A rigorous asymptotic analysis of residual-wave currents in a microstrip line

In this work the problem of approximating the continuous­-spectrum (CS) current excited by a delta­gap source on a microstrip line is addressed, on the basis of a rigorous asymptotic evaluation of the relevant spectral­domain integral representation. It is well known [1] that the CS current can be represented, in terms of the spectral Green's function singularities, as a sum of the discrete contribution of physical leaky­wave (LW) poles, if any, and the contribution of the integrals along the steepest­descent paths through the relevant branch points. The latter contribution is called residual­wave (RW) current. Such a representation is particularly useful in those cases when the contribution of a single LW pole allows us to accurately represent the CS. However, there are frequency regions in which the LW poles are not physical (spectral­gap regions), and therefore the RW is the main contribution to the CS. In any case, since the LW has an exponential decay while the RW has an algebraic asymptotic decay with the longitudinal z coordinate, the latter is the dominant contribution to the CS at large distances from the source and thus deserves a careful investigation. In the case of a microstrip line, the asymptotic behavior of the possible RW currents has been reported. The aim of the present work is to provide a rigorous asymptotic study of the possible types of residual waves in a microstrip line, i.e., bound­mode residual­wave currents related to TM and TE waves of the background structure and free­space residual­wave current related to the k 0 wavenumber. By means of suitable integral evaluations in the complex plane of the transverse spectral variable via Cauchy Integral Theorem, an accurate analytical approximation of the spectral integrand has been obtained in a neighborhood of the relevant branch point. In particular, an explicit expression for the numerical coe#cient arising in the asymptotic expansion obtained through Watson's Lemma has been provided in a closed form. Numerical results will be shown that confirm the accuracy of the presented formulas.