Analytical and numerical optimization of an electronically scanned circular array

by 1962- Stamm, James Matthew

A combined analytical and empirical optimization of an ultra high-frequency (UHF) circular array is presented in this work. This effort can be roughly categorized into three parts. Part 1 is a mathematical and numerical analysis of the general characteristics of circular arrays. The analysis includes a derivation of pattern functions for arrays of isotropic and endfire elements that extends beyond what has been presented in the literature to the limits of manageability (Chapter 2). This includes the derivation of a new closed form expression for the directivity of either isotropic (in the plane of the array) or analytically specified endfire element patterns with a variable elevation beamwidth. A parameter study of the circular array follows (Chapter 5). An empirical approach was undertaken to answer questions left unresolved from the mathematical analysis, which, because of the inherent complexity, is inconclusive. The resulting parameter study documents the relative performance available from a specific array as a function of the physical size, the number of elements, and the element beamwidth. The study demonstrates that the number of array elements is the primary factor limiting the ability to minimize the beamwidth (in the plane of the array) and sidelobe levels. Other discussions include the unavoidable element-element coupling (Chapter 3), relative energy contained in element patterns of various beamwidths, and the observation of a positive relationship between inter-element coupling and element phase center movement towards the array center. Utilizing the results from Part 1, Part 2 discusses the optimization strategy of the element and the array (Chapter 6). Two moment method codes—one of which works directly with a quasi-Newton optimizer—were used to complete the physical design. A complete array was fabricated and tested. Two critically important concepts are presented here. The first is that assuming the pre-1983 IEEE definition of gain is adopted, referred to throughout the thesis as system gain, then the voltage excitation that maximizes the system gain for an array of arbitrary geometry is simply proportional to the field contributions at a given iii beam angle from the respective elements. The second concept is that despite strong interelement coupling, an array with a desirable set of element characteristics can be created by performing an optimization on an isolated element. This is of significance because the optimization of the electromagnetic model of the array can be prohibitive, for the sheer number of unknowns present. Part 3 develops the appropriate beamforming methods. Several techniques are used. The first, based upon a linear least-squares method (LLS), is suitable for reception (Chapter 4). Here it is shown that the LLS method can be used to maximize the directivity. Along this line, the effect that the so-called target null-to-null beamwidth has on the array efficiency (and consequent system gain) is noted and discussed. Both weighted and unweighted versions are considered. With weighting, sidelobes of -40 dB are demonstrated. For transmission, a new means of placing a taper across the aperture while simultaneously operating all amplifiers at full power is introduced. An eight-port vector combiner, which forms the basis of this capability, is explained. The sequential quadratic programming method is employed to permit non-linear array weighting constraints (Chapter 7). Non-linear constraints are needed to maximize the effectiveness of the combiner. This approach to a tapered transmit beam affords the full system gain of a uniform excitation (or more), while reducing peak sidelobes by approximately 11 dB. iv