FAST SOLUTIONS OF ELECTROMAGNETIC FIELDS IN LAYERED MEDIA FEI TING (M.S., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgments I would like to take this opportunity express my most sincere appreciation to my supervisors, Professor Li Le-Wei, Professor Yeo Tat-Soon and Dr. Zheng Yuanjin, for their guidance, supports, and kindness throughout my postgraduate program. I wish to thank the members of Radar Signal Processing Laboratory: Dr. Yao Haiying, Mr. Xu Wei, Mr. Zhang Lei, Mr. Qiu Chengwei, Mr. Feng Zhuo, Mr. Kang Kai, Mr. Yuan Tao, Mr. Hwee Siang Tan, Miss Li Yanan, Miss Wu Yuming, Mr. She Haoyuan, and the lab officer, Ng Jack. Special thanks to my friends Miss Fan Yijing, Miss Zhang Yaqiong, Miss Zhu Yonglan, and Miss Feng Yuan. It is a great time when I live with you through out my Ph.D degree studies. I wish to thank my family and my boyfriend Andrew for enduring my prolonged absence during the doctoral study. i Contents Acknowledgments i Contents ii List of Figures viii List of Symbols xiii Introduction 1.1 Fast Methods for Layered Media . . . . . . . . . . . . . . . . . . . . 1.1.1 Planarly Layered Media . . . . . . . . . . . . . . . . . . . . 1.1.2 Spherically Layered Media . . . . . . . . . . . . . . . . . . . 1.2 Motivation and Research Objectives . . . . . . . . . . . . . . . . . . 12 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 List of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Fields in Spherically Layered Media 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Dyadic Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . 18 ii CONTENTS 2.3 2.4 2.5 iii Dyadic Solution For Spherically Layered Media . . . . . . . . . . . 21 2.3.1 Dyadic Green’s Function in Unbounded Media . . . . . . . . 24 2.3.2 Scattering Dyadic Green’s Functions . . . . . . . . . . . . . 25 2.3.3 Scattering Coefficients for Perfectly Conducting Sphere . . . 26 2.3.4 Scattering Coefficients for Dielectric Sphere . . . . . . . . . 27 2.3.5 Scattering Coefficients for a Conducting Sphere Coated with a Dielectric Layer . . . . . . . . . . . . . . . . . . . . . . . . 29 Eigenfunction Expansion . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.1 A Perfectly Conducting Sphere . . . . . . . . . . . . . . . . 30 2.4.2 A Dielectric Sphere . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.3 A Conducting Sphere Coated with a Dielectric Layer . . . . 31 Accurate and Efficient Computation of Scaled Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Radiation of Vertical Electric Dipole on Large Sphere . . . . . . . . 35 2.7 Continuous Form of Field Expression . . . . . . . . . . . . . . . . . 39 2.8 Radiation Pattern of a Vertical Electric Dipole . . . . . . . . . . . . 45 2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Convergence Acceleration for Spherically Layered Media 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Asymptotic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Convergence Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 Convergence Property of Scattered Waves . . . . . . . . . . . . . . 60 CONTENTS 3.5 3.6 iv Kummer’s Transformation . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.1 Perfectly Electric Conductor Earth . . . . . . . . . . . . . . 65 3.5.2 Dielectric Lossy Spherical Earth . . . . . . . . . . . . . . . . 87 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Fields in Planarly Layered Media 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Sommerfeld Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.2 Singularities in Sommerfeld Integrals . . . . . . . . . . . . . 95 4.3 VED in Three-Layered Media . . . . . . . . . . . . . . . . . . . . . 96 4.4 Comparison of Fields in Thin-Layered Media . . . . . . . . . . . . . 100 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A Comparative Study of Radio Wave Propagation over the Earth Due to a Vertical Electric Dipole 112 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2 Planar Earth Model and Formulation . . . . . . . . . . . . . . . . . 114 5.3 Spherical Earth Model and Formulation . . . . . . . . . . . . . . . 116 5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.5 5.4.1 Asymptotic Methods in Comparison . . . . . . . . . . . . . 119 5.4.2 Asymptotic Computation Compared with Exact Computation131 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 CONTENTS v Conclusions 138 A Asymptotic Representations of Hankel Functions 141 A.1 Debye Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 A.2 Watson Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.3 Olver Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.4 Second-Order Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 146 Summary The fast solutions of a vertical electric dipole antenna radiated fields in the presence of planarly and spherically layered media are studied in this work. For the spherically layered media, the continuity of the field expressions on the spherical surface at r = r in the space is discussed, and the fast solution to the electromagnetic fields due to the presence of a large sphere is presented. Some examples are considered to demonstrate the special properties of the respective field contributions. For the planarly layered media, a comparative study is carried out for the electromagnetic fields radiated by a vertical electric dipole on the surface of a thin dielectric layer. The direct wave and the reflected wave are found to attenuate as ρ−1 in the ρ direction; therefore in the far-field region, the surface wave dominates the total field. It is also found out that the method used in [1] at ρ = 200λ leads to a relative error of 7%, as compared with the result by [2]. The contribution of the pole is compared with that of the branch cut and it is found out that surface wave mode is dominant for ρ > λ. For the radio-wave propagation along the surface of the earth, the electromag- vi Summary vii netic field excited by a vertical electric dipole on the earth is studied. Four sets of formulas for both the planar earth model and the spherical earth model (of large radius) are compared to find out their valid ranges. Numerical computations are also carried out specifically for a three-layered earth model. For the planar earth model, when both the source and observation points are on the surface, and the planar earth covered with a thick-enough dielectric layer, the method by Zhang [1] is more accurate; while for the fields above the surface and the thin-enough dielectric layer, the method by King and Sandler [3] is more accurate. However, the hybrid modes of the trapped surface wave and the lateral wave were exhibited in the curves in [1], but they were not shown in the curves in [3]. Numerical calculations also show that the amplitude of the trapped surface wave by [1] attenuates as ρ−1/2 in the ρ direction as expected. However, the lateral wave given in [1] did not exhibit ρ−2 decay in the ρ direction. For the layered spherical earth model, the exact series summation, which serves as an exact solution to the classic problem, is computed and compared with the residue series. Numerical results show that the residue series gives a good approximation to the field, but the smooth curve illustrates that the hybrid effect due to the trapped surface wave and the lateral wave was ignored in literature. The field strength of the trapped surface wave decreases with the dielectric layer thickness and is affected by the curvature of the earth. The exact series shows the oscillation of the field caused by the hybrid effects, which can be considered as the dielectric resonance between the upper and lower dielectric interfaces when it is guided to propagate, but none of the other three approximations can depict the effects. List of Figures 2.1 Geometry of a multilayered sphere . . . . . . . . . . . . . . . . . . 25 2.2 A dipole over a PEC sphere with dielectric coating . . . . . . . . . 36 2.3 Field strength distribution |E direct | of a vertical dipole in free space obtained using the formula in discontinued field. . . . . . . . . . . . 38 Amplitude and phase of the normalized far field component Eθ each as a function of θ at k0 a = 10 for perfectly conducting and coated spheres. Dashed curve: t = 0.1; dotted curve: t = 0.01; continuous curve: perfectly conductor . . . . . . . . . . . . . . . . . . . . . . . 47 norm Convergence pattern of the radial component Etotal,r (n) of normalnorm ized electric field E total (n) defined in (2.60) as a function of n for a perfectly conducting earth at θ = 0, and for k1 a = 150, k1 b = 151, and k1 r = 154. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4 3.1 3.2 norm Convergence pattern of the normalized scattered electric field Escat,r (n) as a function of n for a perfectly conducting earth at θ = π/3, and for k0 a = 150, k0 b = 151, and k0 r = 154. . . . . . . . . . . . . . . . 56 3.3 Convergence pattern of the radial component of normalized electric norm (n) as a function of n for a lossy earth at ρ = km or field Etotal,r o θ = , and for r = 12, σ = 0.4, and f = 10 kHz. . . . . . . . . . . 58 Convergence pattern of the radial component of normalized electric norm field Etotal,r (n) as a function of n for a lossy earth at ρ = 200 km, and for r = 12, σ = 0.4, and f = 10 kHz. . . . . . . . . . . . . . . 59 Convergence of scattered waves versus observation angle θ for dipole located on the surface. . . . . . . . . . . . . . . . . . . . . . . . . . 62 Convergence of scattered waves versus observation distance r. Dotted curve: a = b; dashed curve: b = a + 2. . . . . . . . . . . . . . . 64 3.4 3.5 3.6 viii LIST OF FIGURES 3.7 ix Relative errors of the r-components of E total (—), E scat (· · · ), and Q(- - -) versus the truncation number n. k0 a = k0 b = 50, k0 r = 52 and θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Relative errors of the θ-components of E total (—), E scat (· · · ), and Q(- - -) versus the truncation number n. k0 a = k0 b = 50, k0 r = 52 and θ = π/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Relative errors of the r-components of E total (—), E scat (· · · ), and Q(- - -) versus the truncation number n. k0 a = k0 b = 50, k0 r = 52, and θ = π/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 norm 3.10 Convergence pattern of the normalized correction component Ecorr,r as a function of n for a PEC sphere at θ = 0, and for k0 a = 150, k0 b = 151, and k0 r = 154. . . . . . . . . . . . . . . . . . . . . . . . 77 norm (n) as 3.11 Convergence pattern of the normalized correction part Ecorr,r π a function of n for a PEC sphere at θ = , and for k0 a = 150, k0 b = 151, and k0 r = 154. . . . . . . . . . . . . . . . . . . . . . . . 78 3.12 Relative errors of the r-components of E total (—), E scat (- - -), and E corr2 (· · · ) versus the truncation number n. k0 a = 50, k0 b = 51, k0 r = 54 and θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.13 Relative errors of the r-components of E total (—), E scat (· · · ), and E corr2 (- - -) versus the truncation number n. k0 a = 50, k0 b = 51, k0 r = 54 and θ = 2π/3. . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.14 Relative errors of the θ-components of E total (· · · ), E scat (—), and E corr2 (- - -) versus the truncation number n. k0 a = 50, k0 b = 51, k0 r = 54 and θ = 2π/3. . . . . . . . . . . . . . . . . . . . . . . . . . 82 norm norm norm , Escat,r , Etotal,r , 3.15 Real and imaginary parts of the electric fields Ecorr,r norm norm Edirect,r and Eimage,r versus the vertical dipole height k0 b for a perfectly conducting sphere. . . . . . . . . . . . . . . . . . . . . . . . . 84 3.16 Truncation errors versus the number of terms for the convergent solution of the normalized field components Etotal (—), Escat (- - ) and Ecorr (· · · ) in the case of a PEC sphere at θ = 0, and for k0 a = 150, k0 b = 151, and k0 r = 154. . . . . . . . . . . . . . . . . . 85 3.17 Relative errors versus the number of terms for the convergent solution of the normalized field components Etotal (—), Escat (- - -) and Ecorr (· · · ) in the case of a PEC sphere at θ = 0, and for k0 a = 150, k0 b = 151, and k0 r = 154. . . . . . . . . . . . . . . . . . . . . . . . 86 3.8 3.9 Bibliography 149 dia,” IEEE Trans. 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[...]... purposes In this method, starting from the spectral representation of the field, an exact Laplace transform was applied to the reflection coefficients in the Sommerfeld integrals The resulting expressions consist of a double integral, one in the original-spectral domain, and the other in the Laplace domain The integral in the spectral domain has an analytical expression and the remaining integral expressions in. .. planarly layered media 1.3 Outline In this work, the fast solutions of the electromagnetic fields radiated by a vertical electric dipole in both planarly layered and spherically layered media are studied First, a literature review is presented, then the various asymptotic and numerical methods are summarized In Chapter 2, the classical problem of the vertical antenna radiation in the presence of a spherically... air in the “shadow region” in terms of exponentially decreasing waves, and gave corresponding attenuation rates as solutions to two uncoupled transcendental equations Fock started with an extension of Watson’s method by neglecting the field that travels through the sphere and not examining the transition to planarearth formulas In a remarkable paper, Wu [50] invoked the concept of the creeping wave in. .. potential GEJ electric type of dyadic Green’s function GHJ magnetic type of dyadic Green’s function g(r − r) free-space scalar Green’s function δ Kronecker delta !! double factorial xiii Chapter 1 Introduction 1.1 1.1.1 Fast Methods for Layered Media Planarly Layered Media The computation of the electromagnetic (EM) fields in planarly layered media has been a classical subject of numerous investigations over... to obtain exact series solutions for the fields For the fields far away from a large sphere and for the wavelength in the air being much smaller than the radius of the sphere, the series converges slowly The terms of the series start to diminish only when the truncation number becomes of the Chapter 1 Introduction 7 order of k0 a, with k0 being the wavenumber of free space and a being the radius of the... The merits of the Watson’s approach are unquestionable: the slowly converging expansion in partial waves was converted to an integral which in turn generated a rapidly converging series Many series representations for problems involving cylindrical and spherical structures can be transformed into the complex integral of this form For a large Chapter 1 Introduction 8 sphere, the convergence of the harmonic... the integral is more useful Since then, research continued in the directions of extending the theory to the case of an earth with finite conductivity, supporting theoretical estimates with numerical calculations and exploring alternative ways for treating this problem [41, 42, 7, 46, 58] The radiation of a horizontal dipole above a finitely conducting sphere was investigated by Fock [48] by use of scalar... Sommerfeld in 1909 [4] in the form of integrals The work is later extended to layered media by other researchers using the generalized reflection coefficients 1 Chapter 1 Introduction 2 However, the closed-form solution to the Sommerfeld integrals (SI) is not known yet Numerous approximation techniques were thereafter developed to obtain more accurate and faster results Generally there are two kinds of solutions. .. coating thickness The amplitude and phase of the vertical component of the fields are significantly effected by the coating compared with the uncoated sphere In addition to the applications in planarly layered media, the image principle has also been applied in electrostatics to problems with charges in front of a dielectric sphere [69] Lindell [70] generalized a theoretical formula for the problem in. .. regardless of the positions of the source and observation points [6] Therefore, in this work, we will investigate the convergence properties of the series Convergence acceleration methods are resorted to enhance the accuracy and to accelerate the solution For the planarly layered media, due to the highly oscillating and slow convergent nature of the integrand of Sommerfeld integral, various solutions . FAST SOLUTIONS OF ELECTROMAGNETIC FIELDS IN LAYERED MEDIA FEI TING (M.S., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING NATIONAL. integral, one in the original-spectral domain, and the other in the Laplace domain. The integral in the spectral domain has an analytical expression and the remaining integral ex- pressions in. 92 4.2.2 Singularities in Sommerfeld Integrals . . . . . . . . . . . . . 95 4.3 VED in Three -Layered Media . . . . . . . . . . . . . . . . . . . . . 96 4.4 Comparison of Fields in Thin -Layered Media