The two-element array, which is investigated in the preceding chapter, may be regarded as the special case N = 2 of an array of N elements arranged either at the vertices of a regular polygon inscribed in a circle, or along a straight line to form a curtain. Owing to its greater geometrical symmetry, the circular array is advantageously treated next. Indeed, the basic assumptions which underlie the subsequent study of the curtain array (chapter 5) depend for their justification on the prior analysis of the circular array.

The real difficulty in analysing an array of N arbitrarily located elements is that the solution of N simultaneous integral equations for N unknown distributions of current is involved. Although the same set of equations applies to the circular array, they may be replaced by an equivalent set of N independent integral equations in the manner illustrated in chapter 3 for the two-element array. Since the N elements are geometrically indistinguishable, it is only necessary to make them electrically identical as well. One way is to drive them all with generators that maintain voltages that are equal in amplitude and in phase. When this is done all N currents must also be equal in amplitude and in phase at corresponding points. But this is only one of N possibilities. If the N voltages are all equal in magnitude but made to increase equally and progressively in phase from element 1 to element N, a condition may be achieved such that each element is in exactly the same environment as every other element.

The preceding chapters are devoted to the development of a theory to predict the characteristics of arrays of physically real dipoles and monopoles. This chapter is concerned with the experimental determination of these characteristics and with the correlation of measured and theoretical results. In practice, the mathematical intricacies of theoretical formulas can be largely avoided by means of a computer programme to which a user need only supply the parameters of a particular array to obtain radiation patterns, driving-point admittances or impedances, and other characteristics. When programmed in this manner, the computer becomes a simulator which can be substituted for the repeated testing and adjusting commonly required in designing an array. However, the value of such a simulator rests entirely on how well predictions agree with observation when the final model is assembled. Comparisons in the preceding chapters between measured and computed results indicate that the theory is capable of describing an actual experimental model with acceptable accuracy. However, in applying the theoretical results to different experimental systems, account must be taken of certain considerations and precautions if such agreement is to be obtained. It is these considerations which are of primary concern in the present chapter.

The required measuring techniques have been discussed in general in several books; the purpose here is to examine difficulties and procedures which apply particularly to arrays of dipoles and monopoles. Although the discussions are not restricted to any particular range of frequencies, most of the procedures have been used in the 100–1200 MHz range.

The general theory of curtain arrays which is developed in the preceding chapter requires all N elements to be identical geometrically, but allows them to be driven by arbitrary voltages or loaded by arbitrary reactors at their centres. If some of these voltages are zero, the corresponding elements are parasitic and their currents are maintained entirely by mutual interaction. In arrays of the well-known Yagi-Uda type, only one element is driven so that the importance of an accurate analytical treatment of the inter-element coupling is increased. In a long array the possible cumulative effect of a small error in the interaction between the currents in adjacent elements must not be overlooked. As an added complication, the tuning of the individual parasitic elements is accomplished by adjustments in their lengths and spacings. This introduces the important problem of arrays with elements that are different in length and that are separated by different distances. In the Yagi- Uda array the range of these differences is relatively small. On the other hand, in frequency-independent arrays of the log-periodic type the range of lengths and distances between adjacent elements is very great.

In this chapter the analytical treatment of arrays with elements that are different in length and unequally spaced is carried out successively for parasitic arrays of the conventional Yagi-Uda type and for driven log-periodic arrays. However, the formulation is sufficiently general to permit its extension to arrays of other types, both parasitic and driven, that involve geometrically different elements.

Fundamentals—field vectors and potential functions

Radio communication depends upon the interaction of oscillating electric currents in specially designed, often widely separated configurations of conductors known as antennas. Those considered in this book consist of thin metal wires, rods or tubes arranged in parallel arrays of circular or planar form. Electric charges in the conductors of a transmitting array are maintained in systematic accelerated motion by suitable generators that are connected to one or more of the elements by transmission lines. These oscillating charges exert forces on other charges located in the distant conductors of a receiving array of elements of which at least one is connected by a transmission line to a receiver. Fundamental quantities upon which such an interaction depends are the electromagnetic field and the driving-point admittance. But these are completely determined by the distribution of current in the elements of an array. In this first chapter the basic electromagnetic equations are formulated and applied to simple antennas and arrays in the conventional manner which is based on assumed rather than actual currents. The limitations of this approach are pointed out as an introduction to the more accurate formulation of the theory of antennas and arrays that is presented in subsequent chapters.

The study of dipole arrays in chapters 3 through 6 has proceeded from simpler to more complicated configurations. In chapters 3 and 4 all elements are physically alike and arranged to be parallel with their centres uniformly spaced around a circle so that when driven in suitable phase sequences all elements are geometrically and electrically identical. Chapter 5 is also concerned with parallel elements that are structurally alike, but they lie in a curtain with their centres along a straight line of finite length; consequently the electromagnetic environments of the several elements are not all the same. In chapter 6 the requirement that the elements in a curtain array be equal in length is omitted and consideration is given first to arrays of elements that differ only moderately in length, then to arrays in which not only the lengths but also the radii of the elements and the distances between them vary widely. The lifting of each restriction introduces additional complications in the approximate representation of the currents on the elements by simple trigonometric functions and in the reduction of the integrals in the simultaneous integral equations to sums of such functions with suitably defined complex coefficients.

The final generalization which is carried out in this chapter is the omission of the requirement maintained throughout the book until this point, that all elements be non-staggered. The removal of this condition leads to the discussion of arrays of parallel elements that are arranged in a plane as in Fig. 7.1 and in three dimensions as shown in Fig. 7.2.

The method of symmetrical components

An array is a configuration of two or more antennas so arranged that the superposition of the electromagnetic fields maintained at distant points by the currents in the individual elements yields a resultant field that fulfils certain desirable directional properties. Since the individual elements in an array are quite close together— the distance between adjacent elements is usually a half-wavelength or less-the currents in them necessarily interact. It follows that the distributions of both the amplitude and the phase of the current along each element depend not only on the length, radius, and driving voltage of that element, but also on the distributions in amplitude and phase of the currents along all elements in the array. Since these currents are the primary unknowns from which the radiation field is computed, a highly complicated situation arises if they are to be determined analytically, and not arbitrarily assumed known, as in conventional array theory.

In order to introduce the properties of arrays in a simple and direct manner, it is advantageous to study first the two-element array in some detail. The integral equation (2.11) for the current in a single isolated antenna is readily generalized to apply to the two identical parallel and non-staggered elements shown in Fig. 3.1. It is merely necessary to add to the vector potential on the surface of each element the contributions by the current in the other element.

To facilitate practical applications of the theory apart from its detailed development and verification, principal points of the two-term theory, together with the steps required in its utilization, are summarized in this appendix. Numerical results for arrays containing only a few elements can be computed by hand with the aid of the tables in appendix I or those in the literature [1]. Calculations for larger arrays or for element parameters not included in the tables generally require a computer. For these applications, the theory can be conveniently packaged as a computer programme to which the user need only supply input data cards specifying the parameters of the elements and array to obtain a numerical evaluation of any properties of the array [2]. In this form, the theory can be used without considering either intricate mathematics or complicated programming steps once the initial programming is completed. In a reasonably general programme, the parameters that can be specified as input data are the number N of antennas in the array, their radius a, length 2h, spacing d, and the relative driving voltages Vi or currents Ii(0).

The theory applies to arrays of thin, identical, parallel, non-staggered, centre-driven, highly conducting dipoles which are uniformly spaced around a circle. If the admittances and currents are multiplied by two, they also apply to arrays of vertical monopoles over a large highlyconducting ground plane.

230.] The methods given by Maxwell for solving problems in Electrostatics by means of Conjugate Functions are somewhat indirect, since there is no rule given for determining the proper transformation for any particular problem. Success in using these methods depends chiefly upon good fortune in guessing the suitable transformation. The use of a general theorem in Transformations given by Schwarz (Ueber einige Abbildungsaufgaben, Crelle 70, pp. 105—120, 1869), and Christoffel (Sul problema delle temperature stazionarie, Annali di Matematica, I. p. 89, 1867), enables us to find by a direct process the proper transformations for electrostatical problems in two dimensions when the lines over which the potential is given are straight. We shall now proceed to the discussion of this method which has been applied to Electrical problems by Kirchhoff (Zur Theorie des Condensators, Gesammelte Abhandlungen, p. 101). and by Potier (Appendix to the French translation of Maxwell's Electricity and Magnetism); it has also been applied to Hydro dynamical problems by Michell (On the Theory of Free Stream Lines, Phil. Trans. 1890, A. p. 389), and Love (Theory of Discontinuous Fluid Motions in two dimensions, Proc. Camb. Phil. Soc. 7, p. 175, 1891).

In Art. 201 of the text there is a description of Perrot's experiments on the electrolysis of steam. As these experiments throw a great deal of light on the way in which electrical discharges pass through gases I have, while this work has been passing through the press, made a series of experiments on the same subject.

The apparatus I used was the same in principle as Perrot's. I made some changes, however, in order to avoid some inconveniences to which it seemed to me Perrot's form was liable. One source of doubt in Perrot's experiments arose from the proximity of the tubes surrounding the electrodes to the surface of the water, and their liability to get damp in consequence. These tubes were narrow, and if they got damp the sparks instead of passing directly through the steam might conceivably have passed from one platinum electrode to the film of moisture on the adjacent tube, then through the steam to the film of moisture on the other tube and thence to the other electrode. If anything of this kind happened it might be urged that since the discharge passed through water in its passage from one terminal to the other, some of the gases collected in the tubes gg (Fig. 84) might have been due to the decomposition of the water and not to that of the steam.