A THEORETICAL STUDY OF LOW SIDE LOBE ANTENNA ARRAYS by Brian Robert Gladman A thesis submitted to the University of London for Examination for the degree of Doctor of Philosophy (September 1975)• Work carried out at the Admiralty Surface Weapons Establishment. -1- ABSTRACT This thesis presents theoretical work undertaken in support of a research programme on low side lobe antenna arrays. In particular it presents results on the effects of mutual coupling on the performance of microwave arrays that consist of a number of identical radiating elements fed by a wide-band power dividing network. Low side lobe distributions are reviewed and the effects of various types of distributional error are derived. A simplified but representative array geometry is described for which a detailed analysis of mutual coupling is possible. Results are given for the element patterns in a finite array which clearly show the influence of mutual coupling and also the distortion in the patterns for elements close to the array edges. Array patterns obtained from low side lobe distri- butions are given. In studying the effects of inter-action between the array and its feed network, a set of 'aperture modes' are discovered that are orthogonal and uncoupled. These modes have a number of interesting properties and are shown to provide a pattern synthesis technique that can take account of the effects of mutual coupling between the array elements. The synthesis of multiple beam networks for producing these modes is covered. A number of potential wide-band feed networks are considered using both directional couplers and shunt coaxial junctions for power division. Their performance is considered both in isolation and in the presence of array effects which can distort the power distribution provided by the feed. Practical results are given for a low power array of 18 elements designed for a side lobe level of -40 decibels over a 20% band. -2- CONTENTS Page ABSTRACT 2 ACKNOWLEDGEMENTS 4 NOTATION 5 CHAPTER ONE DISTRIBUTIONS AND ERRORS 6 TWO MUTUAL COUPLING MIECTS THREE SYNTHESIS WITH MUTUAL COUPLING 98 FOUR ARRAY FEED NETWORKS 142 FIVE A PRACTICAL ARRAY 178 SIX CONCLUSIONS 203 REFERENCES 210 APPENDIX ONE CHEBYSHEV ARRAY DESIGN 215 TWO TAYLOR LINE SOURCE DESIGN 228 THREE THE EVALUATION OF THE FIELD INTEGRAL 247 FOUR COMPUTER PROGRAMS FOR ARRAY SOLUTION 257 FIVE PROGRAMS FOR MATRIX EIGENVALUES AND 282 EIGENVECTORS SIX IMPEDANCE TRANSFORMER AND 287 DIRECTIONAL COUPLER SYNTHESIS SEVEN PROGRAMS FOR SHUNT JUNCTION NETWORK ANALYSIS 300 -3- ACKNOWLEDGEMENTS I would like to thank Professor J Brown and Mr H Page of Imperial College, University of London for their help during this work. Also Professor J Croney of the Admiralty Surface Weapons Establishment (ASWE) who has been a constant source of inspiration and without whose unshakable support and encouragement this work would never have been completed. Thanks are also due to Dr H Salt of ASWE for the many stimulating discussions that preceded the concepts of chapter three, to J Wyatt and F Gregory for their help with the practical work and to Mrs Tandy for typing this thesis. Finally I would like to thank the Director of ASWE for his permission to undertake and publish this work . 0 S NOTATION The numbering of figures, equations and tables is self contained within each chapter but references are numbered across the thesis as a whole. Shorthand notation of F and R followed by a number is used when referring to figures and references respectively. In the main symbols are introduced as required. The symbol i is used throughout for the square root of minus one. Due to an oversight it has also been used as a subscript in chapter three in particular. While this dual usage is unfortunate, it should not cause any confusion in practice as it is always clear from the context which form is in use (the only equation in which both usages occur simultaneously is Eqn. 2.1 of chapter three). The notations ishl and 1 ch1 are used for hyperbolic sines and cosines. Both 'e' and 'exp ( )' are used for the exponential function, 1 121/ is used for natural logarithm and Tr has its normal meaning (v = 3.1415926535898 ....). The function sgn (x) is +1 if x is positive and -1 if x is negative. a 0 -5- CHAPTER ONE - DISTRIBUTIONS AND ERRORS • -6- 1. The Requirement for Low Side Lobes The design of radar systems for ship defence is heavily influenced by the ability of an attacker to interfere with or 'Jam' their normal operation by radiating a noise like signal within the band. in which they function. In this 'contest' the radar designer is at a severe disadvantage in that his signal, the radar echo, varies inversely with the fourth power of the target range whilst that from the jammer suffers only inverse square law attenuation. With this advantage the jammer can often inject sufficient power into the side lobes of the radar antenna to prevent target detection over large azimuth sectors or even over all azimuth angles if the antenna side lobes are poor. Whilst it is unlikely that low side lobe antenna design can completely eliminate this problem, the degradation can be restricted to a small sector (or sectors) by providing antennae in which the side lobe magnitudes fall rapidly with angle from the main beam. This volume presents the authors contribution to a research programme on low side lobe design at the Admiralty Surface Weapons Establishment. It is a largely theoretical study of the factors that limit the side lobe performance (and band width) of linear arrays of identical elements. 2. Performance Criteria for Aperture Distributions Antenna beam width and directive gain are among the more important characteristics of antennae. As is well known, they are functions of the size and shape of the antenna and also of the distribution of electromagnetic energy across the aper- ture. In comparing 'aperture distributions' it is helpful to normalise beam width and gain parameters in such a way that the influence of antenna size is taken out thus leaving values that characterise the performance of the distri- butions alone. In addition, as both continuous 'line source' and discrete 'array' distributions will be considered, it is important, if possible, to ensure that the performance parameters for these two classes are both consistent and directly comparable. This section serves to introduce these parameters and a number of formulae commonly used in line source and linear array analysis. Dealing first with the continuous line source distribution, the relationship between the aperture distribution and its pattern function is normally expressed in the form: E(U) = g(x) exp (iUx) dx 2.1 -1 • -7- where g(x) is the distribution on the standard interval -1 to 1 and U is defined by: U = ITZ = 17-Dsine/X ... 2.2 where D is the actual length of the aperture, A is the wavelength and 0 is the angle of the far field point from the normal to the line source (measured in a plane containing both the point and the line source). Related to the above are the inverse Fourier transform: 00 g(x) - f E(U) exp (-iUx) dU ... 2.3 ••C0 and Parsevals theorem which is a mathematical statement of the conservation of energy: 00 1E(U)12 dU = Ig(xW dx ... 2.4 -00 F1 shows the familiar sin (U)/U pattern obtained when g(x) is a constant. It is plotted in decibels as a function of the parametei. Dsine/A. The latter provides a convenient normalised measure of beam shape since the antenna pattern plotted as a function of this parameter depends only upon the form of the aperture distribution. The quantity DsinO/A. (ie z) measured at a particular level below the peak of the mainbeam will be called the normalised beam width (nbw) at the specified level. The most common measurement levels are at -3 dB (ie half power) and at the first zero, the values for the uniform distribution at these points being 0.))13 and unity respectively. The actual angular beam associated with an nbw value of Z is given by: width 19b b -1 (Zip A/D) ... 2.5 eb = 2 sin 0 and if D/A is large, approximately by: 115 Z A/D degrees ... 2.6 b b The ratio of the normalised beam width of a given pattern to that of the uniform aperture pattern (measured at the same level) is an alternative and often used relative beam width measure that is known as the beam broadening_ factor (bbf). -8- V nuoung ;e xecianx onn JO uoT4ounj w ow uorwaxma TiTmouTs eta jo tappitmeg pormemoN JOKOd nim zi —cram: .::n:. • ....-. -7H----1-7-- PM aw p. ......::::: O MIMINI MINEMINMENSWIMINEIMM•IMMIMILE ■•M andhi 1 lliflt, :Ii_ .,.......milliiii....../t.:L.V - Im0411=133Z3e UTZ _ SAMIUMMI gia.............=1....... -nra ........mmr Tm -...ramii g:ITIMLffro;.' Men il.Clen=MWD: r,===r- a r: no: ...... :::asum meg ,1110Laim ' ffilar.--M= MMMM=MM:1---11 mmIll i • mir.rtal ,m6-1-- : E 9 S I I '.. 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