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).DOI: http://dx.doi.org/10.5755/j01.eie.22.2.14592
Highlights
The ‘Sommerfeld radiation problem’ is a well-known problem in the area of propagation of electromagnetic (EM) waves above flat and lossy ground with applications in the area of wireless and mobile telecommunications [1]–[10]
Under these assumptions, (23) just represents the summation of the Line of Sight (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])
In [18] the authors identified frequency areas where it was not apparent which method, Stationary Phase Method (SPM) or Numerical Integration, is the most appropriate to use for EM field estimation
Summary
The ‘Sommerfeld radiation problem’ is a well-known problem in the area of propagation of electromagnetic (EM) waves above flat and lossy ground with applications in the area of wireless and mobile telecommunications [1]–[10]. In [13], [14] the Stationary Phase Method was proposed (SPM, [15], [16]) and as a result novel, closed-form analytical expressions were derived, for use in the high frequency regime. It is shown that an appropriate selection of the integration variable and subsequent use of the SPM method lead to useful insights regarding the propagation mechanism. Extensive simulations are run that compare the estimated received EM field at an observation point above flat and lossy ground under the following two approaches: (a) SPM method, (b) Numerical integration of the corresponding integral representations. Additional implications for future research, triggered by our results so far, are provided as well
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