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

Abstract

In this paper we examine the problem of radiation from a vertical short (Hertzian) dipole above flat lossy ground, known in the literature as the 'Sommerfeld radiation problem'. Our formulation is in the spectral domain and ends up into simple one dimensional integral expressions for the received electromagnetic (EM) field, representing the exact solution of the problem. The problem can be solved analytically in an approximate sense in the high frequency regime using the Stationary Phase Method (SPM). In this paper the above spectral integrals for the received EM field are also mathematically represented as integrals over the 'grazing angle', a formulation that allows for a more accurate calculation since it avoids the singularities of the integrand expression. Also, a new SPM analytical solution, based on the above novel integral representation is obtained. Numerical comparisons between our SPM solution and the integral representations for the received EM field show that neither the horizontal Transmitter-Receiver distance, nor the frequency of operation are alone sufficient indicators regarding the most appropriate method to use (SPM or Numerical Integration). Instead, such a decision is to be based on their combined effect, given by their product k•r (electric distance).

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

“…The SPM-based analytical method of Section 3 predicts the theoretical field behavior and this is true both for the near field as well as the far field region, as they are defined in [4]. In this regard, the so-claimed in [3], 'high frequency regime analytical method' of [2], is validated for such high frequencies as 1GHz. As another validation of the previous arguments, Fig.…”

“…Application of SPM to (3), 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) [3]: …”

“…Further validations against theoretically driven numerical results, like those of [4] used here, are required to determine the exact frequency limits of the analytic expressions (5) -(7). For this, the perspective described in [3], which is believed to reduce the precision errors in the evaluation of the integral expressions, will be followed, leading also to a novel spectral domain representation for the so called surface wave as well [note that (5) neglects the role of the surface wave]. For this particular issue, ongoing research is carried out by our research group [6], through which the integral representation for the EM field (3) is evaluated by means of a contour integral in the complex plane, as shown in Fig.…”

“…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. 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].…”

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].…”