Based on Duality

The general theory of annular slot antennas is developed in a Z 2 lucid manner. The treatment is novel in that no Green's functions or contour integrations arise. The principle of duality is employed. Topics discussed include formulation of the integral equation for the aperture field and derivations of the leading term in the nearand far-zone fields and effective height. The theory has application in calibrating near-zone / -Y field strength instrumentation and in determining the response of an unsheathed coaxial cable connector mounted flush with the skin of a /// / missile to intense plane-wave electromagnetic field pulses. INTRODUCTINx INTRODUCTION T HE OBJECTIVE of this paper is to obtain in a straightforward manner expressions for H., Ep, Ez, Eo, and he for a flush-mounted annular slot antenna when the assumption is made that the aperture field is proportional to I/p. This aspect of the paper is largely tutorial but provides indispensable background information for later papers treating RF coupling through apertures. An integral equation is derived for Ep(p) in the aperture. The near, intermediate, and far-zone fields, the apparent admittance, and the complex effective height may then be determined to any degree of accuracy desired once the integral equation is solved for the aperture field. MAGNETIC FIELD IN RADIATION HALF-SPACE Consider a coaxial line of inner conductor radius a and sheath radius b truncated in an infinite ground plane as depicted by Fig. 1. The baffle at the open end of the coaxial line lies in the xy plane at z = 0 and the center of the inner conductor is coincident with the z axis. The tangential electric field in the annulus is Ep(p',O). It appears that the simplest way to obtain an expression for the magnetic field valid everywhere in the radiation half-space z > 0 is to employ duality. One replaces the electric field in the slot lying in the xy plane at z 0 by an equivalent but fictitious magnetic current Jm = -z x Eap = -Ep, To find the field in the semi-infinite medium the image theorem is used (Fig. 1). The retarded electric potential dual of the magnetic vector potential is given by M_£ Ab rd 2n d0'2J exp (-jkoIr r'l) 47r Ja dp d Jm r r'I The absolute dielectric constant is s and ko = 27r/2o is the free-space wave number. Evidently, the suppressed time dependence is exp (jcot). Manuscript received July 20, 1970. C. W. Harrison, Jr., is with Sandia Laboratories, Albuquerque, N. Mex. 87115. D. C. Chang is with the Department of Electrical Engineering, University of Colorado, Boulder, Colo. 80302. 2Jm Fig. 1. Replacement of electric field Ep(p) by equivalent magnetic current Jm, using image theorem. Fig. 2. Demonstration that M has only 0 component proportional to cos ( o') and is independent of 4. The most general form of Jr r'l, the distance between two points in a cylindrical coordinate system, is -r r'l = [p2 + p'2 2pp' cos (4 4') + (z _ z')2]12. (2) Let the primed coordinates (p',O',z') represent some point in the annulus where z' = 0. Also, since 4 symmetry in the fields obtains regardless of the position (p,4),z) of the observer, one is at liberty to set 4 = 0 without loss in generality. Expression (2) then becomes Jr r'l = R = (p2 + p'2 2pp' cos ¢' + z2)12. (3) It is clear from Fig. 2 that because of symmetry M is independent of 0 and has only a 0 component proportional to cos (4-4') or simply cos 4'. Hence M = ~ {b dp 2f ) cos4) exp (-jk0R) Ma =a, 8 A Ep(p',O)p' dp' jz dOcos (