ll \ 1 fig LIHIWIHWIIHWWWWWHHHHHWHIW ON THE RADIATEQN {3? THE EQCQNICAL ANYENN‘A Thesis 532' 5‘9 Deg." 90 of 33“. MICHIGAN SMTE NWLRSITY Joseph Alphonse Meier 1957 —- l 1111111111111111111111111111 31293 01743 0301 d. / This is to certify that the thesis entitled "0n the Radiation of the Biconical Antenna" presented by J oeeph Alphonse Meier has been accepted towards fulfillment of the requirements for m.— degree in was Mew-11L.» Major professor Date June 1., 1957 0-169 LIBRAR y L§ T‘llic‘hfi 311 Stat c E Luntrsxt) 1i J ON THE RADIATION OF THE BICONICAL ANTENNA By Jeseph Alphonse Meier ABSTRACT Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1957 Approved by M 17. up“: ABSTRACT The Lebedev integral transform is applied to solve the mixed boundary value problem representing the radiation of a biconical antenna. The problem.is formally solved by use of the conventional Wiener - Hopf technique, and the above transform. This method does not lead to an explicit solution of the problem but to an infinite system of linear equations for the representation of the unknown transform function. ON THE RADIATION OF THE BICONICAL ANTENNA BY Joseph Alphonse Heier A THESIS Submitted to the School of Graduate Studies of Michigan state University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1957 ACKNOWLEDGMENTS The author wishes to express his sincere thanks to Dr. Alfred Leitner under whose constant supervision, and unfailing interest this investigation was undertaken. He also wishes to thank Dr. Charles P. Wells for his assistance and for the interest he has taken in this thesis. The writer is deeply indebted to the Office of Ordnance Research for the financial support which made this investigation possible. DEDICATION To Anita I. II. III. IV. VI. VII. TABLE OF CONTENTS Introduction . . . . . . Statement of the Problem . . Representation of the Solution . l. Preperties of A91) . . . . Solution of the Problem 1. Factorization of P1(coseo) -54}; behavior of the factors 2. Factorization of li(cos90) -§+’J behavior of the factors 5. Preperties of functions in Input Impedance at the origin of Biconical Antenna . . . Discussion of the Infinite System Appendix........ (59) Asymptotic the 15 21 23 27 53 58 1+2 16 INTRODUCTION The symmetrical biconical antenna of small apex angle was devised by Schelkunoff as a model for the simple wire dipole antenna for which no exact theory has yet been provided. The biconical structure permits discussion of the solution of Maxwell's equations in spherical coordinates. Study of this problem.showed that the portion of space between the two cones and bounded by their surfaces forms a transmission line, with principal and higher modes. Thus Schelkunoff reduces the radiation problem to what is essentially a circuit problem, viz. the terminated biconical transmission line "loaded by empty space" with a terminal impedance whose value depends on the apex angle of the cones, their slant height and the driving frequency (Schelkunoff-B). Schelkunoff calculates approximate values for this impedance making use of the simplifications that arise in this problem when the apex angle is nearly zero. By a simple transmission line transformation back to the origin, i.e. common vertex of the cone, the input impedance at the point generator can then be found. .A problem of considerable mathematical interest - even if of less practical significance - is the one involving symmetric biconical antennas of arbitrary apex angles, i.e. of any value between zero and 1809. This is a mixed boundary value problem which can and has been attacked in spherical coordinates by the method of separation of variables using series expansions in terms of the appropriate sets of wave functions (Schelkunoff-7). Being a mixed boundary value problem one is led to an infinite linear system for the coefficients of the expansions rather than explicit values for them. The reason for this is lack of orthogonality over the matching surface between the Open space and the transmission line space. Recent advances in the theory of mixed boundary value problems indicate that such problemm can be solved explicitly by use of integral transforms. We refer here to the Wiener - Hepf technique applied to dual integral equations applicable to two complementary parts of a boundary on which the boundary conditions are not the same. This method is surveyed critically by Karp (1). In the present problem we consider the antenna to be perfectly conducting finite conical sheets of equal apex angle and situated end to end on a common axis of symmetry. The boundary over which mixed conditions apply is the infinite double cone part of which is perfectly conducting the other part free space. In Schelkunoff's theory the conducting surfaces are capped by spherical surfaces. Here, in order to reduce the physical boundary to one coordinate surface we consider the uncapped structure. The integral transform appropriate to the geometry of this structure is the Lebedev - Kontorovich transform (5). It has recently been applied to the problem of radiation from circular disks (Leitner, Wells - 5) which when formulated in spherical coordinates is also a mixed boundary value problem of the two part type. The present thesis is an extension of this work to the theory of biconical antennas, in the hopecfi'obtaining simpler expressions for the input impedance of such antennas at the vertex. II STATEMENT OF THE PROBLEM Consider a finite right biconical shell, c, shown in the figure below. We assume that the generators of the cone make an angle 6 = 6 (O < e < n) with the z axis, and that the slant height 0 of the cone is a.. Mathematically, we propose the following boundary value problem: Find a function R¢(r,6) satisfying the differential equation 2 2 v2 60 csc 6 u-“ .- (1) H“ * 02 2 H¢ 0 where wis the frequency and c the speed of light, subject to the following conditions (a, ___c_3...__ 60°86‘8m93¢)=0, r oo , k2 :93. (Sommerfeld radiation condition). We seek a solution of the time independent Maxwell's equations (Stratton - 9) V x‘g + iiuuig = O V x In: -1we_E_=o subject to the boundary conditions (DE-=0 ,e=e tangential O (2) Continuity of the field functions in free space, (5) Sommerfeld radiation condition 3 )1 and E are inductive capacities, and w the angular wave frequency. It is well known that when symmetry with respect to C is assumed, these equations separate into two independent sets, 6 interested only in the first set of equations, namely one containing Er’ Ea, Hg and the other R¢, Hr' H . We are 1 3 - --- 1911 =iw£E rsine 60 (an ¢) r 1 a ' __ 3-5;” H¢)- nut-:36, as 1 e _ = _ Fla}? ‘1‘ E9) ear} “"f‘na' It is from this set of equations that one can derive the equation 2 7 H + —2- E + 1 a sine fl 4- ‘23 - 28—639. H = 0 81.2 1' Br r2 sine 66 SB 02 r2 9‘ which is equation (I) cited above. Physically, we have two conducting conical sheets fed by an alternating voltage of fixed frequency and amplitude between the two apices of the cone in the limit of zero gap. Such a method of excitation produces electrical currents on the conducting conical sheets which are purely radial. The electromagnetic field of the structure is transverse magnetic, i.e. the only component of magnetic field is H“ where C'is the azimuthal variable about the axis of the antenna. For such an excitation it is also known, from physical considerations, that H¢ has azimuthal symmetry, and planar symmetry about the plane 2 = O. III REPRESENTATION OF THE SOLUTION Writing the differential equation in spherical coordi- nates we have 2 a H 2 aH¢+3J+_&___§_,meEEE.92_-2§eEQ H 31,2 r Er 1.2 .1116 Be Be (32 r2 9‘ = 0' Application of the method of separation of variables yields (5) H¢ _._ oo 1 2:: - 5' - + 1 + 1 - n=o(kr) anJg*£lkr) bnY£+llkr) cnPh(cose) ann( case) 2 2 where J (kr),Y' (kr) and Pl(iicose) are the Bessel, ,1 4.1 n n n - 2 2 Neumann, and first order associated Lengendre functions, respectively. The coefficients ‘n’bn’c and dn are unknown n’ constants whose values, once known, would yield a formal solution to the problem.in terms of a series representation. For physical reasons cited previously, the solution to the boundary value problem is known everywhere provided we obtain the solution to the problem in regions (I) and (II) indicated in the figure below. 24\ I Azimuthal symmetry and planar symmetry about the plane z = 0 provides the solution in the entire space. We shall use a "function theoretic" method of solving the problem Wherein we consider the functions in (5) as functions of their order rather than their argument. Through such an approach we will apply the Wiener-H0pf technique (1) which is essentially an application of the concepts of analytic continuation and Liouville's theorem. From.this point of view, the appropriate representation of the solution in the given regions is not a series represen- tation, but an integral representation on the complex order plane. For the given set of boundary conditions the corresponding eigenfunctions do not form on orthogonal system. The usual methods for obtaining a formal solution in terms of a series representation can not be applied without infinite systems for the unknown quantities an to dn in (5), whose coefficients are of a very complicated structure, as can be found in the work of Schelkunoff (7). We consider the integral representations over a contour L, where L is a contour in a strip of finite width about the imaginary axis in the complex order plane, from o-- 1 co to 0'* 1 oo , a-being a real number. The reason for such a contour becomes evident if one considers the Wiener-Hopf technique which is to be applied. This approach requires a study of the functions involved as functions of their order in overlapping half planes whose common region contains L. The appropriate radial function for our solution, R¢ , is the Hankel function, H£2)(kr), where k is real. This is the function which satisfies the Sommerfeld condition cited above. However, in order that the integral representations of the solution do not diverge we must let k = -17f', where‘y‘is real. Such a substitution for k leads to the Macdonald function, - Im KFVIr). Since for |Im}1| >> 1, 13:2)(kr)~. 2 )1 and u ' a I‘m/“l , this change insures that our integrals converge along the contour L. Such a substitution leads to a transition from.a wave problem to one of ”exponential - decay” with the same boundary conditions. The integral representations over the contour L would diverge if one returned from real'XKto real k under the integral sign. However, these would converge if L were first deformed to surround semi-infinite portions of the real )u axis, and the consequent residue series leads to the correct results in the wave problem. This has recently been demonstrated in a paper by Oberhettinger (6), and verified in a paper by leitner and Wells (3). With these remarks we define our solutions in regions (I) and (II) as follows: —1=Jf A91)1§‘(7fr)Pi(cos 9)d,1, V” L " “'7‘ 2 (6)H¢(r,9) =1 0 5 6 < 90, P1(cos e)+ P1(-cos 6) - l - l 0712)): B()1)151(Tr) ~31? fl " L 1' -.1. sin 5‘?» 2) dp, w 60 5 0 5:5. My) and B(}1) are unknown functions to be determined for a formal solution of the problem. The term.sin §(u - %) appearing in the second integral is introduced in order to explicitly indicate the proper eigenfunction corresponding to the principal mode )1 = l of the biconical transmission line. In the limit/p= l the quantity in the bracket is indeterminate and has the value 1 sine Utilizing the relation (Magnua-h) , which is the appropriate principal mode function. ‘2“? one obtains from (6) the representation 55—:— §in6 19:}: cose) = 19.12 - 1:113” case) 3 ma H ) = (7) “3—0036 fl 2 1 fair 111191 - W “a???“ ° -‘- ° 5 “0 P(cose)-P(-cos6) fijrB‘WTW 1“" -4, 71711“ d)» sin --()1 --—) It follows from Maxwell's equations that inr Er (ineH ). .033 16 Since the tangential component of E, the electric field, is to be continuous at the boundary between regions (I) and (II) -11- cited previously we see that the Jump in Er’ the tangential component of E, across 6 = 60 for all r must be equal to zero. Computing the jump by use of (7) we have Sf“); - 1%) (Yr) Mp) - —§-9‘-% P(cos6o ) "’ L 5‘ ‘1“ 5‘?“% " “"71 + _.$B_B ) 1 P(--cos6o ) d); = 0 n sin Elf‘g)’ gtp for 9 3 60 and all r. Since this is to be true for all r we must have the expression in brackets equal to zero and it follows that P1(coseo ) - -1u <8) ital. =_; My). 31n.39u-l) P1(coseo)-1 P (-coseO ) 2. 2 - Efp' "-7u Before enforcing condition (5) we rewrite (6) in the form Hdlr,0) 3 sine FL}: 1915mm; green): , o 5 e 5 90 .. 2.3.2; P .BJBL 1 x (3‘11) P;(cos9)-P;(-cose) 91, I. L sin USPPEQ’P’ ' zip. ' zip 6 5.9:. “H: O where prime denotes derivative and where we have used the relations (Magnus-LL) P: (4' cose) =+ sine P1(_f cose). - 4;: ~71 Now enforcing condition (3) which states that H¢(r,6) must be continuous for 9 = 60 and r a a we have XL P51(Yr) [A9))-8 m. l] -P1(coaeo) + 1" 5-4401 2 2‘? +£(‘E-L- P(-coseo) c91=0 sin - - ~11 for6=6 and rza. 0 With the aid of the relation between A91) and 891) given by (8) one may rewrite the integral expression just given in the form WlPlgfioseo),-P£(-coseo)l at 1 (Yr) ___a a =0 P )1 5‘ p1(coseo-) Pl ( -coseo) P L "a"? “a”? where W[P1(coseo), P ( noose0 )1 is the Wronskian of the 1 5T 37* functions P (coseo) and P (-coseo), and has the value 1 1 ' '2"? “27‘ l .——2— cos 1: sin 60 P =1.:N -13- Hence, it follows that ( )j cos n 11 101) d e 9 LP P(oos6° )- P(-cos6o ) K(I'l (Xr) )1: -._tp .._ATI P 01 IV 9 Enforcing condition (2) which states that ‘9 (sine 3¢> = O cose for 6 = 60 land r‘g a we have 2 l - - (10) if. MPH}: - E) P1(cos60) KPQ‘rMP - O . 6 - 60 ' 5' r‘g a The expressions (9) and(10) form a set of dual integral equations for the unknown function A91). (1) Preperties of My) By use of certain general properties of the electromag- netic fields of radiating structures such as ours, it is now possible to discuss the behavior of A(P) and certain related “W' ' functions without explicitly knowing them. To do this one makes unseat the Lebedev transform theorem. Such an analysis is necessary in order that the conventional Wiener-Hopf technique can be applied without too much difficulty in later paragraphs. This, itheorem (5) we state as follows: LetA()1) be analytic in a strip of finite width about the imaginary axis containing L and having decay at least as rapid as I)“ Z‘r) in the distant parts of the strip. Then, -provided both integrals converge, mm =JLp/1 ()1) 51mm}: m .. 1 dr A91) " HS ¢(3‘r) KIJXI‘) -;- . 0 Here I (X‘r) is the modified Bessel function whose growth for )1 *Elmfll l (IMF With the aid of condition (2) and Maxwell's equations large 9 Implis we may write = 93;...— )1 A91) (P2 - 11:) 51(Xr) Finesse)? ' r L " '5 )1 One may show that if, 7,1- 1... -47 *7i_4_ a ._ _.___..,7fi_-.i-v_._.e, (11) 1911912 - 13;) P1(cos90) = .- .2.. +P ' m = - ¥J v; Er(r,90) 51(3'1')”. a -15- It is well known that H¢(r,6) need not be continuous (across the conducting conical sheets. If we denote by [H¢(r,60)] the discontinuity in Hd(r,e) when 6 = 60 and r 5 a we have I' _<_ a [H¢(r,60)] r‘z a O __ 2 .cos 1111 A01) - )u ,7 K ()fr)dp. w r sineo A P1(fos60) - Pi-foseo) P L ' 5 P " '5 P .. One may also show A ‘ .i 1, ______._____ _ --_ .3 _ .h . 1 (12) £113) cos 11p a “two IHHWJOH I (Xr)dr . P1(coseo) - P1(-coseo) 21 V—r- )‘ - .2. 4"; - 5+}; 0 Er(r,eo) is a continuous function of r for a _<_ r < oo and has the behavior 1 V r - a -1rr as r approaches a, and at infinity. IP(X‘r) has a r eYr finite value at r =.a and behaves as --—— as r approaches r infinity. Furthermore, Ifler) is an entire function of p . With these facts and theorems concerning integral represen- tations of entire functions it can be shown that the integral in the right member of (ll) defines an entire function of )1 . Similarly, it is known that [H¢(r,60)] behaves as the continuous function 3’; near r - O and is continuous elsewhere in the range of integration in (12). At the lower limit, zero, the integrand in the right member of (12) behaves essentially as Era/2 . Upon further consideration, it follows that the right member of (12) defines an analytic function of); for 1 38,1 ) 2 o 1 Now by (11) A94) has, at most, simple poles at/J = I 2 and at the zeroes, for fixed 60, of P (coseo). If MP) has no ‘ .. le- 2 l’ oo poles at any of these values then I V r Er(r,OO)IP()’r)dr a must have zeroes there. 1 '5': has simple poles at p. 8% , :2?» , i!”- , ... and only there. If By (12) A91) at most, on the right half plane Re}! > 8 not, then Ip(rr)dr has zeroes there. 1/? 0 Since Pl(coseo) = O for): equal to certain irrational 4. "if real numbers, A91) cannot have, on the right half plane Bela >§ , poles at the right hand zeroes of P1890890" since .2)J that would contradict (12). It follows that m J‘ 1/; Er(r,60) 51(Xr)dr = O at the zeroes of_P-1.(coseo). a . 2 I‘ Since by (11) 1191) does not have poles at )1 =3,%, 1321-1 ... , it follows that a [ng'seol] w1/? 1 have shown that, at most, My) may have poles at )5: 2 and at Bu(z‘r)dr = O at these values. In summary, we 0 the negative zeroes of P1930890), for fixed 60 . 2}“ We will now show that with certain assumptions, integrals (9) and (10) can be closed on half planes, thus insuring a formal solution for this set of dual integral equations. Actu- ally we must show that they can be closed on complementary half planes since this is essential to application of the Wiener-Hopf technique. We make the assumption that ‘3')“ '-—> oo in the right half plane RE}: > 5;:- ‘ Ya P H ( ,e ) --- .. S 11.9.3. 1 (fr)dr::: 2 P °‘ 0 Vr P P(1*}1) where Rad> O . This implies that in the distant portions of the strip we have 1'. - .1. - the behavior e+ 2 |T| (t) 2 Re d With this assumption.” 00‘ '9‘ M21) K (Yr) Pl(cos60) - P1(-cos00) )‘1 2 f“ 2 u - “ behaves as («i-T}: ciwhen (1‘)" co and asl’C'I Re on the strip. Hence, (9) converges and can be closed to the right when r Z a. In its present form (10) cannot be closed on either half -18- plane and a decomposition I (7‘ )'-I(X‘) K(X‘r)=g- .341: .P' r .3 up which is the definition of 5“?“ in terms of Ipht‘r), is necessary. It follows that (10) can be written in the form u 2 l I (tr) , (13) '52-ij - K) A()1)_?%(:;seo) in ")1 (1)1. ._ L w 2 l I I (Xr) - — c P = a airy: 1;) My)? %(f;aeo) Ji—-ain "P C)“ 0 We mention at this time that these arguments, involved in closing (9) on the right half plane Rs); >% , restrict the contour L such that a-> :- but finite. In order to carry out the following arguments it is necessary to make another but final restriction that % < cr< l. ’C’ a )1 - plane ‘P V q H Nh‘w H -19 .. The integrand of the second integral in (13) is free of poles in the strip - l < Re); < 1. If we make the transfor- mation): 8 71' , and note that (P2 - i) P1(£oseo) is an even “ P function of )1, we obtain I (Yr) 1 (22-1)AU' P( e)+£P air): I: P)_%f;l80 sinfl): where L' is a contour in the complex order plane from cr" 1 co to o - ice and -l < a~< -%— . (See figure above). With the aid of (11) and the assumption that the integral involved behaves as Iphfa) u-fi, where Rcfi> 0, one can show that the integrand of the above integral approaches zero on distant parts of either end of the strip -1 < Rep < 1. It follows by reversing the sense of L: that the resulting contour may be deformed into L such that one may write (13) in the form )1 (114.) jpgiz- i) a()1)-P%(foseo) 1710’s)? = O L r IA gs where a()z) a g. ME)“: A)?!” . By the use of (11) and the s T! definition of a()1) we have so 2 l _. w ()1 - 11’ a(P)’P%(:;SGO) - 7-75 M; Er(r,ao) gamma. -20- -7fr F at infinity. Furthermore S37?) is an entire function of )1. S‘P’r) has a finite value at r = a and behaves as e Using these facts and those cited previously in connection with Er(r,60) one'can show that the integral in the above expression converges and defines an entire function of )1. With regard to the value of this integral we assume J? E (r e ) K (Yr)dr;:: (Fa) " . r ’ ° 1‘ ’5» P where ReP> O, which, in the distant portions of the strip 27: becomes V ITI With this assumption the integrand of (1)4) behaves as 1T - T 0 2| '|T|-Rep . (3),}..5 when» --> - co and as ‘TVRcFon the strip. Hence (114.) can be closed to the left when r 5 a. Thus we have shown that this integral equation is formally satisfied. To summarize, it has been shown that the dual integral equations are convergent and meaningful statements of the problem when they are written in the form (1 ) “8 "3‘ “Bl—Mm x (rrm = o r > a S LP_Pl(f°860) '_Plg,-°°860) ’1 '- 2? '2'" (16) Jfi(P2- 3-) a()1) P1(coseo) I_ (X‘r)d}1 = O r 5 a L 1‘ " 2’71 ’1 where 9.91) = g “.911; 2:31.). . -21- IV SOLUTION OF THE PROBLEM We are now in a position to attack the revised integral equations by function - theoretical techniques. In order to carry out this procedure it is more convenient to write them I in the form (1; es)» ) ( )d = > a L (P NP) 5‘ fr P o r - P 1 Ta 7“ 7“ .L (2) where <17) {91): my: Asa; . “(12) _P1(fos60) - P1(:coseo) (Lg-DP '5 P -2 (1%" (18) c'( ) = u(2--1-) a< ) rccoae» 2 -- 4. D ()1) is a function analytic on the right half plane R6): > g and of algebraic decay of the order )fo‘in the strip, 2%.. < R9): < 1, and in all directions to its right. c'(;:) is an entire function and of algebraic decay of the order If P in' the strip and in all directions to its left. The superscripts 4‘ and - are to denote ”analytic” on a “plus" or "minus“ half -22- plane with the additional property of at least algebraic decay in those regions respectively. With the aid of the definition of My) we can now + .. establish a relationship between D ()1) and C ()1) of the form C()=( 1)P(cos60)[P (cos90-) P(cose )1 l )1 ’12 E:_ 2 +1” -‘:+ )u 2 cos "I” -2 fi’vn-(Ifi) PMD(;}. 2 [(171) We now define M1(cos90) I Plgcos 890 )- IP1(;coseo) “PIP 5P “I“ d tili th lti ) -1-- anu ze ereaon:—————o:flz8(%r +PF(2I‘) wherein we rewrite the previous expression in the form (19) c “(,n - -—-(}12-71;1)P(—+)1)P(%-p) 3.1.“;“0 o) lecoseo )- 2 2 F D ‘P’ (2‘) 2,11% ”"7“ - )1) Our immediate purpose is to transform this equation into one whose left and right hand members, say, are minus and plus functions respectively. It is at this point where a modicum of investigation reveals that such an equation defines an analytic function in the entire plane and hence by Liouville's Theorem is identically zero. This will be illustrated precisely in later paragraphs. However, in -25. order to carry out this procedure we need first to investi- gate some of the properties of P (0086 ) and M (coseo ). 1+0 1 2" 2"1 (l) Factorization of Plgfoseo) - Asymptotic Behavior of the factors. P190390) is known to possess a countable number of irrational simple zeroes which we shall denote by )1 (60), 3 m = °°° -2,-l,0,l,2,°-- For fixed 60 it is an entire even function of ‘p. and its zeroes are given asymptotically by the expression ) (20) P (90)~r-(m ' K) + m0 ) O ‘65.. 90 i" “E . It can be shownx that we may factor P1(cos6O ) as follows: -_2_+P (21) Pl(coseo) = ' _ (Inf-1‘0) : _Pl(coseo )TTl +#— e 9,130 2 “=1 )1 PE: 0) a: )1 (6 o) ’T—T l.-lL——- e P'mo Let us call the first infinite product appearing in (21) a See note 1 -2 1+. k " 91,60) and the second k‘().1,eo). In order to investigate Pficoseo) as a combination of "plus" and ”minus" functions defined previouslywe will need to know the asymptotic + - behavior of both k 91,60) and k 91,60). For the present we confine our attention to the growth 4. of 1; 91,60) as compared to the growth of the infiniteproduct 9 °° a . - OPX -1) L91) 1:116 11 / LR) ° We can express L91) in terms of gamma functions and obtain (22.) L(u) = FLZ/LLL . e 1' r1 W a) w 1 To carry out this transformation we have made use of the _z + z 6mg m+a where )0 (z) is the logarithmic derivative of the gamma 90" 1 1r expression (Magnus 41) m (25) He + 1) ezwen) a TI' 1 + ["(z + a + 1) m=1 function F”). To obtain (22) we take z = and a = -—- in (25) and compare with the original form of 1.91). 4. Now form the ratio I191), of k 91,60) and L91). We obtain -25- - 11 oo )1 ppwo) ,m k+( 6) 11(1 +P e°>6 P, 0 P,m (21+)R91) = L( ) = 00 P - .A‘..___I 1 + .B..__ m=1 F—(m-l)e %6(m " E) 90 11 Taking the logarithm of both sides and adding corresponding terms we get as log R().1) 8: log m==l 1-+.£L_.___. w _ 1 3117. 1 1- _ 00 PP :0) roe ‘ = 2108 i " Z —£————-1- - JET—3 = .25..— .. IL. .. 9 m 1 1 + l. - .13. m 1 60(m E) Pfimo since each series converges separately for all )1, as a consequence of the asymptotic property of u (60) (See (20)). P,m It follows that as m +p 0) :9): Z 1 - 5—3.... R()l)‘ T) Lu e n “=1 m “t 7,9): (90) . m=1 1 + +11 P,m w l --(m - —) For large lp' , in the region > 11 a- arg P l— 2- , R()1) becomes 1 r; 1 '1 {2-,- Z :71” " 63*“ (m - ) 1 —— (e ) my»; 1T E e m= I; 1' PP,mo m=1 60 ) E‘prgo or —B(e 0)}: R()1)~A(60 )e— where MOO) denotes the infinite product and 8(60) the infinite sum. (From equations (22) and (211) we have then <25) k 91.0 9 )ur‘ Me nag—"OB “6° ) #8)] P719“ 1) ° ’ R941: 0. A similar treatment of {91,00) yields (26) k'(y,eo)~F%)—1 . Moon )1 :OEGO ) ”431,391: 0. ¥ #- See note 2 (2) Factorization of M1 (coseo ) - Asymptotic Behavior of 2 the Factors . M1(coseo) has simple zeroes at the half odd integers - i+p )1 = +(2m - g), m = l,2,5,--- and, for fixed 60, at certain irrational values of .P' which we shall denote by 3;}1 (60), m = 1,2,3,'" . Furthermore, M (coseo) is an u’m - .]_.+ 2 P entire function of )1. The irrational zeroes are given asymptotically by the expression I Zmfl C(60) (7) pfeo) "-2904. Analogous to P1(cos6o ) Just investigated, M1(coseo) has an 2")“ " '2“? infinite product representation of the fonm* (28) M1(coseo) t - 2 4'}: 00 - 2 1 m p - 2 .;g M1(cos60 )I—l 1+ )1 1 e m 2 n l- Lg e m m=l 2m- m=l 2m- 1- ' , ’p.(90)e mel ‘p.(eo) M,m 00 + )1 :Pn(g°)m it PM :10) *See note 5 -28- Let us denote the last two factors of (28) by k;(’.1,90) and k;()z,eo) respectively. We now utilize (23) to rewrite the second and third factors in terms of gamma functions and it follows that we may write (28) in the form (29) H1 (coseo ) = -—l.+P -Ju H1(oosOO)-E—-—E——-——ez Y(E)e E)” (1):“ k M-(flfioh (.90.) 2 Or(e+n)r("2'* 1:) MP For future needs we now develop the asymptotic behavior 4. - of kl(P' 60) and kM(}.1,eo). We begin by considering the growth of k;(p,60) as compared to the growth of the infinite product 00 , - .- pht- 260) ”(I +P(fl 260) 6 2m T! 21111! N91) = A detailed discussion is not necessary in this case since the approach is exactly the same as that used to develop (25). We simply state the result which takes the form 9 l .. .9 ' - (50) k;(P'°o)”A (60) {1(2 1, ){B (90) c} A similar treatment of Iii-401,90) yields 9 a'e) -(l--9- B'e 43} PM?) 0 is the well known Euler constant and has the value .577215 0" . A' (60) is an infinite product given explicitly as oo a Zmu A (e ) = H 0 (6 )(v - 26 ) ' mp1 (”hang O B'(Oo) is an infinite sum of the form. 3'90): _- ( :(i )1” (O Sufi-200) Both expressions Just mentioned are convergent and represent constants for a particular value of 90. Using (21) and (28) we now rewrite (19) as (32) 0'91) = 1 2 l + - 3W ‘ ENC 91:90” 91.90)’ + ,_ 1‘1"? PU +11) 4' {1391) (a) F‘l'F)D(7u)} -50- 8 where 4. (55) x 91.90) - 2 .. 1 \j- P1(60800)M1(00860) r(— +P)Tr(1 + I“ 2)° In 2 2 2 2m - _ N m )1 (90) (D - 9 ) 111; 1 * ° P’m TT 1 + —£—- e PM’mo on)” P “o ) m=1 )1 (e ) ’ Psmo M,m° oo ‘ 2m - 1 \F’l(coseo)ldl(coseo) r095:- -)1)U 1 - )1 e 2 '2 ' E ”’1 2m - % Pfio "fin" -r( ) fl p(OQ°P)> ‘ .7le81( [AME]: LN): e )1 P,mo where HP) is so determined that the growths of 1691,00) and (91,90) are algebraic in the preper regions. To study the asymptotic behavior of 1891,90) and 1591,60) we refer to (25),(26),(29),(50),(§1) and thus Obtain -31- (55) K+(}1,60) = 1 1+) - P P‘ 9)H(0089)-E(fl[‘£2——’ \/-_:.°°a°-%- O l"(&+}.) 2 h )u 1 5%K)+f()1) ) k+()1,80)k;(}1,60)e (56) K-9‘seo) 3 Mare-1») . \/.P;(coseo)l}lcoseo) ( K) 2 2 k'(y,eo)k;()1,eo)o W491, «- Now these factors are, by Stirling's formula , asymptotically 4. (37) K 9:,90)~ - l \k%(ooseo)!%(coseo) P(fi)r(fi) A(60)A'(eo))1 2 ) Hell: 0 K-(P’GO)~ .. l \lfl(cos60)ll(cos60) r(fi)r(fi) MGOM' (eon-)1) 2, Rep: 0 , '2‘ -2 * -l r94)~}if‘/_§;_°.P for 'argr'<fl -52- We now begin the task of transforming equation (52) into one whose left member consists of functions all of which are minus functions and whose right member consists entirely of plus functions. It is important that all these functions, as functions of their order, have a common strip of regularity. Reasons for this restriction were stated previously and will again be emphasized. As will be shown in the following pages this common region of regularity is the strip -1 < Re)u<)aP{60) or Plieo), taking the smallest of these two values. For future reference we shall call this smallest value P911 1 . T , _ g m /‘ -4 '/"'/ ’// grin. - my“ in 8 Hi, We now write (52) in the form (59)0 (a = ()1 - —)()1 + -)K 91.90)!) ()1) - K (p.60 ) 4)-? r(1“‘£) D+('7.l). -9, - -)(}1 + -)K 91.00%}:- m1 -1“, (3) Properties 93‘ functions i_x_1_ (39), (a) The function SLAB-L— . X 91,60) Recall that according to our assumptions 0.91) is an entire function with algebraic decay of the order p.76, Rep >0, in the left half plane. 2%)- has poles, see (314), at): (60 ) K ()1) P,m° and): (60 ),m =l,2,5, "- .—:$&-)— is a function analytic M,m° K ()1) on the left half ju plane R3P ‘2' , 2;Qfl__ is a minus function. K (p) b) The function ()4 - $091 + %)K+()1)D+(p) NIH D+()J) has simple poles at); = , -n, 71 (60),?! (60), P,n ll,n n = 1,2,3, ... , K+94,90) has simple zeroes at )4 =-}1(60), -)1(eo) , n = 1,2,3, "- and simple poles at P,n ll,n ju . -(2n - é), n = 1,2,3, . It follows that ()1 - i.) ()1 4' %)K+(}I)D+()1) has simple poles at); = -n,-(2n - %). n 3 1,2,5, ... and thus is a function analytic on the right half plane Re]: 2- - 1. Furthermore, it is of algebraic decay 5/2 -°< Of - the order); in the strip, 1 < Refi<);m(30), and in all directions to its right provided R2cX> g . 1 _}_ + 134/“ N132) 4’ (c) The function ()3 4- 2MP 2)K 91,60) (2) PM?) D (71), Using information already stated in part (b) and preperties of the gamma function we see that l _ A; + 75‘s .2“ (l 4’ ()1 + 2H): 2)K (p.00) (3-) fl D (71) has simple 8 - C -l a .0. poles atp Pufzobfrfgo)’ :1, (2n 2),n 1,2,5, , 2 + 2 - o< It behaves asITI d‘ 2 in the strip, where 0’ is equal to to the real part of )1. If we let» =(e19 and apply Stirling's formula we find 1 1 + r ’2)‘ [‘(1 + (P + '2") ()1 "' '2")K 91,90) (“'55) Ffi D {-y) behaves asymptotically “5 F2? coso e-HZG - whine - 2e cosO. Hence, it decays in the left half plane and would be a minus function if it did not have simple poles at‘p = en, -(2n - g) n = 1,2,3, 00-, We have found no way in which this mixed term can be avoided such that separation into half planes of analyticity is possible. However, this mixed function can be split into two parts by adding and subtracting Mittag-Leffler series which have the same residues, of the form 00 a co Tn ((+0) 11 9 E Epi'n n==l)‘l + 2n ‘2'? where an and Tn are residues of ‘)-2}1 Pu +21 13" ("<1 I.” (71) ()1 - 49. + in: 91,90 )(1- at the simple poles p = -n and}: '-'-‘ -(2n - -)a n = 13.5: respectively. These residues are 2n -«(n‘°' - flung-‘1) rm H;— + run! Rn: D(n) ( ) 1+1 dam-1 2nfl8(-2n+-)——- 1 n D(2n '5) (2n - 2).(2n - 51P2(2n - is ("(1 +p) cos 11}: AM (Ly) _ u§(:;f°° 0) where D91) 3 .0 ...E._‘ and 8(P)=T‘;l+._£__3>° 2m- . M‘ 2""2 -_£___ - 1“ -36- Using (141) and our assMptions about D91) it can easily be shown that both series in (ho) are uniformly convergent in p. If we consider the functions representing each series in (be) we find that these functions are free of poles on the right half plane Rep > - l. The behavior of these functions or their series representations as | ,1 |—> as is a difficult question. Their nature strongly suggests that they approach zero as lyi—p co and we will assume this fact and consider both series in (110) as plus functions. The difference between . (pr %)(}1+%)K+()1.00) 68%)-? Ft: :13 D+(-)1) and the two Mittag-Leffler series is pole free on the left half plane, R011 <)1Pu(30), and because of previous considerations is in fact a minus function. Hence it may be transposed to the left hand side of (39) . We have -2 (142) __$B_L + (P--)(P+%)x 91,90)(x‘..2‘.2>)‘1 £339.11! p(.}1)+ co as ’E’ +2311 +ZIn 1: 4' n - .— n=lP n==l)1 + 2n 2 co as R T 1 1 + + n n ° = (P - 5) ()1 4' '2')K 9.1960“) ()1) +;1 F + 11 +1; 1 2 -37. The separation of functions is now complete. The result- ing left hand side is a minus function and the resulting right hand side a plus function. The equality states that the two sides are analytic continuations of the same function defined in the common strip, --1 < Re); <}x( e). Since these P 6O continuations decay as p—p oo in the resplegtive half planes, the function so defined has by Liouville' s Theorem the value zero. Hence we have 3—9— 1' 1P n n=1 -_ P’Z" 2 (1+3) (,1 - --)()l + -)x 91:90)” ()1) + :1 where D+(P) a: ELL: P) 308 ")1 ‘Q) , and (3)!“ Mycoseo) 2 “71 59.1) is the unknown function to be determined. When}: is set equal to 1,3, 2, 3, :21, on, it is clear that there arises an infinite system of equations. Hence, (14.5) does not allow explicit evaluation of the unknown function Mp) but only at the values}: = l, 1, 2, 5, 1.14., 5,17, ... 2 which are themselves the roots of the infinite system;2 mentioned above. This set of linear equations may possess a simple explicit inversion, but the present author has not studied this question in detail. -58- V. INPUT IMPEDANCE AT THE ORIGIN OF THE BICONICAL ANTENNA The input impedance, 2(eo,ka), at the origin of the biconical antenna is defined as follows: V(r;6°,ka) W4) Z(Bo,ka) a 1m r-—>0 I(r;00,ka) where 1'. fi 08 90 60 2 '5 W5) V(r;80,ka) =2 Eerde= 3-1-3 aarunfl)“ and (he) 1(r;eo,ka) = 2nr sineO H¢(r,ao). The relation needed to derive the value of the voltage and current at the origin of the antenna is given by um \E aficrmeom) = fl .. E; {1;(eres Mu) 4' 11(‘6‘r)Res M5) m-— 2 2 L]. sine log tan '5 {11(0089) 4' P1 é-cose? - _.+ .. z-j’z)‘ I (X‘r)a(p)P_1-(coseo) P P 5 5"}1 P1(coseo)-_P:cl(os60 ) d}: 2 2 The above expression is derived from Pll(c 5-039)+P1(-cosO? ‘2 P :71 (tn-a) H KPWr) M )P (coaeo ) a 16:71.: )1 P'1+PMP1(°°860)-P_15.(-0°089))h L '5 V“ "P 90.: 9.:f§.a which is obtained from (6) using the representation in the region 60 59 5-3- and the relation (8). In order that (LN-a) be a convergent and meaningful expression in the right half p-plane requires a transition from 151(25‘r) to ZIP(X‘r). As in previous arguments this synthesis leads to a kernel involving a(p) instead of A01). However, in our present case the transformation of contour leads to residues due to simple poles at )1 = I in The two residue terms signify the outgo- ing principal biconical wave, and its reflection from the terminus of the antenna. By substitution of (147) into (L6) and taking the limit as r approaches zero one obtains "a (W (1+8) 11m I(r;90,ka) = - __ 2 T Res 13%). r“'° 2 r(%)log tan-59 The same substitution in U45), assuming that the process of integration and taking the limit can be interchanged, leads to a; (9% W3) Using these results (141;) takes the form (19) lim v(r;e r——>0 - 21 l o,k8) -JEE Res A(""2') . a - 1 (50) Z(Oo,ka) = -——-X1 log cot—9- ________Z__Res A( ) . "WE 2 R 1 es ME) One can write the system (ll-5) in the more convenient form 6’ 5 1 0° °° 2n-- (51)5)1'%-: n +%—: 2 =0: Pn-lpi’n Pn'lp‘FZn-é- =3 1 l .0. P 1,2,233,2’ where 2 1 2 (52) e =£F>>l and arg)i-%|>>l and 2‘srg,1-%|l IEI (If) < 'fm l- |g(m-K)__+1| However , -50- It follows, Irmuf) l _<_ 90 C(60) 11m(m - 11;) We have shown that the series (d) converges uniformly for all g , larg C's-12'- , and hence so does the right hand side of (C)\ 1 Since m?) = M60) mg) we have that 1 _ 1! lim 12(3) --A.(60)J |argp‘_<_§ 5 ‘El—so as was to be shown. The last statement follows from the fact that m5) is continuous for all g , ‘arg §| _<_-g- . -51- Note 5: An Infinite Product Expansion of M1(coseo) . - _. + 2 P Refering to the discussion in Note I, we need only show that the orderf of Ml(cos60) is such that O < f < l. .. _.+ "' " 2 P Recall that Ml(cos60) = 1(coseo) - P1(-cos9 o). "2"“)‘1 ‘5‘!“ 7“!“ We have (Note 1 - (b)) that for ‘p - £21-|>>> 1 and law-w (+1)co‘ e -1 “_P fir()a+l)\l2sineo - It follows that P (-cose ) = P [008(fi - e 1- l O l 0 —.2.+P -§+P 2 P91 + ~21?) cos[}1(n’ - 90)- FL] 1 _. P r [1 + 05-3)]. #7? ()1+1) Vasineo Hence , 3.4.3:.(30860) = 2 + .1.) avg PO) 2 cos 21'9” - %)cos%(260-11)[1 4' 001‘”. n _ )1 r9; + 1N ameo It can easily be shown that the order of both coslyeo '- E] and cos[)1(n - 90) - E] is 1. Recalling the behavior of the gamma functions from Note 1 we have in the region lp-%|>>>l and larg/i-%|