(10) By substituting equation (10) to the general Maxwell-type force equation for normal forces (11), the resultant forces can be concluded as comprising three contributions as shown in equation (12). (11) (12) Controllable excitation force FE prediction with a series of different PM thickness For self-bearing machines, the main contribution to the bearing force is also known as Maxwell normal force due to the interaction between PM field and armature winding field in the airgap region. The magnitude of resultant force represents the fundamental capability of magnetic bearing performance in terms of maintaining rotor position and avoiding the physical contact between rotor and stator. By keeping a constant armature winding field (mmf equals to 250A) in phase 1 of the leading design listed in Table 31, the magnitude of force variation (two-section combined value) with respect to magnet thicknesses in local x- and y-direction predicted from two-dimensional, non-linear finite element calculations is shown in Figure 310. Figure 310 Finite element predicted magnitude of phase-1 force variation with various magnet thickness ( in local x- and y-direction. As would be expected for phase 1, which is aligned along the x-axis, the local of force dominates over the y-component and is the main contributor to the bearing function, especially for small values of magnetic thickness. It is worth noting that the polarity of current in the stator produces an attractive force towards phase 1, i.e. it adds to the attractive force produced by the reluctance force produced by the interaction of the rotor core and the stator winding.
Tooth arc 32 ° (mech) Motor dimensions Stator axial length 200 (2×100) mm Airgap length 1 mm Magnet thickness 2 mm Stator outer diameter 136 mm Stator inner diameter 64 mm Rotor outer diameter 62 mm core thickness 10 mm body width 12 mm Stator core thickness 12 mm Rotor core thickness 10 mm Tooth tip edge thickness 1.0 mm Slot 20.5 mm The predicted variation in the open-circuit flux density along a path at the mean radius of the airgap is shown in Figure 328, for the particular case of a rotor angular displacement of 0º. The mean value of the flux density under the tooth of phase 1 is 0.69T, a value which in reasonable agreement with the 0.65T predicted by equation (3) which forms the basis of the analytical force constant calculations.
Power rating (Total of two sections) P ~6.9 kW Torque rating (Total of two sections) T ~22 Nm Max. Speed ~3000 rpm Max. step changing force load (single section) ~170 N Figure 422 Relationship between torque rating and maximum bearing capability
Step 3 Dual-axis PID tuning and levitation experiment Y-direction static lift-off measurement from back-up bearing position It was demonstrated in section 6.3 that the finite element predicted forces show excellent agreement with the experimental results static uncontrolled lift-off experiments. The simulation results presented previously in section 4.2.1 demonstrated that significant improvements in performance for a given combination of control gains could be achieved by incorporating UMP compensation into the control scheme, in particular the ability to maintain performance with significantly reduced controller gains and hence increased reliance to noise. Hence, for the controller used in the experimental testing of levitation, a look-up table of finite element estimated open-circuit unbalance magnetic pull (UMP) was added into the closed-loop system to compensate for the effect of UMP. Static lift-off is achieved in the Y-direction from the back-up bearing position designed as the step has two major purpose: Identifying the maximum moving clearance, i.e., maximum rotor eccentricities from back-up bearing position at either end. Due to the mechanical tolerance for constructing back-up bearings, the rotor eccentricities at lift-off positions are different as shown in Figure 72 and Figure 73. The corresponding eccentricity values represents the maximum moving clearance at either end of this constructed prototype machine. Determining the minimum value of proportional gain for single-axis (Y-direction) levitation. According to the well-established tuning strategy for PID controllers, proportional gain is usually the first value determined for starting with a marginally stable rotor movement (which means that the rotor is expected to continuously oscillate with a proportional only controller). A higher proportional gain results in increased oscillation frequency, which is unnecessary until an asymptotically stable system achieved. On the other hand, insufficient proportional gain is unable to provide sufficient control action to levitate rotor from back-up bearing position. By using the results from lift-off experiments, the force error at lift-off position can be
Side view Top view Figure 718 Side and top views of the rotor displacement of self-bearing unit 1 at maximum Ys1 and self-bearing unit 2 at maximum Xs2 (Note – rotations exaggerated to aid clarity compared to rotations expected in practice) In order to compare the magnitude and direction of the demand radial bearing force estimated using different coordinate systems of of gb and gs. It is useful to represent the rotor displacement conditions of Figure 718 using radial coordinate in the end-on view of Figure 719. The corresponding radial displacement of coordinate systems of of gb and gs are simply calculated according to equation (70) and (71) respectively. (70) (71) As a result, the corresponding estimated radial force, and , using sensor-measuring coordinate system, gs, are directed in the negative Y and X direction respectively with the maximum value. However, the axial separation between the sensor position and the bearing unit centre, which is just under one third of the total axial length, will results in a significant force error occurs when compared to the actual required force, and, in terms of both magnitude and in particular direction. For such a case, a control system