By R. Belmans
Potentials and formulations
The Maxwell equations represent the physical properties of the fields. To solve them, mainly the differential form of the equations and mathematical functions, the potentials, satisfying the Maxwell equations, are used. The proper choice of a potential depends on the type of field problem. In this section, the various scalar and vector potentials are introduced.
The electric vector potential for the displacement current density will not be introduced here, because it is only important for the calculation of fields in chargefree and currentfree regions such as hollow waveguides or in surrounding fields of antennas.
Various potential formulations are possible for the different field types. Their appropriate definition ensures the accurate transition of the field problem between continuous and discrete space.
Using these artificial field quantities reduces the number of differential equations. Considering a problem described by n differential equations, a potential is chosen in such a way that one of the differential equations is fulfilled. This potential is substituted in all other differential equations, the resulting system of differential equations reduces to nl equations. It is distinguished between magnetic and electric vector respectively scalar potentials:
Table 4.1. Definition of the potentials.
scalar 
potentials 
vector 

electric 

magnetic 
By using the vector identity
and applying eq.(iii), (3.10), introduces the magnetic vector potential A. The magnetic flux density is derived as the curl of another vector field:
The magnetic vector potential is suitable in regions with and without conducting currents. The vector field A is assigned right handed to the direction of the magnetic field B (Fig. 4.1).
Fig. 4.1. Geometrical assignment of the vector potential A with the magnetic field vector B.
A static magnetic problem is described by
By using the magnetic vector potential A, the system of differential equations is reduced to
Applying the vector calculusto eq.(4.4) yields:
withand by assuming a constant permeabilityleads to the Aformulation of a magnetostatic field, a Poisson equation:
To consider quasistationary fields, for example necessary for eddy current calculations, the magnetodynamic formulations have to be employed. In addition to Ampere’s law, the Faraday law (i), (3.26) has to be considered to evaluate the contribution to the field by the eddy currents:
Now employing Ohm’s law to calculate the eddy currents Je yields:
Ampere’s law can now be rewritten, yielding the Aformulation for the quasistationary magnetic field in the time domain:
Substituting againand assuming
results in a similar Aformulation in the time domain for the transient magnetic field:
Assuming sinusoidal excitation currents with an angular frequency and thus substituting
yields the Aformulation in the frequency domain to solve eddy current problems.
This equation is the Aformulation to describe timeharmonic problems. The time dependent components of the vector potential
are expressed by:
The current is expressed in analogy in its complex representation.