Wave propagation in a coaxial system
A solution is obtained for the problem of the propagation of electromagnetic waves in a semi-infinite flanged coaxial line with an infinite center conductor, in terms of an infinite set of coefficients which are determined by an infinite set of linear equations. The solution is discussed, in detail, in limiting cases which illustrate properties both of a thin vertical antenna on a plane perfectly conducting earth, and of a thick antenna fed by a low impedance line. Numerical results are given in these cases. The possibility of a solution for any excitation frequency is also discussed. Introduction. The use of an antenna for the purpose of radiating electromagnetic energy must involve the generation of this energy and its passage to the antenna from the generator along a transmission line. There is, however, very little theoretical work published in which an antenna is considered with its means of excitation. It might appear to be an advantage to examine the radiating system in isolation, and then to regard it as an impedance lumped at the end of a transmission line: the magnitude of such an impedance depends, however, on the transmission line parameters. It is therefore of interest to find whether there is any range of parameters for which the line and the antenna are substantially independent. The work done on this subject by King and others [3] implies that as we reduce the spacing of the transmission line to zero we may extrapolate from a series of experimental measurements of physical quantities to the limiting case of zero spacing; this value may be identified with theoretical results obtained by assuming, in the isolated radiating system, a delta-function excitation (sometimes called a "slice" generator) at the driving point of the antenna. This statement needs qualification, since the question of the existence of the physical quantity in the limit of zero spacing was not considered by King. We find that this limit does not indeed exist. Accordingly, in this paper we are concerned with an idealization of the cylindrical antenna problem in which the complete transmission circuit may be examined mathematically. As an approach to the problem of the antenna of finite length, we shall consider a semi-infinite flanged coaxial line with its center conductor extended to an infinite length. A signal set up at one end of the line is partly reflected at the open end and partly radiated into the half-space outside the line. Since we are interested in antennae, we may describe the radiating system as an infinite vertical antenna of circular cross-section, standing on a horizontal, perfectly conducting ground of infinite extent, with the exciting signal fed in at the bottom. We shall further simplify the system by considering the case in which the field is radially symmetric about the axis of the cylinder, with no magnetic field component parallel to this axis. The field components are related by Maxwell's equations; in this *Received May 25,1958: revised manuscript received January 12, 1959. The research described in this paper was partly supported by the Air Force Cambridge Research Center under Contract No. AF 19(604)-1391, and by the Office of Naval Research and the David W. Taylor Model Basin under Contract Nonr-562(24). 424 V. M. PAPADOPOULOS [Vol. XVII, No. 4 problem all the non-zero field components may be written down in terms of the transverse component of the magnetic field tangential to the surface of the cyclinder, and this component satisfies the scalar wave equation. The normal derivatives of the magnetic field vanish on all the perfectly conducing surfaces. The field components in the halfspace are related to the normal derivative of the magnetic field in the terminal plane of the coaxial line: this relationship may be found by a method in which cosine transforms are used. Since the field within the line may be expressed in terms of an infinite set of discrete modes (Marcuvitz [4]), it is possible by matching orthogonal components across the gap at the open end of the line to set up an infinite set of linear equations involving an infinite set of unknown coefficients. Each coefficient is related simply to the amplitude of the corresponding mode in the line; in turn, the set of coefficients determines all the field components completely and uniquely. To simplify the analysis, the free-space propagation constant is taken to have a small negative imaginary part which is later taken to be zero. The resulting solution is shown to satisfy the conditions of the problem. The solution of the infinite set of equations is simplified when the spacing of the coaxial line is small compared to the wavelength of the exciting signal. This simplifying condition is valid either when the outer diameter of the line is a small fraction of the wavelength or when the spacing between the conductors in the line is a small fraction of the line diameter. With this simplification, we find that the field within the line is very nearly that in an open-circuited line, and this approximate field leads us to a good value both for the admittance of the radiating system and for the energy density at large distances from the antenna. The effect of the simplification is to give us results not only for the thick cylindrical antenna with a small line spacing but also for the thin antenna with a physically plausible method of excitation. It should be emphasized that the results are appropriate to an idealized antenna of infinite length. The energy distribution may not be related to that of a finite antenna. On the other hand, the admittance of the infinite antenna is the limiting value of the admittance of long antennae.