Magnetic scalar potential (Electrical Machine)


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:


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:


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:


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.


Therefore, the boundary condition applied to the cut is


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:


With the vector identity


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


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



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 .


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:


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


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):





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



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


T has to be determined

additional condition element type

gauge edge

cut node