Radiation of a Vertical Dipole over Flat and Lossy Ground using the Spectral Domain Approach: Comparison of Stationary Phase Method Analytical Solution with Numerical Integration Results

Abstract

In this paper we reconsider the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, which represents the well-known in the literature 'Sommerfeld radiation problem'. Particularly, we expand on the problem's solution in the spectral domain, which ends up into simple one dimensional (1-D) integral expressions for the received EM field and represent the exact EM solution to the aforementioned problem. The advantage of the derived expressions is based on the fact that they can be analytically evaluated through the use of the Stationary Phase Method (SPM), which however is valid in the high frequency regime. To our knowledge, the literature lacks specifying the exact frequency range over which the SPM method is applicable. Hence, in this paper numerical integration on the above mentioned integral expressions is applied and the results are compared with those obtained through the SPM. These comparisons are then used as the basis of determining the frequency limits of applicability of the SPM solution. In fact it is shown that due to the specific peculiarities of the integrated expressions, which possess certain singularities, it is often preferable to use the SPM method as the best estimate for the received signal level, especially for most practical frequencies of interest in the area of wireless telecommunications. Additional practical implications that these findings suggest, as well as further research to be triggered, as a result of the overall progress that has been made by our research group so far on the specific subject, are provided as well.

Radiation of a Vertical Dipole Antenna over Flat and Lossy Ground: Accurate Electromagnetic Field Calculation using the Spectral Domain Approach along with Redefined Integral Representations and corresponding Novel Analytical Solution

“…Examining (13) in more detail, an important note can be made at this point: in the present formulation, where variable α is introduced as the variable of integration, instead of kρ, no singularity exists for the integrand within the (0, π) range of α. This is not the case in 3, where singularities exist at points kρ = ±k01, leading to a major problem for our numerical integration procedure, as mentioned in [18]. This appears to be an important advantage of the new proposed formulation, as also described in Section VI below (future research).…”

“…Application of the 'Stationary Phase Method' (SPM) to (3) and (4), leads to the following analytic expressions for the electric field vector scattered from the plane ground, in the far field region and in the high frequency regime (for x > 0) [17], [18]  …”

“…The above SPM closedform analytic result was obtained under the 'high frequency approximation' assumption, as well as the assumption that the 'grazing angle' φ is not very close to zero, as already explained. Under these assumptions, (23) just represents the summation of the LOS field and the field reflected from the 'specular point' [note that the fraction in (23) just represents the usual 'Fresnel reflection coefficient', [14], [18]).…”

“…As stated in [17], [18] determining the necessary conditions for the applicability of the SPM method, an inherently high frequency technique, is essential. There, the issue was investigated for the practical case of transmitterreceiver pairs that are separated by a rather long distance, in which case it was apparent that for most frequencies of interest in the telecommunication area, frequency alone is a good criterion for the selection of the most appropriate method for EM field calculation at the receiver's point.…”

A Novel Asymptotic Solution to the Sommerfeld Radiation Problem: Analytic Field Expressions and the Emergence of the Surface Waves

“…As a result, directly evaluating (19) by the method of saddle points is likely to induce significant errors, since the accuracy of the method depends on the relative position of the pole to the saddle point [17,18]. This was actually revealed in [22,23] in which the evaluation of (19) resulted in a single term, corresponding to the space wave component of the field only, thus missing to describe the surface wave behavior of it. Even, more, at sliding observation angles, the method failed completely to estimate field values, a consequence of the known fact that the space wave component at such circumstances almost vanishes, for the reflection coefficient being almost equal to minus one [19].…”