General approach to prove the existence and uniqueness of the solution in vector potential formulations of 3-D eddy current problems

Gauging of vector potential formulations of 3-D eddy current problems is considered from a new viewpoint. Assuming that the formulation in terms of E and B of the eddy current problem under consideration has a unique solution, gauging can be regarded as setting conditions on potentials, say A, V and ψ, that ensure a one-to-one correspondence between pairs (E,B) and triples (A,V,ψ). This approach decouples two different issues which are usually mixed: gauging properly said and well-posedness of the field problem. Once the above one-to-one correspondence is proved, the existence and uniqueness of (A,V,ψ) will then automatically follow from that of (E,B) whenever the field problem is well-posed. Proving that the correspondence between (E,B) and (A,V,ψ) is one-to-one is a definitely easier task than proving the existence and uniqueness of (A,V,ψ) directly. The approach presented applies to both A,V, and T,ψ formulations and to problems ranging from magnetostatics to high frequency electromagnetics; no additional complexity comes out when dealing with nonlinear materials. Practical impact of the proposed method is discussed