Complex TM models in a generalized Goubau line with arbitrary conductivities

The modal equation for the wavenumbers of the complex transverse magnetic (TM) modes of an open circular-cylinder coaxial waveguide with central conductor is studied for the case when the inner and outer materials have arbitrary conductivity. Analytic approximations for the wavenumhers of all of the modes are obtained for large contrast between the inner and outer materials, and simple numerical algorithms for calculating the wavenumbers are obtained for arbitrary contrast. It is shown that when the conductivities are zero, the wavenumbers group together in a set of four complex values, symmetric in the complex plane, but that the symmetry and the grouping can both be destroyed by adding conductivity to the materials. For fixed conductivities, there are principal modes (with no low-frequency cutoffs) and secondary modes (with low-frequency cutoffs) but one type mode can be converted into the other type by changing the conductivities. A numerical study of the modal equation shows how the modes of a Goubau line can be related to those of a coaxial transmission line. It shows also that the values of individual solutions of the modal equation can depend on the history of the conductivity values of the waveguide.