By R. Belmans A formulation It is sometimes interesting to use a hybrid Aformulation. The formulation is used in regions without current whereas an A-formulation is used in regions with an applied current density. At the interface surfaces between regions with a different potential formulation, the conditions for H and B are applied: The advantage [...]

## AV-formulation (Electrical Machine)

By R. Belmans Using Ampere’s law and the magnetic vector potential and applying the appropriate material equation yields: This equation is called the AV-formulation of the magnetic field. The presence of the derivative in time indicates the magneto-dynamic field problem. By substitutingthis formulation is transferred from the time domain into the frequency domain: [...]

## In-plane formulation (Electrical Machine)

By R. Belmans The inverse problem, to calculate a current density distribution for an imposed magnetic field can be obtained by using an in-plane formulation. Applying the Maxwell equation (3.8) and using the magnetic scalar and electric vector potential yields a particular Finally the in-planevalid in the time domain, is described by: In the frequency [...]

## AV-formulation with vxB motion term (Electrical Machine)

By R. Belmans For the moving system and the resting observer, as introduced earlier, the motion term is considered by applying: For the moving system and the resting observer, as introduced earlier, the motion term is considered by applying: Together with Ampere’s law (3.9), this leads to the AV-formulation considering the motion term v x [...]

## Subsequent treatment of the Maxwell equations (Electrical Machine)

By R. Belmans To obtain solutions for real-life field problems, a subsequent treatment of the Maxwell equations is necessary. The potentials are introduced to reduce the mathematical dimensions of the field problem. This approach results, for example for a line integral, in building a simple difference. To improve understanding of the field equations, a scheme [...]

## Magnetic equivalent circuit (Electrical Machine)

By R. Belmans Field computation and numerical techniques We consider that a variational principle or a boundary problem can describe a given physical-technical problem. Thus, this field problem is given by a differential equation. The problem is now to find a feasible solution of this differential equation. Fig. 5.1 shows the various possibilities for solving [...]

## Point mirroring method (Electrical Machine) Part 1

By R. Belmans advantages disadvantages relatively fast non-linearities only considered by constant factors 3D fields special geometries only special boundary conditions have to be assumed To demonstrate the strength and shortcomings of this method, on the other hand, two examples are worked out. In the first example a ferromagnetic circuit is calculated. The main limitations [...]

## Point mirroring method (Electrical Machine) Part 2

By R. Belmans Magnetic field The magnetic field problem is considered to be linear. Hence, the superposition of partial fields, calculated with the Biot-Savart law, result in the overall three-dimensional field distribution below the line. In this case each segment of the infinitesimally-thin filament (Fig. 5.14) carries a currentThe generated flux density of this part [...]

## The numerical solution of partial differential equations (Electrical Machine)

By R. Belmans Solving a differential equation analytically or by semi-numerical techniques, as demonstrated before, is only possible assuming simplifications in the differential equations that are valid in a domain that can be described in a plain mathematical way. Specific boundary conditions and various material properties in different domains make it difficult to obtain an [...]

## Finite difference method (Electrical Machine)

By R. Belmans 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-distancescan be different. Fig. 5.23. Numerical discretisation for the FDM. Fig. 5.24. Approximation [...]