This article presents new developments concerning three-dimensional (3-D) planar problems, that is, combinations of planar sheets of current oriented along normal and transverse directions in a stratified medium. The conducting structures are analyzed using an improved version of the mixed potential integral equation, including unknown currents along three coordinate directions, and solved through a method of moments. The modifications needed in the integral equation to accommodate simultaneously normal and transverse currents are discussed. Detailed expressions for the Green's functions associated with horizontal and vertical dipoles are presented and cast as Sommerfeld integrals. A convenient integration path is proposed for their estimation. Associated interpolation problems are thoroughly discussed and symmetry and reciprocity considerations are pointed out. Finally, two antennas have been built and studied to demonstrate the validity of the formulation: a short-circuited microstrip antenna, and a two-layer patch antenna.
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Gay-Balmaz et al. (1997) studied this question.