On Two Arbitrarily Located Identical Parallel Antennas

The conventional problem of two arbitrarily located parallel antennas is solved by using an integral equation technique. The two simultaneous integral equations for the two antennas are first decoupled into two independent integral equations and then solved by an approximate method with currents represented by five trigonometric functions, three for the symmetric and two for the a n m e t r i c parts. Typical current distributions and input admittances are obtained for half-wave and full-wIave antennas in nonstaggered, in 45’ echelon, and in collinear arrangements. For the nonstaggered case, the result$ agree with experimental data. For the other two arrangements, no experimental data are yet awilable. However, the current distribution is also obtained by a numerical method. The two theoretical results agree favorably for all three cases. The fiveterm method can be extended to a general array of N-parallel elements. This is resened for a further report. Manuscript received October 9, 1967; revised January 19, 1968. This work was supported in part by the National Science Foundation under Grant GK-273 and in part by the U. S. Air Force under Contract AF19(628)-2406. The authors are with the Gordon McKav Laboratow. Division of Engineering and Applied Physics, Harvard Universit;,. Cambridge, Mass. I. INTRODUWION H E PROBLEM discussed is a classical one, which has been investigated by numerous persons.[’l,[?l In order to study the general N-element array, it is necessary to reexamine thoroughly the problem of two coupled parallel antennas. It is the purpose of this paper to develop a reasonably accurate but simple technique to solve the problem of two coupled antennas. In 1932, Carterrll treated the problem by assuming sinusoidal currents. His method, though simple in nature, does not take into account the asymmetric nature of the array and assumes only sinusoidal currents; its validity is quite doubtful except for a very limited region. Harrison,[*] in 1945, used an iterative method to treat two collinear antennas. His method, however, is too complicated to apply. Fortunately, a simple new technique has been recently made available by King and Wu[9 to deal with a single dipole antenna. This method has been successfully extended to 310 IEEE TRANSACTIONS ON ANrENNM AND PROPAGATION, MAY 1968 apply to the Yagi array141 and also to the log-periodic array.[jl The method is quite similar to that of HallCn,[61 except that the iterative method is not used. The current is arrived at by examining the properties of the kernel and has the simple form of a linear combination of three trigonometric functions. By a slight modification, the method can be extended to apply to the twoelement array. 11. INTEGRAL EQUATION FORMULATION Consider two identical parallel dipole antennas arranged as in Fig. 1. Each antenna has a half length h and radius a. The antennas are separated horizontally by a distance b and vertically by a distance d. If d=O, the array is conventional and nonstaggered; if b= 0 and d> 2h, it is a twoelement collinear array. Each antenna is assumed to be driven by a delta function voltage source of strength Vie. The antennas are oriented in the z direction ; the positive direction is upward from the center C in antenna 1, downward from the center C' in antenna 2. The positive direction of the current is upward in antenna 1 and downward in antenna 2. The whole structure is point symmetric about the point 0, which is the point bisecting the line CC'. Use is made of this property to decouple the integral equations. To simplify the problem, both antennas are assumed to be made of perfect conductors and the antennas are thin so that koa<<l, where ko is the free space propagation constant. If the nearest distance between the two conductors is large compared to the radius of the antenna so that proximity effects are negligible, the current on the antenna can be approximated by an axially distributed current without much error. Since all currents are in the z direction, the vector potential A has a z component A , only. The differential equation for the vector potential AL(z) along the surface of the first antenna id7] It is assumed that the time factor is ejwt. The solution of (1) is readily obtained Where use has been made of the continuity of the vector potential at the driving point and the relations, T I Fig. 1. Two identical parallel dipole antennas. The constant factor j / c o is included for convenience. The vector potential along the surface of the first antenna can also be expressed in terms of all currents in the array, l.e.,