A unified theory on radiation of a vertical electric dipole above a dissipative earth

Abstract

Analytical expressions are derived for the electromagnetic field associated with a vertical electric dipole (VED) above a dissipative earth. The scattered field is shown to consist of a direct contribution from a perfect image source, and a correction due to the finite conductivity of earth which is expressible in terms of an incomplete Hankel function. The resultant expression readily reduces to the well‐known asymptotic solutions in both the space‐wave and ground‐wave regions. Validity of the space‐wave expression in the far zone is found to be dependent upon the observation angle as well as the refractive index of earth. In the near field, alternative approximate formulas in closed form are derived from the incomplete Hankel function expression for different elevation angles. These formulas are shown to yield accurate results whenever the distance from the image source is greater than about 0.2λc where λc is the effective wavelength (or skin depth) of earth. For distance smaller than 0.2λc, however, an analytical expression based upon a quasistatic approximation is generally applicable and has been discussed elsewhere.