On the Radiation from a Short Current-Carrying Straight Wire Oriented Perpendicular to a Stratified Medium

Transcription

1 Progress In Electroagnetics Research, Vol. 159, 49 57, 17 On the Radiation fro a Short Current-Carrying Straight Wire Oriented Perpendicular to a Stratified Mediu Mauro Parise 1, * and Giulio Antonini Abstract In a previous work, iproved full-wave analytical expressions have been derived for the Soerfeld Integrals (SIs) describing electroagnetic radiation fro a short vertical straight wire located in close proxiity to a conductive soil. Such forulas ensure high accuracy of the result of the coputation, as well as tie savings respect to conventional techniques used to evaluate the SIs, but unfortunately ay be used only when both source and field points are located at the air-ediu interface. The scope of this paper is to overcoe the liitations iplied by the previous approach, and provide series-for expressions for the generated field coponents that are valid for an arbitrarily stratified ediu and for any position of the vertical wire antenna and observation point in the air space above it. The expressions follow fro the analytical evaluation of the integral representation for the agnetic vector potential, perfored through contour integration after substituting an equivalent pole set for each branch cut of the integrand. Validity, efficiency and accuracy of the developed forulas are illustrated through nuerical exaples. 1. INTRODUCTION The study of the radiation characteristics of dipole antennas situated in proxiity to stratified edia has attracted the interest of several scientists in the past years 1 ]. This is because electric dipoles are extensively used in a variety of engineering applications, especially in the areas of close-to-thesurface radio counication, geophysical prospecting, radio reote sensing, and hypetheria 7, 11 15,, 3]. Despite the nuber of contributions fro researchers, to date accurate analytical expressions for the fields, valid regardless of the operating frequency, can be derived only under the assuption that both the source and the observation points lie on the surface of the aterial ediu 1, 3]. If this condition cannot be et, closed-for expressions for the fields ay still be obtained, but at the price of introducing approxiations that liit their applicability to electrically dense edia 4 7, 11] or specified frequency ranges 4 6, 17 ]. Exaples of contributions in this direction are the solutions valid for the quasi-static, inderediate, and far-field frequency ranges, which result fro reducing the Soerfeld integrals to fors aenable to asyptotic techniques such as the saddle-point ethod 13, 17 ]. The present paper focuses on the proble of a short current-carrying straight wire oriented perpendicular to a stratified ground. The scope of the work is to overcoe the liitations iplied by the approach described in 1, 3], and deterine exact analytical expressions for the generated fields that are valid for any position of the source and observation points in the air space above the ediu. To accoplish this task, first the direct field and the ideal reflected field are extracted fro the integral representation for the agnetic vector potential. Next, the reaining part of the agnetic vector potential is written as the su of branch-cut integrals and closed-contour integrals around the poles of the integrand. Finally, the hyperbolic branch cuts are extracted fro the integrands of the Received 9 April 17, Accepted June 17, Scheduled 3 June 17 * Corresponding author: Mauro Parise (.parise@unicapus.it). 1 Unit of Electrical Engineering, University Capus Bio-Medico of Roe, Italy. Departent of Industrial Engineering, Coputer Science and Econoics, University of L Aquila, Italy.

2 5 Parise and Antonini branch-cut integrals and replaced equivalent pole sets 3], so as to ake it possible analytical integration. Each field coponent is given as the su of two exponential ters (describing the direct and the ideal reflected fields) and three fast convergent sequences of residue sus, that represent the above surface ground wave, the lateral wave and the surface waves. The obtained forulas for the fields offer advantages in ters of tie savings respect to standard nuerical integration procedures. Moreover, investigating the expressions for the different contributions to the fields, as well as their outcoes, allows to gain useful insight into the physics of the proble, and this feature akes the proposed solution advantageous over nuerical siulation tools eployed to solve electroagnetic boundary value probles.. FORMULATION OF THE PROBLEM The geoetrical configuration under study is sketched in Fig. 1. The eitter, odeled as a vertical electric dipole of oent 1 e jωt A, is located at height h above an N-layer conducting ediu. The nth layer of the ediu has dielectric perittivity ɛ n, electrical conductivity σ n,andthickness d n =z n z n 1,beingz n the position of the upper bound of the layer. The botto or first layer is assued to be a sei-infinite region. The agnetic pereability is everywhere equal to that of free space μ. We introduce a cylindrical coordinate syste (ρ, ϕ, z) such that the air-ediu interface coincides the plane z= and the short wire antenna is placed on the z axis. The syetry about the z axis iplies that the electroagnetic field originates fro a z-directed agnetic vector potential 17]. As a consequence, the field coponents produced in the air space ay be expressed as E ρ = jω A k ρ z, (1) E z = jω ( k ρ + 1 ) A, () ρ ρ H ϕ = 1 A μ ρ, (3) z ρ P z-h μ, ε h z N z N-1 z N- μ, ε N, σ N μ, ε N-1, σ N-1 x ϕ ρ y z z 1 μ, ε, σ μ, ε 1, σ 1 Figure 1. Sketch of a short straight wire oriented perpendicular to a stratified ediu.

3 Progress In Electroagnetics Research, Vol. 159, where A is the non-null coponent of the vector potential, given by ] A (ρ, z, h) = μ e jk R 4π R e jkr R + A s (ρ, z, h), (4) being A s (ρ, z, h) = μ e u (z+h) Z λ H (1) 4π Z + ẐN (λρ)dλ (5) u and R = ρ +(z h), R = ρ +(z + h), (6) u n = λ kn, Reu n ]>, (7) kn = ω μ n ɛ n jωμ n σ n. (8) In Eq. (4), the two exponential ters represent the priary field and the ideal field reflected by the ediu (that is the field of a negative iage), while A s is a correction ter caused by the iperfect conductivity of the layered half-space. On the other hand, Z n is the intrinsic ipedance of the nth layer (n= denotes free space), given by u n Z n =, (9) σ n + jωɛ n while Ẑn is the surface ipedance at z=z n, defined by the recursive forula Ẑ n 1 + Z n tanh(u n d n ) Ẑ n =Z n Z n + Ẑn 1tanh(u, n=,...,n, (1) n d n ) Ẑ 1 =Z 1. (11) The ai of this paper is to analytically evaluate Eq. (5) and, subsequently, the field coponents in Eqs. (1) (3). To this end, it is first convenient to rewrite Eq. (5) as A s (ρ, z, h) = μ 4π G (u,u 1,λ,z,h) H (1) (λρ)λdλ, (1) G (u,u 1,λ,z,h)= e u (z+h) Z, (13) u Z + ẐN where the dependence on the square roots u,u 3,...,u N has not been ade explicit to signify that the function G is even in such quantities 17]. Next, proceeding as in 17, 4] akes it possible to defor the integration contour so that it is wrapped around the singularities of the integrand located in the upper-half of the coplex plane, as shown in Fig. (a). This leads to express A s as A s (ρ, z, h)=...]+...]+...]=i ( 1) + I () + I (1), (14) λ ( 1) k k 1 that is as the su of two branch-cut integrals associated k and k 1 plus a nuber of closedcontour integrals around the poles of G, thatisλ ( 1) 1,λ ( 1),λ ( 1) 3,... Notice that the integral along the upper infinite sei-circle has been neglected in Eq. (14). In fact, this contribution cannot but be identically zero, because the Hankel function in the integrand decays exponentially increasing λ in the upper half-plane. The explicit for of I ( 1) ay be easily obtained through residue calculus. It yields I ( 1) = jμ ] A ( 1) (z,h) H (1) λ ( 1) ρ, (15)

4 5 Parise and Antonini λ- plane λ ( 1) I λ Integration path I λ k 1 λ ( 1) 1 generated poles ( n ) λ 1 ( n ) λ ( n ) λ arc of upper sei-circle k (a) o Re λ suppressed branch line (b) Figure. (a) Defored integration contour and (b) branch cut equivalent pole set. where A ( 1) is the residue of the function λg at λ=λ ( 1) A ( 1) (z,h)=r ( 1) λ λ G(u,u 1,λ,z,h)] = li λ λ ( 1),thatis λ λ ( 1) ] λg(u,u 1,λ,z,h). (16) On the other hand, each of the integrals I () and I (1) ay be evaluated by extracting the relevant branch cut fro the integrand and replacing it an equivalent set of poles. Branch-cut extraction procedure consists of decoposing G into its even and odd parts respect to u or u 1. For instance, I () requires a decoposition of G respect to u, as follows G (u,u 1,λ,z,h)=G () e (λ, z, h)+g () o (λ, z, h) u, (17) where G () e and G () o are even function of u,givenby G () e (λ, z, h) = 1 G (u,u 1,λ,z,h)+G( u,u 1,λ,z,h)], (18) G () o (λ, z, h) = 1 G (u,u 1,λ,z,h) G ( u,u 1,λ,z,h)]. (19) u Since G () e does not exhibit any branch cut running fro k, it will not contribute to I (). As a consequence, it yields I () = μ G () o (λ, z, h) u H (1) (λρ)λdλ, () 4π k and, analogously, for I 1 one obtains I (1) = μ G (1) o (λ, z, h) u 1 H (1) (λρ)λdλ, (1) 4π k 1 G (1) o (λ, z, h)= 1 G (u,u 1,λ,z,h) G (u, u 1,λ,z,h)]. () u 1 Pole sets equivalent to the extracted branch cuts ay be deterined by iteratively applying the square root algorith to the coputation of the square roots u and u 1, as described in 1, 3, 4]. At the lth iteration, the square root algorith generates the following partial-fraction representation for u n 3] ( ) ] l 1 u n = l u n + M l h r n λ λ +, (3) h= =1

5 Progress In Electroagnetics Research, Vol. 159, being M l = l 1 1, and = kn t, (4) r n = 1 M l +1 ( ) l t ] h M l l+1 t h h t ] 1, (5) h= ( π ) t =tan l. (6) Substituting Eq. (3) in the nth branch-cut integral I (n) and closing the integration contour the arc of upper infinite sei-circle coprised between its ends, as sketched in Fig. (b), allows to obtain a closed-contour integral around the s. In fact, such poles lie on the suppressed branch line. Since the residue of Eq. (3) at λ= is equal to one obtains and, after letting A (n) in copact for as I (n) = jμ li λ =1 (z,h)=r n G (n) o λ h=1 h ] u n = r n, n =, 1, (7) r n G (n) o (λ, z, h) H (1) ]λ=λ (λρ), n =, 1, (8) (n) A s (ρ, z, h)= jμ ],z,h,n=, 1, expression (14) for A s ay be rewritten 1 n= 1 =1 A (n) (z,h) H(1) ] ρ, (9) where the first M l poles of the pole set corresponding to n= 1 have been considered. It is understood that the nubering of the λ ( 1) s is arranged so that I λ ( 1) 1 < I λ ( 1) < I λ ( 1) 3 <...and so on, which eans that higher-order poles correspond to highly attenuated waves. Finally, use of Eqs. (4) and (9) in Eqs. (1) (3) gives rise to the following expressions for the fields E ρ = E ρ ζ=z h E ρ ζ=z+h + Eρ s, (3) E z = E z ζ=z h E z ζ=z+h + Ez s, (31) H ϕ = H ϕ ζ=z h H ϕ ζ=z+h + Hϕ, s (3) being E ρ = ρζe jk r ( 3+3jk 4πjωɛ r 5 r k r), (33) r= ρ +ζ,and E z = e jk r 4πjωɛ r 5 ( ζ ρ ) (1 + jk r)+ρ k r], (34) H ϕ = ρe jk r 4πr 3 (1 + jk r), (35) E s ρ = 1 ωɛ E s z = 1 ωɛ H s ϕ = j 1 1 n= 1 =1 1 n= 1 =1 n= 1 =1 A(n) H(1) 1 da (n) ] dz H(1) 1 ρ, (36) ] ] A (n) H(1) ρ, (37) ρ ], (38)

6 54 Parise and Antonini R ( 1) λ u λg(u,u 1,λ,z)], n = 1 da (n) ] dz = r n G () e λ (),z, n = ] r n u G (1) o λ (1),z, n =1 (39) where account has been taken of the identities 1] ( ρ + 1 ρ ρ dh (1) (λρ) dρ = λh (1) 1 (λρ), (4) ) H (1) (λρ) = λ H (1) (λρ). (41) Each of the expressions (36) (38) consists of three sequences of residue sus, which converge as l is increased in virtue of convergence of the square root algorith. The residue sus associated n= and n=1 describe the above-surface ground and lateral waves, respectively. Such contributions, together the direct field and the ideal reflected field, labelled a subscript in Eqs. (3) (3), constitute the space wave radiated by the antenna. The rest of the generated field consists of surface waves, and is described by the residue sus associated n= 1 in Eqs. (36) (38). Expressions (3) (3) allow to overcoe the liitations iplied by the previously published square-root-algorith-based solution to the sae proble 1, 4]. In fact, they allow to relax the assuption that both source and observation points lie at the air-ediu interface. At the sae tie, as will be clarified in Sec. 3, the derived expressions ensure a high level of accuracy and offer advantages in ters of tie savings respect to standard nuerical techniques conventionally used for evaluating Soerfeld-type integrals. 3. RESULTS AND DISCUSSION Priary objective of this section is to show the convergence of the sequences in Eqs. (36) (38) as the nuber of iterations l of the square root algorith is increased. To this goal, we use Eq. (31) to copute the aplitude-frequency spectru of the E z -field generated by a unit-oent straight wire antenna lying on the top surface of a two-layer conductive ground. The field point is at height z=5 above the ediu and at radial distance ρ=3 fro the eitter, while the electroagnetic and geoetric paraeters of the layers of the ediu are assued to be σ 1 =1 S/, ɛ 1 =1ɛ, d 1 =4, σ =1 S/, and ɛ =5ɛ. The results of the siulations, depicted in Fig. 3, are copared the data arising fro nuerically evaluating Eq. () through the Gauss-Kronrod quadrature rule 5]. As is seen, when l is increased the outcoes fro Eq. (31) rapidly converge to the exact nuerical data, denoted by points, and convergence is faster at higher frequencies. Moreover, perfect agreeent between analytical and nuerical data is observed when truncating the sequences of residue sus in Eq. (37) at l=1, that is when each residue su is coposed of M 1 = 11 1=47 residues. It should be noted that, in general, the three contributions to Eq. (37) do not have the sae ipact on the E z -field strength. For instance, if the field point is located at height z=5 fro the top surface of the ediu, one would expect that the effects of surface and lateral waves, associated n= 1 and n=1, respectively, are negligible respect to that of the ground wave (n=). This point is clarified by Fig. 4, which depicts the agnitude of E z and those of the relevant ground-wave, lateral-wave, and surface-wave ters against the radial distance ρ fro the eitter. The paraeters of the ediu are taken to be the sae as in the previous exaple, and the operating frequency is assued to be 1 khz. Again, the sequences of the residue sus have been truncated at l=1. Fro the analysis of the plotted curves it eerges that the aplitude of the E z -field substantially coincides that of the ground-wave field over the considered interval, and that the surface waves suffer fro a ild attenuation increasing ρ. Instead, the agnitude of the lateral wave rapidly goes down below the plot scale as ρ grows up. Finally, it should be observed that direct and ideal reflected fields cannot contribute to the total E z -field, since they cancel out each other because of the position of the eitter at the air-ediu interface.

7 Progress In Electroagnetics Research, Vol. 159, E z (V/) Frequency (Hz) exact l=5 l=6 l=7 l=1 Figure 3. Profiles of E z against frequency. Outcoes fro the proposed solution are copared the data arising fro nuerical integration of the field integral. E z (V/) total gw lw sw Radial distance fro the source, ρ () Figure 4. Profile of E ρ against ρ. Separate contributions of ground wave (gw), lateral wave (lw), and surface waves (sw) to the total field. One would ask whether the derived solution is advantageous in ters of coputation tie over previous solutions and standard nuerical integration procedures. This aspect is illustrated in Table 1, which shows the tie taken by the proposed ethod, the Gauss-Kronrod quadrature rule, and King s well-established solution 7], to calculate the radial distribution of the vertical electric field that a straight wire antenna lying on a hoogeneous ediu produces at z=5. Here, the operating frequency is 1 MHz, while the electroagnetic paraeters of the ediu are taken to be σ 1 =.1 S/ and ɛ 1 =3 ɛ. Table 1 is structured as follows. The second colun contains tie cost data corresponding to Table 1. Coputation ties and accuracy of the proposed and conventional approaches. Approach Tie consuption (s) Speed-up RMS rel. error (%) Gaussian rule King s approach Proposed solution z () ρ () (a) z () ρ () (b) Figure 5. Contour plots of (a) E ρ and (b) E z versus ρ and z, coputed for σ 1 =.1 S/ and ɛ 1 =3 ɛ by applying the proposed approach (solid isolines). Outcoes fro King s approach are denoted by dashed isolines.

8 56 Parise and Antonini siulations running on a single-core 1.8 GHz PC. The third colun shows the ratio of the tie taken by the proposed expression for E z and the Gaussian rule to that required by King s solution (speed-up offered by King s approach). Finally, the fourth colun contains the root-ean-square (RMS) relative errors resulting fro using the proposed and King s solution in place of Gaussian quadrature rule. Fro a glance at the data in Table 1 it is deduced that, accuracy being equal, using Eq. (37) rather than Gaussian quadrature perits to obtain significant tie savings. On the other hand, King s approach iplies iniu tie cost, but the price to be paid for its usage is the loss of accuracy. This is also confired by the contour plots sketched in Fig. 5, which depict the agnitudes of E ρ (Fig. 5(a)) and E z (Fig. 5(b)) versus ρ and z (the antenna is still positioned at height h=). For both the contour plots, a significant discrepancy is observed between the solid isolines associated the proposed solution and the dashed isolines arising fro King s approach. The aount of the discrepancy depends on the conductivity of the ediu, and higher values of the conductivity ake it possible to reduce the discrepancy. For instance, assuing σ 1 =1 S/ and ɛ 1 =1 ɛ leads to the contour plots illustrated in Fig. 6, characterized by solid and dashed isolines alost overlapping. Bad accuracy of King s solution for low values of the conductivity is explained by the failure of the assuptions k 1 ρ 1 andk / k 1 1, underlying its derivation. z () ρ () (a) z () ρ () (b) Figure 6. Contour plots of (a) E ρ and (b) E z versus ρ and z, coputed for σ 1 =1 S/ and ɛ 1 =1 ɛ by applying the proposed approach (solid isolines). Outcoes fro King s approach are denoted by dashed isolines. 4. CONCLUSION This paper presents an explicit solution, in series for, for the fields fro a short current-carrying straight wire oriented perpendicular to a stratified ediu. The solution arises fro the analytical evaluation of the integral representation for the agnetic vector potential, carried out through contour integration upon substituting an equivalent pole set for each branch cut exhibited by the integrand. Each field coponent is expressed as the su of five contributions, naely the direct, the ideal reflected, the ground-wave, the lateral-wave, and the surface-wave fields. The latter three ters are given as convergent sequences of residue sus. Perfored nuerical siulations confir the correctness of the developed theory. It has been shown how the proposed forulas are accurate and significantly less tie consuing than conventional nuerical integration procedures. REFERENCES 1. Soerfeld, A., On the propagation of waves in wireless telegraphy, Annals of Physics, Vol. 81, , Baños, A., Dipole Radiation in the Presence of a Conducting Half-space, Pergaon Press, Oxford, NY, 1966.

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