Both general families of magnetodynamic formulations, i.e. b- and h-conform formulations, are intended to be studied in the frame of the finite element method, as well in 2D as in 3D. These dual formulations are respectively weak forms of the Ampere and Faraday equations, and respectively use unknowns directly associated with the magnetic flux density b (e.g. magnetic vector potential a-formulations) and with the magnetic field h (e.g. magnetic field - magnetic scalar potential h-formulations). When such formulations are applied to systems coupled with electric circuits, not only local quantities, characterizing the unknown field, are involved, but also electric global quantities, i.e. currents and voltages. The physical conducting regions with which these global quantities are associated can be of massive or stranded types, each type necessitating a particular treatment depending on the formulation. The mathematical and numerical tools for naturally coupling local and global quantities will be studied for all these variants. The results of this coupling are circuit relations characterizing the conducting regions. The developed method uses edge finite elements and benefit from their properties to define currents and voltages in strong or weak senses, in accordance with the considered weak formulations, i.e. with no additional approximation. For that, in some cases, they make use of well defined source fields as mathematical tools. In particular, when dealing with a stranded conductor, a h-formulation needs a source magnetic field. The same kind of source field can also be used in a-formulations. Other source fields, source electric scalar potentials, are proposed for massive conductors in a-formulations. A global function of another type is used in h-formulations for massive conductors. Advantages of such dual formulations will be pointed out, in particular concerning the accuracy obtained on local and global solutions and circuit parameters. The use of both formulations can constitute a way to estimate the error on finite element analyses with electric circuit coupling. The consistent approximations proposed for circuit relations appear to be necessary conditions for correct interpretations of the obtained results. A software environment allowing various kinds of coupling has been developed. It will be shown to be particularly well adapted for the multiple analysis undertaken as well for researchers to develop various convenient methods as for users from industry to apply them with a good level of generality.

* Research results of the Belgian programme on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science
   Policy Programming.
** Research Associate with the Belgian National Fund for Scientific Research (F.N.R.S.).