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
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:
(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 .
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 :
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
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.
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 .
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.