In this project we develop an a priori error analysis for a transient eddy current problem when applying the well known Hybridizable discontinuous Galerkin (HDG) method. We introduce a novel curl-curl formulation and analyze both the semidiscrete and the fully discrete scheme. We consider non trivial domains and heterogeneous media which contain conductor and insulating materials. When dealing with these domains, it is necessary to impose the divergence free condition explicitly in the insulator, what is done by means of a suitable Lagrange multiplier in that material. In the end, we deduce an equivalent HDG formulation that includes as unknowns the tangential and normal trace of a vector field. This represents a reduction in the degrees of freedom when compare with the standard DG methods. For this scheme we conduct a complete error analysis.