  We consider systems of nonlinear magnetostatics and quasistatics that
typically arise in the modeling and simulation of electric machines. The
nonlinear problems, eventually obtained after time discretization, are usually
solved by employing a vector potential formulation. In the relevant
two-dimensional setting, a discretization can be obtained by H1-conforming
finite elements. We here consider an alternative formulation based on the
H-field which leads to a nonlinear saddlepoint problem. After commenting on the
unique solvability, we study the numerical approximation by H(curl)-conforming
finite elements and present the main convergence results. A particular focus is
put on the efficient solution of the linearized systems arising in every step
of the nonlinear Newton solver. Via hybridization, the linearized saddlepoint
systems can be transformed into linear elliptic problems, which can be solved
with similar computational complexity as those arising in the vector or scalar
potential formulation. In summary, we can thus claim that the mixed finite
element approach based on the $H$-field can be considered a competitive
alternative to the standard vector or scalar potential formulations for the
solution of problems in nonlinear magneto-quasistatics.
