International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 187806, 8 pages

http://dx.doi.org/10.1155/2015/187806

## On the Surface Fields of a Small Circular Loop Antenna Placed on Plane Stratified Earth

Faculty of Engineering, University Campus Bio-Medico of Rome, 00128 Rome, Italy

Received 29 July 2015; Accepted 8 October 2015

Academic Editor: Jun Hu

Copyright © 2015 Mauro Parise. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analytical method is presented which makes it possible to derive exact explicit expressions for the time-harmonic surface fields excited by a small circular loop antenna placed on the top surface of plane layered earth. The developed procedure leads to casting the complete integral representations for the EM field components into forms suitable for application of Cauchy’s integral formula. As a result, the surface fields are expressed as sums of Hankel functions. Numerical simulations are performed to show the validity and accuracy of the proposed solution.

#### 1. Introduction

The need to evaluate the electromagnetic fields radiated by horizontal loop antennas located in close proximity to layered media arises in a variety of applications of scientific interest, especially in the areas of electromagnetics [1–9], close-to-the-surface radio communication [10, 11], and geophysical prospecting [11–13]. An excellent illustration of such applications is electromagnetic induction sounding technique [12, 14, 15], which allows acquiring information about the subsurface structure of a terrestrial area. The technique consists of two steps. At first, the fields generated by a current-carrying circular loop antenna lying on the surface of the medium to be explored are measured at a discrete set of frequencies. Next, the presence of shallow buried objects, or the geometrical and EM properties of the structure, is deduced by interpreting the recorded experimental data through matching with the theoretical response curves associated with standard homogeneous or layered earth models [12, 14, 15]. It is easily understood how, for correct data interpretation, high accuracy is required when computing such theoretical response curves.

This paper will focus on the problem of an electrically small circular loop antenna (vertical magnetic dipole) lying on the surface of -layer plane stratified earth. To date, the fields of a loop source in the presence of a layered ground are not known in closed form, and evaluation of their complete integral representations has been revealed to be challenging from both analytical and numerical points of view. When available, analytical solutions either pertain to simple earth models, like the single-layer configuration [16], or follow from introducing simplifying assumptions. Examples of approximate solutions are the explicit series-form expressions that result from applying standard asymptotic techniques, such as the method of steepest descent (saddle-point method) [11, 12, 17–19]. The obtained formulas are easy to use but are valid under the restrictive assumption that the source-receiver distance is electrically large, that is, in the far-field region of the antenna. Similar considerations can be made with regard to the low-frequency or quasi-static expressions for the fields [7, 12, 13, 20], which are accurate, subject to the condition that the displacement currents are negligible in both the layered material medium and the air space above it.

On the other hand, numerical computation of the field integrals through standard numerical approaches, like Gaussian quadrature, is often made difficult and impractical by the oscillating behavior of the integrands and the slow convergence of their envelope, especially when both source and field points lie at the air-medium interface [1, 12, 21]. In an attempt to overcome these drawbacks, some papers document extensive usage of the digital linear filter technique in place of the Gaussian quadrature approach [22, 23]. Unfortunately, it has been recently shown [9, 24, 25] that any digital filter is designed and optimized for a specific known integral transform (the so-called test function) and may lead to uncertain results when applied to the calculation of different integrals.

The aim of this work is to derive exact explicit expressions for the fields excited by the loop source at the air-medium interface. The theoretical development is organized as follows. First, the Sommerfeld Integrals describing the fields are split into two terms, that is, the residue contributions from the poles of the integrands and the branch-cut integrals due to the propagation constants in free space and in the bottom layer of the stratified medium. Next, such propagation constants are extracted from the integrands of the branch-cut integrals and replaced with the sums of partial fractions generated by Newton’s iterative method [26, 27]. This makes it possible to apply Cauchy’s integral formula and obtain explicit series representations for the branch-cut integrals. As a result, the Sommerfeld Integrals are converted into infinite sum of Hankel functions. The advantage of the presented approach resides in the fact that the sequence of the sums of partial fractions that are used in place of each propagation constant is convergent, as Newton’s method exhibits quadratic rate of convergence if applied to the computation of a square root [26]. This implies that the derived series representations for the fields are exact.

#### 2. Theory

Consider a vertical magnetic dipole of moment lying on the surface of a flat -layer lossy medium, as shown in Figure 1. Due to symmetry about the axis of the dipole, a cylindrical coordinate system is suitably introduced. The dielectric permittivity and electric conductivity of the th layer are denoted by and , respectively, while the medium is assumed to have the magnetic permeability of free space . The time-harmonic EM field components generated by the dipole in the air space (the zeroth layer, ) may be derived from an electric vector potential which has only component . If the time factor is assumed, it readswhere , withrespectively, being the direct and reflected potentials. In (2) and (3), is the zeroth-order Hankel function of the first kind, whileis the plane wave reflection coefficient at , with and being the intrinsic admittance of free space and the surface admittance at . The intrinsic admittance of the th layer is given bywithwhile, for layers, the surface admittance at is obtained through the recurrence relation withThe reflected field may be split into two parts, that is, the ideal reflected field (the field of a negative image) and a contribution due to the imperfect conductivity of the earth. Use of the identityin (3) allows concluding that the direct and image fields exactly cancel out at all points in the air space. Hence, the total potential readsThe fraction under the integral sign has a number of pole singularities, plus the branch point singularities at and . It is interesting to note that the points are not branch points of the integrand, as the fraction term is an even function of . We now deform the integration path to the upper infinite semicircle centered at the origin of the complex -plane, leaving out all the singularities located in the first and second quadrants, as indicated in Figure 2. The result of the deformation is a new integration path constituted by , the two lines wrapped around the branch cuts, that is, and , and a number of infinitesimal circles around the poles of the integrand. It is not difficult to prove that the infinite semicircle does not contribute to the integral. In fact, for large values of , the Hankel function may be replaced with its asymptotic form [28]and the integrand in (11) becomes suitable for application of Jordan’s Lemma [29], which states the following upper bound:Since, in the air space, factor decays exponentially with increasing , the right-hand side of (13) must vanish, and so does the integral along . As a consequence, may be expressed aswhere is the contour consisting of , , and the infinitesimal circles. It should be observed that the branch-cut integrals along and describe the above-surface ground wave and the lateral-wave field, respectively, while the integrals around the poles are trapped-surface-wave terms [17].