## Details of the Analysis

A line of length 'a' in the r direction in the r-z plane, as defined in Figure 5.10, represents a disc in a three dimensional coordinate system. The total flux linking such a disk in the z direction is equal to the flux density Bz multiplied by the area of that surface. Expressing this as a sum of elemental disks gives ( 5.11).

Gz a

= 2nj Bzrdr

In order to fully account for the axisymmetric nature of the magnetic flux flow, the analysis software solves ( 5.11) in terms of a 'modified radial potential', which gives a value of r x A for each element. Calculation of flux is hence obtained by multiplying this value by 2n.

5.2.3 Details of Results 5.2.3.1 Translator

Figure 5.10: Flux lines of translator

Figure 5.10 shows the magnetic flux lines due to magnetic excitation only. The flux path through the magnets is primarily parallel to the axis. The leakage through the stainless steel central support is negligible compared to that on the translator surface.

The variation of both radial and axial flux density with position along the translator is shown in Figure 5.11. The flux density in the two directions follow a similar pattern but half a pole width out of phase with each other. At p=25, the surface above the centreline of a steel piece, there is no axial flux, which verifies the use of a zero normal flux condition at the boundary of the FEA model. The radial flux reaches a clear peak at the steel / magnet boundary, although the rapid decline of this value is less significant further away from the translator surface.

Figure 5.11: Flux densities moving along the translator
Figure 5.12: Flux densities moving away from translator

Figure 5.12 shows the same data but presented in terms of distance above the translator surface. Regardless of the position above translator or orientation considered, the flux density can be seen to drop off rapidly as the distance from translator increases.

This data can be used to give some confidence in the simple model used to design the prototype. In Figure 5.11 the surface values of radial flux predicted in section 5.1 above were also included and show a reasonable first order approximation to those predicted by the FEA.

The graph of Figure 5.13 shows the average values of radial flux density over the steel and magnet sections of the translator, calculated by the FEA, plotted on the same axes as the flux density as calculated by the simple model using Equations (5.4), (3.16) and (5.5).

 — exponential approx. over steel — exponential approx. over pole — ave. FEA over magnet ----- ave. FEA over pole \ \ v v .. height above translator (mm) 15 20 25 height above translator (mm) Figure 5.13: Comparison of model with 3D FEA averaged data The simple model assumed that the flux density across a surface pole was constant, the limitations of this assumption have been demonstrated, Figure 5.11. Figure 5.13 shows that the average flux density at the surface of the steel segments calculated by the FEA is close to that predicted (over predicting by less than 5%). Over the magnet, however, the simple model assumed no leakage and hence zero flux density, whereas the FEA calculated a value just under half that above the steel. The exponential reduction of flux density with distance above the translator is a reasonable approximation when examining just the area above the steel segments. The two 'dashdot' lines of the graph correspond to the averaged values over an entire surface pole for the two methods. At positions close to the translator, the simplified method over predicts drastically (23% at the surface) but the two methods converge at greater distances, 4% error at 25 mm and less than 1% at 30 mm. For machines with a small airgap there is therefore a need for the more detailed FEA analysis.

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