Magnetic scalar potential (Electrical Machine)

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By analogy to the electric field, the magnetic field strength is calculated as the gradient of a scalar potential. It must be distinguished between current-carrying and current-free regions.

Current-free regions

The magneto-static problem without conducting currents can be formulated in terms of the magnetic scalar potentialtmp10200_thumb

With the vector identitytmp10202_thumband Ampere’s law, a magnetic scalar potentialtmp10203_thumbcan easily be defined by evaluating:

tmp10206_thumb

This potential formulation is not suitable for problems inside regions with conducting currents. A typical application for this type of potential is the calculation of a magnetic shielding.

With the zero divergence condition of the magnetic flux density the tmp10209_thumbof the scalar magnetic potential is introduced:

tmp10211_thumb

This formulation is a Laplace equation.tmp10213_thumbis a scalar and A in the vector potential formulation of the magnetic field is a vector quantity. By using the same numerical discretisation, the scalar potential problem consists of a third of unknown when compared to the formulation using the vector potential. This example makes clear that an appropriate choice of the potential formulation has significant influence on the size of the problem.

This definition of the scalar magnetic potential,tmp10214_thumbcauses problems in multiple connected domains.

Multiple connected domain.

Fig. 4.2. Multiple connected domain.

The domain is free of conducting current. There is a current / carrying conductor leading through the opening in the domain (Fig. 4.2). Applying Ampere’s law on the contour X inside the domain yields:

tmp10218_thumb

The potential differencetmp10219_thumbcan only be non-zero iftmp10220_thumbis discontinuous inside the domain. Therefore, a discontinuity, a cut, is defined in the way that by considering the cut as an outside boundary, the domain is not further multiple connected.

tmp10223_thumb

Therefore, the boundary condition applied to the cut is

tmp10226_thumb

This is a periodic boundary condition.

Current-carrying regions

In the case of current-carrying regions it is not possible to define a magnetic scalar potential. However, by again using an arbitrary vector field T, it is possible to define a similar potential (Silvester & Ferrari’03).

The electric vector potential T and Ampere’s law eq.(ii), (3.27) yield:

tmp10227_thumb

With the vector identity

tmp10228_thumb

the gradient of the scalar magnetic potential d> is now defined by:

tmp10230_thumb

The zero divergence condition of the magnetic fieldtmp10232_thumband the material equation combining B and H yields an equation of Poisson type:

 

tmp10234_thumb

A disadvantage in solving this problem is that the solution of the magnetic field problem has to be obtained in three steps:

1. Determine the auxiliary potential function T .

2. Solve the Poisson equationtmp10237_thumbto find the magnetic scalar potential.

3. Evaluatetmp10238_thumband T to obtain the required overall solution of H .

tmp10241_thumb

is called atmp10242_thumbThe magnetic scalar potential has the dimension [A]. This potential formulation is in common use in magneto-static and diffusion problems. To determine T, Biot-Savart’s law can be evaluated (Hafner47).

By using the finite element method, T can be determined in the following way:

tmp10243_thumb

1. Create a tree of mesh edges.

2. T= 0 for all tree-edges.

3. Apply Ampere’s law to each element and determine T for all co-tree edges.

A tree in the topology of the finite element mesh is defined as a set of edges reaching all nodes of the mesh but forming no loops in this mesh. The associated co-tree is the set of the remaining edges. T is constructed as a field built of edge elementstmp10244_thumb

tmp10245_thumbtmp10246_thumb

The coefficientstmp10247_thumbof the elements associated with the tree edges are 0. The coefficients of the elements associated with the co-tree edges are calculated from Ampere’s law (Fig. 4.3):

tmp10249_thumb

 

 

 

Tree definition and finite element with imposed current.

Fig. 4.3. Tree definition and finite element with imposed current.

With Table 4.2 a comparison of the properties of the magnetic vector and scalar potential is possible.

Table 4.2. Comparison between A- andtmp10252_thumb

A-formulation

tmp10-255

potential formulation

tmp10-256 tmp10-257

implicitly fulfilled equation

tmp10-258 tmp10-259

explicitly fulfilled equations

tmp10-260 tmp10-261
tmp10-262 tmp10-263

source field

tmp10-264

T has to be determined

additional condition element type

gauge edge

cut node