## DW x

4.3.1.4 Demagnetisation model for Cogging Force

The above method accounts for the electromagnetic force of the VHM. At zero current there is no area swept by moving between rotor position lines on a flux linkage vs. mmf graph and hence no change in co-energy and corresponding force. Section 4.3.1.1, the calculation of force by Maxwell stress, demonstrated that significant magnetic forces are present in the absence of current. These forces result from the tendency of the translator teeth to align with the magnets and are called cogging forces. An alternative approach, which considers the co-energy stored within each magnet is required to calculate the cogging force of a VHM. To achieve this the data must be converted into the flux-mmf for a permanent magnet [78]. The flux emanating from a magnet corresponds to the mmf within that magnet according to (4.18), which describes its loading curve.

tm f

MoMr

V Am

The intersection of the load line and the de-magnetisation curve is the operating point of a magnet, which will change according to Figure 4.22. Over one cogging force cycle the operating point moves up and down the PM demagnetisation curve. The area swept out by this line represents the change in co-energy contained within the magnet [78,79].

Figure 4.22: Flux-mmf diagram for a PM

This method has been applied successfully to PM machines with large magnet pole pitch. In the VHM however, the pole pitch is of the order 10 to 20 mm and magnets sit side by side. The cogging force is produced by the interaction of a tooth on the translator and a pair of magnets of opposite polarity on the stator. It is therefore necessary to consider the effect of a pair of magnets and their respective operating points.

Figure 4.23: Divergence of operating points of PMs

When a rotor tooth overlaps 2 PMs equally, the unaligned position, the reluctance seen by each magnet is identical and hence the operating points coincide. As the rotor tooth moves closer to either magnet, the reluctance path of that magnet is reduced and hence more flux will flow. Figure 4.23 shows the tooth moving towards the North pole magnet and the corresponding move up the load curve. Simultaneously the reluctance path for the South pole increases, reducing the flux flow and moving it down the load line. The divergence of the operating points for the VHM is demonstrated by the FEA results shown in Figure 4.24. The area enclosed by the airgap lines for each magnet and the demagnetisation curve is equal to the co-energy, and hence the cogging force is equal to the rate of change of this co-energy with distance.

The operating points are at a maximum divergence when the translator is in the fully aligned position. The area between the two load lines and the demagnetisation curve is then equal to the total energy that has to be overcome in order to move the translator.

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 - magnet pair 2 — magnet pair 3 - total position(mm) 10 15 position(mm) Figure 4.25: Cogging force on one face of the VHM The cogging force calculated in this manner is shown in Figure 4.25. There are several points of interest to note from this graph, firstly that the period of the resulting wave form is over the entire 24 mm rotor pitch, whereas the period of the contribution from magnet pair 2 has a wave length of half this value. Secondly, the two maxima in the leftmost 12 mm of the total wave form are only around % the value of the right hand maxima. Inspection of the two outside magnet pairs, one and three, show that it is these magnets which do not behave uniformly over the two halves of the period. It is clear referring back to Figure 4.11 on page 74 that there are different flux patterns corresponding to the two misaligned positions, depending on whether the edge magnets are above teeth or slot regions of the translator. In the former case, Figure 4.11A and position 18, there is a small amount of flux that flows through the translator back iron. This implies a lower reluctance path allowing a greater flux flow through the iron, which represents more energy stored in the airgap and hence a greater force required to change the position.

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