1 I E> R.AR.Y OF THE UN IVER.SITY Of ILLINOIS 6^l.36 , 5 IZ£55te no. 2- 14 cop-3 Digitized by the Internet Archive in 2013 http://archive.org/details/couplingofantenn05king COUPLING OF ANTENNA ELEMENTS TO A CIRCULAR SURFACE WAVEGUIDE Contract No. AF33(616) -310 RDO No. IM12-110 SR-6f2 TECHNICAL REPORT NO. 5 by H. E. King R. H. DuHamel 30 June 1955 THE LIBRARY OF THE NOV 14 1955 UNIVERSITY CF ILLINOIS ANTENNA SECTION ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UN! VERS STY OF iLL:NO;S URBANA, ILLINOIS 1 1 ABSTRACT A new method of exciting an array of antenna elements by placing them in the field of a surface waveguide is described. Experimental re= suits are given for the specific case wherein the HEi i mode on a dielec- trie coated conducting cylinder is used to excite concentric types of antenna elements. The coupling of the antenna elements to the surface wave was measured by Deschamps' method. The variation of the equivalent admittance of the antenna elements as a function of the antenna dimen- sions is presented in graphical form. The results indicate that a wide variety of radiation patterns may be synthesized by appropriately con- trolling the dimensions and spacing of each element in the array. Ill TABLE OF CONTENTS Page 1. Introduction 1 2= The Coated Conductor Waveguide 3 3= Experimental Procedure and Results 7 4. Experimental Accuracy 16 5. Applications 22 6. Conclusions 25 Bibliography 26 Appendix I Measurement of the Admittance of Antenna Elements Through a Junction 27 Appendix II Alternative Method for the Measurement of the Admittance of Antenna Elements 34 IV ILLUSTRATIONS Figure Numbe- Page 1. Sketches of the Electric Field Lines for the HEn Mode on a Coated Conductor Waveguide Showing Longitudinal and Transverse Planes 4 2. Field Distribution Curves of the HEn Mode on a Coated Conductor Waveguide 5 3. Photograph of the Surface Waveguide,, Launching Horn, Transition from Rectangular to Coaxial Waveguide, Short Circuit Plate, and Antenna Elements with Polyfoam Supports 8 4. Equivalent Normalized Admittance of a Ring Coupled to the HEi i Mode on a Dielectric Coated Conductor as a Function of the Ring Diameter 10 5. Equivalent Normalized Admittance of a Ring Coupled to the HEii Mode on a Dielectric Coated Conductor as a Function of the Cross Sectional Dimensions 11 6. Equivalent Normalized Conductance and Cross Sectional Dimensions for a Ring Coupled to the HE-_i Mode on a Di- electric Coated Conductor as a Function of the Average Diameter 13 7 . Equivalent Normalized Admittance of a Disk Coupled to the HEn Mode on a Dielectric Coated Conductor as a Function of the Width of the Disk 14 8. Equivalent Normalized Admittance of a Disk Coupled to the HEn Mode on a Dielectric Coated Conductor as a Function of the Width of the Disk 15 9. Sketch of the Launching Horn 16 10. Two Views of the Rectangular to Coaxial Waveguide Transition 19 11. Measurement of the Axial Field at the Short Circuit Plate 21 12. Portion of the Longitudinal Plane Pattern of a Seven Element Array with the Elements at One Wavelength Spacing 23 13. Four Terminal Network with the Incident and Reflected Waves Defined 27 14. Short Circuit for the Dielectric Coated Conductor 29 ILLUSTRATIONS (Cont. ) Figu: e Number Page 15. Evaluation of the Scattering Matrix of the Rectangular- to - Surface Waveguide Transition by Graphical Construction 30 16 Illustration of the Procedure for Determining the Antenna Element Reflection Coefficient T x 32 17. Method of Determining the Admittance of Loosely Coupled Antenna Elements 33 18. Four Terminal Networks Representing the Junction and the Antenna Element 34 1. INTRODUCTION The purpose of this work is to investigate new methods of exciting an array of antenna elements. Conventional methods utilize coaxial lines or closed waveguides to excite the antenna elements,, The objective here is to excite the elements by placing them in the field of an open or surface waveguide system, A surface waveguide by itself does not radiate. If an obstacle, such as a piece of metal , is placed near the waveguide, it will radiate and will also excite a reflected wave. By choosing the proper waveguide dimensions and/or choosing obstacles with a certain symmetry, all propa- gating modes except one may be made negligible. The antenna system could then be represented as a single transmission line loaded at the appro- priate point by the equivalent obstacle four pole network. If the axial length of the obstacle is very small, its equivalent admittance will be a simple shunt or series element In the design of an antenna system of this type, the factors of importance are the equivalent admittance presented to the line by the antenna element, the radiation pattern of the element, and the mutual impedance between the antenna elements. For convenience and simplicity, an open waveguide system consisting of a dielectric coated circular conducting cylinder was chosen in order to demonstrate the technique. Concentric conducting rings placed around the cylinder were used as the antenna elements, A hybrid HEi* mode, for which all of the field components vary sinusoidally with the circumferential angle, was excited on the coated cylinder for the experiments. The major part of this paper is concerned with the experi- mental determination of the equivalent admittance that various types of 2 antenna elements present to the surface waveguide. Techniques similar to those described below could be used in the design of antenna arrays excited by other types of open or surface waveguides. Along with this experimental work, a theoretical investigation of the admittance charac- teristics and radiation patterns is in process. Before describing the experimental procedure and results, a brief review of the coated conductor waveguide will be given. 2. THE COATED CONDUCTOR WAVEGUIDE Surface waves are waves which are guided by and along an interface of two different media without losing energy by radiation- Considerable work has been done recently on the dielectric coated conducting cylinder surface waveguide- Goubau demonstrated that the attenuation constant of the lowest order symmetrical mode (the E o mode) on an enameled copper wire was very low compared to that for a coaxial cable of the same relative dimensions.. The Northwestern University Microwave Lab- oratory '*' 5 has conducted detailed theoretical and experimental investigations of the propagation of the principal and higher order modes on the dielectric coated cylinder- For conductors of two wave- lengths diameter or less with dielectric coatings of 2 wavelengths thick or less all except the two principal modes are evanescent- The two dominant modes are called the Ebo mode, a TM symmetric mode, and the HEti mode, a hybrid mode for which all of the field components vary as the sine or cosine of the circumferential angle- For conducting cylinder diameters of less than one wavelength and a thin dielectric coating, the guide wavelength is only slightly less than the free space wavelength, In order to give the reader an idea of how the antenna elements behave when placed in the field of a surface waveguide, the field components of a typical HEi r mode are sketched in Fig. 1- The field distribution curves for a conducting cylinder one wavelength in diameter, with a A/20 thick dielectric coating are shown in Fig- 2- The external fields decrease exponentially in the radial direction which is characv teristic of the surface waveguides- For the waveguide with the dimen- a) Longitudinal Plane b) Transveijse Plane Figure I Sketches of the Electric Field Lines for the HE,! Mode on Coated Conductor Waveguide (From Reference 3) 02 I -0.2 -I >»-0.4 c .2 -0.6 « c o 2 -0.8 -1.0 -1.2 -6 Figure 2 Field Distribution Curves of the HE 58 Mode on a Coated Conductor Waveguide Dimensions are almost identical to those actually used (From Reference 2 S page 44) sions as mentioned aboy^e, approximately 97 percent of the power is confined within a radius of 2A„ For the HE11 mode the radiating element may take the form of a concentric metal ring since this mode has a circumferential component of electric field that will excite currents on the ring The radial currents also exist but are negligible if the cross section of the ring is small A ring with a large radial cross section dimension will utilize both the radial and the circumferential components of the elec- tric field. The E o mode is not suitable for broadside operation with radiating elements, the axial extent of which is small, since the patterns of elements with only radial currents have a null in the broadside direc- tions Elements with axial lengths of a quarter wavelength or more could be used effectively for the E o mode., However, the design would be rather difficult since the coupling to the line could not be represented by a simple shunt admittance . 3. EXPERIMENTAL PROCEDURE AND RESULTS The measurement of the coupling of the antenna elements to the surface waveguide is not an easy task. It is necessary to construct a transition which will excite only the desired mode on the surface wave- guide. Then, to determine the equivalent admittance of the antenna element, field measurements must be made either on the surface wave- guide or in the waveguide which feeds the transition. After some experimentation, the apparatus illustrated in the photo- graph of Fig, 3 was chosen for the measurements. The surface waveguide consisted of a one inch diameter silver plated tube with a teflon dielectric (dielectric constant - 2) coating 0o055 inch thick. At the operating frequency of 9,460 mc the tube diameter is CL801A and the thickness of the dielectric is (L044A The desired hybrid BE tl mode was excited by a conical horn with a mouth diameter of approximately 4X, The horn in turn, was excited by a coaxial line with the TEi 4 mode. This type of feed produces a field at the mouth of the horn which is similar to the transverse field distribution of the surface waveguide, A transition from rectangular waveguide to coaxial waveguide produced the TEi! mode. A variable-position short circuit, consisting of the round flat metal sheet shown in the right side of the picture, was used to terminate the surface waveguide. The antenna elements, shown as concentric rings in the picture, were supported by polyfoam disks. The equivalent admittance of the antenna elements was determined from admittance measurements in the rectangular waveguide, This was made possible by calibrating the transition from rectangular waveguide to the surface waveguide by Deschamps" method Since the coupling -M CO l_ W CO O — c 3 a. 01 3 c CO cO -m e o co a> o a: *- e *- o o L. Q_ H- -C C -H O — — 2 -M — 1- E l_ CO C 3: c 0) 01 -m c c ■- <£ o -o c c 3 CO CO _i a> -M - CO 4) — -o a. 01 — a) 3 > o CO w CD O CO OO CD — 3 O CD > jC CO Cl 3: CO l_ ■ — en co O — -M X O CO -C O Q_ O OO CD 3 cn 9 of a single antenna element to the line was usually quite small, various numbers of identical elements were placed on the line at two or three wavelength intervals to obtain better accuracy. Two or three wave- length spacing was required in order to reduce the mutual impedance between the elements. The short circuit plate was placed an odd number of quarter wavelengths past the last element. The calibration of the junction and the determination of the equivalent antenna admittance is described in Appendix I„ The results of the admittance measurements are shown in Figs. 4 to 80 For each case, the normalized admittance presented is the average value obtained from several measurements with various numbers (from one to seven) of identical antenna elements spaced two or three wave- * lengths apart . Figure 4 is a plot of the admittance components for a ring with a cross section of 1/16 ' by 1/16 ' as a function of the inner diameter of the ring. Loose coupling of the radiating elements to the surface waveguide is indicated here In the design of long arrays this feature would be useful- When the ring is made to just fit over the dielectric coating the conductance is negligible: thus, the admittance has essentially only a susceptance component. Hence these rings could be used for tuning purposes An interesting and surprising feature of these close- fitting rings was disclosed when the axial length was varied from 1/16" to 1/4 ; without an appreciable change in the suscep- tance The variation of the admittance as a function of the cross section of the ring, with the average diameter held constant, was determined for several diameters with some of the results plotted in Fig, 5 The See Appendix I for a brief description of the ayeragmg process. 10 Figure 4 Equivalent Normalized Admittance of a Ring Coupled to the HE M Mode on a Dielectric Coated Conductor as a Function of the Ring Diameter (The cross section of the ring is a I/I6' r x 1/16" square ) 11 o 3 o .02 .01 Figure 5 ife 64*64 1 r Ave. Diam. Ring Equivalent Normal ized Admittance of a Ring Coupled to the HE ±1 Mode on a Dielectric Coated Conductor as a Function of the Cross Sectional Dimensions (The average diameter of the rings was held constant,, ) Cross-section Dimensions (Inches) 12 results indicate that the susceptance of a fixed diameter ring can be tuned out for a limited range of diameters by varying the cross sectional area The curves of Fig, 6 show a plot of the values of the conductance, G, and the cross sectional dimensions as a function of the average diameter of the rings for the condition when the admittance was a conductance;, i.e., when the ring was resonant. Note that a wide range of conductance can be obtained by choosing the correct diameter and cross sectional area. Next, the antenna elements were made with a large radial dimension ^a thin-flat disk^. The wafer shaped antenna element admittance measure- ments are plotted in Fig. 7, with the inner diameter of the wafer fixed at 1 7/8''. This was repeated for another set of disks with the inner diameter made to just fit over the dielectric, with the results plotted in Fig 8 Unlike the other previous results of Figs. 4 to 7, when- ever two or more disks were placed along the surface waveguide at two or three wavelength intervals, inconsistent ^admittance data resulted. Thus the results of Fig, 8 were derived from measurements on one disk and not by the use of the averaging process previously used This indicated that with these close fitting disks the mutual impedance was more pronounced and thus an array design for these elements would be more difficult. 13 Sketch of Ring i_ r rq Averoge Diom .Ub 1 Cross-section Dimensions^, S s ^ .05 \ 0^v v, .04 .03 I/I6X1/I6 2 c 3 E 1/32X1/32 v> s o o x o 1-1/2 1-3/4 2 2-1/4 Z-\k Average Diameter (inches) 1/64X1/64 Figure 6. Equivalent Normalized Conductance and Cross Sectional Dimensions for Ring Coupled to the HEn Mode on a Dielectric Coated Conductor as a Function of the Average Diametero (The susceptance component of the admit- tance of the ring was zerOo ) 14 Dielectric Coated Conductor 1/4 Width (Inches) Figure 7 Equivalent Normalized Admittance of a Disk Coupled to the HE,, Mode on a Dielectric Coated Conductor as a Function of the Width of the Disk (The inner diameter of the disk was I 7/8 inches ) 15 Oielectric Coated Conductor Disk 1/8 1/4 Width ( Inches) Figure 8 Equivalent Normalized Admittance of a Disk Coupled to the HE ir Mode on a Dielectric Coated Conductor as a Function of the Width of the Disk (The inner diameter of the disk was made to just fit over the dielectric coating ) 16 EXPERIMENTAL ACCURACY The major source of error was due to the presence of undesired modes or fields that existed in or around the waveguide, components. This may be better understood by considering the operation of the wave Flange Figure 9 Sketch of the Launching Horn guide system in more detail.. The conical horn illustrated in Fig. 9 serves as a transition between the TE ; - mode in the coaxial line and the HE mode on the surface waveguide. The TE- \ mode is excited from the rectangular guide by the transition shown in Fig. 10 The coaxial i dimensions were chosen so that all modes of order higher than the II were e v anescent Measurement, of the radial electric field in the CoaxiaJ line as a function of the circumferential angle demonstrated that the undesired TEM mode was at least 20 decibels down from the I ! mode Since th< hi I do an ideal transition,, a portion of the input I by rad at -on to the surrounding medium. From the image :le, G' , of F g 15, t he efficiency of the complete transition from 17 the rectangular waveguide to the surface waveguide was determined to be 70 percent Further measurements indicated that approximately 20 percent of the input power was radiated and about 10 percent was dissipated by heat losses In addition, the undesired Eq mode may be excited on the surface waveguide by the horn. It was determined from the measurement of the axial electric field ^.by means of a probe pro truding through a hole in the short circuit plate) near the surface waveguide as a function of the circumferential angle that the Eqq mode was at least 20 decibels down from the desired HE- ; mode. The operation of the plate (the diameter was eight inches) as a short circuit was quite satisfactory. On the other hand, it was large enough so that an appreciable portion of the power radiated by the exciter and the antenna elements was scattered by the plate, A part of this scattered power was returned to the waveguide system. In addition, an appreciable portion of the power radiated by the antenna elements was intercepted by the conical horn. Now, the effect of an obstacle on any mode can always be represented exactly by an equivalent transmission line circuit. However, because of the coexistence of the radiation coupling and the surface waveguide coupling between the horn, antenna elements, and short circuit plate, the coupling of the antenna elements to the KE-p mode alone could not be determined exactly by observations in the rectangular waveguide. Although measurements indicated that a. single transmission line representation was a good first approximation, it was not possible to determine the degree of approximation accurately because of the very slight difference between the surface waveguide wavelength and the free space wavelength. On the other hand, the presence of the rather 18 strong radiation fields near the surface waveguide made it unfeasible to determine the equivalent admittance of the antenna elements by means of standing wave measurements along the coated conductor, For the antenna elements which were loosely coupled to the line, the usual errors arose because of the difficulty of determining accu- rately slight changes in the standing wave pattern As mentioned previously, this error was reduced by measuring several identical elements with two or three wavelength spacing, except for the elements for which the mutual impedance was large. Even though the dielectric constant of polystyrene foam used was 1.037, the effect of the disks on the measurements of loosely coupled antenna elements cannot be neglected. The main effect of the polystyrene foam disk was a slight change in the guide wavelength of the wave while passing through the disk, This factor was accounted for in the data presented in Figs, 4 to 7 Another source of error was due to the assumption that the antenna elements presented a pure shunt admittance to the line. For elements with an appreciable axial extent, it would be necessary to represent the coupling by a general four pole network Since the largest elements reported here had an axial length of A/10 the error involved in the above assumption should be negligible An alternative method for determining the equivalent admittance of the antenna elements was performed by measuring the transmitted wave at the short circuit, plate with and without the antenna in position As demonstrated in Appendix II, the shunt admittance, Y a , is given by v. • — - L - 1 le)- 1 O (0 c ■o 0) > X o o «3 3 c -M o Of 2 O 2 i_ =3 •^1 002* where 20 Field at the short circuit plate without antennas A ■ Field at the short circuit plate with antennas The magnitude and phase of the field was measured by comparing the signal received by a probe protruding through the short circuit plate with a signal taken from the rectangular waveguide. A diagram of the setup is shown in Fig. Ho An advantage of this procedure is the simplicity of calculating the admittance. The accuracy of the method depends upon the precision of the attenuator and the phase shifter. As with Deschamps' method large errors may occur for loosely coupled antennas, A comparison of the results of the two methods of admittance measurements for a few radiating elements is shown in Table I, where Y a is the total normalized admittance presented to the surface wave guide . Number of Elements Ring Dimensions Y a Deschamps ' Method Y - A - 1 a 1 1 Z5 6° DIFFERENCE Conductance Suscept; 3 1 8 IDX f X lV 19 - j 58 43 + j 48 24 10 5 i7 b™ * f x tV" 34 + jl.23 54 + j 87 20 36 7 / ' * " i " 6 jl.9 96 + jl 82 36 08 Table I It is estimated that the maximum error in the measurements reported In Section 3 is less than 0.02 mhos or 25 percent of the measured ralue (the largest limit being used) Although the results shown in M>le I do not substantiate this estimate, it is believed that the major I '■" ' "' the differences was due to the error involved in measuring the r -i t i i, A 21 LU b ■Ka « 2 g to .■9 <5 i o r q- o < x ^Li ~^ SB o to o£ x CD -M <0 Q_ O i. o -M L. o ..c 00 CD -C -M -t-< T3 "cD U_ "to X pattern are a function of the configuration of the antenna elements, th bould make it possible to design a variety of antenna Thi Pad ;ition pattern of this antenna system might be controlled 23 Short Circuit Plate Figure 12 Portion of the Longitudinal Plane Pattern of a Seven Element Array , with the Elements at One Wavelength Spacingo (The rings had an inner diameter of I 7/8 ! with a I/I6 ! x l/i6 cross section ) 24 to give low side lobe, beam-tilt, or "cosecant squared" type patterns by the appropriate amount of coupling and spacing of each element An omnidirectional pattern in the transverse plane can be obtained by exciting two orthogonal HE^ modes in quadrature. The field components then vary as e^p with the circumferential angle, 6, Thus the current which is induced in the antenna elements will vary as e*J , i.e., the magnitude of the current is constant and the phase varies linearly with 9 This results in an omnidirectional transverse plane radiation pattern,. Looking into the future, many channels of development are open to further the progress reported in this paper. An efficient transition and launching device would be desirable . Depending upon the requirements, the dimensions of the dielectric-coated conductor could be changed and other types of circular surface waveguides such as the dielectric rod and dielectric tube could be used. One does not have to be limited to a dominant mode, since the transverse electric modes may have possibil- ities A knowledge of the field distribution and configuration of the various forms of surface waveguides will help the designer to select a surface waveguide. For example, a symmetrical transverse electric mode will yield tight coupling from the line to a ring-shaped antenna element The rffi^ 1 mode of a dielectric tube waveguide will yield tight er coupling than the corresponding dielectric coated conductor. Thus, the design requirement of an antenna system could well dictate the choice of surface waveguide and antenna elements 25 6. CONCLUSIONS A new method was introduced to excite antenna elements by placing them in the field of the hybrid HEj^ mode of a surface waveguide. The coupling of the concentric type of antenna elements to the line was measured by Deschamps' method. Curves were presented to show the variation of the equivalent admittance of the antenna elements as a function of the diameter, width, and cross sectional area. Since the phase and magnitude of the coupling vary with the antenna dimensions, a wide variety of antenna systems can be designed. By appropriately controlling the amount of coupling and spacing of each element in the array, a uniform, low side lobe, beam tilt, or "cosecant squared" type pattern might be obtained. An alternate and also simpler method of measuring the admittance of the antenna elements was presented, whereby the transmission coefficients were measured instead of the reflection coefficients. 26 BIBLIOGRAPHY 1. Goubau., George. "Single-Conductor Surface-Wave Transmission Lines", Proc I.R.E., Vol, 39, pp. 619-624. June, 1951. Beam R. E. Wave Propagation in Dielectric Tubes, First Quarterly Report, Army Signal Corps Contract No. DA36-039 sc-5397, Microwave Laboratory, Northwestern University, Evanston, Illinois. December 1, 1950 to February 28, 1951. 3 Ibid, Second Quarterly Report, March 1, 1951 to May 31, 1951. 4 Ibid, Third Quarterly Report, June 1, 1951 to July 31, 1951 5. Ibid, Fourth Quarterly Report, August 1, 1951 to October 31, 1951. 6. Storer, J. E. , Sheingold, L. S. and Stein, S. "A Simple Graphical Analysis of a Two-Port Waveguide Junction", Proc, I.R.E.., Vol, 41, pp 1004-1013, August, 1953. 7. Montgomery C G„ , Dicke, R. H. and Purcell, E. M. "Principles of Microwave Circuits", Radiat ion Laboratory Series, Vol* 8, pp. 146- 149, McGraw-Hill Book Company, Inc., New York. 1948. 27 APPENDIX I MEASUREMENT OF THE ADMITTANCE OF ANTENNA ELEMENTS THROUGH A JUNCTION A brief description of the graphical analysis based on the work of Deschamps° for obtaining the scattering matrix of the rectangular- to- surface waveguide transition from standing wave measurements will now be presented. In addition, a method will be outlined whereby the admittance of the antenna element can be obtained graphically from the measurement of the reflection coefficient on the rectangular waveguide section. Consider the four terminal structure of Fig. 13, which represents the junction from the rectangular to the surface waveguide. The termi- nals 1-1 are at some reference plane on the rectangular waveguide while Input Terminals 1 Four Terminal Structure z — 0, B^ b| -* -*-b i 2 2 Output Figure 13. Four Terminal Network with the Incident and Reflected Waves Defined terminals 2=2 are located at a reference plane on the dielectric-coated conductor, The a's and the b s s signify the incident and the reflected wave amplitudes at the two terminals. If the network under consider- ation is a linear, passive structure 7 then in matrix form b|| (1) or. expressed as the following equations Di = Oliaj + 1 2 <* 2 D2 " 021&1 "■ 022^2 28 (2a) (2b) The scattering matrix. i>i 1 01 2 ^2i O22 (3) completely characterizes the four terminal network. The coefficient Su is the reflection coefficient at the input terminals 1-1 when line 2 is matched; S 2 2 is the reflection coefficient at the output terminals 2-2 when line 1 is matched: S i2 expresses the ratio of the transmitted wave amplitude at terminals 2-2 to the incident wave amplitude at terminals 1-1, with a matched load on line 2, and vice versa for S21. Furthermore, if the terminal planes 11 and 2 2 are shifted through a lossless trans- mission line, the magnitude of the matrix elements remains unchanged- The impedance matrix Zn z i2 Z2 1 L'2 2 is related to the scattering matrix by iZ - I Z + I -1 (4) where I is the unit matrix and where the characteristic impedances of the two transmission lines are normalized. l)'-'.f hamps has developed a method of determining the elements of the scattering matrix by measuring the reflection coefficient at one terminal 29 pair with the other terminal pair terminated with various lengths (the lengths differ by equal fractions of a half wavelength) of a short cir- cuited line. Eight different positions (1/16 wavelength intervals) of the short circuit, designated by kl = 0, n/& s 2^/8, etc., were obtained by using precise short lengths of teflon that were slipped over the con- ductor,, as illustrated in Fig. 14. The eight measured reflection co™ efficients formed an image circle,, G' 8 with its center at C, as shown Dielectric kl=0 I HUM)} >>>>> > J /JJ.IWTTTf A Conductor >>))),}> nasaaassBBX kl 7r y A i Circular Plote as Reflector Figure 14. Short Circuit for the Dielectric- Coated Conductor in Fig. 15. Following Deschamps 1 procedure, the point Sn was determined by some simple graphical analysis. Once C 1 and Si i were determined, enough information was available to calibrate the junction as shown by the tabulated formulas in Fig. 15. For the admittance measurements, the location of the antenna ele= ments is at the short circuit position kl - 0, i.e., the reference plane on the surface waveguide. The short circuit plate was placed at some odd multiple of a quarter-wavelength from kl 0„ as shown in Fig. 11. Notations used in Figs- 15 and 16 are identical to those of reference 6 30 Open Ci rcuit on Surface Waveguide Terminals Short Circuit on Surface Waveguide Terminals Arg S I (OP 0S W ) I 2 C 811I OS.., 035 S i3 .E' Arg S , W (OP C P, ) 53 5 Arg 5 2a J (S C C P, ) 15 8 l |8,.| s~7F " r 825 0678 Figure 15 Evaluation of the Scattering Matrix of the Rectangular-to- Surface Waveguide Transition by Graphical Construction 31 Once a junction is calibrated it is always possible to compute the output reflection coefficient for a measured input reflection coeffi- cient r ', but such computations are generally laborious. However, by simply adding certain constructions to those used in the calibration of the junctions, as illustrated in Fig. 16, it is possible to determine an unknown reflection coefficient;, T , through a junction readily. Refer- ring to Fig. 16 r ' , is the measured reflection coefficient, and T x is the unknown reflection coefficient of the antenna element on the surface waveguide. Since the reflection coefficient T was plotted on a combi- nation Smith- reflection coefficient chart, Z and T x were identical points. Y x . the admittance, was readily found on the Smith Chart as the inverse of Z . The following example demonstrates how the admittance of the loosely coupled antenna elements was found from the total admittance when several (three, five and seven) identical elements were placed at two wavelength intervals on the surface waveguide. The reflection coefficients measured on the slotted section of waveguide are V 3 ' , r e ' , and T 7 ° , for three, five, and seven elements, respectively. The corresponding total norma- lized admittances are represented by Y 3 , Y 6 , and Y?. The equivalent admittance for each element was the total admittance divided by the number of elements used, as demonstrated in Fig. 17. The close corre- spondence gives an indication of the accuracy of the measurements as well as the fact that the mutual coupling between antenna elements is almost negligible. The method as outlined in Fig. 17 was the procedure used to obtain all the data of Figs. 4 to 7. 32 SlJ V = 823 Arg T - I (U'Y'.S^r ') - Arg S 22 25. 8 C Figure 16 Illustration of the Procedure for Determining the Antenna Element Reflection Coefficient T x 33 Y/ element r ^3 Y/element- 033 + j ,077 03 7 - j 07 01! j 07 Figure 17. Method of Determining the Admittance of Loosely Coupled Antenna Elements 34 APPENDIX II ALTERNATIVE METHOD FOR THE MEASUREMENT OF THE ADMITTANCE OF ANTENNA ELEMENTS A brief description of an alternative method to determine the ad- mittance of antenna elements by u measuring the transmitted wave" at the short circuit plate with and without the antenna in place will now be presented. Consider the antenna element as a four terminal network cascaded with the junction, as shown in Fig. 18„ The terminal pairs 1=1 and 2-2 Short Circuit- Tronsifton Rectangular to Surface Waveguide -o- Antenno Element with only Shunt Admittance Figure 18 Four Terminal Networks Representing the Junction and the Antenna Element (The measurements were made at the probe E ) have the same significance as described in Appendix I„ The terminals 3 3 and 4 4 may be considered identical since the assumption that the antenna element consisted only of a shunt admittance was made, In the actual case then, the three pairs of terminals, 2-2, 3 3, and 4-4, are indistinguishable and are located at the reference position k£ ■ 0. As before, a short circuit was placed at terminals 5 5 which are an odd number of quarter wavelengths from the reference position kZ z 0. Note tMt a : b 2 and with the antenna element removed b 2 ■ a 3 ■ b 4 . A necessary requirement is to hold the incident wave a 3 constant f i Ik- IojhJ on the surface waveguide As shown, in Fig.. 10, ' Ottld I" li'-ld constant with and without the antenna in place, then Mthod *<.uM be Independent of the calibration of the junction This would be trnt if Bgg wr r r frjual tO J.i 35 sufficient attenuation was inserted in the rectangular guide to yield a constant incident wave. With the antenna element in place, the axial electric field (meas- ured with the probe shown in Fig. 11) at the short circuit plate is pro- portional to the incident wave, i.e., E ~b 4 (5) and without the antenna in place, E'~a 3 ' . (6) The primed symbols indicate the condition without the antenna in place. A relation for Y a will be found in terms of the ratio t 04 D4 Because the reflection coefficient T , of the antenna is presented to the transition, one obtains a 2 - r b 2 . Then substituting into Eq. 2b, there results 1 ~ S22 T a With the reflection coefficient T s of the short circuit presented to the antenna terminals, a 4 = Tgb^, and similarly With the antenna removed, the output of the transition sees the reflec- tion coefficient, F s , of the short circuit, thus, b2 < , _JL 2 ^ — . do) 1 S22 ig Equation 7 can now be written as 36 O2 l^: S2 1^1 A =« 1 02 2 -l c S22 r c S^a- '4-3 S21 a l 1 " O44. i „ 1 U44 1 o 1 ■*- O22 J- -: (l - s 44 r s ) (i - s 22 r a ) (ii) \ 1 02 2 1 s / ^4.3 Since the antenna is spaced (2n + l)\ g /4 from the short circuit, T s = -i-1. From the calibration of the junction in Appendix I, S22 = . 068eJ 45,8 . The reflection coefficient V-, in terms of the shunt antenna impedance, Z a> 1S r a = Za - 1/Z a + 1" u Then from Eq. 4 2Z, and '43 O44. 2Z a + 1 -1 2Z a + 1 (12a) (12b) Substituting the known information into Eq. 11, and using Y a = 1/Z a , a - 1 + y. 1 + s gg 1 " " o 2 2 (13) Thus, the unknown admittance of the antenna element is simply 1 - S g 1 + s 2 2 / (A - 1) (14) A - 1 l.leJ 5 - 6 °