Analytical Evaluation of Time-Differentiated Retarded-Time Vector Potentials on SWG Bases

Abstract

It is shown in this letter that the term of the time domain-radiated electromagnetic fields from a non-conductive object in a loss-free medium consisting of the time convolutions of the retarded-time vector potentials sampled on a Schaubert-Wilton-Glisson (SWG) spatial basis and a piecewise-polynomial temporal basis pair can be computed using only analytically-evaluated closed-form expressions. The corresponding impulsively-excited potentials are related to the mean vector and the solid angle measured from the intersection of the tetrahedron and the hyper-cone centered at the observation point. The closed-form expressions that appear after the convolution operations with the temporal-basis result in a novel volumetric integration scheme with a novel geometric partitioning of the above-mentioned intersection. The validity of the obtained formulae is demonstrated through the comparison of the results to those obtained in the frequency domain using numerical quadrature and transformed into the time domain.