62j .365 55te , no. 13-16 co p. 3 Q 62T. 3 CO rt655te no. 16 cop. 3 g wow eg m i wuemiv ANTENNA LABORATORY Technical Report No. 16 THE CHARACTERISTIC IMPEDANCE OF THE FIN ANTENNA OF INFINITE LENGTH by Robert L. Carrel 15 January 1957 Contract No. AF33(616)-3220 Project No. 6(7-4600) Task 40572 WRIGHT AIR DEVELOPMENT CENTER ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA, ILLINOIS Antenna Laboratory Technical Report No. 16. the: characteristic impedance OF THE FIN ANTENNA OF INFINITE LENGTH by Robert L. Carrel 15 January 1957 Contract ;AF33 (6160 ?3220 Project No. 6(7-4600). .Task 40572 WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois 42 1. -16 £ TABLE OF CONTENTS Page Abstract v Acknowledgement vi 1. Introduction 1 2o Formulation of the Problem 3 3. An Example: The Biconical Antenna 7 4. Solution of the Problem 9 5. Concluding Remarks 14 Bibliography 15 Appendix A„ 1° Digitized by the Internet Archive in 2013 http://archive.org/details/characteristicim16carr ILLUSTRATIONS Figure Page Number 1. The Infinite Fin Antenna 1 2. Position of the Fin in the Spherical Coordinate System 2 3. Fin Configuration in the Complex w-Plane 6 4. Cone Configuration in the Complex w -Plane 7 5. Maps of the w, a, and z Planes 9 6. Characteristic Impedance of the Fin and the Cone above a Ground Plane 13 Table Number L Characteristic Impedance of the Fin and Cone 12 ABSTRACT Certain types of multi-conductor structures will support spherical transverse electromagnetic waves. The infinite fin antenna is one of these structures, A general method of attacking the static boundary value problem, as applied to these special structures, is outlined. The problem of finding the characteristic impedance of the infinite fin is solved completely. In addition, a short discussion of spherical TEM waves is appended. ACKNOWLEDGEMENT The author gratefully acknowledges the assistance and directive guidance of Professors V.H. Rumsey and R.H. DuHamel in the preparation of this paper. 1 . INTRODUCTION The infinite fin is a member of the family of antennas whose shapes are defined entirely in terms of angles, hence their electrical characteristics are independent of frequency. See Figure 1. co <4 ► oo Figure I The Infinite Fin Antenna In contrast to the infinite biconical antenna , which is a surface of revolution, the fin antenna is infinitely thin and lies entirely in a plane , Use of the method of images can be made to facilitate the solution of this problem. Thus we need only be concerned with one hemisphere of the spherical coordinate system. See Figure 2. The characteristic impedance of such a structure will be half of the characteristic impedance of the total structure with the image plane removed. 1. S.A. Schelkunoff and H.T. Friis, Antennas: Theory and Practice Wiley, 1952, pp, 104-106 1 ,0=0 Ground Plone f ( ,6,9) Figure 2 Position of the Fin in the Spherical Coordinate System Since the shape of the antenna is clearly independent of r , the radius of the spherical coordinate system, only two parameters, and 4>, are needed to define the structure. Also, since any spherical surface (r=constant) will intersect the antenna and its: image plane in an invariant manner for any r, the problem will be shown to reduce to essentially a two-dimensional problem in electrostatics. 2 FORMULATION OF THE PROBLEM It now becomes necessary to examine the form of the solution of Maxwell's equations in spherical coordinates with the restriction that E r = o = H r . It can be shown that Maxwell's equations reduce to the two-dimensional Laplace equation, sin 9 -3- (sin e ||) + = 0, (1) where T is a function of 9 and qp only. The E and H vectors describing the field contain no radial component, and propagation is in the r direction only. Thus we conclude that the conductors support spherical transverse electromagnetic (TEM) waves, and that these waves permeate all space except that occupied by the conductors. In addition, any solution must satisfy the boundary conditions, namely, that tangential E and normal H must vanish on the surface of the conductors It can be shown that these boundary conditions reduce to those of electro- statics. The reader unfamiliar with this underlying development may refer to the appendix. Thus the problem is drastically reduced to a simpler one, namely, the solving of Laplace's equation in two dimensions subject to the boundary conditions of electrostatics. The use of conjugate function theory in solving two-dimensional electrostatic potential problems has been extensive and solutions of many problems have been tabulated in the literature. It will be convenient then for us to draw a close analogy between our problem and the classical two-dimensional problem. The classical problem postulates that the equi- potential surfaces must be cylindrical surfaces of constant cross section. 4 Mathematically speaking dV/dz = in the cylindrical coordinate system, where V is the potential function. It can be seen that a given conical surface, when defined by angles, will intersect all spheres centered at the apex of the cone in a similar manner. This is analogous to the intersection of a cylindrical surface and any plane (z = constant) in the cylindrical coordinate system. The analogy will now be complete if we can map the spherical sur- face onto a plane in such a way that the identity of the boundaries is preserved and that Laplace's equation remains unchanged. In other words, we are looking for a relation P = f(9) -plane Figure ij- Cone Configuration in the Complex w-Plane Z - Jl log b/a where T^ is the intrinsic impedance of the medium between the conductors Substituting for b and a, z = n- i g (_!—.) 2n tan \j//2 2. Ramo and Whinneay, "Fields and Waves in Modern Radio , Wiley, 1953, p. 119. 7 8 where \|//2 is the half angle of the cone. Therefore, for a cone above its image plane , Z Q = (T\/2rt) log cot \|//2. (10) This is the exact solution given by Schelkunoff and Friis. 4*. SOLUTION OF THE PROBLEM Returning to the original problem, we find that the solution can be written in terms of elliptic functions. First consider the mapping which transforms the w-plane into the z-plane through the use of the intermediate o -plane. See Figure 5. w-plone (a) Pz Qg g. p i -i -\ i j_ k k ij-?lane (b) -k+ik 1 k+ik 1 P 2 Q 2 P. a, -K \ P. K -K -iK' K-i K 1 z-plane (c) Figure 5 Maps of the w-, a-, and z-Planes a = - 1/2 [w +-(l/w)] C da- J oV /(l-a2)(i-k2 a 2) i = L (P1 + i). k 2 l P! (11) (12) (13) F. ObeJrhettinger and W. Magnus, Anwendung der Elliptischen Funkionen in Physik und Technik, Springer, 1949, pp. 50-65. 9 10 Equation 11 maps the w-plane, including the unit circle and its internal slit, into the whole a -plane furnished with the slits as indicated in Figure 5(b). Note the unusual correspondence between the points P^ , P2 , Qi , and Q2. The upper half a -plane may be mapped onto the rectangle QjQ^Pl in the z -plane by the use of the Schwarz-Christoffel transformation Equation 12. Due to the symmetry principle of Riemann and Schwarz, the lower half a -plane is mapped onto a rectangle in the z -plane which is formed by inverting the rectangle Q1Q2P2P1 a ^ out tne line QiQ2- The relation between the modulus k and p is given by Equation 13. With the help of Equation 12 both K and K'can be expressed in terms of k. We have and X 7(l-^)(l- ; k2t2) r 1/k iK' = \ ds j 1 yd-s 2 )(i-k 2 s 2 ) The latter integral can be brought into a more elegant form= If we make the substitution a- (l-k'2 t 2)-l/2 f where k' 2 + k 2 = 1, (15) we obtain 1 K' = \ dt X J (l-t 2 )(l-k' 2 t 2 ) (160 Notice that Equations 14' and 16. are complete elliptic integrals of the first kind of modulus k and k' , respectively. The configuration in the z-plane can be considered as the parallel combination of two parallel plate transmission lines. The parallel 11 plate line has a characteristic impedance given by Z Q ■ Tl d/b, where r\ is the intrinsic impedance of the medium between the conductors, d is the distance between the plates, and b is the width of the plates. Note that this is the exact solution; there are no fringing effects because the total electric field in the a-plane is mapped into the interior of the two period rectangles in the z -plane. Thus the characteristic impedance of the parallel combination of these transmission lines is given by z ° = K k k = ^ I' ■ (17) From Equation 13, k - 2P * Pi 2 -1 ' Upon substituting p^ = tan y/2 (from Figure 3), it is seen that k = sin \j/ k' = cos \j/. (18) The solution of the problem is now complete; see Figure 6> for a graph of the characteristic impedance versus the half angle \j/ of the antenna. UNIVERSITY OF ILUHQ* UBHAHY 12 Table 1. CHARACTERISTIC IMPEDANCE OF THE FIN AND CONE Z - FIN » 5 229 10 187 15 163 20 146, 25 132 30 121 35 111 40 102 45 94.2 50 87.0 55 80.3 60 73.7 65 67.3 70 61.0 75 54'. 4' 80 47.3 85 38.6, 90 Z d CONE 00 188 146, 122 104 90.5 79.0 69.3 60.6, 52.8 45.8 39.2 33.0 27.0 21.4' 15.9 10.5 5.3 L3 15 30 45 60 75 90 \j/ HALF ANGLE (DEGREES) Figure 6 Characteristic Impedance of the Fin and the Cone Above a Ground Plane 5. CONCLUDING REMARKS It should be pointed out that the method here employed to solve the fin antenna is quite general. As outlined in Section 2, this method may be applied to any uniform structure in which 9 = f () . However, the mapping of the w-plane into a more amenable geometry, as in Section 4', may be exceedingly difficult. Furthermore the mapping will, in general, differ from one problem to another. 14 BIBLIOGRAPHY 1. S A. Schelkunoff and H. T. Friis, " Antennas: Theory and Practice", Wiley, 1952 2 S. A. Schelkunoff, M Advanced Antenna Theory", Wiley, 1952 3 S. Ramo and J. R. Whinnery, '*' Fields and Waves in Modern Radio", Wiley, 1953 4. W. R Smythe, "Static and Dynamic Electricity", McGraw-Hill, 1950 5. J, J Thomson, " Recent Researches in Electricity and Magnetism", Oxford, 1893 6. F. Oberhettinger and W.. Magnus, " Anwendung der Elliptischen Funktionen in Physik and Technik", Springer, 1949 7. F. Nehari, " Cbnfoimal Mapping", McGraw-Hill, 1952 8. F. Bowman, 'Elliptic Functions", Wiley, 1953 9. E. T. Whittaker and G. N Watson, " : Modern Analysis", Cambridge, 1952 15 APPENDIX A It will be demonstrated that, under certain conditions, Maxwell's equations reduce to Laplace's equation in two dimensions, and that the boundary conditions on the surface of the conductor reduce to the boundary conditions of electrostatics, Maxwell's equations for a lossless medium can be written as: jweE. = V x h -jwnH = V x E. (19) Let us determine if Maxwell's equations have a solution given in terms of a scalar function of three spherical coordinates. Consider H = V x r n, (20) where r is the unit radial vector and II is a scalar function. If Equation20 is a solution , it must satisfy the vector wave equation V x v x h ■- 3 2 H - 0. (21) Substituting 20 in 21 we find that V x (V xVxrn - 3 2 r II) = 0. (22) Equation 22 will hold if V x Vx r II - 3 2 r n = - V (jweV) (23) where V is any scalar function. (The reason for the use of the arbitrary constant multiplier will become apparent later.) 16 17 Equating the r, 0, and cf> components of each side of Equation 23, we find that -jwe |¥ + 3 2 n + > L 9r ,,-a.iJ 3r30 3 2 n jwe jwe av sin 9 l(sm9 §E)+ §SQ 30 30 3< £| (24') (25) (26,) 0, 3r30 " 30 Equations 25 and 26< will be satisfied if _ sn. -jueV 3r (27 Substituting for V in Equation 24', we find that, for Equation 20 to be a solution of Maxwell's equations, II must satisfy this equation: |?n + 3 2n + _g^T ln e a (sin Q m > + a&TL , 9r ^ r2 sin 2e L 30 30 3<£_ If we assume a product solution of the form II = RT, where R is a function of r alone and T is a function of and 4>, we find that (28) vl ( 3?R + 3 2 R) + __L B 9r2 Tsin 2 9 sin -9 (sin §1) + ^T 30 30 S0 2 (29) Since Equation 29 is in the separated form, both parts can be set equal to a constant. It is obvious that one solution of this equation can be found when 9?£U 3 2 R and 3r< ■ in l Q (sin 91) ♦ m - 0- (30) (31) 18 We shall return to Equations 30 and 31 presently, but first let us examine the class of problems which can be handled by this restricted solution. Equation 20 limits the solution of Maxwell's equations to TM modes, that is, H =0. If we substitute 20 in Maxwell's equations and solve r for E_ we find that, after some algebraic manipulation, j we E> r r^sin^f sine ^-(sin e^H) + ^n ae se a<^ - r a r ae x a2n (32) r sin ara<£ where r,9, and are the unit vectors in the r, 9, 4> directions, respectively. Notice that E r also vanishes due to the assumption of Equation 3l . Using Equation 27, we may now write E = - V t V, (33) where V fc is the transverse gradient operator. Thus we see that the assumptions which have been made are justifiable, and lead to solutions for E and H such that E r = H r = 0. This is the familiar principle or TEM mode for spherical waves. If we solve the differential equation 30 we find tnat R = e^r, (340 We choose to use the minus sign only, since propagation is outward from the origin. Hence II = e"J° r can be substituted in Equation 28, which becomes S1 " e le (sine i )+ S =0 - (35) 19 This is Laplaces equation with 3T/dr = 0. Note that n and V also satisfy this equation. Let us now examine how the boundary conditions on H and E restrict EI and V. At the surface of the conductors E * _N = 0. (N is the outward directed normal). Since E = - V fc V, - V fc V x N = This shows that V and 9EI/9r must be constant on the surface of each conductor. These are the boundary conditions of electrostatics. Thus we have shown that our choice of auxiliary functions satisfies Maxwell's equations and the boundary conditions, and leads to a solution which must satisfy Laplace's equation. 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