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On the Design of Rhombic Antennas

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On the Design of Rhombic Antennas ON THE DESIGN Of RHOMBIC ANTENNAS NEIL IGNATIUS HEENAN Library U. S. Naval Postgraduate School Monterey, California '? / - ON THE DESIGN OF RHOMBIC ANTENNAS by Neil Ignatius Heenan Lieutenant (L), Royal Canadian Navy Submitted in partial fulfillnnent of the requirements for the degree of MASTER OF SCIENCE in ENGINEERING ELECTRONICS United States Naval Postgraduate School Monterey, California 1953 \i.tV> v ; nx This work is accepted as fulfilling the thesis requirements for the degree of MASTER OF SCIENCE in ENGINEERING EJJECTRONICS from the United States Naval Postgraduate School PREFACE Early in 1952, J. G. Chaney, working at the United States Naval Postgraduate School, integrated the generalized circuit of the horizontal free- space rhombic antenna and obtained an express- ion for its radiation irnpedance in terms of tabulated functions. This expression and the results oi an earlier paper, by the same author, on the application of circuit concepts to field problems havf rnabhid this writer to present a design procedure for horizon- tal rhombic antennas which is based, to a great extent, on general- ized circuit theory and which gives results which agree well with the recent experimental work of Christiansen. My thanks are due to The Manager, RCA Review" and E. A. Laport for permission to copy Table II and Figure 4, and to McGraw-Hill Book Connpany for permission to copy Figures 2 and 3. I wish also to express my gratitude for the kind assistance of Professors J. G. Chaney and C. F. Klamm, Jr. , of the U. S. Naval Postgraduate School. Neil I Heenan San Carlos, California February, 1953 11 TABLE OF CONTENTS Page CERTIFICATE OF APPROVAL i PREFACE " TABLE OF CONTENTS "i LIST OF ILLUSTRATIONS iv TABLE OF SYMBOLS AND ABBREVIATIONS v CHAPTER 1 I. TMTnODTir.TION II. iHE DESIGN PROCEDURE 1 III. THEORETICAL CONSIDERATIONS 5 IV. DESinN AIDS 19 V. E.'tAMPLLS 31 VI. CONCLUSIONS 41 BIBLIOGRAPHY 42 111 LIST OF ILLUSTRATIONS Page 1 Figure 1. The Rhombic Antenna vt li^as 21 Figure Z. Height i? actor Nails and Maxima Electrical Heicht 22 Figure 3. Optimum i^aramtters for Horizontal Rhombic Antennas for Maximum Gain and Minimum Side Lobe Amplitudes the Half-Aper- Figure 4. Angle of Fire, A versus a , 23 ture Angle, for Maximum Radiation Intensity at L Figure 4a. Angle of Fire, L versus a.. the Half-Aper- 24 ture Angle. for a DirectionaF Maximum in Radiation Intensity at A Figure 5. Free Space Radiation Resistance versus 26 Half -Aperture Angle for a Side Length of Four Wavelengths Figure 6. Free Space Radiation Reactance versus 27 Half -Aperture Angle for a Side Length of Four Wavelengths Side 28 Figure 7. Free Space Radiation Resistance versus Length in Wavelengths for Constant Aperture Angle Side 29 Figure 8. Free Space Radiation Reactance versus Length in Wavelengths for Constant Aperture Angle 30 Figure 9. Contour of Constant J? and L versus R^ and a ! TABLE OF SYMBOLS AND ABBREVIATIONS antenna f The side length of the rhombic h The distance between the horizontal rhombic and its image t Ihe Angle-of-£ire of the transmitting rhombic X The wavelength 2a s A The aperture or acute angle of the rhombic n i - T^ wnt re n is > 8 of the rhombic r Radius of the wire forming the active element X"^ The wavelength in meters forming the active element d" The diamcLci ... .i.^hes of the wire of the rhombic Z The free-space radiation impedance r R The free-space radiation resistance X The free-space radiation reactance of the Rhombic Antenna Z = tx + i\ The terminating impedance o o "^ o 1 The natural logarithm log The comnnon logarithm oc The attenuation factor m nepers pers meter Z « R + iX. The Driving point impedance in in * in T The db terminal loss g The directivity gain for = 1 G , The power ^j^ G The power gain above ground P with reference to a horizontal half wave dipole G X The power gain P? in the same position fi-^ (1- cos A. cos a) K The radiation intensity in watts per steradian |k| The time peak value of radiation intensity The angle of azimuth The- antle from zenith piULicu.ai S ( The conripiement of 9 (a is a ) Sin e a) 3 = (1 -cos ^,).Co8 = Cos (^ + ' f i It.^ t. = (1 - cos .Cos = Sin e Cos(a - a) 4 ^ 1^ J^ 2) ^^ The time peak driving point current in amperes 1o C Euler'8 constant (0. 577216) = C (x) The cosine integral of X -/ —^j du sin u 1 - J S (x) The sine integral of x = / du I -^ o Z The internal impedance of the circuit in ohms per loop meter ^ (i.e. The mean square f The mean square current modulus "^ magnitude of the normalized < \jrr<nt distribution along the antenna) ^ t^-rmiaatino impedance Z^ f The attenuation factor at thf o The contour integral along the axis ol the wires forming tthe antenna the wires ff The contour integral along the inner periphery of forming the antenna in the circuit in terms of the I 1(P) The current at any point P o This current at an arbitrary reference point P^ . reference point usually being taken at the point of maximum current or at the driving point. P Any point along the axis of the wire VI }l P Any point along the inner pt npiicry of the wire conjugate. Used also to indicate a footnote. ( ) The complex M =y y + k = The steady-state form of the differential operator ^ ^ deltil with the subscript indicating the position at which the differentiations are to be perfornned r,. The vector distance from P^ to P. r,. The diitance between P, and P U = 4ir X 10" henries per niftt. r. 9-1 ptr meter ( = (36 w X 10 ) farads W « The input power ON THE DESIGN OF HORIZONTAL RHOMBIC ANTENNAS I. Introduction Sunnmary: Utilizing generalized circuit theory wherever possible, a design procedure is laid down for the single wire, uni-element horizon- tal rhombic antenna, whose side length is an integral number of quarter wavelengths at the lowest design frequency, suspended above a perfectly conducting ground plane. A brief statement of the design procedure is followed by a discussion of the steps outlined. Following this sonne graphical design aids are introduced. Finally some computed values are compared with the experimental results of Christiansen. In the design of Rhonnbic Antennas there are usually many variables. It is customary, therefore, to design, first of all, for assunned idealized conditions and then to consider the effects of the neglectea variaoies. In general, a theoretical design is followed by experimental xnodification. It is the purpose of this paper to lay down the theoretical design consider- ations for the uni-element horizontal rhombic antenna at a height h/2 above a perfectly conducting ground-plane. II. The Design Procedure For the antenna shown in figure 1 the design procedure can be laid down as follows: Figure 1 '7777777TTZ/^r//T7T7TT77TT7T77T77T77TT7777 ,.xit-i.;JS (a) Select /» ZA for the io.v<.st ircquency to be used. (b) Choose the desired vertical angle of radiation A at the highest frequency to be used. (c) If other engineering considerations permit, pick the height h/2 to yield nnaxinnum reinforcement in A . The height factor is a maximun^ when Sin( --— Sin ^) = 1, h = ^ cos ec t , where h is the antenna height above its innage. (d) The polarization of the radiated fields of the rhombic antenna are horixontal only in the = plane and in the 6 = — plane. It is customary, therefore, to orientate the rhombus such that its principal diagonal is in the direction in which the signal is to be beaxned. Of all possible designs at least three have been named by previ- ous authors. In one, the nriaximum of the principal lobe is ob- tained at an angle A This is called an " alignment design". In the "mavirmim field strength" design fH^ maximum field strength is obtained for a fixed i , in the desired direction A although the directional nr\aximum does not occur at the angle t . 3 In the "maxinnum output design", of Bruce, Beck and Lowry , the values of antenna height, side element length and aperture angle are prescribed to give maximum radiation intensity at an angle A tor a fixed value of the current. In the "alignment design" procedure the angle a is determined from the relationship n- 1.48 5Z Cos a = — n cos A GSitzBi ^ , ' For a maximum field strength" design with a given side length i , a is determined as follows: First, for the given A determine a to satisfy the condition -> cosA-cosa , t * 4 - ii - kl /I . V ZficolP ^ where /3 _{ 1-cos A cos a ) sin^ a cos A nir . , - * — (1 cos A cos a 1 . 4 With this value of a determine R as described in section — o (e). With this value of R recompute a to satisfy the relation: o — ^Q» A - cos a ^ 60 cos a - cos A ^ = j lizoxP J j 2 R sin a cos A, o cos A^2sin a The values of a and i for a nnaxinnum output design are given by the relations 6o ^ cos a - cos A i X ^ * • 6 . ^ Rq- r 2(l-co8' A cos a) o sin a cos^^^ A^ — By varying side elenrient length it is possible to obtain a maxi- mum output alignment design. In this case ^ ^o ^ cos A 1 - cos a / = -371 X ^°® ; Rq — 1 - cos A cos a ' ~ 1 - cos A cos a When either i or li , or both are restricted, so called 2 "conapromise'' designs are obtained.
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