Coaxial cables had once been thought to support only a single transverse-electromagnetic (TEM) propagation mode. But a solution to the Maxwell-Heaviside equations supports the existence of a propagating transverse-magnetic (TM) surface-wave mode as well. This mode is suppressed by the outer conductor in coaxial cables of conventional impedance, but is dominant in higher-impedance coaxial cables and also on a single uninsulated and unconditioned conductor having no outer shield. This non-radiating surface-wave mode, which has practical uses at RF through microwave frequencies, can exhibit very low attenuation and a relative propagation velocity of unity. This first installment of a three-part article will introduce this surface-wave mode and propose some applications using overhead power distribution lines for last-mile communications applications.

Coaxial cable is perhaps the most commonly used transmission line type for RF and microwave measurements and applications. In 1894, Heaviside, Tesla, and others received patents for coaxial line and related structures. A development of coaxial-line theory is often provided as part of basic physics and engineering education,¹ even prior to full development and use of the Maxwell-Heaviside equations, which are generally used for transmission line and macroscopic electromagnetic (EM) analysis. Accordingly, the analysis, measurement and application of coaxial lines are usually considered to be quite mature and complete.

Figure 1 shows the basic concept of a coaxial line. Lossless cylindrical central and outer shielding conductors are separated by a volume of empty space. This structure conveys power between two points. One end is considered an input port and driven with a sinusoidal voltage source of magnitude A at frequency Ω:

This source is applied to the line through a known impedance, Z_S. The other end of the line is terminated by a load of impedance Z_L.

Heaviside’s telegrapher’s equation provides a lumped-circuit equivalent of an infinitesimal length of transmission line. For the lossless case where R = G = 0, Ampere’s Law can be used to find the inductance per unit length:

Gauss’s Law can be used to find the capacitance per unit length:

This line exhibits an entirely real characteristic impedance of

which is dependent only on the geometry of the conductors (b/a). The maximum transfer of power between

the source and the load occurs when all of these impedances are equal and

Current entering the line central conductor produces a real current density, J. By Ampere’s circuit law, this produces an orthogonal magnetic field B which, in vector form, is:

in the region of empty space inside the outer conductor. An equal magnitude by opposite sense current density, -J, returning from the outer shield also contributes to magnetic flux within this region. Beyond this region, the magnetic effects exactly cancel and no fields due to currents are present (the shielding nature of coaxial cable).

Between the conductors, the varying magnetic (B) field produces an electric field according to Eq. 7:

The electric field lines extend between the conductors and are normal to their surfaces. These electric and magnetic fields produce a TEM wave that travels along the line in the space between the two conductors. In the example coaxial line, this wave travels in a vacuum without attenuation and with velocity of light in a vacuum.

Waves propagating on transmission lines can be described in terms of the axes of the electric or magnetic fields and a mode number. One or both of the electric and magnetic fields must be transverse to the direction of propagation. The corresponding modes are transverse electric (TE), transverse magnetic (TM), and TEM in nature. A pair of mode numbers, n and m, represent the order of the mode in the transverse and longitudinal directions, respectively. Values of zero for each describe a principal mode in the corresponding direction.

For a coaxial line of infinite length and for wavelengths that are large compared to the inner circumference of the outer conductor,

there is radial symmetry and the coaxial line exhibits a principle TEM00 propagation mode. The impedance presented to the source by the line can be written as

where

µ = 4 × 10-7 H/m ≈ 1.2566 µH/m (the permeability of a vacuum) and

e = 1/(c2µ) F/m ≈ 8.8542 pF/m (the permeability of a vacuum)

For the matched condition described here, the voltage produced by the wave at a position separated from the source by a distance l along the line can be described as:

where y = α + jβ is the proportionality constant. Parameter  describes the attenuation while parameter  describes the phase per unit length of line. The propagation constant for the principle mode can be shown to relate to the components in Fig. 1 by

which for the lossless case is purely imaginary and the same as that of the enclosed medium.³

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