Viskum, H. H.

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H.Published in:Antennas and Propagation Society International Symposium,Publication date:1986Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Viskum, H. H. (1986). Synthesis and analysis of single shaped reflector antennas. In Antennas and PropagationSociety International Symposium, (pp. 651-654). IEEE.SINGLE SHAPED REFLECTOR ANTENNAS SYNTHESIS AND ANALYSIS OF H.-H. Viskurn Electromagnetia  Institute Technical University of Denmark DK-2800  Lyngby. Denmark Introduction.  Althou h eometrical optics (GO) is an asymptotic theory describ- years constituted  the  foundation  for synthesis of reflector antennas. While  dual ing electromagnetic #e& of infinitely  high frequencies only, it has for several reflectors are usually shaped to  produce a certain aperture  distribution  leading to a  pencil beam type  pattern  with  high  gain  or  low side  lobes,  etc., single reflectors field. In either of  the cases, any design must necessarily be ver!fied by a  diffrac- may be syntheslsed to  generate shaped  beams by using the GO directly in  the  far tion analysis,  because of  the GO assumption made In the synthesis. In this paper we  will demonstrate  a  numerical  method  for solving the synthesis problem for single reflector antennas. The technique has been found to be superior to  what has reviously been applied, in particular because it returns  the solution on a  form  wlich lends itself easily t o  diffraction analysis.  This is accom- plished by solvin Westcott's synthesis problem usin a collocation method and an expansion of t l e  reflector surface in bicubicspline?unctions. The theory will be resumed very briefly, and  we wiU then present some  results figuration,  a  reflector shaped to provide  full  earth coverage from  a  low  orbitting from  applying  the  developed synthesis and analysis programs to  a  particular con- satellite. Formulation of  the synthesis. As stated previously, the synthesis part of this study is based upon the work of Westcott (1983). To formulate the problem we of the  coordinate system.  The half  angle  subtended by the  reflector  at  the  feed is consider the  reflector  shown  in  fig. 1, illuminated by a  feed located at  the centre €Ic. Given the feed power densit pattern I(e,+), we require the reflector t o  G(B. a) inside the  far-field cone of  half angle Of. Westcott showed that by using transform this pattern  into  a  far-fie&  pattern  with  a  power density distribution an  appropriate  parametrization of  the reflectoer surface one can derive a highly non-linear  boundary value problem, the  solution of which overns the  reflector surface. He also showed how  the non-linear problem couqd be linearized and solved by  repeated  solution of linear second order  partial  differential  equations (POE'S). that  the  solution can be  expanded  in  normalised bicubic B-spline functions  [de Numerical solution of  the synthesis problem. To solve the linear PDE's we assume we  now select just as many points, the socalled collocation  points.  Requiring  the Boor (1979)). The expansion involves a  number of sofar unknown  coefficientsand series expansion to  satisfy the PDE at  the  collocation  points leads t o  a set of linear equations  which  upon  solution yields the desired coefficients of  the expansion. obtain  a  continuous expression for  the  reflector surface. For a  detailed  description of this procedure, see Viskum (1985). As a  result  we Analvsis of shaped reflectors. Since single, shaped reflectors usually radiates a using GTD rather  than PO. Thus we have ap lied  the GTD and  found  that  the non-focused beam, the analysis will in general  be faster and  more accurate when continuous expression for  the surface 0btained)from  the synthesis wasa  great  ad- CH2325-9/86/0000-0651$01.00 Q 1986 IEEEE 651 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 01,2010 at 08:34:44 UTC from IEEE Xplore.  Restrictions apply. vantage, both  for  the ray tracing, which is usually the  bottleneck in GTD compu- tations, as for  the necessary calculation of the  differential geometry of  the sur- face. Details of this analysis may also be  found in the previously referenced work by  Viskum. S nthesis and  anal  sisof  a low  orbit sha ed  beam  reflector. The  synthesis and a:alysis techniquls outlined above w:re applied to  a particular example, a situation is shown in fi 2 for an orbit o?app. 800 km above the  ground,  resulting reflector t o  provide full earth covera e from a low orbitting satellite. The in  a coverage angle ot62.4 degrees.  The ideal  far-field  pattern is shown in fig.3 and  the  feed  pattern is of  the  form ~(e,a) =  COS"(^) 00 e e e 100 where  n is chosen to  provide  an edge taper of 10 dB. Note  that  while  the desired far-field  pattern is raised towards the edge of coverage, the feed pattern is ta- pered  at  the  edge of the reflector. Fig. 4 shows the  contour of a  rotationally symmetric design. The reflected rays are disturbed  due to blocking from  the feed. Therefore an offset design was  made, a5 also indicated, and it is seen that the radiation pattern may be significantly shown in fig. 5. Because of the very large coverage area, the  offset  angle  had to  tour  plot  of  the  reflector is shown  in  fig. 6. be  greater  than 73 degrees.  To emphasize the asymmetry of  this design, a con- Finally, one cut of  the radiation  pattern is shown in fig. 7. It follows  the  ideal pattern shape rather well, although it fails t o  reproduce  the  required peak at  the design oal  to extend  be ond 64 2 degrees and to  have a  smooth  transition  in- edge. This is caused by edge  diffraction  and may be remedied by modifying  the stead o?the sharp cutoff sKown in'fig. 3. Conclusion. The  GO-synthesis problem for single reflectors has been solved applyin t i e  method of collocation numericall by expanding the reflector surface in bicubic spline functions and analyse! using the GTD, to  which end the representation of  the surface was . The solution so obtained was diffraction found most convenient. The possibility of designing non-symmetric reflectors has been  demonstrated by the  application of  the synthesis and.analysis programmes to a  particular  reflector  configuration. References de Boor, C., "A Practical Guide t o  Splines", Springer Verlag, N.Y., 1979. yiskum, H.-H., Ph.D.  Thesis, Electromagnetics Institute, Techn. Univ. o f  Denmark, LD 54,1985. pestcott, B.S., Shaped ReflectorAntenna Design", Research Studies Press  Ltd.. UK,. 1983. Synthesisand Analysisof Single Shaped Reflector Antennas", 652 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 01,2010 at 08:34:44 UTC from IEEE Xplore.  Restrictions apply. A x  Fig.  1. Coordinate system for the reflector synthesis.  The feed i s  a t  the centre of the coordinate system and the reflector subtends an angle of 2e, . The far-f ie ld  cone angle i s  ef . 3 Fig. 2. Geometry for  a low orbi t  Fig. 3.  Ideal  pattern  shape for  the s a t e l l i t e .  For a height h = 800 km, configuration in f i g .  2. The absolu- ef i s  app. 62.4'. t e  level i s  arbitrary. 653 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 01,2010 at 08:34:44 UTC from IEEE Xplore.  Restrictions apply. Flg.  4. Reflected rays from a Fig. 5. Offset version of the reflec- c i rcular ly  symetr ic  ref le tor .  t o r  in f ig .  4 .  Offset angle = 73'. Fig. 6. Contour plot of the reflector in fig.  5. Ideal  shape Fig. 7. Radiation pattern for the 73O-offset reflector i n  the plane orthogonal t o  the plane of offset .  - copolar, -.--- cross polar. 654 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 01,2010 at 08:34:44 UTC from IEEE Xplore.  Restrictions apply. 

Synthesis and analysis of single shaped reflector antennas

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Synthesis and analysis of single shaped reflector antennas

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