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

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.


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][2][3][4][5][6][7][8][9], close-tothe-surface radio communication [10,11], and geophysical prospecting [11][12][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 currentcarrying 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 (saddlepoint method) [11,12,[17][18][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.

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  0 .The time-harmonic EM field components generated by the dipole in the air space (the zeroth layer,  < 0) may be derived from an electric vector potential which has only  component .If the time factor   is assumed, it reads where  =   +   , with respectively, being the direct and reflected potentials.In ( 2) and (3),  (1)  0 (⋅) is the zeroth-order Hankel function of the first kind, while is the plane wave reflection coefficient at  =  0 = 0, with  0 and Ŷ1 being the intrinsic admittance of free space and the surface admittance at  0 .The intrinsic admittance of the th layer is given by with while, for  layers, the surface admittance at  =  −1 is obtained through the recurrence relation with The 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 identity in (3) allows concluding that the direct and image fields exactly cancel out at all points in the air space.Hence, the total potential reads (1)  0 () . ( The fraction under the integral sign has a number of pole singularities, plus the branch point singularities at  = ± 0 and  = ±  .It is interesting to note that the points  = ± 1 ,  = ± 2 , . . .,  = ± −1 are not branch points of the integrand, as the fraction term is an even function of  1 ,  2 , . . .,  −1 .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, Γ 0 and Γ  , 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]  (1) 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   0  decays exponentially with increasing ||, the right-hand side of (13) must vanish, and so does the integral along .As a consequence,  may be expressed as where Γ is the contour consisting of Γ 0 , Γ  , and the infinitesimal circles.It should be observed that the branchcut integrals along Γ 0 and Γ  describe the above-surface ground wave and the lateral-wave field, respectively, while the integrals around the poles are trapped-surface-wave terms [17].The scope of this work is to exactly evaluate the field components at the air-ground interface.To this goal, we substitute ( 14) into (1) and take the limit as  → 0 − .It yields with Quantity  ℎ may be rewritten as where is the integral along Γ  , while  () ℎ is the contribution from the residues of the integrand at the poles of  ℎ .The explicit expression for  () ℎ follows from applying Cauchy's theorem [29].It yields where R  [⋅] denotes the residue at the generic pole   ( = 1, 2, . ..).On the other hand, to evaluate  (0) ℎ and  () ℎ , it is convenient to extract the square root terms  0 and   from  ℎ .This task may be accomplished through letting  ℎ () =  ℎ ( 0 ,   ) and then applying the branch-cut extraction procedure: where with It is straightforward to verify that functions  () ℎ ,  () ℎ ,  () ℎ , and  ()  ℎ are even in both  0 and   and, hence, do not exhibit branch cuts.Next, we use Newton's method to convert  0 and   into rational functions, so as to replace the branch cuts running from  = − 0 and  = −  with a set of pole singularities.This will make it possible to exactly evaluate the integrals through contour integration.According to Newton's method, the principal square root of the complex number  2  (i.e., the principal root of the equation  2 −  2  = 0) may be obtained by taking the limit of the sequence as  → ∞.Recursively applying (28) provides the explicit form of ũ()  ; namely, with Thus, substitution of (29) for   's in (22) makes  ℎ become a meromorphic function of , whose poles coincide with those of ũ() 0 and ũ()  .The poles of ũ()  ( = 0, ) with positive imaginary part,   = 2 −1 − 1 in number, are given by with  ()  ( = 1, 2, . . .,   ) being the generic root of the equation Notice that 2 −1 zeros of   () are negative real numbers, and this implies that poles (31) are positioned on the suppressed branch lines, which are described by the equation with   being a negative real parameter.As shown in Figure 3, substitution (29) has the effect of turning the branch-cut integrals  (0) ℎ and  () ℎ into contour integrals around poles (31).It reads where [ () ℎ ũ0 +  () ℎ ũ0 ũ ]  (1)  0 () , [ ()  ℎ ũ +  () ℎ ũ0 ũ ]  (1)  0 () , with Γ  ( = 0, ) being the infinitesimal circle around   , and where the superscript  has been omitted for notational simplicity.It should be observed that   is not a pole of  () ℎ and, hence, it results as Substituting ( 29) for ũ0 and ũ in (35) provides the expressions where we have taken account of the fact that, far from the relevant poles, quantities ũ0 and ũ may be confused with  0 and   , respectively.Since from ( 6), (29), and (31) it follows that lim with applying Cauchy's integral formula [29] provides Thus, (37) are turned into with Combining ( 41) and ( 42) with (34) finally yields the expression where one can clearly distinguish the above-surface ground wave, associated with  0 terms, and the lateral wave, constituted by   terms, the latter traveling in the th layer of the medium along its upper boundary.Substitution of (44) into (19), in conjunction with ( 21), provides explicit expressions, in series form, for integrals  0 and  1 appearing in ( 15)- (17).

Validation of the Formulation
To test the validity of the developed theory, formulas ( 15)-( 17) and ( 19), (21), and (44) are applied to the computation of the  magnitude of the magnetic field components generated on the top surface of a homogeneous medium at distance  from a unit-moment vertical magnetic dipole.In the considered case  = 1, and functions  ℎ (), which reduce to do not exhibit poles.This implies that terms (21) do not contribute to the integrals.Thus, applying ( 24 The numerical results that arise from using ( 16) and ( 17), in combination with (47), are depicted in Figures 4-7 and denoted by lines.Such results are compared with those

Figure 1 :
Figure 1: Sketch of a vertical magnetic dipole on a multilayer ground.

Figure 2 :
Figure 2: Deformed integration contour for the Sommerfeld Integrals.

Figure 3 :
Figure 3: Final integration contour for the Sommerfeld Integrals.

Figure 4 :
Figure 4: Amplitude-frequency spectrum of   , computed at  = 20 m from the source point.