Internatonal Journal of cence and Engneerng Invetgaton vol. 6, ue 6, January 17 IN: 51-8843 Calculatng Poton of eed Antenna and Dmenon of ureflector n the Caegran Antenna Ung the Becton Numercal Method Al-Reza harf aculty of Engneerng, Unverty of Zanjan, Zanjan, Iran (arharf@znu.ac.r) Atract- In th aer Caegran antenna and t geometrc tructure revewed and a relaton for calculatng dmenon of t man and ureflector to mnmze the lockage nvetgated. Th relaton whch a comlcated equaton wth reect to everal varale then mlfed n the degn examle and exreed veru only one varale. The mlfed equaton then olved numercally ung the method of ecton wth rad convergence and hgh recon. The degn examle clarfe the method more clearly. Keyword- Caegran antenna, Becton numercal method, root fndng method, otmzaton, dual reflector antenna zero wth reect to the feld n the aex of reflector. Anyway th cae ha the lowet de loe level n the order of -4 db, and may have advantage n ome alcaton. In lottng th fgure the feld dtruton of the aerture lane condered a a quadratc functon wth reect to r; the radu of aerture lane; and alo a functon of the taerng factor, a the followng equaton: 1 1 1 1 r r r (1) I. INTRODUCTION Dual reflector antenna are wdeand and have encl eam and hgh gan, o they are very common n ace communcaton and mllmeter wave rado telecoe [1]. g. 1 how a rototycal Caegran telecoe whch located n Wet Vrgna[1]. Thee antenna alo have alcaton n the tructure of ntrument whch roduce lane wave a mllmeter wave comact antenna tet range (CATR) devce []. Conderng the large dmenon of thee antenna ;wth man reflector dameter uually n the range of 1 meter and more, ther analy carry out ung the hycal otc (PO) method, and other analytcal method uch a the method of fnte element and the method of moment would e very tme conumng and requre very large comutaton volume. Degn equaton of th tructure are manly aed on the mathematcal equaton of araolc and hyerolc urface. In reflector antenna uually the feld trength n the aex of reflector not the ame a t value at the edge of reflector, and th called taerng the fled trength. Taerng ha drect mact on the eam wdth of the man loe and de loe level. g. dect the calculated ower attern of a rme reflector antenna n logarthmc cale wth reect to u d / n for varou taerng value. the ace angle wth reect to the axe of araolc reflector[1]. Narrowet eam wdth correond to the cae of unform feld dtruton and have larget de loe level value of aout (-17dB), whle the roadet eam wdth related to the cae whch the amount of feld at the edge of reflector taered to gure 1. chematc of the natonal rado atronomy oervatory (NRAO) telecoe, located n Wet Vrgna, UA. The reflector are made from alumnum gure. Varaton of man loe eam wdth and de loe level of rme reflector araolc antenna wth reect to dfferent taerng factor. 194
Other than the aove effect, taerng wll have drect nfluence on aerture effcency and receved noe level y antenna from unwanted ace drecton. Reflector antenna gan calculated from the followng, 4 G A () A Where A the hycal aerture area, the free ace wavelength and A the aerture effcency whch calculated aroxmately from: (3) A In (3), called llumnaton effcency of aerture and relate drectly to the feld taerng and calculated y: r, da r, da urface ntegral taken over the area of aerture lane and r, the feld dtruton on the aerture urface. A een from g., taerng the feld dtruton reduce the gan and ndcate that reduce y ntroducng more taer. In (3), llover effcency and how the rato of feed energy that reache to the reflector urface to total energy radated y the feed. o y ncreang the edge taer value ncreae too and le energy would e dated n unwanted drecton. and act n revere manner y changng the edge taer value ut the roduct taer of aout -1 db. (4) eak to.8 for the edge In (3), the cro olarzaton effcency and deend on the unwanted comonent of feld erendcular to the man feld drecton. Increang thee comonent would reult n reducton of. Cro olarzaton reducton one of the challenge every one encounter whle degnng reflector antenna ytem. ecally n trackng alcaton, the reence of unwanted erendcular feld comonent would ncreae nothng ut extra receved noe. Th factor deend on everal arameter lke the rato of focal length to the antenna dameter, feld dtruton of feed antenna, roughne of the reflector urface and etc. Dual reflector antenna lke Caegran antenna have le cro olarzaton comonent nce they have larger effectve rato of focal length to man reflector dameter and have larger. In (3), the lockage effcency and due to lockng the wave y otacle laced n front of the man reflector, lke the feed antenna or ureflector, etc. or accetale n reflector antenna, dameter of ureflector or feed antenna laced n front of the man reflector mut e le than.1 dameter of the man reflector antenna. To otan otmum n Caegran antenna, ureflector dameter may e equal to the hadow made y feed antenna on the reflector urface. In th aer after ntroducng the mathematcal relaton that govern the concal urface and Caegran antenna, method of ecton n olvng nonlnear equaton revewed n ecton and 3 reectvely. Degn examle of th knd of antenna reented n ecton 4 to llutrate the rolem more clearly. II. CONICAL ECTION Paraola, hyerola and elle are created when a lane croe a cone n dfferent drecton. Generally eakng, a concal urface or curve the locu of all ont that ther dce from a fxed ont called focal ont e e tme ther dce from a fx lne, called drectrx. Coeffcent e called eccentrcty. In elle e le than 1, whle n araola t equal 1 and n hyerola greater than 1. elle and hyerola have two focal ont whle araola have one. g. 3 dect a concal curve. In th fgure, concal ont reented y. the dce of to drectrx lne aumed equal to c, whle the dce of curve aex to drectrx lne taken equal to a, a ndcated n the fgure. Paraolc or hyerolc equaton n olar coordnate ytem ung the aove defnton exloted here. Aume that the orgn ont e located on and the axe of the curve e laced on x axe. Conderng g. 3, we have, X exz r e c r co r 1 e co ec o, r ec 1 e co gure 3. A hyerolc or araolc concal ecton whch t focal ont,, located on the Orgn, O, and t axe laced on x-axe, drectrx lne arallel to the y-axe. The concal ecton a araola f e=1 and a hyerola f e>1. (5) (6) Internatonal Journal of cence and Engneerng Invetgaton, Volume 6, Iue 6, January 17 195 www.ijei.com IN: 51-8843 Paer ID: 6617-8
r In a araola, e=1, c=a. o the aove equaton ecome, a 1 co whle n the hyerola, (6) turn nto, 1 ae e e c a / a c a e 1 r (8) 1 e co By rotatng araola or hyerola along t axe y 36 degree, araolc or hyerolc urface create, whch have wde alcaton n degnng Caegran antenna. III. GEOMETRIC TRUCTURE O CAEGRAIN ANTENNA In g. 4, geometrc tructure of Caegran antenna ndcated. Th tructure created from a araolc man reflector wth dameter D P and focal length P, a hyerolc ureflector wth dameter D and focal length and a feed antenna. Phae center of the feed antenna reented y the left cro gn n the fgure and located on one of the focal ont of the hyerolc urface. another focal ont of the ureflector llutrated y the rght cro gn n the fgure and located on the focal ont of the araolc man reflector. The angle of edge ray ht from feed antenna to the ureflector ndcated y, and that of man reflector tated y. eed antenna aumed here a concal corrugated tye, whch ha le cro olarzaton feld comonent wth reect to ordnary crcular horn antenna. It ha almot the ame E and H attern. In the fgure, a arameter whch reent the amount of dlacng the hae center of the feed wth reect to the feed aerture. It deend on the lant radu of horn and wavelength, etc. wave leavng the feed antenna, ht the ureflector a lane wave. (7) mlarly, D 4 e (1) The rato of effectve focal length e to called antenna magnfcaton, M. relaton etween M and eccentrcty, e,, M 1 e M 1 eccentrcty alo may e otaned veru and a, 1 n e 1 n (13) (14) In degnng antenna, the man reflector dameter uually calculated from communcaton lnk udget and mechancal conderaton. The rato of / D uually choen etween.5 and.5 and M elected larger than 1 to reduce cro olarzaton feld comonent. Knowng M, the / D raton wll e known too. rom (11) and (1), and gotten. o the arameter, D, and n g. 4 are all known. e rom g. 4 and equ. (7), and can e otaned veru / D and / D reectvely. In the man reflector, e dce etween focal ont and the edge of reflector otaned from (7) a, r 1 co co on the other hand, D r n (1) from (9) and (1), D 4 (9) (11) gure 4. Geometrc tructure of Caegran antenna ytem. Now a roer feed antenna mut e choen wth le cro olarzaton feld comonent and wth alty to llumnate the edge of reflector wth fled taer equal to aout -1 db wth reect to reflector center. Conderng g. 4, t clear that Internatonal Journal of cence and Engneerng Invetgaton, Volume 6, Iue 6, January 17 196 www.ijei.com IN: 51-8843 Paer ID: 6617-8
for fxed and, whatever the feed e laced farther than ureflector, the ureflector dameter D mut e larger to e ale to cover the eam lmt, and that wll lead to greater lockage agant the man reflector and reduce. On the other de f the feed antenna e located very cloe to the ureflector, D, can e made maller, ut the hadow of the antenna feed created on the man reflector would e large enough to ncreae the lockage effect agan, o n otmum cae, the feed oton mut e determned n uch a way that ureflector dameter and the hadow of feed antenna agant the man reflector ecome equal. To do th, we need ome extra nformaton aout feed antenna, lke feed antenna aerture dameter and the hae center oton of radated feld. Phycal hae of a tycal feed antenna hown n g. 5. In Caegran antenna, the feed may e located n a rotecton cavty agant humdty and ran effect, o t effectve dameter may e larger than antenna aerture dameter. In g. 5, 7, the um of feed aerture dameter lu rotectve cavty thckne named a the feed dameter, A. Corrugaton n the nternal urface of the horn generate hyrd mode whch have lower cro olarzaton comonent wth reect to domnant TE11 mode n ordnary concal horn. Dmenon of the tructure deend on the antenna gan, the requred edge taerng off the feld, degn frequency and mechancal conderaton and aed on g. 6 [4]. In th fgure arameter defned a: gure 6. radated ower of the corrugated horn antenna wth reect to radaton angle and for varou arameter. a (15) R where a the radu of feed aerture, the free ace wavelength and R the radu of curvature of the horn. In degnng the feed, arameter may e choen artrarly, knowng the requred amount of edge taerng of a the eam and, from g.6, the amount of n wll e known too, and o the feed aerture radu, a, wll e otaned. gure 7. Geometrc decrton of the lockage effect made y feed hadow on the man reflector. gure 5. Phycal tructure of a rototycal concal corrugate horn antenna gure 8. Geometrc arameter of hyerolc ecton. Internatonal Journal of cence and Engneerng Invetgaton, Volume 6, Iue 6, January 17 197 www.ijei.com IN: 51-8843 Paer ID: 6617-8
Knowng arameter and feed aerture radu, lant radu of the horn, R, determned from (15) and the rato of hae center dlacement of the feed, to R determned from [4]. By otanng A and, ung the theory of hycal otc, a relaton can e otaned to equalze the ureflector dameter wth the feed hadow on the man reflector. In g. 8, hyerolc concal ecton dected whch cont of two curve and two focal ont. ocal ont dce c, and the dce etween aex of curve a. The left hyerolc curve whch hown y dahed lne n th fgure doe not ext hycally n g. 4. The eccentrcty coeffcent, e, n g.8 otaned a: c a e (16) a To determne the geometrc hae of hyerolc, two arameter, a and c mut e known. To otan a relaton etween ureflector dameter, and angle, we have from (8): 1 D, ae e (17) 1 e co D o, D n (18) ae e 1 n (19) 1 e co To determne dameter of the hadow of the feed on the man reflector, B, one may ue (7) together wth g.7. angle calculated a: ung (7): A B / n 1 co o, B 4 n 4 1 co () (1) () y equatng () and (19), ae e 1 1 e co n A a 4 n (3), c a 1 e f a, o (3) may e wrtten a ae e 1 1 e co A a a 1 e n 4 (3) (4) A exlaned earler, n (4), all arameter excet for a known. equaton. (4) a nonlnear equaton wth reect to a and cannot e olved mly. Among dfferent numercal method for olvng th equaton, one mut ue a mle method wth low calculaton volume and rad rate of convergence. The method of Becton ued here for th uroe. The Becton method one of the racketng method n whch we eek the root of functon n a ecal nterval. The algorthm a follow [5]. At frt y ketchng the functon n a ecal nterval we eek for any ont that functon croe the horzontal axe. Then a utale nterval choen [j,k], where j and k are ont at the left and rght of the aroxmate root reectvely and f(j)f(k)<. The dce etween j and k then dvde nto two ecton wth the mddle ecton called c. the amount of functon n th ont, f(c), then calculated. c wll e cloer than j and k to the root ont. Now f f(c) atfe the requred recon for the rolem under hand, we to at th ont, otherwe we chooe the new nterval [c, k] f f(c)f(k)<, or [j,c] f f(c)f(k)< to eek for the root oton agan. Th roce reeated untl the atfed recon otaned or the numer of teraton exceed than maxmum requred. In contnue we exlan all the aove mentoned y a numercal examle. Internatonal Journal of cence and Engneerng Invetgaton, Volume 6, Iue 6, January 17 198 www.ijei.com IN: 51-8843 Paer ID: 6617-8
f(a) IV. NUMERICAL EXAMPLE IN DEIGNING CAEGRAIN ANTENNA YTEM Aume that the man reflector dameter, D P, e 1 m conderng communcaton lnk udget and mechancal lmtaton. The center frequency of oeraton for a X-and radar taken a 1 GHz. We chooe /. 5, o the D focal dce of araolc urface would e 5 (m). electng antenna magnfcaton, M, equal to 4: / 4 m e e from (13): 4 1 5 e 4 1 3 from (11), (1), 53.1 and 14.. By takng =., for concal corrugated horn feed, and aumng -1 db edge taer for llumnaton of reflector, oervng g. 6, a n 3.5, 8 9 c / f 3 1 / 1 1.3 m feed aerture dameter then : cm a 3.5.3 / ( n 14. ) 6.8 the thckne of held urroundng the feed taken equal to cm o, A ( a ) 17.6 cm from (15), R otaned. A mentoned earler, R ued n relaton for determnaton of the hae center dlacement of the feed,. We have: 4 a 6.8 1 R 38.3. from [4], R.14 4.7 cm cm Now all arameter n (4) are ecfed excet for a. To olve th equaton y the method of Becton, y f a lotted y [6], n the nterval [,5] n g. 9. The nterval choen accordng the hycal tructure of the rolem. nce we aume that feed antenna laced n the ace etween the man reflector and ureflector, conderng the orgn ont located at the focal ont of the man reflector, referrng to g. 8 the amount of a would e greater than zero and le than 5 (m) n th rolem. To nvetgate the olty of extng zero of y f a, eyond th dce, we may lot th functon n [, 5]. A een from g. 9, the root laced n [.1,.4] nterval, where f(.1)f(.4)<. The algorthm for root fndng y the method of Becton mlemented n MATLAB oftware [6]. Whle runnng the rogram, t By aumed that the recon e etter than.1 or the teraton numer doe not exceed tme. After 7 teraton the aroxmate root, a=.386, otaned. te of th roce are ummarzed n Tale 1. After determnaton of a, other hycal arameter of the ureflector lke D, wll e otaned. rom (19), D.31 5 / 3 5 / 3 1 n 53.1 1.1 1 5 / 3 co 53.1 1-1 m -.5 1 1.5.5 3 3.5 4 4.5 5 a (m) gure 9. Varaton of (4), n the degn examle veru varale a whch half the dce etween hyerola aexe, hown n g. 8. TABLE I. teraton 1 3 4 5 6 7 and nce c a 1 e ITERATION O THE IMPLEMENTED ALGORITHM IN OLVING (4) OR THE DEIGN EXAMPLE. f c.4737.1138.1585.18.489.157.11, we have: m.31 1 (5 / 3) 1.65 ) c.5.35.875.363.3156.319.386 o the hae center of feed antenna mut e 3.35 (m) aart from the man reflector aex. The te of degnng antenna dmenon n th examle, now over. V. CONCLUION In th aer Caegran antenna ytem ecfcaton are revewed. Ung the hycal otc method and amng mathematcal relaton of araolc and hyerolc urface, we get a nonlnear and comlex functon wth reect to Internatonal Journal of cence and Engneerng Invetgaton, Volume 6, Iue 6, January 17 199 www.ijei.com IN: 51-8843 Paer ID: 6617-8
antenna arameter. Th equaton then aroxmately olved ung the method of Becton. A degn examle for radar alcaton n X-and reented whch exlan the method clearly. REERENCE [1] Baar, J. W. M., The Paraolodal Reflector Antenna n Rado Atronomy and Communcaton, Atrohy. ace c. Lr., rnger, 7. [] Parn, C., Theory and Practce of Modern Antenna Range Meaurement, IET Electromagnetc Wave, 14. [3] Vareg, D., Purcell, E.J., Rgdon,.E., Calculu wth dfferental equaton, Prentce Hall. Inc., 9th ed., 7. [4] B. M. Thoma, "Degn of corrugated concal horn", IEEE Tranacton on Antenna and Proagaton, vol. AP-6, no., March 1978,. 367 37. [5] Chara,., Aled Numercal Method Wth MATLAB for Engneer & centt, McGraw-Hll, 3rd Edton, 11. [6] MATLAB 7.4..347 (R9a) oftware, The MathWork, Inc., eruary, 1, 9. Internatonal Journal of cence and Engneerng Invetgaton, Volume 6, Iue 6, January 17 www.ijei.com IN: 51-8843 Paer ID: 6617-8