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

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