Propagation of Electromagnetic Field From a Pulsed Electric Dipole in a Dielectric Medium

Transcription

1 CHINESE JOURNAL OF PHYSICS VOL. 39, NO. 2 APRIL 2001 Propagation of Electromagnetic Field From a Pulsed Electric Dipole in a Dielectric Medium Osama M. Abo-Seida 1 and Samira T. Bishay 2 1 Department of Mathematics, Faculty of Education, Kafr El-Sheikh Branch, Tanta Uniersity, Kafr El-Sheikh, Egypt 2 Department of Mathematics, Faculty of Science, Ain Shams Uniersity, Cairo, Egypt (Receied July 31, 2000) The transient electromagnetic fields of a ertical electric dipole antenna with an impulsie current in a dielectric medium are expressed in an analytical form. It is found that the first part of the electromagnetic fields of the excited electromagnetic pulse is an impulsie wae, which propagates with the speed 1= p ¹" and decays exponentially. The other part builds up gradually and propagates slowly, and attenuates with a much lower rate than exponential decay. PACS Jb Electromagnetic wae propagation; radiowae propagation. PACS Hr Electromagnetic waes. I. Introduction Modification of propagated electromagnetic waes and their distortion greatly affects radio naigation, identification of targets by means of radar and generally other telecommunication systems. The sinusoidal electromagnetic wae undergoes a strong attenuation of an exponential decay nature when it propagates in a dielectric medium. Howeer, transient electromagnetic fields excited by a pulsed antenna may attenuate rapidly and may propagate oer a moderate distance from the antenna because it contains a wide band of low frequency components. The propagation of a transient electromagnetic wae in a dielectric medium has been studied by seeral authors [1-5]. In considering our problem, the exact solutions for the transient electromagnetic fields excited by a pulsed electric dipole in a dielectric medium are deried. The transient field responses due to a step-function current antenna are then calculated under the assumption that the displacement currents are insignificant for reasonable alues of time after the initial antenna pulse. Consequently, by conolution, the propagation of electromagnetic pulses excited by an antenna with currents of arbitrary waeforms can be ealuated. II. Description of the configuration and method of solution The geometry of the problem is illustrated in Fig. 1. To specify the position of the source in the configuration, we employ the coordinates (r; µ; Á) as a spherical coordinate system with origin O and three mutually perpendicular base ectors (~e r ;~e µ ;~e Á ) of unit length each. The source is a ertical electric dipole antenna immersed in a dielectric medium of infinite extent and excited by 177 c 2001 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 178 PROPAGATION OF ELECTROMAGNETIC FIELD FROM A VOL. 39 FIG. 1. Pulsed Electric Dipole and Coordinate Systems. an impulsie current. The electric permittiity is", the magnetic permeability¹ is taken equal to that of the free space¹ 0 eerywhere, and the conductiity¾ is assumed to be zero. The electric dipole antenna, located at the origin O and oriented ertically in the z-direction, is small in length,d`, carying a current If(t). I is the amplitude andf(t) is the temporal function of the antenna current. Without loss of generality, we can restrict attention to situations where f(t) = 0 for t < 0. The corresponding frequency domain representation for the current is then F(s) = Z 1 0 f(t)exp( st)dt =Lff(t)g (1) wheres =i! in terms of the angular frequency!, where the operator L denotes the Laplace transform. In the frequency domain, Maxwell s fundamental electromagnetic equations are written as curl ~ E(~r;s) = s¹ ~ H(~r;s); (2) curl~h(~r;s) =s"~e(~r;s) +~J(~r;s); (3) where ~E and ~H are the electric and magnetic field intensity ectors, respectiely and ~J is the impressed current density ector of the dipole antenna which can be expressed as ~J(~r;s) =Id`±(~r)F(s)~e z ; (4) ~e z is the unit ector in the z-direction. Each component of the electric and magnetic fields is related to a Hertz ector ~Q. In the frequency domain, these well-known relations are gien by [2] ~E(~r;s) = 2 ~ Y (~r;s) + grad di (~r;s); (5) ~H(~r;s) =s" curl (~r;s); (6)

3 VOL. 39 OSAMA M. ABO-SEIDA AND SAMIRA T. BISHAY 179 where 2 =¹"s 2 : (7) The ector electromagnetic wae equation is (r 2 2 ) J (~r;s) = ~ (~r;s): (8) s" A solution of this equation is gien by Stratton [6] Z ¹ (~r;s) = ~J(~r 0 ;s) e R 4¼s" R dv0 (9) V where ~Q (~r;s) is ealuated at the obseration point (r;µ;á) and ~ J(~r 0 ;s) at the source point (r 0 ;µ 0 ;Á 0 ). The distancer is equal tor =j~r ~r 0 j, and the integration takes place oer all the ~r 0 space. (The medium is infinite and homogenous). As the dipole antenna is situated at the origin of the spherical coordinate system (r;µ;á), as indicated in Fig. 1; and oriented in the z-direction, on applying of equation (9) it is found that the Hertz ector has only a z-component which is gien by (~r;s) = Y z ~e z = Id`F(s) 4¼s" e (s)r r ~e z : (10) The operations (5) and (6) are applied to obtain the field components [2]: E µ (~r;s) = Id`F(s) 4¼"sr 3 f1+ (s)r + 2 (s)r 2 ge (s)r sinµ; (11) E r (~r;s) = Id`F(s) 2¼"sr 3 f1+ (s)rge (s)r cosµ; (12) H Á (~r;s) = Id`F(s) 4¼r 2 f1 + (s)rge (s)r sinµ: (13) Now choosingf(s) = 1, that is to say takingf(t) =±(t), then equation (10) becomes, (~r;s) = Id` 4¼s" e (s)r r ~e z = Y z (~r;s)~e z : (14) Using the tables of Laplace transfonns [7, 8], we hae L 1 e asª =±(t a); (15) wherel 1 denotes the inerse Laplace transform. This leads to Y (~r;t) = Id` ³t 4¼rs" ± r ; (16) z

4 180 PROPAGATION OF ELECTROMAGNETIC FIELD FROM A VOL. 39 where± t r 1 is the Dirac-± function and= p. Applying Maxwell s equations in the time ¹" domain, we can find the magnetic fields H Á (~r;t) = Id`sinµ 4¼r 2 h ³ ± t r + r ±0³ t r i ; (17) where± 0 (x) = dx d ±(x). To obtain the electric field, let s define G 1 (r;s) [1 + (s)r]e (s)r ; (18) and then compareg 1 (r;s) withh Á (~r;s). In iew of equations (13), (17) and (18), we find the inerse Laplace transform ofg 1 (r;s) to be ³ L 1 fg 1 (r;s)g =g 1 (r;t) =± t r + r ±0³ t r : (19) Then the time domain electric field can be obtained if we rewritee r (~r;s) as E r (~r;s) = Id`cosµ 2¼r 3 G 2 (r;s)g 1 (r;s); (20) whereg 2 (r;s) = s" 1. Applying the conolution theorem of the Laplace transform [7], L 1 fg 2 (r;s)g 1 (r;s)g = to equation (20) leads to E r (~r;t) = Z t 0 g 2 (r;t)g 1 (r;t )d =g 1 (r;t) g 2 (r;t); (21) Id` cosµ h r ³t 2¼"r 3 ± r ³ +u t r i ; (22) whereu(t r ) is the Heaiside step function. Similarly,E µ(~r;s) can be rewritten as E µ (~r;s) =s Y z (r;s)sinµ + Id` sinµ 4¼r 3 G 1 (r;s)g 2 (r;s): (23) Then, we hae E µ (~r;t) = Id` sinµ 4¼"r 3 r ³t ± r + r2 2±0³ t r ³ +u t r : (24) Equations (17), (22) and (24) gies us the exact time domain solutions for an electromagnetic pulse excited by a dipole antenna with an impulsie current in a dielectric medium. These forms are considered to be original and new. These findings may lead to some useful applications in an enironment such as the atmosphere.

5 VOL. 39 OSAMA M. ABO-SEIDA AND SAMIRA T. BISHAY 181 III. Conclusions The exact time domain solutions for an electromagnetic field excited by an electric dipole antenna with an impulsie current in a dielectric medium hae been deried. It is found that electromagnetic fields of the excited electromagnetic pulse can be diided into two parts. The first part is an impulse wae, which propagates with the speed = 1= p ¹" and decays exponentially. The other part builds up gradually and propagates slowly, and attenuates with a much lower rate than the exponential decay. These findings may lead tosome useful applications in anenironment such as the atmosphere. References [ 1 ] J. R. Wait, J. Appl. Physics., 24, 340 (1953). [ 2 ] J. R. Wait, in Antenna Theory, (Electromagnetic fields of sources in lossy media) Eds. R. E. Collin and F. J. Zucker, (Mc Graw-Hill, 1969), part 2, Ch. 24. [ 3 ] J. R. Wait, IEEE Trans. Antennas Propagation, AP-18, 714 (1970). [ 4 ] S. T. Bishay, Canad. J. Phys. 65, 376 (1987). [ 5 ] S. T. Bishay, Indian Journa1 of Radio and Space Physics, 20, 22 (1991). [ 6 ] J. A. Stratton, Electromagnetic Theory, (Mc Graw-Hill Book Company, New York, 1941). [ 7 ] G. E. Roberts and H. Kaufman, Table of Laplace Transforms, (W. B. Sounders Company, 1966) p [ 8 ] G. Doetsch, Guide to the Applications of the Laplace and Z-transforms (Van Nostrand, London, 1971), p. 37.