April 15, 2014

It is well known (Stratton 1941, Sommerfeld 1952) that an infinite straight cylindrical conductor embedded in a homogeneous dielectric medium supports an infinity of guided electromagnetic modes. Of these modes, only the circularly symmetric TM0 mode with propogation constant nearly equal to that of a plane wave in the external dielectric, or the

Field values shown are normalized to the total conduction current in the wire cross-section. While the Eρ(ρ) and Hφ(ρ) field values are comparable for the coated and uncoated cases near the wire up to radial distances of ~ 100mm and drop off as ~ 1/ρ for these closer radial distances, for greater distances these field components for the coated wire drop very quickly compared to the uncoated wire. In the final medium (ε=1) the ratio of Eρ/Hφ which is the longitudinal (axial) impedance is very nearly 376 Ω, the free space intrinsic impedance indicating that the field phasefronts outside the wire are very nearly normal to the wire.

The mode propogation loss (β_imaginary) is roughly twice as high for the coated case compared to the bare wire case but is about 16 times lower than that of coaxial line (ignoring any loss in the dielectric) with the same thickness dielectric but with outer shielding conductor: For the example above:

loss (bare wire wave): 0.012 dB/m loss (coated wire wave): 0.021 dB/m loss (coax): 0.33 dB/m

Note that in the small ρ region, Pz(ρ) only depends on the geometry factor ρ/a and not the mode transverse propogation constant λ. However for large values of ρ for which γ'ρ ~ 1, the field dependence becomes exponential with radial distance and the radially integrated total axial power quickly approaches a finite limit. Simplifying the expression provided by Goubau (1950) for the lossless coated conductor case, the total power is:

In this lossless approximation for the coated wire case, the transverse propogation constant in the final dielectric medium is pure imaginary (λ3 = jγ'). As an estimate for the radial distance containing the full power, the small γ'ρ approximation Pz(ρ) can be equated to the total power value Pt which leads to:

where β is the complex mode propogation constant, and bulk propogation constants given by:

The mode propogation constants for the axially symmetric TM modes for the bare wire are given by the solutions of:

The mode propogation constants for the axially symmetric TM modes for the Goubau coated wire configuration are given by the solutions of:

where the transverse propogation constants are:

**Electromagnetic Theory**, J. Stratton, 1941, McGraw Hill, pp. 524-536, 546-554

**Surface Waves and Their Application to Transmission Lines**, G. Goubau, J. Appl. Phys, V21, 1950, p.1119

**Electrodynamics**, A. Sommerfeld, Lectures on Theoretical Physics, 1952, VIII Academic Press, pp 177-185

**Field Theory of Guided Waves**, R. E. Collin, 1991, IEEE Press, pp.697-700, 718-720