In this paper, we consider a dense array of metallic circular patches printed on a electrically thin metal-backed dielectric substrate. Since the sub-wavelength dimensions, the array and the metal-backed substrate can be described in terms of a lumped capacitance and a lumped inductance, respectively. Around the resonant frequency, the structure, known as high-impedance surface, reflects totally an incident electromagnetic wave with zero shift in phase. Due to this property, it is widely employed in antenna systems as compact back reflector with improved performances with respect to typical metal reflector. Starting from the concept of the grid capacitive reactance of a planar array of squared patches and its related formulas, we investigate on the field distribution on the array plane and properly modify the formulas for the case of the circular patches. We present two new analytical formulas which can be effectively used for the fast design of 2D-isotropic circular HISs. In order to validate the models, we compare the resonant frequency of the array obtained through our approaches to the one resulting from full-wave numerical simulations and from other analytical methods available in the open technical literature.
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