Theory of antennas of arbitrary size and shape

In this paper there are presented (1) a quite general method of antenna analysis; (2) a physical picture of transmission phenomena in antennas, based on this method; and (3) an expression for the input impedance of antennas of any shape, whose transverse dimensions are small compared with the wavelength. In a brief historical sketch of the antenna problem the factors which must be taken into consideration in solving the problem are discussed. While in ordinary transmission lines the voltage is proportional to the charge, this is not the case in antennas. The explanation lies in the fact that antennas are multiple transmission lines (like wave guides) and not simple, that is, single-mode transmission lines. Our present theory is based on the voltage-current equations since these appear to be considerably simpler than charge-current equations. The latter are considered only briefly. In the absence of dissipation and in so far as the total voltage wave and the "principal" current wave are concerned, radiation is strictly an end effect. In so far as the total current and the total charge waves are concerned, radiation effects are distributed (nonuniformly) along the entire antenna. In the first approximation, regardless of the shape of the wire the charge is proportional to the voltage and waves are sinusoidal, the current wave having nodes while the voltage wave and the charge wave antinodes at the ends of the antenna. The second approximation depends on the shape of the longitudinal cross section of the antenna as well as on the size of the transverse cross section. Our analysis is based on Maxwell's equations but the final results are quite simple and the physical picture growing out of this mathematics is attractive to an engineer. It is permissible to think that a wave emerging from a generator in the center of an antenna is guided by an antenna until it reaches its "boundary sphere" passing through the ends of the antenna and separating the antenna region from the external space; at the boundary sphere some energy passes into the external space and some is reflected back--a situation existing at the juncture between two transmission lines with different characteristic impedances. We may also think of the antenna as the wall of an electric horn with an aperture so wide that one can hardly see the horn itself--just like a Cheshire Cat: only the grin can be seen. In fact, the mathematics that we use is that appropriate to wave guides and electric horns. The antenna problem is stated in Section I and its history is briefly discussed in Section II; Section III contains a summary and a discussion of the results for antennas with uniformly distributed capacitance (conical antennas); Section IV is devoted to antennas with non-uniformly distributed capacitance; Section V presents a derivation of the formulas contained in Section III; Section VI reviews the induced-electromotive-force method of computing radiation and its use in the present problem; Section VII is devoted to the current-charge equations; Section VIII is devoted to wave propagation along parallel wires; in Section IX an expression is given for the impedance of an infinitely long cylindrical wire, and Section X deals with an approximation needed in our discussion of the problem.