## Details of the model

The flux flow of a VHM may be assumed to be two dimensional and modelling can hence be undertaken using a two dimensional FEA program. The software used here is Vector Field's PC Opera 2 D [74]. Within this program a single phase, two pole double sided linear VHM was simulated. Each pole has six magnets attached in such a way that the flux return path passes across, not along, the axis of the translator back iron. The pattern it traces is hence symmetrical about the centreline of the translator, thus allowing the model to consist of one half of the machine, the second half being implied by a boundary condition of symmetry applied to this centreline. Flux flow is not symmetric about a plane at 90 degrees to this however, and so it is still necessary to model two poles. The basic mesh of the model is shown in Figure 4.6.

The model consists of 15 000 elements of varying size, the smallest of which are within the airgap and magnetic regions. Predicting the behaviour of a variable reluctance machine using FEA generally requires the model be run at various different operating positions, typically different translator positions. In the case of the linear VHM, however, this means the model be solved at various different relative magnet and stator tooth positions. In order to maintain boundary integrity between the model regions, the smallest increment of movement is limited to the fineness of the airgap mesh. In the model presented here this is 1mm. Noting the repetitive nature of both the translator and the stator it can be seen that it is only necessary to model the movement of the rotor through the distance of the rotor pitch, or twice the magnet pitch, in order to obtain the behaviour of the machine at any subsequent position. The magnet pitch of the model was 12 mm, resulting in 24 different positions being available. In all the results quoted in this section, the relative position, p, is defined such that zero corresponds to the magnet and tooth being fully aligned, Figure 4.7.

The model is surrounded by a background region of air in order to reduce the influence of artificial boundary conditions necessary for the running of the software. To facilitate more accurate modelling, the non linearities of both the steel and magnetic materials are provided, shown in Figure 4.8.

Figure 4.7: Definition of relative position

4.2.2 Details of the Analysis

The software solves the FEA model as a non-linear magnostatic problem, it therefore does not take account of time dependent effects, such as the losses due to eddy currents. The equations solved relate the magnetic field strength, H, to the current density in the conductors, J, according to (4.8). The flux density, B, has zero divergence as in (4.9) and a magnetic vector potential, Q, is defined as (4.10).

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