving it an initial velocity. The control then comes into
operation, and causes the rotor to seek a stable equilibrium.
A version of the bearing system in fig. 7 with velocity feedback and an air gap of 0.25 mm has an average dissipation of only 5 mW at an axial stiffness of 6000 N/mm and a radial stiffness of 345 N/mm.
