ANALYSIS
OF
RADOME COVERED
CIRCULAR REFLECTORS BY COMPLEX
SOURCE-DUAL SERIES APPROACH
Tailer O&zer*, Ayhan AltiiitaS
Dqmrtnient of Electrical antl Electionics Eiig., Bilkent ITniversity
065.13 Bilkent, ANKARA, TURKEY Alexander.I.Nosich
Ciiiiently Visiting Prof. at Department of EE-CS, Kuiiiainoto Univeisity Kuiiiainoto 8G0, JAPAN
1.
Introduction
Radiation from a two dimeiisional reflector antenna covered by a rylindrical ratlome is analyzed by complex source-dual series approarli. It is only performed for the electrically polarized incideiit field. Tlie approach is not based on the moment niethotl biit on the analytical-iiumerical type regularization terhniqrie. Tlie methotl gives the exact solution wit11 any desired accuracy and the directivity of the feed antenna rail be modelled by using the complex source method [l]. I n niinierical resiilts. tlie far field radiation patterns are obtained and tlie effect of tlie radome is vrrifietl.
2.
Formulation
Tlie circular reflector and the radoiiie geometry is shown i i i Figure 1. Tlie reflector is modelled by a part of a circular, zero-thickness, perfertly coil-
ducting material of t l i v radius a and angiilar witltli 28,,,. A coniplex line source is located at the focal point (r,=a/2,
40=O).
It simiilates the physical directivity of a feed a n t m n a by the directivity factor Iil, ( k is waveniimller). The problem is two dirnensional iii geometry and the radome is a co-axial dielectric cylinder of inner radius r, oiiter radiiis li,tliickness tl=li-c antl the relative perniitivity is t v .Tlie requirements for the rigoroiis solutioii of the present I,oiiiitlary value problem can be stated as the satisfartion of the Helmlilotz wave equation, Suiniiierfeld radiation condition. edge conditions antl tlie boiintlary condi- tiuiis for the dielectric and rontliicting I,ountlnries.
The total electric field expansions in tlie foiir regions rail he written as
where
k
= k o m , k, is free space wavenumber,
4
= x-
'p and theincident field expausion coefficient is given as :6 = J , , ( ~ o r , ) H ~ l ) ( k ~ ~ ) e - ~ ~ ' 8 ~
wliere 0, and r, is defined i u [3].
Buuiidary conditions are imposed a t the dirlectric boundaries at c and r+d, and further a t the metal and slot part of the circular interface r=a. T l ~ e o , the following dual series equations are obtained in terms of the inner legion expausion coefficient.
where x,, = A,,J,,(k,n)
+
b f , g,, = $,,a,, -m,,b,, and R,, = -P,,C,,+
ff,,cL.
Further, e,,,/3,L,n,,,6,,,c,, and d,, are defined as follow.
Tlie dual series equations are tlieu ronvrrted to a certain canonical form and regularized by the Riemann Hilbert Problem techniqiie. Tlie static part of the scattering operator is invertrd analytically and the remaining part is further inverted numerically by an accurate algorithm. This partial inver- sion of the scattering operator finally gives a matrix equation. Resulting matrix equation is Fredliolm 2nd liind, so the ronvergence and accuracy is
guarenteed and it is possible to ol)taiii a solution wit11 any desired accuracy
PI.
3.
Numerical Results
Tlie effect of the radome on tlir racliatioii pattern of a circular reflector
aiitrtitia is verified by the present method. T h e far field radiatiou pat- ten1 is obtained for the thin-lossless dielectric radome material. It c m be
serii from Figure 2 that the transmission coefficient froin a dielectric slab reniains almost unity for tlie Iialfwavelengtli of the ratlome thickness for
sinall incident aogli3.s. Figure 3 shows that tlie radiation pattrrn changes
in furin due to radomv reflections but for tlie selection of ratloiiie thickness
as a half of the wavelcngtli in tlir dielectric medium, thrn the tlistortioii in the radiation pattern is iiiiniiiiized. The free space radiatioii pattern of a
ciri-ular reflector aiitriiiia was also coiiiparetl 1)y tiit. Iiish frrqueiiry solution of .lull and Suedan i n [3].
4.
Conclusions
Coinplex Source-Dual Serics approach is applied to a radome covered cir- c u h r reflector antenna wliicli is rxcitetl by a feed h a s a directive radiation pattern. The present accurate results can be tliougl~t as a reliahle data for
the validity of aproxiniate solutions. Fiirtlier, lossy rase and the multi layer rarloiiie problems can be solved by tlir same nwtliotl.
Acknowledgement
This research was spoiisorrd by NATO’s Scientific Affairs Divisiou i n tlie fraiiiework of the science for stahility programme.
References
[l] L.B. Felseii ” Cornplex sourcr point solutions of the field equations ant1 their relation to the propagation antl scattering of Gaussian beams,” Symp. Math., vo1.18, PI> 39-56, 1975
[‘L]
A.I. Nosicli ” G ~ P C I I ’ S Function-Dual Series ApproacIi iii wave scattering by combined resonant scatterers,” in hl.Hasliiinoto, M.Idenien ant1 O.A.Tretyakov (eds.), Analytical and Niiiiirrical Methods in
EM
Wave Theory, Tokyo, Science Hoiise, 199“.[3] T. O$uzer, A. Altinta.~ antl A.I. Nosicli ”Analysis of circular reflectors by complex source-dual srries approa~.Ii”, in IEEE AI’-S International Syinp.,
proc, vo1.2, pp. 9’22-925, 1993.
Figure 1: a) Circular reflector and radome geometry
b)
Power transmission coefficient of a dielectric slab(c,=4)0.0 -20.0 -40.0 -60.0 -80.0 - Fr.. I p O O . d=0.22 Iwnbda d=O 25 Iwnbda I
I
-0.0 50.0 100.0 150.0 PHI (DEC)Figure 2: Radiation pattern of a circular reflector antenna in the presence of a co-axial cylindrical radome(ka=62.8, kb=2.6, kc=100, t,=4)
