■h wm^- iiii ^^^^H ^^H 621.365 I i G55te no. 50-57 cop. 2 Digitized by the Internet Archive in 2013 http://archive.org/details/inputimpedanceso56tang Antenna Laboratory Technical Report No. 56 INPUT IMPEDANCES OF SOME CURVED WIRE ANTENNAS by C. H. Tang Contract AF33 (657) -8460 Project No. 6278, Task No. 40572 June 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana. Illinois Antenna Laboratory Technical Report No. 56 INPUT IMPEDANCES OF SOME CURVED WIRE ANTENNAS by C. H. Tang Contract AF33 (657)-8460 Project No. 6278, Task No. 40572 June 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ACKNOWLEDGEMENT The author wishes to express his deep appreciation of the guidance of Professor G. A. Deschamps whose advice and suggestions were always illumi- nating. Many helpful discussions with members of the antenna laboratory are acknowledged. The computer program used in this work was done by Mr . V H. Gonzalez. It is a pleasure to acknowledge the financial support of Wright Air Development Division of the United States Air Force. ABSTRACT The problem of the input impedance of curved wire antennas is formu- lated in terms of an integral equation. A stationary formula is used in evaluating the input impedances of arc antennas and helical antennas of which the cylindrical antenna is a special case. The computational results are presented in graphical form. The impedance characteristics of these curved antennas are discussed. iv CONTENTS Page 1. Introduction 1 2. Integral Equation Formulation for Wire Antennas 4 2.1 Integro-Dif f erential Equation for Thin Wire Antennas 4 2.2 Linearization 9 2.3 Transformation of the Integro-Dilf erential Equation 11 2.4 The Singular Integral Equation and the Approximate Kernel Function 13 2.5 Structures Which Give Closed Cycle Type Kernels 17 3. Variational Formulation of the Input Impedance of Wire Antennas 20 3.1 Derivation of the Variation Function 20 3.2 The First Order Solution and the Assumed Current Distribution 23 3.3 Higher Order Solutions 25 3.4 First Order Solution for Arc Antennas and Helical Antennas 28 4. Computational Results and Discussions 35 4.1 Computational Method 35 4.2 The Input Impedances of Cylindrical Antennas 36 4.3 The Input Impedance of Arc Antennas 44 4.4 Electromagnetic Resonance of Thin Wire Conductors 54 4.5 The Input Impedances of Helical Antennas 59 4.6 Equivalent Helical Antennas 72 5. Summary of the Results and Conclusions 75 Bibliography 78 Appendix A 79 Appendix B 82 Appendix C 89 ILLUSTRATIONS Figure Number Page 1 An Idealized Antenna Structure 5 2 Bent Cylindrical Coordinate System 6 3 A Helical Antenna 8 4 An Arc Antenna 9 5 A Center Driven Idealized Cylindrical Antenna 15 6 The Input Impedances of Cylindrical Antennas (ki = 1.5 to 3.7) 40 7 The Input Impedances of Cylindrical Antennas (ki = 4.7 to 6.9) 41 8 The Input Impedances of Cylindrical Antennas (ki = 7.9 to 9.9) 42 9 The Input Impedances of Arc Antennas 45 10 Resonant Resistances of Arc Antennas 47 11 Antiresonant Resistances of Arc Antennas 48 12 Resonant Lengths and Antiresonant Lengths for Arc Antennas 49 13 Shortenings of Resonant Length and Antiresonant Length of Arc Antennas 50 14 The Ratio of Resonant Wavelength to Arc Length as a Function of C — Comparison Between Englund's Measured Results and That Obtained From the Impedance Graphs . 51 15 Resistive Part of the Input Impedance Along Equi-ki Contours 53 16 The Input Impedances of Helical Antennas C = .5 60 17 The Input Impedances of Helical Antennas C = 0.75 61 18 The Input Impedances of Helical Antennas C = 1 62 19 Resonant Resistance as a Function of Pitch in Wavelength 63 vi ILLUSTRATIONS (Cont'd) Figure Number Page 20 Antiresonant Resistance as a Function of Pitch in Wavelength 21a The Input Impedances of Helical Antennas 21b The Input Impedances of Helical Antennas 22a The Input Impedances of Helical Antennas 22b The Input Impedances of Helical Antennas 23 Resonant Resistance as a Function of Circumference in Wavelength 24 Antiresonant Resistance as a Function of Circumference 71 in Wavelength 25 Equal Resonant Resistance Contours 73 26 Equal Antiresonant Resistance Contours 74 64 C 2 = 0.25 65 C = 0.25 66 C 2 = 0.5 67 C 2 = 0.5 68 tf erence 70 1 1. INTRODUCTION It is of practical importance to know the characteristics of the input impedance of an antenna, as a function of the frequency of the source and the geometry of the antenna structure. As far as wire antennas are concerned, the input impedance of cylindrical antennas has been investigated to a great 1,2,3,4 extent by many authors ; but practically no theoretical results have been obtained in the case of curved wires. The problem to be considered in this report is that of evaluating the input impedance for some particular curve wires . The theoretical model used in formulating the problem is the idealized wire antenna with a 6-source excitation, which is usually called a slice generator. This idealization avoids the problem of the actual transition between the trans- mission line and the radiating part of the antenna. With proper end zone correction, the input impedance computed from the theoretical model becomes the terminal impedance of the transmission line. In this report, only the idealized antenna problem will be considered. The integral equation formulation for the electromagnetic problem is an important technique in antenna analysis, especially for cylindrical antennas. 5 ^2 It was first used by Pocklington and later developed by Hall e n . The method has the advantage of generality and is conceptually simple, although rigorous solutions are in general difficult to obtain. A large number of papers have been written on its application to cylindrical antennas using various means of approximation, much less attention was paid to the discussion of curved wire structures 6 ' 7 # With the aid of the integral equation formulation, it can be shown that the input impedance of an antenna can be expressed in such a way that it is 2 stationary with respect to the current distribution in the antenna. It is recognized that the integral equation for the current distribution serves as an Euler equation of the variational problem of the input impedance. The well known Hallen iterative solution to the integral equation has been shown to give a good result for thin antennas, for which the asymptotic solution of the current distribution is known to be sinusoidal. The asymptotic so- lution, when applicable, can be used in the variational expression for the 8 9 input impedance. Storer and Tai have calculated the input impedance of thin cylindrical antennas by such a method. Their results agree well with those obtained by the iteration method. We shall extend these solutions to some curved wire antennas . Those curved wire antennas which we shall discuss are (1) arc antennas — the antenna arms are bent into the shape of circular arcs with a given radius. (2) Helical antennas — the antenna arms are sections of a helix with a given radius and pitch. Cylindrical antennas are special cases of these two classes The common property of being invariant under a one dimensional Abelian group of congruent transformations sets these structures apart. This particular symmetry implies that the kernel of the linearized integral equation has a special form. K(s,s') = K( . I s - s'l ) This is known as a closed cycle type kernel. The above mentioned structures are the only ones which lead to such a kernel. In this work, the computation of input impedances have been performed for both arc antennas and helical antennas . Results are shown in graphical forms . For arc antennas the input impedances are given for various radii 3 of the arc while those of helical antennas are shown as functions of the radius and the pitch of the helix. The computational results show the following general trend: in comparison with the cylindrical antenna, the curved wire antenna has a higher quality factor Q and radiates less with respect to the same input current. In many respects, a curved wire antenna behaves like a corresponding thinner cylindrical antenna. The input impedance graph for the arc antennas also exhibits the extreme phenomena observed by Englund in his measurements for the natural period of linear conductors Based on the computed input impedances of helical antennas the contour lines of equivalent helical antennas are given and they are closely described by the circles for the constant radius of the curvature of the helix. 4 2. INTEGRAL EQUATION FORMULATION FOR WIRE ANTENNAS 2.1 Integro-Dif f erential Equation for Thin Wire Antennas It is well known that in electromagnetic problems one can find the electric field strength E produced by current source J through the use of the vector potential function A -} 2 •- jw e E = [grad div + k ] A (1) with -> 1 ^ * A(x) = — f J(x') G(x,x» ) dv (2) In Equation (1) k is the free space propagation constant. The integration in Equation (2) is performed over the source region v and G(x,x') is a Green's function of the vector wave equation 2 ~b 2 "$ -) V A + k A = -J (3) with appropriate boundary conditions. However, in" a general case it is not possible to find the Green's function for the region outside the antenna, with specified boundary conditions on the antenna surface, therefore we re- duce the antenna problem to the problem of currents in free space and let -jkr(x,x' ) G(x,x')=- 7 pr— (4) ' r(x,x') where r(x,x') is the linear distance between the source point x' and the observation point x. Equations (1) and (2) are the basic relations for the antenna analysis If the current distribution J was given, the field strength E could readily be evaluated. If, on the other hand, the tangential electric field strength is specified on a close surface enclosing the antenna, then Equations (1) and (2) lead to an integro-dif f erential equation for the current distri- bution on the antenna. For convenience, the surface is usually taken on the boundary of the antenna, and we shall discuss only antennas made of perfect conductors . GAP SOURCE Figure 1. An idealized antenna structure Since the tangential electric field vanishes on the antenna surface, the boundary value problem can be formulated in terms of an integro-dif f erential equation with the relation "*t " " 3 1 <5) where E is the tangent electric field deduced from Equation (1) E is the given impressed field which is assumed to be a 6-function The integro-dif f erential equation so formulated will give a unique solution if the support for the current distribution is specified. For a general antenna surface, the vector integro-dif f erential relation leads to two coupled scaler integro-dif f erential equations for two components of the current distribution. However, for thin antennas of unifrom cross- section, the current flow is mostly in the axial direction; therefore we have a single two-dimensional integro-dif f erential equation, instead of two. For the thin wire antennas of circular cross sections, the integro-dif f erential equation can be expressed with the aid of a bent cylindrical coordinate system. Figure 2 describes a bent cylindrical coordinate system. C (s) is a A A given smooth curve, s is the unit tangential vector at s, n is the unit princi- A pal normal vector and b is the unit binomal vector. A point p in the A A n-b plane is described by the polar coordinates (P, (p) and hence any point in the neighborhood of the curve can be described by (P, , s) in the direction parallel to axial tangent at s and 3 2 2/V A -Jkr(s,s •,*>,*>•) K(S ' S '' W ' ) = [ -3-^ tMS - 5,)1 .C,.',W) (?) The expression for r (s, s ' ,<£>,<^' ) is known when the geometric configuration of the antenna is given. For cylindrical antennas, we have r = [(Z-Z') 2 + 4a 2 sin 2 (&2L) ] 1/2 (8) where Z = s and a is the radius of the wire For helical antennas, we can write r Jr 2 [2- 2 cos 2=2L + tan 2 4; ( £Z£1) \ o R R o o + 2a 2 [1 + sin ^ sin () ] + 2aR [ (cos

o o (9) where ^ is the pitch angle tan ^ = 27T R and R is the radius of the helix o p is the pitch of the helix a - RADIU3 OF I it GAP SOURCE Figure 3. A helical antenna For arc antennas, *\> = 0, and Equation (9) becomes r = J [2R 2 + 2aR (cos s, > = - 7577 hrr d4) 'hen Equation (6) becomes (15) 12 where s 2 c(s) = / M(s,s') I(s') ds» (16) and M(s,s') = k 2 [(s • s') - N(s,s')] G (S,S') - 3- ^ G (s,s')| (17) [€•.<""] c(s) in Equation (15) can be considered as a secondary source representing the effect of the curvature. It is seen that for cylindrical antennas N = 1 (z • £•) = 1 Therefore M = and there is no curvature effect. The function in Equation (13) becomes the axial component of the vector potential function which satis- fies the one dimensional wave equation ^-5- + k 2 | A(z) = -juc E + 1 (z) (18) dz 2 I * Therefore for center-fed cylindrical antennas, A(z) can be expressed either in terms of standing waves A(z) = C cos kz + C Q sin k I z J (19) 1 ^ or in terms of traveling waves A(z) = B 1 e" JkZ + D 2 e JkZ 4- D 3 e" Jk ' Z ' (20) 13 The constants C and C ( or D D and D ) are determined by the source condition and the boundary condition of the current I(z). If the source condition is given as E t 1 (z) = V 6( Z ) (21) then °2 = "J if ° r D 3 " if (22) where t, is the characteristic impedance of free space. The integral equation for cylindrical antennas of length 2£ is thus £ - .^ , _ . _ ) G & (z, z') dz' = c 1 cos kz - j — sin k(z) (23) for the standing wave type solution. In case of curved wire antennas, we solve A(s) from Equation (15) which gives for symmetric antennas s v • „ 1 r ■ — sin ks + — / sin A(s) = C cos ks - j 777 sin ks + - / sin k(s-s ' ) C(s') ds ' (24) Equations (24) and (13) also give an integral equation for curved wire antennas 2o4 The S ingular Integral Equation and the Approximate Kernel Function The integral equation for the antenna problem so formulated has a singu- j larity in its kernel function; when the observation point is approaching the 1 source point, the value of the kernel function increases indefinitely. It 14 can be shown that in the neighborhood of the singularity the kernel function 11 behaves as (.,.., _5i£L-> ^ inr |a Jk (25) As far as the singularity is concerned, the kernel function for the curved wire and that for the cylindrical wire have the same asymptotic behavior, since any smoothly curved wires are locally straight. The function in Equation (25) is square integrable, for ss- 2 s-s'l ds'ds<°o < s < i, < s' < i hence the kernel is of the Fredholm type. Fredholm's alternative theroem is therefore applicable. Either the inhomogeneous equation is solvable, or else the corresponding homogeneous equation has a non-trivial solution. However, if the size of the wire becomes inf initesimally small, the singularity of the kernel function is of higher order than that in Equation (25), G(s,s.)-^^-> v ^ n (26) The integral of this function does not exist even in Canchy's principal value | sense. This fact is known in potential theory; the attraction of a line of mass upon one of its own points does not have a definite meaning, while for surface distributions and volume distributions the potential function together jwith its derivative exists inside the source region (see The Theory of the (Potential by W . D. Mac Mil Ian) . i 15 In the light of these considerations, one realizes that the problem should be formulated with the finite size model for which the solution is predicted by the Fredholm theorem. It was indicated in Section 2.2 that the rigorous expression of the averaged function G (s,s') is rather complicated, even for cylindrical wires k 1 ^-jklz-z'l G a (z > Z?) = S F ( 2 *> h) " lz-z'l + ° (k a > (27) where 2 2 h = 2a[ (z-z') + 4a -1/2 is the modulus of the elliptic integral F. Usually, the simplification is obtained by introducing an approximation to the averaged kernel function of the integral equation. For cylindrical antennas, Hallen derived the linearized equation i 4(j 2 - z 2 ) log g + +i k I z-z' I r i(z>)( -a I z-z' I 1^- dz 1 = 477[C cos hz -j-7 sin k I z I ] 1 2C, (28) 2 a SLICE \ / 1 GENERATOR ! 1 i Figure 5. A center driven idealized cylindrical antenna 16 >y splitting the kernel function into two terms; one depends on the size of :he wire and has the static characteristic, while the other term is indepen- lent of the size. +! 27T m a(z) = i(z) _i p p d(jP' dz' r I 2 *J J A /^^ 2 TTJ r 7T n 2 ^ + J +i -jk I z-z' I (z')e -I(z) I z-z' ! dz -i ~b 2 -i (29) t is readily seen that the following approximation is good except very close o the end of the wire 27T +! 1_ n p dz' dff 1 log 4(i 2 -z 2 ) (30) x J. dz (i-z) +Vq-z) 2 + 4a 2 sin 2 » ^ J Ks')l-9^- r+ k 2 (^ -s') G a (s,s') ds« (6') Symbolically, it can be written as E = H(l (37) where and E = E t 1 (s) lj\±s an integro-dif f erential operator K l = 4^3T f 2 I ( s '>[w " * 2 (s ' s^jGjs^s') d.< The kernel K can be shown by the Lorentz Reciprocity Theorem, to have the symmetry property expressed by the following equality of inner products K h > - < V K 1 ! > <38) where < I..K I. > - I (I. HC I.) ds (39) 21 Equation (39) is a bilinear from with an infinite number of variables, the corresponding quadratic form is called the Hilbert's double integral. It is known that when K is real and symmetric, the absolute value of the Hilbert double integral is bounded under the constraint I i > = 1 The kernel function /fy in our case, is complex symmetric. Instead of the above —mentioned property, the double integral can be shown to have a stationary point under the similar constraint condition. When I I satisfies h - K.*2 (40) respectively, the function '12 " < E x , I 2 X E 2 , ^ > (41) which gives the input impedance of an antenna when E = E , is stationary. This can be seen as follows . When functions I and I vary, the first variation of Equation (41) can be written as /j(6l 2 > - < 1 ± , lKj- 2 > < V 6I 2 > 6z I 2 > L < E. , I > < E .. I < b h> ^ 6l 2 > < E 2 > h > ~ < h> lHh > ^2' 6l l I l > ■] 22 which vanishes when I and I satisfy Equation (40) and /ff has the property expressed by Equation (38) . The input impedance of a wire antenna, when referred to the input current I(s ), is defined as Z in = 7(fT (42) o where V is the voltage across the gap, and it can be expressed as J*' (s) ds for a 6-function excitation, we write E (s) = V 6 (s-s ). Then Equati< (6') becomes i " S o>=/^ 1 Z in I(S o } 6 (S " S ~ ) = K I multiplying both sides of Equation (43) by I and integrating between the limits s to s we obtain Z in l2 < s > =< I /)(_I > hence 1 <■„> Equation (44), which is a special case of Equation (41), has been shown g to have the stationary property by Storer for cylindrical antennas. Storer 9 and Tai have calculated the input impedance for these cylindrical antennas 4 and the results checked closely with that obtained by King and Middleton who used the Hallen's iterative method. 23 3.2 The First Order Solution and the Assumed Current Distribution It is worthwhile to point out that the variational problem described in Chapter 3.1 is equivalent to the boundary value problem expressed by the integro-differential Equation (6). This equation can be considered as the Euler equation of the variation problem, in the sense that the variational formula is derived from the integro-differential equation and the integro- differential equation can be recovered from the variational formula by taking its first variation to be zero. The equivalence between these two problems is the basis of a method of solution to many physical problems. The well known Rayleigh-Ritz method is an example. For the solution of the Euler equation, one expands the unknown function in terms of a linear combination of known functions with unknown coefficients; the set of unknown coefficients are then determined by employing the stationary (or extremal) property of the varij tional formula. In the wire antenna problem, the procedure consists of using the asymp- totic solution of thin wire antennas as a zeroth order solution, for symmetric antennas, we write I 0) (s) = I i (s) = I sin k (i- I s I ) (45) o o o where I is the amplitude of the current distribution and the zeroth order solution to this wire antennas is thus z o = o'^-o ( i 2 (o) o This leads to the same result as the EMF method. The first order solution is obtained by adding a correction term to the zeroth order sinusoidal current distribution 24 I (1) (s) = I [i (s) + A 1,(3) ] (47) o o 11 where i (s) is a chosen function which satisfies the boundary condition required by I(s) and A is a variational parameter to be determined by the condition 9z (1) in ^— = (48) In terms of the assumed current distribution functions, the input impedance formula can be written as (1) V l + 2A 1 V 2 + A l V 3 Z U) . =j30^ L* 13 (49) [i (o) + AijCor where s s „ 2 2 If S ^ V^'Ka ds ds f (50 o iya " ' i ' i V 2 = V ol = 1 J J^ i o (3) 4 l (fl,) C (S ^ !) dS dS? = V 10 S l S l 2 2 V ll=iS S il(S> 1 l (s, >^a pq k U sj P s) i (s T ) /V (s,s f ) ds' ds = v q v a qp (60) and the A. 's are determined by 5z. (n) = i = 0,1,2 6A. > -> —j (61) Equation (61) gives a set of equations as follows SA i (o) o p p 2 A v . o p pi 2 S A A v o o p q pq i. (o) = l (62) i = 0,1,2, n From Equations (62) and (59) we have :. (n) fsA i (cjl i in I o p p i L. (o) = j30 S A v o p pi 27 (63) M These are n + 1 equations for the set of variational parameters i = 0,1,2, n. To solve these equations, we first express the feeding condition n Z (n) i A i (o) = JL o p p I o (64) in where V is the given voltage difference at the input terminals, and thus Equation (63) becomes n V i. (o) S A v . = * = o p pi j30 I In matrix form it can be written as vA = J30 I (65) where 10 ol 11 In nO nl is a symmetric matrix and 28 A = the solution of A is i (o) o i 1 (o) i (o) n T V =-1t (66) substituting Equation (66) into Equation (64), we obtain Z. (n) ^^O;^ (67) This reduces to the form (55) in the case of two parameters. 3.4 First Order Solution for Arc Antennas and Helical Antennas In this section we shall apply the first order formula to some symmetri- cally curved wire antennas of half lenth i , and curved wire antennas which we will consider are those which possess the closed cycle type kernel, namely, arc antennas and helical antennas, with cylindrical antennas as special cases The set of trial functions for the current distribution will be i (s) = sin k (i- I s I ) o and i (s) = k Q- I s I ) cos k a- I s I ) 29 For these antennas, the kernel f unction \_ _ (s, s ' ) becomes G ( I s-s' I ) (68) f/ds-s'l) = (-4 + k 2 C^ ■£•)} This particular property can be used to simplify the set of double integrals V V 2 and V We first convert these integrals into the following forms. 1 * (~di (s) di (s») ~| V I = k J J Mb 57T- + k f (S > S,) l o (B)1 o (B, -2j V 8 ' 8 '* dS ' dS -i -i (69) i ' Pdi (s) di^s') -I *2 = k J J - "di ds^" + k f(S ^ S,) V S)i l (s,) -t -1 *- -i i i | ai^s; a^ts"; 9 | G (s.s 1 ) ds' ds a (70) i r r I di i (s) di i (s,) 2 1 *J J Q"5S ^ + kf(s,s.) i 1 (s)i i (s'J where -jkr(s.s') (71) G ) r cos kv + cos k(2i-u)] (82) ds ds ' 2 di (s) di (s*) k 3 R 2 — -: — — ; = -— - (2i-u+v)[sin kv + sin k(2i-u)] + — [cos k(2i-u) + cos kv us as 4 ^ (83) di (s) di (s>) 4 3 — ^ — — — = |- (2i-u+v)(2i-u-v)[cos kv - cos k(2i-u)] - — (2i-u-v) k 3 k 2 [sin k(2i-u)-sin kv ] - — - (2i-u+v) [sin k(2i-u) + sin kv ] + — [cos kv + cos k(2i-u) ] (84) 32 In u-v plane, double integrals v , v , and v can thus be written as single integrals (see Appendix B) . If we normalize the length in radians, i.e. let L = ki and with x = ku = kv as new variables, then the set of integrals v , v and v Q become 2L v i=J F n (x) dx+ S F_(x) dx + | F 12 (x) G(x) dx 2L v 2 = f F 2i (x) G(x) dx + P F 22 (X)G(X) dX (85) 2L v 3 = P F 31 (x) G(x) dx + j P F 32 (x) G(x) dx o L where F n (x) = [x cos (2L-x) - 2(L-x) cos x] [l-f(x)] + [2 sin x - sin (2L-x) ][ 1+f (x) (86) F 12 (x) = [l-f(x)] (2L-x) cos (2L-x) + [l+f(x)] sin (2L-x) (87) F 21 (x) = J|x-L-(L-|) f(x) cos (2i-x) + [l-f(x)] (| Lx ~2j sin ( 2L -x) - (x -xL-1) sin x + [ (3x-2L) + x f(x)] cos x (88) 33 F 22 (x) = (L-|) [3 + f(x)] cos (2L-x) - I*- -2Lx + 2L 2 ~) [l-f(x)] sin (2L-x) (89) F 31 (x) = [(l-f(x)) (-|-+ Lx 2 " L 2 x +| - L ) + x| cos (2L-x) + x 2 - 2Lx + (1+f (x)(-Lx+L -|) sin (2L-x) + j-2x 2 + L(l-f(x))x + (l+L 2 )(l+f (x))J sin x + [(l-f(x)) (^|- + L 2 x + x - | L 3 ^ + 2x -2lJ J -x+2L -x + 4Lx - 4L 2 + -| + -| f (x) J sin (2L-x) (91) F 32 (x) = [(l-f(x)) f|- -Lx 2 + 2L 2 x - | - | L 3 + L J and where -jr(x) G(x) = - r(x) (90) f(x) =\[ Cl 2 cosf + C 2 2 j (92) / N 2 ~| 1/2 7(x) = kr(u) = j^ 2 sin 2 (§-) M 4c/ (|J + 6 2 J (93) with 6 = ka 34 C = kR 1 o c 2 = kb V^ 2 —2 + C 2 For arc antennas C = f(x) = cos £- (94) r(x) = |4C 1 2 sin 2 (-£-\ + e 2 l (95) and for cylindrical antennas C > °o f(x) = 1 r 2 2i 1/2 r(x) = |^x + € J (96) The curvature of a helix can be expressed as R K = ° 2 2 R + b o and the normalized radius of curvature of the helix is thus ^2 2 C l + C 2 C = 1 2 (97) C l I The Equation (97) will be useful in describing some equivelent helical antennas ■ j which will be explained later. 35 4. COMPUTATIONAL RESULTS AND DISCUSSIONS 4.1 Computational Method The numerical computations were carried out on ILLIAC, the digital com- puter of the University of Illinois. Some of the procedures used in computing the integrals and performing the algebraic operations of the impedance formula are described in Appendix C. The computer program was prepared for computing the input impedances of helical antennas for a given set of parameters C , C , L and Q, . The first 1 ■£ three parameters are the normalized physical dimensions of a helix, as de- fined previously. C, = kR 1 o id £1 is defined as c 2 = kb L = k£ fl = 2in 21 For arc antennas, C = 0. and for cylindrical antennas C = 0, C = °°. The computations for the input impedance of cylindrical antennas have been performed for £2=8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14, 14.5, 15. While the parameter Q, was held constant (£2 = 10) for the impedance computation for the arc antennas and the helical antennas . In case of the arc antennas, a series of computations were performed for the parameter C in the range 0.5 < C < 3.5 36 If the dimensions were such as to make the antenna arms meet, the computation outline would not be valid thus the largest values of L used in the compu- tation were limited by the relation L < 7TC 1 for a given value of C . For helical antennas, the impedance computations were performed for the values of C and C in the following ranges . 0.3 < C < 3 0.3 < C < 2 4.2 The Input Impedances of Cylindrical Antennas Although our main purpose was to evaluate the input impedances of curved wire antennas, particularly the arc antennas and the helical antennas, the impedance formula derived in Chapter 3 also gives the input impedances of cylindrical antennas. This will be discussed first for the following reasons. (a) A cylindrical antenna is a special case of curved antennas. For curved wire antennas of small curvature, their input impedances approach that of cylindrical antennas. Furthermore, when ki is small, the input impedance of all wire antennas should be practically the same. As ki increases, the effect of the curvature begins to appear. The input impedance of a cylind- rical antenna is therefore useful in extracting the effect of the curvature in the input impedance of a curved wire antenna. (b) It is the nature of the approximation of the problem that it jives a better result for structures of relatively small curvature. Yet, \ve have no way of determining what exactly is the upper bound of the curvature 37 for which the approximation is still valid, there is no explicit condition for limiting values of C and C in the impedance formula. Hence in inves- tigating the effect of curvature on the input impedances of these wire antennas, it is important for the purpose of comparison to compute the input impedances of the corresponding cylindrical antennas with the same method. (c) The computing program can be checked by allowing the comparison of our results with published, experimentally confirmed data for the cylindrical antenna case. (d) It will be shown later that in many respects the curved wire antenna behaves like a corresponding thinner cylindrical antenna. In case of cylindrical antennas f(x) = 1 The set of functions F. . 's are simplified ' (x) = 4 sin x - 2 sin (2L - x) F (x) = 2 sin (2L - x) F (x) = 2(X-L) cos (2L - x) + 2(2x - L) cos x F (x) = 2(L - x) cos (2L - x) (98) ' 31 (x) = x cos (2L-x) + (x 2 - 4Lx + I? - 1) sin (2L-x) + (-2x + 2L +2) sin x + (2x-L) cos x F 32 (x) = (2L-x) cos (2L ~ x ) + (-x 2 + 4Lx " 4Ij2 + D sin (2L _x) 38 For the thin wires ka « 1, i » a the following approximation is valid P e ' Jr(x ' €) dx .f_dx_ P i - e -J r <^° J r(x,e) J r(x,e) J r(x,o) ) - dx hen the set of integrals v , v , and v for cylindrical antennas can be 9 xpressed in terms of known integrals, as given by Tai . ^-{ (2L) - 2 \(4L)) -j | * e" j2L [ L ( - |+ 2in2) +jl ] (99) i £(2L) + L(- i - ft+^(2L) ) -jL 2 (- i + ^ (2L) )] -Z 5 1 „^<2L) + L(- i + e J2L [j-j (- | - 2 in2 + ft- 4^\(2L) + 2X(4L)) + L(l + in2 +J^ (2L)-^T£4L)) + jL 2 (2in2 - | «+ 3^(2L) - 2 ^(4L)| + e" j2L [j i ( ~2in 2 - Si) + L(l + in2) + jL 2 (- | - 2in2 + | ft) I ft = 2in — 39 and ae-vT^ dx (100) which can be expressed in terms of sine and cosine integrals. The input impedance of cylindrical antennas has been calculated for the following values of the parameters. a. k! = 1.5 ^ 3.7 ft = 8 —7 15 b. k£ = 4.7 > 6.9 ft = 11.5 — ) 15 c. ki = 7.9 > 9.9 ft = 12 — > 15 These results are presented in Figure 6, Figure 7 and Figure 8, respectively. For wire antennas, thier impedance characteristics resemble that of a lossy transmission line; the impedance curve plotted in a rectangular chart would circle around a point which represents the characteristic impedance of the structure. For this reason, the impedance curves of wire antennas are some- times called circular graph. In these figures, the dotted lines indicated the equi-ki contours. For Qi = 10 and 15, our results coincide with that given by Tai and Storer. The properties of the input impedances of cylindrical antennas have been in great detail by King . We shall put down only a few useful defi- nitions and mention some notable effects to facilitate the discussion for the input impedances of curved wire antennas. The terms, input resonance and input anti— resonance, will be used as in the following defined sense. !• Input resonance is characterized by 40 1000 (OHMS) -1000 Figure 6. The Input Impedances of Cylindrical Antennas (kl = 1.5 to 3.7) 41 iooo- -1000 Figure 7. The Input Impedances of Culindrical Antennas (kl = 4.7 to 6.9) 42 1000 -1000- Figure 8. The Input Impedances of Cylindrical Antennas (kl = 7 9 to 9.9) 43 a) The reactive part of the input impedance vanishes b) k£ = (ki ) res is near wn/2 with n = 1,3,5... The resistive part of the input impedance in this case will be called the resonant resistance, (R) res and (ki ) res, the resonant length. 2. Input antiresonant is characterized by a) The reactive part of the input impedance vanishes b) ki = (ki ) antires is near n7r/2 with n = 2, 4, 6 The resistive part of the input impedance in this case will be called the antiresonant resistance, (R) antires and (ki ) antires, the antiresonant length , It is known that for cylindrical antennas, (R) res decreases as to increases This indicates that for a same input current the thinner wire radiates less power at their resonant lengths, also at these lengths the thinner wire has more stored energy. The quality factor Q of a cylindrical antenna increases as to increases at the resonant length. On the other hand (R) anti— res increases as to increases , Another known fact for cylindrical antennas is that the formula for the natural wavelengths 4i \ = — , n = 1,2,3 n n' ' ' is valid only for wires of the infinitesimal size. For finite size conductors, lit is replaced by n ^i^L+Il, n . M>a n n ' ' ' where 6 is always greater than zero and decreases as to increases . The shortening ,af resonant lengths and antiresonant length are expressed by 44 n7T a = — - - (ki )res > n 2 (3 = |= - (kO anti— res > n = 2,4. Furthermore, it is seen from the circular graphs of the input impedance (Figures 6 through 8) that a. decreases as fi increases, i.e. (ki) res is larger for thinner (3 also decreases as fi increases; i.e. (ki ) anti— res is larger for thinner wires. 4.3 The Input Impedance of Arc Antennas In order to see the effect of the curvature, we computed the input im- pedance of arc antennas with constant £1 = 10 and C = 0. A series of impedance graphs were obtained by varying the parameter C , i.e. o/\, the circumference in wavelength The half length of the arc antenna, i , is limited for a given R by the relation i < 7T R or ki<7TC o 1 The input impedances for C = 0.5 to 3.5, together with the input impedances of the corresponding cylindrical antenna are shown in Figure 9. The dotted lines are contours of equal length (equi ki ) . It is seen that the larger the curvature of the arc antenna, the larger is the circular graph of its input impedance. This enlargement effect of the circular graph for arc antennas is very similar to the effect for cylindrical 1000 45 -400 Ci = THE CIRCUMFERENCE I N WAVELENGTH Figure 9. The Input Impedances of Arc Antennas antennas as we increase £2, i.e., decreasing the thickness of the wire, in which case the thinner wire has lower resonance resistance and higher anti— reso- nance resistance. In the case of arc antennas of the same thickness the cur- vature produces a similar effect on the input impedance as that due to the thickness of cylindrical antennas. Therefore the curved antenna has a lower resonance resistance and yet a higher anti— resonance resistance. Resonant resistances, (R)res, range from 17 ohms for C = 0.5 to 70 ohms for C =3.5 (Figure 10) , While antiresonance resistances, (R) antires, range from 945 ohms for C = 3,5 to 1800 ohms for C = 0.875 (Figure 11). For smaller values of C , the antenna arms do not reach the resonant and the antiresonant length, respectively, before meeting each other. The resonant length, (ki ) res and the antiresonant length, (ki ) antires as a function of the normalized radius of curvature as shown in Figure 12 . The shortening effect expressed by a = | - (ki)res (3 = 7T - (ki ) antires are shown in Figure 13 as a function of the normalized radius of curvature for the arc antennas. The variation of these lengths have maximums at C = 1,175 for the anti resonant lengths and at C = 0.575, for the resonant length. The former corresponds to the structure 2-jiR = 1.175 \ and later corresponds to 2fl-R = 0.575 X. This particular effect was observed by o Englund in 1928 when he performed some measurements on the natural period of curcular arcs of various radii of curvature. He obtained an extremum in | his measured values of the ratio of resonant wavelength to arc length. The 47 0.5 1.0 1.5 2.0 2.5 3.0 3.5 CIRCUMFERENCE IN WAVELENGTH, C,=27TR /\ Figure 10. Resonant Resistances of Arc Antennas 48 ^ 1800- I O y 1600 z < (f> (/) Ixl X 1400 z < z o [3 1200 o: z z UJ _l 2.1 2.0 Englunds ^ Measurement t Computed from the Impedance Graph fc W 0.375 CIRCUMFERENCE 0.5 0.625 IN WAVELENGTH, 0.75 C. = 277-Ro/X Figure 14. The Ratio of Resonant Wavelength to Arc Length as a Function of C — Comparison Between Englund's Measured Results and That Obtained from the Impedance Graphs 52 computed result from the impedance graph give values oi(— Jres very close to Englund's measured values (Figure 14). In both results, the extremum occur at C = 0,575. The variation of the resonant resistance, (R) res, and the antiresonant resistance, (R) antires, and also those of the resonant length, (ki ) res, and the antiresonant length, (ki ) antires, can be divided into two regions of different characteristics. For structures of large curvature, the variation shows a larger slope and becomes slowly varying in the region where the cur- vature of the structure is small. In the resonance case, the length of the antenna is short, and the effect of the curvature increases gradually as we decrease C . For small values of C where the arms of a resonant antenna almost meet each other, the effect of the curvature increases much more appreciably. TABLE I c l 0.5 0.75 1 1.5 2 2.5 3 3.5 * c l 1.57 2.36 3.14 4.71 6.28 7.85 9.43 11 (ki )res 1.508 1.513 1.501 1.493 1.4905 1.4985 1.489 1.4885 (ki)res /77C 1 0.96 0.642 0.478 0.317 0.238 0.19 0.158 0.135 Table I gives the resonant length in terms of the half circumference length of the loop, all in radians for (ki )res/7TC > 0.5 where C < 1. 'There is more variation in the curves of (R)res and (ki)res than in that for the region (ki)res/77C < 0.5, where C > 1. The set of corresponding values [of (ki)antires and TfC are shown in Table II. 53 0.5 1.0 15 2.0 25 30 3.5 CIRCUMFERENCE IN WAVELENGTH, C. = 277-R /\ Figure 15. Resistive Part of the Input Impedance Along Equi-ki Contours * c l (k£ )antires 54 TABLE II 0.875 1 1.25 1.5 2 2.5 3 3.5 2.65 3.14 3.93 4.71 6.28 7.96 9.43 11 2.58 2.64 2.66 2.63 2.57 2.55 2.54 2.536 (kl ) antires ! 0.938 0.844 0.703 0.58 0.409 0.32 0.269 0.211 Again, for higher values of the ratio (ki )antires/7rC the curves of (R)antires and (ki)antires change more rapidly. Particularly, there appears a reso- nant phenomena at e =1, i.e. when the circumference of the loop 277R equals to one wavelength (277R = \) . o The values of the resistance along various equi-ki contours are plotted in Figure 15. For small ki , the variation is small except for C < 2. As ki increases, the resistances along the equi-ki contour show more variation. For ki > 2.2, the curves begin to give a minimum in the neighborhood of C = 1. This minimum shifts toward larger C for larger ki . There is also a maximum In these curves which moves toward smaller C as shown in the figure. These naximum and minimum effects of the resistance and the extremum effect observed oy Englund are closely related, yet th^ay are difficult to explain in terms of simple physical reasonings. i.4 Electromagnetic Resonance of Thin Wire Conductors The resonant wavelengths and quality factors of thin wire conductors can 11 j»e determined by equating the stored electric and magnetic energies of the system . ;ince the loss of power by radiation has only a second-order effect on the atural frequencies, the formulation is essentially on the static basis. 55 Associated with the current distribution on the wire there is certain quantity 6m of stored magnetic energy e .K- m 877 I I r X 1(3)1(3') (s ' s') ds' ds = ?- < i/Q m 1 > ( 10 i) //^m = (s • a«)/r Similarly, associated with the charge distribution the stored electric energy = i kfJ Vlq(S)q(S ' )dS ' ds=i ^ (102) /£- 1/r e by the equation of continuity € can be written as e 877600 < I'/A I' > where u> is a natural frequency . Equating the Equations (101) and (102), we obtain the formula „ 2 < 1 1 IJC I ' > -rr For thin wires, the double integrals in Equation (103) can be written as jf = < I' /|l£ I' > =] |j [I'(s)] 2 |(v (s,s«) ds' ds J J JTm = < I /AC m I > = I [I (s)] 2 /\ m (s,s') ds' ds 56 The approximation is justified for thin wires due to the large contribution of the integrand at the neighborhood of s = s'. To evaluate resonant wavelength and the quality factor Q, we assume I(s) =1 sin ~ s = I i(s) where 2i is the antenna length. From Equation (103), we have 1/2 1/2 i' \K i and the quality factor Q associated with the nth natural mode can be expressed n s,-ft--?[] va -?ar.ff.] v> n n n n when R is the radiation resistance referred to a current antinode n For cylindrical antennas, we have where ^■IF ^ - . ± (2n„ JL ° - i 2t [A + s. (2„„) m 2 n77 l A = 2n7T |j + log 2 - l| 57 and s. is a sine integral. For arc antennas, we write and j-ir^i 1 ^ . , v 2 . , ^ -l 1 . R o ° R o 2 3 R Q o R Q (nvr) For n = even i i 3 R (n7T) for n = odd 3 OI--^ i (106) 3 „2 . ,2 R (n77) o Therefore, it is seen that in comparison to cylindrical antennas (ki)res becomes larger for arc antennas and helical antennas while (ki)antires becomes smaller for curved structures . Since . A - s. (2n77) .2 (ki)res = SE X (n . [1+| 1 ] (107) 9 \ I A + s (2n7r) 3 2-1 2 \| A + s. (2n7T) 3 ^ R 2 ^ n = 1,3,5. 58 and (kOant: /A - s.(2n7T) 1 17 I 2 A + s. (2n7T) 1 3 2 A Ro n77 ■ 2,4,6 (108) They therefore agree with that indicated in Figure 9 for kR > 2.5. The approximation for deriving Equation (103) holds only for relatively small curvatures . For the quality factor Q, we have, From Equation (105) 'res ft = -TpVtA - s (2n.7T)][A + s (2 n7 r) ] 1 1 K n W X X L a R 2 mr J (109) n = 1,3,5 and wJ^ - s i (2ni7)][A + s i (2nir) ] 1 + 3 [■ 16 i 2 A Ro n 7T- (110) n = 2,4,6 Q is larger for curved structures while Q is larger for cylindrical res ant ires antennas- since R at resonance decreases as the curvative is increased at a 7 n 2 2 faster rate than the decrement due to the term £ /A R Q nj[ . At antiresonance R increases as the curvature is increased and its order is also higher than the increment due the term — .£ /A R n7T • The results obtained from this analysis agree qualitatively with those .obtained from the input impedance graphs. Quantitatively speaking, the dis- i crepancy between these two results exist even for the cylindrical antenna 59 case. This,porha.(.o.. is due to the different assumption on the current distr] bution,. 4.5 The Input Impedances of Helical Antennas For helical antennas, input impedances were calculated as functions of C and C with £1 = 10, where C = kR , R is the radius of the helix 1 o> o p C = kb, b = ~— } p is the pitch of the helix Results are presented in the form of circular graphs. Figures 16, 17, and 18 give the input impedances of helical antennas of constant C (C = 0.5, C = 0.75 and C =1), while Figure 21a and 21b give the input impedances of constant pitch helical antennas for which C = 0.25 and Figure 22 is that for C = 0.5. When the size of the helix is fixed and the pitch is varied, the cir- cular graph of the input impedance enlarges as we decrease the pitch. The input impedance of the helical antennas approaches that of the arc antenna for the small pitch and approaches that of the cylindrical antenna for the large pitch as expected. The range between these two limits is a function of C , namely the size of the helix in wavelengths. The range is larger for smaller C and is narrowed down to zero for large values of C . The ! rate of this convergence is shown in Figure 19 and Figure 20. Figure 19 is the variation of the resonant resistance as a function of C and with C as a parameter while Figure 20 indicates that of the antiresonant re- sistance . 60 800 600 400- 200 -200 -400 •600 C.= THE CIRCUMFERENCE IN WAVELENGTH C 2 =THE PITCH IN WAVELENGTH Figure 16. The Input Impedances of Helical Antennas C = 0.5 6] CO X o 1200 800 400 -400 - -800 -1200 Figure 17. The Input Impedances of Helical Antennas C = 0.75 62 800 •800- kl 00 (OHMS) C, ***'* Cz= p/X Figure 18. The Input Impedances of Helical Antennas c 1 = 1 RESONANT RESISTANCE IN OHMS 64 o o o o «• o CO CO -&■ +-> be a rH 0) *< ^ T > o tr a & m CO — ■a •p o v. a. j: \- o z UJ -I UJ O c o +-> o G 3 o 5 £ c8 (1) SIAIHO Nl 30NV1SIS3U 1NVN0S3U - I1NV 65 1000 0.75 — Ci 1000 2000 \ , (OHMS) G=»d/X Figure 21a. The Input Impedances of Helical Antennas C = 0.25 2 66 kl 1000 500 1500 (OHMS) 2000 ■500 Cl = 7rd/X C 2 = P/X Figure 21b. The Input Impedance of Helical Antennas C^ 0.25 67 800 400 400 1600 R (OHMS! Figure 22a. The Input Impedances of Helical Antennas C 2 = 0.5 68 800 R (OHM) Figure 22b. The Input Impedances of Helical Antennas C ? = °- 5 69 The characteristics of the input impedance as a function of the antenna arm length is exhibited by the equi-ki contours. Generally speaking, when the antenna arm length is not long, say ki < 2.1, the characteristic of the equi- ki contours behaves very much like that for the arc antenna; the reactance is almost constant while the resistance increases monotonically as the pitch is increased. For larger ki , the characteristic deviates both from that of the arc antenna and that of the cylindrical antenna. When the pitch of the helix is held constant and the radius of the helix is increased, the circu- lar graph of the input impedance is first enlarged from that of the cylindrical antenna. It reaches a certain maximum and then starts to shrink to the input impedance of a cylindrical antenna again. This is because of the fact that helices of zero radius and infinite radius both become cylindrical antennas. For example, Figures 21a and 21b show the impedance for a con- stant pitch helical antenna for which C = 0.25; in Figure 21a the circular graph becomes larger for smaller values of C and it reaches the maximum at about C = 0.75. For even smaller values of C the circular graph shrinks and approaches to the impedance of the cylindrical antenna. Figures 22a and 22b give the impedance for C = 0.5. The convergence of the input impedance of helical antennas toward that of cylindrical antennas is shown in Figures 23 and 24. It is seen that there is a minimum in (/2) and a maximum in (Py ) for each given '^ res ' antires C . The locations of these extrema are shifted toward larger C as we increase (a) The location of minimums of (/P,) are at V c l 0.4 0.625 0.825 1.05 1.125 C 2 0.25 0.5 0.75 1 1.05 1.25 V C 2 1.6 1.25 1.1 0.9 1.325 1.5 0.883 70 ^< a V. u a? 0) . N s 3 * C\J O fn o U c8 < O ^ O CM S3 Z en UJ o 2 S3 in UJ o3 — : (Z LU en Ll ?H 2 ~) CI o _l c Ld O £ en CM £ z Ctf u •H C < cd 3 LO Ixl O w ■z. CO Ixl CM cr Ixl Sh lx 3 o S •H — Z) o CE \/d = *0 'H19N313AVM Nl HOlId 75 C o C circles with center at( — f OJand of radius — . Since it can be written as C l " 2" + C 2 o I 2 Examining Figures 25 and 26, one finds that for large values of C the contour lines of the input impedances of equivalent helical antennas are circles described by Equation (97'), that is to say, equivalent helical an- tennas are helical antennas with the same radius of curvature for the rela- tively large size of the helix. It is also seen from Figures 25 and 26 that when the size of the helix is relatively large, the equivalent helical antennas with respect to the resonant resistance are also equivalent helical antennas with respect to the antiresonant assistance. 76 5. SUMMARY OF THE RESULTS AND CONCLUSIONS The integral equation formulation for the problem of thin wire antennas has been examined. The linearized integral equation has been derived by using the thin wire approximation, ka « 1, i » a. It was shown that there are three special types of wire antennas having a simplified kernel function, namely the difference kernel. They are the cylindrical antennas, the arc an- tennas and the helical antennas. The latter obviously includes the first two as special cases. The input impedances of wire antennas were expressed in terms of a variational formula which involves the Hilbert's double integral. For the helical antennas, the set of double integrals in the impedance formula was reduced to single integrals . The evaluation of the input impedances of the cylindrical antennas, the arc antennas and the helical antennas was performed for a large range of the parameters. Although the input impedances of the cylindrical antennas are now considered as a classical result, no results on the input impedance of the arc antennas and the helical antennas was available to date as far as the author is aware. The input impedances for these antennas are shown in graphical form. For the arc antennas, the circular graph of the impedance enlarges as the curva- ture is increased, i.e. the resonant resistance increases while the antireso- nant resistance decreases. This effect is similar to that of the cylindrical antennas as the thickness is decreased. The input impedances for the helical i antennas are computed and the region where the input impedances of helical antennas approach that of cylindrical antennas is shown. 77 The shortening effect of the resonant wavelength and the antiresonant wavelength for the arc antennas are also investigated. It is interesting to note that the extreme value of the ratio of the resonant wavelength to the arc length measured by Englund is in good agreement with our computed result . The equivalent helical antennas were defined and given. The contour lines are closely related to the set of circles for the constant radius of the curvature of helices . 78 BIBLIOGRAPHY 1. L. V. King, "On the Radiation Field of a Perfectly Conducting Base In- sulated Antenna Over a Perfectly Conducting Plane Earth, and the Calcu- lation of Radiation Resistance and Reactance", Phil. Trans. Roy. Soc . (Lond.) Ser. A. 236, 1937, pp. 381-422. 2. E. Hallen, "Theoretical Investigations into the Transmitting and Receiving Qualities of Antennas", Nova Acta (Uppsala) 11, 1938, No. 4. 3. R. King and D. Middleton, "The Cylindrical Antenna; Current and Impedance", Quart. Appl. Math. 3., 1946, pp. 302-335. 4. S. A. Schelkunoff, "Theory of Antennas of Arbitrary Size and Shape", IRE Proc. 29, 1941, pp. 493-521. 5. H. C. Pocklington, "Electrical Oscillation on Wires", Camb . Phil. Soc. Proc. 9, 1897, pp. 324-332. 6. J. E. Storer, "Theoretical Discussion of Circular Loops," TR212, Cruft Lab. Harvard U., 1955. 7. Olof Brundell, "Current and Potential Distribution on a Circular Loop Antenna", Trans. Royal Inst, of Tech., Stocholm, Sweden, No. 154, 1960. 8. J. E. Storer, "Variational Solution to the Problem of the Symmetrical Cylindrical Antenna", TR-101, Cruft Lab., Harvard U., 1950. 9. C. T. Tai, "A Variational Solution to the Problem of Cylindrical An- tennas", TR-12, SRI Project No. 188, Stanford Research Institute, 1950. ■10. C. R. Englund, "The Natural Period of Linear Conductors", B .S.T.J. Vol. 7-B, 1928, pp. 404-419. 11. S. A. Schelkunoff, "Advanced Antenna Theory", John Wiley & Sons, Inc. , New York, 1952, pp. 141 and 169. 12. Co T. Tai, "A New Interpretation of the Integral Equation Formulation of Cylindrical Antennas", IRE Trans. AP-3, 1955, pp. 125. L|. R. Wo Po King, "The Theory of Linear Antennas", Harvard University Press, Cambridge, Mass., 1956 . 79 APPENDIX A STRUCTURES WHICH GIVE CLOSED CYCLE TYPE KERNELS Since the kernel function /C is expressed in terms of the function r (s.s 1 ), the Euclidean distance between two points in the curve. There- o fore the structures giving rise to the closed cycle type kernel are those leading to the special property r (s.s') = r (I s-s' I ) (A.l) o o To show that the Euclidean distance between two points on a straignt line, a circular arc and a helix lead to the expression (A.l), we express an arbitrary curve by the following set of parametric equations x = x(s) (A. 2) y = y(s) (A. 3) z = z(s) (A. 4) The Euclidean distance between two points on the curve is thus .... ' ■(s,s«) = \ [X(s) - X(s')r + [y(s) - y(s')] + [z(s) - z(s') 1 (A. 5) which becomes the expression (A.l) in the following cases (a) when x, y, z are linear functions of s. This describes a straight line. 2 2 (b) when the expression [X(s) - X(s')] + [y(s) - y(s')] has the pro- perty expressed in (A.l). A special set of transcendental functions satisfying this condition are 80 y = cos s which describes a circle. (c) The combination of cases (a) and (b) is a circular helix. The fact that the Euclidean transformation includes only translation and/or rotation indicates that the circular helix is the most general structure, in- variant under a one-dimensional Abelian group of congruent transformation. The statement that a curve with the property r (s, s ' ) = r ( I s-s ' I ) o o is invariant under a one-parameter Abelian group of congruent transformation, will be elaborated as follows (1) One Dimensional Case When all points of the curve are on a straight line, it is trivial that they are invariant under the translation, which is a group transformation. (2) T wo Dimensional Case - plane curve There exist three points of the curve forming a triangle. Any other points of the curve lie in the plane of the triangle. Let P , P and P be three points on the curve with arc lengths s , s and s respectively and consider three other points P ', P ' and P ' with arc length s ' = s + a, s ' = s + a and s ' = s + a . Then the triangle (P n P P Q ) is congruent ^ 2i 3 3 ± A o to the triangle (P 'P 'P ') and this defines a displacement T . 12 3 3- AP i P 2 P 3— ► ^ p i'Y p : 81 This displacement transforms the whole curve onto itself. Proof : Take any other point P on the curve. The image of P under Ta is P' . Then the Euclidean distances from P' to P ' P ' and P ■ are equal to those from P to P P and P respectively. Therefore P' is also on the curve. Furthermore, the displacement T is a rotation; it is a plane displacement and since case (1) is excluded it is not a translation. The plane rotation with a fixed origin is a group transformation. (3) Thre e Dimensional Case There exist four points not coplanar, say P , P , P and P with arc lengths s , s , s and s respectively. Then similar reasoning to (2) gives points P ' , P ' , P ' and P ' , with arc lengths s ' = s + a, s ' = s + ; X A O ~E X X . &t A s ' = s + a and s ' = s + a respectively, P 'P 'P 'P ' forms a tetrahedron •do 44 l^o4 congruent to the first one and this defines a congruent transformation T (P lV 3 V ►

-) I = / M o v F o< v ) r . r F o( v > = f F 1 (u) ( y -^— dv) du + A P 1 (u) ( j -^2— dv) (B.6) Applying these formulas, a list of double integrals which are useful to the derivation of input impedance formulas are given below. 84 >2! 1. / jsin u G(u) — du dv = / u sin u G(u) du + / (2i-u) sin u G(u) du rr r 1 r 2i 2. / / cos u G(u) — du dv =/ u cos u G(u) du + / (2i-u) cos u G(u) du u v 3. / / U V sin u G(u) — du dv = / [ cos v - cos (2i-v) ] G(v ) dv / cos u G(v) — du dv = / [sin (2i-v) - sin v] G(v) dv | sin v G(v) — du dv = / cos v G(v) - du dv = 2(i-u) cos v G(v) dv 7. i / sin v G(u) — du dv = | cos v G(u) - du dv =1 sin u G(u) du + sin (2i-u) G(u) du f " i / | u sin u G(u) — du dv =/ u sin u G(u) du 4 ff U V ). u cos u G(u) ~ du dv = I u" cos u G(u) du + (2i-u) u cos u G(u) du 85 11. I I u sin u G < v ) — du dv = J [sin (2i -v)- (2i-v) cos (2i-v)-sinv+v cos v ]G(v)dv 12. I I u cos u G(v) - dudv =] [ cos (2i -v)+(2i-v)sin(2i-v)-cos v-v sin v]G(v) dv | u sin v G(v) - du dv = l! V 14. / u cos v G(v) — du dv = | u v .5. I] u sin v G(u) — du dv = ff x r 1 f 6. j u cos v G(u) — du dv = I u sin u G(u) du + I u sin (2i-u) G(u) du u J v ' ^o J I v sin u G(u) — ■ du dv = io / / v cos u G(u) — du dv = I I v sin u G(v) - du dv = v cos u G(v) — du dv = 86 | v sin v G(v) - du dv = U V s v G(v) - du dv = 23. . ' v sin v G(u) — du dv = | [sin u - u cos u]G(u)du + u v o 2i [sin(2i-u)-(2i-u) cos(2i-u) ] G(u) du \ } v cos v G(u) — du dv = ■IP I u sin u G(u) — du dv = | u sin u G(u) du + \ u (2i-u)sin u G(u) du I j u cos u G(u) - du dv = J u cos u G(u)du + / u (2i-u)cos u G(u) du ■IP : // u sin u G(v) ^ du dv = I [ 2 (21 -v)sin(2i-v)- (4i 2 -4i v+v -2) cos (2i-v) 2 ■2v sin v + (v -2)cos v] G(v) dv ffi j ;u cos u G(v) - du dv = I [2(2i-v) cos (2i-v) + (4i -4iv+v -2)sin(2i-v) Jo 2 -2v cos v -(v -2) sin v] G(v) dv 87 2 1 29. / j u sin v G(v) — du dv = u v \U- 1 '13 2 2 3 | u"cos v G(v) — du dv = I -[81 -12i v+6iv -2v ] cos v G(v) dv 31. I / u sin v G(u) — du dv = 21 32. | I u" cos v G(u) ^ du dv = j u sin u G(u) du + u sin(2i-u)G(u) du J J u cos v G(u) — du dv = I u v 2£ 2 1 13 I 1 3 / / v sin u G(u) — du dv = I — u sin u G(u) du + / — (2i-u) sin u G(u)du fr f r i I I 2 1 I 1 3 J 1 3 34. / / v cos u G(u) - du dv = — u cos u G(u)du +i — (2i-u) cos u G(u) du u v ^o " 1 [35. j j v sin u G(v) — du dv = I v [cos v-cos (2i-v) ] G(v) dv u^v Jo 36. / / v cos u G(v) - du dv = v [sin (2i-v)-sin v] G(v) dv 2 1 v sin v G(v) — du dv = KK 38. / / v cos v G(v) — du dv = J 2v (i-v) cos v G(v) dv f 2 1 39. J v sin v G(u) — du dv = ; | v cos v G(u) — du dv = I [ 2u cos u + (u -2) sin u] G(u) du u v 21 2 2 [2(2i-u)cos(2i-u) + (4i -4iu+u -2) sin(2i-u) G(u) du 89 APPENDIX C COMPUTATIONAL PROCEDURES The numerical computations were carried out on ILLIAC, the digital computer of the University of Illinois . The computation involved two parts. (a) The integration of the set of integrals v f v , and v . They can be written as L 2L f ™ / \ cos T ^ x ) ^ I t, ( ^ cos ~r(x) 1 F (x) — — — r dx + I F. (x) =^7— - I ll r(x) J i2 r(x) Jr, T. dx [ _ . . sin r(x) , F. n (x) — _. ; dx + ll r(x) 2L /„ , . sin r F. (x) _. i2 r(x (x) 1,2,3. where F 's and F 's are given by the Equations (86) through (91) and r(x) is given by Equation (93) for the helical antennas and Equation (95) for the arc antennas . The integration was performed by using Simpson's rule. The set of functions F. .'s were first evaluated by the auxiliary routines; the re- sults then entered the integration routine with the assigned number of intervals for the integration « This number was determined by the convergence of the result^, which will be explained later a (b) The algebraic operations for the impedance formula. After the set of integrals v , v and v are evaluated, the set of 1' 2 3 complex numbers = j30 L_J 2 m a ± jp (sin 2 L) v„ - (L sin 2L) v. + (L 2 cos 2 L) v, Y + j5 «J ^ 1 aY + (36 . p\ - a Y 2 ♦ 6 2 V + 6 2 where • = -sotVa + Vs - 2a 2 T 2 J M* 1 V T lV! !t, 2 !l \ = CT sin 2 L - (7 L sin 2L + (J L cos L 2 2 2 6 = t sin L - t L sin 2L + t L cos L The computer program was prepared for computing the impedance of the helical antennas for a given set of parameters C , C , L and £1. These 1 2i parameters are defined as C = kR = 27TR A loo C 2 = kb = P/> L = ki ft _ 2 in 2i_ [where | R is the radius of the helix o i is the half length of the antenna a is the thickness of the antenna wire 91 and b = p/277 where p is the pitch of the helix The arc antennas and the cylindrical antennas are special cases of the helical antennas . For the arc antennas c 2 = o and for the cylindrical antennas C l = *»> C 2 = ° The program was coded in such a way that it could compute automatically the impedances for several values of L. The published results of the input impedances for cylindrical antennas were used as guides in checking the computation program. Since for those integrals involving Cos r(x)/r(x), the integrand gives a large value when x = 0, the integral / , . cos r(x) F._(x) dx r(x) was split into A L f F (x) COS r(x) dx + f Ffa) cos r(x) J iX Too / xl TOO dx 92 where the value A together with the number of points in the Simpson's integra- tion rule were determined by the stability of the results. A fixed number n = 20 for the integration interval was used to determine the stable value for A- The computation results for a series of integrals indicate that A = 0.25 was an optimum value . The value A = 0o25 was then used in the splitting of the above mentioned integrals „ To determine the number of the integration intervals the integrals were evaluated with a set of different number n. After analyzing these results,, we chose the value n = 30. For this combi- nation of values of A and n, the impedance for the cylindrical antennas were compared with that obtained by Tai, for i2 = 10 and i2 = 15. The comparison showed that in the range L < 6, the difference between these two results is 3 ^ at the most . 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Box 188 Mountain View, California Technical Research Group Attn: Librarian (Antenna Section) 2 Aerial Way Syosset, New York Ling Temco Aircraft Corporation Temco Aircraft Division Attn: Librarian (Antenna Lab) Garland, Texas mders Associates |:tn: Librarian (Antenna Lab) IS Canal Street ishua, New Hampshire imthwest Research Institute :tn: Librarian (Antenna Lab) )00 Culebra Road m Antonio, Texas Texas Instruments, Inc. Attn: Librarian (Antenna Lab) 6000 Lemmon Ave. Dallas 9, Texas A. S. Thomas, Inc. Attn: Librarian (Antenna Lab) 355 Providence Highway Westwood, Massachusetts New Mexico State University Head Antenna Department Physical Science Laboratory University Park, New Mexico Aeronautical Systems Division Attn: ASAD - Library Wright-Patterson Air Force Base Ohio Bell Telephone Laboratories, Inc. Whippany Laboratory Whippany, New Jersey Attn: Technical Reports Librarian Room 2A-165 National Bureau of Standards Department of Commerce Attn: Dr. A. G. McNish Washington 25, D. C. Robert C. Hansen Aerospace Corporation Box 95085 Los Angeles 45, California Dr. Richard C. Becker 10829 Berkshire Westchester, Illinois Dr. Harry Letaw, Jr. Raytheon Company Surface Radar and Navigation Operations State Road West Wayland, Massachusetts Dr. Frank Fu Fang IBM Research Laboratory Poughkeepsie, New York Mr. Dwight Isbell 1422 11th West Seattle 99, Washington Dr. Robert L. Carrel Collins Radio Corporation Antenna Section Dallas. Texas Dr. A. K. Chatterjee Vice Principal & Head of the Department of Research Birla Institute of Technology P. 0. Mesra District-Ranchi (Bihar) India ANTENNA LABORATORY TECHNICAL REPORTS AND MEMORANDA ISSUED i rntract AF33<616)-310 Synthesis cf Aperture Antennas,, 111 ' Technical Report No, 1„ C»ToAo Johnk, itober, 1954„* i Synthesis Method .for Broad-band Antenna Impedance Matching Networks," ' tcfanical R eport No. 2, Nicholas ¥aru, 1 February 1955,,* AD 61049 „ 'be Assymmetrically Excited Spherical Antenna,," Technical Report No. 3 , M>ert C. Hansen,, 30 April 1955 „* \nalysis of an Airborne Homing System,," Technical Report. No u 4, Paul E„ lyes, 1 June 1955 (CONFIDENTIAL) . 'oupling of Antenna Elements to a Circular Surface Waveguides," Technical [port No, 5, H„ Eo King and I. H. DuHamel,, 30 June 1955 ,* Vxially Excited Surface Wave Antennas /° Techni cal Rep ort NOo 7 9 Do E Royal, ! October 1955 o* ™_ _«=__«_=_______ 'oming Antennas for the F-86F Aircraft (450-2500 mc)'° ' Techni cal Report No 8, ] Eo Mayes,, Ro F„ Hyneman,, and R Co Becker, 20 February 1957, (CONFIDENTIAL), 'round Screen Pattern Range/ Technic al Memorandum No. 1 ,-, Roger R» Trapp„ 1 July 1955 o* C atract A F33 (616)) -3220 Vffective Permeability of Spheroidal Shells/" Technical Report No 9, Eo J. ifbtt and Ro Ho DuHamel „ 16 April 1956 „ 'n Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report jo 10. Do Go Berry and J» B„ Kreer, 10 May 1956, AD 98615 ':' Technique for Controlling the Radiation from Dielectric Rod Waveguides/ 8 I chnical Report No, 11, J, ff„ Duncan and R„ Ho DuHamel,, 15 July 1956 c * 'irectional Characteristics of a U-Shaped Slot Antenna , " Technical Report j'^^12,, Richard Co Becker, 30 September 1956 »** Ifflpedaiace of Ferrite Loop Antennas/ Technical Repo rt NOo 13, V. H. Rumsey aid Wo Lo Weeks, 15 October 1956 „ AD 119780 'jlosely Spaced Transverse Slots in Rectangular Waveguide/ Technical Report Ko 14, Richard F. Hyneman, 20 December 1956 „ Dist'ibuted Coupling to Surface Wave Antennas/' Te chnical Re port No., 15, ;alph Richard Hodges, Jr.,, 5 January 1957, 'The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical teport No. 16 , Robert L. Carrel, 15 January 1957 „* 'On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17 , 'alter L. Weeks, 10 April 1957.* AD 143989 A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source ,ntenna," Techni cal Report No. 18 ,, Walter L„ Weeks,, 20 April 1957 „ Broadband Logarithmically Periodic Antenna Structures," Technical Report No. 19 , U H« DuHamel and D. E, Isbell, 1 May 1957. AD 140734 'Frequency Independent Antennas,' Technical Rep ort No. 20 , V. Ho Rumsey, 25 ictober 1957. 'The Equiangular Spiral Antenna,,'" Technical Report No. 21, J. D. Dyson, 15 eptember 1957, AD 145019 'Experimental Investigation of the Conical Spiral Antenna, " Technical Report to. 22, R„ L. Carrel, 25 May 1957.** AD 144021 'Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical teport No. 23., E. J. Scott, 10 August 1957. 'Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," ecfanical Report No. 24, J. W» Duncan and R„ H, DuHamel, August 1957. ■'The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary Toss Section," Technical Report No. 25, Robert L. Carrel, August 1957. i'Cavity-Backed Slot Antennas,," Technical Report No. 26, R„ J. Tector, 30 'ctober 1957. I' Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical Report No. 27, W. L. Weeks, 1 December 1957. Thase Velocities in Rectangular Waveguide Partially Filled with Dielectric," technical Report No. 28, W. L. Weeks, 20 December 1957. Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary i'ross Section/ Technical Report No. 29, J. D. Dyson, 10 January 1958. j'Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. 30 , ! 'v E. Isbell, 20 February 1958. AD 156203 ! Electromagnetic Fields in Rectangular Slots/" Technical, Report No. 31 , N. J. i:uhn and P. E. Mast, 10 March 1958. The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder,' echnical Report No, 32,, J. W Duncan, 25 May 1958 „ A Unidirectional Equiangular Spiral Antenna" 1 Technic a l Re port No. 33„ , Do Dyson, 10 July 1958 . AD 201138 Dielectric Coated Spberiodal Radiators," 1 Technical Report No. 34 , W„ L„ eeks, 12 September 1958 . AD 204547 A Theoretical Study of the Equiangular Spiral Antenna," Technical Report o. 35, P. E. Mast, 12 September 1958. AD 204548 ontract AF33(616)-6079 Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Technical eport No. 36, R. Ho MacPhie, 30 April 1959. AD 215558 On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and nfinite Ranges,," Technical Report No. 37,, Raj Mittra, 15 May 1959 . Prolate Spheroidal Wave Functions for Electromagnetic Theory/' Technical eport No, 38 , W. L. Weeks, 5 June 1959 „ Log Periodic Dipole Arrays," Technical Report .No. 39,, Do E. Isbell, 1 June 1959 . D 220651 A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical eport No. 40 ,, So Dasgupta and Y. To Lo, 17 July 1959. The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical eport No. 41 , K, Go Balmain, 1 August 1959. the Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem n a Waveguide," Technical Report No. 42, Raj Mittra,, 1 October 1959 . Impedance Properties of Complementary Multiterminal Planar Stnactwes, " schnical Report No. 43 ,, G. A. Deschamps, 11 November 1959 „ On the Synthesis of Strip Sources," Technical Report No. 44 , Raj Mittra, December 1959. Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,' ,echnical Report No, 45 ,, Co H. Tang and Y„ T. Lo, 11 March I960. New Circularly Polarized Frequency Independent Antennas with Conical Beam or .unidirectional Patterns," Technica l Report No. 46, J. D. Dyson and P„ E. Mayes ; June I960. AD 241321 Logarithmically Periodic Resonant -V Arrays,," Technical Report No. 47 , P. E. Mayes , j ad Ro L. Carrel, 15 July 1960. AD 246302 Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Technical port Noo 48, Y, To Lo, June 1960 valuation of Cross-Correlation Methods in the Utilization of Antenna Systems," chni cal Report No. 49 , R. H„ MacPhie, 25 January 1961 ynthesis of Antenna Product Patterns Obtained from a Single Array," Technical !port NOo 50 , Ro Ho MacPhie, 25 January 1961 „ )n the Solution of a Class of Dual Integral Equations," Technical Report No, 51 , Mittra s 1 October 1961 . AD 264557 inalysis and Design of the Log-Periodic Dipole Antenna/* Technical Report No, 52 , >bert Lo Carrel, 1 October 1961 .* AD 264558 I Study of the Non-Uniform Convergence of the Inverse of a Doubly-Infinite itrix Associated with a Boundary Value Problem in a Waveguide," Technical Report ). 53, R„ Mittra, 1 October 1961 . AD 264556 Copies available for a three-week loan period, * Copies no longer available . Antenna Laboratory Technical Report No. 57 POLYGONAL SPIRAL ANTENNAS by C. H. Tang and 0. L. McClelland Contract AF33(657)-8460 Project No. 6278, Task No. 40572 June 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois