The nature of the spectral gap between bound and leaky solutions when dielectric loss is present in printed-circuit lines

摘要

For lossless printed-circuit transmission lines, it is known that a spectral gap is present at the transition in frequency between bound and leaky modal solutions, and that the solutions are nonphysical within that gap. This paper examines the corresponding modal behavior when the dielectric substrate material is lossy and reports several new interesting and important results. The new features become particularly clear when the data are displayed on the steepest-descent plane, which itself changes when loss is present and contains a new narrow spectral region that vanishes when the dielectric becomes lossless. Among the other new results we find that the leaky solution reverts to a bound solution at still higher frequencies, in contrast to the lossless case, and that the modal pole solutions are captured by the steepest-descent path at all frequencies, so that an explicit spectral gap is not found in the lossy case examined here.