ELECTROMAGNETIC FIELD IN THE PRESENCE OF A THREE-LAYERED SPHERICAL REGION

Progre In Electromagnetic Reearch, PIER 45, 103 11, 004 ELECTROMAGNETIC FIELD IN THE PRESENCE OF A THREE-LAYERED SPHERICAL REGION K. Li and S.-O. Park School of Engineering Information and Communication Univerity 58-4 Hwaam-dong, Yuung-gu, Daejeon, 305-73, South Korea H.-Q. Zhang Shaanxi Atronomical Obervatory Chinee Academy of Science Xi an, Shaanxi, 710600, P. R. China Abtract In thi paper, the region of interet conit of the pherical dielectric earth, coated with a dielectric layer under the air. The imple explicit formula have been derived for the electromagnetic field of a vertical electric dipole and vertical magnetic dipole in the preence of three-layered pherical region, repectively. Next, baing on the above reult, the formula for the ix component of the field in the air generated by a horizontal electric dipole are derived by uing reciprocity. The computation how that the trapped urface wave of electric type can be excited efficiently and the trapped urface wave of magnetic type can not be excited when the thickne l of the dielectric layer i omewhat and atifie the condition 0 < k1 k 0 l< π. When the thickne l of the dielectric layer i omewhat and atifie the condition π k < 1 k 0 l<π, the trapped urface wave of magnetic type can be excited. Thee formula and the computation can be applied to the urface communication at the lower frequencie. 1 Introduction The Field in the Air Generated by a Vertical Electric Dipole over the Surface of the Dielectric-Coated Spherical Electrically Earth Alo with China Reearch Intitute of Radiowave Propagation, Xinxiang, Henan, 453003, P. R. China

104 Li, Park, and Zhang 3 The Field in the Air Generated by a Magnetic Dipole over the Surface of the Dielectric-Coated Spherical Electrically Earth 4 The Field in the Air Generated by a Horizontal Electric Dipole over the Surface of the Dielectric- Coated Spherical Electrically Earth 5 Concluion Acknowledgment Reference 1. INTRODUCTION The old problem of the electromagnetic field generated by a vertical electric dipole and horizontal electric dipole in a three-layered planar region ha been tudied and debated by everal invetigator becaue of it ueful application in everal cae [1 5]. It i neceary to reanalyze the old problem again. Only everal year ago, the complete, imple and accurate formula were derived for the electromagnetic field generated by a vertical electric dipole in a three-layered planar region. In [6], it ha been drawn a concluion that the trapped urface wave can be induced by a vertical electric dipole in a three-layered planar region. The wave number of the trapped urface wave i between that of the wave number k 0 in the air and the wave number k 1 in the dielectric layer, which i different with that preented in [4]. The amplitude of the trapped wave attenuate a r 1/ in the r direction near the urface and attenuate exponentially a e gz in the z direction. Thi development, naturally, rekindled the interet in the pherical problem. Recently, Pan and Zhang [8] and Li et al. [9] had derived the accurate and imple formula for the electromagnetic field of a vertical electric dipole and horizontal electric dipole over the urface of a pherical conducting earth coated with a dielectric layer, repectively. When applied to practical ituation uch a the wave propagation over the dielectric-coated earth, the earth hould be regarded a an electrically dielectric phere. It i neceary to tudy the problem furthermore. In the preent tudy, the imple analytical formula for the electromagnetic field in the air are outlined when a dipole ource (vertical electric dipole, vertical magnetic dipole, or horizontal electric dipole) i located near the urface of the dielectric-coated pherical electrically earth. Firt, along uch reearch line of [8] and [9], we will preent the formula for the field of vertical electric dipole and

EM field in three-layered pherical region 105 vertical magnetic dipole over the urface of a dielectric-coated pherical electrically earth. Next, baing on thee reult, we will derive the formula for the ix component of the electromagnetic field generated by a horizontal electric dipole over the dielectric-coated pherical electrically earth. It hould be noted that the condition of the trapped urface wave of electric type and thoe of the trapped urface wave of magnetic type are preented clearly in thi paper. Thee formula and the computation can be applied to the urface communication at the lower frequencie. Figure 1. Geometry of a vertical electric dipole in the preence of a three-layered pherical region.. THE FIELD IN THE AIR GENERATED BY A VERTICAL ELECTRIC DIPOLE OVER THE SURFACE OF THE DIELECTRIC-COATED SPHERICAL ELECTRICALLY EARTH Firt, we will analyze the field of a vertical electric dipole in the preence of three-layered pherical region. In the preent tudy, the geometry of the problem conit of the air (region 0, r > a), the dielectric layer with it thickne l (region 1, a l < r < a), and the earth (region, r<a l), a illutrated in Figure 1. The wave number in the three region are k 0 = ω µ 0 ε 0, k 1 = ω µ 0 (ε 0 ε r1 + iσ 1 /ω),

106 Li, Park, and Zhang k = ω µ 0 (ε 0 ε r + iσ /ω) (1) where ε r i the relative permittivity, σ i the conductivity. Auming that the dipole i repreented by the current denity ẑidlδ(x)δ(y)δ(z b), where b = z + a, and z > 0 denote the height of the dipole above the urface of the dielectric-coated earth, and ue i made of the time dependence e ( iωt), the nonzero component of the field E r, E θ, and H ϕ, which the field i defined a the field of electric type, can be expreed a follow, ( ) E r (i) = r + k i (U i r) () E (i) θ = 1 r θ r (U ir) (3) H ϕ (i) = iωε i r θ (U ir). (4) Here the potential function U i i the olution of in the calar Helmholtz equation ( + ki )U i = 0 (5) where i = 0, 1,. In pherical coordinate, becaue of ymmetry, the expanded form (5) i written a ( 1 r r U ) i 1 + r r r in θ ( θ in θ U i θ ) + k i U i =0. (6) Uing the eparation of variable method, we aume that a olution for U i can be written in the form of U i = 1 r Z i(r)φ(θ) (7) then [ 1 d in θ Φ(θ) ] + ν(ν + 1)Φ(θ) = 0 (8) in θ dθ θ d [ ] Z i (r) dr + ki ν(ν +1) 1 ki Z i (r) = 0. (9) r The equation (8) i cloely related to the well-known Legendre differential equation whoe olution i taken a the Legendre function of firt kind P ν (co(π θ)) [8]. In Region 0, the olution of (9) ha been addreed in [8]. Z 0 (r) =AW (t y) (10)

EM field in three-layered pherical region 107 where W (t) i the Airy function of the econd kind, and t and y are defined a [ a 3 ] /3 [ ] ν(ν +1) t = ν(ν +1) a k0 (11) [ ] ν(ν +1) 1/3 ( ) 1/3 y = a 3 z k 0 z. (1) k 0 a In (10), r = a + z and z a, z>0denote the height above the urface of the dielectric-coated earth. We aume that both the dipole and the point of obervation cloe to the pherical interface between the air and the dielectric layer. In the whole text we take z a and z r a. In Region 1, a l r a, taking into account l a, we take r a. Then (9) i implified a d Z 1 (r) dr +(k1 k0)z 1 (r) = 0 (13) The olution of (13) i Z 1 (r) =Be i k1 k 0 [r (a l)] + Ce i k1 k 0 [r (a l)]. (14) According to the impedance boundary condition at r = a l, weget E (1) θ η 1 H ϕ (1) = g (15) r=a l At lower frequencie, the urface impedance g can be approximated a, g k ( ) 1 k1 1 (16) k k Then, e = C B = k1 k 0 k 1 g. (17) k 1 k 0 + k 1 g At the boundary r = a, with the condition E (0) θ z=a = E (1) θ z=a and H ϕ (0) z=a = H ϕ (1) z=a,weget ( )1 AW 3 k0 = ib k1 k 0 a k 0 (ei k1 k 0 l e e i k1 k 0 l ) (18) A k0w = B k1(e i k1 k 0 l + e e i k1 k 0 l ). (19)

108Li, Park, and Zhang Then, the differential equation i obtained a follow where q = ik 0 k 1 k 0 k 1 W (t) qw (t) = 0 (0) ( k0 a ) 1 3 1 e e i k1 k 0 l 1+ e e i. (1) k1 k 0 l In the pherical conducting cae g 0, then e = 1, the above formula reduce to q = k 0 k1 k ( ) 1 0 k0 a 3 tan( k1 k 0 l). () k 1 The expreion of () coincide with that given in the literature [8]. If t are the root of the differential equation (37). Then the potential function U 0 in the air i expreed a U 0 = 1 A F (z r )P ν (co(π θ)) (3) r where the height-gain function F (z r )i F (z r )= W (t y). (4) W (t ) With the imilar method in [8], the coefficient A in (3) will be determined. ( ) 1/3 A = k0idl F(z ) e iνπ ka ωε 0 t q. (5) When ν i very large and θ i not cloe to 0 and π [8]. 1 P ν (co(π θ)) = πk 0 a in θ exp [i(k 0aθ + t x)] exp [i(ν + 1 ] 4 )π (6) where x =( k 0a )1/3 θ. Finally, the accurate and complete formula for the component E r, E θ and ηh ϕ of the electromagnetic field generated by a vertical

EM field in three-layered pherical region 109 electric dipole at (a + z, 0, 0) over the dielectric-coated pherical earth are expreed a follow. E r(a + z r,θ,ϕ) E θ (a + z r,θ,ϕ) = iidl η η 0 H ϕ (a + z r,θ,ϕ) λa F (z ) F (z r ) t q e itx ei(kaθ+ π 4 ) πx if (z ) F(z) k 0 z z=z r [ ( ) ] /3 e itx θ in θ k 0 (t q ) 1+ t (7) ka F (z ) F (z r ) [ ] (t q ) 1+ t ( e itx ka )/3 where r = a + z r. To the aphalt- and cement-coated pherical earth or ice-coated pherical eawater at the lower frequencie, it i eaily to compute the component of the electromagnetic field veru the propagating ditance at different thicknee of the dielectric layer. If the thickne of the dielectric layer i not zero, the parameter t can be computed directly by uing the method preented in [8]. Auming that the radiu of the earth i taken a a = 6370 km, the dielectric layer a l < r < a(region 1) i characterized by the relative permittivity ε r1 = 15, and conductivity σ 1 = 10 5 S/m, the earth in the region r < a l (Region ) i characterized by the relative permittivity ε r = 100, and conductivity σ = 4 S/m, t can be calculated eaily. Fig. and 3 how both the real part and the imaginary part of t 1 and t t 5, repectively. It hould be noted that q i alway imaginary and will increae with the thickne l from zero when the thickne l of the dielectric layer atify the condition 0 < k1 k 0 l < π. From Fig., it can be een that the real part of t 1 increae ignificantly when l>40 m and the imaginary part of t 1 approache zero when the thickne of the dielectric layer l i between 40 m and 50 m. It mean that the firt propagation mode will attenuate very lowly along the urface of the pherical ground when the thickne of the dielectric layer. Therefore, the firt term may be conidered a the trapped urface wave [8]. In thi paper, we will define the trapped urface wave generated by a vertical electric dipole a the trapped wave of electric type. Taking into account the conductivity σ 1 of the dielectric layer, the imaginary part of t 1 increae ignificantly when l>50 m.

110 Li, Park, and Zhang 50 40 30 Re t 1 0 10 0 0 0 40 60 80 100 10 l (m) (a) The real part of t 1 50 40 30 Re t 1 0 10 0 0 0 40 60 80 100 10 l (m) (b) The imaginary part of t 1 Figure. The value of t 1 veru the thickne of the dielectric layer l.

EM field in three-layered pherical region 111 In contrat with the effect on E r and H ϕ by the thickne l, we will compute E r and H ϕ at different thicknee l. Auming a = 6370 km, f = 100 khz, ε r1 = 15, ε r = 100, σ 1 = 10 5 S/m, σ = 4 S/m, and z r = z =0, E r in V/m and H ϕ in A/m due to unit vertical electric dipole at l =0m,l = 50 m, and l = 100 m are plotted in Fig. 4 and 5, repectively. From Fig. 5, it can be een that the trapped urface wave of electric magnetic can be excited efficiently when the thickne l of the dielectric layer i between 40 m and 50 m and atifie the condition 0 < k1 k 0 l<π. Compared with the computation preented in [8], it i concluded that the electromagnetic field i affected largely by the conductivity of the earth and the electromagnetic field attenuate rapidly for the dielectric-coated earth with high conductivity. In the cae of the perfect conducting earth, the attenuation of the electromagnetic field over the urface of the dielectric-coated pherical conducting earth will be larget. 3. THE FIELD IN THE AIR GENERATED BY A MAGNETIC DIPOLE OVER THE SURFACE OF THE DIELECTRIC-COATED SPHERICAL ELECTRICALLY EARTH If the excitation ource i replaced by a vertical magnetic dipole with it moment M = Ida, and da i the area of the loop, the nonzero component of the electromagnetic field in the air are E ϕ, H r, and H θ (TE mode), which the field i defined a the field of electric type. E ϕ (i), H r (i), and H (i) θ can be expreed a follow, E ϕ (i) = iωµ 0 r θ (V ir) (8) ( ) H r (i) = r + k (V i r) (9) H (i) θ = 1 r θ r (V ir). (30) where the potential function V i i the olution of in the calar Helmholtz equation ( + ki )V i =0, i =0, 1. (31) It i noted that the urface impedance of magnetic type i E (1) ϕ η 1 H (1) θ = g (3) r=a l

11 Li, Park, and Zhang 4.0 3.5 Re t 5 Re t - Re t 5 3.0.5.0 Re t 4 Re t 3 1.5 Re t 1.0 0 0 40 60 80 100 10 l (m) (a) The real part of t t 5 4.0 3.5 Re t 5 Re t - Re t 5 3.0.5.0 Re t 4 Re t 3 1.5 Re t 1.0 0 0 40 60 80 100 10 l (m) (b) The imaginary part of t t 5 Figure 3. The value of t t 5 veru the thickne of the dielectric layer l.

EM field in three-layered pherical region 113 1e-7 l = 0 m l = 50 m l = 100 m The abolute value 1e-8 1e-9 0 50 100 150 00 50 300 The propagating ditance (km) Figure 4. E r in V/m due to unit vertical electric dipole veru the propagating ditance ρ at l = 0 m, l = 50 m, and l = 100 m, repectively: f = 100 khz, a = 6370 km, ε r1 = 15, ε r = 100, σ 1 =10 5 S/m, σ = 4 S/m, and z r = z =0. 1e-9 l = 0 m l = 50 m l = 100 m The abolute value 1e-10 1e-11 1e-1 0 50 100 150 00 50 300 The propagating ditance (km) Figure 5. H ϕ in A/m due to unit vertical electric dipole veru the propagating ditance ρ at l = 0 m, l = 50 m, and l = 100 m, repectively: f = 100 khz, a = 6370 km, ε r1 = 15, ε r = 100, σ 1 =10 5 S/m, σ = 4 S/m, and z r = z =0.

114 Li, Park, and Zhang Follow the ame tep in Section, we will obtain the formula of the component E ϕ, H r, and H θ in the air generated by a vertical magnetic dipole over the dielectric earth. E ϕ η 0 H r η 0 H θ where G (z )G (z r ) t (q h ) = ωµ 0Ida e i(kaθ+ π 4 ) G (z )G (z r ) πx λa θ in θ t (q h ) i G (z ) G(z) k 0 z z=z r t (q h ) q h = i k 1 k 0 k 0 ( k0 a m = ) 1 3 e it x e it x e it x. (33) 1 m e i k1 k 0 l 1+ m e i. (34) k1 k 0 l k 1 k 0 g + k 1 k 1 k 0 g k 1. (35) In the pherical perfectly conducting cae g 0, then m = 1, the above formula reduce to ( ) 1 q h k0 a 3 k1 = k 0 ]. (36) k 0 tan [k 1 k0 l t are the root of the following differential equation W (t ) q h W (t ) = 0 (37) Here, W (t ) i the Airy function of the econd kind. The height-gain function G (z r )i G (z r )= W (t y) (38) W (t ) Auming that all the parameter of the dielectric layer a l< r<a(region 1) and the earth in the region r<a l (Region ) are ame with thoe in Section. Both the real part and the imaginary part of t 1 t 5 are calculated at f = 100 khz and plotted in Fig. 6, repectively. Fig. 6 how both the real part and the imaginary part of

EM field in three-layered pherical region 115 4 Re t 5 Re t 4 Re t 1 - Re t 5 3 Re t 3 Re t Re t 1 1 0 50 100 150 00 l (m) (a) The real part of t 1 t 5 8 7 Im t 5 6 Im t 4 Im t 1 - Im t 5 5 4 3 Im t 3 Im t Im t 1 1 0 50 100 150 00 l (m) (b) The imaginary part of t 1 t 5 Figure 6. The value of t 1 t 5 layer l. veru the thickne of the dielectric

116 Li, Park, and Zhang t 1 t 5. It hould be noted that q i alway negative and will increae from negative infinite to zero with the thickne l when the thickne l of the dielectric layer atify the condition 0 < k1 k 0 l< π. From Fig. 6, it i een that the urface wave cannot be excited by a vertical magnetic dipole when the thickne l atifie the condition 0 < k1 k 0 l<π. When the thickne l of the dielectric layer atifie the condition π k < 1 k 0 l <π, q i alway poitive and will increae from zero to infinite with the thickne l. When the thickne of the layer i larger than a certain value, the real part of t 1 increae ignificantly and the imaginary part of t 1 approache to zero. From the dicuion in Section, it i een that the urface wave can be excited by a vertical magnetic dipole when the thickne l of the dielectric layer i larger than a certain value and atifie the condition π k < 1 k 0 l<π. In thi paper, the trapped urface wave generated by a vertical electric dipole i defined a the trapped urface wave of magnetic type. In fact, the thickne l of the electric layer uch a ice or permafrot i le than 00 m, the above condition cannot be atified. In contrat with the effect on E ϕ and H r by the thickne l, we will compute E ϕ and H r at different thicknee l. Auming a = 6370 km, f = 100 khz, ε r1 = 15, ε r = 100, σ 1 = 10 5 S/m, σ = 4 S/m, and z r = z =0, E ϕ in V/m and H r in A/m due to unit vertical electric dipole at l =0m,l = 50 m, and l = 100 m are plotted in Fig. 7 and 8, repectively. 4. THE FIELD IN THE AIR GENERATED BY A HORIZONTAL ELECTRIC DIPOLE OVER THE SURFACE OF THE DIELECTRIC-COATED SPHERICAL ELECTRICALLY EARTH From the above expreion of the electromagnetic field of the vertical electric dipole and thoe of the vertical magnetic dipole in the air, uing reciprocity theorem, the accurate and complete formula of the vertical component Er he and Hr he for the electromagnetic field generated by a horizontal electric dipole upon the urface of the dielectric-coated pherical conducting earth in the air can be developed eaily. E he r = iηidhe ei(k 0aθ+ π 4 ) πx co ϕ λa θ in θ if (z r ) F(z) k 0 z z=z (t q )[1+ t ( k 0 a )/3 ] eit x (39)

EM field in three-layered pherical region 117 1e-10 1e-11 1e-1 l = 0 m l = 50 m l = 100 m The abolute value 1e-13 1e-14 1e-15 1e-16 1e-17 1e-18 0 50 100 150 00 50 300 The propagating ditance (km) Figure 7. E ϕ in V/m due to unit vertical electric dipole veru the propagating ditance ρ at l = 0 m, l = 50 m, and l = 100 m, repectively: f = 100 khz, a = 6370 km, ε r1 = 15, ε r = 100, σ 1 =10 5 S/m, σ = 4 S/m, and z r = z =0. 1e-1 1e-13 1e-14 l = 0 m l = 50 m l = 100 m The abolute value 1e-15 1e-16 1e-17 1e-18 1e-19 1e-0 1e-1 0 50 100 150 00 50 300 The propagating ditance (km) Figure 8. H r in A/m due to unit vertical electric dipole veru the propagating ditance ρ at l = 0 m, l = 50 m, and l = 100 m, repectively: f = 100 khz, a = 6370 km, ε r1 = 15, ε r = 100, σ 1 =10 5 S/m, σ = 4 S/m, and z r = z =0.

118Li, Park, and Zhang Hr he = iidhe G (z )G (z r ) λa t (q h ) e it x (40) In pherical coordinate, from Maxwell equation, it i eaily to expre the ret four component Eθ he, ϕ, Hθ he and Hϕ he in term of Er he and H he r ( r + k ( ) r + k ei(k 0aθ+ π 4 ) θ in θ πx in ϕ ) r + k (rhϕ he Er he ) = iωε 0 θ + 1 in θ ( ) r + k (reθ he ) = Er he r θ + iωµ 0 in θ ( ) (rhθ he ) = iωε 0 Er he in θ ϕ From (41), we get (re he ϕ ) = 1 in θ H he r ϕ H he r r ϕ + Hr he r θ E he r r ϕ iωµ 0 H he r θ (41) (4) (43) (44) ν(ν +1) (H he Er he ϕ )=iωε 0 r θ + 1 in θ H he r r ϕ (45) Taking into account the relation ν(ν +1)/r k a, the component H he ϕ i formulated a follow. Hϕ he = iidhe λa ei(k 0aθ+ π 4 ) θ in θ if (z r ) F(z) k 0 z t q 1 + k 0 a in θ πx co ϕ z=z e itx G (z) k 0 z z=zr G (z ) t (q h ) e it x (46) Similarly, the ret three component Eθ he, Hhe θ and Eϕ he are written a Hθ he = Idhe λa k 0 a in θ ei(k0aθ+ π 4 ) πx in ϕ θ in θ i F (z r ) F(z) k 0 z z=z (t q )[1 + t ( k 0 a )/3 ] eit x

EM field in three-layered pherical region 119 G (z) k 0 z z=zr G (z ) t (q h ) e it x (47) Eθ he = ηidhe λa i ei(k 0aθ+ π 4 ) θ in θ F (z) k 0 z 1 + k 0 a in θ Eϕ he = iηidhe λa ei(k 0aθ+ π 4 ) θ in θ i k 0 a in θ + πx co ϕ z=zr F(z) k 0 z t q G (z r ) G (z ) t (q h ) z=z πx in ϕ F (z) k 0 z G (z r ) G (z ) t (q h ) e itx ] e it x z=zr F(z) k 0 z z=z (t q )[1 + t ( k 0 a )/3 ] eit x ] e it x (48) (49) where upercript he deignate the horizontal electric dipole ource. From the above derivation, it can be een that the electromagnetic field generated by a horizontal electric dipole include the electromagnetic field of electric type and the field of magnetic type. Obviouly, for the electromagnetic field of a horizontal electric dipole, the trapped urface wave of electric type can be excited efficiently when the thickne l of the dielectric layer i omewhat and atifie the condition 0 < k1 k 0 l<π and the trapped urface wave of magnetic type can be excited when the thickne l of the dielectric layer i omewhat and atifie the condition π k < 1 k 0 l<π. 5. CONCLUSION In the above analyi, the imple explicit formula have been derived for the electromagnetic field over the urface of a dielectric-coated pherical electrically earth when a dipole ource (a vertical electric dipole, vertical magnetic dipole or horizontal electric dipole) i in the air. The computation how that both the trapped urface wave of electric type and that of magnetic type can be exited efficiently under the different condition. Alo, the electromagnetic field i affected largely by the conductivity of the earth.

10 Li, Park, and Zhang ACKNOWLEDGMENT Thi work wa upported by the National Reearch Laboratory (NRL) of Minitry of Science and Technology, in Korea under contract M1-003-00-0015. REFERENCES 1. King, R. W. P., The electromagnetic field of a horizontal electric dipole in the preence of a three-layered region, Journal of Applied Phy., Vol. 69, No. 1, 7987 7995, 1991.. King, R. W. P., The electromagnetic field of a horizontal electric dipole in the preence of a three-layered region: upplement, Journal of Applied Phy., Vol. 74, No. 8, 4845 4548, 1993. 3. King, R. W. P. and S. S. Sandler, The electromagnetic field of a vertical electric dipole in the preence of a three-layered region, Radio Sci., Vol. 9, No. 1, 97 113, 1994. 4. Wait, J. R., Comment on The electromagnetic field of a vertical electric dipole in the preence of a three-layered region by R. W. P. King and S. S. Sandler, Radio Sci., Vol. 33, No., 51 53, 1998. 5. King, R. W. P. and S. S. Sandler, Reply, Radio Sci., Vol. 33, No., 55 56, 1998. 6. Zhang, H. Q. and W. Y. Pan, Electromagnetic field of a vertical electric dipole on a perfect conductor coated with a dielectric layer, Radio Science, Vol. 37, No. 4, 10.109/000RS00348, 00. 7. Fock, V. A., Electromagnetic Diffraction and Propagation Problem, 191 1, Pergamon Pre, Ltd., Oxford, 1965. 8. Pan, W. Y. and H. Q. Zhang, Electromagnetic field of a vertical electric dipole on the pherical conductor covered with a dielectric layer, Radio Sci., Vol. 38, No. 3, 10.109/00RS00689, 003. 9. Li, K., S. O. Park, and H. Q. Zhang, Electromagnetic field generated by a vertical magnetic dipole upon the urface of a dielectric-coated pherical conducting earth, Journal of Electromagnetic Wave and Application, Vol. 1, No. 1, 1719 173, Dec. 00. 10. Gradhteyn, I. S. and I. M. Ryzhik, Table of Integral, Serie, and Product, Academic Pre, New York, 1980.

EM field in three-layered pherical region 11 Kai Li received the degree of B.Sc. degree in Phyic, M.Sc. degree in Radio Phyic and Ph.D. degree Atrophyic from Fuyang Normal Univerity, Anhui, China, in 1990, Xidian Univerity, Xi an, Shaanxi, China, in 1994 and Shaanxi Atronomical Obervatory, the Chinee Academy of Science, Shaanxi, China, in 1998, repectively. From Aug. 1990 to Dec. 000, he had been on the faculty of China Reearch Intitute Radiowave Propagation (CRIRP). From Jan. 001 to Dec. 00, he wa a potdoctoral fellow at Information and Communication Univerity, Taejon, South Korea. Since Jan. 003, he ha been a reearch fellow in the School of Electrical and Electronic Engineering, Nanyang Technology Univerity (NTU), Singapore. Hi current reearch interet include electromagnetic theory and radio wave propagation. Dr. Li i a enior member of Chinee Intitute of Electronic (CIE) and a member of Chinee Intitute of Space Science (CISS). Seong-Ook Park wa born in KyungPook, Korea, on December, 1964. He received the B.S. degree in electrical engineering in 1987 from KyungPook National Univerity, Korea. In 1989, he received the M.S. degree in electrical engineering from Korea Advanced Intitute of Science and Technology, Seoul. He i currently working toward the Ph.D. degree from Arizona State Univerity. From March 1989 to Augut 1993, he wa an reearch engineer with Korea Telecom, Korea, working with microwave ytem and network. Later, he joined the Telecommunication Reearch Center at Arizona State Univerity. Hi reearch interet include analytical and numerical technique in the area of microwave integrated circuit. Mr. Park i a member of Phi Kappa Phi Scholatic Honor Societie. Hong-Qi Zhang received the degree of B.Sc. degree in Radio Phyic and Ph.D. in degree Atrophyic from Lanzhou Univerity at Lanzhou, China, in 1986 and Shaanxi Atronomical Obervatory, the Chinee Academy of Science, Shaanxi, China, in 001, repectively. From 1986, he ha been on the faculty of China Reearch Intitute Radiowave Propagation (CRIRP). At 1995 he advanced to hi preent rank of enior engineer at CRIRP. Hi reearch interet include wave propagation, thundertorm location technology, and tranient electromagnetic field. Dr. Zhang won the Chinee Academy of Science award for hi excellent Ph.D. thei in 001.