The radiation resistance of aerials whose length is comparable with the wavelength

Abstract

In this paper the electric force is calculated at any point of a vertical aerial, and also the power radiated from it. It is shown that the power can always be expressed as a factor multiplying the square of the field strength at an assigned radius in the equatorial plane. This is called the “height factor,” denoted by k, and the problem is to find the height of aerial which makes this factor a minimum; for then a given direct-ray communication can be produced with the minimum power and external annoyance. The expression for the height factor is very general and is applicable to any distribution of current: it is expressed as an infinite series of certain functions of the current distribution. These functions, however, are simple ones, being the first, second, third, etc., moments of area of the current-distribution curve.The radiation resistance is calculated for a sinusoidal distribution of current, and the values agree, to within a few parts in a thousand, with those calculated in 1923 by Ballantine, who used the process of viewing the aerial from a far-distant point. It is shown that k is insensitive to the form of the current-distribution curve, provided this has a constant area. Hence it follows that the output of any aerial can be predicted with certainty provided that the value of the “metre-amperes” is known; this can be deduced if the field strength is measured accurately at a comparatively close point in the equatorial plane.The height factor is calculated for a straight aerial inclined at any angle to the ground and it is found that, for a given length of aerial, the power is always a minimum when the aerial is vertical. It follows that if a single mast of given height is available, then the most economical aerial is the straight vertical wire.A general expression is derived for the height factor of an aerial with a flat horizontal roof, of either the inverted L or the symmetrical T type. If the current distribution along the up-lead and roof is part of the same sinusoid, this expression shows that the height factor of a roofed aerial is greater than that of a vertical aerial of the same total length; and that with masts of given height it can be an advantage to omit the roof. The analysis of the roofed aerial shows that it would be economical of power to obliterate the radiation from the roof by folding up the roof. This might be done by coiling the roof into a flat spiral centred on the top of the up-lead, an arrangement which would be reminiscent of the Franklin phasing coils used in beam arrays. A given length of wire disposed in an up-lead and folded roof is never as economical of power as the same total length of wire disposed vertically, but in general a mast of given height can be used more economically if fitted with a suitable folded roof.The problem of reception is discussed briefly, and it is shown that the current distributions in reception and transmission are not the same. Finally, equations are developed for solving the current-distribution problem, and a second approximation is made to the distribution in a vertical half-wave aerial.