  This paper presents a novel finite-element approach for the electromagnetic
modeling of superconducting coated conductors. We combine a thin-shell (TS)
method to the $H$-$\phi$ formulation to avoid the meshing difficulties related
to the high aspect ratio of these conductors and reduce the computational
burden in simulations. The interface boundary conditions in the TS method are
defined using an auxiliary 1-D finite-element (FE) discretization of $N$
elements along the thinnest dimension of the conductor. This procedure permits
the approximation of the superconductor's nonlinearities inside the TS in a
time-transient analysis. Four application examples of increasing complexity are
discussed: (i) single coated conductor, (ii) two closely packed conductors
carrying anti-parallel currents, (iii) a stack of twenty superconducting tapes
and a (iv) full representation of a HTS tape comprising a stack of thin films.
In all these examples, the profiles of both the tangential and normal
components of the magnetic field show good agreement with a reference solution
obtained with standard $2$-D $H$-$\phi$ formulation. Results are also compared
with the widely used $T$-$A$ formulation. This formulation is shown to be dual
to the TS model with a single FE ($N=1$) in the auxiliary 1-D systems. The
increase of $N$ in the TS model is shown to be advantageous at small inter-tape
separation and low transport current since it allows the tangential components
of the magnetic field to penetrate the thin region. The reduction in
computational cost without compromising accuracy makes the proposed model
promising for the simulation of large-scale superconducting applications.
