4. Dynamic stability of the control loops

4.1 Control with displacement feedback

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

(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 .

a
b
c
Fig. 12. Control of the height of the bearing gap by rotor-displacement feedback (see fig. 6).
a) Block diagram of the control loop. v0 desired signal value. PID controller (proportional, integrating and differentiating action) with transfer function  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.
b) Corresponding Nyquist diagram. Re real axis. Im imaginary axis. In this diagram the complex value  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.
c) Simplified control loop for calculating the stiffness  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.

4.2 Control with velocity feedback

a
b
c
d
e
Fig. 13. Control of the height of the bearing gap by means of rotor-velocity feedback (see fig. 7).
a) Block diagram. The transfer function of the controller R is equal to  time constant. The transfer function of the velocity sensor Se is equal to sK. See also the caption to fig. 12a.
b) Diagram of the electronic circuit with operational amplifiers which produces the transfer function mentioned above in the controller.
c) Simplified block diagram of the circuit in (b). It is assumed that the operational amplifiers have an infinitely large negative gain. With K2 = 1 and K3 = 2 this diagram gives the required transfer function.
d) Nyquist diagram corresponding to the complete control loop. For  the end  describes a circle with centre (, 0) and radius , with  and . The continuous curve applies for  and ; the dashed curve applies for  and .
e) Comparison of the behaviour of the control with the equilibrium of a 'ball on a hill'. 1 stabilised equilibrium - the resultant FPM of the forces due to the permanent magnets is zero. Fs perturbing force. 2 the hall is about to roll down the hill with increasing velocity - FPM+Fs increases in magnitude, but the control produces a repelling force FCC from the control coils. 3 stabilised equilibrium state - the ball is driven over the top of the hill by the increase in FCC and finds a position on the other side such that FPMFs,. 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.