There are several ways of coupling field and circuit models but, in our opinion, the most natural one uses Electric Circuit Element (ECE) boundary conditions (BC) for the field formulation. In our previous work we have  successfully implemented full-wave (FW) frequency-domain E- and H-based formulations with ECE BC in the finite element method (FEM). Our interest was only in high frequency (HF) FW applications. At HF these formulations with an unknown field strictly inside the domain and a scalar potential with support only on the boundary, are stable and accurate. Motivated by the current desire of having all-purpose Maxwell’s solvers at hand, in this contribution we investigate their numerical stability when used at very low frequencies (LF). We show that these FW formulations can be used at LF provided that techniques to ensure the numerical stability are applied. If only global (energy based) stability is sought for, then just using the excitation which is essential for FEM might be enough. If a local (field spatial distribution) stability is needed, we show that scaled hybrid versions of these formulations ensure LF stability. Results are shown for 5 models of increasing complexity: a homogeneous cylinder (2Daxi/3D), a coaxial cable (2Daxi /3D), a coplanar waveguide (3D).