Finite difference method (Electrical Machine)

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The field domain of interest is discretised by a grid, where the grid-lines are in parallel to the co-ordinate axes. This type of mesh is called an orthogonal grid and must not consist of equidistant grid-points (Fig. 5.23). The grid-distancestmp10458_thumbcan be different.

 Numerical discretisation for the FDM.

Fig. 5.23. Numerical discretisation for the FDM.

Approximation of the potential distribution by a Taylor series.

Fig. 5.24. Approximation of the potential distribution by a Taylor series.

The potential distribution over the studied domain is approximated by the first terms of a Taylor series (Fig. 5.24):

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The differential equation of the particular field problem is locally transferred into a difference equation. For example, calculating the value of a potential distribution at point 0, by using the 4-point approach (Fig. 5.23), and a grid distance d with a derivative

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is expressed by a finite difference with a known approximation error T](x). By considering only the first terms of the Taylor series, this error is known and dependent on the grid distance. Forward,

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backward

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and central difference

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are used to assemble a system of linear equations to calculate the potentials at all grid points. This leads to a large linear system of equations to be solved. The potentials at the mesh points represent the approximated field solution. To obtain an accurate field solution a fine discretisation is required.