ELECTROMAGNETIC FIELD IN THE AIR GENERATED BY A HORIZONTAL ELECTRIC DIPOLE LOCATED IN THE SPHERICAL ELECTRICALLY EARTH COATED WITH A DIELECTRIC LAYER

Transcription

1 J. of Electromagn. Wave and Appl., Vol. 1, No. 1, 199{11, ELECTROMAGNETIC FIELD IN THE AIR GENERATED BY A HORIZONTAL ELECTRIC DIPOLE LOCATED IN THE SPHERICAL ELECTRICALLY EARTH COATED WITH A DIELECTRIC LAYER K. Li and S.-O. Park School of Engineering Information and Communication Univerity 8- Hwaam-dong, Yuung-gu, Daejeon, -, South Korea Abtract In thi paper, the region of interet conit of the pherical dielectric earth, coated with a dielectric layer under the air. Firt, we will derive the formula for the electromagnetic eld in the earth of a vertical electric dipole located in the air and thoe of a vertical magnetic dipole, repectively. Then, the formula for the component of the eld in the air generated by a horizontal electric dipole are obtained by uing the reciprocity theorem. Finally, the computation will be carried out. 1 Introduction The Field in Region Generated by a Vertical Electric Dipole Located in Region The Field in Region Generated by a Magnetic Dipole Located in Region The Field in Region Generated by a Horizontal Electric Dipole Situated in Region Computation and Dicuion Concluion Acknowledgment Reference

2 1 Li and Park 1. INTRODUCTION The problem of the electromagnetic eld generated by a vertical or horizontal antenna ituated on or cloe to the urface of a dielectric-coated planar conducting or dielectric half-pace have been invetigated becaue of it wide ueful application [1{]. The problem ha been invetigated by King and Sandler [] and a comment wa given by Wait []. Recently, Zhang and Pan [9] re-analyze the problem and conclude that the electromagnetic eld include the direct wave, re ected wave, lateral wave and trapped urface wave. Thi development, naturally, rekindled the interet in the pherical earth problem. In thi paper, intereting and practically important three-layered pherical region conit of the air (region, 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 of the three region are k =! p ; q k 1 =! ( r1 + i¼ 1=!); q k =! ( r + i¼ =!) (1) where r i the relative permittivity, ¼ i the conductivity, and ue i made of the time dependence e i!t. Auming that the dipole i repreented by the current denity bxid he (x) (y) (z b), where b = a d, and d denote the depth of the dipole below the urface of the dielectric layer. The problem addreed in thi paper parallel the analyi carried out in the reference [1, 18]. Pan et al. [1] have derived the accurate and imple formula of the eld of a vertical electric dipole over the urface of a dielectric-coated pherical conducting earth. Lately, Li et al. [18] have derived the imple analytical formula for the eld of a dipole ource ituated in the air, including vertical electric dipole, vertical magnetic dipole and horizontal electric dipole, over the dielectric earth coated with a dielectric earth. The purpoe of thi paper i to outline the analytical formula for the eld in the air when the horizontal electric dipole ource i located in the pherical earth. Thee formula and computation can be ued to the uburface and cloe-to-the-urface communication at the lower frequencie.

3 Electromagnetic eld in the air 11. THE FIELD IN REGION GENERATED BY A VER- TICAL ELECTRIC DIPOLE LOCATED IN REGION In order to analyze the eld in the earth when the horizontal electric dipole ource i located in the pherical earth, it i neceary to analyze the eld in the earth of a vertical electric dipole located in the air and that of a vertical magnetic dipole. Firt, Auming that a vertical electric dipole i repreented by the current denity bzidl (x) (y) (z b), where b = z +a, and z > denote the height of the dipole above the urface of the dielectric-coated earth, the nonzero component E r, E µ, and H of the electromagnetic eld in the air have been derived in [18]. Taking into account that both the dipole and the point of obervation are cloe to the pherical interface between the air and the dielectric layer, the formula for the component E r, E µ and H are expreed a follow; E r (a + z r ; ; ) E µ (a + z r ; ; ) H (a + z r ; ; ) = iidl ² a ei(µ+ º ) p in where r = a + z r and x = ( k a )1=. The \height-gain function F (z r ) i p º x X F (z ) F (z r ) t q e itx X if (z (z) j z=z r k (t q ) 1 + t µ # = e itx ; 1 X F (z ) F (z r ) ² (t q ) 1 + t µ # = e itx F (z r ) = W (t y) ; () W (t ) ( + 1) 1= µ 1= y = a z º k z; () k a where t are the root of the following di erential equation () W (t) qw (t) = : ()

4 1 Li and Park Here q q = ik k1 k µ 1 k a 1 e e ip 1 k1 k l 1 + e e ip A ; () k1 k l q e = C B = k1 k k 1 g : () qk 1 k + k 1 g The urface impedance g at r = a E (1) µ ² 1H (1) l i de ned a = g : (8) r=a l At lower frequencie, the urface impedance g can be approximated a, g º k µ 1 k1 1 : (9) k k When the point of obervation lie on the urface of the ) coated pherical earth, r = a z r =, F () = 1, k@z zr = = p à p! ik k 1 k 1 ee i k 1 k p l, and the formula for the component k1 1+ ee i k 1 k l E, E µ, and H of the eld generated by a vertical magnetic dipole can be impli ed. They are E r E µ H zr = = iidl ² a e P 1 ² º e i(µ+ ) p p º x in X P (t q ) (t q ) F (z )e itx t q F (z )e itx 1 + t µ F (z )e itx 1 + t µ = # = # ; (1)

5 Electromagnetic eld in the air 1 where e = k qk1 k 1 e e ip 1 k1 k l 1 + e e ip A : (11) k1 k l If the point of obervation i in the dielectric layer (Region 1, a l < r < a), the electromagnetic wave due to a vertical electric dipole travel along the earth urface and travel down from the air to the point of the obervation in the dielectric layer. With the aid of (1) and the boundary condition at r = a, the formula for the component E r, E µ, and H in the dielectric layer are expreed a follow; E r (r; ; ) E µ (r; ; ) = iidl ² e i(kaµ+ º ) p p º x a in H (r; ; ) k k1 e P e ² X 1e X F (z )e itx t q F (z )e itx (t q ) (t q ) 1 + t µ F (z )e itx 1 + t µ = # = # ; (1) where 1 + e e ip k1 k [r (a l)] 1 + e e ip k1 k l 1e e = e ip k1 k (r a) 1 e e ip k1 k [r (a l)] e 1 e e ip k1 k l e : 1 + e e ip k1 k [r (a l)] 1 + e e ip k1 k l (1) When the point of obervation lie on the interface between the dielectric layer and the earth, r = a l, the formula for the component

6 1 Li and Park E r, E µ, and H can be impli ed a E r (r; ; ) E µ (r; ; ) = iidl ² e i(k aµ+ º ) p p º x a in H (r; ; ) r=a l k k1 e P e ² X 1e X F (z )e itx t q F (z )e itx (t q ) (t q ) 1 + t µ F (z )e itx 1 + t µ # = ; # = (1) where 1 + e 1 + e e ip k1 k l 1e e = e ip k1 k l k qk1 k 1 e k1 e 1 + e e ip : k1 k l 1 + e 1 + e e ip k1 k l (1) If the point of obervation (a l d r ; ; ) i in the earth (Region ), the electromagnetic wave due to a vertical electric dipole travel along the earth urface and travel down from the air to the point of the obervation in the earth. With the aid of (1) and the boundary condition at r = a l, the formula for the component E r, E µ and H of the eld are expreed a follow; E r (a l d r ; ; ) E µ (a l d r ; ; ) = iidl ² e i(µ+kdr + º ) p p º x a in H (a l d r ; ; )

7 Electromagnetic eld in the air 1 k k 1e X F (z )e itx t q e P F (z )e itx µ # = (t q ) 1 + t : e X F (z )e itx # ² = (t q ) 1 + t µ. THE FIELD IN REGION GENERATED BY A MAGNETIC DIPOLE LOCATED IN REGION (1) 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 eld in the air are E, H r, and H µ (T E mode), which have been addreed in [18]. where E H r H µ =! Ida a q q h = i k1 k k m = e i(µ+ º ) p p º x in P G (z )G (z r ) t (qh ) e it x 1 X G (z )G (z r ) ² t (qh ) e it x ; (1) i X G (z (z) j z=zr ² t (qh ) e it x µ k a 1 m e ip k1 k l 1 + m e ip A ; (18) k1 k l q k1 k g + k 1 : (19) qk1 k g k 1 1

8 1 Li and Park t are the root of the following di erential equation W (t ) q h W (t ) = : () 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) W (t ) : (1) It i noted that the urface impedance of magnetic type i E (1) ² 1H (1) = g : () µ r=a l When the point of obervation lie on the urface of the coated pherical earth, r = a z r =, G () = 1, (z r ) = zr p à p! = i k1 k 1 k m e i k 1 k l, and the formula for the component 1+ m e i p k 1 k l E, H r and H µ of the eld generated by a vertical magnetic dipole can be impli ed. They are E H r = Idl ² e i(µ+ º ) p p º x a in H µ zr = 1 ² m ² X X P (t q ) (t q ) G (z )e it x t q G (z )e it x 1 + t µ G (z )e it x 1 + t µ = # = # ; () where m = q k1 k 1 m e ip 1 k1 k l 1 + m e ip A : () k1 k l

9 Electromagnetic eld in the air 1 If the point of obervation i in the dielectric layer (Region 1, a l < r < a), the electromagnetic wave due to a vertical magnetic dipole travel along the earth urface and travel down from the air to the point of the obervation in the dielectric layer. With the aid of () and the boundary condition at r = a, the formula for the component E r, E µ, and H are expreed a follow; E (r; ; ) H r (r; ; ) = Idl ² e i(k aµ+ º ) p p º x a in H µ (r; ; ) where 1m m m m ² m ² G (z )e it x t q 1m P X G (z )e it x = (t q ) 1 + t ; () X G (z )e it x # = (t q ) 1 + t µ 1 + m e ip k1 k [r (a l)] 1 + m e ip k1 k l = e ip k1 k (r a) 1 + m e ip k1 k [r (a l)] 1 + m e ip k1 k l :() 1 m e ip k1 k [r (a l)] 1 m e ip k 1 k l m When the point of obervation lie on the interface between the dielectric layer and the earth, r = a l, the formula for the component E, H r and H µ can be impli ed a E (r; ; ) H r (r; ; ) = Idl ² e i(kaµ+ º ) p p º x a in H µ (r; ; ) r=a l

10 18 Li and Park 1m P G (z )e it x t q m X G (z )e it x µ # ² (t q ) 1 + t = ; m X G (z )e it x µ # ² (t q ) 1 + t = () where 1m m m = e ip k 1 k l 1 + m 1 + m e ip k 1 k l 1 + m 1 + m e ip k1 k l q k1 k 1 m k 1 + m e ip k1 k l : (8) If the point of obervation (a l d r ; ; ) i in the earth (Region ), the electromagnetic wave due to a vertical electric dipole travel along the earth urface and travel down from the air to the point of the obervation in the earth. The formula for the component E r, E µ, and H are expreed a follow; E (a l d r ; ; ) H r (a l d r ; ; ) H µ (a l d r ; ; ) = iidl ² a e i(µ+k dr + º ) p p º x in

11 Electromagnetic eld in the air 19 1m P G (z )e it x t q m X G (z )e it x µ # ² (t q ) 1 + t = : m X G (z )e it x µ # ² (t q ) 1 + t =. THE FIELD IN REGION GENERATED BY A HORIZONTAL ELECTRIC DIPOLE SITUATED IN REGION (9) From the above expreion of the eld in region of the vertical electric dipole and thoe of the vertical magnetic dipole, uing reciprocity theorem, the complete formula of the vertical component Er he and Hr he of the eld in region generated by a horizontal electric dipole ituated in the region can be developed eaily. E he r = H he r = H he = i² Id he a (t q ) X iid he a p in ei(k aµ+k d+ e F (z r ) 1 + t µ k a º ) p º x co X = # e itx ; () ei(kaµ+k d+ º ) p in p º x in 1m G (z r ) t (qh ) e itx ; (1) iid he a X ei(kaµ+k d+ º ) p in p º x co e F (z r ) t q e itx

12 11 Li and Park 1 + X (z) j z=zr k a in t (qh ) e it x ; () Hµ he = Idhe º d+ a ei(kaµ+k ) p p º x in in 1 X e F (z r ) k a in (t q ) 1 + t µ # = e itx k a X (z) j z=zr t (qh ) e it x ; () Eµ he = ² Idhe aµ+kd+ º ei(k ) p p º x co a in X (z) j z=zr t q e itx 1 + k a in X 1m G (z r ) t (qh ) e it x # ; () E he = i² Idhe a i k a in + X ei(kaµ+kd+ º ) p in p º x in X (t q ) 1m G (z r ) t (qh ) e it (z) x # 1 + t j z=zr µ # = e itx k a () where the upercript he deignate the horizontal electric dipole ource, the horizontal electric dipole i located at (a l d ; ; ), and z r i the height of the point of obervation.

13 Electromagnetic eld in the air 111 Figure 1. Geometry of a horizontal electric dipole located in the earth for three-layered pherical region.. COMPUTATIONS AND DISCUSSIONS In [18], both the excitation condition of the trapped urface wave of electric type and thoe of the trapped wave of magnetic type have been addreed peci cally. when the dipole ource i placed in the earth (region ), the excitation condition are ame with thoe addreed in [18]. For the eld of a horizontal electric dipole, the trapped urface wave of electric type can be excited e ciently when the thickne l of the dielectric qlayer i larger than the peci c value and ati e the condition < k1 k l < ¼. The trapped urface wave of magnetic type can be excited when the thickne l of the dielectric layer i larger than the peci c value and ati e the condition ¼ qk < 1 k l < º. We aume that the radiu of the earth i a = km, the dielectric medium in the region a l < r < a (Region 1) i characterized by the relative permittivity r1 = 1, and conductivity ¼ 1 = 1, the earth in the region r < a l (Region ) i characterized by the relative permittivity r1 = 1, and conductivity ¼ 1 =, f = 1 khz, z = and d r = : m. The computation will carried out for the component of the eld in region generated by a vertical electric dipole or a vertical magnetic dipole ituated in region. je r j in V/m and jh j in A/m due to unit vertical electric dipole at l = m and l = 11 m are plotted in Fig. and, repectively. Similarly, je j in V/m and jh r j

14 11 Li and Park Figure. je r j in V/m due to unit vertical electric dipole veru the propagating ditance at l = m and l = 11 m, repectively: f = 1 khz, a = km, r1 = 1, r = 1, ¼ 1 = 1 S/m, ¼ = S/m, z =, and d r = : m. Figure. With the ame parameter a hown in Fig., jh j in A/m due to unit vertical electric dipole veru the propagating ditance.

15 Electromagnetic eld in the air 11 Figure. With the ame parameter a hown in Fig. and, je j in V/m due to unit vertical magnetic dipole veru the propagating ditance. Figure. With the ame parameter a hown in Fig. {, jh r j in A/m due to unit vertical magnetic dipole veru the propagating ditance.

16 11 Li and Park Figure. jer he j in V/m due to unit horizontal electric dipole veru the propagating ditance at l = m and l = 11 m, repectively: f = 1 khz, a = km, r1 = 1, r = 1, ¼ 1 = 1 S/m, ¼ = S/m, z r =, and d = : m. Figure. Geometry of a horizontal electric dipole located in the earth for three-layered pherical region.

17 Electromagnetic eld in the air 11 in A/m due to unit vertical magnetic dipole at l = m and l = 11 m are plotted in Fig. and, repectively. Next, we will compute the component of the eld in region generated by a horizontal electric dipole ituated in region. If we aume that the parameter of the earth and the dielectric layer are ame with the above computation in Fig. {, the dipole i ituated in region with the depth d = : m, and the point of obervation i on the urface between the air and the dielectric layer, z r =, jer hej in V/m and jhr he j in A/m due to unit horizontal electric dipole at l = m and l = 11 m are plotted in Fig. and, repectively.. CONCLUSION In the above analyi, the imple explicit formula have been derived for the electromagnetic eld in the earth when the vertical electric dipole and the vertical magnetic eld located in the air. Baing on thee reult, the formula of the eld in the air generated by a horizontal electric dipole located in the earth have been developed readily. Finally, the computation have been carried out for the eld in the earth generated by a vertical electric and magnetic dipole ituated in the air and the eld in the air generated by a horizontal electric dipole ituated in the air. Thee formula and computation can be ued to the uburface and cloe-to-the-urface communication at the lower frequencie. ACKNOWLEDGMENT Thi work i upported by the National Reearch Laboratory (NRL) Program of Minitry of Science and Technology, in Korea under contract M REFERENCES 1. King, R. W. P., M. Owen, and T. T. Wu, Lateral Electromagnetic Wave, Springer-Verlay, King, R. W. P., \The electromagnetic eld of a horizontal electric dipole in the preence of a three-layered region, Journal of Applied Phy., Vol. 9, No. 1, 98{99, King, R. W. P., \The electromagnetic eld of a horizontal electric dipole in the preence of a three-layered region: upplement, Journal of Applied Phy., Vol., No. 8, 8{8, 199.

18 11 Li and Park. King, R. W. P. and S. S. Sandler, \The electromagnetic eld of a vertical electric dipole in the preence of a three-layered region, Radio Sci., Vol. 9, No. 1, 9{11, Wait, J. R., \Comment on `The electromagnetic eld of a vertical electric dipole in the preence of a three-layered region by Ronold, W. P. King and S. S. Sandler, Radio Sci., Vol., No., 1{, King, R. W. P. and S. S. Sandler, \Reply, Radio Sci., Vol., No., {, Margeti, D. and T. T. Wu, \Exactly calculable eld component of electric dipole in planar boundary, J. Math. Phy., Vol., 1{, Margeti, D., \Radiation of horizontal electric dipole on large dielectric phere, J. Math. Phy., Vol., 1{1,. 9. Zhang, H. Q. and W. Y. Pan, \Electromagnetic eld of a vertical electric dipole on a perfect conductor coated with a dielectric layer, Accepted by Radio Science, Waton, G. N., \The di raction of electric wave by the earth, Proc. R. Soc. London, Vol. A9, 8{99, Bremmer, H., Terretrial Radion Wave, Elevier Publihing Co., New York, Bremmer, H., \Application of operational calculu to groundwave propagation, particularly for long wave, IRE Tran. Antenna Porpag., Vol. AP-, {, Fock, V. A., Electromagnetic Di raction and Propagation Problem, 191{1, Pergamon Pre, Ltd., Oxford, Hill, D. A. and J. R. Wait, \Ground wave attenuation function for a pherical earth with arbitrary urface impedance, Radio Sci., Vol. 1, No., {, Houdzoumi, V. A., \Two mode of wave propagation manifeted in vertical electric dipole radiation over a phere, Radio Sci., Vol., No. 1, 19{9,. 1. Houdzoumi, V. A., \Vertical electric dipole radiation over a phere: character of the wave that propagate through the phere, J. Appl. Phy., Vol. 8, 99{9, Pan, W. Y. et al., \Electromagnetic eld of a vertical electric dipole on the urface of a dielectric layer overlay a global conductor, Submitted to Radio Science, April. 18. Li, K., S. O. Park, et al., \Electromagnetic eld in the preence of a three-layered pherical region, Submitted to Radio Sci., Aug.

19 Electromagnetic eld in the air Gradhteyn, I. S. and I. M. Ryzhik, Table of Integral, Serie, and Product, Academic Pre, New York, 198. Kai Li received the degree of B.Sc. degree in Phyic, M.Sc. degree in Radio Phyic and Ph.D. in degree Atrophyic from Fuyang Normal Univerity, Anhui, China, in 199, Xidian Univerity, Xi an, Shaanxi, China, in 199 and Shaanxi Atronomical Obervatory, the Chinee Academy of Science, Shaanxi, China, in 1998, repectively. From 199, he had been on the faculty of China Reearch Intitute Radiowave Propagation (CRIRP). Since 1, he ha been a potdoctoral fellow at Information and Communication Univerity, South Korea. Hi current reearch interet include electromagnetic theory, radio wave propagation in variou media, and radio phyic. 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, in December, 19. He received the B.S. degree from KyungPook National Univerity, KyungPook, Korea, in 198, the M.S. degree from Korea Advanced Intitute of Science and Technology, Seoul, Korea, in 1989, and the Ph.D. degree from Arizona State Univerity, Tempe, AZ, in 199, all in electrical engineering. From March 1989 to Augut 199, he wa an Reearch Engineer with Korea Telecom, Taejon, Korea, working with microwave ytem and network. He later joined the Telecommunication Reearch Center, Arizona State Univerity, until hi departure in September 199. Since October 199, he ha been with the Information and Communication Univerity, Daejeon, Korea, a an Aitant Profeor. Hi reearch interet include analytical and numerical technique in the area of microwave integrated circuit and MMIC. Dr. Park i a member of Phi Kappa Phi Scholatic Honor Societie.