## Model Development

4.3.3.1 Approximation of Flux Results for Open Circuit

To model the flux accurately at positions between consecutive 1 mm positions provided by the FEA, approximating the results to a continuous function could provide a convenient way to obtain data.

The fourth order polynomial given in (4.30), calculated using the least squares method, is shown plotted on the same axis as the FEA calculated flux in Figure 4.32. The results show good correlation and hence imply that the polynomial could be used to generate a continuous value of flux.

Where the coefficients are given in Table 4-2.

a |
b |
c |
d |
e |

9633365136x105 |
-4.623714952x103 |
0.05757048809279 |
-0.05014353241196 |
-0.86614961061008 |

(KiSiliOn(rWfl)

Figure 4.32: Comparison of Polynomial approximation and FEA results

Figure 4.31: The look up table model

(KiSiliOn(rWfl)

Figure 4.32: Comparison of Polynomial approximation and FEA results

The non periodic nature of this polynomial makes it necessary to convert translator displacement into relative tooth and magnet position. During a displacement of greater amplitude than the rotor tooth pitch requires the use of p=23 and p=0 in consecutive time steps. In theory these values should be smoothly separated, yet the FEA, and hence polynomial, gives a very small discrepancy between them. In Figure 4.33 this manifests itself as a small discontinuity at the trough of the flux curve.

20 25 30 35 40 45 50 55 60 65 70 absolute translator position (mm)

Figure 4.33: Flux position obtained with polynomial approximation

20 25 30 35 40 45 50 55 60 65 70 absolute translator position (mm)

Figure 4.33: Flux position obtained with polynomial approximation

The coil emf is obtained by numerical differentiation of this signal during which this discontinuity becomes a significant spike in the results, Figure 4.34. Furthermore, if the simulation is extended to encompass inductive and capacitive elements this effect has a significant impact on the results.

Higher order polynomial approximations also exhibit this property. Polynomial substitution of the flux-position results is hence unsuitable.

An alternative approach, which has the distinct advantage of being periodic and continuously smooth, is to approximate the flux linkage to a sinusoidal function. Using an amplitude of 0.898 and a period of n /24 gives a good correlation.

4.3.3.2 Approximation of Flux Results for y-I Map

Although the flux mmf plots of Figure 4.14 implies that the flux data could be modelled by a function of the form ( 4.31), this is avoided here. Inspection of the differential of this equation and comparison with (4.22) demonstrate that using this approximation is simply an alternative approximation to the constant inductance model.

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