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The work presented in this paper concerns a theoretical study on frequency selective surfaces (FSS) with application to artificial magnetic conductors or high-impedance surfaces (HIS). Current realizations of HIS are based on a planar FSS at the interface of a metal-backed dielectric slab either including vertical vias or not. A stable resonance was found for the case of series-resonance grids without vias in the slab. The resonance turns out to be unique in theory for all angles of incidence and both polarizations of plane waves illuminating the HIS. It was shown that vias destroy the stabilization effect and introduce a frequency shift. The analytical model was validated by HFSS simulations.