Two-dimensional equations for microwave planar circuits

A planar circuit [1] is defined as an electric circuit having dimensions comparable to the wavelength in two directions but a much smaller dimension in the third direction. Microwave integrated circuits, which have replaced the conventional microwave circuits [2], are constructed in a planar form. The transmission structures contain striplines, the most used being the microstrips. The two-dimensional equations for the microwave planar circuits, first presented in [3], were derived by considering a quasistationary TEM mode as the fundamental mode of propagation. In this paper we show that these equations can be employed for analyzing not only the fundamental mode but also the higher-order modes of propagation. Indeed, for a rectangular waveguide model with hybrid electric and magnetic walls [4], [5] the higher-order TE modes are identical with those derived from the two-dimensional equations for lossless circuits. This suggests the possibility of applying the two-dimensional equations for lossy planar circuits [3] to analyze the propagation of higher-order TE modes along lossy structures, for which there are only crude models available in the literature.