Asymptotic analysis of bound-mode and free-space residual-wave currents excited by a delta-gap source on a microstrip line

In this work, an asymptotic analysis is presented for bound-mode and free-space residual-wave currents, with the aim of obtaining an analytical representation of the continuous-spectrum current excited by a delta-gap source on a microstrip line. The proposed approach is valid in a frequency region which extends through the entire spectral gap of the dominant leaky mode. The analysis proves in a rigorous way the different asymptotic behaviors of the TM0 bound-mode and free-space residual-wave currents. The relevance of the nature of the involved branch points to these asymptotic behaviors is also discussed. An explicit expression for the numerical coefficient arising in the asymptotic expansion obtained through Watson's Lemma is provided in a simple closed form for both types of residual-wave current. Analytical details are supplied, together with numerical results which confirm the predicted asymptotic behaviors and the accuracy of the closed-form representation also in proximity of the source, if the presence of singularities of the relevant spectral integrands is properly taken into account.