The Pennsylvania State University The Graduate School College of Engineering ANALYTICAL AND NUMERICAL OPTIMIZATION OF AN ELECTRONICALLY SCANNED CIRCULAR ARRAY A Thesis in Electrical Engineering by James Matthew Stamm 2000 James Matthew Stamm Submitted in Partial Fulfillment Of the Requirements For the Degree of Doctor of Philosophy December 2000 We approve the thesis of Mr. James Stamm. Date of Signature _________________________________ ________________ James K. Breakall Professor of Electrical Engineering Thesis Advisor/Committee Chair _________________________________ ________________ Raymond J. Luebbers Professor of Electrical Engineering _________________________________ ________________ Akhlesh Lakhtakia Professor of Engineering Science and Mechanics _________________________________ ________________ Douglas H. Werner Associate Professor of Electrical Engineering _________________________________ ________________ Anthony J. Ferraro Dist. Professor Emeritus of Electrical Engineering _________________________________ ________________ W. Kenneth Jenkins Professor of Electrical Engineering Electrical Engineering Department Chair ABSTRACT 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 inter- element 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 TABLE OF CONTENTS LIST OF FIGURES…………………………………………………………………….. vi LIST OF TABLES……………………………………………………………………… ix ACKNOWLEDGEMENTS……………………………………………………………...x 1. INTRODUCTION…………………………………………………………………….1 1.1 THESIS OBJECTIVES………………………………………………………...… 3 2. ANALYTICAL PATTERN FUNCTION STUDY………………………………….. 5 2.1 INTRODUCTION……………………………………………………………...… 5 2.2 ARRAY FACTOR REVIEW…………………………………………...………... 6 2.3 ARRAY FACTOR DECOMPOSITION…………………………………………. 9 2.4 DIRECTIVITY…………………………………………………………………...21 2.5 EXAMPLES……………………………………………………………………...27 2.6 SUMMARY….…………………………………………………………………..32 3. THE EMBEDDED (ACTIVE) ELEMENT……………………………...….…… 33 3.1 INTRODUCTION…………………………………………………………....…..33 3.2 BACKGROUND………………………………………………………….…… 33 3.3 NETWORK BASICS AND EQUIVALENCE TO MoM…………………..…. 36 3.4 RADIATION PATTERNS………………………………………………….…. 46 3.5 INCLUSION OF LOADS……………………………………………………... 48 3.6 INVARIANCE TO ELEMENT/SOURCE MISMATCH……………….….….. 49 3.7 SENSITIVITY TO SOURCE LOAD VARIATION………………………..…. 53 3.8 SUMMARY ………………………………………………………………..….. 54 4. RECEIVE MODE BEAMFORMING………………………………………...…… 55 4.1 INTRODUCTION………………………………………………………………. 55 4.2 CALCULATION OF THE EXCITATION VECTOR…………...…………....... 56 4.3 EQUIVALENCE TO DIRECTIVITY MAXIMIZATION…………………...… 59 4.4 EFFECT UPON ACTIVE IMPEDANCES………………………………...…… 61 4.5 LINEAR LEAST-SQUARES BEAMFORMING CAPABILITIES……………. 72 4.6 SENSITIVITY TO AMPLITUDE AND PHASE ERRORS……………………. 79 4.7 SUMMARY ………………………………………………………………… 84 5. A PARAMETER STUDY OF THE CIRCULAR ARRAY………………………..85 5.1 INTRODUCTION………………………………………………………………..85 5.2 RELATIVE PERFORMANCE STUDY…………………………………………86 5.3 GRATING LOBE FORMATION………………………………………………..99 5.4 RELATIVE ENERGY IN ELEMENT PATTERNS…………………...……….101 5.5 PHASE CENTER MOVEMENT………………………………...…………......102 5.6 SUMMARY OF RESULTS………………...…………………………..………106 v 6. ARRAY DESIGN, FABRICATION, AND MEASUREMENT…………………. 109 6.1 INTRODUCTION…………………………………………………...…...….. 109 6.2 SELECTION OF PROTOTYPE ARRAY PARAMETERS…………………... 109 6.3 ELEMENT SELECTION…………………………………….....……………... 111 6.4 ELEMENT DESIGN………………………………………………….……….. 113 6.5 OPTIMIZATION WITH NEC-OPT…………………………………………....118 6.6 ELEMENT EVOLUTION AND MODELS……….……………………….….. 120 6.7 ARRAY MODELS…………………………….…………………….………….131 6.8 ARRAY FABRICATION……………………………………..……………….. 134 6.9 MEASUREMENTS………………………………………………….………… 137 6.10 SYSTEM GAIN MAXIMIZATION…….. …………………………..…..….. 146 6.11 COMPARATIVE CRITIQUE OF NEC AND WIPL…………………………149 6.12 SUMMARY OF RESULTS……………….. ……………………………..…..156 7. TRANSMITTER POWER MANAGEMENT…………………………………... 159 7.1 INTRODUCTION…………………………………………………………….. 159 7.2 VECTOR COMBINER THEORY…………………………………………….. 161 7.3 EXCITATION OPTIMIZATION……………………………………………… 167 7.4 REQUIRED MODULE ACCURACY………………………………...……..... 171 7.5 SUMMARY ……………………………………………………..………….. 179 8. CONCLUSION... …………………………...…………………………………..…..181 APPENDIX A DERIVATION OF F()φ FOR THE HALF-CIRCLE WITH ENDFIRE ELEMENTS ……………………………………………….……….……… 185 APPENDIX B DERIVATION OF RECEIVE-MODE EXCITATION VECTOR…... 188 APPENDIX C NEC GEOMETRY FILES FOR SELECTED FIGURES…………….191 REFERENCES…………………………………………………………..……………. 197 vi LIST OF FIGURES Figure 1.1—The E-2C is catapulted off the carrier deck…………………………………. 1 Figure 2.1—Coordinate system of circular array………………………………………….7 Figure 2.2—Inter-element separation…………………………….……………………... 11 Figure 2.3—Bessel functions of the first kind and order 0, 30, and 60..………………. 12 Figure 2.4—Bessel functions with order and argument related to number of elements…13 Figure 2.5— cotπ () 2qM+ 1 / for M = 40, 60……………………………………………... 18 Figure 2.6—Contributions to field pattern from parts of decomposed pattern function… 19 Figure 2.7—Analytical endfire element patterns…….…………….……………………. 20 Figure 2.8—40-element circular array of endfire elements and emb. element pattern…..25 Figure 2.9—Directivity vs. array radius for isotropic elements with M = 20, 40, 60….... 28 Figure 2.10—Directivity of 60-element array of isotropic elements λ = 1 meter, p = 2….29 Figure 2.11—Directivity of 60-element array of isotropic elements λ = 1 meter, p = 9….29 Figure 2.12—Directivity of 40-element endfire array λ = 1 meter, p = 2 ……..………… 30 Figure 2.13—Directivity of 40-element endfire array λ = 1 meter, p = 7 ……..………… 30 Figure 2.14—Directivity of 80-element endfire array λ = 1 meter, p = 2 ………..……… 31 Figure 2.15—Directivity of 80-element endfire array λ = 1 meter, p = 7 ……..………… 31 Figure 3.1—Array with arbitrary voltage sources, loads, and geometry………………... 35 Figure 3.2—Two-port network………..………………………………………………… 36 Figure 3.3—Method for measuring Z11, Z21 …………………………………………... 44 Figure 3.4—Two-element array.………………………………………………………… 45 Figure 3.5—Radiation patterns for array of Fig. 3.4………………………………..….