THE UNSYMMETR‘luLLY FED PROLATE' sPHERomAL ANTENNA » Thesis for m Dogma of Ph. D. mcmcm STATE some; Rugs) A. Myws 1954 THESIS LIBRARY Michigan State University This is to certify that the thesis entitled The Unsymnetrically Fed Prola’oe Spheroidal Antenna presented by Hugo Alexander Myers has been accepted towards fulfillment of the requirements for PhD degree in Mathematics Wfimxflk Major professor Date MW MSU LIBRARIES .-,‘-. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wiII be charged if book is returned after the date stamped beIow. THE UNSIMMEI’RICAILI FED PROL‘LTE SPHEROIDAL ANTENNA b ll Hugo A 1. Myers A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requiranentc for the degree or DQ313011 OF PHILO SOPHI Department of anthem atice 19514 ACKNWS The author wishes to express his sincere thanks to Professor C. P. Wells, under whose constant supervision and unfailing interest this investigation was undertaken and to when the results are herewith dedicated. He is also greatly indebted to Professor R. D. Spence and Mr. B . C . Hatcher for the use of unpublished tables of the prolate spheroidal wave functions. Grateful acknowledg- ment is also due Professor F. neraog and many others in the Mathematics and Electrical Engineering Departments for their kind help and encouragement . The writer deeply appreciates the financial support of the Office of Ordnance Research for the past six months which made it possible for him to complete this investi- gation. ‘ 'll!|l Jail-.1 Ill is‘i 'Illl‘ll‘l VITA Hugo Alexander layers candidate for the degree of Doctor of Philosophy Final examination, September 15, 1951;, 10:00 A. 11., Physics Conference Room Dissertation: The Unsysnetrically-fed Prolate Spheroidal Antenna Outline of Studies hajor subject: Applied Mathematics liner subjects: Electrical Engineering, Physics Biographical Itens Born, July 11, 1925, Lansing, nichigan Undergraduate Studies , University of Oklahoma 19M-l9h6 Graduate Studies, Michigan State College, 19145-1950 continued 1950-1951: Experience: Engineer, International Telephone & Telegraph, 19h6, Graduate Assistant in Electrical. Engineering, 1950-1951, Graduate Assistant in asthmatics, 1953-1951:, Graduate Research Assistant in asthmatics, 195k Member of Ian Beta Pi, Phi Kappa Phi, Pi nu Epsilon, Eta Kappa Nu, Sign Tan, Signs Pi Sign; Associate Ito-her of the Society of the Sign Xi. ABSTRACT The problem studied in this thesis is the unsymetrically-fed prolate spheroidal transmitting antenna . The prolate spheroidal functions are expressed in the form of power series and Laurent series. Radiation patterns have been obtained for antennas of three different lengths up to about one wave length long, for length/thickness ratios of about 5/1, 10/1, 22/1, and 316/1, and for nine unsymmetrical gap locations as well as for the symmetrically-fed cases . Two methods have been develOped for computing antenna impedances that take into consideration the width of the gap and the geometry of the transmission ’ line feeding the antenna. One method is based on the usual assmption that the current is driven by the field in the gap . The other method is based on the asstnnption that the current in the antenna elasents is driven by the component of the electric field of the transmission line that is tangential to the elements. Further refinements in the impedance theory will probably have to depend upon experimental evidence . | 1. II it. ,I 1' llllx‘ll‘.‘ TABIE 0? CW8 Section Page I mmm’CTIw O O O C O O O I O C O O O O 000000 O O O O O O O O O O O O o O O ........ O 1 11 moms}; smsaomu FUNCTIONS.. . ............. .. .......... 9 III assures cmurrons Aim RADIATION PATTERNS .......... '. 28 n commsxoxs O O O O I O O ..... O O O O O O O ...... O ...... O ...... C O C O O 97 BELIOWOOOOOOOOOOOOOOOOOO. OOOOOOOOOOOOOOOOOOOOOOOOOO O. 00000 1m ‘l I III [.1 I'll-'1‘" LIST OF FIGIRES Figure ' Page 1 . The Prolate Spheroidal Coordinate mater: . . . . . . . . . . . . . . . . . . 8 2. Prolate Spheroids........................... .......... .... l9 3. Delta-fed Antenna... 31 h. Spheroidal Antenna at the end'of a Two Wire Line. . . . 31 5. Two WireTrananission Line... 32 6. Tangential Component of the Applied Field............... .. 3h 7. Typical Plot of Current Distribution and Phase Angle. . . .. . 39 8 . Plot of Current Distribution and Phase fingle (Narrow gap) . no 9. Plot of Current Distribution and Phase Angle (Wide gap) . . . bl 10-60. Radiation Patterns.......................................h6-96 Table 1. Impedance Constants. 37 illii‘lll‘llllll l I INTRODUCTION The problem of the radiating antenna when studied mathematically becomes an exterior boundary value problem. By this the following is meant: Given a region R with finite or infinite boundary 8, a solu- ticn of a differential equation or, as in the case of the antenna, a cysts: of differential equations is sought which 1) takes on given values on the boundary 8, 2) is regular everywhere exterior to 3, and 3) satisfies some special regularity condition at infinity. In the case of the antenna, the system of differential equations is naxwen's electrmagnetic equations , the solution of which will give the electric field 2 and/or the magnetic field H. The boundary conditions that must be satisfied on S are well known1 and will be discussed for the particular problem later. The regularity of the solution at infinity is usually described for radiation problans as the radiation condition. In contrast to potential problas , vanishing at infinity is not sufficient for solutions of radiation problaas , whether electrodynanic or acoustic in origin} 1 J. A. Stratton Electrcnamtic Theog nearest-Hill Dev fork pp. 3h-37, 19m. ’ ’ ’ ’ 3 A. J. H. Sousarfeld, Partial Differential a ations in ice, Tr. by E. o. Straus , Academic—Press, use—ToTk, p" '. $335; 153. For those cases when Maxwell's equations can be reduced to find- ing the solution of the scalar equation Vzp + 13¢ I 0, the radiation condition for time variation of the form a" hat is 113 H32 - 11¢) - 0. r->oo 3 r - This condition can also be stated more generally for the 3' and H fields themselves as has been shown by Synge .3 For an arbitrary finite surface S the exterior , or for that matter the corresponding interior, boundu-y value problem presents great dif- ficulties . In fact , for a simple finite cylinder with ends , so that S is composed of the lateral surface plus the ends , no solution has so far been found for the exterior problaa except by numerical approxi- nations. The masher of choices that can be blade for 8 for which explicit solutions can be found is extremely limited. Conditions can be stated that will detemine surfaces 8 for which solutions can be found. For ample, let u;(x,y,s) - c1, u,tx,y,a) - c3, u,(x,y,a) - c3 be triply orthogonal funnies of surfaces. .Then '1) if the scalar wave equation ‘7'” + In? - 0 when written in the coordinate system uz, u., n, has a separable solution p - ftu1)g(u.)htu,) and 2) if the 1' J. L. Synge and c. 3. Albert, The General Problan of Antenna Radiation and the Fundunentsl Integ-al Equation, with Application to an Antenna of Revolution Part 1, Quarterly of Applied Hath. vol . VI , no. 2, p. 119, was). surface S is m of the times surfaces 11; - c1, u3 - ea, or u, , c., then the exterior boundary value problen can be solved. For regions with finite boundaries this restricts the possibilities to such simple surfaces as spheres and spheroids .‘ Of these a thin prolate spheroid , which in the limiting case becomes a rod, or a cement of a straiglt line, offers an excellent approximation to the circular cylindrical antenna. Problems related to the free oscillations of prolate spheroids have been discussed frequently in the literature .‘ The problem of the forced oscillations of a prolate spheroid was first considered by Page and Adana.‘ They treated the case of the thin spheroid driven by a plane wave whose electric field was parallel to the major axis. This would correspond to the case of the receiving antenna with its gap or feed point shorted out. Some of the ideas used here in constructing radial spheroidal functions were first presented by Page and ideas. ‘ Exterior problems for other finite surfaces have been solved for Laplace's equation, see, e.g., S. N. Karp, Se aration of Variables and Wiener-Ho f Techni ues, Research Report No. fi HES New Iork Univ. ., New Iort, 15$. ‘ See, e.g., M. Abraham, Die electrischen Schwingungen us einen stabfomigen Leiter behandelt nach der Maxwell 'schen Theorie, An. der Pmik 66, pp. 1135-572, 1898. J. Meixner has recently published maxw papers on heroidal functions. See Math. Nachrichten Band 3, Heft 11, April (1950, Band 5, Heft 1, March ( 9 , & Ban ,Heft 6, August (1951) Archiv der mum, Band 1,Heft3 (19118/119) & sand 1, Heft 6 (19118 957mm:j::j;:d:er:1>j1§slk, Band 6, (191:9)41 Band 7, Heft 3-h, (1950f, Zeitschrift fur snggwandte 951511;, Band 1,1133 12, Dec (19119) a Band 3, l; Zeitsc t fur an ewandtc hathematik und Mechanik, Band 28, Heft 10, October EBB. L. Page and N. I. Adsns, The Electrical Oscillations of a Prolate Spheroid, I, Pm, Rev., 53, pp. 819-831, may 15,1938). ~ 111‘. \ll-Il‘llll‘l‘lll‘ Chu and Stratton" attacked the case of the center-fed prolate spheroidal transmitting antenna by quite different methods and ob- tained curves from which the impedance at the gap could be estimated. The basic electrmagnetic theory used in this thesis follows the analysis of Chu and Stratton as elaborated by Sohelkunorrf n. 1!. Ryder“ extended the work of Page and Adm by investigating the behavior of the harmonies due to forced oscillations by perturb- ation methods. In a second and third paper Page" treated the more general vector wave equation and extended his previous results for the thin receiving antenna. The problem of determining the current distribution and impedance of an unsymetrically driven cylindrical antenna was forwulated by Ronald King.11 He nade use of an integral equation which he solved by the method of successive approximations to obtain general expressions for the current and the impedance. He was able to find a simple approxi- nats expression for the impedance of the unsynetricslly driven antenna 7 L. J. Chu and J. A. Stratton, Forced Oscillations of a Prolate Spheroid, Jour. Appl. .M, 12, pp. 2111-2118, March (19111). . S. A. Schelkunoff, Advanced Antenna Theory, John Riley & Sons, New Iork, pp. 111-125, (19%): . 9 R. 14. Ryder, The Electrical Oscillations of a Perfectly Conduct- ing Prolate Spheroid, Jour. Am, gm” 13, pp. 327-310, new (191:2). '9 L. Page, The Electrical Oscillations of a Prolate Spheroid, Ii tnd III, Pgs. Rev., 65, pp. 98-117, February 1 and 15, (19111;). n Ronold King, A etricall Driven Antennas and the Sleeve Ikllmle , unclassified tecmmcfi report no. 93 for Office of Naval search, Gruft Lab., Harvard, 1919. cl 1|‘l‘ Ill .1 II‘ 1.11 involving a series combination of the knows: impedances of symmetric- ally driven antennas . The impedance and current distribution for a cylindrical antenna of length 3 7“/h driven 73/}; true one end were evaluated and the broad-band properties were discussed . The theory and tables of prolate spheroidal functions was first published by Stratton, horse, Chu, and thither,m hereafter referred to as Stratton at 31. The equations they solved were more general than the unwell equations in spheroidal coordinates in that they include the latter as a special case. This work was extended by Spence“ and nest recently by Hatcher.“ new of the values for the noras and the radial functions used in this thesis were taken from the tables that had been worked out by Spence and Hatcher. The particular problem studied in this thesis is the unsymetric- ally-fed prolate spheroidal transnitting antenna . Radiation patterns have been obtained for antennas of three different lengths up to about one wave length long, for length/thickness ratios of about 5/1, 10/1, 22/1, end 316/1, end for nine unsymmetrical gep locations as well es for the metrically-fed cases . Two methods have been developed for 1’ J. A. Stratton, P. )1. horse, L. J. 61111, and a. A. anther, gigtic Cilinder and mercidal Wave. Functions , John Wiley and Sons , New ork, . , n R. D. Spence, The Scattem of Sound Pro- Prolate Spheroids , Final Report, Office 0 Na searc , ll , . u E. c. Batcher, Jr., Radiation of a Point Di le located at the Tip of a Prolate Spheroid, H. S. thesis, iichigan State aoilege, . s thesis hes recently been published in the August 1951; issue of the Journal of Applied Physics . computing antenna impedances which take into consideration the width of the gap and the geometry of the transmission line feeding the antenna. Further refinements in the impedance theory will probably have to depend upon experimental evidence . The reason for the choice of this thesis tapic was a very practi- cal one. It was desired to design and construct a TV antenna which would receive.Detroit on channels two and four and Kaluasoo on channel three without having to rotate the antenna. Prom radiation pattern honours-onto of various unsynetricslly-fed cylindrical antennas it was found that an antenna very similar to the one analysed by King had a radiation pattern consisting of two lobes whose directions of maximum radiation were about 130 demos apart . This was the pattern that was desired. The integral equation which King used in this problem is extraely complicated and a large amount of computing is required to find even a first approximation to the solution. From a purely nethe- naticsl point of view it has not even been proved that successive approximations will converge to a solution of the integral equation. Substituting a prolate spheroid for the circular cylinder permits one to proceed directly from Maxwell's equations and use the exact spher- oidal functions . Moreover the approximation involved in replacing the cylinder by the spheroid is probably nuch better than the approximation one must use in solving the integral equation.“ 'In addition, prolate spheroids are good approximations to the bodies of .niss flea and so 1' Per a discussion of this see Ryder, op, cit., p. 327. i From Lansing have a plume; effect! vhicb : mum is e. mun; the. “‘9 Metric have a pmsical counterpart in their own right. The problem of and effects which enters in the cylindrical antenna and is of considerable importance is eliminated by going to the prolate spheroid. Also, eliminating these and effects permitted more attention to be given to the unsymmetrical effects and the spheroidal functions themselves . Figure l . The Prolate Spheroidsl Coordinate System A ! The prelim he mm and WWI! .“ \ crunm varied X ' Luv, 3 a \fi L 13 the 'mUOC 91 ‘ L\“3.Y3‘ For a honor-,9... E In 1; fire 1 "Mn-11y field. “‘0‘ to “mine C! RE\‘L 1 6 “einen. 18 17 1 1113’ 1‘ II PROIATE MORAL FUNCTIONS The prolate spheroidal coordinate system is shown in Figure 1. The radial and angular variables we taken as 2, 1, and d, after Schelkunoff.“ The relations between the spheroidal variables and the cartesian variables .x_, 1, and 3 are l 1 i l x - Luv, y - L1u3-1)‘\l-v’)’eos¢, s - L1u3-1)'5u-v3)'1sin¢. L is the semifocal distance . The metric coefficients are A 1 1 1 a .2 1 e1 - L(u'-v‘) 2(u3-1)1, e. - Ltu3-v3)’(1-v3)", ea - Lu: -1) (l-va)’. For a homogeneous medium limell's equations are mI curl Ell-fgfi, cur-107.5: . (1) C" E and? m the electric and magnetic field vectors; ,u and e are the permeability and dielectric constant of the medium . For sources and fields whose time variation is harmonic , it is possible and convenient to eliminate the time variable by the use of complex vectors." Then a - Re£ge1wt), end 33:: . Retiwgsiu’t). e2- 2W1, where g is the a u) oscillation frequency of the source. The factors 33 and 61 t A. 1‘ s. A. Schellcunoff, op. cit., p. 111. 1" J. Aharoni, Antennae, Clarendon Press, Oflord, mgland, pp. 12 e 13, 19116. then drapped bx d \ equation. 1 Rotational 33m for thin "item‘s. A!“ in this cast 1: osmium, «z lent: Ire 800d (1 the meat ”18% t0 the current. 1,. "m ‘5 dishing 10 then dropped by convention, because they appear in each term of the equation. Thus equations (1) become curlE--iw/IH, curlH-iwel. (2) Rotational symmetry is seemed. The basis of this assinnption is that for thin antennas the diameter is small compared with one wave length, and in this case the potential of any point on a given circumference is essentially the same as that of any other because the antenna ele- ments are good conductors. In other words, the assmption is that the current that flows around the elements is negligible in comparison to the current that flows along the elements, and that the latter cur- rent is distributed uniformly sround the surface of the elements . This is a good approximation for the antennas treated in this thesis, and the thinner the antenna, the better the approximation will be. With the assxmption of rotational symmetry the equations and their solutions are independent of the 1 coordinate, and i can be taken as any convenient value; 1 is taken equal to zero. here, after Schelkunoff. In prolate spheroidal coordinates equations (2) become . la _ la) ’3‘“ m4#’ at “I‘m-é? (3) BSeIEJ) - aged?!) - - lad/618311;, an . a v The field intensities may thus be expressed in terms of the auxiliary wave function, A 2 fig. Thus mere fl s 23} my or the equation , It is come Men “Mm; c “Want a 01' Y‘Afim (v I .5 11 - 7 Eu ' - 22%} [(112-1) (113-13))- % 9A, .1 u- an a- 2') ’é 9A, (u) 3' 7%! ' u v I "a“? H; - + t(u’-1)(l-v3)]-%A, 7 =1 11/7;- where ,6 - 2 11’ /7\ ; 7t is the wave length corresponding to the fre- quency of the source , and where A satisfies the following differential equation, (ua-l) i2 + (l-v’) 3:}; + p aLama-want a O. (S) . . a u2 - e v3 . It is convenient to make the substitution flL - g. Anpere's circuitsl law,1° equation (6) , states that the line integral of magnetic intensity around a closed path 1: equal to the current enclosed by the path. the: . x, (6) 01' . 2w 0H,,» dd - I. (7) Hence the current in the antenna is 1hr) - 2 1" M1103). (8) The substitution I M ' l A - mqu) h (9) 1" A. B. Bronwell, 8: R. E. Bean, Theo 3nd Applicationjf Micro- waves, McGraw—Hill, New York, p. 21:8, 1W7. \ ”paste: the ‘. than for U m. BothUamV: u'tlandu onesatil regulumr UM , VM This to? M 12 separates the variables in equation (5), and the differential equa- tions for U and V are (ha-1mm. + (cauZ-RNJ so, (10) ‘33“ . (l-v')_d_'_V_ + (k-c'irzw a O. (11) dV' Both U and V satisfy the same equation. There are three singularities: u - t 1 and u - on. The singularity at infinity is irregular, but the ones at I l are regular.19 Therefore there exists one solution which is regular everywhere and one which is not.” Substituting U(u) - (uz-l) Um) , Vkv) - (l-v‘ka) , gives tut-l) 9:? + 1m dfi + (c‘u'—k+2)U - O, (12) - (in2 u , (143) an - 1w dV + (k-Z-c’v‘w - o. (13) '33 a? . - This form of the equation for spheroidal wave functions was used by Stratton _e_t _s_l_. Page and Adans and Ryder used the form (11) . At the ends (v - :1) of the spheroid the current vanishes; hence, V0.1) - 0. (114) This condition also follows from the fact that at the ends the field components must be finite. For most values of 3, equation (11) is E. L. Inca Ordin Differential_ Equations Dover Publications New York, p. 160, £955. , , ’° Ibid., pp. boo-hoz. ha no solution] \ Mimi t}! pro-i1 K eigehhmions a the tpheroid u 1 it muse dist: Bum.“ the t “billion ll < WeAp’ ”lunch. 0: Tms ) the the What 13 has no solutions satisfying the above condition; thus, equation (11) defines the proper or eigenvalues of E. The corresponding set of eigenfunctions are designated by V]: and U At large distances from k. the spheroid g is large, and equation (10) becomes d'u e c’U - o. (15) 1E: _ At these distances also . 1 r - u' + ya)! - Lu. (16) Because the time variation is of the form ei ““3 and a diverging wave solution is desired, at such distances the field must be prOportional to e ‘1 P r - e '1“. Hence, the proper Ukm) functions are those solutions of equation (10) that satisfy the following condition: Uk(u) a: 5-1“ as u —o oo. (17) Thus, the general solution satisfying the requirements at the ends of the spheroid and at infinity is A - szkm) kav). (18) The solution of the angular equation (11) was obtained in the form of a power series about the origin. Vkflv) = 20:0 "3;, V” (19) .. - n-O,l . The prime on the sumation sign indicates that the sunnation is taken o'er even powers of _v_ if k's with even subscripts are used, and over odd power: \19) mm This plu matter of k) '1: “1%. the 1h odd powers of _v_ if 108 with odd subscripts are used. mbstitution of {19) into {11) leads to the recursion formula - (n+2) (n+l)cn“ + [um-1) - k]cn + c'cna - 0 (20) This plus the boundary condition uh) then leads to e transcendental equation \i.e. , an equation with an infinite number of positive powers of k) in 5, the roots of which are the eigenvalues k 1 . Thus C°+Ca+...+c‘n +c2n+2+...80 . ‘21) fork .kO’ k2, h, 3150. and c1 " cs " ... * cant-1 + can-rs + ... '0 (22) for k - 1:1, k,, k., etc. The reason for this particular choice of subscripts for the k's is that the roots of equations (21) and (22) are all positive,“'and the mmerical value of It; lies between the values of kc and k., k, between 1:. and it” etc. - i.e., the sequence 51:9} is monotonically increas- ing with v . Now it happens that by taking the first seven terms of (21) or (22) one obtains values for the k's which are accurate to from 81:: to nine-places, the higher accuracy obtaining for the We with lower subscripts. This is to say that if one were to take eight terms 01‘ (21) as the approximation, the It, so obtained would be the sane as the 1:. obtained with seven toms to eight place accuracy - at least an. R. Courant and D. Hilbert, Methods of Mathematical Lhygics, Inter-Science Publishers, New fork, v. ,p. , . fer the ca: 5—1 15 for the case c - l, for this was the case that was computed in detail during the period of preliminary investigation. Thus for the prOper values of k, the contribution of cu and higher terms to the value of ZEO'~‘.:,.,vn is very small indeed, and so the physical boundary condition {1:13) leads to the mathematical condition that the on approach zero very rapidly for the preper values of k. Actually, since kau) is a solution of the 5592:; equation for u > 1 (see below), equation (19) converges for all v, and this implies that ling—[El 1-6. Otherwise stated, given [ >0, lcnl<€n for n > I£ . n... By swing the equations (20) for n a O, 2, ..., 2n-2, seaming op; - O, and defining 02,, by 00 111-1 omcan " Z car " 'goczr (23) r-n r one finds 030m-3 4* [b(2fl‘l) " °‘(k ‘02)] Can '0. (214) If, as it appears for k - kl and for r .3 m 7 R , the coefficients ca. decrease in absolute value and alternate in sign, then 92m in (2b) is less than unity and 16 - 3 m C can-.2 uncaCm‘Iz ; K‘wa) . ('16,) c3“ ' (2m)(an:l‘)r::zm_GIR-c2) ;' finmn-l) (2m)! (I This result is consistent with the assumed monotone decreasing char- acter of (’1)mCm for r 2 m. A similar argument holds for the c's with odd subscripts . ' When c is zero the values of kg are exactly 1( Q -l), and when c is mall they are found to be k_3- “1-1) + car-é -§(—29+fi(2£_-35=] + 0(c‘), (26) for - 2, 3, ... However, small inaccuracies in the values of k .2 completely upset the assumed rapid convergence of cm. The values of the separation constants k} used in this thesis were taken from the eleven place tables recently computed by the Bureau of Standards. The power series representation (19) of the angular function Vk(v) was chosen for three reasons. First, it led to the relatively simple recursion fomula (20). Second, the Vk(v) can be easily evaluated for g particular value v1, whereas tables of Associated Legendre Poly- nomials, as used by Stratton gt _a_l_. , are published only for a limited 17 number of values of the argunent. Finally, the power series can be easily multiplied by an expression for the applied electric field intensity and integrated by elementary methods to yield a value for the coefficient 3k in {18) . This representation has the disadvantage that values for the normalis ations can be more easily computed by other methods. (See Section III). The solution of the radial'equstion (10) subject to the condition at infinity (1?) can be expressed as 0(a) - e'icu ganUuYn, ‘0 - l. (27) 11-0 The argument in the power series was taken as (in) so that the an would all be real. Substitution .of this expression in (10) leads to the recursion formula for the an: 2c(n41) ‘nu‘i (n+1) n+c'dc]a#+2c(n-1) 54411-1) (n-2)an_,- 0 . (28) McCrea and Nearing“ have proven the existence of series solutions satisfying the boundary conditions for the generalised spheroidal wave equation, and therefore the expression (27) can converge and represent U(u) for lul .’. 1. Computations indicated that the series did in fact converge in every case for k - k} . _ _ For thin antennas it is necessary to evaluate the radial functions Mu) for values of “O in the neighborhood of one - i.e., uO - 1.020, as W. H. McCrea and R. A. Having, "Boundary Conditions for the Wavelgquation' , Pros” London Math. Soc., v. 37, London, pp. 520-5314, (193 . ...- M- 1335 , 1.00 the converg figure 2. 1n the for The sow. m g... H 18 1.005, 1.001, and 1.00001 - and this is just the region for which the convergence of (27) is very slow. The antennas are shown in Figure 2. One solution of the radial equation (10) can be obtained in the form- ' (1) U (u) - an (u ._1)n+ , for to, kg, 1“, (29) use The solution of the second kind can then be written as" (2) (1 1) u (u) - u (u) met-1) + {tonne-1)”. (30) The general solution is then (11) (2) Um) - C; U (u) + C; U (u), (31) and the 01 and 0. must be detemined so that the condition at infinity is satisfied. This was accomplished by expressing both (27) and Q 31) as Laurent series expansions in powers of g and then comparing co- efficients. The expansion of (27) was perfectly straightforward. One merely multiplies the power series expansion for 6 4°“ by the series 2 “sn(iu)'n to obtain n-O . "(11) " ... " ill-1‘3; " can + c": " c334 * 0‘ - ...) (32) . _ ‘2"? 3'. FE ' + (a0 "C‘l + 03‘; ' 63" § C‘a‘ " so.) 51' ‘3'? F? as E. L. Inca, op. cit., p. 16h. 2.1.. Figure 2 . _—— o» 7/ Prolate Spheroids u. = 1.00/ U0 1: LOOy do =/'020 19 20 + in (- cao + c'a; - c3a. + c4a. - c'a, + ...) '2'! 3".- E 3!— - u‘ (c'ao - c3a1 +c4s. ~61, + ...) + . 53% 3. 131' '55: Thus, although the an themselves do not converge rapidly, the g factors do, and so each term can be computed readily. However, the co- efficients of the u's with negative exponents converge very slowly, and so the computation of Mu) can not be carried out directly using (32). For the purposes of expanding U((21) (eq. 29) in powers of u it is important to note the following fact. U (31) is the only solution of (10) that is regular in the neighborhood of the singular point u - 1. Similarly, thr) is the only solution of (11) that is regular in the neighborhood of the singular int v - 1. But (10) and (11) are the same equation} Therefore U(](.u) and Vk(v) represent the same function, and by the identity theorem“ for power series the series representations of U 3(1)) and Vk(v) must be the some (for the same k, of course). When the (at-1)n+1 terms are expanded and multiplied by the bn to obtain a series in powers of 3 for Uka) it is found that the resulting co- efficients are exactly preportional to the °n for the corresponding thv) . Still amther way of obtaining an expression for the radial function is to write it in the form of a Laurent series: m“ fivw 3‘ K. Knapp, Theory of Functions, Tr. by F.3agemih1, part I, Dover Publications, p. 81, New York, 19145. \ \ i ‘l R 1 H Slbstitution of formula \23) . converge Very r exponent: is 3n efficients of t “11813 the re: 01), since th an expression “ms with no “i. Conside (”a Pllce ‘ U (11) Ink“! Theresa e “$.30, 21 ...” Uku) - Zrnun. (33) undo Substitution of this expression in £10) leads again to the recursion formula (23). The coefficients of thelterns with positive exponents converge very rapidly - in fact the series of terms with positive even exponents is just a multiple of kav) for k0, k3, ... - but the co- efficients of the terms with negative exponents converge very slowly. This is the result that was obtained also for the expressions (2?) and k 31) , since they represent the sane function. The problqn was to find an expression of Uku) that would not involve the coefficients of the terms with negative exponents directly. This was done in the following my. Considering Uku) as written in the fem (31), we notice that the 0 place powers of '2 with negative exponents can appear is in the U. U (u) lntu.‘,—l) term. low lntu‘—l) - ln u-l) + 2 1n(u+l) (31.) u and + ln(u+1) - ln(2+u-l) - ln 2 + ln(1+ u-l) , T or 1n (n+1) - ln(2) + gu-lz - % gu-l)' + 9: u-l ’ -... (35) 3 for-1(1). é 30 Therefore, ln(u+1) can be expanded in positive powers of u in a neigh- borhood of u - 1. Also, Therefore, the amount: is th e m that the Q 18 given by 2'4, n-0\ because Ra en $\ be Obtained from The series) ()6) comm b0. ‘w ”mm: the CC; mg” “‘7 Ilowli. Lf‘c \ln 1!) u-ou part of U‘\ D’% 2?. 1n 3%: - - 2(1/11 + 1/311’ + l/Su‘ + 1/7117 +...) (36) foru > 1. Therefore, the part of 1n(u’-1) that contains the terms with negative exponents ie the ln—-%- term. Remembering that 0‘81) - Vk(u) - g'cn un , we see that the part of U(u) containing terms with negative expgnents is given by Z'cnnanfi-i'f- This expression can be readily canputed because the 2:) converge very rapidly and the value of the logarith'n can be obtained from tables. The eerie: that reeulte as the product of }‘I,?'¢:nun and the eerie: (36) contains both poeitive and negative powera 2:02 and, as was to be expected, the coefficients of the terms with negative exponents con- verge very alowly. However , the part of the product containing the positive powers of 3 is given by (io'cnu 11’1“) .-2[u(03+%‘+%.+%°*%10+...) +u’(c. +‘31s +"a +310 " ...) 7 +u'(§_fi +c. +o1o + ...) + +uukgu +2“ + on + ...) + ...] = - 25%) un ; U7) “'1 . and theee terms converge rapidly because the en do. Therefore, the part of Uku) containing terms with negative exponents can be evaluated . w by aubtractigg the above expression from Z'cnunlnEE. n-O The {-111 part 0 export. | Mai “SD The 23 0° 00 mar - genufiu‘gfi + ZZ'Dnun. (38) n-l The minus and plus signs in the superscripts indicate that only the part of the series containing. the terms with negative or non-negative exponents is meant . ‘ Thus um) for k - kg, ha, k., can be evaluated in the neighbor- hood of u e l'in the following way: From (33) the terms with positive m even exponents can be expressed as a multiple of Vk - i.e., as AZ'cnun. n-O The constant 5 can be evaluated and checked by comparing the coefficient Acn with the coefficient of the corresponding positive even power of 2 in (32) . Thus, since co was taken as unity for convenience, ACO . A . (80 " 681 + 038.3 " 03" + C“‘ " 0". 4" ...) (39) E? ‘5': ET B"! and a check is 26 A03 . (C'ao " Caal ‘0' 3‘83 " 0.8: ‘5 C.“ " ...)o. (’40) 2'! '5‘.‘ E‘.’ 3‘.’ ‘6? It is slinost a necessity to have a simple and accurate check at every step in numerical computations. Again from (33) the coefficients of the terms with positive odd exponents can be obtained by evaluating three corresponding coefficients in (32) and checking that they satisfy —_‘ ‘° In some cases, for We with higher subscripts - e.g. k. - it may be necessary to equate corresponding coefficients of higher powers of 2 - u' and u', say - in order to maintain the desired accuracy, but the procedure is the sane. Since, for kg, ha, etc. there are nc negative even exponents in (33), a check on the an is to evaluate the coefficient of u" in (32) to be sure that it is sero. the recursio: positive odd (Milt. T}: can be em The first ‘ native e mgtthe < “Pming Dimmer: 391m. 31: ‘ mi ‘. 2h the recursion formula (23) . The remaining coefficients of terms with positive odd exponents can be evaluated by means of the recursion formula. Thus the complete expression for U(u) for Ra, 1:3, 1:4, ... can be written as 1 Mn) - Age 11“ + Zoo'c 11“ + B(§'c unlrfi—g + 231) un). (’41) . . mo 11 n-l n n-O n + n=1 :1 The first two terms on the right represent the terms in U(u) with non- negative exponents, and the last expression represents the terms with negative exponents. The constant B can be evaluated and checked by expanding (Bonunlfir and comparing the coefficients of l/un so obtained wiztlhothe corresponding coefficients from (32). Thus w - - 8(nZ-o'cnunlngfi) - - ZB [ .11; (co 4- %3 a» fi‘ +9]. + ...) (142) +l‘(C° * c. + c‘ + so.) * see] u Equating the coefficients of 1/11 in (32) and (1:2) , one expression for B is i(a -ca +c'a3-c’a + ...) B - 1 1' I: 3: ‘ , Koo +93 + c‘-+ e: +53. 4» ..J (143) and by equating the coefficients of 1/1:3 a check is 1(a, - ca‘ + c'a. - 9:“ + ...) . m If. 2', . (ht) -2C3° + c, + c‘ + c. + .j 3 B- Tc {MESS Int 30: tbs ‘LJ 25 To obtain the derivative of the radial function U'(u) , it is only necessary to differentiate each tone in (bl) . Thus 0° 00 no n-1 n-1 n-1 . u-l U' (u) . AZ'non “n + Z'ncu + ZBZ'nD u + BZ'nc u .ln—I - n-0 n-1 nu n-l n n-O n ‘1‘” + BZ'c nu n~2/(113-l) (145) n-Orl In this expression all the constants have already been detemined, and so only the series need to be aimed again because of the factor of 3 that was introduced by the differentiation. The radial functions for the It's with odd subscripts are canputed in exactly the sane way, the only difference being that the summations over the odd powers in the previous case are now taken over the even powers, and vice-versa. Thus, for k1, ka, 1:“, etc., 0(u) - AXI'cnun + Sl'cnun + B(nX'cnunlr€-§ + 2Z'D n.un) (146) n-l n0 :10 where Ac; - A - i(-cao 4- ca; - c‘a. + c‘a, - ...), (b7) and a check is Ac, - i(c’ao - c‘a1 + gs, - 35a, + ...). (b8) a. - ca, tong/2! - Cains/3'. 4- c‘ag/h'. - (149) B- c +c, +c +c, t...) and a check is B. " (N - cae o c‘ae/Z! - c’aq/B! + c‘u/M - (50) 2 5f + c377 + 0.79 «- c-Jfie ...) In this (:85 The mic acu- 1" 26 In this case Do - (C; + 03/3 ‘0' 05/5 + 01/7 + see), (51) D. - (c, + c5/3 «5 37/5 + c9/7 4- ...), (52) D‘ - (c. + c,/3 4- 09/5 + 011/7 + ...), etc. (53) The derivative of the radial function Ugu) for this case is than as 00 an U'(u) - AZ'nc u”.1 + Z'ncnun'1 + ZBX'nDnun":l n-l n n-O n0 (Sh) °° n-1 ‘u-l °° ' + BZincnu olfi + BXicnunZ/(u'él) n- n- for k -kx, k3, kc, so. Finally, if sufficient accuracy has been maintained in the coapu- tation of the radial function and its derivative, a good check on the accuracy of U(u) and U'(u) can be obtained by the use of the Wronskian. If one writes U(\l) ' U; '* “a (55) were U, is the real part and U. is the imaginary part, including the i, substitution of u1 and 0, into (10),zmd1tip1ying'the first equation by B; and the"second':by U 1, and subtracting the two resulting equations leads to 0.0; - 010; - 0, (56) or, 0:51; ' ”a”; " cs (57) there C r“. ¢.I.‘....fil4t. . F s ! |. liltltl|$r II...) P..- tlbrirl ,~ I . . . . .. 27 where Q is a constant that can be detemined from the behavior of the radial function at infinity. Thus, as g approaches infinity Ulm cos(cu), U, m - i sin(cu) 0; m -c sin(cu), U; m -ic cos(cu), and therefore 0:0, - 0,0; - i_o. (58) 28 III IMPEDANCE CALCULATIONS and RADIATION PATTERNS Prom eQuations (h) , the . electric intensity tangential to a typical spheroid confocal with the given spheroid is ,)-1 1-.) .-.)“ u'()v(). (59) E'(uv fl.[(v (uvl'lzkgkku k' At the perfectly conducting surface of the given spheroid u - no the total electric intensity should vanish; Mao,» + 2301.,» - o, (to) where 3:010 ,v) is the tangential component of the 222‘ lied electric field . Therefore, $3: [use (as ea) 1"} zkgkugeomm - - use”). (on The orthogonality of the 1 functions is easily proven: a. 1 I l(l—v‘) -‘vk'nd' - 0 for k f n. (62) Multiplying (61) by ( fl L'/ir7)(u3-v3)‘;'n(v) , integrating, and using (62) yields “ S. A. Schelkunoff, op. cit., p. 115. 29 fix I L1 I 1E3 1&6 ‘5 0-1,) I ...a " ._. ...-....h- ,_. I My I 'nl. ." E‘ ,. 1 . , \ _ - /,. / ,2” .0/ ‘ fl . \“ 6 O o f” r‘ M .\ xx _\ r O ‘ \\ x I'. O O “4% \\ \\ . x \ \‘~ . 4/", \9 3 “ // b“. / \‘ \ '4 \ \ 1 3' x ' A :rvs'\ Irv ’9’ w ’0’ ’0 ’ 3 ."0 ..--" 0 , _,.r—" /‘9 Figum ll. he 4!)! £0 31:, u.=/,a-o/ «...-£9 4).: 7 Km ?0‘ {1° ,/ 5 E1;— j'th -?0 a Figure , 1h . Fa :« 53 a.” 4 e=fi/*’ JJO =/ 005'- r1), = Og\ ....u. . W6 Figure 17 . fa ' H._I ...-_ 05"? m ado “m. ,. ‘ A , g 2‘ 7‘ ..I 1' l s O <" ‘17 ‘I/ - . .’_I - "Us..-" “ ’- Figure 16 , 427° L-U'UL Figure 19. 06 SS *“ "rm.” *“ ‘ 0 ‘EW‘ mgr) -. .. .~, A I” z/a W1; / f’ 0 h 772.2% ¢I .. c’j \\ a 0 hf? ‘00 .Y \ :fl/‘ 2 . 3 /- c~M~0/ (R ‘I ' . o. ‘SWM'--_. - Figure 21 . 7/ , .% 0 Figure 22 . C’-/9’/rA #:s/oooor/ 26’ .1) , 59 IO)- ‘Wu ‘. -9" ’Kr.””‘ 1|) lnl/IIH|.IEPE:I.I«1I.I.IJI1' 7h,“ LII-"EH r r (“fl/w A *“%r7‘"' ”5 3 I: 00/ ”a: 0 rt 5 74" ' . I?"7 00 "re”, r f \ , , ,3 d 1/3 A l . / ‘ \ /’ / t. Figure 25 . e . fl! . 5 h 1/. ' /’ M / 7"?“ 4): '7 ’9‘ <=:::_=:l::=- W" 290 “Mo/arr ‘ “\fl\ ‘0 6 \‘ 9" __,_ \ /5’U’ ”of," W ch” \ / ’1’ ..Pl I I 'F if 63 7a” 2/ : .Cfl.m. I I. I‘“ . ’3 163 ,7” 700 to, 640 G=fll= 2 ”03/,0—0! @030 . , ...- n°__ a.“ 9 ’/’. ' ., s / ‘. j 1 l I ' 2/ i112: x... Q . VJ?) W; '99 1 .3 a; I f4 ‘3 J—a%?k ———b CT}. 10 a \ \ _\\ ~\. 67 Figure 31 . -‘-"-‘- 2.x- ‘3 an... .- , . <12. “sfifif /V. a). __ _“ “" ‘s- . .., (ml/’1 ado’flhf ’U. .3 70' 69 0% _ I3 '9‘ v/ \ T’s, "\ V \ 220 o /‘\ 1% i 2 I I 1 . / . ll u" l',/ - a 5° 1/ ‘ I 6:0 J a : mob/5W1 . ’2‘ 5' '05" ‘— £1” '/0 ,5“ C=fl/=2 M'J'o-as’ 4):“; ’0} 0 /?’0° p “#39 f/ ad / \ V 71 r ”‘5! 105' aim #- n ‘q. “Win... i .. I” U .. T ...» ..w .. 1"]: lfl' "‘9. ‘. "3:. 72 73 . . . 5‘," r[ y. w H‘ i!‘.b“ . . ' 7h 0 IO QQY Figure 38. 75 C-‘fif‘j’ o———%_A --—> M.- AMj’ m /U = O 0 qo' (yo ‘o‘ 0 /€‘o ° ‘4’ .7 we, . .J‘ /" . 3 o \ x (‘0‘. ‘ f0 0 Figure 39. . Tr?! . 5% a /. (my *"" # A —--> . / Q23 . . 7/ (7° ‘00 in“ II vs, e& . \' a f '3’ :3 “a I ’ 0" >400 ’00. ~70° Figure 10. 76 C Al. =/.rv.5’ éo . 77 0Q/ 78. o NnV J4 7.3.... ti F igure h2 cvflh 3 ....TW, 1/. 2/. 0133’ 0.2.4 Km 79 61 Figure 145 . azga ‘- c ,3]. 3 /€’0° of" it 82 I / /I/ 0 o .. , 3 1/ / 83 C’ = fl]: 3 #1 A 1/ =/.m5’ :: 4/ “ .7 o 900 ‘9” a... x” 8h 86 v- =0 . <1) 13., \ mo ° 9’ A C -. V \ L 05” /, \\ \ / / Cc lab” 6: w: 3 do 2/, 0A0 A). :.l ¢0 0 Figure 51 . 87 fififl.fl.| .l .l n... 5.1 ..‘. >nmtsnbl§ iii K. . ....WI 1 T . 88 a 00 70° Figure 52 . 89 -900 Figure 53 . 6=//: 3 ”0:’a0)o .. w ”h-‘ 49" 3“. viii... BI I IV |l1 . ... .. . I. a.. .. ..rl.h.-Cv ... 5! / ’90:. O ’00 Figure 55 . -‘.' 1 ‘ “ r' ““_IEJ I /(90 Figure 56- Mp~ 'v“-m ’ JO 0 / h 93 =3 C751 1/. /0 L010 .7 Figure 57 . .... ... . I fir" I....|l.l‘.llln 9W ... pd 9h (Hg/’5 9%}.pr A/o= /,0;L() may 95 .-.. :ry he mu‘flr i 1 fa ..H1.‘.. n.. . .'.m. n!9rll'..rl.|0.rl.uru.‘ la .rnl. J- 0 ‘44 96 O 97 IV 001C111 SIGNS The angular functions were obtained in the form of a power series expanded about the origin. This form has the following advantages: 1) The angular functions in this form have a simple recursion formula and are therefore easily computed. 2) The functions can be easily evaluated for any value of the variable, in contrast to angular functions that depend upon tables of associated Legendre polynomials. 3) If the applied field can be expressed in am of a nuuber of simple closed forms, use of the orthogonality preperties leads to an integral expression for the coefficients in the series expansion of the solution that can be evaluated by elementary methods . The disadvantage of the power series representation is that the norms are not as easily computed as they are by other methods . The radial functions were expressed in the form of a Laurent series. This method is valuable in that it furnishes an independent check on the values of the radial functions obtained by other methods. Its main advantage over other methods is that there is a complete check at almost every step of the computations. A disadvantage is that, in its present form, it is not as convenient to compute radial functions for a large number of L/D ratios for the sane g and g values. However, usually only two or three different L/D ratios are required, and the computation 98 of radial functions for extra L/D value: would require only a small fraction of the time spent in computing the radial constants. Two methods have been developed for canputing antenna impedances that take into consideration the width of the gap am the geometry of the transmission line feeding the antenna. One method is based on the usual assumption that the current is driven by the field in the gap. The other method is based on the assumption that the current in the antenna elements is driven by the component of the electric field of the transmission line that is tangential to the elanents. Using the L..- first method, the result is that, even for fairly wide gaps, the L . ’i impedance values obtained are essentially the same as those obtained L; under the assumption of a uniform step-function applied field over a ”small" gap. Using the second method, the impedance values obtained are slightly higher (on the order of 5%) than those calculated by the first method. In this case , the wider gap yielded a slightly higher value of input impedance than the narrow gap. To find out what the best approximation to the applied field is , calculations and experimental evidence vdll have to be compared for antennas less than or equal to a half wave length long. Radiation patterns have been obtained for antennas of three different lengths up to about one wave length long, for length/thickness ratios of about 5/1, 10/1, 22/1, and 316/1, and for nine unsymmetrical gap locations as well as for the symmetrically-fed cases. It was found 99 that the radiation patterns, like the impedances, were relatively insensitive to gap widths or'methods of feeding, but that they de- pended primarily on the frequency of the source, gap location, and length/thickness ratio. The tables for the angular and radial functions and their related constants will be included in a forthcoming report for the Office of Ordnance Research. ffi-‘r’ ‘ , . ” “ IL. , I, ‘.‘!1 ‘fi. '-——.. in! ,F ‘5'.“ M ‘r’ 10. 12. 13. 100 81131.10th Abraham, M. , Die electrischen Schwingungen an einen stabformigen Leiter, behandelt nach der Maxwell'schen Theorie, Ann. der Phfiik 66, (1898). . Aharoni, J ., Antennae, Clarendon Press, Oxford, England, 19%. Attwood, S. 3., Electric and Magnetic Fields, John Wiley 8: Sons, 191:8. . Beatty, R. W., Eggerimentel Check at 2,000 Megacygles of Calcu- 1 lated Antenna pedance, . . thesis, ass. Inst. of ech., 1. 1953 . ‘A.‘* v... 4 ‘-.¢. ‘. . Bronwell, A. B. , & Bean, R. 13., Theo and Application of Micro- Waves , McGraw-Hill, New Iork, 19E. Chu, L. J. and Stratton, J. A., Forced Oscillations of a Prolate L4 Spheroid, Jour. AppL 111323., 13, May k19h2). Courant, R. , and Hilbert, D. , Methods of Mathematical Physics , Inter-Science Publishers, New York, . l, 9 3... Hatcher, E. 0., Radiation of a Point Difigle Located at the Tip of a Prolate Spheroid, M. S. hes a, ichigan State ollege, . Ince, E. L. , Ordinary Differential Equations, Dover Publications, New Iork , Research epor so. , New ork niv., new ork King, Ronald, As etricall Driven Antennas and the Sleeve Di ole, unclassif e tec c report no. 3 or ice 0 Naval Research, Cruft Lab. , Harvard, 191:9. Knapp K. Theory of Functions Tr.,by F. Begemihl part I Dover Publications, Few Iork , 19145 . , ’ McCrea, W. H. , and uewing, R. A., Boundary Conditions for the Wave Equation, Free. London hath. Soc., v. 37, London, (1931;) . l.‘ _ “.2".' '_.._ Ty 101 114. Meixner, J. , Math lachrichten, Band 3, Heft 1;, April (1950) , Band 5, Heft 1'""""'fi, Marc ('1—5’9 17,‘ a Band 5, Heft 6, August (1951); Archiv der nun, Band 1, Heft 3, (19h8/h9) a Band 1, Heft 6, (1958;59); Ann. der Pmsik, Band 6, (l9h9) & Band 7, Heft 3-h, (1950 3 Zeitsc ft fur wandte P sik, Band 1, Heft 12, Dec. (1959) 5 Band 5, Refit E, (1951,; Zeitschrift fur wandte Mathematik und Mechanik, Band 28, Heft .15, October (19%; . 15. Myers, H. A., Fundsnentals of Antenna Radiation, Unpublished M. S. thesis, chiglan ate 0 age, . 16. Page, L. and Adana, N. 1., "The Electrical Oscillations of a Prolate Spheroid, I, P1113. Rev., 53, My 15, (1938). 17. Page, L., The Electrical Oscillations of a Prolate Spheroid, II 8: 111, Pm. Rev.,.65, Feb. 1 and 15, (19th). 18. Ryder, R. M. , The Electrical Oscillations of a Perfect Conducting . Prolate Spheroid, Jour. Applng” 13, May (1912 . 19. Schelkunoff, S. A. , Advanced Antenna Thegy, John Wiley & Sons New Iork , 1952 . 20. Somerfeld, A., Electrodynamics, Academic Press, 1952. 21. Sonnerfeld, A. J. W. , Partial Differential Eguations in Pmics, Tr. by E. G. Strauss ,Tcademic Press, ew Iork, 9r 22. Spence, R. D., The Scatterigg of Sound From Prolate $heroids, Final Report, ice 0 av esearch, - , 51. 23. Stratton, J. A. , Electromggnetic Theory, McGraw-Hill, New York, 19“. 2h. Stratton, J. A., Morse, P. 14., Chu, L. J., and Hutner, R. A. myth Cylinder and Spheroidsl Wave Functions , John Wiley ons, ew ork, . . . . . 25. Synge, J. L., and Albert, G. E., The General Problem of Antenna Radiation and the Fundamental Integral Equation, with Appli- cation to an Antenna of Revolution Part I, Quarterly of Applied Math. vol. VI, no. 2, \19hé). ‘ .. :7 :~ '1“?!- W “..-”. ‘ Cd’Imf! ‘W-Lm an—L... .- . . E I“ am A 1 . muummm' w..." wlll ILI VI U" THL 1293 03196 7031 3 IIIILILIILHIIWI I l