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1 This dissertation has been microfilmed exactly as received CARVER, Keith R oss, A CAVITY-FED CONCENTRIC RING PHASED ARRAY OF HELICES FOR USE IN RADIO ASTRONOMY. The Ohio State U niversity, Ph.D., 1967 Engineering, electrica l University Microfilms, Inc.. Ann Arbor, M ichigan

2 C o p y rig h t by Keith Ross Carver 1968

3 A CAVITY-FED CONCENTRIC RING PHASED ARRAY OF HELICES FOR USE IN RADIO ASTRONOMY DISSERTATION P resented in P a rtia l F u lfillm en t of the Requirements fo r th e Degree Doctor o f Philosophy in th e Graduate School o f The Ohio S ta te U n iv ersity by Keith Ross C arver, B.S.E.E., M.S. The Ohio S ta te U niversity 1967 Approved by A dviser Department o f E le c tr ic a l Engineering

4 ACKNOWLEDGMENTS During the course of my graduate stu d ie s a t Ohio S ta te U n iv e rsity, i t has been my good fo rtu n e to have P ro fesso r John D. Kraus as both academic advisor and research su p erv iso r at the Radio O bservatory. I wish to express my sin c e re ap p reciatio n to Dr. Kraus fo r h is p ersonal in t e r e s t in my research work, p a r tic u la r ly in th e e a rly phases when the experim ental r e s u lts were not encouraging. In ad d itio n to h is prov id in g the idea which served as th e nucleus of th is d is s e r ta tio n. Dr. Kraus has c o n trib u te d much to th is work through p riv a te d iscu ssio n s. His e n th u s ia s tic encouragement o f my work on the h elico n e antenna and s tu d ie s o f the p o la riz a tio n p ro p e rtie s o f the h e lic a l beam antenna was o f g re a t h e lp. A lso, h is deep in s ig h t in to the physics o f antennas and ra d ia tin g systems has been of in estim ab le v alue in designing experim ental programs and th e o r e tic a l stu d ie s o f antennas fo r use in radio astronomy. I a lso wish to thank P. N. Myers, C hief Programmer of the Ohio S ta te U niversity Radio O bservatory fo r h is t i r e l e s s e f f o r ts in w ritin g sev eral computer programs which were used in num erical stu d ie s o f the h e lic a l beam antenna and of c o n cen tric rin g a rra y s. Without h is h e lp, the num erical data summarized in Figs , and F igs would not have been a v a ila b le. i i

5 A lso, the help o f th e follow ing s t a f f members of th e Ohio S ta te U niversity Radio Observatory is g ra te fu lly acknowledged: J. G. Cox, L. T. F itc h, G. C. M ikesell, E. T eiga, R. E, Townsend, and W. T russ, W, L, Stutzman provided v aluable d iscu ssio n s on phased array th eo ry. L ast, but c e r ta in ly not le a s t, I wish to thank my w ife, P a t, fo r h er help in connection with th e d is s e r ta tio n re sea rc h. Her constant encouragement and i n te r e s t has made i t p o ssib le to view the work in p e rsp ec tiv e and to ev alu ate i t s p ro g ress. I l l

6 VITA May 18, Born - Beech Creek, Kentucky B.S.E.E., U niversity o f Kentucky Lexington, Kentucky Research A ss is ta n t, Department of E le c tr ic a l E ngineering, The Ohio S ta te U n iv ersity, Columbus, Ohio M.S., The Ohio S ta te U n iv e rsity, Columbus, Ohio PUBLICATIONS "Wave V e lo c itie s on the G rid -S tru ctu re Backward A ngle-fire A ntenna," I.E.E.E. T ransactions on Antennas and Propagation, Vol. AP-12, pp , Ju ly, communication. (coauthored w ith Dr. J. D. Kraus) "The H elicone - A C irc u la rly P olarized Antenna with Low Sidelobe L evel," Proceedings o f I.E.E.E., v o l. 55, p. 559, A p ril, communication. FIELDS OF STUDY Major F ie ld : E le c tr ic a l Engineering S tudies in Antennas. H. W alter P rofessors John D. Kraus and C arlton S tudies in Radio Astronomy. Hsien Ching Ko P rofessors John D. Kraus and S tudies in Communication Theory. P ro fesso r Claude F.. Warren IV

7 TABLE OF CONTENTS Page ACKNOWLEDGMENTS... i i V IT A... iv LIST OF TABLES. ix LIST OF ILLUSTRATIONS x Chapter I, INTRODUCTION... '... 1 I I. THE ARRAY ELEMENT... 7 A D iscussion o f C irc u la rly P o larized Array Elem ents.. 7 The co n ical lo g -s p ira l antenna The p la n a r s p i r a l The crossed d i p o l e The a x ia l mode h e lic a l beam a n t e n n a The h e lic o n e The Svennerus m o d ificatio n of th e h e lix The Helicone - A C irc u la rly P o larized Antenna w ith Low Sidelobe L e v e l D escription of experim ental r e s u lts C o n stru ctio n Measured power p a tte rn s Measured impedance c h a r a c te r is tic s Axial r a tio c h a r a c te r is tic s o f the h elico n e.. 38

8 D iscussion Comparison of helicone with conical horn The h elico n e as an array elem ent The H elix - A Second Look Phase c e n te r of the h e l i x P o la riz a tio n c h a r a c te r is tic s of th e h e lix and i t s relevance to a phased a r r a y D erivation of an e x p lic it r e la tio n fo r the a x ia l r a tio o f an e l l i p t i c a l l y p o la riz e d electro m ag n etic f i e l d A pplication to an a x ia l mode h e lic a l beam antenna M odifications of the h e lix to improve phase c e n te r and p o la riz a tio n s t a b i l i t y I I I. CONCENTRIC RING ARRAY DESIGN CONSIDERATIONS Array Theory fo r C oncentric Ring Arrays o f Is o tro p ic R adiators Uniform amplitude and phase..... ' Uniform am plitude and nonuniform phase D iscussion of C oncentric Ring Arrays C oncentric Ring Arrays o f H elical Beam Antennas IV. THE ARRAY FEED SYSTEM: A RADIAL TRANSMISSION LINE.. 97 The Theory of Radial Transm ission L ines A R adial Transm ission Line Feed System D iscussion of the b a sic p ro p e rtie s of th e feed s y s te m " L im itations on th e maximum guide rad iu s I l l Probe-loaded ra d ia l g u id es v i

9 V. EXPERIMENTAL MODELS OF CAVITY-FED PHASE-SCANNING ARRAYS OF HELICAL BEAM ANTENNAS D iscussion o Measurement Techniques A 24-H elix Ring Array Fed by a C irc u la r R adial C a v i t y D e tails of geometry and c o n stru ctio n Far-zone power p a tte r n s, measured and c a l c u l a t e d An 8-H elix Ring Array Fed by a C irc u la r Radial Cavity D e tails of elem ent c o n stru ctio n Far-zone power p a tte r n s, measured and c a lc u la te d A 14-H elix C oncentric Ring Array Fed by a Quadrant C avity D e tails of c a v ity co n stru ctio n and array geometry Far-zone power p a tte r n s, measured and c a lc u la te d Input impedance of the quadrant array Radio observ atio n of th e sun w ith th e 14 -h elix array VI. SONE DESIGN NOTES ON A LARGE PHASED ARRAY FOR RADIO ASTRONOMY The Feed System Amplitude and Phase Adjustments o f the Elements,. 177 Automated C ontrol o f Phase S e ttin g s, M echanical and E le c tr ic a l Array S teerin g The Array Element The Array G e o m e tr y via

10 VII. CONCLUSIONS APPENDIX A: A DERIVATION OF THE FIELDS OF A HELIX APPENDIX B: ARRAY FACTORS FOR CONCENTRIC RING ARRAYS APPENDIX C: DERIVATION OF THE FIELDS OF A RADIAL TRANSMISSION LIN E APPENDIX D: TWO UNSUCCESSFUL EXPERIMENTAL MODELS BIBLIOGRAPHY V l l l

11 VII. CONCLUSIONS APPENDIX A: A DERIVATION OF THE FIELDS OF A HELIX APPENDIX B; ARRAY FACTORS FOR CONCENTRIC RING ARRAYS APPENDIX C: DERIVATION OF THE FIELDS OF A RADIAL TRANSMISSION LIN E APPENDIX D: TWO UNSUCCESSFUL EXPERIMENTAL MODELS BIBLIOGRAPHY V l l l

12 LIST OF TABLES Table Page 1. Dimensions o f a Typical H e l i c o n e Dimensions o f Two H elicone M o d e ls Roots o f J (x ) = o on 4. Roots o f = Maxima and Minima o f 0^( r/a ) C alculated Probe Depths fo r 14-Element Array Geometry o f Thinned A r r a y IX

13 LIST OF ILLUSTRATIONS Figure Page 1. I l l u s t r a t i o n of a ty p ic a l c a v ity -fe d co n cen tric rin g array o f h e lic a l beam antennas as proposed in memorandum to au th o r from Dr. J. D. Kraus [1964] (a) c ro s s-s e c tio n view; (b) plan view I l l u s t r a t i n g a lin e a r phased array o f 2N+1 is o tr o p ic elem ents I l l u s t r a t i n g th e co o rd in ate systems f o r a c ir c u la r ly p o la riz e d wave, (a) Coordinate system fo r antenna in i t s o rig in a l p o s itio n. (b) Coordinate tran sfo rm atio n through th e angle A The h e l i c o n e Comparison of se v e ra l re la te d antenna types (a) Conical horn fed by h e lix -e x c ite d c ir c u la r waveguide, (b) Conical horn fed by c ir c u la r ly p o la riz e d TE mode, (c) Axial mode h e lix with co n ica l co u n terp o ise, (d) H elicone Dimensions o f the h elico n e Support arrangement fo r h e lix in sid e c o n e O rie n ta tio n of h elico n e in sp h e ric a l coordinate s y s t e m Logarithm ic p o la r power p a tte rn s fo r a ty p ic a l h elico n e (a) * = 0 plan e, (b) <p = 90 p lan e Logarithm ic p o la r power p a tte rn s fo r a ten tu rn 14 p itc h angle a x ia l mode h e lix in the * = 0 p lan e Logarithm ic p o la r power p a tte rn s fo r a ten tu rn 14 p itc h angle a x ia l mode h e lix in th e * = 90 plane I l l u s t r a t i n g th e p a tte rn bandwidth o f a ty p ic a l h elico n e I l l u s t r a t i n g th e beamwidth of a ty p ic a l h elico n e as a fu n ctio n of frequency and as compared to a simple h e lix X

14 14, HPBW and sid elo b e le v el versus cone included angle fo r the h elico n e , HPBW and sid elo b e le v el as a function o f cone a ltitu d e fo r th e h e l i c o n e , Cutaway drawing of coaxial in p u t to h elico n e, , Input impedance of a h elico n e as a function of cone a n g l e...,,, 40 18, Input impedance of a h elico n e as a function of frequency , Comparison o f far-zone power p a tte rn s o f a conical horn with power p a tte rn of a h elico n e with s im ila r dimensions , I l l u s t r a t i n g the far-zone p o in t in the sp h e ric a l coordinate s y s t e m , The p o la riz a tio n e llip s e at the far-zone p o in t P ( R,8, , The a x ia l r a tio as a fu n ctio n o f r fo r sev eral values o f , A h e lix w ith i t s m idpoint at th e o rig in of th e coordinate system , The angular dependence of the a x ia l r a tio on the mode am plitude co n stan ts I^ and I, The dashed curve is from the experim ental work o f Kouyoumjian and Chu; perm ission to use th is d ata was k in d ly granted by Dr. R. C. Kouyoumjian , The angular dependence of the a x ia l r a tio on the number o f tu r n s , The angular dependence of th e a x ia l r a tio on the p itc h angle , (a) A h e lix with ground plan e, (b) A h e lix w ith tru n c a ted conical co u n terp o ise, (c) A h e lix with ta p ered ends and w ith coaxial lin e fla re d in to conical counterpoise , (a) A co n cen tric rin g array o f is o tro p ic ra d ia to r s. (b) Array geometry fo r mn elem ent in c o n ce n tric rin g a r r a y x i

15 29. HPBW vs, scan angle fo r a 24 elem ent sin g le rin g a rra y o f is o tro p ic ra d ia to rs I l l u s t r a t i n g the use of d ire c tiv e elem ents to reduce sid elo b es and g ra tin g lobes A c ir c u la r ra d ia l c av ity operated in the TM mode (a) plan view, (b) sid e v i e w A ra d ia l tran sm issio n lin e term inated in i t s c h a r a c te r is tic impedance and feeding a h e lic a l antenna element Amplitude and phase o f outgoing tr a v e lin g wave in dominant mode matched ra d ia l waveguide Main components o f heavy duty antenna t e s t p latfo rm and asso c iate d d riv e components Main components o f antenna t e s t r a n g e Block diagram o f antenna p a tte rn range Cross se c tio n view o f ra d ia l c a v ity showing e l e c t r i c f i e l d, h e lix probe p o s itio n s, and p rin c ip a l d im e n s io n s (a) The c av ity c e n tra l feed arrangem ent. (b) A probe feed fo r a ty p ic a l h e lix A comparison o f measured and c a lc u la te d lin e a r power p a tte rn s f o r a 2 4 -h elix rin g array phased fo r a broadside b e a m Ten turn h e lix in sid e sm all co n ical co u n terp o ise mounted on upper c av ity p la te Photograph o f the 8 -h e lix array mounted on antenna te s t p l a t f o r m Measured and c a lc u la te d far-zone power p a tte rn s fo r an 8 -h elix rin g array phased fo r a broadside b e a m Measured and c a lc u la te d far-zone power p a tte rn s fo r an 8 -h e lix rin g array phased fo r a 5 t i l t a n g l e Xll

16 44. Measured and c alc u la te d far-zo n e power p a tte rn s f o r an 8 -h e lix rin g array phased f o r a 10 t i l t a n g l e Measured and c a lc u la te d far-zone power p a tte rn s fo r an 8 -h e lix rin g array phased fo r a 15 t i l t angle (a) A 56-elem ent quadrant c a v ity -fe d co n cen tric rin g a rra y, (b) One 14-elem ent quadrant a rra y. C avity w alls and feed p o in t are s h o w n Cutaway view o f quadrant c a v ity showing a contour map o f the e l e c t r i c f ie ld (a) D e tails of c e n tra l feed probe fo r quadrant c a v ity, (b) V ariable probe depth feed arrangement f o r a ty p ic a l h e lix.... ' I l l u s t r a t i n g 't h e numbering system fo r elem ents of the quadrant a r r a y Measured and c a lc u la te d far-zo n e power p a tte rn s fo r a 14-helix quadrant rin g array phased fo r a broadside beam Measured and c a lc u la te d far-zo n e power p a tte rn s fo r a 14-helix quadrant rin g array phased fo r a 5 t i l t angle Measured and c a lc u la te d far-zo n e power p a tte rn s fo r a 14 -h elix quadrant rin g array phased fo r a 10 t i l t angle Measured and c a lc u la te d far-zo n e power p a tte rn s fo r a 1 4 -h elix quadrant rin g array phased fo r a 15 t i l t angle Measured and c a lc u la te d far-zo n e power p a tte rn s fo r a 14 -h elix quadrant rin g array phased fo r a 20 t i l t angle C alcu lated far-zone power p a tte rn fo r a 5 6 -h elix array o f three c o n cen tric rin g s (fo u r 14-helix quadrant a r r a y s ), phased f o r a broadside beam Measured in p u t VSWR vs. frequency f o r th e quadra n t c a v ity -fe d 14-helix rin g array xiii

17 57. Photograph o f the quadrant c a v ity -fe d 14 -h elix array mounted above the prime focus la b o ra to ry o f th e O.S.U. rad io te le sc o p e D rift record taken through th e sun u sin g the 14-helix quadrant array a t a frequency of 1378 MHz (a) Plan view of ra d ia l c a v ity cut in to six 60 wedges, (b) C ro ss-sectio n view o f main c a v ity showing su b -cav ity used fo r e x c ita tio n o f a l l s ix wedges D etail of step p in g motor drive fo r h e lic a l elem ents I l l u s t r a t i n g the geometry o f the m echanically t i l t a b l e c a v ity -fe d rin g a r r a y Ring array geometry s u ita b le fo r use w ith a six-wedge ra d ia l feed cav ity I l l u s t r a t i n g d e te rm in is tic d en sity ta p e rin g technique f o r approxim ating a 1 - r^ d is tr ib u tio n w ith a 4 1 -rin g thinned array (a) Left-handed h e lix in c a rte s ia n co o rd in ate system, (b) Right-handed h e lix in c a rte s ia n coordinate system Far f i e l d p a tte rn of 8 element rin g array phased f o r a broadside beam Far f i e l d p a tte rn of 24 elem ent rin g a rra y phased f o r a broadside beam Far f i e l d p a tte rn of 80 elem ent rin g a rra y phased fo r a broadside beam Far f ie ld p a tte rn of 168 elem ent rin g array phased f o r a b roadside beam Far f i e l d p a tte rn of 168 elem ent rin g array phased f o r a broadside beam and w ith 0.5X rin g spacing Far f ie ld p a tte rn of 168 elem ent rin g array phased f o r a broadside beam and w ith 2.OX. rin g spacing. 207 XIV

18 71, (a) A 42 elem ent array o f s ix co n cen tric rin g s, = 2m. (b) A 84 elem ent array o f s ix co n c e n tric rin g s. N(ra) = 4m , Far f i e l d p a tte rn of th re e array s (168 elem ents, 84 elem ents, and 42 elem en ts), a ll phased fo r a broadside beam and with one wavelength spacing between r i n g s , Far f ie ld p a tte rn of 168 elem ent rin g array phased fo r a broadside b e a m , Far f i e l d p a tte rn of 168 elem ent rin g array phased fo r a 5 beam t i l t. S olid lin e in d ic a te s is o tro p ic elem ents, dashed lin e in d ic a te s h e lic a l e le m e n ts , Far f i e l d p a tte rn of 168 elem ent rin g array phased fo r a 10 beam t i l t , Far f i e l d p a tte rn o f 168 element rin g array phased fo r a 15 beam t i l t , Two types o f ra d ia l tran sm issio n lin e s, (a) C irc u la r ra d ia l tran sm issio n lin e, (b) Wedge r a d ia l tran sm issio n l i n e , (a) Plan view o f 80-elem ent square array of h e lic e s fed by square c a v ity, (b) Side view, showing coaxial feed lin e w ith l o o p , (a) Plan view o f 80-element square array o f h e lic e s fed by a g rid tran sm issio n lin e. (b) Side view, showing d ir e c t connection o f elem ents to conductive g r i d XV

19 CHAPTER I INTRODUCTION Contemporary o b serv atio n a l rad io astronomy programs have led to th e c o n stru ctio n o f a number o f larg e antennas, capable o f high re so lu tio n scanning over much of the sky. One technique fo r achieving a s te e ra b le high re so lu tio n beam is th a t o f co h eren tly e x c itin g a la rg e array o f phase c o n tro lla b le elem ents. By a s u i t able phasing o f each elem ent, th e response o f the composite antenna can be maximized fo r a wave in cid en t from a p a r tic u la r d ire c tio n. However, the problems asso c iate d with the phase c o n tro l o f the elem ents are u su ally not sim ple and in fa c t c o n s titu te one o f the m ajor d i f f i c u l t i e s in designing such phased array system s. The purpose of th is d is s e r ta tio n is to study a new type of phased array antenna in which a l l elem ents are c ir c u la r ly p o la riz e d and are fed from a common ra d ia l c a v ity. T his technique was sugg ested to the author in a memorandum from Dr. John D. Kraus, P rofe s s o r of E le c tr ic a l Engineering at The Ohio S ta te U n iv ersity [Kraus, 1964] in which the b a sic p rin c ip le s o f operation were enum erated and in which some p o ssib le design techniques were proposed. B a s ic a lly, th e proposed antenna was to be an array o f h e lic a l beam antennas (o r co n ical h e lic e s ), a l l of which were fed from a common, c e n tr a lly -e x c ite d ra d ia l c a v ity by means of coupling through holes

20 2 in one f l a t w all of th e c a v ity, as shown in F ig. 1. The memorandum p o in ted out th a t the main beam could be s te e re d by a x ia lly r o ta tin g each in d iv id u a l element a s u ita b le amount so th a t proper phasing was o b tain ed. I t was suggested th a t th e elem ents be a r ranged in c o n cen tric rin g s o f a p p ro p riate sp acin g, so th a t a l l h e lic e s were fed in phase; th is arrangement would then be capable o f expansion to a very larg e a rra y, w ith only one c e n tr a l feed p o in t, thus e lim in a tin g the high lo sses and com plexity norm ally a sso c ia te d with the feeding of larg e a rra y s. Also, the fa c t th a t th e beam s te e rin g can be achieved by a sim ple a x ia l ro ta tio n of each elem ent would o bviate the need f o r m echanically t i l t i n g * the e n tir e a rra y. This memorandum by J, D, Kraus [1964] has served as the b a sis fo r an ex ten siv e experim ental and th e o r e tic a l study of se v e ra l c a v ity -fe d c o n cen tric rin g phased arrays o f c ir c u la r ly p o la riz e d antennas. The in v e stig a tio n includes stu d ie s of th e pro p er ra d ia l waveguide s tru c tu re s to be used, p ro p e rtie s o f the c ir c u la r ly p o la riz e d elem ents, feed tech n iq u es, p o la riz a tio n In another memorandum, J, D, Kraus [1967] has p o in ted out th a t a sm all amount of m echanical t i l t i n g in d e c lin a tio n might be d e sira b le on a m eridian t r a n s i t array in o rd er to measure th e p a tte rn in d e c lin a tio n fo r a given phased array co n d itio n. This memorandum w ill be d iscu ssed more f u lly in C hapter VI,

21 Feed Cable- Resonant C avity / 1 (a) Id e n tic a l H e lica l Beam Antennas Feed Sections A ll in Same O rien tatio n fo r Broadside Beam F ig. 1. I l l u s t r a t i o n of a ty p ic a l c a v ity -fe d c o n ce n tric rin g array o f h e lic a l beam antennas as proposed in memorandum to author from Dr. J. D. Kraus [1964] (a) c ro s s-s e c tio n view; (b) p lan view

22 p r o p e r tie s, beamwidth and sid elo b e b eh av io r, th e amount of beam 4 s te e r in g f e a s ib le, and bandwidth and impedance b eh av io r. An im portant by-product of th e in v e s tig a tio n was th e discovery o f a new and im portant c la s s of e n d -fire antennas with low sidelobe le v e ls ; th e "h elico n e" d escrib ed in C hapter II is one such antenna. In a d d itio n, th e o r e tic a l stu d ie s of the h e lic a l beam antenna undertaken as a p a r t of the design o f the array have made a v a ilab le new inform ation concerning the p o la riz a tio n p ro p e rtie s of th e h e lix and the im portance o f th e h e lix co u n terp o ise s tr u c tu r e. Several s p e c ific q u estio n s must be answered in th e design o f such an a rra y. Some o f th e se are lis te d below. 1. Ubat type of c ir c u la r ly p o la riz e d elem ent should be used? 2. hbat are the p o la riz a tio n p ro p e rtie s of the element and how much can the a x ia l r a tio be expected to vary over th e main beam? 3. Does th e elem ent appear to be a p o in t source over the expected range of scan, i. e. is th e phase c e n te r of the elem ent w ell defined? 4, How much mutual coupling can be to le r a te d between elem ents? 5, What are the g en eral c h a r a c te r is tic s of c o n ce n tric rin g arra y s o f c ir c u la r ly p o la riz e d antennas? 6, Over what angle of scan can the sid elo b e le v el be kept under a s p e c ifie d lev el? 7. How does th e choice of the in d iv id u a l elem ent depend on the p a r tic u la r array geometry? 8, What is th e optimum co n fig u ra tio n o f th e ra d ia l

23 5 tran sm issio n lin e feed system? 9. How a cc u ra te ly can th e am plitude and phase of each elem ent be c o n tro lle d from the c av ity and how do the antennas in te r a c t with th e c av ity and w ith each o th e r through the c av ity? 10, How i s th e bandwidth o f th e antenna re la te d to the c o n fig u ra tio n of the feed system? 11, What are the th e o r e tic a l lim ita tio n s on the s iz e of a r a d ia l tran sm issio n lin e due to d is s ip a tio n o f power in f i n i t e l y conducting w a lls, and due to a tte n u a tio n by ra d ia tin g elem ents? 12, What th e o r e tic a l lim ita tio n s are th e re on the gain o f a very la rg e array o f th is type? 13, What type o f geometry is optimum in th e design o f a very larg e c a v ity -fe d array of c ir c u la r ly p o la riz e d antennas? 14, What type of coupling network between the elem ent and th e tran sm issio n lin e would be s u ita b le fo r use in a very la rg e array o f c ir c u la r ly p o la riz e d antennas? 15, How should the p a r a lle l p la te s in such a larg e array be m echanically supported and how does th is a f f e c t the design of the r a d ia l tran sm issio n lin e? These q u estio n s and o th er re la te d problems are discussed in the follow ing c h ap te rs. Chapters I I, I I I, and IV d iscu ss re s p e c tiv e ly th e array elem ent, th e array geom etry, and the array feed system. Chapter V summarizes th e experim ental r e s u lts obtained on th re e d if f e r e n t models which were b u i l t and te s te d to v e rify the g en eral th eo ry ; Chapter VI proposes some techniques to be

24 . used in designing a large phased a rra y f o r use in ra d io astronomy. Appendix A is a d e ta ile d d e riv a tio n of th e f ie ld s o f a th in 6 h e lic a l w ire carry in g a nonuniform c u rre n t. Appendix B is a sy stem atic c o lle c tio n of a rra y fa c to rs p lo tte d fo r a number of c o n ce n tric rin g array s and Appendix C tr e a ts th e f ie ld s tru c tu re o f a ra d ia l tran sm issio n lin e. In any experim ental program th e re are f a ilu r e s as w ell as su ccesses; Appendix D d iscu sses two array s which were ex perim entally found n o t to work and gives the reasons why.

25 CHAPTER I I THE ARRAY ELEMENT A. A D iscussion o f C irc u la rly P o larized Array Elements Consider the lin e a r array of 2N+1 is o tro p ic elem ents shown in Fig. 2. A ll elem ents are eq u ally spaced (d) and are e x c ite d w ith uniform am plitude but with v a ria b le phase, where the phase referen ce p o in t is a r b i t r a r i l y taken to be the c e n te r o f the arra y. The elem ent is fed with phase \p^ and am plitude E^ such th a t th e fa r e l e c t r i c f ie ld at angle 0 can be w ritte n as; where 2(8) = E " f ' (2-1) n=-n 0 = angle from broadside k = 2ir/X d = spacing between elem ents According to the P rin c ip le of S ta tio n a ry Phase [W alter, 1965, pp ], th e maximum value of the to t a l e l e c t r i c f ie ld can be made to occur a t angle 0 i f = knd s in 0 (2-2) Thus th is id e a liz e d phased array can be made to scan 180 by c o r re c tly c o n tro llin g the phase of each elem ent. There are sev e ra l id e a liz a tio n s involved in th is array which 7

26 To Far Field Point - o o -N KH nd sin 0 Fig. 2. I llu s tr a tin g a lin e a r phased array of 2N+1 is o tro p ic elem ents. 00

27 9 are very d i f f i c u l t to achieve in p ra c tic e. F ir s t of a l l, i t is assumed th a t the elem ents are is o tro p ic and m aintain p e rfe c t p o l a riz a tio n s t a b i l i t y w ith angular s h i f t s ; in p r a c tic e, th is co n d itio n i s never achieved. Secondly, the elem ents are assumed to be in dependent o f each o th e r in e le c tr i c a l c h a r a c te r is tic s, i. e. th ere is no mutual coupling; again th is is n ev er observed to be the case. Furtherm ore, the elem ents are assumed to have one w e ll-d efin ed phase c e n te r which is independent of angle. This co n d itio n is very d i f f i c u l t to achieve in p ra c tic e and i s often overlooked in array th eo ry l i t e r a t u r e ; some antenna elem ents are u n su ited f o r use in a phased array because t h e i r in h eren t phase c e n te r s h i f t with angle may cancel the s p a tia l phase s h i f t w ith an g le, thus o b v iatin g any p o s s ib il ity o f accu rate phase scanning. Of a l l th e lim itin g fa c to rs in designing a phased a rra y, mutual coupling is almost always the most im portant. For example, i f a n e a rly is o tro p ic element (such as a sm all dipole) is chosen, th e mutual coupling can be q u ite high and se rio u s feed impedance problems can a ris e [O liner and Malech, 1966, pp ]. The mutual coupling can be reduced by in c re asin g th e elem ent spacing, b u t th is can lead to one or more g ra tin g lobes in th e v is ib le reg io n. Often the answer to th is problem is to use more d ire c tiv e elem ents and s e t t l e fo r a sm aller angle of scan. L inearly p o la riz e d elem ents can be used as components o f a phased a rra y by feeding each element through a phase s h i f t e r. The necessary phase co n tro l o f each elem ent can be re a liz e d by e le c tr o n ic a lly o r m echanically c o n tro llin g th e phase s h i f t e r s.

28 By e le c tr o n ic a lly c o n tro llin g f e r r i t e phase s h i f t e r s, very fa s t 10 beam scanning is p o s s ib le. There is much inform ation in the l i t e r a t u r e on th is ty p e of phased array and a co n sid erab le amount o f development work is s t i l l in p ro g ress. A d i s t i n c t disadvantage o f u sin g e le c tro n ic phase s h if te r s is th a t larg e array s may c a ll fo r thousands o f such phase s h if te r s with the atten d an t problems o f u n ifo rm ity, r e l i a b i l i t y, and economy. C irc u la rly p o la riz e d antennas such as log co n ical s p ir a ls, crossed d ip o le s, o r h e lic e s are in h e re n tly capable of phase s h if tin g by a sim ple a x ia l ro ta tio n of th e elem ent. To see t h i s, c o n sid e r th e far-zo n e e l e c t r i c f ie ld on the axis o f a c ir c u la r ly p o la riz e d antenna as shown in Fig, 3 (a ). The e le c tr i c f ie ld on axis o f the antenna can be w ritte n as E = E^C 8 : j *) (2-3) where Eg = complex co n stan t 8 = u n it v e c to r in the 6 -d ire c tio n * = u n it v e c to r in the ^ -d ire c tio n t in d ic a te s r ig h t- and le ft-h a n d c ir c u la r p o la riz a tio n re sp e c tiv e ly Now suppose th a t the antenna is p h y s ic a lly ro ta te d on i t s axis through an angle A (see Fig, 3(b) ) so th a t i t s e l e c t r i c f ie ld in term s o f th e new co o rd in ate system (8*, < >') can be w ritte n as E* = Eg (e Î j $') (2-4)

29 (a) Fig. 3. I llu s tr a tin g the coordinate systems fo r a c irc u la rly p o larized wave. (a) Coordinate system fo r antenna in i t s o rig in a l p o sitio n. (b) Coordinate transform ation through the angle A,

30 I A f where 9 and (j> are u n it v ecto rs in th e new co o rd in ate system. 12 But 8 = cos A ê' - sin A (2-5) *. "I At, * = Sin A 0 + cos A * (2-6) S u b s titu tin g (2-5) said (2-6) in to (2-3), i t i s seen th a t E = e*j* ( ê' t j Î ') (2-7) = e efja so th a t p h y sical a x ia l ro ta tio n of th e elem ent by an angle produces a phase change of A in th e f a r - e l e c t r i c c ir c u la r ly p o l a riz e d f i e l d. I f th e antenna is not p e rfe c tly c ir c u la r ly p o la riz e d, then ro ta tio n by A w ill produce a phase s h i f t le ss than A. An example of a phased array w ith c ir c u la r ly p o la riz e d antennas as phase s h if tin g elem ents is th e lobe sweeping h e lix array used in 1956 a t th e Ohio S tate-o hio Wesleyan Radio O bservatory [Kraus, 1958, 1966], A more recen t example is the feed array fo r th e p a ra b o lic c y lin d e r ra d io telesco p e antenna a t the V erm illion Radio O bservatory operated by The U n iv ersity o f I l l i n o i s [Swenson and Lo, 1961], The feed system f o r th is telesco p e is a lin e a r array of co n ical s p i r a l s. By ro ta tio n of th e in d iv id u a l co n ica l s p i r a l elem ents, beam s te e rin g in the m eridian plane of ±30 from zen ith is o b ta in a b le. I t is obvious th a t the use of c ir c u la r ly p o la riz e d antennas as elem ents of a phased array can r e s u lt in a sim ple, economical

31 13 d esign. Furtherm ore, since no f e r r i t e phase s h if te r s are used, power handling problems are g re a tly reduced in th e case of a tra n s m ittin g a rra y. However, f o r use in ra d io astronom y, the elem ents must s a t i s f y o th e r c r i t e r i a in a d d itio n to t h e i r phase s h if tin g c a p a b ility. For example, i f p o la riz a tio n measurements are to be made, th e re are two im portant r e s tr ic tio n s which must be imposed on th e elem ent: (1) the p o la riz a tio n e llip s e must e x h ib it a minimum of v a ria tio n from c i r c u l a r i t y over th e main beam of any one elem ent in i t s array environm ent, and (2) the p o la riz a tio n e llip s e must e x h ib it a minimum o f v a ria tio n from c i r c u l a r i t y over th e main beam of the array as th e beam is phase scanned across th e sky. This means th a t in th e f i r s t case the a x ia l r a tio v a ria tio n across th e main beam of th e elem ent in i t s arra y environment must be minimized and in the second case th a t the mutual coupling between c ir c u la r ly p o la riz e d elem ents must be low enough th a t the p o la riz a tio n p ro p e rtie s of any one element in fre e space determ ines th e p o la riz a tio n p ro p e rtie s of th e a rra y. In c o n stru c tin g a larg e a rra y, i t is d i f f i c u l t to make d ir e c t measurements o f the far-zo n e a x ial r a tio of th e array ; th e re fo re, i t i s n ecessary to be able to p re d ic t i t s p o la riz a tio n p r o p e rtie s, i. e. i t is necessary th a t the mutual coupling be minimized. In ad d itio n to th e p o la riz a tio n s t a b i l i t y requirem ents on th e element ju s t d iscu ssed, i t is a lso d e sira b le th a t the phase c e n te r of the elem ent be w ell-d efin ed and frequency s ta b le, sin ce conventional array th eo ry is based on t h i s. Furtherm ore, the sid e lobe lev el of th e array elem ent must be low enough to reduce wide angle g ra tin g

32 14 lobes o f the array to a minimum. F in a lly, i t is d e sira b le th a t the feed arrangem ent to the in d iv id u a l elem ent be as sim ple as p o s sib le and th a t th e feed impedance of any one elem ent be m inim ally dependent on the a rra y scan angle. I t is u s e fu l to d iscu ss th e se p ro p e rtie s in terms of some s p e c if ic c ir c u la r ly p o la riz e d antennas. 1, The Conical L o g -sp iral Antenna The co n ical lo g -s p ira l antenna [Dyson, 1959] ra d ia te s a c ir c u la r ly p o la riz e d f i e l d over a bandwidth o f ty p ic a lly 10:1. The ra d ia tio n f i e l d is maximum in th e a x ia l d ire c tio n and o ff th e apex (feed) end. The half-pow er beamwidth (HPBW) of a ty p ic a l model is approxim ately 100 and th e sid e lobe and backlobe le v els u su a lly vary from -3 db to -30 db depending p rim a rily on the included cone an g le, th e angular w idth of th e ex p o n en tially expanding arms, th e ra te o f wrap o f th e arms, and the frequency. The an g u lar p o la riz a tio n s t a b i l i t y of th e co n ical lo g - s p ir a l is q u ite good; fo r example, a ty p ic a l model may e x h ib it an a x ia l r a t i o under 1.5 over a t 70 angular swing from the d ire c tio n of maximum r a d ia tio n. A unique c e n te r o f phase does n o t e x is t fo r th e co n ica l lo g - s p ir a l; however, an apparent phase c e n te r does e x is t over a p o rtio n of th e main beam, in a region v ery n e a r th e apex of th e s tr u c tu r e. One disadvantage o f th e co n ica l lo g -s p ira l antenna from th e larg e array stan d p o in t i s th a t i t must be fed by a balanced lin e ; a s lig h t imbalance in th e feed lin e can cause sq u in t in the p a tte rn with consequent d e le te rio u s e ff e c ts on the

33 15 am plitude d is tr ib u tio n of th e a rra y. One o f th e most su ccessfu l feeds uses co ax ial lin e s where th e o u te r conductor of the coax a cts as an arm of the s p i r a l. The c e n te r conductor t i e s to th e o th e r arm a t the apex, 2. The P lan ar S p ira l P lan ar s p ir a ls can tak e sev e ra l form s, such as eq u ian g u lar, re c ta n g u la r, or Archimedean [O liner and Malech, 1966, pp ], The e q u ian g u lar p la n a r s p i r a l [Rumsey, 1957, pp ; 1966] is capable o f very wideband o peration (ty p ic a l bandwidths are 20:1) and has a b id ir e c tio n a l ra d ia tio n f i e l d. I t i s capable of producing very pure c ir c u la r p o la riz a tio n over th e u sab le beamwidth, which may ty p ic a lly be 100*. However, sin ce u n id ir e c tio n a l p a tte rn s are d e sire d fo r th e elem ents of a rra y s, th e p la n a r s p ir a ls are u s u a lly Archimedean and are cavity-backed. The c av ity imposes a bandwidth lim it o f the o rd er of 2 :1, thus o b v ia tin g th e need f o r a wideband eq u ian g u lar s p ir a l. Bawer and Wolfe [1960] have d escribed such a cavity-backed b a lu n -fed Archimedean s p i r a l. However, the need fo r balun feeding p re sen ts a disadvantage in designing a very larg e a rra y, sin ce thousands of such baluns would be re q u ire d, one f o r each ra d ia tin g elem ent. The co ax ial s p ir a l p rev io u sly mentioned could be used to elim in ate th e need f o r a balun. 3, The Crossed Dipole Two d ip o les a t r ig h t angles to e a c h.o th e r and fed in time phase quadrature can be used to generate a c ir c u la r ly p o la riz e d beam [Kraus, 1950, pp ], Allen and Diamond [1965] have

34 noted th a t crossed d ip o les can be used in an a rra y environm ent w ith good g a in, impedance, and p o la riz a tio n s t a b i l i t y f o r scan angles 16 up to approxim ately 40, The mutual coupling between crossed d ip o les can be h ig h, however, and the p re d ic tio n of the p o la riz a tio n v a ria tio n across th e main beam of a crossed d ip o le in an array environment is d i f f i c u l t. In a d d itio n, a 3 db quadrature hybrid must be used to feed th e crossed d ip o le, thus p re se n tin g the same kind of economic and r e l i a b i l i t y drawback as th a t encountered in using conical lo g -s p ira ls and p la n a r s p ir a ls, 4, The A xial Mode H elical Beam Antenna A h e lix of p itc h angle between 10 and 18 and o f circum ference approxim ately equal to a wavelength is sa id to o p erate in the a x ia l mode and is capable of producing a m oderately d ir e c tiv e u n id ire c tio n a l beam which is very n e a rly c ir c u la r ly p o la riz e d on axis [Kraus, 1950, Chapter 7 ], Most axial-mode h e lic e s which have been te s te d have from two to twenty tu rn s and are operated over a ground plane whose maximum dimension is comparable to a w avelength. The HPBW of a ty p ic a l ten tu rn h e lix is about 38 and the f i r s t sid elo b es may be from 8 to 14 db down. Mutual coupling w ith re sp ec t to p a tte rn and impedance between axial-mode h e lic e s is q u ite low [B la si, 1966] when the elem ents are sep arated by one wavelength or more. The h e lix i s very sim ple to feed, since no balanced.cu rren ts are necessary fo r p ro p er o p eratio n. Most h e lic e s are fed by a co ax ia l lin e w ith o u te r conductor connected to the co u n terp o ise (u su a lly a

35 17 ground plane) and the in n e r conductor connected to th e h e lic a l w ire. However, a h e lix fed over a ground p lane is a d iffu se r a d ia to r, i. e. i t does n o t have a w ell d efined phase c e n te r. In a d d itio n, the h e lix does n o t m aintain c ir c u la r p o la riz a tio n over i t s main beam, i. e. i t is n o t p o la riz a tio n s ta b le ; fo r example, a ty p ic a l h e lix may have an a x ia l r a tio of 2:1 a t only 30 o ff a x is. This w ill be discussed in more d e ta il in Section C of th is ch ap ter. 5. The H elicone When an axial-m ode h e lix is fed in sid e a co n ica l h o rn, a co n sid erab le improvement in the ra d ia tio n f i e l d c h a r a c te r is tic s can be made w hile s t i l l m aintaining good c ir c u la r p o la riz a tio n on ax is [C arver, 1967]. This antenna, c a lle d the h e lic o n e, w ill be discu ssed in co n sid erab le d e ta il in Section B of th is ch ap ter. 6. The Svennerus M odification of th e H elix Svennerus [1958] has noted th a t by m odifying th e usual ground plane in to a sm all co n ical c o u n terp o ise, th e angular p o la riz a tio n s t a b i l i t y and the an g u lar phase c e n te r s t a b i l i t y can be con sid erab ly improved. This antenna, a s p e c ia l case of th e h elico n e antenna, has been used by th e au th o r in sev e ra l experim ental phased array s and has been found to s a t i s f y many o f the d e sira b le p ro p e rtie s of c ir c u la r ly p o la riz e d antennas p re v io u sly d iscu ssed. This m odified h e lix w ill be discussed in S ection C o f th is ch ap ter.

36 I t should be po in ted out th a t although the au th o r independ e n tly te s te d h e lic a l antennas with sh o rt co n ical counterpoises in conjunction w ith experim ental work on the h e lic o n e, he was not in a p o s itio n to n o te the improvement in p o la riz a tio n and phase c e n te r s t a b i l i t y as re p o rted by Svennerus, sin ce th e h elico n e experiments, were p rim a rily concerned w ith improvements 18 in sid elo b e le v el and beamwidth. It is f e l t by th e au th o r th a t Svennerus made a very im portant c o n trib u tio n in h is work, p a r t i c u l a r l y with regard to the p o la riz a tio n and focusing improvements o b tain ab le by rep lacin g th e f l a t ground plane w ith a c o n ica l co u n terp o ise. This o b serv atio n has led to th e o r e tic a l work on the p o la riz a tio n p ro p e rtie s of the h e lix, p a rt of which is d e ta ile d in Section C. B. The H elicone - A C irc u la rly P o larized Antenna With Low Sidelobe Level This sec tio n describ es a new c ir c u la r ly p o la riz e d antenna w ith low sid elo b e le v e l, c a lle d the h elico n e [C arver, 1967], The antenna c o n sis ts of an a x ia l mode h e lix (en d fire) in a co n ical h o rn, where the a x ia l length o f the h e lix is ty p ic a lly th e same length as the " a ltitu d e " of the tru n cated cone, as shown in Fig, 4. The h e lix serves as a c ir c u la r ly p o la riz e d e x c itin g elem ent w ithin th e co n ica l horn and u n lik e previous h e lix e x c ite d h o rn s, the h e lix extends to th e mouth of th e horn. T y p ic a lly, th e mouth diam eter of the horn is two to fo u r w avelengths, although the horn dimensions are n o t p a r tic u la r ly c r i t i c a l. These dimensions w ill be discussed in more d e ta il l a t e r. The HPBW of th is antenna

37 Truncated Conical Horn Ground Plane Axial Mode H elix Coaxial Feed Line Fig, 4, The helicone. «0

38 can be as low as h a lf th a t of an ordinary h e lix, and the sidelobe le v e l is q u ite s u b s ta n tia lly reduced over th a t o f an ordinary 20 h e lix. The ra d ia tio n f ie ld is nom inally c ir c u la r ly p o la riz e d on the axis o f the h e lic o n e, with approxim ately th e same a x ial r a t i o o f a h e lix alone. than th a t o f a h e lix. The input impedance is s lig h tly lower The p a tte rn and impedance bandwidth is approxim ately 2:1 fo r the optimum dim ensions. The a x ia l mode h e lix has been d escrib ed by Kraus [1950, Chapter 7] and the co n ical horn by Southworth and King [1939]. A co n ical horn fed by a h e lix was f i r s t proposed by Kraus [1949]; th is antenna used a h e lix -e x c ite d c ir c u la r waveguide feed but the h e lix was not extended beyond the waveguide-hom tr a n s itio n, as shown in Fig. 5 (a ). A co n ical horn w ith a c ir c u la r ly p o la riz e d ra d ia tio n f ie ld can also be co n stru cted by feeding i t w ith a c ir c u la r waveguide supporting a c ir c u la r ly p o la riz e d TE^^ mode; th is mode can be obtained by o rie n tin g a q u a rte r wave d ie le c tr ic phase delay p la te at an angle of 45" to th e in c id e n t lin e a rly p o la riz e d TE^^ mode wave, as shown in Fig. 5 (b ). Svennerus [1958] has d iscu ssed the case of a h e lix in a sm all co n ical co unterpoise; he found th a t the use o f th e co n ical co u n terp o ise, as shown in Fig. 5 ( c ), g re a tly improved the c ir c u la r p o la riz a tio n s t a b i l i t y w ith aspect angle fo r h e lic e s o f two or th re e tu rn s, b ut th a t only a sm all reduction in sidelobe le v el was o b tain ed. Fig. 5(d) shows a h elico n e fo r purposes o f comparison.

39 2 1 Coax Side View H elix Conical Horn Front View H elix C irc u la r waveguide (a) Conical Horn Q uarter Wave Delay P la te s C irc u la r waveguide (b) Coax C onical Counterpoise (c) H elix Coax Conical Horn (d) H elix F ig. 5. Comparison of sev e ra l re la te d antenna ty p e s. (a) C onical horn fed by h e lix -e x c ite d c ir c u la r waveguide. (b) C onical horn fed by c ir c u la r ly p o la riz e d TE mode. (c) Axial mode h e lix w ith co n ica l co u n terp o ise. (d) H elicone.

40 2 2 1, D escrip tio n of Experim ental R esults A sid e view of the helicone is shown in Fig. 6 w ith the dimensions o f i n t e r e s t. These dimensions a re : < )q = included angle of the cone d D = diam eter of th e base p la te = mouth diam eter & = cone sid e length a A C S a N = cone " a ltitu d e " = I cos(( >^/2) = a x ia l length o f h e lix =,circum ference o f h e lix = spacing between tu rn s of h e lix = p itc h angle o f h e lix = number o f tu rn s X = free space wavelength a) C onstruction The experim ental r e s u lts rep o rted here were taken from se v e ra l models b u i l t and te s te d at the Ohio S ta te U n iv ersity Radio O bservatory and a t the Ohio S ta te U n iv ersity E lectro scien ce Labo ra to ry. The design frequency was 1500 MHz. Except where i t is n o te d, i t can be assumed th a t the a x ia l length o f the h e lix was equal to the cone a ltitu d e (a = A), Most of the d ata were taken from two b a sic models: (1) a fiv e tu rn h e lix in sid e a cone and (2) a ten turn h e lix in sid e a cone. I t was found th a t the cons tru c tio n of the cones was extrem ely n o n c r itic a l in s o fa r as t h e i r su rface accuracy was concerned. The cones which were b u i l t fo r

41 to w Fig, 6, Dimensions o f the helicone.

42 24 th e fiv e tu rn h e lix were made of p o s te r board covered with aluminum f o i l. For one of the ten tu rn h e lix models, th e cone was made from inch aluminum sh ee t curved in to a cone and jo in e d w ith masking ta p e. w ith both o f th ese models. E xcellent power p a tte rn s were measured L ater cones f o r the ten tu rn h e lix were made more p re c is e ly from fib e rg la s formed over a la th e generated co n ical wood mold and then were lin e d with aluminum f o i l. However, th e power p a tte rn s measured fo r th ese cones were n o t markedly b e t t e r than those measured f o r e a r l i e r and much more roughly c o n stru cte d m odels. A ll of th i s in d ic a te s th a t the su rfa ce accuracy and c ir c u la r symmetry of the cones is not extrem ely c r i t i c a l. The h e lix was fed from th e c e n te r of th e ground plane as in d ic a te d in Fig. 6. the h e lix were te s te d. S everal d if f e r e n t support arrangem ents fo r From the stan d p o in t o f p a tte rn symmetry, the b e st arrangem ent was found to be th a t shown in Fig, 7, b) Measured power p a tte rn s In g en eral, f u l l knowledge of the f a r f i e l d from an antenna re q u ire s th e measurement of two orthogonal f i e l d components and the phase between them a t a l l p o in ts of a sp h ere. However, as a p r a c tic a l expedient s u f f ic ie n t inform ation may be obtained by m easuring two orthogonal lin e a r p o la riz e d power p a tte rn s in any p lan e of i n t e r e s t. This was done fo r th e h elico n e antenna. The two power p a tte rn s w ill be designated and when th e horn axis is o rie n te d along th e z -a x is, as shown in Fig. 8. The antenna is o rie n te d in the co o rd in ate system such th a t the q u a rte r wave

43 25 Nylon Guy Polystyrene y Dowel y y Rods Nylon Guy Fig. 7. Support arrangem ent f o r h e lix in sid e cone. q u a rte r wave feed stub X F ig. 8. O rie n ta tio n of h elico n e in sp h e ric a l coordinate system.

44 feed stu b (sometimes re fe rre d to as the "feed fin g e r") from the c e n te r of the ground plane to th e s t a r t of the h e lix proper lie s in the $ = 0* p la n e. By th e "P power p a tte r n," i t is meant th a t the power p a tte rn was measured w ith a lin e a r te s t d ip o le in the 6 -d ire c tio n and by th e expression "P power p a tte r n," i t i s meant th a t th e power p a tte rn was measured by a lin e a r t e s t d ip o le in the ^ -d ire c tio n. Fig, 9 shows fo u r power p a tte rn s measured on a d ecib el p o la r sc a le f o r a h elico n e with the dimensions given in Table 1, 2 6 TABLE 1 DIMENSIONS OF A TYPICAL HELICONE Horn Dimensions H elix Dimensions D = 3.05X C = l.oox I = 3.05X a = 14 d = 0.76X N = 10 44* a = A = 2.83X The average HPBW ( a f te r a sm all re c o rd e r non lin e a r ity co rre ctio n ) is 20 and the sid elo b e le v el v a rie s from -23 db to -33 db, depending on th e p a r tic u la r p a tte r n. This re p resen ts a s u b s ta n tia l improvement over the o rdinary h e lix p a tte r n. A comparable s e t of measured lo g arith m ic power p a tte rn s f o r the same h e lix w ithout the horn is shown in Fig. 10 and F ig. 11. The dimensions of the h e lix

45 E ' (b) Fig. 9. (aj ^ = 0 plane (b} ^ = 90 plane

46 / Fig. 10, Logarithm ic p o la r power p a tte rn s fo r a ten tu rn 14* p itc h angle a x ial mode h e lix in the 4= 0* plane.

47 Fig. 11. Logarithm ic p o la r power p a tte rn s f o r a ten turn 14 p itc h angle a x ia l mode h e lix in th e if) = 90 plane,

48 30 and ground plane are th e same as those fo r the p a tte rn s o f Fig. 9. From a comparison o f F ig s. 9, 10, and 11, i t is seen th a t the h elico n e n o t only reduces th e HPBW from th a t o f a h e lix alone, but i t also has exceedingly low back ra d ia tio n with a fro n t to back r a tio g re a te r than 35 db, and a fro n t to sid e r a t i o g re a te r than the dynamic range of the measuring equipm ent, which was 40 db. Feed cable ra d ia tio n may be resp o n sib le fo r the sm all back lobe. The h elico n e w ith th e dimensions li s t e d above has a nominal 2:1 p a tte rn bandwidth. Fig. 12 shows measured p o la r lin e a r power p a tte rn s fo r th e antenna from 900 MHz. to 2100 MHz., w ith a c e n te r design frequency o f 1500 MHz. [The h elico n e of Table 1 would be a t 1500 MHz.) This c o n se rv a tiv e ly defined p a tte rn bandwidth, which is u n u su ally wide fo r most c ir c u la r ly p o la riz e d h o rn s, is due p rim a rily to th e bandwidth of th e h e lic a l e x c itin g elem ent. The h e lix m aintains a h ig h ly ta p ered ap ertu re d is tr ib u tio n acro ss th e horn mouth over the bandw idth, but the exact form o f the d is tr ib u tio n changes s lig h tly w ith frequency. These measurements are shown in the graph of Fig. 13, along w ith the frequency b eh av io r of th re e assumed ap ertu re d is tr ib u tio n s [S ilv e r, 1964, p. 195; Kraus, 1966]. The frequency-beamwidth behavior o f a ten tu rn h e lix w ithout th e horn is also shown f o r both measured and c a lc u la te d v a lu es. I t is e a s ily seen th a t the ad d itio n o f the horn s u b s ta n tia lly reduces th e beamwidth across th e bandwidth of o p e ra tio n. Furtherm ore, th e HPBW v a rie s more slow ly as a function o f frequency fo r the h elico n e (HPBW ~X) than fo r a h e lix alone

49 A 900 MHz MHz MHz MHz MHz MHz MHz. Fig. 12. I llu s tr a tin g the p a tte rn bandwidth of a ty p ic a l helicone.

50 Measured fo r ten turn 14 h elix C alculated fo r ten turn 14 Kraus s formula h e lix using HPBW O Measured fo r helicone C alculated, assuming E(r) C alculated, assuming E[r) C alculated, assuming E(r] Aperture diam eter = 61,0 cm Frequency in GHz. Pig. 13. I llu s tr a tin g the beamwidth of a ty p ic a l helicone as a function o f frequency and as compared to a simple h e lix. (X N)

51 3 /2 (HPBW '^X ). A lso, th e sid elo b e lev el does not exceed - 20 db fo r ( 2.03 < D^< 3.96 ) in the case of th is p a r tic u la r model. The beamwidth as a fu n ctio n of cone angle is a lso of 35 i n t e r e s t. Fig, 14 is a graph of the measured HPBW v s. cone angle (<j)^) f o r two b a sic models o p eratin g at a frequency such th a t = 1.00, a fiv e tu rn h e lix w ith a sso c ia te d co n ical h o rn s, and a ten turn h e lix with a sso c iate d co n ical h o rn s. The dimensions o f th e fiv e tu rn m odel.(#1) and the ten tu rn model (#2) are given in Table 2. The optimum cone angle co n sid erin g both beamwidth TABLE 2 DINENSIONS OF TWO HELICONE MODELS Model #1 Model #2 d = 0.76X d = 0.76X v a rie s (j)^ v a rie s a = A 1.20X a = A = 2.41X C = l.oox C = l.oox o = 14 o = 14 N = 5 N = 10 and sid elo b e lev el is between 45 and 60. I f th e cone angle is to o sm all, th e re is in creased in te ra c tio n between the horn w alls and th e h e lix ; in p a r tic u la r, th e mouth edges have la rg e r f i e l d s, thus in c re asin g th e d iff ra c te d power appearing as high sid e lo b e s.

52 Level of H ighest Sidelobe in db 34 HPBW 60 HP_BW o f 5 tu rn h e lix 50 HPBW of Model # HPBW of Model # Sidelobe le v el o f Model # Optimum region f o r te n tu rn h e lix O c y lin d e r Included Cone Angle <t> F ig, 14. HPBW and sid elo b e le v el versus cone in clu d ed angle f o r the h e lic o n e.

53 35 A lso, sin ce th e ap ertu re d e crea se s, th e beamwidth r i s e s. For la rg e an g les, th e in te ra c tio n between th e c o n ica l horn and the h e lix is sm all and the HPBW is p rim a rily determ ined by the h e lix alo n e; however, sin ce the h e lix cannot ra d ia te ap p reciab ly in the back d ire c tio n due to the sh ie ld in g actio n o f th e h o rn, th e backlobes and sid elo b es are s t i l l sm all. For la r g e r a p e rtu re s, the phase is not c o n stan t across the a p e rtu re, so th a t the d ir e c tiv ity d ecreases fo r 6^ >70. o I t is n a tu ra l to ask w hether the a x ia l length o f the h e lix should be the same as th e a ltitu d e of the cone fo r an optimum p a tte rn o f lowest p o ssib le sid elo b e lev el and beamwidth. In o th e r words, should th e cone n e c e s s a rily extend to th e f u l l length of th e h e lix? This q u estio n has been p a r t i a l l y answered by the c o n stru c tio n of a h elico n e in which the horn was then g rad u ally c u t back u n t i l only th e h e lix was l e f t. The c u ttin g was done in ste p s of X/8, w ith th e HPBW and sid elo b e le v e l measured a f te r each c u t. The e f f e c t o f the cone a ltitu d e on the HPBW and sid elo b e le v e l is shown in Fig. 15, along w ith th e p e r tin e n t antenna dimens io n s. As th e a ltitu d e of the cone is reduced, the HPBW r is e s u n til th e HPBW o f th e o rd in ary h e lix is reached. The sid elo b e le v e l also r i s e s (both f i r s t sid elo b e le v e ls are shown), although i t appears to ta p e r o ff a f te r th e f i r s t h a lf of the cone has been removed. This may in d ic a te th a t the f i r s t h a l f of th e cone is p rim a rily re sp o n sib le f o r the extrem ely low sid elo b e le v e l of th e h e lic o n e, although th e evidence is n o t yet com plete. However, lowest beamw idth and sid elo b e le v e l are obtained fo r a f u l l cone. About one

54 36 14,10 tu rn h e lix HPBW 2.51 HPBW 30 Sidel obe le v e ls y / of o rd in ary h e lix 03 "O F ir s t Sidelobe Levels - u I0) * 4 V TS H tn a (cone a ltitu d e ) F ig. 15, length o f h e lix HPBW and sid elo b e le v e l as a fu n ctio n o f cone a ltitu d e fo r the h e lic o n e.

55 37 month a f t e r th ese in v e stig a tio n s were com pleted, th e a u th o r's a tte n tio n was c a lle d to the re p o rt of Svennerus [1958] which d iscu ssed the experim ental r e s u lts obtained by using a cone of a ltitu d e a = 0.40X w ith a h e lix. This cone a ltitu d e is shown as a dashed lin e in Fig. IS fo r comparison. However, Svennerus* num erical d a ta are p rim a rily taken from models using h e lic e s of only two o r th re e tu rn s and th e re fo re th e rep o rted d ata cannot be compared e x a c tly w ith th e d a ta of F ig. 15. No sy stem atic attem pt has been made by th e au th o r to in v e s tig a te th e case where th e cone a ltitu d e extends beyond the h e lix a x ia l len g th ; however, c e r ta in s e le c te d models te s te d have in d ic a te d th a t th is extension may have a d e le te rio u s e f f e c t, p a r tic u la r ly on th e sid elo b e le v e l. When th e cone a ltitu d e is appreciably longer than th e h e lix a x ia l le n g th, th e re can be a nonuniform phase fr o n t across th e mouth o f the horn as w ell as a stro n g e r illu m in a tio n of the horn edges which thus ra is e s th e le v el of back r a d ia tio n. c) Measured impedance c h a r a c te r is tic s The input impedance of the h elico n e has been found to be only s lig h tly lower than th a t of a h e lix alone. The in p u t impedance was measured as a fu n ctio n of cone angle a t a f r e q uency such th a t = 1.00 (1500 MHz.), and then as a fu n ctio n of frequency fo r a cone angle o f 44, using a h elico n e of the dim ensions o f Model #1 in Table 2. The measurements were made w ith a s lo tte d lin e connected d ir e c tly to the in put of the antenna, A cutaway drawing of the co ax ial input to th e h elico n e is shown

56 38 in Fig. 16. The input impedance as a fu n ctio n of cone angle is shown in Fig. 17, where i t is seen th a t th e r e s is ti v e p a rt ( 'V 100 n) is v ir t u a lly independent of cone angle; the in p u t reactan ce is about h a lf the in p u t re s ista n c e except fo r th e cone angle o f 0 (cy lin d er) where the reactance n e arly doubles due to th e in creased sto re d energy of the c y lin d e r over a cone with non-zero included angle. Fig. 18 i l l u s t r a t e s th e 2:1 impedance bandwidth of th e h e lic o n e, where re s is ta n c e and reactan ce are shown as a fu n ctio n of frequency fo r a h elico n e w ith a 44 cone an g le. The impedance is seen to vary by * 20 p ercen t over th e frequency range with a mean of about j35 ohms, fo r th is p a r tic u la r model. However, sin ce th e impedance of a h e lix is known to show a second-order dependence on the exact c o n stru ctio n of th e in p u t [Stegen, 1964], i t is reasonable to expect th a t i f th e in p u t co n fig u ratio n to the helicone is not id e n tic a l to th a t shown in Fig. 16, th e impedance may change. d) Axial r a t i o c h a r a c te r is tic s of the h elico n e The v o ltag e a x ia l r a tio of the p o la riz a tio n e llip s e of the h elico n e on axis is s u b s ta n tia lly th e same as th a t on th e axis o f th e h e lix alo n e, i. e. (2N+1)/2N [Kraus, 1950, Chapter 7]. However, th e a x ia l r a t i o r is e s ra p id ly fo r angles o ff a x is, so th a t the ax is of the h e lix is n o t an asp ect of p o la riz a tio n s t a t i o n a r i t y ; a x ia l r a tio s as high as 2:1 have been measured at only 20 o f f a x is. This in d ic a te s th a t th e a p ertu re d is tr ib u tio n is not symmetric across any two orthogonal ap ertu re planes of th e

57 Polystyrene Support Legs 016X H elical Conductor Aluminum Conical Surface Type N Connector Fig. 16. Cutaway drawing of coaxial input to helicone.

58 100 - o Input Resistance 5 tu rn h e lix 1.00 V) 6 O 40 c o u 4-> U a> cc Cone Angle ) o s 4J W Input Reactance in o CL -100 Fig. 17. Input impedance of a helicone as a function of cone angle. A. o

59 Input R esistance turn h e lix IX = 0.76 = 1.73 Frequency in GHz « Input Reactance Fig. 18. Input impedance of helicone as a function of frequency.

60 42 horn [Chu and Kouyoumjian, 1962] which is in tu rn due to th e asymmetric n ear f ie ld s tru c tu re o f th e f i n i t e h e lic a l e x c itin g elem ent (see S ection C of th is c h a p te r). I f th e mode could be reduced and some accep tab le end-loading arranged [S p rin g er, 1949; Angelakos and K ajfez, 1967] in o rd er to prevent r e fle c tio n s from th e end of the h e lix, the p o la riz a tio n s t a b i l i t y could undoubtedly be improved. 2. D iscussion I t would be a form idable ta sk to c a lc u la te e x ac tly th e a p ertu re d is tr ib u tio n of the h elicone f o r a l l p o ssib le co n fig u r a tio n s. Since th e n e ar f i e l d s tru c tu re of the f i n i t e a x ia l mode h e lix is unknown in exact form, i t is d o u btful t h a t the p e rtu rb a tio n a l e f f e c t of th e horn on the h e lix can be estim ated in a n a ly tic a l term s, although recen t advances in th e i t e r a t i v e so lu tio n method of so lv in g in te g ra l equations [Andreason, 1966] may y e t make i t p o ssib le to p re d ic t the ap ertu re d is tr ib u tio n. The low sid elo b e le v e l of the h elico n e im plies th a t the a p ertu re d is tr ib u tio n is ta p e re d, as suggested in Fig. 13. At th e high frequency end of the band, th e f ie ld s are r e la tiv e ly tig h t l y bound t o th e h e lix and th e coupling to the horn is weak; th e re fo re the HPBW a t the high end of the band is determ ined p rim a rily by th e h e lix i t s e l f. However, the presence o f th e co n ical sh ie ld in g su rface m aintains th e sid elo b es and backlobes a t a low le v e l. At th e low frequency end of the band, th e coupling between th e h e lix and th e horn is h ig h e r and the a p ertu re d is tr ib u tio n becomes le ss tap ered. The bandwidth of th e helicone

61 43 could be extended by using a w ider bandwidth e x c itin g element such as a tap ered h e lix [S p rin g er, 1949], or p o ssib ly some type o f log p e rio d ic s tr u c tu r e. The low sid elo b e le v e l o f the h elico n e is a lso found in r e la te d lin e a rly p o la riz e d co n ical horn antennas such as th e dual-mode co n ical horn f i r s t rep o rted by P o tte r [1963] and more re c e n tly by Nagelberg [1966]; th e TE ^^-excited co n ical horn [Southworth and King, 1939] also has a low backlobe le v el a l though th e main lobe in th e E-plane u su a lly has sidelobe "sh o u ld ers" at about -15 db to -20 db, When th e sid elo b e lev el of th e h elico n e is too h ig h, i t may be p o ssib le to obtain a f u r th e r red u ctio n by using th e choke s l o t technique [Lawrie and P e te rs, 1966]. The h elico n e d isp lay s a nominal a x ial symmetry in i t s fa r-zo n e power p a tte r n s. As th e p a tte rn s of F ig. 9 in d ic a te, th i s nominal symmetry is m aintained only to approxim ately th e -23 db le v e l, where th e sid elo b e s tru c tu re becomes asymmetric. I t has been found experim en tally th a t th is type o f asymmetry is ty p ic a l o f the h elicone and is due to the fa c t th a t a f i n i t e h e lix o f a moderate number of tu rn s does not have an a x ia lly symmetric n ear f ie ld. The d iffe re n c e in HPBW in any two o rth o gonal p lan es o f the h elico n e v a rie s from 3% to 9% of the mean HPBW and th e -10 db width may vary as much as 12% over any two orthogonal p la n e s. I t may be p o ssib le to improve th e a x ial symmetry o f the p a tte rn s o f the h elico n e by reducing the mode and by end-loading the h e lix [S p rin g er, 1949; M innett and

62 44 Thomas, 1966; Rumsey, 1966]. No co nclusive experim ental d ata is a v a ila b le on h elico n es o f la r g e r dimensions than those discussed h e re, e i t h e r fo r cone angles g re a te r than 90" o r cone a ltitu d e s g re a te r than about 2.80 X, Furtherm ore, the wave gen eratio n process in sid e th e h elico n e is n ot w ell understood, p a r tic u la r ly w ith regard to the in te r a c tio n between th e h e lix and th e horn w a lls. T h erefo re, i t i s im possible to a c c u ra te ly p re d ic t th e behavior o f la rg e r h orns. The d ir e c t i v i t y of the helicone can be c a lc u la te d approxim ately according to [Kraus, 1950, p. 25] jo = 41,253/(HPBW)2 (2-8) I t has been found e m p iric a lly th a t fo r a h elico n e of moderate a ltitu d e and fo r cone angles between 15 and 75, HPBW = 73 /D^ ( Î 15% ) (2-9) T h erefo re, s u b s titu tin g equation (2-9) in to equation (2-8 ), D = 7.8 (2-10) Since th e e ffe c tiv e ap ertu re of an antenna can be w ritte n as Ag = (X^/4tt)JÜ5, a sim ple c a lc u la tio n shows th a t th e e ffe c tiv e a p ertu re of th e h elico n e is about 80% o f the p h y sica l ap ertu re under th e assum ptions p re v io u sly d iscu ssed. This in d ic a te s th a t th e phase d is tr ib u tio n across th e ap ertu re is very n e arly u n i form. For a h e lix o f given len g th, th e a d d itio n o f an optimum cone can in c re ase th e d i r e c tiv ity by a f a c to r o f 4,o r 6 db,

63 45 I t i s o f in t e r e s t to compare th is d ir e c tiv ity in crease to th a t o f a h e lix alone. The d ir e c tiv ity of th e h e lix n ear design frequency can be approxim ately w ritte n as [Kraus, 1950, Chapter 7] JO = 15 A (2-11) where = NS^ = a x ia l length of the h e lix. T herefore th e h e lix a x ia l length must be in creased by a f a c to r of 4 fo r a 6 db in c re ase in d ir e c tiv ity. For example, co n sid er the d iffe re n c e in d ir e c tiv ity between a 14 p itc h angle te n tu rn h e lix w ith circum ference equal to one w avelength and the same h e lix in a 44 included angle cone. The d i r e c t i v i t y of the h e lix alone is approxim ately 33; th e d ir e c tiv ity o f th e comparable h elico n e is approxim ately 75, or 3.6 db h ig h e r. By using a 52 cone, the HPBW of th e h elico n e can be cut to almost h a l f th a t of the h e lix alone (w ith a sm all in c re ase in sidelobe le v e l), i. e. an in c re ase of about 6 db in d i r e c tiv ity. I t i s not known y et how larg e th e h elico n e can be made w ith s a tis f a c to r y p a tte r n s, and thus th e upper lim it o f in equation (2-10) cannot be s e t. The lower lim it, however, is imposed by th e coupling between th e h e lix and the horn; i t has been e m p iric a lly found to be about 1.9 X. 3. Comparison of H elicone w ith Conical Horn I t is a lso of in t e r e s t to compare th e p a tte rn s o f a h elico n e w ith those of th e cone when the h e lix i s removed, Southworth and King [1939] have measured p a tte rn s on lin e a rly p o la riz e d co n ical

64 46 horns e x c ite d by a dominant mode c ir c u la r waveguide. These p a tte rn s are u sefu l fo r comparison purposes with the h e lic o n e. Fig. 19 compares the e l e c t r i c f i e l d p a tte rn s o f a co n ical horn of 40 included angle (from th e d ata o f Southworth and King) w ith th o se o f a h elico n e of 44 included angle and a ltitu d e comparable to th a t of the co n ical horn. The ap ertu re diam eters of th e two antennas d i f f e r by about 0.5 X, b u t th is does not a f f e c t th e r e la tiv e d iffe re n c e in th e shape o f th e two p a tte r n s. The p a tte rn s are very s im ila r, although th e helicone d isp lay s a more narrow beam p a r tic u la r ly a t th e 0.1 power lev el (-10 db) and below ; the h elico n e is s lig h tly la rg e r th an th e comparable co n ical horn, thus accounting to some e x te n t fo r the d iffe re n c e in HPBW. Southworth and King found th a t th e gain of a conical horn could not be in creased much above 22 db. This type of upper lim it a lso holds fo r th e h e lic o n e. In e i t h e r case, fo r a given horn len g th, in c re a sin g th e mouth diam eter beyond about 2.5 gives r i s e to s ig n if ic a n t phase v a ria tio n s across th e a p e rtu re, thus lim itin g the gain. The co n ical horn d isp lay s a nominal 2:1 p a tte rn bandw idth, as does th e h e lic o n e. However, when the co n ical horn i s energized by a waveguide carry in g th e c ir c u la r ly p o la riz e d dominant mode (TE^^), th e bandwidth is considerably reduced (depending to some ex ten t upon the bandwidth of the feed) when the p o la r iz a tio n c ir c u la r ity is taken in to account. T h erefo re, th e h elico n e has su p e rio r bandwidth p ro p e rtie s over the comparable c ir c u la r ly p o la riz e d co n ical horn.

65 % Helicone 3.0SA 2.84 A 0.4 Conical Horn 40' Helicone 2.57A E-Plane-Conical Horn 0.2 H-Plane-Conical Horn Angle from Axis Fig. 19. Comparison of far-zone power p a tte rn s of a conical horn with power p a tte rn of a helicone with sim ila r dimensions.

66 48 4, The H elicone as an Array Element Although the helicone has been dem onstrated as a very u se fu l antenna, c e rta in d i f f i c u l t i e s a ris e when i t is used in an a rra y. Although the low sid e ra d ia tio n im plies low mutual co u p lin g, the p h y sical s iz e o f the cone is u su a lly at le a s t two w avelengths, thus p rev en tin g clo se elem ent sp acin g. However, a more se rio u s drawback is th e angular p o la riz a tio n i n s t a b i l i t y o f the ra d ia tio n f ie ld of th e h elico n e. As p re v io u sly p o in ted o u t, the a x ia l r a tio r is e s ra p id ly w ith angle from a x is, so th a t good c i r c u l a r p o la riz a tio n would n o t be o b tain ab le over a very larg e angle in a phase scanning array of a x ia lly fix e d h e lic o n e s. Although th is d if f ic u l ty could be overcome by m echanically t i l t i n g each elem ent, th is approach d ev iates from the s ta te d o b je ctiv es o f th is d is s e r ta tio n. C, The H elix - A Second Look Although the b a sic p ro p e rtie s o f the a x ia l mode h e lic a l beam antenna are w ell known [Kraus, 1950, C hapter 7 ], i t is u sefu l to examine th e h e lix in more d e t a i l. In p a r tic u la r, a knowledge of the p o la riz a tio n and phase c e n te r b ehavior of the h e lix is in d isp en sab le when the antenna is to be used in an a rra y, 1. Phase C enter o f the H elix Sander and Cheng [1958] have made experim ental and th e o re t i c a l stu d ie s of the p o s itio n of the phase c e n te r of the h e lix. T h eir conclusions are enumerated in p a rt below.

67 a) For a seven tu rn h e lix of p itc h angle 11 to 13, th e phase c e n te r is w ell defined over the main beam when a c ir c u la r 49 ground plane of diam eter.88x is used. The lo c atio n of the phase c e n te r is a t.255x from the c e n te r o f the h e lix (toward th e ground plane) fo r th e 13 h e lix and a t,230x from th e c e n te r (tow ard the ground plane) fo r th e 11 h e lix. b) As the ground plane is made la rg e r, th e phase c e n te r moves toward the ground p lan e. c) The phase c e n te r moves away from th e ground plane in p ro p o rtio n to in creased length when th e number o f tu rn s i s in c re ase d. d) As the p itc h angle is d ecreased, the phase c e n te r moves toward the ground p la n e, e) The HPBW decreases w ith in c re asin g number o f tu rn s, d ecreasin g ground plane diam eter, and in c re a sin g p itc h angle. f) As th e ground plane is made la r g e r, th e sid elo b e lev el in c re a s e s. g) In creasin g ground plane diam eter in c re ase s the on-axis a x ia l r a t i o, but in c re a sin g p itc h angle and number of tu rn s improves th e on-axis a x ia l r a tio. h) For an angular swing of * 25 to * 35 from the h e lix a x is, th e phase s h i f t is le s s than 10 from th e on-axis referen ce phase. This is fo r h e lic e s of 5 to 7 tu rn s, Sander and Cheng a lso found th a t when a c ir c u la r ground plane (only c ir c u la r ground planes were te s te d ) of diam eter 1.52X is used, th e phase c e n te r can move as much as.5 0 X along the h e lix a x is, depending on the p a r tic u la r plane co n tain in g the ax is in which

68 5 0 the observ er is s itu a te d. This is im p o rtan t, sin ce i f th e h e lix is to be ro ta te d on axis fo r phase s h if tin g p u rp o ses, th e fa c t th a t the apparent phase c e n te r p o s itio n a lso changes w ith ro ta tio n means th a t accurate beam s te e rin g may be im possible when th e wrong siz e ground plane is used. These experim ental o b servations provide a stro n g in d ic a tio n th a t the ground plane can be very im portant when the h e lix is to be used as an array elem ent. 2. P o la riz a tio n C h a ra c te ris tic s of the Helix and i t s Relevance to a Phased Array In many a p p lic a tio n s o f a nom inally c ir c u la r ly p o la riz e d r a d ia to r, i t is d esired to know th e complete p o la riz a tio n s ta t e over a s e c to r of angles of ra d ia tio n. The p o la riz a tio n s ta te can be found as a locus on the Poincare sphere [Deschamps, 1951]. The p o la riz a tio n e llip s e thus found can be s p e c ifie d by i t s a x ia l r a t i o and i t s t i l t angle. In th is sec tio n an e x p lic it expression fo r the a x ia l r a t i o of an antenna at any angle is developed under the assum ption th a t the two orthogonal far-zo n e e l e c t r i c f ie ld components are known. The p o la riz a tio n c h a r a c te r is tic s o f a h e lic a l beam antenna are then examined with p a r tic u la r emphasis on the e f f e c t of th e counterpoise on the p o la riz a tio n s t a b i l i t y of the antenna. The p o la riz a tio n c h a r a c te r is tic s of a broad c la s s o f c i r c u la r ly p o la riz e d ra d ia to rs have been e le g a n tly tr e a te d by Chu and Kouyoumjian [1960, 1962], T h eir treatm en t has included general theorem s on p o la riz a tio n s t a b i l i t y and a p p lic a tio n s to a number of nom inally c ir c u la r ly p o la riz e d r a d ia to r s, in clu d in g the h e lic a l

69 beam antenna, Morgan and Evans [1951] have d iscu ssed the p o la r 51 iz a tio n e llip s e in terms o f the orthogonal e l e c t r i c f ie ld components and the a x ia l r a tio of the p o la riz a tio n e l l i p s e. In the follo w in g, an e x p lic it sin g le expression is obtained f o r th e a x ia l r a tio of an antenna a t any aspect in terms of i t s s p a tia lly o rthogonal e l e c t r i c f ie ld components. a) D erivation of an e x p lic it r e la tio n fo r th e a x ia l r a t i o o f ah e l l i p t i c a l l y p o la riz e d electro m ag n etic f ie ld C onsider an antenna s itu a te d at the o rig in as shown in Fig. 20. Assuming a tim e-harm onic v a r ia tio n, the complete e l e c t r i c f i e l d a t th e far-zo n e p o in t P can be w ritte n as 8 = Re [ CEgê + E^î)e^' ^ ] ' (2-12) where Re = " r e a l p a rt of" E 6 = component of e l e c t r i c f ie ld in th e 0 -d ire c tio n = component o f e l e c t r i c f ie ld in the ^ -d ire c tio n 0 = u n it v e cto r in the 9 -d ire c tio n * = u n it v e c to r in the ^ -d ire c tio n Ü) = angular frequency t = time Since E and E are complex numbers, they can be w ritte n as 0 (j) Eg = legle^^e (2-13) C2-14)

70 52 X F ig, 20. I l l u s t r a t i n g the far-zo n e p o in t in th e s p h e ric a l co o rd in ate system.

71 53 so th a t. Re 1 H ^ e j ( V ^ 5. = I Eg I COS (wt +7g)e + E^ cos(wt +Yg+ 5)3 = 6 g 6+ E 3 (2-1 5 ) where ^e~ l^glcoscwt +Yg) (2-16) ^= E ^ cos(w t +Yq+ 5) (2-17) 6 is the tim e phase d iffe re n c e between the two orthogonal f i e l d componentse and 8. By combining equations (2-16) and (2-17) 6 SO th a t (bit +Y ) is e lim in a te d, the fa m ilia r equation fo r the 6 p o la riz a tio n e llip s e is obtained [Kraus, 1950, p. 467] _ z St, n c o s{ s = sin ^ 6 (2-1 8 ) 1 ^ 1 ' I ^ I I V E, P The p o la riz a tio n e llip s e at the far-zo n e p o in t P(R,0,^) is shown in Fig. 21. The a x ia l r a tio a t the p o in t P(R,0,<}>) is Imaximum e l e c t r i c f i e l d in te n s ity A.R.(R,0,*) = [minimum e l e c t r i c f i e l d in te n s ity = OA/OB I t i s n ecessary, th e re fo re, to fin d th e sem i-m ajor axis (OA) and the sem i-m inor ax is (OB) of the p o la riz a tio n e llip s e in term s of th e two orthogonal f ie ld components and E^, which are assumed

72 54 Fig. 21, The p o la riz a tio n e llip s e a t the far-zo n e p o in t PCR,6,4i).

73 55 to be kno\vn. For most p h y sica l antennas th e accu rate d e sc rip tio n of E and E from a rig o ro u s a p p lic a tio n o f f i e l d theory is ex- 6 <P trem ely d i f f i c u l t. However, many p r a c tic a l antennas do have approxim ate so lu tio n s fo r th e two orthogonal f i e l d components which agree w ell w ith measured r e s u lts. I f a co o rd in ate tran sfo rm atio n is made so th a t the f i e l d components (6_, E ) a re transform ed to (, ) with th e (n,ç) o <p n Ç axes as shown in Fig. 21, then equation (2-18) can be re w ritte n as (2-19) The d e sire d tran sfo rm atio n is [Kraus, 1950, p. 476] 3 - ^ s in X + 5 cos X (2-21) * n S When equations (2-20) and (2-21) are s u b s titu te d in to equation (2-1 8 ), equation (2-19) w ill r e s u lt such th a t no cro ss-p ro d u ct e x is ts i f the angle x is chosen such th a t [B om, 1933]

74 56 tan 2t 2 - L î ^ L è l _ I:el - (2-22) s u b s titu tio n. (sin^t k ' ^ 2 ^ COS T 1 2 EJ - E I, » s in 2 t tan 2t ( 2 CCS ^ VI%1' s in 2t tan 2t sin^6 (2-23) T herefore the a x ia l r a t i o a t P(R,8,4») is ; A.R. = [ 2. 2 ] COS T Sin T 1 * 2 s in 2t ^ 1sin T COS T LV:/ -T 1 s in 2t + 1^ ~2 (2-24) Let r be defined as r = E I (2-25) Equation (2-24) then becomes A.R. = (1 + r ^ ) COS 2t + (1 - r^) (1 + r ^ ) COS 2t - (1 - r ') ( )

75 57 But since 2 _ 1 cos 2 t = (1 + tan 2 t) T = ll + 4r^ cos^6 equation (2-26) reduces to ; A.R. = 1.. s/j + 2r^cos r cos *1^- \/? ~ (2-27) The + sign is chosen to follow the convention th a t the a x ia l r a tio is equal to or g re a te r than u n ity. The + sign w ill be p roper fo r a l l r when 6 f Ti/2. lyhen 6 = ir/2, th e + sign holds fo r r > 1 and the - sign holds fo r 0 <r< 1. Since (1 + r^) > ^ r^ + 2r^cos 26+1 fo r a l l ô, the q u a n tity in the b rack ets o f equation (2-27) is p o s itiv e d e f in ite and th e re fo re th e absolute value signs of equation (2-26)can be d isreg ard ed. A sp e c ia l case of in te r e s t is th a t fo r which the two orthogonal f ie ld components are in time phase quadrature (5 = 90 ). Equation (2-27) then reduces to I I f 1^ A.R. = (r) ± I tl (2-28) y\h\j This could have been seen from equation (2-22) where fo r Eq ^ E^ and 5 = i 90, the t i l t angle o f the p o la riz a tio n e llip s e is zero and th e re fo re the components je I and E [ coincide re s p e c tiv e ly with 6 ({)

76 58 th e semimajor and the sem i-m inor axes. Furtherm ore, when Eg = E^ and 6 = + 90, c ir c u la r p o la riz a tio n is obtained. To c a lc u la te the a x ia l r a t i o a t any p o in t in space fo r a given antenna, the r a t i o of the absolute magnitudes o f th e two orthogonal f i e l d components must be known along with the time phase angle between them. Equation (2-27) is then used fo r the com putation. F ig. 22 is a graph o f the a x ia l r a tio as a fu n ctio n of r fo r se v e ra l values o f 6. The curves are u sefu l in determ ining the e f f e c t o f the v a ria tio n of the time phase angle and the e f f e c t o f the r a tio r on the a x ia l r a t i o. By knowing th e fu n ctio n r ( G, ^) and 6 ( 0,(ji) and by using Fig. 22, the a x ia l r a tio v a ria tio n can be "mapped out over th e an g u lar range of i n t e r e s t. The graph in d ic a te s th a t fo r any value of 6, th e a x ia l r a tio is minimum fo r r = 1. This indeed can be shown rig o ro u sly by employing th e usu al m inim ization techniques of the d i f f e r e n t i a l c a lc u lu s. By s e ttin g r = 1 in equation (2-2 7 ), th e minimum value is found to be: A.R.. min ^ + 1 co t(6 /2 ) (2-29) The u t i l i t y o f equation (2-27) extends beyond th a t o f c a lc u l a tin g the a x ia l r a tio v a ria tio n of a p r a c tic a l antenna w ith w ell known far-zo n e orthogonal f ie ld components. Even when the source c u rre n t d is tr ib u tio n is incom pletely known, equation (2-27) can be used to analyze the e ff e c t of d iff e r e n t modes on the angular d e te r io r atio n o f th e a x ia l r a t i o. This can lead to a design procedure to improve th e a x ia l r a t i o v a ria tio n by reducing or e lim in a tin g those modes which c o n trib u te most to the lin e a rly p o la riz e d p a rt o f the

77 Axial Ratio Fig. 22. The a x ial r a tio as a function of r fo r sev eral values o f 6. en O

78 ra d ia tio n f i e l d ; o f co u rse, t h i s mode red u ctio n procedure must be commensurate w ith requirem ents on th e ra d ia tio n p a tte rn and th e in 6 0 p u t impedance. This w ill be i l l u s t r a t e d in th e next se c tio n fo r the case of th e h e lic a l beam antenna? Chu and Kouyoumjian [1960, 1962] have d iscu ssed the co n d itio n s which a source d is tr ib u tio n must s a ti s f y i f i t is to support a c i r c u la rly p o la riz e d f ie ld a t a l l angles fo r which th e f i e l d is nonv an ish in g. This necessary and s u f f ic ie n t co n d itio n is given by: J (x,y ) =! j KCx,y) (2-30) where J (x,y ) = volume e l e c t r i c c u rre n t d e n sity K(x,y) = volume m agnetic c u rre n t d e n sity = free space adm ittance Rumsey, Cheo, and Welch [1959] have shown th a t th is co n d itio n holds in the th e o r e tic a l model o f th e eq u ian g u lar p la n a r s p ir a l. Measurements by Dyson [1959a, 1959b] have in d ic a te d an a x ia l r a t i o o f le ss than 1.5 over an an g u lar range of * 70 (measured from the normal to the plane) o f the eq u ian g u lar p la n a r s p ir a l and less than 1.5 over an angular range of * 50 fo r th e co n ical s p i r a l. b) A pplication to an Axial Mode H e lica l Beam Antenna The a x ia l mode h e lix is a w ell known antenna w ith moderate d i r e c t i v i t y and w ith n e a rly c ir c u la r p o la riz a tio n on a x is. I t has been th e su b je c t of in te n siv e a n a ly tic a l and experim ental a n aly sis sin ce i t was f i r s t re p o rted by Kraus n e a rly two decades ago. Maclean and

79 Kouyoumjian [1959] have d iscu ssed the use of th e phase v e lo c ity r e s u ltin g from S e n sip e r's th e o r e tic a l so lu tio n f o r an i n f i n i t e h e lix [S ensiper, 1955] in the f i n i t e h e lix model to p re d ic t the bandwidth 61 o f th e h e lix. MacJean and F arvis [1962] have d iscu ssed th e phase v e lo c itie s r e s u ltin g from the s h e a th -h e lix approach t o th e a x ia l mode h e lix. Also Bagby [1948] has obtained a s o lu tio n fo r th e phase v e lo c ity along an in f i n i t e h e lix by so lv in g M axwell's equations in h e lic o id a l c y lin d ric a l.c o o r d in a te s. However no exact so lu tio n of the general wave equation has y et been obtained f o r a f i n i t e h e lix ta k in g in to co n sid eratio n th e s iz e and shape of the co u n terp o ise, w ire d iam eter, e tc. A review of S e n s ip e r's and Bagby's work serves as an in d ic a tio n of th e p ro h ib itiv e com plexity o f th is u n d ertak in g, although re c e n t developments in the use of the in te g ra l equation method with i t e r a t i v e techniques fo r a high speed d i g ita l computer in d ic a te th a t a so lu tio n fo r the f i n i t e h e lix may y et be p o ssib le [Andreason, 1966]. In order to c a lc u la te th e angular a x ia l r a tio v a r ia tio n of the h e lix, i t is necessary to know th e two orthogonal f i e l d components E and E where Band (pare measured as shown in Fig. 23. Kraus has c a lc u la te d E and E fo r a h e lix of an in te g ra l number 8 9 o f tu rn s by using the p rin c ip le of p a tte rn m u ltip lic a tio n and by cons id e rin g th e p a tte rn of any in d iv id u a l tu rn to be re p re se n ta b le by a square tu rn approxim ation [Kraus, 1950, pp ]. The array fa c to r is th a t o f N CO-lin e a r is o tr o p ic elem ents w ith the phasing between elem ents dependent on the phase v e lo c ity o f the wave propagating on th e h e lix. He has shown th a t i f the Hansen-Woodyard in creased

80 62 X Fig. 23. A h e lix w ith i t s m idpoint a t the o rig in of the co o rd in ate system.

81 63 d i r e c t i v i t y co n d itio n is used to c a lc u la te th e phase v e lo c ity, then the measured p a tte rn s and the c a lc u la te d p a tte rn s agree w e ll. In h eren t in th is a rra y method is th e assumption th a t the h e lix supports a uniform c u rre n t d is tr ib u tio n, Kom hauser [1951] has a lso c a lc u la te d th e f ie ld s o f a h e lix assuming a uniform cu rre n t d is tr ib u tio n by u sin g the v e c to r magn e tic p o te n tia l in te g r a l. Chu and Kouyoumjian [1960] have used K om hauser's f i e l d expressions and ex perim entally obtained phase V elo city values to c a lc u la te th e angular a x ia l r a tio v a ria tio n of two s ix tu rn h e lic e s o f p itc h angles 12 and 18 re s p e c tiv e ly ; they a lso made a x ia l r a tio measurements on a 12 six tu rn h e lix and an 18 s ix tu rn h e lix. These c a lc u la tio n s and measurements were made up to 30 o f f the axis of th e h e lix. T heir d ata show th a t th e c a lc u la te d value of th e a x ia l r a tio is always lower than the measured value with the discrepancy g e ttin g worse as th e aspect angle in c re a s e s. The disagreem ent is due to the assumption of a uniform c u rre n t. I t is of in te r e s t to re c a lc u la te th e f ie ld s of a h e lix using th e v e c to r m agnetic p o te n tia l in te g ra l method, but not assuming a uniform c u rre n t. Marsh [1951] has made ex ten siv e measurements of th e c u rre n t d is tr ib u tio n on an a x ia l mode h e lix and has been able to in te r p r e t the c u rre n t d is tr ib u tio n as being due to two modes when th e h e lix circum ference is approxim ately one wavelength. The f i r s t mode, T^, is th a t of a wave propagating along th e h e lic a l conductor a t the speed of lig h t and being ra p id ly a tte n u ate d as i t

82 leaves th e feed p o in t and as i t is r e f le c te d from th e open end of th e h e lix. The second mode, T^, is th e w ell known " h e lic a l 64 mode" and is th e one which K om hauser used in h is work. This mode propagates as a slow wave w ith a phase v e lo c ity c a lc u la b le from th e Hansen-Woodyard in creased d ir e c tiv ity co n d itio n and experiences no a tte n u a tio n as i t propagates along the h e lix. However, i t is a tte n u a te d on re f le c tio n from the end of th e h e lix. The T mode o is norm ally not considered in the c a lc u la tio n of th e f ie ld s of a h e lix because i t is im portant only n e a r th e ends o f the h e lix. However, th is mode can be q u ite im portant when c a lc u la tio n s of th e a x ia l r a t i o are made. In previous th e o r e tic a l treatm en ts of th e a x ia l mode h e lix, a th in h e lic a l wire in fre e space c arry in g a constant am plitude slow tra v e lin g wave (T^ mode) has been assumed; i f th e in creased d ir e c t i v i t y co n d itio n is presumed to hold over the a x ia l mode bandw idth, then the c a lc u la te d beamwidths agree w ell w ith measured beamwidths and the a x ia l r a tio on axis of th e h e lix is p re d ic te d to be independent of frequency over th e bandwidth. In order to account fo r th e observed angular dependence of the a x ia l r a tio o f a h e lix, th e c u rre n t d is tr ib u tio n on the h e lix must include the T^ mode and the asso c iate d e f f e c t of th e ground p lan e as w ell as th e T^ mode. Maclean [1963] has observed th a t the su rface c u rre n ts in th e ground plane of a h e lix are predom inantly c irc u m fe re n tia l and th a t when the ground p lane is too la rg e, i t can have a d e le te rio u s e f f e c t on even the main lobe of the antenna. This e f f e c t has also been observed independently by th e au th o r. I f the

83 ground p lane has a maximum dimension of th e o rd er o f one w avelength, then i t does not have an ap p reciab le e f f e c t on the main lobe shape 65 o f th e h e lix. However, i t can degrade th e angular p o la riz a tio n s t a b i l i t y o f the h e lix. The mode is a sso c ia te d w ith d is c o n tin u itie s on the h e lix, both at th e feed end and a t the open end. At th e feed end, the ev o lu tio n of the tran sm issio n lin e conduction cu rren t to th e displacem ent c u rre n t between the ground plane and the f i r s t one o r two tu rn s r e s u lts in a nonuniform c u rre n t d is tr ib u tio n. The r e f le c tio n s from th e open end of th e h e lix are experim en tally observed to be sm all f o r h e lic e s o f more than about s ix tu rn s and w ill be n e g lec te d in the follow ing a n a ly s is. The ground plane is also n e g le c te d, due to th e m athem atical com plexity of p re d ic tin g an a cc u ra te su rfa ce c u rre n t d is tr ib u tio n ; n o n e th e le ss, u n t i l th e ground p lane is considered a n a ly tic a lly, only lim ited agreement can be expected between measured and c a lc u la te d a x ia l r a tio s as a fu n c tio n o f angle. However, th e purpose here is sim ply to show n u m erically t h a t any contam ination of th e T^ mode on the h e lix tends to degrade the an g u lar p o la riz a tio n s t a b i l i t y of th e antenna. In th e sense of m ain tain in g c ir c u la r p o la riz a tio n p u rity over th e main beam, th e T^ mode can be considered as sp u rio u s, although i t is in e x tric a b ly lin k ed w ith th e ground plane and appears to always be p re se n t in experim ental models. The c u rre n t d is tr ib u tio n along the h e lix conductor is w ritte n as m=0 m (2-31)

84 6 6 where (m = 0,1) = complex mode am plitude co n stan ts a = a tte n u a tio n c o n sta n t of T mode o o Oj = a tte n u a tio n co n stan t o f mode = 0 6 = 2 tt/x o Bj = 2ir/(p^A) p = r e la tiv e phase v e lo c ity of T mode [Kraus, 1950, Chapter 7] -1 = [sin a + (cos a/c )(2N+1/2N)] i f in creased d ir e c t- ^ i v ity co n d itio n holds a = h e lix p itc h angle = circum ference of h e lix r e la tiv e to one wavelength N = to t a l number o f tu rn s (not n e c e s s a rily an in te g e r) s = d ista n c e measured along h e lix conductor L = t o t a l len g th o f h e lix In equation (2-31) re fle c tio n s from the end o f the h e lix are assumed to be n e g lig ib le, i. e. only outgoing waves are considered. By u sin g th e v e c to r magnetic p o te n tia l in te g r a l, by assuming the c u rre n t flow to be e n tir e ly in the d ire c tio n o f the h e lic a l conductor and by making th e usual far-zo n e approxim ations, the two orthogonal e l e c t r i c f ie ld components can be w ritte n as (see Appendix A) r ikk L/2 T r si E,(8,* ) = -jwye (Xcosa) / I( s ) cos(* _ s c o sa /r )e ds 4ttR -L/2 (2-32) r -ikr jk y (s) E (6,tp) = -jmye [-s in a sin 0 / I(s ) e ds 4ttR -L/2 L/2 jky(s) T cosa cose / I(s ) sin((j) - s c o s a /r )e ds] (2-33) -L/2

85 w ith Ÿ(s) = s sin a cose + r sin e cos(* - s c o s a /r ) o where th e s u p e rsc rip t Z denotes a left-h an d ed h e lix and the su p ers c r ip t r denotes a right-handed h e lix, IVhen a sig n choice is in d i c a te d, th e upper sign is taken fo r a le ft-h a n d e d h e lix and the lower 67 sign fo r a rig h t handed h e lix. Also, y = k = p erm eab ility 2ir/X (R,8,40 = f a r f i e l d p o in t (see Fig. 23) r c = rad iu s o f h e lix Here the h e lix is assumed to be s itu a te d w ith i t s m idpoint at the o rig in of the co o rd in ate system as shown in Fig. 23. Next equation (2-31) is s u b s titu te d in to equations (2-32) and (2-33). Use is also made of the fa c t th a t and th a t gjk cos X = ^ ( j ) j (K )e^"^ (2-34) n=-oo = ( - 1 ) \ ( K ) (2-35) so th a t ik cos X = JL(K) + " n jnx -jn x I (j) J (K) [e + e ] (2-36) n=l The in te g ra ls o f equations (2-32) and (2-33) are thus reduced to in te g r a ls o f exponential fu n ctio n s with complex arguments. The r e s u lt of the in te g ra tio n is given below:

86 6 8 I 9 - Jo sincv +l)nir n».. gj* + sincv t1)nit m ^-](j> ' mi 1 ' m Î 1 r n + I (j) J (C sine) n=l n X sin (v m + n L '»m + n )Ntt sin (v ^ m + n + l)nir V m + n + 1 s in (u m + n + 1)Ntt V m + n + 1) a - j( - n - l) 4» (-n+ 1)^ - j ( n - 1) ({) 5 ln (v ^ + n + 1)Ntt "m ' equation (2-37) I EgCe,<i>) I E -ta n a sino m=0 ^ + I ( j ) " j (C sin e) " X n=l J (C sin0) s i n ( v N ir) o ' X m m sin(v i n)nn. sin(v T n)nn., m in* m + _-jn* = =------e % - " + COS0 j J^(C ^sin0) sin (v 1 1)Ntt i r n - e ' * m sin (v I 1)Ntt ' m + - 1* ' - + i ( j ) " * \ ( C 5i n 0) n=l " A sin (v t n 2 l)nir m - -i(-n-l)* V ^ n + 1 m s in (v + n ; l)nn. m -jc -n+ D * ^ V : n - 1 m * s ln (v ^ ; n ; 1)N s in (v m + n Î l)nir equation (2-38)

87 6 9 where E = jioh Ce'-'*'-' I e-( m + m _Q m 8 tt2r (2-39) V = C m X tan a cos0-1 m cosa 2 it cosa (2-40) I t w ill be noted th a t the f ie ld equations fo r a left-h an d ed and a rig h t-h an d ed h e lix are very s im ila r. The f ie ld s are sep arab le functio n s o f 0 and ; i f th e f ie ld s o f a rig h t-h an d ed h e lix are w ritte n as : E^Ce,<j>) = I I f Ce) g ($) * m n m,n n (2-41) Eg(8,*) = I I R (6) X W (2-42) then i t follow s from equations (2-37) and (2-38) th a t th e f ie ld equatio n s f o r th e corresponding left-h an d ed h e lix can be w ritte n as E (8,4) = <P I l f ( 8) g (4,) m n m,n n (2-43) Eg(8,4i) = ' I I h (0) %. C4) 8 m n n mi, n n (2-44) where * denotes th e complex conjugate. The fa c t th a t th e assumed cu rre n t d is tr ib u tio n uses a complex propagation co n stan t gives r is e to an i n f i n i t e s e r ie s of sin X terms X where X is complex. For num erical e v a lu a tio n, th e sin e o f a complex number can be expanded in to another complex number by using c ir c u la r and h y p erb o lic sin e s and co sin es. However, i t is obvious th a t the

88 mode complex propagation c o n sta n t makes a num erical e v alu a tio n o f th e f ie ld s co n sid erab ly more d i f f i c u l t. I f th e mode is negle c te d, then the f i e l d eq uations d escrib ed by Kom hauser fo r a r ig h t- handed h e lix are obtained as a s p e c ia l case. 7 0 Furtherm ore, i f th e mode is n e g le c te d, the components and are in time phase quadrature on axis [G = 0 ) and th e a x ia l r a t i o i s found as ; A.R. ( 6= 0 ) = C, ta n a - 1 (2-45) P j cosa I f th e phase v e lo c ity is assumed to e x a c tly s a t i s f y the Hansen-Woodyard in c re ase d d ir e c tiv ity c o n d itio n, then so th a t f \ s in a + cosa 2N+1) C, 2N L ^ X \ I L -1 (2-46) A.R.( 6= 0") = 2N+1 2N (2-47) which i s th e w ell known form ula f o r th e a x ia l r a t i o o f a h e lix [Kraus, 1950, p. 206]. Although equation (2-47) p re d ic ts th a t th e a x ia l r a tio on axis w ill be frequency independent fo r (. 8 <C^<1.2), i t is known experim en tally th a t th e a x ia l r a tio shows a sm all frequency dependence; th is is p rim a rily due to th e ground p lane e f f e c t and the mode c u r re n t on the h e lix. Numerical e v alu a tio n of th e e l e c t r i c f ie ld s (equations 2-37 and 2-38) was made p o ssib le by a program w ritte n fo r use on the IBM 7094 computer. The program has made i t p o ssib le fo r the f i r s t time

89 to make d e ta ile d num erical c a lc u la tio n s of th e f ie ld s o f a h e lix c a rry in g any c u rre n t d is tr ib u tio n re p re se n ta b le as th e su p erp o s itio n of tra v e lin g waves. Both T and T waves were used in 0 1 th e c a lc u la tio n s, although th e equations (2-37) and (2-38) e v a l u ated by the program are not r e s t r i c t e d to such an a x ia l mode 7 1 re p re s e n ta tio n. The m-summation can be expanded to include any a d d itio n a l modes req u ired to re p re se n t a c u rre n t d is tr ib u tio n. I t was found th a t a l l in f i n i t e s e r ie s in th e expressions fo r o rd in ary h e lix param eters could be term inated a f te r n = 5 with an accuracy b e t t e r than 0. 1% re g a rd le ss o f angle e The s p e c ia l case of a s ix tu rn 12" rig h t-h an d ed h e lix (C^ = 1. 00) is o f co n sid erab le i n t e r e s t, sin ce a d e ta ile d analys is o f the c u rre n t d is tr ib u tio n fo r th is h e lix has been made by Marsh [1951]. M arsh's technique was to measure the c u rre n t d is tr ib u tio n on a h e lix over a larg e square ground p la n e, and then to fin d the m athem atical form of th e c u rre n t d is tr ib u tio n by a curve f i t t i n g procedure. However, i t was found th a t the use of th e mode co n stan ts re s u ltin g from M arsh's d a ta in equations (2-37) and (2-38) d id not r e s u lt in clo se agreement between c a lc u la te d and measured e l e c t r i c f ie ld p a tte r n s. This is undoubted ly due to th e e f f e c t o f th e ground p la n e, which was ignored in th e c a lc u la tio n, and which must be c lo se ly a sso c ia te d w ith th e mode cu rre n t on the f i r s t one or two tu rn s o f the h e lix. F ig, 24 shows th e e f f e c t of th e mode am plitude co n stan ts and on the angular dependence of the a x ia l r a tio fo r a s ix tu rn 12" h e lix. The a tte n u a tio n co n stan t o f the T mode was o

90 A xial R atio N = 6 C = l.oox a = H alf Power Point y. 1 / I / I = 3.3 / / / / X r X / -< / / / / v ''..."T""... I / I, = 1.0 o 1 _ X ^ [ / I, = I H alf Power Point I 1 30' 40' Angle from Helix Axis Fig. 24. The angular dependence o f the ax ial ra tio on the mode amplitude constants I and I. The dashed curve is from the experim ental work o f KSuyoumjian and Chu; perm ission to use th is data was kindly granted by Dr, R. G. Kouyoumjian.

91 73 assumed to be 1 neper/a, in agreement w ith M arsh's r e s u lts. The to p curve is c alc u la te d by using M arsh's mode c o n sta n ts. The bottom curve is th a t c a lc u la te d by n e g le c tin g th e T^ mode ent i r e l y ; i f = I j i s assumed, then th e c a lc u la te d power p a tte rn s agree w ell w ith measured power p a tte r n s, and the I^ /I^ = 1.0 curve is o b tained as shown. The dashed curve i s from d ata measured by Kouyoumjian and Chu [1960] and shows b e t t e r agreement w ith th e I q/ I j = 1.0 curve than w ith th e o th e r two. The c a lc u la te d angular dependence of the a x ia l r a tio on th e number of tu rn s is shown in Fig. 25. I t can re a d ily be seen th a t h e lic e s of a larg e number o f tu rn s n o t only have a lower on- ax is a x ia l r a t i o than those of a few tu rn s, b u t tend to be more p o la riz a tio n s ta b le as w e ll. The black dots on th e curves of F ig s, 24 and 25 in d ic a te th e h a lf power p o in ts. Fig, 26 shows the c a lc u la te d angular dependence o f the a x ia l r a t i o on the p itc h angle fo r a ten tu rn h e lix. Changing th e p itc h angle has only a sm all e f f e c t, b u t a h e lix of sm aller p itc h angle is shown to be s lig h tly more p o la riz a tio n s ta b le. The e f f e c t of the p itc h angle is more pronounced f o r h e lic e s o f fewer tu rn s than te n. The a n a ly sis also p re d ic ts th a t the h e lix w ill be lin e a rly p o la riz e d a t roughly 90 from th e a x is. This has been observed ex p erim en tally by the author on a l l models te s te d with a f l a t ground p lan e. Sander and Cheng [1958] have noted th a t c o n sid e ra tio n of th e Tj^ mode alone did not lead to good agreement between c alc u la te d

92 2.0 A xial R atio 1.6 l.oox = H alf Power Point /r 1.0 O " Angle from Helix Axis Fig. 25. The angular dependence of the a x ial ra tio on the number of tu rn s.

93 2.0 Axial Ratio 1.6 l.oox Angle from H elix Axis Fig. 26. The angular dependence of the ax ial r a tio on the p itc h angle.

94 and exp erim en tally derived p o s itio n s o f the phase c e n te r of the 76 a x ia l mode h e lix. However, when the phase c e n te r of a ty p ic a l h e lix i s c a lc u la te d from th e e l e c t r i c f i e l d expressions of eq u atio n s (2-37) and (2-38) with th e a p p ro p ria te mode c o n sta n ts, much b e t t e r agreement w ith the experim ental p o s itio n s o f the phase c e n te r i s obtain ed. The fa c t th a t the phase c e n te r lo c atio n depends, among o th e r p aram eters, on th e siz e of the ground plane em phasizes th e importance o f in clu d in g the ground plane and th e a sso c ia te d mode in th e a n a ly s is. 3. M odifications of th e H elix to Improve Phase C enter and P o la riz a tio n S ta b ilit y Since th e T mode has been shown to cause a d e te rio ra tio n o o f the c ir c u la r p o la riz a tio n s t a b i l i t y n e ar th e axis of the h e lix, i t is o f in te r e s t to see i f the T o mode can be reduced r e la tiv e t o the T mode. I t is known th a t the T mode i s a sso c ia te d p r i - ^ o m arily w ith th e counterpoise o f the h e lix and the f i r s t few tu rn s as a wave which e x p o n en tially decays away from th e feed p o in t. This mode is ap p aren tly a r e s u lt of the abrupt d is c o n tin u ity between th e feed lin e (which is u su a lly co ax ial) and th e c o u n terp o ise -h e lix s tr u c tu r e. I f th is is t r u e, then a co u n terp o ise which makes a more gradual tr a n s itio n between th e feed lin e and the h e lix p ro p er would tend to reduce the magnitude of the T^ mode and thereby improve the p o la riz a tio n s t a b i l i t y. indeed shoim t h i s to be the case. Experim ental r e s u lts have C onsider, f o r example, the th re e h e lic e s shown in Fig. 27 w ith d if f e r e n t counterpoise s tr u c tu r e s. By f a r the most fa m ilia r counterpoise is the ground

95 7 7 lïm r r m Ca),40X,75X T.40X nrmrrnrrr Cb) (c) Fig. 27. (a) A h e lix w ith ground plan e. (b) A h e lix w ith tru n c a te d co n ical co u n terp o ise. (c) A h e lix w ith tap ered ends and with coaxial lin e fla re d in to conical counterpoise,

96 p la n e, f i r s t used by Kraus [1950, Chapter 7 ], and shown in Fig. 2 7 (a ). Almost a l l of th e experim ental d a ta a v a ila b le fo r the h e lix has been taken from models with th is type of co u n terp o ise. 7 8 The diam eter D is u su a lly g re a te r than 1/2 w avelength. For th is type of c o u n terp o ise, a x ia l r a tio s as high as 2 :1 have been measured a t only 30 o f f axis [Chu and Kouyoumjian, 1962]. In F ig. 27(b) i s shown a tru n c a te d conical co u n terp o ise which was developed by Svennerus [1958]. Measurements made by Svennerus and independently by the author have in d ic a te d th a t th is co u n terp o ise can make a s u b s ta n tia l improvement in th e angular a x ia l r a t i o v a r ia tio n over th a t of a h e lix with an o rdinary ground p la n e. The improvement obtained is due to th e red u ctio n of th e T^ mode which in tu rn is a r e s u lt of th e more gradual tr a n s itio n between th e feed lin e and the h e lix p ro p er. The impedance of th i s antenna is s u b s ta n tia lly unchanged over th a t obtained w ith th e ground p lan e. Fig. 27(c) shows a h e lix w ith beginning and ending se c tio n s of ta p ered diam eter and w ith the o u te r conductor of th e co ax ial lin e f la re d in to a sm all co n ical counterpoise lau n ch er. This type of c o n stru ctio n was shown by Angelakos and Kajfez [1967] to r e s u lt in a lower on-axis a x ia l r a tio and a w ider a x ia l r a t i o bandwidth.

97 CHAPTER I I I CONCENTRIC RING ARRAY DESIGN CONSIDERATIONS In C hapter I I, a sim ple lin e a r phase-scanning array o f 2N+1 elem ents was d iscu ssed. I t was p o in ted o u t th a t by prope r ly c o n tro llin g the phase of each elem ent, th e array could be made to scan in angle. By the same tech n iq u e, a p la n a r array can be made to scan, and when th e array geometry is p ro p erly chosen, a sin g le main beam can be formed a t a d e sired angle and sid e lobes may be h eld to low le v e ls. Roughly speaking, the beamwidth o f an array is dependent on the maximum a rra y dim ension, w hile the sid elo b e lev el is c o n tro lle d by th e number o f elem ents w ith in the array p e rim e te r, and th e am plitude and p o s itio n o f the elem ents. T h erefo re, when the sy n th esis problem is to design an a rra y having a s p e c ifie d beamwidth and sidelobe le v e l, th e maximum a rra y dimension is chosen in accordance w ith the beamwidth requirem ent, while the sid elo b e lev el w ill be determ ined p rim arily by th e number o f elem ents. These ru le s o f thumb hold in phase-scanning array s as w ell as p h ase-fix ed a rra y s. The ex act geom etrical d isp o sitio n o f the array elem ents and t h e i r am plitude can be used to co n tro l the p o s itio n s o f the sid e lobes and the ra te o f sid elo b e f a l l - o f f. 7 9

98 8 0 A, Array Theory f o r C oncentric Ring Arrays o f Is o tro p ic R adiators C onsider a co n cen tric rin g array o f is o tro p ic ra d ia to rs on a p la n a r su rface such as shown in Fig, 2 8 (a ). The rin g s are lo c ated a t r a d ii ( r. r_, r,,,., r,,..) as shown and the m is o tr o p ic elem ent a t an angle in rin g m can be denoted as the mn^^ elem ent. In ord er to c a lc u la te the a rra y f a c to r fo r th i s c o n ce n tric rin g a rra y, the d istan ce from the mn^^ elem ent to the f i e l d p o in t P(R,G,*) must be known. Fig. 28(b) shows the q u a n titie s necessary to c a lc u la te th is d ista n c e. These q u a n titie s are defined as fo llo w s: R = d ista n c e from o rig in to f ie ld p o in t P. 6 = c o la titu d e angle o f f ie ld p o in t P. ((( = azim uthal angle o f f ie ld p o in t P. r^ = rad iu s o f m^^ rin g. ^ mn ~ angle th a t n^^ elem ent in m rin g makes w ith x -a x is. = d ista n c e from source p o in t S(elem ent mn) to f ie ld p o in t P. Q = p o in t defined by th e in te rs e c tio n of th e x-y plane w ith the plane (ji = co n stan t such th a t angle SQO is 90. b = d ista n ce from 0 to source p o in t S. mn ' ^ a = d ista n c e from Q to o rig in 0. mn d mn = d ista n c e from Q to f ie ld p ^ o in t P. The follow ing r e la tio n s can be deduced from Fig. 28(b): cosc* - (3-1) sin((j) - * ) = b / r (3-2) mn mn m

99 mn mn mn mn (a) (b) Fig, 28. (a) A concentric rin g array of iso tro p ic ra d ia to rs. (b) Array geometry fo r mn^ element in concentric rin g array. 00

100 (d ) = (R sin e - a ) + (R coso ) ' mn mn"^ = R^ sin ^ e - 2Ra s i n G + a^ + R^ cos^q ran ran (3-3) = - 2R s ln e [ r^ COSC+ - * [ r ^ c n s ^ t - * ^ ) ] (3-4) T h e re fo re, p =. / r sin^(< >-(t) ) + R^ - 2 R (sin 6 )r c o s( 6-6 ) + r^ cos^(d)-d) ) ran U rn ^mn m ran m ^ ^mn^ (3-5) o r ran r - 2Rr sing cosf* - 6 ) (3-6) m ra ^ran By expanding equation (3-6) in to a binom ial s e r ie s and re ta in in g only the f i r s t ord er terras, the far-zo n e approxim ation to (3-6) can be w ritte n as; p = R - r sin e cos($ - (j) ) (3-7) mn HI ran The d iffe re n c e in phase between a wave th a t p ro g resses from the o rig in 0 to th e f ie ld p o in t P and one th a t p ro g resses from th e source p o in t S to th e f ie ld p o in t P is %mn = - "m = k r^ sin e cos(<^ - *^^) (3-8)

101 83 I t i s now assumed th a t the mn^ elem ent is fed with am plitude A and phase il» so th a t the e x c ita tio n of the mn^^ elem ent can mn mn i ^ be w ritte n as A e^ mn. T herefore the far-zo n e e l e c t r i c f ie ld mn o f th e array can be w ritte n as: M NCm) jk r_ sin 0 cos(* - * ) E(0»< >)=y I A e m n e mn (3-9) m=l n=l where M denotes the number of rin g s and N(m) denotes the t o t a l number o f elem ents in the m^^ rin g. I t is im p lic it th a t th ere can be d if f e r e n t numbers o f elem ents in each rin g. Suppose now th a t i t is d esired to have the maximum value o f E^(0,*) a t the angular p o in t Then i t follow s from the P rin c ip le of S ta tio n a ry Phase [W alter, 1965, pp ] th a t ^ = -k r sin 0 co s(é - * ) (3-10) mn m s mn Therefore the array f a c to r fo r th is phased a rra y can be w ritte n as : M NJm) jk r^ [s in e co s(,^ -* ^ ) - sine^ =0 5 ( 85-4^^) A _ e m=l n=l ^ (3-11) E (8,*) = 1 1 Equation (3-10) is used to c a lc u la te th e phase s e ttin g of each elem ent fo r a beam maximum a t the angular p o s itio n (8g**g) and equation (3-11) can then be used to compute th e array f a c to r fo r th a t case. R eferring to Fig. 28, the array a n a ly sis can be sim p lifie d f o r c e r ta in sp e c ia l cases. For example, i f i t is d e sired to know th e p a tte rn in th e 4 = 0 p la n e, then th e elem ents in the

102 84 rin g s can be p ro je c te d onto th e x -a x is. By seg reg atin g p ro je c tio n s which are e q u id is ta n t on the x -ax is in to groups, each group having uniform ly spaced p ro je c tio n s can be considered as a uniform lin e a r a rra y. The t o t a l array f a c to r i s then the su p erp o sitio n of th ese lin e a r array fa c to rs. The d ir e c tiv ity o f th e c o n cen tric rin g array can be shown to be dependent on the scan angles (6,(j> ). s s th e scan angles (8,(p ) is c alc u la te d by s s The d ir e c tiv ity a t ir 2 tt * / / E (8,40 E (8,4) sin 8 d8 d4> 0 0 ^ a (3-12) S u b s titu tin g equation (3-11) in to equation (3-1 2 ), D = M N(m) 2 4ir I I A m=l n=l IT 2 IT M Nfm) j* 1 / / 0 0 I X r M Nfp) -1 I I A e p q p=l q=l sin 0 d8 d4> (3-13) where i t is im p lic it th a t a l l A are r e a l, and th a t 4> is defined mn mn as $ = k r [sin8 co s(6-4 ) - sin8 cos(4> -4> )] (3-14) mn m*- s mn"^ When th e denominator o f equation (3-13) is expanded, i t is found th a t th e dependence on the angles (8, 4^ ) cannot, in g e n eral, be removed. Thus, th e d ir e c tiv ity is dependent on the scan an g les, as i s the d ir e c tiv ity fo r a p la n a r re c ta n g u la r array [ E l l i o t t, 1966]

103 8 5 However, the a ctu a l c a lc u la tio n o f th e d i r e c t i v i t y fo r a conc e n tr ic rin g array i s co nsiderably more complex than f o r a r e c t an g u lar a rra y ; th is is p rim a rily due to th e fa c t th a t element p o s itio n s in c o n cen tric rin g array s are s p e c ifie d by trig o n o m e tric fu n c tio n s, whereas the elem ent p o s itio n s in re c ta n g u la r array s are s p e c ifie d by sim ple lin e a r a lg e b ra ic a d d itiv e fu n c tio n s. Equation (3-11) can be re w ritte n as [Das, 1966]: M.pN(m) -pn (m) 6 E (e,(f>) = 1 1 N(m) (j) e (3-15) m-1 p=-«-pn(m) m where cos 6 = (1 /a)[sin G cos* - sine^ cos*^] (3-16) * = 2 n/n(m) (This im plies th a t every rin g has an (3-17) elem ent on the x -a x is.) I 2 2~ a = w (s in e cos* - sing^ cos*^) (sing sin * - sing^ sin*^) (3-18) N(m) = t o t a l number o f elem ents in the m rin g of rad iu s r^ This way o f w ritin g the array f a c to r can be u sefu l when sy n th e siz ing an array from beamwidth and sid elo b e le v e l requirem ents. Das [1966] has used such a sy n th esis technique to design a 10-ring array having 759 t o t a l elem ents w ith a 2 beam and no sid elo b e above -20 db, 1, Uniform Amplitude and Phase For th is c ase, a l l A = 1 and the beam w ill be broadside mn to th e plane o f the a rra y, i. e. G^ = 0". From equation (3-11) E,CO.«. Ï - W C3-19) m=l n=l

104 86 But ^jk cos X ^ ~ ( j) P j (K)eiPX (3-20) p=_oo P or ^jk cos X ^ + 2 Î ( j) P j (K) cos px (3-21) p=l P Therefore equation (3-19) can be w ritte n as M N(m)» p E (6,4») - I J {J (k r sin0) + 2 % (j) J (k r sin e) cos p(< >-é )} m=l n=l o m. p^j p m mn S everal im portant deductions can be made from equation (3-2 2 ), F i r s t, i t is n o tic e d th a t fo r angles n e ar the 0= 0 a x is, the dominant term in the braces is J (k r s in 0 ). o m A ll the terms in the p -s e r ie s involving non-zero order Bessel fu n ctio n s are sm all n ear the o rig in, but ju s t how sm all depends on th e rin g ra d iu s. Second, the azimuth dependence becomes more pronounced as the angle 0 in c re a se s; th is follow s in tu i t i v e l y from the fa c t th a t th e re is a w ell-d efin ed beam maximum, but the exact dependence on th e azim uthal co o rd in ate is shown to be co sin u so id a l fo r each term of th e s e r ie s. (3-22) I t is of in te r e s t to study the case of one rin g. Equation (3-22) becomes N E (0,4») = NJ (k r sin0) + 2 (j) J (k r sin0) % cos p (* -* ) ^ ^ p=l P=' P 1 n=l " (3-2 3 ) C onsider, fo r example, the case o f an a rra y having one rin g of

105 elem ents, with the rin g radius equal to 3.74 w avelengths. The elem ents are eq u ally spaced in angle so th a t = n(1 5 ). The p a tte rn in the *= 0 plane can be examined fo r purposes of i l l u s t r a t i o n. Equation (3-23) becomes E (8,0 ) = 24J (23.5sin0) * 2 oo 24 \ (j)p j (2 3.5 sin 0 ) \ cos(pnl5 ) p=l n=l (3-24) But \ cos(nl5 ) = ^ cos(n30 ) = \ cos(n45 ) = 0 n=l n=l n=l T herefore sin ce the Bessel fu n ctio n term s o f o rd e r h ig h er than 3 are sm all, equation (3-24) can be approxim ately w ritte n as E (6,0 ) = 24 J (23.5sin6) (fo r 23.5 sin 0 < 3) (3-25) a o T herefore the HPBW can be estim ated by s e ttin g J^(23.5sin0j^) = This re q u ire s th a t 23.5sin0^ = 1.12 (3-26) T herefore or -1 8^ = Sin (1.12/23.5) = 2.74 HPBW = 5.5 S im ilar procedures can be used w ith o th e r sin g le and m u ltip le rin g array s f o r th e estim atio n of the beamwidth in th e case of uniform am plitude and phase.

106 88 2. Uniform Amplitude and Nonuniform Phase When a l l elem ents are e x c ite d w ith uniform am plitude, but with a phase d is tr ib u tio n such th a t the main beam p o in ts a t the angular p o s itio n then the array f a c to r can be w ritte n from equation (3-11) : E ( e, «= f ' V ^ m=l n=l (3-28) Suppose th a t i t i s d e sired to fin d th e beamwidth in th e <^= 0 plane when the phase is s e t fo r a beam maximum in th a t plane (4»^ = 0 ). Then equation (3-28) becomes E (6.0 -, =! T ( 3. 2g) ^ m=l n=l which can be w ritte n as [see equation (3-2 1 )]: M Nfm) E (6,0 ) = y ) { J [k r (sin e - s i n 0 )] m=l n=l s " P + 2 y ( j) J [kr ( s in 0 - s in e )]co s p(j> (3-30) «1 p=l ^ P m s *^^mn C onsidering again the s p e c if ic case of a 24-elem ent sin g le rin g array with rad iu s 3.74X, equation (3-30) can be approxim ately w ritte n as E (0,0 ) = 24J [2 3.5 (sin 0 - sin0 )] (3-31) The half-pow er angle 0^ can be found from HPBW E (e^+, 0 ) = 24(0.707) = 24 [2 3.5 (sin (6 +HPBW) - sin e )] a s ^ s ^ s (3-32)

107 89 T h erefo re, or o r 23.5[sinC 0 +HPBW ) - sin6 ] = sin (0 + HPBW ) - sin 0» s 2 s HPBW = 2 [s in " ^( sin0^) - 0^] (3-33) This shows th a t the beamwidth is a fu n ctio n o f scan angle which is to be expected sin ce th e p ro je c te d ap ertu re changes. At a scan angle of 60*, th e p ro je c te d ap ertu re is reduced by a f a c to r of 2 and so th e HPBW should be doubled. F ig. 29 is a graph o f th e c a lc u la te d beamwidth versus scan angle fo r th e 24 elem ent sin g le rin g array w ith ra d iu s = 3.74X. The v a r ia tio n fo r a rad iu s o f one wavelength and ten wavelengths is a lso shown. I t can be seen th a t the la rg e r rad iu s rin g produces le ss v a ria tio n of beamwidth with scan angle than does th e sm aller rad iu s a rra y. Also i t can be seen th a t the r = 3.74X and 10X curves confirm th e p re d ic tio n made p re v io u sly th a t at an angle o f 60*, the HPBW should be doubled. The c a lc u la tio n of the array f a c to r f o r a number of rin g s with uniform am plitude elem ents eq u ally spaced in each rin g can be made by using equation (3-2 8 ). Appendix B d iscu sses a number o f s p e c if ic array geom etries and the a rra y f a c to r f o r sev eral s p e c if ic cases.

108 30 HPBW 20 Fig scan angle 0 HPBW vs. scan angle fo r a 24 element sin g le rin g array of is o tro p ic ra d ia to rs. 60 to o

109 91 B. D iscussion of C oncentric Ring Arrays When an a c tu a l c o n ce n tric rin g array i s designed, the p a tte rn f a c to r o f th e in d iv id u a l elem ent must be considered as w ell as the array fa c to r. By using th e p rin c ip le of p a tte rn m u ltip lic a tio n, th e to t a l e l e c t r i c f ie ld can be computed as E(0,*) = E (6,*) E (8,40 (3-34) where 2^(8,*) is th e elem ent f a c to r of the elem ent ( a l l elem ents are assumed to be id e n tic a l) and Eg^6,4) i s th e c o n ce n tric rin g array f a c to r. The choice of th e array geometry to be used may depend on which elem ent is to be used. Elements which have low mutual coupling can be used in a c lo s e r spaced arranagement than can elem ents w ith high mutual coupling. The p h y sical s iz e of the elem ent also imposes a lim ita tio n ; f o r example, an array of co n ica l horns would have i t s minimum spacing d ic ta te d by th e horn mouth diam eter which could be sev e ra l w avelengths. The array geometry must a lso be chosen w ith th e d e sired beamwidth and sid elo b e le v e l in mind. For larg e array s which use only m oderately d ire c tiv e elem ents, the beamwidth w ill be p rim a rily determ ined by th e array f a c to r. However, th e fa c t th a t the elem ent is m oderately d ire c tiv e may be used to reduce both sid elo b es and g ra tin g lobes o f the t o t a l f i e l d p a tte r n. Fig. 30 shows th e array f a c to r, th e elem ent f a c to r, and the re s u lta n t f ie ld f o r a ty p ic a l case. The a rra y f a c to r i s seen to have fo u r main lobes; however, when the elem ents are made d ir e c tiv e, the th re e un d esired lobes are g re a tly reduced.

110 9 2 E (0) Array F acto r Element Factor E(0) R esu ltan t F ield Fig. 30. I l l u s t r a t i n g the use o f d ire c tiv e elem ents to reduce sid elo b es and g ra tin g lobes.

111 93 G enerally, when wide elem ent spacing i s used, g ra tin g lobes can be expected to appear. In re c ta n g u la r a rra y s, g r a t ing lobes can appear f o r elem ent spacings g re a te r than about one w avelength, when th e elem ent is is o tro p ic. The same type of behavior might be expected o f c o n ce n tric rin g a rra y s. However, no c o n sis te n t g ra tin g lobe theoiy has been worked out fo r conc e n tr ic rin g a rra y s, and much le ss is knoivn about g ra tin g lobes o f n o n -rec ta n g u lar array s than is know fo r re c ta n g u la r a rra y s. Sharp [1961] has worked out a g ra tin g theory f o r a tr ia n g u la r array o f c ir c u la r a p e rtu re s, but h is r e s u lts are not in general a p p lic ab le to the co n cen tric rin g c o n fig u ra tio n. As a phased array is scanned o ff b ro ad sid e, g ra tin g lobes can appear which may not have been v is ib le when th e array was phased fo r a broadside beam. These lobes are always p re s e n t, b u t they may be nonpropagating. In th is c a se, they are sa id to be in the " in v is ib le " region [Rhodes, 1964] and re p re se n t re a c tiv e sto re d energy. As th e array is scanned, th ese in v is ib le lobes can move in to the v is ib le or p ropagating region. Since i t i s d e sira b le to have only one main beam, th e array should be designed so th a t no propagating g ra tin g lobes appear over the d e sired angular scan range. This means, in g e n e ra l, th a t element spacings o f the o rd er o f one wavelength o r le ss should be used. When the array elem ent has s u f f ic ie n t d i r e c t i v i t y, w ider elem ent spacings than one wavelength can be used sin ce the elem ent fa c to r w ill reduce th e array g ra tin g lobes as shown in Fig. 30. However, i f th e o rie n ta tio n of th e array elem ent is fix e d, the angle of

112 94 scan w ill be lim ited to approxim ately th e HPBW o f the elem ent f a c to r. Therefore i t may be d e sira b le to use an elem ent o f only m oderate d ir e c tiv ity and then design th e array geometry such th a t no g ra tin g lobes w ill appear over the d esired scan range. C. C oncentric Ring Arrays of H e lica l Beam Antennas In Chapter I I, i t was shown th a t a h e lix with a p ro p erly designed counterpoise can be used as a m oderately d ire c tiv e a r ray elem ent. This sec tio n d iscu sses th e use of a h e lix w ith a sm all c o n ica l counterpoise [Svennerus, 1958] as an elem ent of a c o n ce n tric rin g arra y. The HPBW of a h e lix is in v e rse ly p ro p o rtio n a l to the square ro o t of the number o f tu rn s, when the diam eter, frequency, and p itc h angle are h eld c o n sta n t. For a h e lix of only 3 tu r n s, the beamwidth may be around 70 and fo r a 40 tu rn h e lix th e beamwidth w ill be approxim ately 18 when the p itc h angle is about 13. U su ally, a h e lix o f fiv e t o ten tu rn s is used. When th e number o f tu rn s is le ss than fiv e, th e a x ial r a t i o can be high ( p a r t i c u la rly o f f axis) and when th ere are more than ten tu r n s, i t is d i f f i c u l t to p h y sic a lly support the h e lic a l conductor and in an a rra y, th e scan angle is reduced. T herefore a beamwidth of 35 ( f o r th e ten tu rn h e lix ) to 50 (f o r th e fiv e tu rn h e lix ) can be expected. When a h e lix of from fiv e to ten tu rn s is used as an elem ent of the a rra y, i t can be expected th a t the scan angle w ill be lim ited to roughly + 20 when the axis o f th e h e lix is fix e d. This angle of scan can be in creased to about t 50 by

113 t i l t i n g the axis o f each h e lix so th a t each elem ent m echanically 95 scan s. An example o f a m echanically scanning array is the S tanfo rd U n iv ersity Radio O bservatory M ills C ross. This crossed array o f p a ra b o lic d ish es is used fo r tra c k in g the sun and studyin g s o la r sunspots in the radio range; a l l elem ents are mechanic a lly driven by a common ro ta tin g s h a ft so as to ro ta te to the p ro p er angle in the sky. However, in the case of a co n cen tric rin g array c o n fig u ra tio n a mechanical d riv e arrangem ent fo r each elem ent would be much more complex, due to th e array geometry. htien scan angles in excess o f t 30 are d e sire d, i t might be d e sira b le to use w ider beamwidth elem ent w ith fix ed axis r a th e r than to m echanically scan each elem ent. A log co n ica l s p ir a l could be used, fo r example, although i t would be n ecessary to provide a balanced e x c ita tio n fo r each s p i r a l. T h erefo re, when a h e lix is to be used, the choice depends on the requirem ents o f p o la riz a tio n c i r c u l a r i t y and scan angle. A h e lix of only a few tu rn s can be used fo r scanning over Î 40 o r m ore, b u t the a x ia l r a tio may be h ig h, p a r tic u la r ly f o r angles o f f a x is. The p o la riz a tio n c ir c u la r ity can be improved by u sin g a h e lix o f more tu r n s, b ut the scan angle w ill be reduced. When a sm all co n ical counterpoise is used w ith th e h e lix, th e phase c e n te r and p o la riz a tio n s t a b i l i t y improve as p ointed out in Chapter I I. For example, when th e mouth diam eter of the cone is 0.75 X, th e minimum spacing of th e se elem ents in the array w ill be 0.75 X. A lso, i t is expected th a t the use of the co n ical co u n terp o ise w ill reduce even fu r th e r th e mutual coupling

114 between elem ents, sin ce th e re is no common f l a t ground plane f o r 96 a l l h e lic e s. The beamwidth of th e h e lix w ith sm all counterpoise i s approxim ately th e same as th a t using a f l a t ground p la n e, a l though th e cone reduces th e sid elo b e le v e l s l i g h t l y. T herefore, an approxim ation f o r the far-zo n e f i e l d p a tte rn of th is m odified h e lix w ill be the approxim ate form ula used f o r a h e lix over a ground p lane [Kraus, 1950, p. 202]: sin(n #/2) E^(6) = cos 0 (3-35) sin(i i/2) where Ip = 2v[S ^(l - cos 0) + 1/2N] (3-36) and is the spacing between tu rn s in term s o f one w avelength. Since th e a rra y f a c to r has a much more narrow beam than th e elem ent f a c to r, the fa c t th a t equation (3-35) does n o t give a clo se approxim ation to th e sid elo b e behavior of the m odified h e lix w ill lead to only sm all e rro rs in th e com putation of the t o t a l f i e l d p a tte r n. At 30 or more o ff a x is, where the h e lix sid elo b es e x is t, the arrgy f a c to r may be down by 20 db o r more, fo r array s o f se v e ra l w avelengths diam eter. Chapter V d iscu sses some measured p a tte rn s o f experim ental co n ce n tric rin g array s and compares these p a tte rn s to ones c a l c u la te d on the b a sis of the techniques d iscu ssed in th is c h ap te r.

115 CHAPTER IV THE ARRAY FEED SYSTEM: A RADIAL TRANSMISSION LINE As mentioned in C hapter I, i t was proposed by Kraus [1964] th a t the co n cen tric 'r in g array be fed from a common ra d ia l c a v ity. This ch ap ter d iscu sses the th eo ry o f ra d ia l tran sm issio n lin e s as i t ap p lies to the design of a common ra d ia l feed system fo r a c o n c e n tric rin g a rra y. A fter the unloaded lin e is examined, cons id e r a tio n w ill be given to the e ffe c t o f loading the ra d ia l lin e w ith probes, A, The Theory o f Radial Transm ission Lines Appendix C d eriv es the f ie ld s o f two types o f ra d ia l tr a n s m ission lin e s, a c ir c u la r type and a wedge ty p e. The c ir c u la r r a d ia l tran sm issio n lin e w ill how be examined in more d e ta i l. There are two cases of p rin c ip a l i n t e r e s t. The f i r s t is th a t in which the ra d ia l lin e is e x c ite d at th e c e n te r and has some non-zero impedance term in atio n around i t s p e rip h e ry. This is c a lle d a tra v e lin g wave ra d ia l tran sm issio n lin e. The second case involves th e r a d ia l c a v ity, i. e. a ra d ia l tran sm issio n lin e w ith a sh o rt c i r c u i t a t i t s edge. C onsider a c ir c u la r ra d ia l tran sm issio n lin e having a rad iu s R and a spacing b between p la te s. Assume th a t around the p erip h ery o f the lin e th e re is some a r b itr a r y te rm in a tin g impedance. I t 97

116 98 i s o f in t e r e s t to examine the f ie ld s fo r th is case. I f th e lin e is operated in the dominant mode (TEM to the r a d ia l d ire c tio n ) then the f ie ld s are given by [see equations (C-23) and (C -24)]: = (k^/jwe) [A^ H^^^(k^r)] (4-1) H = -[Ad H ^ ^ ^ k r ) + A. ^ (k r) ] (4-2) Equations (4-1) and (4-2) can be a lte r n a te ly w ritte n as: E, = (k2/jwe)[b J (k r) + B N (k r) ] (4-3) là l o r ^ u r W 1 (4-4) T h erefo re, the v o ltag e between p la te s at any ra d ia l p o in t r can be w ritte n as: V(r) = (k^b/jto )[B jj^(k^r) + B^N^(k^r) ] (4-5) The cu rre n t in one p la te a t rad iu s r w ill be: I ( r ) = -2irr[B, d J (k r) + B. d N (k r) ] (4-6) 1 ^ o r z ^ o r Suppose now th a t th is c ir c u la r ra d ia l waveguide is e x cited by a c e n tr a lly lo cated f i n i t e l y conducting c y lin d ric a l m e ta llic probe o f rad iu s r^ so th a t the v o ltag e across the probe i s V(r^) = V^. Also the c u rre n t a t the feed p o in t is I( t^ ) = Then i t can be shown [M arcuvitz, 1964] th a t the v o ltag e and c u rre n t can be w ritte n as :

117 V = VCr) CsCkr^.kr) - jz I ( r ) s n (k r,k r) (4-7) = Z I ( r ) c s(k r^,k r) - j V(r) S n(kr^, kr) (4-8) where CsCkr^.kr) =(Trkr/2;[Jj(kr)N^(kr^) - N^(kr) J^(k r^ ) ] (4-9) c s(k r^,k r) =(Trkr/2) [N^(kr) J^(k r^ ) - J^(kr)N ^(kr^) ] (4-10) Sn(kTg,kr) = (irkr/2) [J^Ckr)N^(kr^) - N ^(k r)j^(k r^)] (4-11) sn(kx kr) = (Ttkr/2) [J (kr)n ( k r j - N (kr)j (k r )] (4-12) O O o o o Z = 377(b/2ïïr) (4-13) Z = 377(b/2irr ) (4-14) o o assuming th a t the medium in sid e the ra d ia l guide is a i r. I t can be seen from the form of the v o ltag e and c u rre n t in equations (4-7) and (4-8) th a t the ra d ia l tran sm issio n lin e behaves in a s im ila r fashion to the sim ple lin e a r tran sm issio n lin e, w ith two im portant ex cep tio n s. F i r s t, the wavelength in sid e th e guide i s a fu n ctio n of radius and second the c h a r a c te r is tic impedance is also a function of ra d iu s. I t follow s from equation (4-13) th a t the c h a r a c te r is tic impedance n e a r th e c e n te r of th e ra d ia l guide is q u ite high and th a t i t s value is in v e rse ly p ro p o rtio n a l to the ra d iu s. Therefore, a matched term in atio n fo r a ra d ia l guide of one rad iu s would have a d if f e r e n t value than i t would fo r a ra d ia l guide of another ra d iu s.

118 I t can a lso be shown [H arrington, 1961, p. 209] th a t the phase co n stan t fo r the TEM c ir c u la r ra d ia l guide is given by By = (2/Trr)[ / ( k r ) + N^(kr) ' (4-15) To ex p lain more f u lly th e meaning o f equation (4-1 5 ), the d efin itio n of phase co n stan t is re c a lle d. I f th e w avefunction of th e ra d ia l guide is w ritte n as j4 (r,(b,z) Y = A (r,4,z ) e (4-16) then the v e c to r phase c o n stan t is d efined as g = -?$ (4-17) For th e TEM case th e re is n e ith e r any v a r ia tio n of th e wave- fu n ctio n in the (j)- o r z -d ir e c tio n s. T herefore the s c a la r phase c o n stan t is a measure o f the ra te a t which th e phase decreases in th e ra d ia l d ir e c tio n, i. e. g 3$ (4 18) ^ 3r As in th e case o f a uniform c ro s s-s e c tio n tra n sm issio n lin e, the phase v e lo c ity o f a wave p ro g ressin g in th e ra d ia l d ire c tio n is found as Vr = w/gp (4-19) o r from equation (4-15) v = (nw /2)r[ / ( k r ) + / ( k r ) ] (4-20) r o o

119 1 0 1 T herefore the phase v e lo c ity n e ar the c e n te r o f th e guide is h ig h e r than i t is n e a r th e o u te r edges. The "wavelength*' is thus lo n g er n e a r the c e n te r than n e a r th e edge. I t would be i n tu iti v e ly expected th a t th e wavelength would be v i r t u a l l y co n stan t a t large r a d ii sin ce the w avefronts would be very n e a rly plane and tih -.s s im ila r to the re c ta n g u la r p a r a lle l p la te guide. This i s, in f a c t, easy to show. The larg e argument approxim ations to J^ (k r) and N (kr) are re s p e c tiv e ly ^ (2 /irk r) c o s(k r - ir/4) and \J(2/vkT) s in ( k r - ti/4). Thus th e phase v e lo c ity becomes fo r larg e r a d i i, Vy = (itaj/2) (r) (2/irkr) = w/k (4-21) which is independent of r and is id e n tic a l to the phase v e lo c ity o f a re c ta n g u la r p a r a lle l p la te guide. I t i s o f i n t e r e s t now to examine th e s p e c if ic case o f a ra d ia l c a v ity, i. e. a ra d ia l tran sm issio n lin e w ith a sh o rt c i r c u it te rm in a tio n. The c a v ity rad iu s w ill be denoted by r. I f, c as b e fo re, th e c a v ity i s operated in the TEM mode, the f ie ld s can be found from equations (4-3) and (4-4 ). Since th e e l e c t r i c f i e l d must be f i n i t e a t the c e n te r o f th e c a v ity, and must go to zero at the o u te r edge, the f ie ld s w ill be given by E = (k ^ /ju e) J (x r / r ) 2 o on c (4-2 2 )

120 102 This is the TM mode (TM means tra n sv e rse m agnetic to the z- ono d ir e c tio n, in accordance with c ir c u la r waveguide mode n o ta tio n convention) where denotes the p o s itio n of the n^^ zero of the zero o rd er Bessel fu n ctio n o f the f i r s t kind. The sep a ra tio n co n d itio n re q u ire s th a t (4-23) so th a t the resonant frequency is (4-24) where c is the v e lo c ity o f lig h t. This in d ic a te s th a t fo r any given c av ity rad iu s r ^, th ere is an in f i n i t e ^number o f resonant freq u e n cies, with one resonant frequency fo r every zero o f the B essel fu n ctio n. For example, co n sid er a c ir c u la r ra d ia l cav ity w ith a rad iu s o f 100 cen tim eters. Suppose th a t i t is d esired to operate in the TM^^_ mode. The root x.^ can be found from Table 3 where UoU 08 th e ro o ts o f J q(^qj^) = 0 are lis te d [Abramowitz and Stegun, 1964]; TABLE 3 ROOTS OF J (x ) = 0 o on n *on n X on n X on n *on n X on

121 Since = and r = 1 m eter, the resonant frequency can 08 c ^ be computed from equation (4-24) as: 103 f = (3x10^/6.28) (24.353/1) r = MHz. (4-25) The e l e c t r i c f ie ld f o r th is resonant frequency is found from equation (4-22) as : = (k^/jwg) J^( r /r ^ ) (4-26) and th e m agnetic f ie ld as: = (24.353/r^) J^( r /r ^ ) (4-27) Fig, 31 shows two views o f the c av ity w ith e l e c t r i c and m agnetic f i e l d lin e s superposed on the c av ity o u tlin e. Except fo r the region around the feed p o in t, the fie ld s show an almost sin u so id a l beh av io r. Since th e s tru c tu re is a c a v ity, the fie ld s change in phase by 180 at every zero of the B essel fu n ctio n. I t w ill be shown in the next se c tio n th a t th is is a u se fu l p ro p erty when using the c a v ity as a common feed to an a rra y of antennas. B. A R adial Transm ission Line Feed System 1. D iscussion o f the Basic P ro p e rtie s o f th e Feed System As mentioned in Chapter I, a ra d ia l tran sm issio n lin e can be used as a feed system fo r an array of elem ents. In th is s e c tio n two such feed systems w ill be d iscu ssed ; the f i r s t type

122 104 A (b) F ig. 31. A c ir c u la r ra d ia l c a v ity operated in th e TM0 8 0 mode, (a) plan view (b) sid e view

123 105 i s a ra d ia l tran sm issio n lin e term in ated in i t s c h a r a c te r is tic impedance and the second type is the ra d ia l c a v ity. For th is d i s c u s s io n,.it w ill be assumed th a t both types are operated in th e dominant TM mode (TEM to the ra d ia l d ir e c tio n ), Fig. 32 i l l u s t r a t e s the cross se c tio n o f a ra d ia l tr a n s m ission lin e term inated in i t s c h a r a c te r is tic impedance so th a t only outward going waves are p re se n t (no re fle c tio n s from the o u te r w a ll). Also a h e lix is shown as fed by a sh o rt probe in s e r te d in to the guide at rad iu s r^. I f, as a f i r s t approxim ation, the h e lix is assumed to in tro d u ce n e g lig ib le r e f le c tio n s, then the r a d ia l v a ria tio n o f the e l e c t r i c f ie ld w ill be described by a zero o rd er Hankel function of th e second k in d, i. e. 2 (2) E, = (k /juie) h'' ''(k r) (4-28) z 0 and s im ila rly fo r th e magnetic f i e l d, H. = k (kr) (4-29) 9 1 A Hankel fu n ctio n o f the second kind and of the m^^ o rd e r can be w ritte n as where (kr) =, / J^ (k r) + N^(kr) (4-30) m V m m 0(kr) = tan ^[ -N ^ (k r)/j^ (k r) ] (4-31) Therefore i t can be seen th a t both am plitude and phase o f the

124 Probe Helix Coaxial Feed Fig. 32. A ra d ia l transm ission lin e term inated in i t s c h a ra c te ris tic impedance and feeding a h e lic a l antenna elem ent. o o\

125 1 0 7 e l e c t r i c f i e l d are continuous fu n ctio n s of the ra d iu s. Since th e h e lix probe is assumed to introduce only sm all p e rtu rb a tio n s in th e f i e l d, the am plitude of the v o ltag e induced on the probe can be assumed to be p ro p o rtio n a l to th e am plitude o f the unperturbed f i e l d and to the probe depth, i. e. ^ ~ (4-32) where is th e vo ltag e induced on th e h e lix probe o f depth d^. The phase w ith which th e probe is e x c ite d ( r e la tiv e to th e c e n te r of the guide) is computed as G(kr^) = ta n "^ [ -N ^ (k r^ )/J^ (k r^ )] (4-33) The r a d ii r (p = 2, 3, 4,... ) a t which th e phase d if f e r s from P (kr^) by in te g r a l m u ltip les of 2 t t is found by 0(krp) = G(kr^) + 2piT = ta n " l[ - N ^ ( k r^ )/J ^ ( k rp i + 2pir (4-34) o r tan ^ [ -N (k r ) / J (k r )] = ta n ^[-N (k r ) / J (k r )]+2pn (4-35) o p o p o 1 o 1 Thus, once r^ has been chosen, a d d itio n a l r a d ii r^ (p = 2,3,4,... ) can be found from the tra n sce n d e n tal e q u atio n (4-3 5 ), such th a t the phase a t a l l r^ is id e n tic a l (or d if f e r s by m u ltip les of 2 t t ). Tberefore rin g s of elem ent probes can be in s e r te d a t r^, r ^, r^, e tc. and a ll elem ents w ill be e x cited in phase. For ra d ia l p o s itio n s which are large in comparison to a w avelength, the

126 108 asym ptotic approxim ations to N (k r ) and J (k r ) can be used, i. e. o p O p Jp(kr^) = /c2/irkrp) cosckr^ - r/4 ) (4-36) NoCkXp) = ^(2/TTkTp) sincktp - n/4) (4-37) so th a t equation (4-35) becomes r = r + px ( p large) (4-38) P 1 which sim ply means th a t the "in -p h ase" ra d ii d i f f e r by m u ltip le s o f one wavelength when th e se r a d ii are la rg e. Thus th is tra v e lin g wave ra d ia l waveguide feed system o ffe rs a means o f c o n tro llin g both am plitude and phase of the elem ents o f a co n cen tric rin g a rra y. The am plitude o f the v o ltag e can be v a rie d by a d ju stin g the probe depth and the elem ents can be a l l fed in phase by choosing s u ita b le rin g r a d i i. F ig. 33 is a p lo t of the amplitude and phase behavior fo r a matched ra d ia l waveguide, based upon equations (4-30) and (4-3 1 ). I t can be seen from an exam ination o f the fig u re th a t except fo r r a d ii which are sm all in terms of a wavelength th a t the am plitude of th e e l e c t r i c f i e l d f a l l s o ff approxim ately as 1 //F and th a t th e phase i s a lin e a r fu nction of the ra d iu s. T h erefo re, the outgoing power in a ra d ia l guide is approxim ately in v e rse ly p roportio n a l to the ra d iu s. A graph such as th a t o f Fig. 33 is u se fu l fo r fin d in g those r a d ii a t which th e phase d if f e r s by a m u ltip le o f 2tt. Of course i t is not necessary th a t the array elem ents be fed in phase u sin g th is feed scheme when i t i s p o ssib le to in tro d u ce

127 1.6 Amplitude Phase " 1 4 it -- 12t t Amplitude Phase tt tt k r 3 0 Fig. 33. Amplitude and phase of outgoing tra v e lin g wave in dominant mode matched ra d ia l waveguide.

128 a phase c o rre c tio n at the in d iv id u a l elem ents by a sim ple a x ia l r o ta tio n, such as discussed in Chapter I I, I t should be p o in ted out th a t a matched ra d ia l waveguide has one prim ary disadvantage from the stan d p o in t o f use in radio astronomy. Since the matched load d is s ip a te s power and i s th e r e fo re "hot" in the sense o f black body r a d ia tio n, the antenna tem perature may be high. This is a d e f in ite disadvantage when th e design goal i s a high s e n s itiv ity or cold ra d io te le sc o p e antenna. Turning now to a ra d ia l cav ity feed system, i t can be seen th a t th e re are sev e ra l im portant d iffe re n c e s between th e c a v ity and th e matched guide. F i r s t, sin ce a c av ity i s tru ly resonant only a t d is c re te fre q u e n c ie s, i t follow s th a t a ra d ia l c a v ity feed system w ill be more narrow-band than a comparable matched ra d ia l guide. Second, th e am plitude v a ria tio n o f the e l e c t r i c f i e l d in sid e the c av ity w ill have d is c r e te p o s itio n s where th e re w ill be maxima and minima, which i s u n lik e th e mono- to n ic a lly decreasing am plitude of th e matched guide. A lso, the phase v a ria tio n of a ra d ia l cav ity jumps in ste p s of 180* w ith in c re a sin g ra d iu s, whereas th e phase v a ria tio n of the matched guide is a monotonie fu nction of the ra d iu s. Therefore when i t is d esired to feed a l l elem ents in phase w ith a common ra d ia l c a v ity, the rin g r a d ii should be placed at the p o s itio n s of e ith e r the maxima o f J^ (k r) o r the minima of J^ (k r) but not b o th. This im plies th a t the minimum rin g spacing w ill be approxim ately one free space w avelength. I f i t is d esired

129 I l l to space th e rin g s approxim ately every h a l f w avelength, then the rin g r a d ii should be placed at the p o s itio n s of the maxima of IJ^ (k r)i, w ith the understanding th a t any rin g w ill be fed 180 out of phase with i t s n e a re s t neighbor rin g [s ) and th a t a 180 a x ia l ro ta tio n of a l l elem ents in every o th e r rin g w ill have to be made in o rd er to achieve a b roadside beam. To give an example where th is ra d ia l guide feed technique has been used, Goebels and Kelly [1959] have d escribed a conc e n tr ic rin g array (non-phase s h iftin g ) o f crossed s lo ts cut in to one face of a ra d ia l tran sm issio n lin e. T h eir experim ents in clu d ed both matched r a d ia l guide feeds and ra d ia l c a v ity fe e d s, o p erated n e a r X-band. They observed bandwidths of th e o rd e r of 3% f o r th e c av ity feed and about 5% fo r the matched guide feed. 2. L im itatio n s on th e Maximum Guide Radius I t is o f in te r e s t to know how larg e th e ra d ia l guide can be made in p ra c tic e w ithout ap p reciab le d is s ip a tio n of power by th e guide w alls and by the element probes. F ir s t, th e problem o f wave a tte n u a tio n by a f i n i t e l y conducting ra d ia l waveguide w ithout any in s e rte d probes w ill be considered. Assume f o r th e p re sen t th a t th e r a d ia l guide supports an outward going TEN wave only and th a t th e guide i s i n f i n i t e in e x te n t. The a tte n u a tio n due to power d is s ip a tio n in the w alls i s given f o r any waveguide by [Kraus, 1953, p. 474]:

130 1 1 2 Re[Z ] / H 1^ d l a = ^ tl (4-39) 2 Re[Z^] ds where Re[Z ] = re a l p a rt o f th e i n t r i n s i c impedance of the ^ conducting guide w alls Re[Z^] = re a l p a rt of the tra n sv e rse impedance o f the guide H til = a b so lu te value of the component of H tangent to th e conducting su rface of the guide w alls (in te g ra te d around the i n t e r i o r su rface o f the guide) H I = ab so lu te value of the component o f H tangent to the plane of cro ss se c tio n through the guide (in te g ra te d over the c ro s s -s e c tio n a l area) Thus f o r the TM mode, where 00 * and E = (k^/joje) (kr) (4-40) z o = k H p \ k r ) (4-41) 9 1 i t can be seen th a t l t l l ' k l H p ckdl (4-42) Furtherm ore, dl = r d(j) (4-43) and ds = r d<j) dz (4-44)

131 T herefore i f the ra d ia l guide has a p la te spacing of b, the a tte n u a tio n becomes 2n «f21 2 Re[Z ] 2 r f k^ H, '( k r ) ( d* a = ^ 0 ^ b 2ïï 2 R e[z J r / / (kr) d<^ dz T o o 1 2 C2irr) Re[Z ] c 2 (2itr) b Re[Z^] Re[Z 3 = F (4-45) Re[Z^l where th e f a c to r o f 2 in the num erator o f the f i r s t equation of (4-45) accounts fo r th e fa c t th a t th ere are two w alls in which conductive lo sses take p la ce. The re a l p a r t o f th e su rfa ce impedance is found as Re[Z^3 = /uu/2a (4-46) where o is th e co n d u ctiv ity of th e w a lls. The tra n sv e rse wave impedance is found from Zt = " ^ 2 % = -[(k V jto c )H ^ ^ ^ (k r)]/[k H (2)(kr)3 (4-47) = -(k /jw e )[J (k r )- jx ( k r ) ] / [ J ( k r ) - jn ( k r ) ] o u JL 1 = -(k /jw c ){[J ^ J ^ + N ^ N j + j [ J ^ N j - J j N j } /( J j + N ^ ) J N -J.N J J +N N, = -(k/wg){ ^ ^ - j ^ ^ } (4-48)

132 114 T herefore R e [Z j = -(k/w E )[J N, - J-N ] / [ / + N^] (4-49) T o 1 1 o 1 1 so th a t th e a tte n u a tio n as a function of the rad iu s becomes J^Ckr) + N^(kr) a (r) = -(toe/kb)/ytii/2a nepers/m J (kr)n fkr) - J fkr)n (kr) (4-50) o 1 1 o The t o t a l a tte n u a tio n o f an outgoing wave out to ra d iu s R w ill be given by R o = / a (r) dr nepers (4-51) " 0 I f R is much la r g e r than one w avelength, then a good approxim ation to e q u a tio n (4-50) can be found by ta k in g the asym ptotic approxim ations to the v ario u s B essel fu n ctio n s involved. Then 2 2 J^ (k r) + N^(kr) = 2 /n k r (4-52) Also the computation of the ap p ro p ria te Wronskian o f B essel s d i f f e r e n tia l equation gives J fkr)n (kr) - J (kr)n (kr) = 2/trkr (4-53) 1 o o 1 Thus fo r larg e r a d ii th e a tte n u a tio n is independent o f the ra d ia l co o rd in ate r. Thus an approximate value o f th e t o t a l atte n u atio n o f an outgoing TEM tra v e lin g wave at radius R is (Xp = (R/b)(wE/k) /yw/2o = (R/b) /we/2o (4-54)

133 For example co n sid er an in f i n i t e aluminum (a = 3.5x10 mhos/meter) ra d ia l guide operated at 1400 MHz. (X = 21 cm.) with a p la te spacing of 1*'. Suppose th a t i t is d e sire d to know the to t a l a tte n u atio n at 100 w avelengths from the c e n te r. = (21.0/.0254) = nepers = 0.24 db Then X 1.4x10 X 8.85x10 2 X 3.5x10' Thus th e w all loss can be considered as n e g lig ib le even fo r very larg e ra d ia l guides, which th e re fo re p laces v i r t u a l l y no re s t r i c t i o n on the s iz e o f the guide as f a r as conductive power lo sse s in the w alls are concerned. For a ra d ia l guide operated as a c a v ity, s im ila r r e s u lts are found f o r th e a tte n u a tio n b ehavior. I t is obvious t h a t when elem ent feed probes are in tr o duced in to th e ra d ia l guide, they w ill in c re ase the a tte n u a tio n co n sid erab ly due to the ra d ia tio n o f power by th e elem ents. I f th e probe depths are p ro p erly ad ju sted (o u te r elem ents probing deeply in to guide and in n er elem ents probing lig h tly in to guide) then a l l elem ents can be made to ra d ia te the same amount of power. 3. Probe-loaded Radial Guides I t has been assumed in a l l of the previous d iscu ssio n th a t the elem ent probes have a n e g lig ib le e ff e c t on the fie ld s of ra d ia l guides. This assumption w ill now be discu ssed in more d e ta i l.

134 For most p r a c tic a l a rra y s, th e element spacing w ill be no le ss than one h a l f w avelength, so th a t the probes in sid e th e guide w ill also be spaced a t le a s t one h a lf wavelength a p a rt. I f the probes are th in, then the r a tio o f to ta l probe volume to guide volume w ill be very sm all and the p e rtu rb a tio n s of the f i e l d w ill be lo c a liz e d to the region around each o f th e probes. In a re c ta n g u la r waveguide with probes in s e rte d p e rio d i c a l l y along i t s le n g th, an eq u iv alen t c i r c u i t can be drawn which in c o rp o ra te s both the eq u iv alen t Tee re p re se n ta tio n o f the unloaded guide and the eq u iv alen t Pi c i r c u i t o f a probe [S ilv e r, 1964, pp ]. By examining the combination eq u iv alen t c i r c u i t, the c h a r a c te r is tic impedance and guide wavelength o f the loaded guide can be computed. These q u a n titie s are in g eneral d if f e r e n t from those of the unloaded guide. I t is expected th a t the loading of a ra d ia l guide w ill a lso change i t s guide wavelength and c h a r a c te r is tic impedance b eh av io r. However, b efo re e n te rin g in to a d e ta ile d c a lc u la tio n o f th e p e rtu rb in g e f f e c t o f probes on a ra d ia l guide, i t must be decided as to what are the d e sired q u a n titie s to c a lc u la te. F i r s t, the d iffe re n c e in phase between any two ad jacen t rin g s must be known; however, i t is not n ecessary t o know the ab so lu te phase o f a rin g r e la tiv e to the c e n te r. Second, the am plitude o f the f ie ld at the p o sitio n of each probe must be known. And l a s t, i t may be d e sira b le to know the c h a r a c te r is tic impedance when the lin e is to be term inated in a matched load.

135 117 However, a fu r th e r look a t the lo a d ed -lin e c i r c u i t te ch nique [S ilv e r, 1964, pp ] shows th a t a ra d ia l lin e loaded by c o n ce n tric rin g s o f probes has an exceedingly complex c i r c u i t re p re se n ta tio n which depends to a larg e e x ten t on th e exact manner in which the probes are arranged. F ir s t of a l l, sin ce th e r a d ii (in a ra d ia l cav ity ) are lo cated according to the maxima o f J ^ ( k r ), then th e rin g s are n o t evenly spaced. Second, the number o f probes in each rin g w ill be d if f e r e n t, in g en eral; although th e probes in any one rin g are eq u ally spaced in angle, th e lo c atio n of these angles may be d if f e r e n t fo r every rin g. Thus, a wave p ropagating out from the c e n te r does n o t see a p e r io d ic a lly loaded ra d ia l guide. This b asic fa c t makes a lo a d ed -lin e a n aly sis v ir t u a lly im possible fo r any general case. In th e absence of a s u ita b le scheme fo r c a lc u la tin g the p e rtu rb in g e f f e c t of p robes, th e re are a t le a s t two a lte r n a tiv e s. F i r s t, as p rev io u sly m entioned, i f the probes are sm all then the guide w ill not be s e rio u sly p e rtu rb ed and m eaningful c a lc u la tio n s o f am plitude and phase can be made on the b a sis o f the unloaded guide. This technique has been used in a ll o f the a c tu a l models co n stru cted and t e s t, with good su ccess, as w ill be shown in C hapter V. Second, i f i t is d esired to place a more d ire c tiv e probe fo r each elem ent, then i t is lik e ly th a t the guide w ill be co n sid erab ly p ertu rb ed and the c a lc u la tio n s made on the b asis o f th e unloaded guide may lead to serio u s in a c c u ra c ie s. In such a c ase, the only a lte rn a tiv e is to measure the am plitude and phase in the a c tu a l antenna. C orrections fo r am plitude in accu racies

136 can then be made by varying the probe depth w hile phase c o rre c tio n s can be made by a x ia lly r o ta tin g the c ir c u la r ly p o la riz e d antenna 118 elem ent. S p e c ific techniques fo r th i s w ill be discussed in C hapter VI, In the case o f a ra d ia l c a v ity feed system, when th e probes are only lig h tly coupled the c a v ity Q w ill be high and th e bandw idth w ill be sm all. However, as the probes are more h e av ily coupled, th e power lo s t through ra d ia tio n w ill be in c re a se d, the Q lowered and thus the bandwidth in c re ase d. I f th e elem ents are very h e av ily coupled, then i t may be p o ssib le to ra d ia te v i r t u a l l y a l l of the outgoing wave, thus e f f e c tiv e ly changing the feed system from a c a v ity to a matched guide.

137 CHAPTER V EXPERIMENTAL MODELS OF CAVITY-FED PHASE-SCANNING ARRAYS OF HELICAL BEAM ANTENNAS This ch ap ter d iscu sses th re e d if f e r e n t models o f the c a v ity -fe d co n cen tric rin g array and th e instrum ents used f o r th e measurements. A ll ra d ia l feed lin e s were made in the form of c a v itie s because i t was d e sire d to keep th e antenna tem perature as low as p o s s ib le. When using a complex feed system such as th e one d esc rib e d, no amount of th e o r e tic a l c a lc u la tio n can take th e p lace of c a re fu l and m ethodical experim ental work. More feed systems were b u i l t and t r i e d th a t d id n 't work than those th a t did work. Appendix D is a d iscu ssio n o f some of these feed systems and the reasons why they did not operate as planned. However, a f te r se v e ra l models were tr i e d (and found n o t to work) c e r ta in p a tte rn s began to emerge, p a r tic u la r ly concerning the f i e l d behavior in sid e the c a v ity and also the elem ent in te ra c tio n o u tsid e the c a v ity. I t is c e r ta in ly n o t claim ed th a t th e feed system and array geometry used in the models to be d escribed rep resen t the optimum; however, the models did perform w ell enough to dem onstrate th e v a lid ity o f the p rin c ip le s s e t fo rth in the o rig in a l memorandum [Kraus, 1964]. 119

138 120 A. D iscussion of Measurement Techniques The measurements made on th e models were p r in c ip a lly f a r - zone p a tte rn measurements, w ith some impedance measurements. The b a sic goal o the experim ents was to c o n stru c t a c o n ce n tric rin g array o f h e lic a l antennas fed by a common c a v ity and to measure p a tte rn s over a reasonable range of scan. By reaso n ab le, i t is meant th a t th e scan angular range should be o f th e o rd e r o f the beamwidth of one h e lic a l elem ent. I t was decided at the o u tse t to work around the frequency o f 1400 MHz,, sin ce th e re was re a d ily a v a ila b le equipment such as o s c i l l a t o r s, co n n ecto rs, e tc, fo r use a t th a t frequency. Moreover, 1400 MHz, was considered to be a p o ssib le f in a l design frequency in a larg e working a rra y. Even a t 1400 MHz,, where one wavelength is about 8 1/2 in ch es, th e array dim ensions become la rg e f o r only two o r th re e rin g s, with a ty p ic a l maximum dimension of s ix fe e t in the case of the a ctu a l array s te s te d. Therefore i t was c le a r th a t in o rd er to make meaningful p a tte rn measurements on r e la tiv e ly larg e antennas, a very heavy duty r o ta tin g te s t p latfo rm would be needed. I t would need to be capable o f supporting the weight o f the antenna and se rv ic e personnel under a m oderately heavy wind load. A fte r c o n su lta tio n w ith se v e ra l m achinists a t the Department of E le c tr ic a l Engineerin g, i t was decided to e re c t a square p latfo rm (ten fe e t on a sid e) supported by a s te e l frame and a heavy duty ro ta ta b le s te e l pipe. This t e s t platfo rm and drive assembly was co n stru cted during the

139 w in te r of The s t e e l pipe is driven by a spur g e a r-p u lle y motor com bination at th e c o n stan t ra te o f 1 rpm. The e n tir e assembly is welded to a heavy s te e l base frame which is b o lte d to fo u r larg e concrete p ie rs sunk in to the ground. This arran g e ment, shown in Fig, 34, has provided a very s ta b le co n stan t speed r o ta to r capable o f supporting at le a s t 600 lb s. under winds g u stin g to 50 mph. A house fo r th e receiv in g equipment is b u i l t around the p latfo rm supports as shown in Fig. 35. No servo system of speed c o n tro l has been in s ta lle d on th e r o ta to r sim ply because money and time have n o t p erm itted. The m echanical d rive system is designed so th a t the r o ta tio n speed is h eld c o n sta n t, even when the wind g u sts to 25 or 30 mph. This provides a system angular recording accuracy of t 1/4 in s o fa r as p a tte rn r e p e a ta b ility is concerned. The t e s t antenna, which is operated as a re c eiv in g antenna, is mounted on the p latfo rm so th a t the forward beam p o in ts in th e h o riz o n ta l plane as shown in Fig. 35; th e i l l u s t r a t i o n a lso shows th e b a sic components o f the p a tte rn range. A wooden shed p rovides w eather p ro te c tio n fo r th e gear d riv e and a lso houses th e re c e iv e r and reco rd in g equipment. This is shown on the l e f t o f Fig. 35. The in c id e n t wave is provided by a 21 tu rn h e lix fed by a General Radio tunable o s c i l l a t o r, modulated by a 1000 cps audio to n e. This tr a n s m itte r is housed in a wooden box mounted on the A-frame wooden tower as shown in tbe rig h t of Fig. 35; t h i s minimizes the length o f the feed lin e from the o s c i l l a t o r to th e tra n s m ittin g h e lix, thus reducing any feed lin e ra d ia tio n.

140 122 main support p illo w block lo 'xlo ' wooden p l a t form pillow block support pipe ro ta ta b le s te e l pipe mam sp u r gears s t e e l base frame 1/2 hp. -d riv e motor concrete p ie r underground concrete counterw eight slab Fig. 34. Main components o f heavy duty antenna t e s t p latfo rm and a sso c ia te d drive components.

141 cav ity support s tr u t h e lic a l element of array tran sm ittin g h e lix ^ a n s m itte r r ^ o u s i n g wooden platform (lo 'xlo ') motor house and receiv er room lilt Mi.i. J 1iJ*.- ll.alt. ali U * ro ta tin g t e s t platform m e ta llic screen 4^ d iffra c tio n screen A-frame tower Fig, 35. Main components of antenna te s t range.

142 124 Also a d if f r a c tio n fence is e re c te d approxim ately halfway between tra n s m ittin g and receiv in g s ite s in o rd er to reduce ground r e fle c te d waves at the re c eiv in g s i t e. The d istan ce between tra n s m itte r and re c e iv e r is about 120 f e e t. At 1400 MHz., th is p laces the tra n s m ittin g antenna in the far-zone of th e.r e c e iv e r and thus the wave in c id e n t on the re c e iv in g antenna is very n e arly p lan e. In one experim ent, the in c id e n t f ie ld was probed in the volume occupied by th e te s t antenna and was found to be am plitude c o n stan t to * 0.67 db. F ig, 36 is a block diagram o f the p a tte rn range equipment. The sig n a l source (General Radio Type 1218-A o s c illa to r ) is tu n able from 900 MHz. to 2100 MHz. with a nominal power output of 100 m illiw a tts, and is modulated by a 1000 cps square wave. The tra n s m ittin g antenna, a 21 turn 14 h e lix, provides a c ir c u la r ly p o la riz e d in c id e n t plane wave a t the re c eiv in g t e s t antenna lo c a tio n. The received audio tone is d e tected by a tuned 1N416 c r y s ta l and is fed to a Hewlett Packard Model 4158 SWR In d ic a to r which operates as a tuned audio a m p lifie r. The output of th is a m p lifie r d riv es a Sanborn two-channel s t r ip - c h a r t reco rd er c o n sis tin g of a DC a m p lifie r d riv in g a galvanom etric s ty lu s. One channel records th e antenna p a tte rn and the o th e r channel is used f o r recording a referen ce spike which in d ic a te s When the tu rn ta b le is p o in tin g at th e tra n s m ittin g h e lix (0 ). An an aly sis o f p o ssib le e rro rs in the p a tte rn s follow s. I t has been found th a t the heavy duty r o ta to r used fo r th e te s t p latfo rm along w ith the Sanborn s tr ip - c h a r t re c o rd er provides a

143 te s t antenna tran sm ittin g h e lix te s t platform reference marker microswitch mechanical drive GR 12 ISA RF o s c ill. coaxial / ro tary jo in t galvanom etric sty lu s movements HP 415B audio amp 1000 cps square wave o s c illa to r Sanborn DC am plifie Sanborn DC am plifie] Fig. 36. Block diagram of antenna p a tte rn range. Dual channel s trip -c h a rt N) Kn

144 126 recorded angular accuracy o f ty p ic a lly - 1/4* to - 1/2*. A s ig n a l lev el range can be found whereby th e 1N416 tuned d e te c to r w ill be lin e a r to db over a 20 db e x cu rsio n. The HP 415B a m p lifie r is q u ite lin e a r over the e n tir e working range. The Sanborn re c o rd er e x h ib its some n o n lin e a rity, ty p ic a lly - 1 d iv isio n out o f 50, when the sig n a l le v el is low. This non- l i n e a r i t y can be tra c e d to the galvanom eter movement i t s e l f. T herefore a very co n serv ativ e e r r o r a n a ly sis would place the am plitude accuracy o f th e recorded p a tte rn s a t - 5% and the an g u lar accuracy a t - 1/ 2. Impedance measurements were made by u sin g the conventional s l o t t e d lin e tech n iq u es. The s lo tte d lin e is connected d ire c tly t o th e antenna in p u t and is driv en by an audio-m odulated RF source as d escrib ed above. In o rd er to minimize o s c il la to r " p u llin g " by re a c tiv e lo ad s, an i s o la to r is in s e r te d between the o s c i l l a t o r and the s lo tte d lin e. B. A 24-H elix Ring Array Fed by a C irc u la r Radial' C avity 1. D e tails o f Geometry and C onstruction This se c tio n describ es an experim ental 24-elem ent ring array o f fiv e tu rn 14* p itc h angle h e lic e s fed by a c ir c u la r ra d ia l c a v ity. Although th is array did n o t scan as p re d ic te d, i t d id e x h ib it a w e ll-d e fin e d main beam f o r broadside element p hasing and led to some im portant d isco v e rie s concerning the use o f h e lic a l antennas in an array.

145 127 A sid e view o f the c a v ity is shown in Fig. 37 w ith th e ab so lu te value of th e e l e c t r i c f ie ld ( I e^ ) shown superimposed on th e c a v ity o u tlin e ; th e h e lic a l elem ent probes are in s e rte d a t th e p o s itio n of the maximum o f e^ n e a re s t th e o u ter cav ity w a ll, which is very clo se to one q u a rte r of a fre e space wavelength from the w a ll. The c a v ity is operated in th e mode; th e c a v ity rad iu s is inches (0.845 m e te rs). The resonant wavelength o f the unloaded c a v ity is found from the sep a ra tio n equation fo r the TM^gg mode, i. e. k = Z'/Xp = (S-1) where r = 0,845 m eters c Xgg = (from Table 3) T herefore = 2 (0.845)/ = meters which corresponds to a resonant frequency of 8 f = 3x10 / = 1375 MHz. (5-3) From equations (4-26) and (4-2 7 ), th e fie ld s in the c a v ity are J^( r /r ^ ) (5-4) H = K J, ( r / r ) (5-5) ^ ^ 1 c where r = m eters and K, and K_ are c o n sta n ts. A p lo t of c 1 2

146 Helix probe Helix probe I T.025 Feed line 845 m.791 m Fig. 37, Cross sectio n view of ra d ia l cav ity showing e le c tr ic f ie ld, h e lix probe p o s itio n s, and p rin c ip a l dimensions. NJ 00

147 129 E^[ gives th e e l e c t r i c f i e l d shown in Fig. 37. The elem ent probes are in s e rte d at the la s t maximum of e^. From a ta b le o f J ^ ( x ), the la s t maximum of E^ befo re th e zero x^_ is found a t x = , which corresponds h ere to a 08 probe rad iu s of (0.845 m eters) R - = m eters (5-6) Since 24 elem ents are used, eq u ally spaced in an g le, th e angular s e p a ra tio n A<(i is 15*, which gives an elem ent spacing of d = 2R sin (6 * /2 ) = 2(0.791 m eters) sin 7.5* = m eters (5-7) Since th e o p eratin g wavelength at resonance is m eters, th e elem ent spacing in terms o f a wavelength is found as d^ = / = (5-8) Thus, from th e array sta n d p o in t, th e antenna c o n sists of a rin g of 24 h e lic a l beam antennas spaced approxim ately one w avelength a p a rt. According to the work of B lasi [1966], th is spacing should be wide enough so th a t mutual coupling is very low. The c av ity was co n stru cted from two 3/16 inch aluminum p la te s sep arated by 1 inch w ith an aluminum band around the o u ter edge to form the c ir c u la r c a v ity. I t was probe fed in the c e n te r, as shown in Fig, 3 8 (a ); Fig. 38(b) i l l u s t r a t e s th e d e ta ils o f the feed f o r a ty p ic a l h e lic a l elem ent. Since th e h e lix had only fiv e

148 130 aluminum upper c a v ity p la te 3/16 copper 3/16 aluminum Type N c h assis connector lower c a v ity p la te Formvar coated copper wire polyethylene plug upper c a v ity p la te ////// / / / / / / / / / / / > - 4 M - 1/ 2 " 3/16" copper tubing lower c a v ity p la te mrmjnriiiiiiii inn iiiiii! nn f i/iivniiiiiitmth (b) Fig. 38. (a) The c a v ity c e n tra l feed arrangement. (b) A probe feed fo r a ty p ic a l h e lix.

149 131 tu rn s, i t was made to be se lf-s u p p o rtin g ; th e copper tu b in g probe was so ld ered to th e h e lix and was in s e rte d in to the c av ity through th e polyethylene plug shown. The probe depth could then be v a ried by s lid in g th e probe through th e plug. I t was found th a t th e probe depth had very l i t t l e e f f e c t on the p a tte r n, but th a t the in p u t impedance to th e c av ity was m oderately a ffe c te d by th e probe depth, as would be expected. 2. Far-zone Power P a tte rn s, Measured and C alculated Fig. 39 compares th e measured and c a lc u la te d power p a tte rn s o f th e 2 4 -h e lix rin g array when a l l elem ents are o rie n te d in the same manner, i. e. fo r a broadside beam. The measured p a tte rn was o btained by th e techniques discu ssed in the f i r s t p a r t of th i s c h ap te r and th e c a lc u la te d p a tte rn was obtained by using the c o n ce n tric rin g array theory o u tlin e d in Chapter I I I. The a c tu a l c a lc u la tio n was made by a sp e c ia l S catran computer program used on the OSU IBM 7094 computer. Both lin e a r power p a tte rn s are shown norm alized to 1.0 f o r ease in com parison. The agreement between the measured and c a lc u la te d p a tte rn s is good, p a r tic u la r ly with regard to th e lo catio n of th e minima and maxima o f the s id e - lo b es, the h e ig h t of th e sid elo b es and th e HPBW. This agreement in d ic a te s th a t the feed system is o p eratin g as p lanned, i. e. a l l elem ents are being fed w ith very n e a rly th e same am plitude and phase. The probe depth p o f the elem ents was s e t to 3/4 inch fo r th e measurements. The measured reso n an t frequency was found to be very clo se to th e c a lc u la te d resonant

150 1.0 Normalized Power 0.6 Measured - Ca;.cula :ed Angle from Broadside Fig. 39. A comparison o f measured and calcu lated lin e a r power p a tte rn s fo r a 24-helix rin g array phased fo r a broadside beam.

151 133 frequency so th a t fo r a moderate probe depth, the f i e l d in sid e the c a v ity is not se rio u sly p e rtu rb e d. I t can be seen th a t the measured power p a tte rn of Fig, 39 does not have deep n u lls. This is an in d ic a tio n o f sm all phase and am plitude e rro rs in th e s e ttin g o f each elem ent and is due p rim a rily to phase c e n te r p o s itio n s of each elem ent. sm all d iffe re n c e s in the This d iffe re n c e occurs because a l l elem ents were not c o n stru cte d e x a c tly a lik e, and the ground plane environment of each element was not ex ac tly i d e n tic a l. I t was p o in ted out in Chapter II th a t the s iz e and shape o f the ground plane can be an im portant fa c to r in the lo c a tio n of the phase c en te r of a h e lix. Although th i s array did work as p re d ic te d fo r th e broadsid e case, i t did not phase scan p ro p e rly. The scanning experim ents were c a rrie d out fo r th e h o riz o n ta l (<(i = 0") p lan e. From the array th eo ry o f Chapter I I, the c o rre c t phase f o r the th n elem ent is found as * = kr sin 0 cos d> (5-9) But R = 3,66X and ((i = n(15*) where <(* = 0 denotes th e rig h t- n n most elem ent in the h o riz o n ta l plane as th e o b serv er faces the a rra y. T h erefo re, t ;^ = 1318" sin 0^ cos n(15") (5-10) For example, i f the beam is to be p o in ted at 10" o f f b ro ad sid e, th e h e lix a t 30" up from the h o riz o n ta l plane (n = 2) would be given an angular s e ttin g of;

152 134 \l>^ = 1318 (sin 10 ) (cos 30 ) = 197 (5-11) However, when th e h e lic e s were phase se t by t h i s method, the beam maximum did n o t occur a t th e p re d ic te d angle. In f a c t, th e maximum beam s h i f t o b tain ab le was only about - 11, when approxim ately - 25 was expected. The c a lc u la te d and measured p a tte rn s did not agree as th e antenna was phased fo r a beam t i l t. No s e t of elem ent phasing could be found, e i t h e r by c a lc u la tio n o r by experim ent, whereby th e beam would s h i f t more than about 11 from b ro ad sid e. At th is p o in t, a sy stem atic in v e s tig a tio n was begun to determ ine why th e array was not phase scanning p ro p e rly. A care fu l study o f the in d iv id u a l h e lic a l elem ents (see Chapter II) showed th a t when a h e lix op erates over a f l a t ground p la n e, i t s apparent phase c e n te r as seen by a fixed o b serv er can s h i f t along the ax is o f th e h e lix as the h e lix is a x ia lly r o ta te d. This means th a t the phase s h i f t obtained by a x ia l ro ta tio n o f the h e lix may be p a r t i a l l y o r com pletely can celled by th e phase s h i f t due to th e movement of th e apparent phase c e n te r. No sy stem atic d ata is a v a ila b le to in d ic a te which shapes and s iz e s o f ground planes worsen the phase c en te r s h i f t e f f e c t, b u t the experim ents o f th e au th o r as w ell as those o f Sander and Cheng [1958] in d ic a te th a t th e phase c e n te r s t a b i l i t y i s d e f in ite ly dependent on th e type o f counterpoise used. I t is n a tu ra l to ask i f the mutual coupling between h e lic a l elem ents was resp o n sib le fo r the anomalous phase s h if tin g b eh av io r.

153 In an attem pt to answer th is q u e stio n, every o th e r h e lic a l elem ent was removed, so th a t th e r e s u lt was a 12-elem ent rin g array with 135 an elem ent spacing o f approxim ately two w avelengths. Thus, a doubling of the in te re le m en t spacing would co n sid erab ly reduce any mutual coupling. However, a f t e r th is array was te s te d, th e beam t i l t was s t i l l lim ite d to approxim ately - 11, in d ic a tin g th a t th e mutual coupling was n o t p rim a rily resp o n sib le fo r th e phase s h if tin g beh av io r. I t was found th a t the use o f a sm all co n ical counterpoise f o r each h e lix such as describ ed by Svennerus [1958] s ta b iliz e d the phase c e n te r s u f f ic ie n tly so th a t much la rg e r angular scan ranges could be obtain ed. An array u sin g these m odified h e lic a l elem ents is describ ed in the next se c tio n. C, An 8-H elix Ring Array Fed by a C irc u la r R adial Cavity This s e c tio n d escrib es a rin g array o f e ig h t m odified h e lic e s fed by a c ir c u la r ra d ia l c av ity o f the same dimensions as the c a v ity used f o r the 24-elem ent array. This array perform ed very much as p re d ic te d, and a beam t i l t of about - 20 was obtained, 1, D etails of Element C onstruction The b a sic elem ent of the array is a ten tu rn 14 p itc h angle h e lix with a sm all conical counterpoise as shown in Fig, 40. The co unterpoise has th e dimensions suggested by Svennerus [1958] and is made of sh ee t b ra s s. The ten tu rn h e lix is wound o f 1/8 inch

154 136 wood a c r y lic tube 1/ 2" a c ry lic rod 25" upper and lower c a v ity p la te s Fig. 40. Ten tu rn h e lix in sid e sm all conical co unterpoise mounted on upper c av ity p la te

155 137 copper wire on a 2 1/2 inch O.D. (1/8 inch w all) a c r y lic (p la s tic ) tube and is coupled to th e c a v ity by means o f a 3/4 inch probe, as shown. The upper end of the tube is lo o sely sea te d in a p o ly eth y len e plug supported by a p l a s t i c tr ip o d, so th a t the e n tir e tu b e -h e lix com bination is fre e to r o ta te in sid e the co n ical co u n terp o ise. Although th ere is a co n sid erab le amount o f p l a s t i c in the n e a r f i e l d o f the h e lix, i t does n o t s e rio u s ly a ffe c t th e operatio n o f the a rra y as a whole. Any spurious su rfa ce waves s e t up on the p la s t ic tu b e, fo r example, would be m anifested only in the p a tte rn o f the in d iv id u a l h e lix and would be v i r t u a lly masked o u t by the much narrow er array fa c to r. The trip o d support could be elim in a te d by desig n in g th e a c ry lic tube so th a t i t is s e l f - su p p o rtin g from the b ase. Fig. 41 is a photograph o f the 8-elem ent array mounted on the antenna t e s t p la tfo rm. The c a v ity p la te s are sq u are, a l though the i n t e r i o r is c ir c u la r. The e ig h t h e lic a l elem ents can be seen along w ith th e p l a s t i c trip o d support fo r each elem ent and the sm all co n ical co u n terp o ises. The low er edge o f the c a v ity is 2 1/2 fe e t above the p latfo rm, so th a t the antenna is immersed in a very n e a rly plane wave. The feed p o in t w ith tuned d e te c to r is lo cated in the c e n te r o f the r e a r c a v ity p la te and cannot be seen in the photograph.

156 Fig, 41, Photograph of the 8-h e lix array mounted on antenna te s t platform. w 00

157 Far-zone Power P a tte rn s, Measured and C alculated Figs, compare th e measured and c a lc u la te d far-zo n e power p a tte rn s fo r th e 8- h e lix a rra y, fo r t i l t angles o f 0, S, 10, and 15 re s p e c tiv e ly. The rin g ra d iu s o f the a rra y is th e same as fo r th e 24-elem ent array p rev io u sly mentioned (R = 3.66X ) and th e element angular spacing is 45. This gives an in terelem en t spacing of d = 2R sin(a<{i/2) X * = 2(3.66) sin 22 5 = 2.8 (5-12) For an elem ent spacing th is w ide, high sid elo b es and g ra tin g lobes would be expected. However, the purpose in b u ild in g an array w ith th is wide an elem ent spacing was to elim in ate any mutual coupling between elem ents in an attem pt to obtain a phase scanning a rra y, even a t the expense of high sid elo b es and g ra tin g lo b es. I t was l a t e r found th a t such a wide elem ent spacing is no t necessary fo r proper phase scanning perform ance. The use of the co n ical counterpoise s ta b iliz e s the phase c e n te r and reduces in terelem en t coupling v ia common ground p lan e c u rre n ts so th a t elem ent spacings o f the order o f one wavelength can be su ccessfu lly used. The measured and c a lc u la te d p a tte rn s ( a l l norm alized to 1.0) show good agreement fo r a l l fo u r scan a n g les. The p o sitio n s

158 Measure' * Angle from Broadside P ig. 42. Measured and c a lc u la te d fa r-z o n e power p a tte r n s f o r an 8 - h e lix r in g a rra y phased f o r a b ro a d sid e beam.

159 IH 5 O CL. -a o>n 0.8 Measure Iculated (0 6 o 0.4 z Angle from Broadside F ig M easured and c a lc u la te d fa r-z o n e power p a tte r n s f o r an 8 -h e lix rin g a rra y phased f o r a 5 t i l t a n g le.

160 0.8 Calculated 0.6 % Angle from Broadside F ig. 44, Measured and c a lc u la te d fa r-z o n e power p a tte r n s f o r an 8 -h e lix rin g a rra y phased f o r a 10 t i l t a n g le. N)

161 u 3 o CL ^ 0.6 0) N Measureil - Ca cula :ed * -28* * -16* -12* - 8-4* 0 4* 8* 12* 16 20* 24* 28* 32* Angle from Broadside F ig M easured and c a lc u la te d fa r-z o n e power p a tte r n s f o r an 8 -h e lix r in g a rra y phased f o r a 15* t i l t a n g le. A. W

162 144 o f the sid elo b e minima and maxima g en erally agree to a few p e rc e n t; th e measured and c alc u la te d HPBWs show very good c o r r e la tio n. In g e n e ra l, th e measured sid elo b e le v el is h ig h e r than the c a lc u la te d sid elo b e le v e l, in d ic a tin g th a t sm all phase e rro rs e x is t in the angular s e tt in g of each h e lix. No s o p h is tic a te d method was used fo r the angular s e ttin g o f an elem ent and the type of s e tt in g e rro rs which were known to e x is t e a s ily accounts f o r th e in crease in sid elo b e le v e l. The main beam can be s h ifte d to about i 20 from broadsid e and s t i l l be id e n tif ie d as "the" main beam. However, as th is beam is s h if te d away from b ro a d sid e, high sid elo b es s t a r t to appear so th a t fo r th e 0^ = 15 case (F ig. 4 5 ), one sid elo b e i s alm ost equal in am plitude to the main lobe. At the scan angle where th is sid elo b e becomes equal to th e am plitude of the main-- lobe, i t is c a lle d a g ra tin g lo b e. As p re v io u sly m entioned, th e se g ra tin g lobes are p rim a rily due to the wide in terelem en t spacing and to the fa c t th a t the rin g ra d iu s is la r g e r than one w avelength. Viewed from the stan d p o in t o f a continuous a p e rtu re, th is antenna has a larg e "hole" in the ap ertu re d is tr ib u tio n, thus lead in g to a high sid elo b e le v e l. Although p a tte rn s t i l t e d in the +0-d ir e c tio n are the only ones compared h e re, measured and c a lc u la te d p a tte rn s t i l t e d in the - 0-d ir e c tio n a lso show good agreement. The c a lc u la te d and measured p o sitio n s of the beam t i l t angle g en erally agree to b e t t e r than one degree over the -20 an g u lar scan range. This array thus has dem onstrated th a t a rin g

163 145 array o f h e lic e s fed by a common ra d ia l c a v ity can be su ccessf u lly used as a phase scanning antenna. D. A 14-Helix C oncentric Ring Array Fed by a Quadrant Cavity 1. D e tails of Cavity C onstruction and Array Geometry This se c tio n d escrib es a 3 rin g array of 14 h e lic e s fed by a quadrant c a v ity, i. e. one fo u rth of a complete c ir c u la r r a d ia l c av ity. I t was p o in ted out to the au th o r by P ro fesso r J. D. Kraus th a t in a larg e array fed by a sin g le la rg e c a v ity, i t might be d e sira b le to cut the c ir c u la r c a v ity in to a number o f wedges, each wedge being fed s e p a ra te ly. The metal w alls form ing the "spokes" of the c av ity would then provide a convenie n t means fo r su pporting the upper cav ity w all. An example of th is type o f co n stru ctio n is shown in Fig. 4 6 (a ), where a 56-elem ent array o f 3 rin g s is shown. The elem ents in one quadrant are shown darkened. I f each such 14-element quadrant is fed s e p a ra te ly, as shown, and the four quadrants are p ro p erly combined, then the t o t a l o f 56 elem ents can be made to behave as though fed by a sin g le common c ir c u la r ra d ia l c av ity. F ig. 46(b) shows a 14-element quadrant along w ith feed p o in t and c a v ity w a lls. The feed p o in t is lo cated a t rad iu s r^ and an angle o f 45*. Ring ra d ii are denoted as r^, rg, and r^. T ransverse c a v ity w alls are located at <j) = 0, <{> = 90*, and r = a. An array o f th is type has been c o n stru cted and su c c e ssfu lly te s t e d. Each o f the 14 elem ents was a 10 tu rn 14* p itc h angle h e lix

164 146 oo ty p ic a l elem ent o u te r c a v ity w all P o oq ty p ic a l c e n tr a l feed p o in t (a) r o r r 2 r (b) Fig (a) A 56-elem ent quadrant c a v ity - fed c o n cen tric rin g a rra y. (b) One 14-elem ent quadrant array, C avity w alls and feed p o in t are shown.

165 147 w ith a sm all co n ica l c o u n terp o ise, such as used in the 8- h e lix a rra y p re v io u sly d escrib ed. The rad iu s o f the quadrant feed c a v ity (a) fo r th is arrangem ent was in ch es, and the c av ity was o perated in the dominant mode. The f ie ld s in sid e th is cav ity can be found from equations (C -26), (C -27), and (C -28), where 4 = it/ 2 : 0 E = (k ^/ju e)[a H Î ^ \k r ) + A. H ^ ^ \k r ) ] sin(2q*) (5-13) z o 2q 1 2q = (2q/r)[A ^ (kr) + A. H ^ J ^ k r)] cos(2q*) (5-14) H = -[A d (H (k r)} + A. d { H y -'(k r)} ] sin(2q*) (5-15) * o ^ Zq 1 Applying the boundary co n d itio n th a t th e e l e c t r i c f i e l d must go to zero a t r = a, i t follow s th a t Eg = (k^a/jwc) *J2q^^2n sin(2q*) (5-16) = (2qA/r) J2q(*2n cos(2q*) (5-17) = CkA) [J^^_^(X2^ r/a ) - (2q /k r) J 2q^^2n s i n ( 2q 4,) (5-18) Here q is understood to be a mode number, and A is a co n stan t. For th e dominant mode, q = 1. This is th e sim p lest mode to e x c ite and is th e most u sefu l fo r the purpose of feeding an a rra y of h e lic a l elem ents. The q = 1 mode can be e x c ite d by a sim ple probe in s e r te d at the p o s itio n o f the f i r s t ra d ia l f ie ld maximum (see Fig. 47). For th is case,

166 148 E = (k^a/jwe) J - ( x - r/a ) s in Z* z 6 /n = (2A /r) 2(j> (5-19) (5-20) % " r/a ) - C2/k r) J 2(x2j^ r /a ) ] s in 2<j> (5-21) where th e sep a ra tio n condition re q u ire s th a t (5-22) th and where X2^ is the n ro o t of Jg(x) = 0, Table 4 l i s t s the ro o ts x^^ [Abramowitz and Stegun, 1964]: TABLE 4 ROOTS OF = 0 n *2n n *2n n *2n n *2n n *2n , , I t was d e sired to o perate the c a v ity a t resonance n e ar 1400 MHz., (X = 8.45 in c h e s). Thus, from equation (5-2 2 ), th e root Xg^^ must be n ear Xzn = ka = (6.28/8.4 5 )( ) = (5-2 3 ) From Table 4, th e c lo se s t ro o t to is

167 = (5-24) For th e given c a v ity, th is corresponds to a resonant frequency o f» (3xlO )C49.442)/C6.28)(1.71) h i. = 1380 MHz. (5-25) T h erefo re, at a resonant frequency o f 1380 MHz., th e e l e c t r i c f i e l d in sid e th e c a v ity can be w ritte n as ^z ^o ^2( r/a ) sin 2* (5-26) where E i s a c o n sta n t, o A co n to u r map o f the ab so lu te value o f i s shown in Fig. 47, in sid e a cutaway view of th e quadrant c a v ity n e ar the o rig in. Only the f i r s t s ix maxima of E^[ are shown, but the f ie ld s tru c tu re in the rem aining p a r t of the c a v ity is q u ite s im ila r. I t can be seen th a t in accordance with the boundary c o n d itio n s, the f i r s t maximum of the f ie ld does n ot occur at th e o rig in. For any ra d iu s, i t is seen th a t the f ie ld e x h ib its a maximum on th e lin e b is e c tin g th e two o u te r edges (ip = 45 ). When the e n tir e c a v ity is examined, i t is found th a t th e re are 15 such maxima, a l l occuring a t ( ) =45. As the rad iu s in creases from zero, the phase of the e l e c t r i c f ie ld jumps by 180 at every zero o f J2( r / a ). The p o s i t i o n s o f th e maxima and minima of ( r /a ) are shown in Table 5:

168 tran sv erse cav ity w all y tran sv erse cav ity wall cavity bottom p la te Fig. 47. Cutaway view of quadrant cav ity showing a contour map of the e le c tr ic f ie ld in o

169 151 TABLE 5 MAXIMA AND MINIMA OF ( r/a ) Max. No. P o sitio n (r/a ) Value Min. No. P o sitio n ( r /a ) Value , ,130 7 ' Since i t was d e sire d to feed a ll th re e rin g s in phase, th e rin g r a d ii were chosen to coincide w ith th e 4 th, 6th, and 8th maxima o f th e unloaded c a v ity, i. e. at r a d ii 31.0 in ch es, 48.0 in ch es, and 65.1 inches re s p e c tiv e ly. This gave a spacing between rin g s o f approxim ately 2 free-sp ace w avelengths. The c e n tr a l feed probe was lo cated a t the rad iu s o f the f i r s t maximum o f J2( r / a ), i. e. a t r / a, o r 4.17 in ch es, and a t an angle o f 45 from one c a v ity edge. The c e n tra l feed probe, a 3/4 inch O.D. copper tu b in g, is shown in Fig. 4 8 (a). length o f 3/16 inch The length 3/4 inch was a r b i t r a r i l y chosen, b ut i t was a fo rtu n a te ch o ice; i t turned out in la t e r measurements th a t the in put impedance at resonance w ith th a t p a r tic u la r arrangem ent was almost e x a c tly 50 ohms, when the c av ity was loaded w ith elem ent probes.

170 152 upper c a v ity p la te vniniinnnniimrriiiirmiiiifirmiitiiiiiin-m T3/4" iz mrrrrnn rmiirii im^ Type N ^ c h assis connector >*7~ /n/nn/n lower cav ity p la te (a), h e lic a l conductor 2" 6-32 b ra ss screw 3/16" O.D. threaded copper tubing polyethylene plug Î w m m T P //////////2 t. upper c a v ity p la te l _ i lower c av ity p la te ÿ/z/z///////////////////// / n /ïtrnrnn/tnj n fnn (b) Fig, 48. (a) D etails o f c e n tr a l feed probe fo r quadrant c a v ity. (b) V ariable probe depth feed arrangement fo r a ty p ic a l h e lix.

171 The h e lic a l elem ents were id e n tic a l to the ones used in 153 th e 8-elem ent array p re v io u sly d escrib ed. However, th e probe arrangem ent was changed s lig h tly ; the probe length was made a d ju sta b le by using a 2 inch 6-32 screw in sid e a sh o rt threaded p ie ce of copper tubing so ld ered to the h e lic a l conductor, as shown in Fig. 4 8 (b ). This c o n stru ctio n p e rm itte d the probe depth o f each elem ent to be e a s ily a d ju sted from 0 inches to a f u l l 1 inch. I t i s o f in te r e s t to know what probe depths w ill be re q u ire d fo r each elem ent in order to give a uniform e x c ita tio n of th e a rra y. I t is obvious th a t sin ce the e l e c t r i c f ie ld is weakest fo r elem ents n e a r th e cav ity w a lls, th a t the probes fo r those elem ents must be s e t deeper than those clo se to the c e n tr a l feed p o in t. Fig. 49 shows the elem ent numbering system f o r elem ents of the quadrant a rra y. The m,n^^ elem ent is understo o d to be th e n^^ elem ent counting counterclockw ise from the h o riz o n ta l lin e in te r s e c tin g rin g m. A f i r s t estim ate o f the probe depths can be obtained by assuming th a t the probes do not p e rtu rb the f ie ld s in sid e th e unloaded c a v ity. I t is d e sire d to induce th e same v oltage on a l l pro b es, i. e. '' ' ^. n Pm.n < ^ - 5 f o r a l l m = 1,2,3,...M, n = 1, 2, 3,...N(m) and where V = v o ltag e induced on any probe

172 \3,2 C entral Feed Ring Ring No. 2 Ring Fig. 49. I l l u s t r a t i n g the numbering system fo r elem ents of the quadrant a rra y.

173 155 E m,n p m,n = e l e c t r i c f i e l d a t m.n^^ probe = depth o f m,n^^ probe Assuming now th a t the e l e c t r i c f i e l d is unperturbed. E = E J ( r /a ) sin 2* (5-28) n»,n o 2 m mn where i t is re c a lle d th a t r is th e rad iu s o f th e rin g, m <()^ is th e azimuth angle of th e n^^ elem ent in the rin g, and Eq i s a c o n stan t which can fo r th is purpose be s e t to 1. I t is a lso re c a lle d th a t r^ /a = 0,460, r ^ /a = 0.713, and r ^ /a = Thus fo r th e innerm ost rin g. E, = sin 2* (5-29) 1,n In For th e middle rin g, E2 n = 0*137 sin 2*^^ (5-30) For the o u te r rin g, E = sin 2<^ (5-31) 3,n Since th e elem ents 3,1 and 3,8 must be probed d eep est, suppose th a t these depths are a r b i t r a r i l y s e t to 3/4 inch. Then the vo ltag e induced on these elem ents is V = (0.125) sin 2(5:62) (3/4) (5-32) = Thus th e probe depth o f the m,n^^ elem ent is found in inches as

174 156 p = /CO. 168 s in 2<fi ) inches l,n In p = /CO. 137 s in 2if> ) inches 2 ^n 2n p = /CO. 125 sin 2<j> ) inches 3,n 3n CS-33) (5-34) (5-35) Table 6 l i s t s the probe depths c a lc u la te d on th is b a sis, TABLE 6 CALCULATED PROBE DEPTHS FOR 14-ELEMENT ARRAY Probe No. Depth Cinches) Probe No, Depth Cinches) , , , , , , ,8.750 I t was found th a t when the probe depths were s e t according to Table 6, the measured power p a tte rn s d isp lay ed a c o n sis te n tly h ig h e r sid elo b e lev el than would be c a lc u la te d on the assum ption o f uniform e x c ita tio n. This was an in d ic a tio n th a t the o u ter elem ents were being probed too deeply, producing too larg e an in v erse ta p e r in th e ap ertu re d is tr ib u tio n. The probe depths were then a d ju sted so th a t the c e n tr a l elem ents were more h eav ily e x c ite d. This t r i a l and e r r o r procedure brought the sidelobe le v e l and HPBW in to values commensurate w ith those expected on

175 157 th e b a s is o f a uniform e x c ita tio n. The conclusion from th is is th a t the probes do p e rtu rb th e f ie ld s to some extent^ and the deeper th e probe p e n e tra tio n, th e more serio u s the p e rtu rb a tio n. The p e rtu rb a tio n is most lik e ly o f a lo cal n a tu re, since i t has been experim en tally found th a t th e in tro d u c tio n of the probes a ffe c ts th e c av ity resonant frequency very l i t t l e. An a n a ly tic a l e stim atio n of th e p e rtu rb in g e f f e c t o f a probe is a m athem atical task o f gargantuan p ro p o rtio n s and is co n sid erab ly com plicated by th e in te ra c tio n between probes in s id e the c av ity as w ell as the h e lic a l elem ents on the o u tsid e of the c a v ity. Although the f i r s t o rd er approxim ation technique ju s t o u tlin e d does n o t give a uniform e x c ita tio n, i t can be used as a s t a r t i n g p o in t in an i t e r a tiv e procedure. Chapter VI d iscu sses a measurement technique f o r determ ining the p ro p er probe depths in the case of a larg e array where the adjustm ent procedure p rev io u sly mentioned is im p ra c tic a l. 2. Far-zone Power P a tte rn s, Measured and C alculated F igs compare the measured and c a lc u la te d far-zo n e power p a tte rn s fo r the 14 -h elix quadrant a rra y, fo r beam t i l t angles o f 0, 5, 10, 15, and 20 re s p e c tiv e ly. The probe depths were s e t to 1/2 inch fo r the two elem ents in the in n e r rin g, to 5/8 inch fo r th e fo u r c en te r rin g elem ents, and to 3/4 inch fo r the e ig h t o u te r rin g elem ents. The c a lc u la tio n s were made by u sin g the c o n c e n tric rin g array theory p rev io u sly discussed and by assuming the elem ent p a tte rn to be th a t o f a

176 0.8 <u o Cu Ca] cula:ed * * 20 25* Angle from Broadside Fig. 50. Measured and calcu lated far-zone power p a tte rn s fo r a 14-helix quadrant ring array phased fo r a broadside beam. tn 00

177 Ca cula:ed ^ 0.6 o CL <U N H^ 0,4 (d B u o z Angle from Broadside Fig. 51. Measured and c alcu lated far-zone power p a tte rn s fo r a 14-helix quadrant rin g array phased fo r a 5 t i l t angle. tn «0

178 Measured C alculated I o a. -n 0)N H rhno 2: ' - 20 ' -15' ' ' 10' 15' 20' 25' Angle from Broadside Fig. 52. Measured and c alcu lated far-zone power p a tte rn s fo r a 14-helix quadrant ring array phased fo r a 10 t i l t angle. o\ o

179 1.0 \V UU 0.8 %0.6 C L T3 0) M CO / SUT6C // / / - Cal culat ed / / / / / / / I 1 t J' ; / / / / \\r \ s. \ V / / XvC> h / / \l \ \ u 11 n \ \ \i u \ \\,\ 1 1 \ \ \ \ \ \ \ \ o - 5 C1 5 o V Angle from Broadside Fig. 53. Measured and calcu lated far-zone power p a tte rn s fo r a 14-helix quadrant ring array phased fo r a 15 t i l t angle. Ov

180 Me asured C alculated IS -10 Angle from Broadside Fig. 54. Measured and calcu lated far-zone power p a tte rn s fo r a 14-helix quadrant ring array phased fo r a 20" t i l t angle. o\ M

181 sim ple 10 tu rn h e lix, describ ed by equations (3-35) and (3-3 6 ). Both measured and c a lc u la te d p a tte rn s were in th e h o riz o n ta l 163 p la n e, with the array mounted as in Fig. 49. All elem ents were assumed to be uniform ly e x c ite d. The a c tu a l num erical. c a lc u la tio n s were c a rrie d out by a s p e c ia l F o rtran computer program w ritte n by P. N, Myers f o r th e OSU Radio O bservatory s IBM 1130 computer. The measured and c a lc u la te d p a tte rn s ( a l l norm alized to 1. 0) show g en erally good agreement fo r a l l fiv e scan an g les. I t i s noted th a t the p o s itio n s o f the sid elo b es o f the measured p a tte rn s show a f a i r l y sy stem atic s h i f t of 1. 8 from the c a l c u la te d p o s itio n s. This d iffe re n c e i s due to a s lig h t m isa lig n ment of the zero degree re fe re n ce m arker used in the measured p a tte rn s (see Fig. 3 6 ), which occurred a f t e r th e broadside beam p a tte rn had been tak en. The agreement in sid elo b e le v el and sid elo b e p o s itio n s is not as good f o r the 15 and 2 0 t i l t angle p a tte rn s (F ig s. 53 and 54). However, as th e beam i s phase t i l t e d fo r la rg e r an g les, th e accuracy o f the angular s e ttin g s o f th e elem ents becomes more im portant in s o fa r as sid elo b e s tru c tu re i s concerned. Phase s e ttin g e rro rs ten d to r a is e th e sid elo b e le v el as w ell as s h i f t the sid elo b e p o s itio n s from t h e i r th e o r e tic a l v a lu es. I t can be seen th a t the maximum beam t i l t is about 18, as shown in Fig. 54. For th is case, even though the a rra y is phased fo r a 2 0 beam t i l t, the p a tte rn f a c to r of th e h e lic a l elem ent prevents t h i s f u l l t i l t from being re a liz e d. A s lig h tly

182 164 g re a te r t i l t could be obtained by reducing the number of tu rn s on the h e lic e s, b u t th is would in c re ase the a x ia l r a tio. The beamwidth v a rie s from 7.0 to 7.5 as th e array is phase scanned and the sid elo b e le v e l g e n erally in creases fo r in c re a sin g scan angle, although th e measured p a tte rn fo r th e 2 0 t i l t angle case shows a lower sid elo b e lev el than p re d ic te d. However, th e p a tte rn measured fo r th e -20 t i l t angle case (not shown) showed much b e t t e r agreement w ith th e c a lc u la te d curve. The im portant conclusion from th ese p a tte rn s is th a t a number o f c o n cen tric rin g s o f elem ents can be fed from a common quadrant c a v ity, and th a t power d iv isio n among the rin g s can be c o n tro lle d by varying the probe depths. The probes s e t up a lo c a l p e rtu rb a tio n of the f i e l d in th e v ic in ity of each p robe, so th a t the probe depths must be determ ined by em p irical p ro ced u res. F ig. 55 i s th e c a lc u la te d power p a tte rn fo r a 5 6 -h elix 3 rin g array which would be obtained from th e combination of fo u r 1 4 -h elix quadrants o f th e type describ ed above (see Fig. 4 6 (a )). The phasing of the elem ents is fo r a broadside beam. The HPBW is 3.0 and th e f i r s t sidelobe le v e l i s at -12 db. 3. Input Impedance o f the Quadrant Array The in p u t impedance o f the quadrant array was measured by th e techniques d escribed in th e f i r s t sec tio n of th is ch ap ter. A graph o f th e in p u t VSWR vs. frequency is shown in Fig. 56.

183 1.0 h 0.6 G _ Angle from Broadside Fig. 55. C alculated far-zone power p a tte rn fo r a 56-helix array of three concentric rings (four 14-helix quadrant a rra y s ), phased fo r a broadside beam. o\ in

184 ce Frequency in MHz Fig. 56. Measured input VSWR vs. frequency fo r the quadrant cav ity -fed 14-helix rin g array. O' o\

185 167 The probe depths f o r these measurements were 1/2 in ch, 5/8 in ch, and 3 /4 inch f o r th e in n e r, m iddle, and o u te r rin g s, re s p e c tiv e ly. The minimum VSWR occurs a t 1378 MHz. which is the c av ity resonant frequency. This is very close to th e reso n an t frequency o f 1380 ^Mz. c a lc u la te d on th e unloaded c av ity assum ption. This minimum VSWR value is very close to 1.0, which in d ic a te s th a t the in p u t impedance is v i r t u a l l y 50 ohms a t resonance. The in p u t VSWR i s seen to r i s e very ra p id ly as th e frequency moves away from resonance. This in d ic a te s a sm all impedance bandwidth and conseq u en tly a high c a v ity -fe d antenna Q [Rhodes, 1966]. Thus the 14-elem ent probes do not p e rtu rb the c a v ity f i e l d on a la rg e -s c a le b a s is. 4. Radio O bservation o f the Sun with the 14-H elix Array The 1 4 -h elix quadrant array has been used f o r the observatio n o f th e sun a t 1378 MHz. The array was p laced on th e ground p lane d ir e c tly above the prime focus la b o ra to ry of the larg e ra d io te le sc o p e antenna o f th e Ohio S ta te U n iv ersity [Kraus, 1963] so th a t the L-band radiom eter of the rad io telesco p e could be used w ith a sh o rt tran sm issio n lin e connection to the a rra y. Although the radiom eter norm ally uses a liq u id n itro g e n - cooled p aram etric a m p lifie r [Uenohara and. Elward, 1964] as a f i r s t s ta g e, i t was not p o ssib le to use th is h ig h ly s e n s itiv e f i r s t stag e w ith th e a rra y, since th e quadrant c a v ity resonant frequency was out o f the pass band of the p aram etric a m p lifie r.

186 168 However, th e rem ainder o f th e radiom eter was used. The input was through a c r y s ta l m ixer and the Dicke type radiom eter was used to sw itch the a rra y ag ain st a liq u id n itro g en -c o o le d 50 ohm te rm in a tio n. The analog output was recorded on a Honeywell s t r i p ch art re c o rd er. The tran sm issio n lin e connecting the array to th e re c e iv e r was composed o f a s e c tio n of waveguide w ith a sh o rt fle x ib le p ie ce o f co ax ial lin e on e ith e r end fo r connection to the re c e iv e r sw itch and to th e c a v ity in p u t connector, re s p e c tiv e ly. The to t a l lin e length was about 25 f e e t. The c a lib r a tin g noise tube was lo cated about 5 tran sm issio n lin e fe e t from th e input to th e re c e iv e r. A p o rtio n o f the waveguide and co ax ial lin e can be seen in th e lower l e f t o f Fig, 57 which also shows the 1 4 -h elix array mounted in place fo r the s o la r o b serv atio n s. In th e background can be seen a se c tio n o f the t i l t a b l e f l a t r e f l e c t o r o f th e large rad io te le sc o p e antenna o f the O.S.U, Radio O bservatory. The array was phased fo r a broadside beam and was t i l t e d so as to p o in t at th e d e c lin a tio n o f th e sun, which was about +20* a t the dates o f th e o b serv atio n s. The array was used as a m e rid ia n -tra n s it instrum ent fo r fo u r ob serv atio n s o f th e sun on fo u r su ccessiv e days. The record o b tained on May 21, 1967 i s shown in Fig, 58, fo r which th e d e c lin a tio n s e tt in g was 20*00'. A long b a se lin e has been drawn in to in d ic a te th e general d ire c tio n o f d r i f t of th e record. To the l e f t of the main beam can be seen a sm all r is e corresponding to a sid elo b e of the antenna.

187 Fig. 57. Photograph of the quadrant cav ity -fed 14-helix array mounted above the prime focus laboratory of the O.S.U. radio telescope. O' to