LI p

Where y = total flux linkage due to both coil and PM excitation yPM = flux linkage due to PM excitation only Li,p = inductance considering effect of current and position Figure 4.27 shows the distribution of inductance with current and rotor position calculated using this method. The value of inductance defined in this manner is greater than the electrical excitation only, Li, and varies greatly. The 'inverse' nature of the position inductance graph across the zero current mark represents the model accounting for the direction of armature driven flux with respect to the aligned tooth magnet flux. Also highlighted in this graph are the inaccuracies of the FEA, which manifests itself as the rough nature of some of the curves, particularly at low values of current. Slight errors in these low values of flux are highlighted by division of low currents.

Figure 4.27: Predicted self inductance per phase for different rotor positions and coil currents

If the inductance is calculated at this value for each time step, the equivalent circuit of Figure 4.28 and its governing equation (4.24) results.

Figure 4.28: Step variable inductance equivalent circuit model

4.3.2.4 Continuous Variable Inductance Method - Model C

The value of inductance LI>p does not change significantly with low values of current (around 10%) so it can be predicted that the performance of the variable inductance model will not differ greatly from the simple fixed inductance model. A more accurate model than either of these is to restate (4.20) as (4.25), which accounts for the rate of change of inductance during each time step.

dl T dLi,p

dt dt

The equivalent circuit is given in Figure 4.29 and described by ( 4.26).

Figure 4.29: Continuous Variable Inductance Method

Within this model, the inductance is numerically calculated by cubic spline interpolation from a look up table and its rate of change is taken as the difference in inductance value between consecutive steps, divided by the time step.

4.3.2.5 Lumped Reactance Model - Model D

The two inductance terms on the right hand side of ( 4.26) are the resulting terms of differentiating (4.27) in two parts.

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