Application of Vilenkin´s additional theorem in the analysis of spherical antennas and periodic structures

This article demonstrates a novel approach to the mutual coupling calculation within the Moment Method (MoM) analysis of spherical arrays. By expressing the patch/dipole current in terms of two suitable potential-like auxiliary functions, it is possible to avoid the use of Euler´s formulas for coordinate system rotation and the related lengthy integrations. Instead, the rotation in θ-direction can be done in closed form with the help of Vilenkin´s additional theorem for associated Legendre functions. It is shown that the new approach results in significant acceleration of the analysis of both spherical antenna arrays and frequency selective surfaces. The algorithm is validated by comparing the results to the measurements performed on the developed laboratory model.