Theory and experiment have run hand-in-hand leading the way toward new concepts and new radiating and receiving structures throughout the history of antennas. Sometimes new antennas were developed through experiment with supporting analysis coming later and perhaps pointing the way to optimization. In other instances, the analysis pointed the way with experiment showing that the analytical model could be achieved in the physically real world. Regardless of the origin of the idea, the most effective results have occurred when the theoretical analysis and the experimental measurements come together. For the experimental and theoretical results to agree, both must describe a model that has exactly the same electromagnetic boundary conditions. Since it is rarely possible to have identical models in the theoretical and the real world, the models must be critically examined, their differences identified, and at least approximately taken into account. Theoretical concepts such as “less than” and “thin” must be reconciled in each experimental model. Similarly, the measurement process must be critically examined to understand exactly what is being measured. Are sampling probes extracting enough power to distort the interference pattern on the transmission line? Are the fields being sampled too close to a discontinuity? Are loop probes being excited only in their basic mode? Finally, there is always the basic question in antenna experiments: “To what in the world are the elements really coupling?”

Measurement procedures on uniform sections of transmission lines have been discussed many times [1–7].

This set of tables is abridged from the report “Tables for Curtain Arrays” by Ronold W. P. King, Barbara H. Sandler and Sheldon S. Sandler, Cruft Laboratory Scientific Report No. 4 (Series 3), Harvard University, May 1964.

The calculations for the individual elements are given for Ω = 2 ln 2h/a = 8.6138 for β0h = π/4 and for Ω = 10 for all other electrical lengths. The admittances are given in millisiemens and the impedances in ohms. The vertical listings begin with the first element at the top. The unilateral endfire patterns are prescribed to point in the direction away from the first element toward the last element.

Over three decades have passed since the publication, in 1968, of “Arrays of Cylindrical Dipoles” by R. W. P. King, R. B. Mack, and S. S. Sandler. The present volume is a revised and enlarged second edition of that work. The objectives of “Cylindrical Antennas and Arrays” are similar to those of “Arrays of Cylindrical Dipoles”: to present approximate but efficient theoretical methods for determining current distributions, input admittances, and field patterns of arrays of cylindrical dipoles; to use such methods to analyze particular types of arrays; to describe experimental methods for determining current distributions, input admittances, and field patterns; and to correlate and compare theoretical and experimental results.

The most fundamental quantities, and the ones most difficult to determine theoretically, are the current distributions on the array elements. Rather than postulating the current distributions, perhaps the most common treatment in the literature of the 1960s, “Arrays of Cylindrical Dipoles” sought to determine the current distributions on the array elements by solving integral equations. Today's antenna and engineering literature is quite different from that of the 1960s: even elementary textbooks include discussions on determining current distributions from integral or similar equations.

As an example, consider the simplest configuration discussed in this book, the single, isolated cylindrical antenna of given length and radius. The integral equation treated in Chapter 2 is but one of the several integral or integro-differential equations that are encountered in the present-day literature.

Studies of coupled antennas in arrays may be separated into two groups: those which postulate a single convenient distribution of current along all structurally identical elements regardless of their relative locations in the array and those which seek to determine the actual currents in the several elements. Virtually all of the early and most of the more recent analyses are in the first group in which both field patterns and impedances have been obtained for elements with assumed currents. Pioneer work in the determination of field patterns of arrays of elements with sinusoidally distributed currents was carried out for uniform arrays by Bontsch-Bruewitsch [1] in 1926, by Southworth [2] in 1930, by Sterba [3] and by Carter et al. [4] in 1931. Early studies of non-uniform arrays are by Schelkunoff [5] in 1943, by Dolph [6] in 1946, and by Taylor and Whinnery [7] in 1951. The self- and mutual impedances of arrays of elements with sinusoidally distributed currents were studied especially by Carter [8] in 1932, by Brown [9] in 1937, by Walkinshaw [10] in 1946, by Cox [11] in 1947, by Barzilai [12] in 1948, and by Starnecki and Fitch [13] in 1948. A thorough presentation of the basic theory of antennas with sinusoidal currents was given by Brüuckmann [14] in 1939.

In Chapter 11, the phenomenon of resonances in large circular arrays of dipoles is discussed. It is assumed throughout that the dipoles are perfectly conducting. The effect of ohmic losses is considered in this chapter. It is assumed here that one dipole is driven. The case where two dipoles are driven is a very simple extension. In Sections 12.1–12.6, an array of highly conducting dipoles in free space is examined. The two-term theory is extended so that it applies to this theoretical model. The model is mathematically equivalent to the physically unrealizable one of an array consisting of highly conducting monopoles over a ground plane which is perfectly conducting: one can determine the currents in the latter model by multiplying those of the former by a factor of 2.

The case of a circular array of highly conducting monopoles over a highly conducting ground plane is examined briefly in Section 12.7. An approximate method is outlined which allows calculation of the admittances in this case by slightly modifying the theory of Sections 12.1–12.6 in which a lossless ground plane is assumed. The model of Section 12.7 closely approximates experimental conditions. The theoretical curve (driving-point admittance as a function of frequency) obtained after the effect of the imperfectly conducting ground plane is taken into account is compared in this section to a corresponding experimental curve and the agreement is very good.

Linear antennas

Wireless 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 arrays. 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 which describe such interactions are the electromagnetic field, the driving-point admittance, and the driving-point impedance. These can be easily determined if the distributions of current on the array elements are known. The determination of the currents on the array elements is the main concern of this book. In this first chapter, the basic electromagnetic equations are formulated and applied to a single antenna in free space. The simplest approach of assuming the current rather than actually determining it is reviewed first. Then, integral equations for the current distributions are derived, and determining the current by numerical methods is discussed. These discussions serve as an introduction to the analytical theory of antennas and arrays based on the solution of integral equations that is presented in subsequent chapters.

Figures 1.1a and 1.1b show two simple practical radiating systems. In Fig. 1.1a, a section at the open end of a two-wire transmission line has been bent outward to form a dipole antenna.

This appendix presents a set of five tables of (1) Ψd R, Φu and Φυ for single elements as functions of Ω and h/λ, and (2) Φu and Φυ for off-diagonal elements as functions of (K − i)(b/λ) and h/λ. The left column, i.e. (k − i)(b/λ), is the electrical distance between elements k and i. The two columns under each Φkiu and Φkiυ are the real and imaginary parts.

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 centers. 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.

The two-element array, which was 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 analyzing 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.

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 centers 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 centers 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.

Introduction

Dipole antennas located parallel to the plane boundaries of a layered region have numerous important applications over a wide range of frequencies. Examples include horizontal-wire (Beverage) antennas in air close to the earth, insulated antennas on or below the surface of the earth or sea, cellular telephone transceivers close to the human head, and patch antennas on microstrip.

The electromagnetic field of a dipole antenna parallel to the surface of a layered region is more complicated than the field of the same dipole when perpendicular to the boundaries. This is a consequence of the fact that all six components of the electromagnetic field are involved. In the cylindrical coordinates ρ, Φ′, Z′ shown in Fig. 9.1, there are three components of electric type, namely, Eρ, EZ′, and BΦ′, and three components of magnetic type, namely, Bρ, BZ′, and EΦ′. The dipole with the length 2h and electric moment 2he I is located at the height d′ in the air (region 0, wave number k0) over the surface of the electrically thin layer (region 1, wave number k1, thickness l). This coats a dielectric or conducting half-space (region 2, wave number k2 = β2 + iα2). The vertical Z′ = –Z axis passes through the center of the dipole. The field in the air, Z′ ≥ 0, is expressed in terms of the coordinates ρ, Φ′, Z′.