The Radiation Problem from a Vertical Hertzian Dipole Antenna above Flat and Lossy Ground: Novel Formulation in the Spectral Domain with Closed-Form Analytical Solution in the High Frequency Regime

Abstract

We consider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the wellknown "Sommerfeld radiation problem" in the literature. The problem is formulated in a novel spectral domain approach, and by inverse three-dimensional Fourier transformation the expressions for the received electric and magnetic (EM) field in the physical space are derived as one-dimensional integrals over the radial component of wavevector, in cylindrical coordinates. This formulation appears to have inherent advantages over the classical formulation by Sommerfeld, performed in the spatial domain, since it avoids the use of the so-called Hertz potential and its subsequent differentiation for the calculation of the received EM field. Subsequent use of the stationary phase method in the high frequency regime yields closed-form analytical solutions for the received EM field vectors, which coincide with the corresponding reflected EM field originating from the image point. In this way, we conclude that the so-called "space wave" in the literature represents the total solution of the Sommerfeld problem in the high frequency regime, in which case the surface wave can be ignored. Finally, numerical results are presented, in comparison with corresponding numerical results based on Norton's solution of the problem.

The Radiation Problem from a Vertical Short Dipole Antenna Above Flat and Lossy Ground: Comparison of Exact Analytical Solutions with Empirical Terrain Propagation Models and Investigation of Surface Waves

“…In [2] it is shown that the electric field at the receiver's position above the ground level (x>0) can be expressed with the following integral formula (E LOS denotes the direct field), where:…”

“…The last part of this work is devoted to comparing the above mentioned SPMbased analytical expressions to a wellknown empirical model for path loss prediction, particularly Okumura -Hata [5]. This, as well as similar comparisons to be performed in the future, will eventually determine the extent to which the easily implemented model of [2] can be used for the purposes of radio wave coverage prediction in real life scenarios.…”

“…The original Sommerfeld solution to this problem is provided in the physical space by using the 'Hertz potentials' and it does not end up with closed-form analytical solutions. In [2], the authors considered the problem from a spectral domain perspective, which led to relatively simple 1-D integral representations for the received EM field. Then with the use of the Stationary Phase Method (SPM, [2]) novel, closed-form analytic formulas were derived, applicable in the high frequency regime.…”

“…In [2], the authors considered the problem from a spectral domain perspective, which led to relatively simple 1-D integral representations for the received EM field. Then with the use of the Stationary Phase Method (SPM, [2]) novel, closed-form analytic formulas were derived, applicable in the high frequency regime. The comparison between the analytical expressions of [2] and their integral counterparts was performed in [3] where the requirements for the applicability of the SPM method were extracted.…”

“…The comparison between the analytical expressions of [2] and their integral counterparts was performed in [3] where the requirements for the applicability of the SPM method were extracted. However, as mentioned there, due to the peculiarities of the integrands, which possess particular singularities, the integral expressions of [2] are not easily evaluated using standard numerical integration techniques, as for example the adaptive Simpson's method used in [3]. As a result, a confirmation of the results by using alternative integral evaluation techniques is justified.…”