May 24, 2014

It is well known (Stratton 1941, Sommerfeld 1952) that an infinite straight cylindrical homogeneous medium embedded in a homogeneous medium supports an infinity of guided electromagnetic modes. The various types of modes for different media types have been discussed in detail by these authors. For the case of a typical conducting cylindrical medium embedded in a medium with low loss, only the lowest order symmetrical TM0 mode or

where the values for the constants vacuum permeability μ

where the radial field dependence

Interface matching of tangential E and H fields leads to an

where

and if the wire is a good conductor so that the conduction current is much larger than the displacement current:

For the TM0 principal mode :

and the left hand side of the equation above is a constant, independent of β. If in addition the wire radius is not too small, the equation above simplies further:

Either equation can be used in a rapidly converting successive approximations iterative algorithm (Sommerfeld) to solve for

where

|u| can have arbitrary value (i.e. the wire radius can be arbitrarily small) and Jo(u)/J1(u) is calculated directly during iteration.

The alternating-current complex impedance (Stratton 1941) or

At high frequency, the real and imaginary parts of the surface impedance are nearly identical, with the imaginary part being inductive. This means that at high frequency, the surface axial electric field and the total conduction current are out of phase my 45 degrees (as are Ez and Hphi at the surface of the wire). For a detailed discussion of the frequency dependence of Z

At 1 GHz (λ

The time-averaged Poynting vectors (power flow) for the axial z component

and are displayed below normalized to the total current squared (in units of Ω/m^2). The radial component of power flow is directed INWARD (negative). For a typical good conductor, all the axial power flow is carried outside the wire. The inward radial flow of power enters the wire and is rapidly attenuated as dissipation in the wire conductor:

Although the power density Sz(ρ) outside the wire appears to be confined near the wire, in fact only 50% of the total axial power flow in the principal mode is contained in a radius of 50 mm, 75% in a radius of 390 mm and 90% within a radius of 1.4m. The power

where ξ and therefore ν are solutions of the transcendental equation above for the TM wire wave mode. For the example here P

The propogation loss of the TM mode is described by β

For the example above, the total axial power is 238 W/A^2 and the inward radial average power density S

leading to the simple result for the planar TM0 surface wave:

and substituting for u and v, the following elegant symmetric expression is obtained for the propogation contant of the TM0 mode for an ideal infinite planar interface, in agreement with the result obtained from direct analysis of the planar case:

For the example above with copper at 1GHz, the planar interface propogation loss result is 8.8e-8 dB/m, much lower than the result for the wire propogation loss.

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

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

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

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

**Fields and Waves in Communication Electronics**, S. Ramo, J. Whinnery, T. Van Duzer, 1984, J. Wiley & Sons. pp. 279-283