300 MHz ARRAY USING PHYSICALLY SMALL SLOT ANTENNA ELEMENTS M.A.H. Ibrahim and J.A. M. Lyon This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with the prior approval of AFAL (AVWE), WrightPatterson Air Force Base, Ohio 45433

FOREWORD This report, 1770-2-T, was prepared by the Radiation Laboratory of The University of Michigan, Department of Electrical Engineering, 201 Catherine Street, Ann Arbor, Michigan 48108, under the direction of Professor Ralph E. Hiatt and Professor John A. M. Lyon on Air Force Contract F33615-68-C-1381, Task 627801 of Project 6278, "Study and Investigation of UHF-VHF Antennas." The work was administered under the direction of the Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433. The Task Engineer was Mr. Olin E. Horton and the Project Engineer, Mr. Edwin M. Turner, AVWE. This report was submitted by the authors in October 1969. The prefatory material to this report, Sections 01, 02, and 03, covers the general aspects of the utilization of miniaturized slot elements in antenna arrays. The remainder of the report, Chapters I through V and the Appendices, presents detailed background information and an analysis of the coupling of loaded slot elements. Chapters I through V and the Appendices have been written entirely by Dr. M.A.H. Ibrahim and constitute his recent doctoral dissertation. ACKNOWLEDGEMENT The following individuals contributed eithez r analytical or experimental effort for some parts of this report: Dr. C.J. Digenis, Y-P Liu, D.G. Berry and B. Whitney. This report has been reviewed and is approved. / OS<EOI A. DOMBROWSKI t Colonel, USAF / hief, Electronic Warfare Division ii

1770-2-T ABSTRACT The coupling between two loaded rectangular waveguides opening in an infinite conducting ground plane has been investigated. The slots are equal in size and placed so as to have the broadsides parallel. The analysis is made in two steps: i) the coupling between the apertures is derived; ii) the effect of the backing cavities on the coupling is determined. The reaction concept, a form of the "variational technique", was used in the analysis. This method helped to simplify the mathematical work and provided very reasonable results. Substantial experimental work on the behavior of coupling versus frequency and coupling versus slot separation was performed. In order to simplify the experimental investigation the standard empty X-band waveguide (0. 9"xO. 4") was filled with dielectric loading of Emerson and Cuming Stycast Hi-K dielectric material. Materials having dielectric constants of 5, 7, 11 and 15 were used. Most of the measurements for the loaded case were in the S-band range of frequency (2. 6 - 3. 95 GHz). The choices of the S-band frequency range and the dielectric constant were made to permit use of the anechoic chamber in the first instance with reasonable accuracy and small waveguide in the second instance. An extensive comparison of coupling for the case of standard waveguides with and without loading has been made and explanations are given for the observed differences. An iteration procedure has been used to determine the effect of the electromagnetic interaction between the slots. Detailed computer programs were developed for the analytical expressions. Numerical results for many important cases were then obtained. IveiLhods and results can readily be extended to other frequencies and other slot sizes. iii

TABLE OF CONTENTS PREFATORY MATERIAL 1 Section 01: Introduction 1 Section 02; Arrays and Interaction 2 Section 03: Conclusions 23 I INTRODUCTION 25 1.1 Approach to the Problem 26 II THEORETICAL ANALYSIS 28 2. 1 Assumptions 28 2. 2 Formulation of the Fields in the Different Regions 28 2. 3 Evaluation of U 36 2. 4 Iteration Procedure 50 2. 5 Consideration of the Short-circuited Waveguide Backing the Slots 53 III NUMERICAL ANALYSIS 66 3.1 General 66 3. 2 Calculation of Reflection Coefficients at the Aperture 66 3. 3 Calculations of the Aperture-to-Aperture Coupling Between Slots versus Frequency 70 3. 4 Coupling Aperture to Aperture versus Spacing 86 3. 5 Evaluation of the Cavity Effect 96 3. 6 Evaluation of Coupling 120 IV EXPERIMENTAL INVESTIGATION 121 4. 1 General Discussion 121 4. 2 Coupling Measurements 126 4. 3 The Far-field Patterns 148 V SUMMARY AND CONCLUSIONS 152 5. 1 Areas for Future Investigation 155 REFERENCES 157 APPENDIX A: The Reduction of Quadruple Integrals to Double Integrals. 159 APPENDIX B: Examples of Computer Output Data on Coupling 162 iv

LIST OF TABLES 3.1 Relation between FN and f for ur=l, er=15, a=0.02286m, b=0.01016m. 78 3.2 Relation between FN and f for,r=1, cr=11, a=0.02286m, b=0.01016m. 78 3. 3 Relation between FN and f for,r=l, er=7, a=0.02286m, b=0.01016m. 78 3.4 Relation between FN and f for /r= 1, er=l, a=0.07136m, b=0.034036m (S-band). 79 3. 5a Relation between FN and f for Ur=1, er=l, a=0.02286m, b=0.0106m (X-band). 79 3. 5b Relation between FN and f for,r=l, er=l, a=0.0157988, b=0.0078994 (Ku-band) 79 3. 6 Cutoff Frequency vs Dielectric Constant of the Loading Material; a=0.02286m, b=0.01016m. 80 v

LIST OF ILLUSTRATIONS 0. 2-1 Experimental Power Pattern (linear) of Ferrite Slot in Ground Plane, H-plane Cut. 7 0. 2-2 Calculated Power Pattern (linear) for 3-element Ferrite Slot Array (H-plane). 7 0. 2-3 Experimental Power Pattern (linear) for 3-element Ferrite Slot Array (H-plane). 7 0. 2-4 Broadside to Broadside Coupling of Closely Spaced Rectangular Slots Showing a l/r2 Trend. 9 0. 2-5 Far-field Coupling for Representative Angles. 10 0. 2-6 End to End Coupling of Closely Spaced Rectangular Slots Showing a l/r4 Trend. 11 0. 2-7 Comparison between Experimental and Theoretical Results. 12 0. 2-8 Coupling of Air-filled S-band Waveguide Slots in Common Ground Plane, 0. 5X spacing, Broadside to Broadside Arrangement. 13 0. 2-9 Coupling of Ferrite Loaded Slots in Conducting Plane, 0. 5X Spacing, Broadside to Broadside Arrangement. 15 0. 2-10 Coupling of Ferrite Loaded Slots in Conducting Plane, 0. 5X Spacing, End to End Arrangement. 15 0. 2-11 Interaction of Certain Chosen Groups of Slots Corresponding to Three Radii; A, B and C. Polarization Aligned Left to Right. 18 0. 2-12 Returned Power Level to Feed Generator for Many Element Array as Function of Number of Slots Used in Computing Interaction. 21 2-1 Arrangement and Slot Configuration. 29 2-2 Method of Images. 32 2-3 Source and Scatterer. 36 2-4 Coordinate Transformation. 47 2-5 Coordinate Transformation for Reduction of Quadruple Integral to Double Integral. 48 2-6 Multiple Reflection between Transmitter and Receiver to Represent Interaction between them. 52 2-7 Short-circuited Cavity. 55 2-8 Standing Electric and Magnetic Fields in Cavity. 56 2-9 Input Impedance to the Cavity at z=0. 57 2-10 Configuration of Waveguide when it is Closed at Both Ends. 58 2-11 Equivalent Circuit for Cavity Slot Antenna. 59 2-12 Resonance and Half-power Bandwidth. 64 3-1 Aperture Normalized Conductance and Susceptance, Er=5. 71 3-2 Aperture Normalized Conductance and Susceptance, Er=6. 72 3-3 Aperture Normalized Conductance and Susceptance, Er=10. 73 3-4 Aperture Normalized Conductance and Susceptance, er=ll. 74 3-5 Aperture Normalized Conductance and Susceptance, r=15. 75 3-6 Variation of Cutoff Frequency vs Permittivity of Loaded X-band Waveguide.. 81 VI

3-7 Standard S-band Coupling vs Normalized Frequency. 84 3-8 Standard X-band Coupling vs Normalized Frequency. 85 3-9 Standard Ku-band Coupling vs Frequency. 87 3-lOa Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=2. 88 3-10b Aperture to Aperture E-Plane Coupling of Loaded Slots, er=3. 89 3-10c Aperture to Aperture E-Plane Coupling of Loaded Slots, r=4. 90 3-lOd Aperture to Aperture E-Plane Coupling of Loaded Slots, r=5. 91 3-lOe Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=7. 92 3-lOf Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=ll. 93 3-1g Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=15.94 3-10h Aperture to Aperture E-Plane Coupling of Loaded Slots, r=20. 95 3-11 Broadside Coupling between Empty Slots (Separation Center to Center). 97 3-12 Aperture to Aperture Coupling vs Distance in Wavelength. 98 3-13a Resonance of the Cavity (Digital Plot), d=0.031, er=7. 105 3-13b Resonance of the Cavity (DigitalPlot), d=0.031, r=ll1. 106 3-13c Resonance of the Cavity (DigitalPlot),d=0.031, r =15. 107 3-14a Resonance of the Cavity (Digital Plot), d=0.1015, er=7. 108 3-14b Resonance of the Cavity (Digital Plot), d=0.1015, r=11. 109 3-14c Resonance of the Cavity (Digital Plot), d=0.1015, er=15. 110 3-15a An Expanded Digital Plot for the Resonance Peaks, n=l. 111 3-15b An Expanded Digital Plot for the Resonance Peaks, n=2. 112 3-16a An Expanded Digital Plot for the Resonance Peaks, n=3. 113 3-16b An Expanded Digital Plot for the Resonance Peaks, n=4. 114 3-16c An Expanded Digital Plot for the Resonance Peaks, n=5. 115 3-16d An Expanded Digital Plot for the Resonance Peaks, n=6. 116 3-16e An Expanded Digital Plot for the Resonance Peaks, n=7. 117 3-16f An Expanded Digital Plot for the Resonance Peaks, n=8. 118 3-16g An Expanded Digital Plot for the Resonance Peaks, n=9. 119 4-1 Anechoic Chamber Ground Plane. 122 4-2a Test Antenna Mount. 123 4-2b Arrangements for Loaded X-band Waveguides. 124 4-2c Arrangements for Standard S-band Waveguides. 125 4-3a Resonance Cavity and Dielectric Filling plus Coaxial to Waveguide Feed. 127 4-3b The 0.031m and the 0.1015m Dielectric Loaded Cavities. 128 4-3c Comparison between Loaded and Unloaded Cavity. 129 vii

4-4 Standard S-band Waveguide Coupling vs Frequency 131 4-5 E-plane Coupling vs Frequency for Two Slots Spaced 11. 4 cm 132 4-6 Coupling vs Frequency for Loaded X-band Waveguides, r= 11, d=0. 031 133 4-7 Coupling vs Frequency for Loaded X-band Waveguides, Cr 15, d=0. 1015 134 4-8 Gain vs Frequency of Dielectric Loaded Cavity 135 4-9 Anechoic Chamber Set-up for Far-field Pattern Measurements 136 4-10 Coupling vs Frequency for Loaded Cavity 138 4-11 Gain vs Frequency for Loaded Cavity 139 4-12 Coupling vs Frequency for Loaded Waveguides 140 4-13 Gain vs Frequency for Loaded Cavity 141 4-14 Experimental Set-up for Coupling with Point by Point Tuning 142 4-15 Coupling vs Frequency with Tuning both Transmitter and Receiver at Each point 143 4-16a Coupling vs Frequency Point by Point, er=15 144 4-16b Coupling vs Frequency Point by Point, r =ll 145 4-16c Coupling vs Frequency Point by Point, er=7 146 4-17 Coupling vs Frequency Point by Point, xl=0. 03, Er=7 147 4-18a E-plane Radiation Pattern of a Loaded X-band Waveguide 149 4-18b H-plane Radiation Pattern of a Loaded X-band Waveguide 150 5-1 Explanation of Non-Uniformity of Separation of the Resonance Peaks 156 viii

LIST OF SYMBOLS a Slot dimension in H-plane b Slot dimension in E-plane d Cavity length dI Aperture to feeding probe dimension in the cavity d2 Probe to short side dimension of the cavity E Electric field intensity f Frequency F Electric vector potential FX, Fy F, Cartesian coordinates of F FN Normalized frequency (FN = ka / 7r) J Surface current in amp/meter H Magnetic field intensity k Wave number ko Free space wave number I Cavity length M Magnetic moment n An integer n Unit vector perpendicular to the aperture U Amplitude of the received illumination v Velocity of waves in material medium x Cartesian coordinate XI Center to center separation of slots A x Unit vector in the x-direction y Cartesian coordinate Y Admittance A Unit vector in the y-direction z Cartesian coordinate Z Impedance z Unit vector in z-direction ix

ao Attenuation constant P Phase constant y Propagation constant, or a variable of integration e Dielectric permittivity Co Free-space permittivity er Relative dielectric constant or permittivity rlT ]Second cartesian coordinate (source point) First cartesian coordinate (source point) X Wavelength Xo Free-space wavelength Xc Cutoff wavelength Xg Waveguide wavelength p Permeability PQo Free-space permeability wur Relative permeability.T Hertzian electric vector potential T"' Hertzian magnetic vector potential p First cylindrical coordinate w Angular frequency x

1770-2-T PREFATORY MATERIAL Section 0.1: Introduction This report covers the work to date on a task devoted to the utilization of loaded slot antenna elements in arrays. The objective of the task has been to develop the advantages and criteria for the use of physically small slot antenna elements in a 300 MHz array. It was recognized early that one of the chief advantages of such an array would be the possibility of reduced interaction between elements. A large part of this report is devoted to an analysis of loaded slot elements and an evaluation of the coupling between such elements. The reduced level of coupling can easily be ascertained from these analytical methods for specificinstances through the use of an appropriate computer program. The detailed studies, including the essentials for programming on a computer, are found in later chapters. This report includes the study of simple three-element linear arrays. The data obtained clearly show that with miniaturized slots the coupling involved from one element to another is sufficiently small so that the interaction can frequently be considered negligible. This fact has been deduced by comparing the array patterns obtained analytically based on assumed element illumination with experimentally determined patterns. It has been found that the experimentally derived patterns are extremely close to the theoretically calculated patterns. This means that the illumination of each element has been very close to the assumed illumination neglecting interaction and as used in the calculations. No modifications in the feed network were made in order to compensate for the driving point impedance level of each element. For instance in the case of the broadside three-element array, the feed network was arranged to give three in-phase values of illumination of equal magnitude. Equal lengths of lines were used so that the phasing and feed current magnitude were all the same before each of the feeds was connected to a loaded slot element. After the feeds were connected to the individual slots and the array pattern tests made, it was found that the patterns so derived were very close to those 1

predicted by calculation. The spacing of the elements was very nearly the usual half wavelength for a broadside beam. Section 0. 2 of this chapter describes the nature of the interaction problem among various elements of an array. This section also gives the experimental results on simple three-element linear arrays. Section 0. 3 indicates the conclusions obtained through the use of the information contained in this report. The remaining chapters contain the detailed analytical work on the interaction of elements. Chapters I through V and the Appendices comprise in their entirety the doctoral dissertation of M. A. H. Ibrahim of which he is the sole author. Section 0. 2: Arrays and Interaction 0. 2a: General Discussion The interaction of elements in an array of elements can be expressed in at least two ways. One is by means of the mesh equations for coupled antennas. These mesh equations contain impedance parameters which theoretically can be evaluated by means of appropriate line integrals. The most important impedance for each element is the driving point of impedance. The driving point of impedance itself is a function of the self impedance as well as each of the mutual impedances which come into being through the interaction of each element on every other element. In the analytical work which has been done and is contained in Chapters II and In, the analysis of the interaction of one loaded slot upon another has been on the basis of a consideration of the electromagnetic field boundary value problem Results have frequently been given in terms of the power level in dB received at one antenna referred to the power level input at the other antenna. The coupling then gives the power level of the receiving antenna as so many dB below the power level of the transmitting antenna. The dB coupling level data can be directly interpreted in terms of mutual impedance. However, thiisis not always necessary. The main objective of the analytical work has been to show that 2

miniaturized slots are accompanied by much lower interaction than air-filled slots at the same frequency and of standard size. On this basis, the work in Chapters I through V is back-up or supporting information for the proposition that miniaturized slots used in phased array antennas are accompanied by the very considerable benefit of decreased interaction. In fact, the interaction level is now sufficiently low to allow the frequently used compensation schemes for the driving network to be eliminated or greatly simplified. In the consideration of phased arrays the problem of driving point impedance becomes strikingly evident especially in the case of using the array for transmitting. In general, power from each element may be coupled back into every other element. If there is a relatively high level of interaction of one element upon another, then it is necessary to have appropriate matching techniques at every frequency and for every beam position. Obviously, it is extremely difficult to obtain a good impedance match over a wide bandwidth. It is even more difficult when matching must be good for wide ranges of scan and for various scanning modes. A convenient measure of the need for matching is obtained by considering an array of elements with respect to the total returned power from all the elements normalized to the total power fed to the array. This return of power to all elements will be a function of frequency and position of scan for the array beam. The advantages of miniaturized slots would then be associated with a major reduction of the returned power summed over all elements over a useful range of frequency and over the desired range of scan. There is some restriction on the bandwidth associated with a need to keep the interaction low. This is brought out in the analysis given in the later chapters. A possible bandwidth goal for phased arrays might be 15 percent. It is less easy to generalize on a desirable scan angle. The requirements for scan are very closely associated with the system requirements of the antenna. Certainly for many anticipated uses it is desired that the scanning angle from the normal to the array approach 90~. The indications given in this report are that interaction may be so considerably reduced that the influence of scanning angle on the interaction, although present, is relatively 3

insignificant since the overall level of interaction is always small. This fact will be explored in some detail in succeeding paragraphs. The use of loading material in a slot in order to reduce the size of the slot for a given operating frequency points to the possibility of an additional advantage not mentioned heretofore. There are available high quality dielectric materials, some of which are of ceramic nature. Such materials can be utilized to improve the hardening aspect of the antenna array, where hardening is used to denote capability of withstanding the effects of nuclear detonations. On the other hand, there are also useful materials for loading such as the ferrites. Ferrite material also may be useful for hardening purposes. However, there is a temperature limitation on ferrite which should be taken into full consideration. 0. 2b; Reduction of Coupling with Dielectric Loading For miniaturized slots the nature of the reduction in coupling from one element to another is due to the reduced area of aperture of each element and to the impedance mismatch at the aperture associated with the loading material. The reduced aperture means that a receiving element, or more properly an element receiving energy from another element, is capturing less energy. This advantage would be true for either linear polarization of each element or for circular polarization for each element. However, in this discussion, major attention is being placed upon linearly polarized elements. Each element investigated in this study was a dielectrically loaded element. The size of the element was so reduced that a slot formed by the open end of an X-band waveguide when loaded with dielectric material within the waveguide constituted a dipole slot suitable for S-band operation. This means that slots 2 13/16 " x 1 11/32" formed by the use of WR-284 S-band waveguide, will be supplanted by loaded slots 29/32" x 3/8" formed by WR-90 standard X-band waveguide filled with Emerson and Cuming stycast high-K solid dielectric. Various values of relative dielectric permeability were used; 5, 7, 11 and 15. 4

The relative sizes of these slots are illustrated in Chapter IV (Fig. 4-2a) The miniature slots as just described and as used in the experimental arrangement utilized a short length of waveguide behind each slot. Each slot was, of course, mounted in a common large conducting plane. In actual array use, it is very likely that the many elements would also be mounted in a common conducting plane. The use of the common conducting plane provides a 3 dB increase in the directivity of each slot element. Furthermore, the plane defines the physical field situation much more precisely than if the slot elements are merely an arrangement of cut off waveguides without a common boundary metal plane surface. Without such a plane surface, considerable uncertainty is introduced into the effective radiative structure of each element. Currents will tend to run back down the outside of the waveguide walls. For this reason it is believed that the use of the common ground plane has been justified in the experimental effort which is to be described. It was convenient in undertaking the experimental work on the interaction of elements to have the experiments performed at S-band. The analysis of the interaction between two slots based upon an iterative procedure as described in Chapter II is applicable to slots at any frequency and with any size as well as any type of loading material. 0. 2c: Experimental Ferrite Loaded Slot Arrays The experimental work done on simple three-element linear arrays utilized slots having ferrite loading and the size of each slot element was different from that described in the comparison of loaded and air-filled slots. For these experiments on the three-element arrays the slot was 5"x2". The depth of the cavity backing of each slot was 1 1/2". The center frequency of each slot was 350 MHz. The loading of filling material of these slots was ferrite EAF-2; the relative permeability of this ferrite was 6. 6, the relative permittivity, 12. 6 and the magnetic qualify factor, 30. As a preliminary in the experimental work it was appropriate to take the radiation pattern of a typical ferrite loaded slot. For this purpose Slot No. 108 5

was mounted in a 5'xl-1/2' aluminum ground plane A radiation pattern taken at 345 MHz is shown in Fig. 0. 2-1. Use of standard procedures for the calculation of a broadside array of three slots similar to No. 108 and fed equally both in magnitude and phase resulted in the pattern shown in Fig. 0. 2-2. An array of three slots was then fabricated with the slots having the magnetic fields collinear (slots arranged end to end in a line). The experimentally determined radiation pattern is shown in Fig. 0. 2-3. Since the patterns of these three figures are so similar it was concluded that there was comparatively little interaction between slots. This means that the illumination of the slots was not appreciably modified by mutual coupling. The feed network to the three slots was made with three identical branches, one being connected to each slot. Further work on the same three-element array with uniform progression of phasing of the elements also showed agreement between calculated and experimental patterns. Thus it appeared that for ferrite loaded slots the interaction of coupling of slots was relatively low. This indicates that the driving point impedance of each slot in an array is relatively independent of scan angle. 0. 2d: Observed Coupling Effects of Dielectric Loading Utilizing the information shown in Chapter III, it is possible to make some simple comparisons which are helpful in evaluating the return power. For example, in the consideration of a phased array of elements assume a requirement of 15 percent bandwidth. If two slot elements are made using the usual air-filled S-band waveguide, it can be found from Fig. 3-11 (Ch. III) that the coupling for a spacing of 0. 5X will be approximately -15 dB. On the other hand, the same frequency band can be used with X-band waveguide if it is filled with appropriate loading material such as with a permittivity of 7. 0. In Fig. 4-17 (Ch. IV), it can be seen that X-band waveguide so loaded as to be used for S-band frequencies can provide for a 15 percent bandwidth with a coupling of the order of -30 dB. It is observed that the use of dielectric loading material has made the waveguide feed components somewhat frequency sensitive whereas the airfilled waveguide slots did not exhibit this degree of sensitivity. However, 6

350 (o 10o. -10o 1o 9 350 FIG. 0. 2-1: EXPERITIENTAL POWER PATTERN (LINEAR) OF FERRITE SLOT IN GROUND PLANE, H-PLANE CUT. 350' ~ lo' 350: 10~o 3-00 04 0,,100 I20 3' o 3 0 10 0.350-0.. FIG 0 2-2: CALCULATED POWER PATTERN (LINEAR) FI 0. 2-3: EXPERIMENTAL POWER PATFOR 3-ELEMENT FERRITE SLOT ARRAY (H-PLANE) TERN (LINEAR) FOR 3-ELEMENT FERRITE 7 SLOT ARRAY (H-PLANE). 7 SLOT ARRAY (H —PLANE).

increased isolation can be obtained using the smaller slots and the loading material over a useful frequency bandwidth. In order to make this comparison still more apparent, attention is called to Fig. 4-4 (Ch. IV) which shows the experimentally determined coupling for S-band air-filled slots. From this curve it is observed that for a frequency of 3. 0 GHz, the coupling is approximately -17 dB. It is interesting to note that the coupling between the air-loaded slots can also be determined from previously published reports (Lyon et al, 1963; 1966). A number of graphs from these earlier reports are introduced here as Figures 0. 2-4, -5, -6 and -7. Figure 0. 2-4 shows that for a spacing of 0. 5X, two broadside slots would have a coupling level of -14 dB. This figure was based on X-band but is valid at other frequencies, if the slots scale exactly with frequency. Figure 0.2-5, when applied to the same physical situation, would yield about the same result. For this case, 0=0~. Read along the vertical axis for a separation of 2X, the value of -26 dB. Then for a separation of 1X, add 6 dB. Then for a separation of 0. 5 X add still another 6 dB. The total is: 026 dB + 6 dB + 6 dB = -14 dB. The graphs in Fig. 0. 2-5 were also obtained for X-band but are applicable to other frequencies, upon scaling. This presupposes the size of the slot has been changed proportionally to wavelength. This of course is the case for standard size waveguides, and it is true in comparing S-band waveguide in size with X-band waveguide. Figure 0. 2-8 shows coupling for S-band slots at various frequencies but at a spacing of 0.5 X. The frequency bandwidth associated with dielectrically loaded waveguide slots is quite different from that with ferrite material loading in a waveguide. This presupposes that the ferrite material has approximately equal values of relative permittivity and permeability. This being the case, the addition of such material does not change the susceptance at given points of discontinuity in the waveguide and the waveguide fittings as much as a dielectric material. A dielectric material tends to increase the capacitive susceptance at points of discontinuity whereas the ferrite material involves increases of both kinds of susceptance and thus there is compensation. 8

f - 9.20 KMC X = 3.26 cm (1.283") 5 kg= 4.67 cm a * 0.4" b = 0.9" 10 \ ^\ THEORETICAL 3n \ ^^ EXPERIMENTAL ~ 20 z o 25 30 35.1.2.3.5.7 Ik 2 3 5 7 lox d FIG. 0. 2-4: BROADSIDE TO BROADSIDE COUPLING OF CLOSELY SPACED RECTANGULAR SLOTS SHOWING A 1/r2 TREND. 9

-25 ~- ~ ~. EXPERIMENTAL THEORETICAL \ / f 9.20 KMC \~~* ^~X = 3.26 cm (1.283") ~_. 2 \ * ^s.~~Xgr 4.67 cm -30 a = 0.4" \s. \ b = 0.9" EXPERIMENTAL- \ THEORETICAL - -—'- N -35* \\ ~rd \ \ * * i -' - i d = 2 X 50 I X ~.,, I I \. 8 -40 0O 10' 20* 300 40~ 50~ 60~ 70~ 80~ 90~ FIG. 0. 2-5: FAR-FIELD COUPLING FOR REPRESENTATIVE ANGLES. 10

20 25- 30 -- *1 \ —THEORETICAL 35 d 40 -o 40- 0 0 EXPERIMENTAL - 4550- 5S- ---—.- - I-I I 1 1 1 t I I I I I I iii.1.2.3.5.7 IX 2 3 5 7 IOX d — FIG. 0. 2-6: END TO END COUPLING OF CLOSELY SPACED RECTANGULAR SLOTS SHOWING A 1/r4 TREND. f = 9.20 GHz, X= 3.26 cm (1. 283"), Xg= 4. 67 cm, a = 0. 4", and b = 0. 9" 11

~N^^ <>~~~~A &Experimental } Theoretical *, 2 a = 45~ c (db) =90 a =90~ f = 9.20 KMC X = 1,283 in. at = b = 0.9 in. bt = b 1 0.4 in. trJX~~\ rta80 o I _ I Ii. I I I tI It0 0 Id I I I/ I I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 7 08t9 I d/x 1.5 2 3 4 5 6 7 8 1( FIG. 0. 2-7: COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS. 12

-10 Coupling (dB) o15 20 I 2.5 3.0 305 4.0 Frequency (MHz) FIG. 0. 2-8: COUPLING OF AIR-FILLED S-BAND WAVEGUIDE SLOTS IN COMMON GROUND PLANE, 0. 5X SPACING, BROADSIDE TO BROADSIDE ARRANGEMENT. 13

0. 2e: Observed Coupling Effects of Ferrite Loading In the consideration of the advantages of miniaturized elements in arrays, some of the experimental work was done with dielectric loading material merely for convenience. Other parts of the experimental work used ferrite material. The type of ferrite used was EAF-2 characterized by a relative permittivity of 12. 6 and a relative permeability of 6. 63. The contention that a ferrite loaded slot is less frequency sensitive than a dielectric loaded slot has been borne out by experimental evidence of other studies (Adams, 1964). One of the slots used in this current study (5"x2"xl 1/2"), has a bandwidth of 19 MHz based on a VSWR of 3. A comparable dielectrically filled slot of the size 12" x 3" x 5" has a bandwidth of 10 MHz on the same VSWR basis Additional experimental measurements have been made for the coupling from one ferrite loaded slot to another. These coupling values have been obtained for spacings of approximately 0.5 X and are values for the slots being end to end and also for the slots broadside to broadside. The data are plotted in Figs. 0. 2-9 and 0. 2-10. It is observed, as expected, that the broadside to broadside level of coupling is somewhat higher than the end to end coupling. However, both levels of coupling are very much reduced from what they would be if air-filled slots were used. Again, in this case the comparable values for air-filled slots would be -14 dB (Fig. 0. 2-4) for broadside to broadside coupling and -20 dB (Fig. 0. 2-6) for end to end coupling for this spacing. Note that the experimental data on S-band slots as shown in Fig. 0. 2-8 shows -15 dB for broadside to broadside coupling. In an array of slots, the end to end coupling is relatively insignificant. 0. 2 f: Array Returned Power One of the important measures for an array of many elements is the total returned power from all active elements. For instance an array of many elements may be fed with illumination to give a desired steer angle. Then for a given position of the beam a measure of the returned power coming from all 14

0 -I0 -20- 30 - 40 o -50 -60 -70 Frequency (MHz) I I I I I I i I I I 320 325 330 335 340 345 350 355 360 365 370 FIG. 0. 2-9: COUPLING OF FERRITE LOADED SLOTS IN CONDUCTING PLANE, 0. 5X SPACING, BROADSIDE TO BROADSIDE ARRANGEMENT. 0-10 -20-30 0 _-40 50 - -60 -70Frequency (MHz) I I 1 1 I I I I I I 320 325 330 335 340 345 350 355 360 365 370 FIG. 0. 2-10: COUPLING OF FERRITE LOADED SLOTS IN CONDUCTING PLANE, 0. 5X SPACING, END TO END ARRANGEMENT. 15

of the elements can be made. A total returned power of 10 percent or less would be considered reasonably good. A returned power of 10 percent would mean the returned power is 10 dB below the incident total feed power. If there were 100 elements in an array, each of them coupling at the level of -30 dB to each of the others, and then all of the returned powers in each of the elements added together in going back to the feed network, the returned power level in the feed network would be 10 dB above the incident feed power to one element. However, this would be -10 dB below the total incident feed power to all elements. Obviously, it is assumed that each element is fed the same power as every other element, and also the phase of each element is the same. This would correspond to the beaming being broadside. Such crude calculations tend to be conservative. Fields are assumed to be in phase and, therefore, additive. 0. 2 g: Interaction of Elements in an Array Consideration of the interaction of every element of an array on every other element can assume somewhat sophisticated levels. At the time of writing this interim report, the ultimate in detailed analysis of this interaction has not been accomplished. However, methods are available for this. If, in the future, additional effort is required on this project, it is expected that it will assume the form of detailed machine calculation of this interaction problem. Two significant sources of information on this general analytical approach are Lechtreck (1968) and Amitay et al (1968) Approaching the problem of interaction in a relatively simple manner corresponds to the method which is outlined in the following paragraphs. For this purpose, interaction is considered on the basis of an array with all elements equally illuminated and in phase. This corresponds to having a broadside beam for the array. Of course, as the beam scans, the interaction will change. However, having evaluated the interaction in terms of returned power for the broadside beam, it is possible by comparing with results of others (Butzien, 1968) to get a reasonable assessment of interaction and to predict the change in 16

the interaction level due to having miniaturized slots. Miniaturized slots using dielectric loading will give a somewhat different level of interaction from miniaturized slots using ferrite loading, as will be discussed in succeeding paragraphs. 0. 2 h: Examples of Returned Power for Full-Sized Air-Filled Slots The returned power compared to the input power per single slot is considered. Figure 0. 2-11 indicates a two-dimensional array of rectangular slots with uniform separation of 0. 5X. Assume that all of the slots are fed with the same power level. Also, assume that each slot is in phase with every other slot. The returned power to slot No. 0 is considered. Referring again to Fig. 0.2-11, the contribution due to the slots inside circle "A" is as follows: (1) Slots 2 and 4: The coupling between 0 and 2 or 0 and 4 is of the order -14 dB. This can be obtained by looking at Fig. 0.2-4. Therefore, the coupling to slot 0 is of the order -14 +10 log 2 = -11 dB. (2) Slots 1 and 3: The coupling between either 0 and 1 or 0 and 3 is of the order -20 dB (see Fig. 0. 2-6). Therefore, the power received by slot 0 from elements 1 and 3 is of the order: -20 + 10 log 2 = -17 dB. Then the power received by element 0 from all four slots in circle "A" can be computed. antilog - 1.7 = 0. 01995 antilog - 1.1 = 0. 07943 sum = 0. 09938 10 log 0. 09938 = -10. 03 dB power received from all four slots. The contributions due to the slots in a larger circle "B" can be found as follows: (3) To calculate the power received by slot ), take all the slots enclosed by circle "B". The contributions due to 5, 6, 7 and 8 must now be added to the contributions found for slots inside "A". The separations of these slots from slot 0 is: 2' x 0. 5X = 0. 707 X. From Fig. 0. 2-7, the coupling level between each of these slots (5, 6, 7 and 8) and slot 0 can be obtained for a = 450; 17

3 B B I D / / \ u/ o/ n- D60' \ /; \ \ v4 / I ] \4 \ \ 8 \ ^B^3\ 80 ^3EL 70/ FIG. 0. 2-11: INTERACTION OF CERTAIN CHOSEN GROUPS OF SLOTS CORRESPONDING TO FOUR RADII A, B, C AND D. POLARIZATION ALIGNED LEFT TO RIGHT.

the value of this coupling is -25 dB. Therefore, the total of the contributions from the four elements is: 10 log 4-25 - -19 dB antilog- 1.9 =.0127 antilog- 1.003 =. 0998.0127+. 0998 =. 1125 10 log 0.1125 = -9. 5 dB. The -9. 5 dB indicates 11. 25 percent of the incident power per slot is returned to slot 0 considering all slots in circle "B". The contributions due to the slots in the still larger circle'C" can be found: (4) For the enclosed elements in circle "C", the additional elements will be 9, 10, 11 and 12. The received power from elements 9 and 11 is obtained from Fig. 0.2-4. Each of these has a coupling to slot 0 of -20 dB. 10 log 2 -20 = -17 dB contribution from 9 and 11. The coupling from slot 10 or 12 to slot 0 can be found from Fig. 0. 2-6 to be -32 dB. 10 log 2-32 =-29 dB contribution from 10 and 12 antilog -2. 9 = 0. 00125. Then the total power level from all elements enclosed in circle "C" can be found: 10 log (0.07943+0.01995 + 0.0127+0.01995+ 0.00125) = 10 log 0. 13328 = -8.75 dB power received from all elements in circle "C". The contributions of all elements within circle "D" may be obtained now by including elements 13, 14, 15, 16, 17, 18, 19 and 20: (5) Slots 14, 15, 18 and 19 are spaced 2. 23X from slot 0 and the angle a = 63020' using Figs. 0. 2-11 and 0. 2-7. Interpolation on Fig. 0. 2-7 will yield -31 dB of coupling each slot to slot 0. 10 log 4 - 31 = -25 dB. For elements 13, 16, 17 and 20 the spacing is still 2. 23X but the angle a is 26030'. Again, using Fig. 0.2-7 the coupling of each slot to slot 0 is -44 dB. 19

10 log 4 - 44 = -38dB antilog - 2. 5 =. 003145 antilog - 3.8 =. 000137 10 log (0.13328+0.003145+0.000137) = 10 log 0.136562= -8.65 dB power received from all elements in circle "D". The results of the interaction for various numbers of nearby elements considered are shown in the graph of Fig. 0.2-12. As shown, the returned power of an array of many elements (say, 1000) will be on the order of -8. 65 dB as judged by consideration of the first 20 elements surrounding any one element. 0. 2 i: Returned Power for Dielectrically Loaded Slots As examples of returned power for air-filled slots it is to be noted that the two great contributions are from the two nearest slots which couple broadside to broadside with the slot under consideration. For air-filled slots this coupling level is -14 dB. For dielectrically loaded slots the corresponding level is -30 dB, (see Fig. 4-17, Ch. IV). Although complete information is not yet available it appears with good certainty that for bandwidths of approximately 15 percent or less, the returned power for an array of dielectrically filled slots will be substantially less than for airfilled slots. 0.2 j: Returned Power for Ferrite Loaded Slots Recent experimental data on the coupling of ferrite filled slots are shown in Figs. 0.2-9 and -10. The lower levels of coupling would then result in a substantially lower returned power level. Reading from the figures for a frequency of 350 MHz and spacing of 0. 5X the broadside to broadside coupling is -35 dB; the end to end coupling is -37 dB. It is interesting to observe that these two coupling values are very close. This contrasts with the unloaded slot case with corresponding values of -14 dB and -20 dB. Again, making use of Fig. 0.2-11, the returned power to slot number 0 is considered, assuming all slots are ferrite loaded and have the above coupling behavior. The contributions due to the slots inside the circle "A" are as follows: 20

0 -8I A C 4 elements 8 elements 12 elements 20 elements FIG. 0. 2-12: RETURNED POWER LEVEL TO FEED GENERATOR FOR MANY ELEMENT ARRAY AS A FUNCTION OF THE NUMBER OF SLOTS USED IN COMPUTING INTERACTION. Array has all elements illuminated with same phase and magnitude. Spacing of elements is 0. 5X and the coupling level is in terms of input power level to any one slot.

(1) Slots 2 and 4: the coupling between 0 and 2 or 0 and 4 is of the order -35 dB. This can be obtained from Fig. 0. 2-9 for ferrite loaded slots. Therefore the coupling to slot 0 is of the order -35 dB + 10 log 2 = - 32 dB. (2) Slots 1 and 3: the coupling between either 0 and 1 or 0 and 3 is of the order -37 dB as can be obtained from Fig. 0. 2-10. Therefore, the power received by the slot 0 from elements 1 and 3 is of the order -37 dB + 10 log 2 = - 34 dB. Then the power received by element 0 from all four slots in the circle "A" can be computed: antilog -3.2 = 0. 00063 antilog -3. 4 = 0. 00040 sum = 0. 00103 10 log 0. 00103= -29.9 dB This last value is the power received in terms of the incident power level to one slot. The contributions due to the slots in a larger circle "B" can be found: (3) To calculate the power received by slot 0 take all the slots enclosed by circle "B". The contributions due to 5, 6, 7 and 8 must now be added to the contributions found for slots inside "A". The separation of these slots from slot 0 is: I x0.5X = 0. 707X Since tests have not been made for the dependence of coupling of ferrite loaded slots on separation in echelon, assume the most conservative dependence of 6 dB per octave of distance as for broadside to broadside alignment (this is well established). This is conservative since 12 dB is the value for end to end alignment. 0. 707X 20 log 70 = 20 log. 414 = 3 dB 0. 5X The coupling level between each of these slots, 5, 6, 7 and 8, and slot 0 can be obtained as: -35 - 3 = -38 dB. Note that -35 dB has been chosen arbitrarily rather than -37 dB, again being conservative. Therefore the total of the contributions from the four elements is: 22

10 log 4-38 = -32 dB antilog - 3.2 = 0. 000629 antilog - 2. 99 =0.00102 sum = 0.001649 10 log 0. 001649 = - 27.9 dB. This -27. 9 dB is for all slots 1 through 8. It indicates that 0. 16 percent of the incident power in a slot is returned considering all slots in circle "B". This contrasts with 10. 99 percent for air-filled slots. The contributions of slots in the still larger circles "C" and "D" can readily be included by similar calculations. In this way it is found that the returned power level using contributions from all slots in the "D" dircle is -26. 9 dB. This value -26. 9 dB means the returned power per element from 20 elements surrounding this element is 0. 2 percent of the incident power fed to each slot. Section 0. 3: Conclusions The work covered in this report has considered a simple linear array of three elements. The experimental radiation patterns of this array indicated a low level of interaction of elements. This low level of interaction is attributed to the use of miniaturized slot elements. The miniaturization has been accomplished by either ferrite or dielectric loading. Such loading imposes some bandwidth restriction on the elements but this constitutes no real limitation for the usual application of phased arrays. Of course, the loading material in slots creates an additional material interface which is of importance both as to the interaction of the elements, and the bandwidth. The following chapters give considerable analysis of the slot-to-slot interaction with loaded slots. The net result of the work reported here is that the use of smaller loaded slots reduces element interaction; the driving point impedance of each slot in an array of other active elements remains more nearly constant than for unloaded slots. It has been shown that the use of ferrite loaded slots rather than air-loaded slots at a given frequency can result in a returned power level of 0. 2 percent instead of 13. 6 23

percent. This example was based on conservative calculations for an array of 21 elements. This great reduction in returned power or level of interaction would carry over to much larger arrays using ferrite loaded slots. References Adams, A. T. (1964), "The Rectangular Slot Antenna with Homogeneous Isotropic Loading, " The University of Michigan Cooley Electronics Laboratory Technical Report No. 147 (5549-7-T). Amitay, N., P. E. Butzien and R. C. Heidt (January 1968), "Match Optimization of a Two-Port Phased Array Antenna Element, ". IEEE Trans., AP-16, No. 1, 47-57. Butzien, P. E. (1968), "Antenna Element Match Design for Planar Phased Array Antennas, " Proceedings of the 18th Annual USAF Symposium on Antenna Research and Development, 22 pp. Lechtreck, L. W. (1968), "Effects of Coupling Accumulation in Antenna Arrays, " IEEE Trans., AP-16 No. 1, 31-36. Lyon, J. A. M. and R. M. Kalafus (March 1963), "Aerospace Antenna Coupling Factor Interference Prediction Techniques as Related to the Electromagnetic Compatibility of Different Systems, " presented at the Contractor's Conference on Electromagnetic Compatibility Research, Wright-Patterson Air Force Base, 3 pp. Lyon, J. A. M., R. M. Kalafus, Y-K Kwon, C. J. Digenis, M. A. H. Ibrahim and C-C Chen (1966), "Derivation of Aerospace Coupling-Factor Interference Prediction Techniques - Final Report, " The University of Michigan Radiation Laboratory Report 6633-1-F, AFAL-TR-66-57, AD 483051, UNCLASSIFIED, 334 pp. 24

Chapter I INTRODUCTION The interest in the problem of coupling between two loaded slots backed by loaded waveguide cavities arose when a study was initiated on the use of arrays of miniaturized slot antenna elements. The interest was based upon prior work of the writer on the coupling between unloaded slots and also by the writer's early association with the design problems of antennas of reduced size. The problem of electromagnetic coupling between loaded slots is considerably more complicated than the coupling between unloaded slots. The problem can be considered from a physical viewpoint as a microwave network of considerable complexity due in part to material interfaces. Such a picture is helpful in the consideration of incident and reflected waves at various locations in the physical arrangement. A thorough search of prior literature was made in three directions: i) A survey of the literature was made for information on loaded rectangular cavities including the characteristics of these antennas as individual elements: aperture admittance data were sought. There were several papers on this subject such as Adams and Swift and Hatcher. There were some reports on the properties of rectangular waveguides covered by dielectric or plasma 19 13 1 slabs as in Cockrell and Galejs. The report by Adams was studied very carefully. He used the variational technique to obtain the normalized conductance and susceptance. Some slight changes in his formulas were made and then used as indicated in Section 3.2 of Chapter III. It was possible to obtain the aperture reflection coefficient using simple relations involving normalized conductance and normalized susceptance. ii) Information was sought on the radiation from empty rectangular waveguide openings in a ground plane and also coupling between such guides. There were several papers and reports in this area in addition to work in which the writer 25

had participated at an earlier date. Some of these are; the final report by Archer and Hardie, Galejs and Mikenas and Mayes. Both Galejs and Mayes have used the reaction concept in their formulations. Some reports from The Ohio State University such as Jennetti, Svoboda, OSU 0 21 have some discussions on an empty or unloaded cavity-backed rectangular slot antenna. Finally, the report of Lyon,et al, where integral equation formulations were used to predict the coupling of empty slots flush mounted in a ground plane proved helpful. iii) A search in the literature for information on coupling between loaded slots proved fruitless. There were no papers which have treated the coupling between loaded slots except one recent paper by Wu which appeared in the March 1969 issue of Radio Science. The paper describes a theoretical investigation whereby an integral equation method is used for an analysis of coupling between parallel plate waveguides loaded with dielectric plugs. This present investigation is different in several ways. The analysis is completely different from Wu's since in the present investigation there is a short-circuit at the bottom of each cavity. Also in this investigation, using rectangular waveguide, both theoretical and experimental results have been achieved. The resonance phenomena in this investigation has been explained very carefully; Wu16 did not do this but indicated more information was needed. This literature search was continued at the same time the experimental work was being conducted on the coupling of loaded slots with cavity backing. In the meantime several papers on the reaction concept such as Rumsey5 12 and Richmond were read carefully and the reaction concept was applied in the theoretical analysis. 1. 1 Approach to the Problem The coupling was sought by first obtaining the coupling between loaded slots on an aperture-to-aperture basis and then to ascertain the effect of the 26

backing cavity later. From these two analytical steps the total coupling can be obtained. An iterative procedure has been used to compensate for the interaction between the two slots especially when they are very close to each other in spacing. Some of the graphs in the text will show that the effect of interaction is more pronounced in the case of near field spacing. Interaction is of much less significance for large spacings of slot elements. The situation closely parallels the inductive coupling of two wire circuits. 27

Chapter II THEORETICAL ANALYSIS 2. 1 Assumptions In order to formulate the problem there are several assumptions which are appropriate. The exact solution of this problem is very complex. Some of these assumptions are going to be listed at the beginning and the rest will be mentioned at the point where the assumption is made. To simplify the analysis, it will be assumed that only the dominant TE10 mode is propagated at both transmitting and receiving waveguides. The assumption is justified by the experimental work, as we shall see later. Of course, for operation above the cutoff frequency there may be more than one mode propagating. It will be assumed at the start that the scattered field from the receiving slot is not going to affect the excitation at the transmitting slot but compensation for this will be made later in the analysis, by means of an iterative technique. Also it will be assumed that the apertures are equal in size with the broad sides parallel as shown in Fig. 2-1(a). 2.2 Formulation of the Fields in the Different Regions The proposed method of solving the problem is to assume an aperture illumination at the transmitting slot and then solve to obtain the scattered field inside the receiving aperture. The analysis is that of a three region boundary value problem with Regions I, II and III as shown in Fig. 2-l(b). Region I The fields are formulated in general as shown in the equations: -- 2 * E= V(V- t) + X A E X -jWPu Vx X (2.1) H =V(V V 7r)+k2 r-+jwe Vx ri (2.2) 28

y, r) Trans. Rcvr 0 b xi x2 x,: (a) z z II Ground Plane (b) FIG. 2-1: ARRANGEMENT AND SLOT CONFIGURATION 29

where k2=u2 E c (2.3) 7T is the Hertzian electric vector potential and " is the Hertzian magnetic vector potential. From the vector identity, Vx Vx L= V(V- L)-V2L (2.4) and knowing V2L+k2 L =0, (2.5) then E and H equations can be rewritten in the form E= Vx VxT -j Mu Vx;r (2.6) H= Vx Vx'+ jw E Vx F. (2.7) For E =0, then set wr = r^=z"= 0 and substitute in (2. 6) and (2. 7) obtaining EI and HI in the waveguide which are: E = - ij Vx (7 r Y) (2.8) and H =VxVx (7r* ) (2.9) I y where the subscript I is taken to denote the fields in Region I. Therefore, there results -jz jpz+Re )sin o mry nrx mn z =(e - - A sin cos - e I a mn a b x m= n=o (2.10) where the prime on the double summation indicates that (1, 0) term has already been removed and where the subscript x is used to denote the xcomponent of the field, 3 1- ( where f > f (2.11) V (2 C11) 30

f is the cutoff frequency, C v- (2. 12) and R is the reflection coefficient of the dominant mode at the aperture. Region II Region II is the free space upper hemisphere over the ground plane (see Fig. 2-l(b) ). The method of images is going to be used to derive the fields. To formulate the equations of the fields in Region II take a b -- b f ~ e-jkp F = I Ir M ds (2.13) e 0 a0 From Fig. 2-2 an application of the equivalence principle yields (b) from (a) and according to Huygen's principle the result as (c) is obtained and this is going to be used in (2.13). Therefore - 1 a b - eko F= 2E x n - dx dy, where M is the magnetic moment vector P = /(x)2 +(y-7) 2+z2 (2.14) and F is the electric vector potential. Since for the problem at hand Exn Exz = -ExY (2.15) then a b i- ejkop F=F =F - EJ - dxdy) (2.16) and F =F = 0, (2.17) x z 31

z=0 z=0 z=0 E,H I E,H E,H t E, zero' I fields.. Il | Ms=Exn I Ms=2Exn sources I I matter I I _ at. _ n n t n (a) (b) (c) FIG. 2-2: METHOD OF IMAGES

where Ex is the x-component of the field at the transmitting aperture or, in this case, the aperture illumination. The fields in Region II could be represented, in general, by these equations: E= - Vx F-j A + V(V A) (2.18) 0 J Eo and H =VxA -jwe F+ —- V(VF). (2.19) The radiating source in this case is the rectangular slot denoted by TR and located in an infinite ground plane (see Fig. 2-1). Since only a dominant mode is going to be assumed then the aperture illumination can be given as: E 0) =0 (1+R) sin - x (2.20) where a where R is the reflection coefficient of the dominant mode at the aperture and e, rl are the source coordinates as shown in Fig. 2-1(b). Therefore from (2.20) and (2.16) a b2 e-jko (x-) +(y-rl) +z F= - I J J 2(1+R)sin a e ddn2 (2. 21) 4 4 ^x-?)+(y-7) 2+z2- (2.21) where x, y, z are the field points of the coordinate system and k = t i. (2.22) From eqs. (2.18) and (2. 19) by setting A = 0, then E=-VxF, (2.23) and H = -jE + F + - V(V F ). (2.24) o jL4~ Therefore 33

- aF aF- aF aF \ y A YJA yA y E= —X +z -J=X -— Z _ az ax_ az ax and a b -JkoP E1I= a f r 2(1+R)sin ( ) 3 (-jp-l) dr (2. 25) X 00 p which is the x component of the field in Region II. Notice that E O=0at z=0 or the tangential component of the electric field is zero over the ground plane (perfectly conducting). Also the other components can be obtained inthe same manner For the H field we can write a b Hay = - 4 j ( o ClY e -d de 1 k a 2 (7r7)/ e -jkop H= — I 2(1+R)sin( -)2 e d-jp d (2.26) HII 4 rjw axay a 2(l+R)sin(T ) e d.d.) 1nd 1 I r f 0 r7 P + 2= (l+R)sin ) e — d d (2.27) 4r \ a p and a b a2 7T) eJkoP H1 ____ 2(1+R)sin(-) d d) (2.28) II 4ir jc azay\ a p z ~ Region III In Region III again it is to be assumed that the only mode existing is the dominant mode. Therefore as in Region I: E (x, y, z)= U sin a e p, (2.29) IIx_ a where U is the complex amplitude of the electric field at the receiving aperture 34

and from Maxwell's equations, aE a =-/ jwHy. (2.30) y Therefore: _H- 1 a (UsinYeJZ) Hm j4Lu az a and HO =- Usin yej. (2.31) III to a y At the surface of the ground plane or at z = 0, H _ - Usin( —) y (2.32) y where as before B =- - ( 2 ) (2.33) v or 27rf L fc 2 =(1 2 1 2 v/X 2 hx~f f h X v/X or r 2 ) 2 / ) (2.34) where X is the wavelength in the material medium. Therefore, from (2. 33) and (2. 32), H -U _ sina y (2. 35) III 2a a where jL is the permeability of the material e is the permittivity of the material used C = COer = Jolr a 35

co, e are the free space and relative permittivity of the material r respectively, and Mo',r are the free space and relative permeability respectively. The objective now is to use the formulations of the fields in Regions I, II and II to obtain an expression for U the complex amplitude of the electric field at the receiving apertures corresponding to a given excitation of the transmitting aperture. 2. 3 Evaluation of U To obtain an expression for U there are several ways, including an integral equation formulation or variational technique. For this analysis the reaction concept of Rumsey is going to be used since e it simplifies the formulations. As stated by Rumsey, J(c) generates the same field as "g" on the source free side of S as shown in Fig. 2-3, where g is the source, J(c) represents the surface distribution of the electric current which is induced on the scatterer by g, and S is the scattering body. J(c) S Source S > Scatterer FIG. 2-3: SOURCE AND SCATTERER. 36

For the specific problem at hand g is the same as the transmitting aperture with a given illumination and the scatterer is the receiving slot. The boundary conditions with the tangential magnetic fields at the receiving aperture states: A _ A A - ZxHi= ZxH +ZxH (2.36) in s I' where H is the scattered field inside the waveguide, in H is the scattered field in the half-space over the ground plane, and HII is the field due to the illumination of the transmitting aperture. Also A _ ZxH =CZx Hin (2. 37) S in' and A _ 1 A and ZxH = ZxH (2. 38) in 1+C II For the free space case C is taken to be unity as in Lyon. Therefore, A - 1 _ in 2 ZxH (2. 39) Also from the reaction concept which states (a,b b= fff (b) dJ(a)-H(b) d K(a), (2. 40) V where V is the volume which contains the sources, J (a), K(a) are the electric and magnetic current densities of source "a", and J (b), K(b) are the electric and magnetic current densities of source "b". It is also helpful to note that the reaction formula has the properties of a scalar product. The reciprocity theorem of all the sources that can be contained in a finite volume is expressed by (a,b) =:b,a). (2.41) 37

The linearity of the fields is implied by the identities, <a, b+C> = <a, b> + <a, C>, (2.42) and <A a, b> = A a, b> = <a, A b>, (2.43) where the notation "A a" means the "a" field and source are multiplied by the number A. Also <x,a> 4x, C>, (2.44) where x is the test source. Equation (2. 44) expresses the condition that "a" and "C" should "look" the same to an arbitrary test source "x". Choosing the test source to be "a" itself, then (2. 44) contains "a" as the assumed value of sources and C is the correct value of it. <a, a> = <a, C> (2. 45) from (2. 40), (2. 41) and <a, C> C,a>= a f-J(c) E iidS (2.46) S From the boundary conditions it is known that j= x H. (2.47) Therefore A J(c) = -H11 x, (2. 48) t where the subscript t is to denote the tangential component at the aperture. From Eqs. (2. 46) and (2. 48) there results (a, C> = fI HIk EItdS, (2.49) and <a, a> f J(a) Et dS (2. 50) S 38

in which J(a) is the current assumed when the scatterer is removed or J(a)= -H + H (2. 51) I It t where HMit is the tangential component of the magnetic field in the Region III at 15 z = 0. Following the same procedure as given by Rumsey, let a=Uu. (2.52) Then apply (2. 50) and (2. 52), < a, a> = Jf (-HI t+HI ) (U EDut) dS (2. 53) S where E t= sin -Y. (2. 54) IIIut a From Eqs. (2. 45), (2. 49), (2. 52) and (2. 53), u u m En ^u dS=U J2 (- +H U J HII EudS=U f (H +HmuIIt)EIIIutdS S t t S Simplifying the above to get an expression in U, 2U I HIItEIutdS=U JHInU Emut dS S S or 2 H E dS UV- — ^7 —------- S (2.55) ff Hmniu dS S t t where E =sin T at z IL u, atz= H =- sin -, at z = 0, IIIut /u a S is the area over the aperture of the receiving slot, and 3 is defined as before in Ea. (2. 34). 39

A special case of the above would be that of free space slots and it can be analyzed as follows. Again <a,a> =/f(-Hit+Hit)UEmutdS (2.53) where Em ut=sin a x UIt a From (2. 39), which is A - 1 - ZxH = ZxH in 2 II 2 and using C = 1 as in Lyon then in this case Hin = HI r H =2H. (2.56) II III t t Substituting in Eq. (2. 53): <a, a> = J(-2HIt+Hit) UEm ut dS, or <a, a> = -U IfHIItEEut dS However HI =UHI t t Therefore <a, a>=-U2 /H uEmu dS (2.57) S Illt It from Eqs. (2. 45) and (2. 46). 40

Therefore -U IfJ(C)EI dS = -U2 H UII EudS S ut S t t or j, JJHIE EI dS U =. (2.58) ((H E -dS II utEI II utdS This is for the free space case where u = u and e =, where S is the receiving 0 o aperture area, and E = sin z= 0 IIIu a H sin, Z = 0. IIIu tqL a To obtain the value of U it is necessary first to obtain HIt. From (2.27), again write a b -jkaP H =- ( 2(1+R)sin 2 dd II 4w a p y 0 2 a b -jkop k p 47-r jwj0Ay2 I |2(1+R)sin --- ded7, (2.27) where P x-()2+ y-r) 2+z2 To simplify (2.27), consider the second term on the right hand side, a b -jkp L = a2 ( 2(1+R)sin l e dd) Jy2 4a 41

Interchanging the order of differentiation and integration and substituting a2 a2 ay2 ayarj a y2 yields a b e2 -jkop 7rr 7a e L= - 2(1+R)sin r ( ) dedrl. (2.59) a ayarl P Integrating once by parts with respect to rl in the manner ju(x)v'(x)dx=u(x) v(x)-Jv(x)u(x)dx. Let u7u=cos, Let u = sin U' - -cos,7 a a a e-jkp a a e-jkop v = (, and v' =a (P) a ay p aray p Hence _ b-jk p a a b -jk p L=-J 2(1+R)sin r dr + 2(1+R)cos -(e )ddr, a P a " ay P or L= a 2(l+R)cos - (- ) dedrl (2.60) a I a ay p Therefore, Eq. (2. 27) reduces to a b -j 1 r e "jkop 14 j~ a J 2(1+R)cost1(e- ) djdr7. (2.61) 4u2+Oi w a I aay p 42

Next, return to evaluate U using (2. 61). At z = 0, P = P' = x 2+(yt-r)2. (2. 62) Then the numerator of Eq. (2. 55) can be written in the form Numerator = { a a ab -jkop a 4 I a p 1 a b(1+R)s r) a' 2 - f I 2((l+R) )sin(- sin-ye ddrndydx x2 a a b.-jk p' 1 w ffr raab7nry e k t 2(T f ff 2(l+R)cos-sinT s(e-,)ddrdydx. 4jwuo a I a a y p' Then simplify the second part on the right hand side of (2. 63) by means of integration by parts as before. Take ry r ry u =sin-, u'= - cos-, us a a a e-jkoPt and ejkop' v=- aynd v'= - P' ay pa 43

Let a a b jkop''I= f fJI 2(l+R)cos( 2U1 )sin ( dy da e d x 0 a 0 0 Then integrate by parts I=| ff(2(l1+R)cos sin aY e dedridx Sfa pt X2 a a b -jkop - a J 1; 2(l+R) cos 77 os Ty e, ddrldydx. a Ja a p'1 Again the numerator can be written as follows Numerator = x2 a a b -jklp' jWE _k?Ty e 2 -J 2( +R) sin2 sinry e_ ddrndydx 47r a a p' 11 2 X2 a a b -jk pt 1 7Tr2y + 4j. w 2 | I 2(1+R)eos- cos - ddrd —yddddx 4?rjwp a 2 a a P Some simplification yields: Numerator =.2.a a b jwto (i+R) -jkp, o |n| [n ry+ (1+R) i rl i y e 0sin- sin -- COS — cos — dg drtdydx JoL r a a 2. a a P' 0 & 0a j (.a (2. 64) 44

Use the trigonometric identity below: 7r Ty al a7r ClCSa a osa a a a- 2sinl- si -+ C aos - cos aYC an a 2y CL-C2)cos (y-r))+(Cl+C2)cos a (y C1 and C2 would be defined as: and w7 (1+R) C = r l (2.65) and ~ (1+R) C -. (2.66) 2 j 7 Finally U can be rewritten in the somewhat simpler form: x2 a a b -jkoP' U= 2~C l+C2 cos" (p —adrjdydx, 2C 3 IO r f j1C C2)cos a (y-r')+(C+C2)Cs a (Y+r (2.67) where C3 iHiut Eit dS px2 a T.a.27.y - | 1-( ) sin dydx 2 a a x *'0 12 =2V _2a (x2-xl) In Eq. (2. 67) C= - 1- ^ - (2.68) since from Fig. 2-l(a), x2-x1 = b. X in this case is the wavelength in material medium corresponding to a certain operating frequency. 45

Since this integration of (2. 67) could not be obtained in a closed form, computational techniques are needed. From the point of view of computational techniques a quadruple integral is impractical to use. Therefore, the aim here is to reduce Eq. (2. 67) to a simpler form. The simplification is going to be achieved by means of transformation of coordinates. First change the limits of integration for x from x1->x2, to the new limits 0 -* b using a new variable x'. This corresponds to a shift of axis as in Fig. 2-4. The new variable x' is defined by: Xl = X-X1 Thus dx' = dx. Also, take 21' 2 2 p'= W~x-)2+(y-r)2 = /x,+x1 -Ky-r)2 (x+x) 2+(y-r) To reduce (2. 67) to a double integral which is practical in numerical calculation, another coordinate transformation is now performed. y-I7 =, x'-~ = a, y+r77= a+, and x' + = b+ The above are equivalent to a rotation by 450, translation and magnification by eI1 (see Appendix A and Fig. 2-5). Notice that e o ~)'-jkk /(o4xl)2+' f(%y, u v)= (CC2)cos 7+(C1+C2)cos (a+) a a ^(,+x )2+,y2 This simplifies to v){(l, 2co~(lC2co ) - jk /(^+x1)2+^2 f(%y, Ua, V)=((C-C2)coSa-(C )cos2+2, (2. 69) \ 4. ~ - a- - " - - 91 _ a - //. \2.92' where f is an even function in y, jI and v. 46

y Y' Trans. Rcvr _ _. ___.._ x10 x2 x' FIG. 2-4: COORDINATE TRANSFORMATION

\ y y 1ft' ~ 71l, Y2 \ ^0 FIG. 2-5: COORDINATE TRANSFORMATION FOR REDUCTION OF QUADRUPLE INTEGRAL TO DOUBLE INTEGRAL.

Then: a b -y (b- -c)jk - III I I~c^-C~cos I(C +C )COS7]- dvd/-tdcydy u=_ ) lW oi (-L) 1 C2 a 2 aed ~-~-~ 0 a (b-a) 2C3 a 0 ~a-7 (b1a) -jkV V(+x1) 2-jk2' + I c cC ) F cos7W-(C+C )cos eAh dzdijudd7 1f fo J-(b) -2 a 1 2 a 4 (2.70) Performing integration in both v and,u the expression for U can be reduced to f a b _-jk pt d ( 2(b-ca) l-C)(a-)cos7 -(C1+C) sin a e p dcd. 3 a 0 _jk p' aCT a ny~e0 + j2(b4'A)-C )(a-y)cos- y —(C+C )sin-t dardy. (2.71) 1 2 a e 2 P' This is for X < 2a, p' =vkc+xl)2+~, (1+R) |7 u j-z J k-o ko(l+R) O C2- and ab _ _ ab 2/ 2 C3=- 2-2 2 1 — ab R is the aperture reflection coefficient. A further simplification can be made: 49

Consider: a 0 -jk p' Il=1 2(b+)C )-C2) (a- ) 4cos 7 (C ) dody a b -jkoP" = 2(b-)[ C 2)(a-y)cos 7 t -(Cl+C2)sin j e (2.72) fC 2 aa 7 p2" where Ptt =(xl-f)Y+Y 2 Therefore, Eq. (2. 71) reduces to: ab a, e-jk t -jll a 1 r^ f ^V1 ""-ko p' u= J C C2)(a- cos-'Y (CC2)sina ]y(b -) + e- d -o.C a 1 2 a PEc0IIa 7 (2. 73) 2. 4 Iteration Procedure It was assumed at the beginning of the analysis that the illumination of the transmitting aperture is not affected by the scattered field from the receiving aperture. This is a very good assumption if the two apertures are sufficiently far apart. The assumption is not justified for very close spacings of the aperture. The interaction between a transmitting and receiving antenna as a result of scattering can be described as follows. If a transmitting antenna is considered completely isolated and matched to its line, then when a receiving antenna is introduced into the field of this antenna there will be a scattered field as can be seen from Eq. (2. 36). This scattered field from the receiving aperture when intercepted by the transmitting slot, in turn gives rise to a wave transmitted down the feed line causing a mismatch. Also the transmitting slot gives rise to scattered fields that again will be partly intercepted and partly scattered at the 50

receiving aperture. Thus the interaction is due to a multiple scattering and absorption process as can be represented by Fig. 2-6. An analysis corresponding to the above explanation can be made. From Eqs. (2. 73), (2. 65) and (2. 66) then: /iC>\ E = (1+R) l+R) C 2r(1+R) o (I(+R) R ~1 C1 2 jcooa2 j R)L a2 j~r.' C -C2=(1+R)C4, and C +C2=(1+R) C, where toe C = -~ (2. 74) 4 jwj a2 jT and C5 = 22 + j~ (2. 75) 5 jwtAa 2 j 7 1 b -jkoj' _jko, JkoP" U=(l+R) 5( 4 (a-y)coss aIC sin (b-a) C-(b ) + e dody=(I+R), Ua+R f E4 i ae e -j0ip" (2.76) where f 4(a -jkoP' -, e o F= 4 E(a-,y) cos- P'y- - C5sin +(b-a) { Ip + - dady. (2.77) 00 Now assume that the magnitude of the aperture illumination is (1+R). Then the received aperture illumination will be (1+R)F for the first iteration. For the second iteration there will be (1+R)F3 added to the results for the first iteration and so on. Therefore the received aperture illumination is: UI= (l+R)F [l+F +F,4+F6+..., or UI=(1+R) F2 (2. 78) 51

This is the upper hemisphere in the problem Incident wave Received Waves Trans. V Rcvr FIG 2-6: MULTIPLE REFLECTION BETWEEN TRANSMITTER AND RECEIVER TO REPRESENT INTERACTION BETWEEN THEM.

where UI stands for the modified U. This iterative process has been described rather simply due to the fact that the reciprocity property has been used. The factor F of coupling is the same from transmitter to the receiver as from the receiver to transmitter. The absorbed power or energy in the transmitter due to scattering from receiver has been ignored since it is small compared to the original excitation. 2. 5 Consideration of the Short-Circuited Waveguide Backing the Slots 2. 5. 1 General Discussion The problem under study uses a short-circuited waveguide section at both transmitter and receiver. The shorted waveguide section is fed by means of a coaxial-to-waveguide probe. Consider first the effect of the cavity alone as shown in Fig. 2-7. The arrangement of the axis for formulation is taken as in Fig. 2-1 except for taking z to be positive in the reverse direction. Note that the input impedance seen at z=0 or at the aperture of the homogeneously filled waveguide with material described by 1l and e1 and shorted at ""l is: Zg1(0) Zgltanh 7l I, (2. 79) with the subscript "g" referring to the field in the guide and the subscript "1" referring to waveguide Number 1. Also, 1'gl agl+ gl' (2. 80) and 3 =27r/Xgl (2.81) ~g= = 2 /gl where X is the guide wavelength in the filled guide. If the attenuation gl could be neglected, Eq. (2. 79) could be rewritten 2,K Zgl(O)=Zgl tan i (2.82) ggl where Zg1=.. =...._ (2.83) gl l-(f/Z /l-(fc/f2853 53

y lAl, El Short 0 - -. z FIG. 2-7: SHORT-CIRCUITED CAVITY.

The real parts of the fields in the waveguide are: Re(E )=Usinwt sin 2 (z- (2. 84) and Re(Hx)= - cos t cos -(z-)]. (2.85) Re( g g The standing electric and magnetic waves in front of the short-circuit are displayed in Fig. 2-8 in terms of Xgl the guide wavelength for Section 1. The impedance z (0) which is the input impedance at z=0 varies as shown in Fig. 2-9. It is noticed from Fig. 2-9 that the impedance is zero at z=which is at the shorted end of the waveguide. It varies from -oo to +oo at X-kgl/4 and then returns to zero again at t-Xgl/2 and so on. The pattern shows resonance points at n Xgl/2 where n = 1, 2, 3,..., from the shorted end and antiresonance at points n Xgl/4 where n=l, 2, 3,..., from the shorted end. It is to be remembered that these resonances and anti-resonances do not take into consideration perturbations due to the material or the perturbation due to the feed probe into the waveguide. These effects can be included as was done by Adams1 and will be mentioned later. For more physical insight, it is to be noted that if the waveguide is tuned to a multiple of a half wavelength, then in this case if a short is placed on the open end of the guide, as in Fig. 2-10, at z=0, nothing will be changed inside the guide. The wave pattern in this special case can satisfy the boundary conditions at both ends. On the other hand for the case of any arbitrary length I, a phase mismatch results between the standing wave building up in front of the shorted end at z=e and that required by the boundary at z=0. Destructive interference takes place between the incident waves and the reflected waves at the aperture (z=0) and therefore the amplitude of the standing wave decreases. Note that the impedance expression (2.79) remains the same since the only requirement is that a wave entering from the aperture at z=0 is totally reflected at 55

E H Short FIG. \ \ / / 2G I z FIG. 2-8: STANDING ELECTRIC AND MAGNETIC FIELDS IN CAVITY.

Inductive jx } I v - Reso ance -Xgl \-(3Xgl/4) IQ -(Xgl/4) O. JCapacitiv \ -(Xgl/2 -jx -L... AntiResonance FIG. 2-9: INPUT IMPEDANCE OF THE CAVITY AT z=0.

short end or z=1 and thus superposes with the returning wave to form a standing wave pattern. FIG. 2-10: CONFIGURATION OF WAVEGUIDE WHEN IT FIG. 2-10: CONFIGURATION OF WAVEGUIDE WHEN IT IS CLOSED AT BOTH ENDS. 2. 5.2 Feed Perturbations Perturbation of Metal Probe: The equation for the correction of the frequency of resonance is w-w 2xV 2 2 2 1 wo =abd os d k2a2 (2.86) where r is the resonance frequency obtained without perturbation V is the volume of the probe and dl, d are as shown in the simplified equivalent circuit of a cavity slot antenna (Fig. 2-11). a and b are the dimensions of the cross section perpendicular to the z-axis as shown in Fig. 2-1 at the beginning of the analysis. 58

Aperture Probe G B — Short GA BA G 0 - Bp co I t —--- d! ---.... —---— 3 - FIG. 2-11: EQUIVALENT CIRCUIT FOR CAVITY SLOT ANTENNA.

Material Perturbation: The equation of material perturbation compensation is given by: W-to 2Vlrk 22 2 - d 7rd o 1 k a-7r Am Cos2 1 e+.s 2 1 (2. 87) abd 2 2 d, d O ka and for dielectric material, r =1 degenerates into o 2V A~ 2 1 3- sin2 (2. 88) W abd e d where V1 is the volume of a cylindrical hole in the material, and ao is the resonance frequency without perturbation. It is to be noted that the resonance frequency of the cavity corresponds to the resonance frequency for the coupling. This fact will be shown in a subsequent chapter on experimental work. The Smith chart can also be used to obtain frequencies as was done by Adams. 2.5.3 Resonance Frequencies and Bandwidth As can be seen in Section 2. 5.2 where are several ways to obtain the unperturbed resonance frequencies. The resonance frequencies could be obtained from the equation Xg = n 2. (n=1,2,3,...). (2.89) Another way to do this is by consideration of Eq. (2. 79) which is: Z l(O)= Zg tanh g. gl ) gl gl Then 1 1 Ygl(0 () Z(= zg coth (gl+ igl) (2.90) Zgi Z g l (0) zl60 60

where Ygl(0) is the input admittance to the cavity, Z g = — 2 as defined before, 81 f and 3 J # i - [-( ]. (2. 92) Then a gi n -1 (2. 93) Pgl [/l+tan26' +lJ where tan 6 is the loss tangent of the material and d.E e" =' -je". (2.94) From (2. 91), since O o4/'oo then agl 42Per r {/1+tan26 -1} [r-f)2] (2. 95) gl o/2 r r — ] where wr and e are the relative permeability and permittivity of the material. For the case of dielectric material with a low loss tangent, Eq. (2. 95) can be rewritten as: gl o/tan6 [ ce) 2 gi V T k 0-Mer 4 f since k La ~JI k (O / reC (2.96) 61

Therefore a = (ktan6/2) -(). (2.97) gl Now Eq. (2. 90) becomes 1 cos(agl +gl ) Y(0) = -z zgl sin( gl +jgl)f but /cosh2 a -ecos2 3 +sinh2ot a sin2 s |coth(oa+j|3) = / g 22 gl12 gl2 (2.98) sinh a -lcos j3 1+cosh a -lsin 2 1I gl gl g l Therefore e e f f cosh2 a glcos23 jl+sinh2a % sin2 -glB 1Y 2(0)1 1_(02 c 02 g gggl gl gl 1 ] |y20 ) L 2 ~o/r g1 —--- 2. 2. 2.(2.99) 1lA01Ar (I sinh a 2l cos glf +cosh a glsin 2glf Since the energy delivered to the waveguide varies with I Y (0) | then by computing Eq. (2. 99) for various frequencies it is possible to obtain the unperturbed resonance frequencies and also the bandwidth as will be shown in Chapter llI. To approximate, roughly, for the bandwidth of a cavity with low loss material in it and for small phase deviation A from resonance, insert in Eq. (2. 99) the approximate forms; 2 2 2 sinh a -gl (agl)' cosha o -g 2 1 gl gi' g1, sin2 Pbl -* A2 and cos23gla - 1. Then Y(O) 01 1- -2 2 1 (2.100) 2 odr f 2 + 2(a( 1) gl But for resonance, the approximation condition is that should be an integral multiple of the half wavelength or, 62

a nX a _ gi 2 g gn?r. (2.101) gl 2 g Consider that IY(0)2 falls to its half value, when the section is detuned from resonance by a change in phase. A z ag1 gl + n?r (2. 102) 2 -g This is equivalent to a change in the resonant length by T A Zh/2. iut AZh X A. ag __h - hgl 41 -+ S1 I (2. 103) 2 27 2 - g Alternatively, the phase shift can be produced by changing the input frequency from its resonance point f to f +. Therefore o o 2 A Z Af a h h n _ 2gl h 2 (2. 104) fo gl From another viewpoint Eq. (2. 104) indicates that the bandwidth for the case using low loss material is approximately equal to the loss tangent (see Fig. 2-12 for symbols defined above). 2. 5. 4 Waveguide Wavelength Xg In general there are multiple modes in the waveguide with dielectric material filling of relative permittivity Er. The general formula for waveguide wavelength is: n o, (2. 105) no n Xo 2 TrET ( 2a ) 63

1.0 Ir3 "I.0. 5 rT- -A li "0-. AAfh I] I f.S f o FIG. 2-12: RESONA NCE AND HALF-POWER BANDWIDTH.

where X is the wavelength of the TE mode, no no X is the free-space wavelength, and 0 a is the dimension of the broadside of the waveguide. For example, for the assumption of the dominant mode TE1 in the section filled with the dielectric material, Xo XlO-r)g / f is the cutoff frequency of the filled guide and c f =(2. 106) c 2a s T 65

Chapter III NUMERICAL ANALYSIS 3. 1 General Since some of the formulation was not obtainable in a closed form, extensive use was made of The University of Michigan's Computing Center for the evaluation of these integrals. The IBM 360/67 digital computer has been used in most of these calculations. The Fortran programs for each case were written and compiled using the -Fortran compiler and both the Fortran programs and the object programs were stored in the IBM 360/67 memory. After the specific program was cleared of errors and made operational, the Fortran program was destroyed from the file leaving only the object program, thus releasing storage space. Then the object program was available for either a'batch job' or for MTS (Michigan Terminal system through teleprinter). 3.2 Calculation of Reflection Coefficients at the Aperture The first program was written to find the reflection coefficient, R, at the aperture. Since there was already an analysis of this in the literature, rather than duplicate that of Adams, a careful examination was made and some slight variations were introduced. The formulas which have been used in computations of the normalized conductance and susceptance are: 222 2 a b 2/r(ka r) C 2N,2 j J2' - (b-) 3r(a —y)cos -L+a 1 sin ] o X a bk a -?r Ja I 1 sin(ko / 2a) -2 — dody (i) 66

22 a2 -a b B 2/lr(ko af C) el Y] Y 2(b-) 22 (a-y) +a - sin r Yo o r a b k2a2,-r2 E a cos kVT2 d d y (ii) Since both (i) and (ii) have a singularity at the origin, a small rectangle was considered with sides of ratio Q to the original dimensions of the slot, or, namely, Q a and Q b. An approximate formulation was then developed for that area and these were: Cl C 22 2 -2 5.57+ 2G 2 (koa b 1 D 1_ Q4_ Yo 7rk2a2_7r2 a'r r Q3( ) 4 + 2.06Q-.616 Q (iii) B r(koa - a b lb 1 a 2 Yo 02 ka-2 2 2b 2 2 a b C1 ( —1) 3 D f 22 2 2 g b 22 2 1 -+ -(2r +k a )+ g- (k2a -7r2) /a, b2,a 21 bd 2 2 (d-l)+ —( d-1 -T (2 a -7r Q +a2 b 2 a (iv) -1 b where 0, = tan - 0 7 f= log tan (-+ -) 67

01 g = log tan - jb2 I d = - +1 = sec 0 2 2 C= ka +iT and 2 2 D = k a -7 1 o The change was mainly in Eq. (iii) where 1/ JA was used rather thanV/Er. The contribution for the small area around the origin was calculated by (iii) and (iv) and was compared with the polar coordinate integration on the same area. A good agreement was obtained. The rest of the integration was performed by cartesian integration using Eqs. (i) and (ii). The two results were added together to give G/Yo and B/Yo The change in Eq. (iii) has helped to stabilize the integration with respect to both the parameter Q and the number of divisions to be taken in using Simpson's rule. An example from the computer calculations is shown below using Adams' formula (2. 54) and the corrected one (iii). Using Eq. (2. 54), Adams, with other equations to calculate G/Yo, B/Yo gave these results. Case (a): For b/a = 0. 444, Lr=3. 0, Er=3. 0, NS=30, NL= 50 where NS and NL are the divisions for the integrations and Q = 0. 05, G = 0.2542843 B = -0.9999304 G' = -0. 1218292 B' = -0.9656234 1 1 G2= -0. 3980840D-01 B = -0.5975625 2 2 Then FN =1. 100000, G/Yo = 0. 9264673D-01, B/Yo= -2. 563116, where G and Gt are for the integration outside the small rectangle around the 1 1 origin. Similarly, B and Bt give the susceptance for outside the small rec1 1 tangle. G2 and B2 are for the integration over the small rectangle around the origin, and FN is the normalized frequency. FN = ka /. (3. 1) Note: -0. 3980840D-01 - -0. 03980840 68

Case (b): For the same parameters as in Case (a), except for the change Q = 0. 01; G1 =0. 1476497 B1= -2. 064409 G' = -0. 2789158D-01 B' = -0. 3865364 1 1 G = -0. 1711596D-02' B = -0. 1254613 Then 2 FN=1. 100000, G/Y=O0. 1180465, B/Yo=-2. 576407 Case (c): For the same parameters as in Case (a), except for Q=0. 1; G1 = 0.3695354 B = -0.2567721 G' = -0.2024414 B' = -1. 190663 ~~~~~~1 1 Then G = -0. 1437410 B = -1.118208 Then 2 2 FN=1. 100000, G/Y=O0.2335299D-01, B/Yo=-2. 565642 Note that corresponding to the change in Q, there is a significant change in G/Yo. This should not be so distinct. However, using Eq. (iii) along with the others, the computer gave these results. Case (1): For b/a = 0. 444,'r= 3. 0, er = 3. 0, NS = 30, NL = 50 and Q=0. 05. G1= 0.2542843 B = -0. 9999304 G'- -0. 1218292 B' -0. 9656234 1 1 Then G2 = -0. 1326947D-01 B2= -0. 5975625 FN=1. 100000, G/Y =0. 1191857, B/Yo=-2. 563116 Case (2): For the same parameters as Case (1) except Q = 0. 01 G1= 0. 1476497 B = -2.064409 G' = -0. 2789158D-01 B' = -0. 3865364 1 1 G2= -0. 5705321D-03 B2= -0.1254613 Then FN=1. 100000, G/Yo=0. 1191876, B/Yo=-2. 576407 -0. 1711596D-02 - -0. 001711596. 69

Case (3): For the same parameters as Case (1) except Q = 0. 1, G1= 0. 3695354 B1= -0. 2567721 G= -0. 2024414 B' = -1. 190663 1 1 G2= -0. 4791366D-01 B2= -1. 118208 Then FN=1. 100000, G/Yo=O. 1191803, B/Yo=-2. 56542. Note that G/Yo and B/Yo are more stable with the change of Q. Also, in choosing Q to be very small, both results come very closely together. Also, a variation of the number of the divisions for the integration has been made according to formula (iii) rather than (2. 54), Adams1, and thus gave these results. For b/a=0.444, Er=10. 0, Pr=10. 0, NS=30, NL=50 and Q= 0. 05, then FN = 1. 5 gives G/Yo=0. 1134798D.01 and B/Yo=-4. 186671. Keeping all the parameters the same as before except NS=100 and NL=100, then FN = 1. 5 gives G/Yo=0. 1134785D-01 and B/Yo= -4. 189243. These indicate that there is no error of 2. 5 percent in the value of G/Yo which has been claimed when Eq. (2. 54) of Adams1 was used. The reflection coefficient can be obtained from G/Yo and B/Yo: 1R = G/Yo +j B/Yo (3.2) Using the computer, several values of Cr were considered as well as different values of the ratio b/a as shown in Figs. 3-1 through 3-5. 3. 3 Calculations of the Aperture-to-Aperture Coupling Between Slots versus Frequency 3. 3. 1. General Discussion A computer program has been written to calculate the coupling between two parallel slots in an infinite conducting ground plane. Since the formulas to predict the coupling as shown in Chapter II could not be obtained in a closed form they 70

b/a.8 YO.2.02 __.02 __.._____________ —---.-.12-,b/a. -. -_ _,,,,,,, /-.27,,444;,,/ / I / "7 - — I.1 / 15 1 FOR l./ ~,/r/ ---'/.41 B,,,;','"k/ I i''I'///'' II I' I Ii I/ I i~'i ~ ~ Normalized Frequency FN -.85..-. —-_ —----- - --— + —-- -.4- 1.10 1.30 1.50 1.70 1.90 FIG. 3-1: APERTURE NORMALIZED CONDUCTANCE AND SUSCEPTANCE FOR r=rl. 0, Cr=5.0, — (B/Yo) — (G/Y) 71

b/a -^..8 -.07 b/a. 6 Yo' - - -06g9/''a. - -.444 - -. /./. —'.5.37 —/ /\.7I __ l[ / 6"'. 8 — d. -'I.,, —..0.'.- __ldl''' --.' J w1 *..../.. ~ CO f. /N / -.0 82 - I,,0, /, I, 67 fii', nil 2 hg /l / //l'I/ // 1.10 1.30 1.50 1.70 1.90 FIG. 3-2: APERTURE NORMALIZED CONDUCTANCE AND SUSCEPTANCE FOR.0 c=7., — (B/Y -(G/Y) I / 8~',Ss72 I I /' / II I' Ill!I I(I I I t, orirl,'::,:' I 1.10 13 1 72

b/a G 02. 1 4 r.02 - 2-i - -.., ~ ~ ~ ~ 0 o... -*13".'~~~~-&.''. Z ——.. --- i -. ~, / /so -.91 / v'. /f, -.29" 3 - v O, -- /' -. ",.444 60/-" / / / I / // -1 I Z,,, 9 I f'Il -.6 0 1.3 15 1/7 1.9 /0 I gI II -/.7//._//.7 #fI! / / 8 "I I II/ If I / 1 11', i1 F1 I 3i I I, -.91 qII /' i, ss ~'r~ ~ Normalized Frequency FN 1.10 1.30 1.50 1.70 1.90 FIG. 3-3: APERTURE NORMALIZED CONDUCTANCE AND SUSCEPTANCE FOR Pr=1.0, erlO=10. O, - - - (B/Y), - (G/Y). 73

b/a G e r-s-. 0 - -.: —-- --. - -- - -, _ 4 444 W.3.1 15s 0 -. - i'..i -.6- -,X0 11' -..1............,0, ",,.' 5' I /,* -.31' /'/- # ~ ~</'/ \ ".7 I - - - // /' \ —- *9 /-/79,',';"- O,~ 01.11 SO. 944 I I" -1 lN m eFeey, III /s I st t7 F O gill ig -.95 If FOpr1. 1 30 1. 5 0 1. 70 1( g0

b/a G A9 8 0 6 444 ~~~~~~~~~~~~~^-. 02 - -.18" b/a 1.,-' ^-'"- -'': 5. P-5. - -,V -' -.98~ - -, - z -9 J. O' Ol", O r. r w - f I II I R, - I / i ill',fit =.50' 301'.-50 1.70 1.901 -- 75 i; 1.1 II_ |iI 1.10 111 +''/,,/''75

were evaluated by numerical calculation using Simpson's rule of integration. The program was devised at the beginning to calculate the coupling versus frequency (normalized frequency, FN = ka/7r). The parameters which were considered variable in this case were the dimensions of the aperture'a' and'b', separation of the slots from center to center, the permeability and permittivity of the material loading. The object program was stored in the memory of the IBM 360/67 in such a way that the input data required by the program were a, b, ur, Er, x1 (separation of center to center) and the range of frequencies over which the calculations are required. For example, if the frequency range is between a normalized frequency FN1 = 1.1 (which is just over the cutoff frequency of the guide), and FN2= 2.0 (which is the upper range of the frequency), then the program would call for repetitive calculations for the desired frequencies. The desired frequencies can be chosen with a uniform separation DFN (divisions of FN). The cutoff frequency corresponds to the normalized frequency of 1.0. For example, if it is desirable to make calculations at intervals of 0. 1, from 1. 1 to 2. 0 (normalized frequency), then DFN = 0. 1. Then the number of repetitive calculations = FN2-FN1/DFN = (2.0- 1.1)/0.1 = 9 and FN2-FN1 NFN= +1. 5 DFN where NFN is an integer, FN = FN1 + (NFN-1),DFN, where - is the Fortran multiplication symbol and NFN takes the values of 1 up to 10. The relation between the normalized frequency and frequency depends on the size of the guide. It also depends on the parameters of the material loading. 76

Since FN = ka/7r (3.3) where k = 2/X (3.4) and 2- Fo' f (3. 6) 0 x 2 0f Then f = GHz (3. 7) 2aV r" r The cutoff frequency 0. 3 f = — ~3 GHz,(3.8) c 2a, r r where a is taken to be in meters, and FN is the normalized frequency which is dimensionless. A scale can be obtained between FN and f for different values of waveguide parameters and material loading. Consider the case of dielectric loading where M.= 1, Er= 15, a = 0. 02286 meter and b = 0. 01016 meter (the dimensions of standard x-band waveguide). The cutoff frequency in this case from Eq. (3. 8) is 0.3 f -= ~3 = 1.6942 GHz. 2x 0. 02286 x i. x 15. In the following tables, correspondence between FN and f in the S-band range is shown for different parameters of material loading. 77

TABLE 3.1: r= 1, r=15, a = 0.02286m., b=0.01016 m. ka FN =- f (GHz) 1.0 1.694 1.1 1.864 1.2 2.033 1.3 2.202 1.4 2.374 1.5 2.541 1.6 2.711 1.7 2.880 1.8 3.050 1.9 3.219 2.0 3.388 2.1 3.558 2.2 3.727 2.3 3.897 TABLE 3.2: jr=l, r= 11, a=0. 02286m., b=0. 01016m. ka FN =- f (GHz) 1.0 1.978 1.1 2.176 1.2 2.374 1.3 2.572 1.4 2.770 1.5 2.968 1.6 3.165 1.7 3.363 1.8 3.561 1.9 3.759 2.0 3.957 TABLE 3. 3: r=l, Cr=7, a=0. 02286m., b=0. 01016 m. ka FN =- f (GHz) 7T 1.0 2.480 1.1 2. 728 1.2 2.976 1.3 3.224 1.4 3.472 1.5 3.720 1.6 3.968 1.7 4.216 78

TABLE 3. 4: /,r=l, Er=l, a=0. 07136m., b=0. 034036m. (S-band) ka FN= - f (GHz) 1.0 2.102 1.1 2.312 1.2 2.522 1.3 2.733 1.4 2.943 1.5 3. 153 1.6 3.363 1.7 3.573 1.8 3.784 1.9 3.994 2.0 4.204 TABLE3. 5a: /r=l, Er=l, a=0. 02286m., b=0. 01016m. (X-band) FN=ka f(GHz) 1.0 6.562 1.1 7.218 1.2 7.874 1.3 8.530 1.4 9.186 1. 5 9.843 1.6 10.499 1. 7 11. 155 1.8 11.811 1.9 12. 467 2.0 13. 123 TABLE 3.5b: ur=l, er=1, a=0.0157988, b=0. 0078994 (Ku-band) FN=ka f (GHz) 7r. 1 10. 444 1.2 11.393 1.3 12.343 1.4 13.292 1.5 14.242 1.6 15. 191 1.7 16.140 1.8 17.090 1.9 18.039 2.0 18.989 S-band (2. 6 - 3. 95 GHz); X-band (8.2 - 12.4 GHz); Ku-band (12. 4 - 18. 0 GHz) 79

Note that FN=1. 0 corresponds to the cutoff frequency and the Table 3. 6 indicates how the cutoff frequency depends upon the values of Er of the dielectric material for a fixed waveguide with transverse cross section of a=0. 02286, b=0. 01016 meters (standard X-band waveguide). TABLE 3. 6: wr = 1. 0, a=0. 02286m., b=0. 01016m. (X-band) er fc (GHz) 1.0 6.5617 2.0 4. 6398 3. 0 3.7884 4.0 3.2808 5.0 2.9345 6.0 2.6788 7.0 2.4801 8.0 2.3199 9. 0 2. 1877 10.0 2.0750 11.0 1.9784 12.0 1.8942 13.0 1.8199 14.0 1. 7537 15.0 1. 6942 16.0 1. 6404 Therefore the cutoff frequency for standard X-band waveguide filled with dielectric material versus Er is as shown in Fig. 3-6. The coupling was calculated without the iterative procedure of Section 2. 4. To compensate for the interaction between the two slots, especially if they are very close to each other, the iteration factor was included in the evaluation. On this basis the coupling was obtained. As expected the effect of the interaction becomes more noticeable if the two slots are very close to each other in wavelengths. Of course, the separation in wavelength depends on the separation of the guides in the physical layout. For a given physical separation it also depends on the operating frequency. For example, if the physical separation were kept constant and the operating frequency were swept higher, in effect, the separation 80

8.0 6.0 C O f c (G Hz) 4.0 2. 0..|11 13.15 Er FIG. 3-6: VARIATION OF CUTOFF FREQUENCY VS PERTTY OF LOADED XBAD WAVEGUIDE, a=O. 02286m., b=0. 01016 m.

in wavelengths of the two apertures increases. Also it was noticed that the value obtained with the iteration converges to the one obtained without iteration for large separation of the slot antennas in wavelengths. To formulate the coupling from an assumed illumination of the transmitting aperture and to evaluate the corresponding illumination of the receiving aperture consider the Poynting vector P P=ExH, P = Re (Ex H. av Therefore, the average received power in the receiving slot is: b a Pav=Re f (Usiny x) x (- - U'sin -Y) dydx Sav I a CL a b a =-Re r f lu2 sin2 7 dydx, where the negative sign indicates that the power flow is in the negative direction of the z-axiL shown in Fig. 2-1. ab U12 av 2 wu where f3=2r 2 Similarly, the transmitted power can be obtained. 2IU2 C (coupling in dB) = 10 log10 - =10 log10 amplitude 2 T amplitude of TR aperture ilum. I glo0 amplitude of TR aperture illum 82

For the calculations the amplitude of the aperture illumination was assumed to be 1+R. Several curves of coupling versus frequency were evaluated for both empty and loaded operation of the waveguide cavities. Of course, it should be noted here that this is the coupling from aperture to aperture; the effect of the backing cavities is not taken into consideration at this point but will be later on. 3.3.2 Empty Waveguide Case For the case of empty waveguide the effect of the cavity is less pronounced than in the case of loaded cavity. This is because in the case of the empty cavity the bandwidth is wider than in the case of the dielectric loaded cavity. The bandwidth for a low loss material is approximately equal to the loss tangent. The graphs obtained for empty or air-filled cases show very good agreement to the order of a fraction of dB with the theoretical counterpart as in Lyon et al 2. Also it is within approximately 1 dB of the experimental results as will be seen in Chapter IV. But for the case of the loaded slot, the effect of the cavity has to be added and then the comparison can be made. The coupling versus normalized frequency for the case of empty S-band waveguide of dimensions a=0. 07136 m., b=0. 034036 m., is shown in Fig. 3-7. The separation of the two slots (center-to-center) is 0. 075 m., and the range of variation of frequency is from 1. 1 to 2. 0 on a normalized scale. The relation between the normalized frequency and frequency in GHz is given in Table 3. 4. In Fig. 3-8 the coupling versus frequency is shown for the case of the empty X-band waveguide of dimensions a=0. 02286 m., b=0. 01016m., for a separation of 0. 114m., and a range of frequency between 1. 3 and 1. 9 normalized frequency (see Table 3. 5 for corresponding real frequencies). Notice in both cases that the coupling pattern drops with an increase of frequency according to a 6 dB/ octave frequency characteristic. Also, the differences between curves, with and without interaction between the two slots, decreases as the separation between the two slots increases (see Appendix B ). 83

-15 -16 -17 -18 v, -19 I -20 U -21 -22 -23 1.1 1.2 1.3 1.4 i5 16. 7 1.8 1.9 2.0 FN FIG. 3-7: STANDARD S-BAND COUPLING VS NORMALIZED FREQUENCY, a=0. 07136 m., b=0. 034036 m., xl= 0. 075 m.,,r=l. 0, er= 1. 0

-28 -29 -30 | -31 Q -33 -34 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN FIG. 3-8: STANDARD X-BAND COUPLING VS NORMALIZED FREQUENCY, a=0. 02286m., b=0. 01016m., xl=0. 114m., Ar=1. 0, er= 1. 0.

Then the coupling for a pair of Ku-band slots spaced at 0. 114 m. from each other was studied. The coupling was calculated at different normalized frequencies(see Fig. 3-9). The results agree with Fig. 4-6(b) in the chapter on experimental investigations. 3. 3. 3 Dielectric Loaded Case Several graphical curves have been evaluated for standard X-band waveguide with dimensions a=0. 02286 m., b=0. 01016m., filled with dielectric material of er= 2, 3, 4, 5, 7, 11, 15 and 20 (see Fig. 3-10, a-h ). Notice that x1 = 0. 013m., corresponds to two slots put next to each other or metal-to-metal contact. It is very interesting to note that the coupling in this case does not follow the 6 dB/octave of frequency trend as in the case of empty slots. The variation depends on the material loading parameters. It is to be noticed that the lower the permittivity number, the more the variation is restricted; in some cases it is about constant, as in Fig. 3-9. For the higher electric permittivity the curve goes to a minimum with increasing frequency and then rises. For a large portion of the frequency band the variation is confined to about 2 dB. Also since the coupling level is low in this case the iteration process has little effect (see Appendix B). Of course, it should be remembered that this coupling is aperture-to-aperture and does not take into consideration the resonance cavity which is feeding it. 3. 4 Coupling Aperture to Aperture versus Spacing 3.. 4. 1 General Discussion The computer program of Section 3. 3. 1 was next modified to permit variation in frequency for a fixed spacing and to obtain the corresponding coupling level. Also for a fixed frequency the program provides for variation in spacing between the slots and then the evaluation of the corresponding coupling level. Therefore, upon supplying the proper data, the computer will evaluate coupling versus frequency or coupling versus distance according to need. 86

-34 -35 co' -37 0 -38 -39 1. 3 L:4 11.5 1.6 1.7 1. 8 1.9 2.0 2. FN FIG. 3-9: STANDARD Ku-BAND COUPLING, xl=0. 114m., VS FREQUENCY, a=0. 015988m., b=O. 0078994m.,,ur=1.0, Cr=1. 0.

-11 -12 u -15 1.1 1.'2 1.3 1 4 15 1:.6.7 1.8 19 2'0 FN1 FIG. 3-10(a): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS a=0. 02286m., b=0. 01016m., x1=0. 013m.,r=l. 0, Er=2. 0.

68 Aperture Coupling (dB) 00', 0' C.. oh O tO 00i - 3 X, vo 0 P U) o P I m o \ tl I 1 r d

-17 -18 -19'. -20 bJ -21 0 -22 ~ -23 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN FIG. 3-10(c): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS a=0. 02286m., b=0. 0 -016m., x1=0. 013m., ir=1. 0, er=4. 0.

-18 -19 -20 ^ g> -21 * < 4 o -22 0 -23. -24. -'1 ~............. __ L 1. _ ~,,;, ___ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN FIG. 3-10(d): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS a=0. 02286m., b=0. 01016m., x1=0. 013m., ur^=1. 0, er=5. 0.

-19 -20 -21 P -22 X\ =fl -23 -25 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 17 1. 8 1. 9 2. o 2 1 FN FIG. 3-10(e): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS a=0.02286m., b=0.01016m., xl=0. 013m.,,r=l, er=7.

-19 -20 -21 -22 0 -23 -25 -26 -27 -28 1. 1 1.2 1.3 1:4.5 1.6 17 1.8 9 2.0 2.1 2.2 F FIG. 3-10(f): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS a=0.02286m., b=0.01016m., x1=0.013m., r=l, Er=11.

Aperture Coupling (dB) CD NC 10 I IS N M I3! N I0 CD o0 J C) Cr1,4 X IN 1- 0 CD CO 6 t - 0 0 l_3 - A M ~ W O C 0 0 oo t / M r /

-1 -20o -21 \ -22 -23 E -24..g -25.C? \ 0 o -26 -27- -28 -29 4 -30 -31; 1.1 ^+2 its it f'5 rt I^T I^S 1*^ 1^ t _4 i6 N 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2:2 2.3 2:4 2.5 FN FIG. 3-10(h): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS a=0.02286m., b=0.01016m.,x=0.013m.,.=l, er=20.

The increment in spacing for calculations was taken to be smaller in the range from 0. IX up to 2X, namely every 0. 1X to show in more detail the effect of interaction between the slots which is more pronounced for closer spacings. From a spacing of 2X up to 10X it was varied in steps of 1X. Also the modified program included the program of Section 3. 2 as a subroutine which could be called upon to evaluate the aperture conductance and susceptance; also the aperture reflection coefficient can be obtained as shown in the examples of the computer output given in Appendix B. 3.4.2 Graphs 3. 4. 2(a): Figure 3-11 which is drawn on semi-logarithmic paper represents the coupling in dB versus spacing in wavelength. (see also Appendix B ). As has been mentioned before the effect of the interaction between the two slots is noticeable for closer spacings. For slots further apart in wavelength the differences due to interaction is very small. This curve agrees within a fraction of a dB with Fig. 4-4 of Lyon et al2. 3. 4. 2(b): For the loaded case of aperture-to-aperture coupling it is shown in Fig. 3-12 that the approximation of 6 dB/octave of spacing is quite good Notice in this case spacings can be smaller than 0. 4X as in Fig. 3-11; for this case the parameters of the guide are a=0. 02286m., b=0. 01016m., and it is filled with dielectric material with relative permittivity of 11. 0. The operating frequency in this case is S-band; therefore it was possible to go down to 0. 1 X0, where Xo is in the free-space wavelength (Fig. 3-12). 3. 5 Evaluation of the Cavity Effect Due to the loaded cavities which are backing both transmitter and receiver slots, as shown in Fig. 4-2 of Chapter IV, the resonance and anti-resonance points become very distinct. These waveguides can be considered as a five-sided cavity as was explained in Chapter II. An approximate way for rapid calculations of the resonance and antiresonance points can be made according to the analysis in Chapter II. 96

For resonance points: Q 2 =n-g (n= 1,2,3,... ) where Q is as defined before (length of the cavity) and X is the waveguide wavelength. For anti-resonance: x I = n-, (n= 1,2,3,... ). A formula has been developed to predict the resonant frequency; likewise another one for anti-resonance could be developed. For resonance: x, Q=n 2 but O__ 1 3 x 1010 X P i/UrTTU'' 2 41-( ) c 3x 10 1 n- =n 2 ^ Finally 2 fres!L54n + f2 GHz (n=l, 2, 3...) (3.9) res. J — r j where Q is in meters, 1 0. 3 0.15 f _1 0.3 0.15 GHz, (3.10) 2a4T' 2a 7! a Prcr where a is in meters. Therefore 0.15 n2 1 f 0.15 In2 + 1 (3.11) res. 2 ^ a2 99

It is to be noted in the empty or unloaded cavity,r=1 and er=l. Therefore Eq. (3.11) becomes f = 0.15 +- (n = 12,3,...) (3.12) res.'2 2 a2 a By comparing (3. 11) and (3. 12) note that the resonance frequency is determined by the broadside length of the aperture'a' and the depth of the cavity. It also depends on the parameters of the material loading. Another way to find the resonance of the cavity and at the same time be able to predict the bandwidth is by means of Eq. (2. 79) in Chapter II which is: Z(0) = Zg tanh (alg +j ) or Y(O) = coth (1 a+ j g), and = 1 i ~osh2a lg cos2 31g +sinh a Slg sin 2lg] Y2(0)1 __lg =g Fg Ig' j ~ I 2 2 i 2.-2 -- 2 — Z1 2 sinh a S cos 31 +cosh a1 g sin 1 Ig| ig 1Ig lg Ig9 This last equation is proportional to the power received. A computer program has been written for this equation to evaluate IY(O)2 versus frequency. The output of the computer is best presented by means of the plots shown in Figs. 3-13 to 3-16. These graphs were plotted directly by means of the University of Michigan Digital Plotting System. An example for calculating the resonant frequency corresponding to the half-wavelength method is given below. Example 1 For a = 0. 02286m., b=0. 01016m., -= 0. 031m., Er= 7. 0, = 1. 0, f = 2. 48008 GHz. c 100

For n = 1 0.15 1 fres. + = 3. 08163 GHz. I1res./2 ~ a2 For n= 2 f2 res = 4. 41935 GHz. Example 2 For a=0, 02286 m., b=0. 01016m., e=0. 031m.,,r= 1. 0, er = 11. 0 f = 1.97842. C For n = 1 fl res. = 2. 4583 GHz. For n= 2 f = 3. 5254 GHz. 2 res. Now consider the perturbation formulas applying to this case. For the metal probe and Lr=l, Er= 11.0, diameter = 0. 1143 cm. (metal probe) projected length inside the guide = 0. 47498 cm. d = 2. 48792 cm d2= 0. 60706 cm d = 3. 09498 cm. Here dl, d2 and d are as shown in Fig. 2-11 of Chapter II. Then W-wo 2V r 2 2rd1 2 1 -- -+- Icos - cos2 030 abd k d 2 s Now take f = 2. 4583 then 2 x f ten k k = 3x010 E = 17. 0674876. 101

Now 2 V = 7r (da) x length = 0. 0060943 cm3 and 2V 2x 0. 0061 2 2 2 abd 2. 286x1. 016x3.10 o2(29 39)- 072286)2 os44 (17. 07 x 2. 286)2 = 0. 0017 [sin2(19. 39)-0. 00645 sin2(54. 69] = 0. 0017E0. 33199)2-0. 00645 x (0. 816)2] = 0. 00018 (this is very small). For the material filled hole or cylinder around the probe consider it to be filled with polystyrene having a dielectric constant of 2. 54 and at the range of frequencies considered. W)-o 2V Ae 2 s id "^ ax _sin -1 abd e d'Jo diameter of the cylinder = 0. 55118 x 102 m. length of the hole = 1.016 x 10-2 m. In the above A e is negative or positive corresponding to changing to a lower or higher dielectric constant respectively. The volume is easily found to be V da. =0.2423 cm3. Also 2V -- = 0.0675. abd Also A _ 2. 54 - 11 -=ks - 11 = -0. 7691. e 11 Therefore W-W = -0. 0675 -0. 7691 x cos2 (54. 69) = 0.0173435'o The total perturbation = 0. 0175 or 1. 75 percent. 102

Therefore fl should be = 2. 4583 (1+0. 0175)= 2. 501 GHz. For the other peak the material perturbation is the same but the probe perturbation changes because of the factor k in the formula. The metal probe perturbation at 3. 5254 GHz is now found: 0. 00017 x. 33199)2 9. 8596 x (0. 816)2'o L(24. 48)2 x2.286 Then for 3. 5254 GHz the total perturbations are 0. 017522. This is not much different from that at 2. 4583 GHz. Therefore, f2 = 3. 5254 (1+0. 017522) = 3. 587 GHz. Notice that these values are very close to the experimental measurements (see Fig. 4-7 in Ch. IV). There is good agreement considering the errors in measuring the dimensions and the experimental errors of instrumentation. Also, there might be some variation due to a slight change in er with a change of frequency. Example 3 For a = 0. 02286m., b=0. 01016m., I = 0. 031 m., ur = 1. 0, r = 15. 0 fc = 1. 6942 GHz (cutoff frequency). Using Eq. (3. 11) the resonance frequencies are: For n = 1 f = 2. 105 GHz 1 res. For n = 2 fre = 3. 01899 GHz 2 res. 103

Example No. 4(a) For a = 0. 02286 m., b= 0. 01016 m., I = 0.1015 m., jur=l. 0, er=15.0 the following corresponding resonant frequencies using Eq. (3. 11) are: n = 1 f= 1. 7368 n = 2 f2 = 1.8583 n = 3 f3 = 2.0448 n = 4 f4 = 2.2804 n = 5 f5 =2.5516 n = 6 f6 = 2.8482 n = 7 f7 = 3.1631 n = 8 f8= 3.4913 n=9 f9=3.8293 n = 10 f0 = 4. 1750 Example No. 4(b) For a = 0. 02286m., b=0. 01016m., 1=0. 1015m., lp=1.0, r=11. 0, the resonance frequencies using (3. 11) are n = 1 f1 = 2.027 n = 2 f2 = 2.169 n = 3 f3 = 2.388 n = 4 f4 = 2.663 n = 5 f5 = 2.979 n = 6 f6= 3.325 n = 7 f7 = 3. 694 n = 8 f8 = 4. 077 Example No. 4(c) For a = 0. 02286m., b=0. 01016m., Q=0. 1015m., p,=1. 0, Er= 7. 0, the resonance frequencies using (3.11) are: n = 1 fl = 2.54 n = 2 f2 = 2. 72 n = 3 f3 = 2.99 n = 4 f4 = 3.338 n = 5 f5= 3. 735 Note that these resonance frequencies agree with the digital plots shown. 104

Co 0o 0 3.08 4.418'2 48 2.98 3.47 3.97 4.46'4.96 5. 46 5.95 Frequency (GHz) 0o Co a=0. 02286m d= cavity length, 0. 031m. n=l n=2 Ar=1 0 0. I I E =7.0 o C1 I - ka/yr) So bD 0'.4 0 \J Normalized Frequency (FN = ka/7r) ~- -- t. ~:...j m. I I. J I, ~' — \ —— I ——.'1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 FIG. 3-13(a): RESONANCE OF THE CAVITY (Digital Plot).

or ) 2.46 3.53 o S. 4.. I u1[98 2.3 2.77 3.17 B.56 3.96 4.35 4.75 Frequency (GHz) o n=1 n=2 a= 0. 02286m. d= 0. 031m.? - - ) 1 A tr = 1 - - er=11 0 o0 0 0 do. 0 CT IsNormalized Frequency (FN = ka /r) 0,I 4.'1o0 1.20 1.40 1.60 1.80 2.00 2.20 2.40 FIG 3-13(b): RESONANCE OF THE CAVITY (Digital Plot). FIG 3-13(b)- RESONANCE OF THE CAVITY (Digital Plot)

0o o 2.105 3.02._ _ 4 - -. — _t. -4 +- - _ — - - ___ + - "I 69 2.03 2.37 2.71 3.05 3.39 3.73 4.07 Frequency (GHz) Co so=^~~~~~~~~~~0.~ ~~a=0. 02286m. d=0. 031m. /r=l. 0 e=15.0 o n=1 n=2 cq ~ 0o I \ 0 L) ~; L + Normalized Frequency (FN=ka/7r) 1.00 1.20 1.- - 1.60 1.80 2.00 2.20 2.40 00FIG. 3-13(c) RESONANCE OF THE CAVITY (Digital P 2.20 2. FIG. 3-13(c): RESONANCE OF THE CAVITY (Digital Plot).

C3 C) C — 4 - ---- ---- --- - ----- - -— 4 — ---- ------ ------ ---- --- 2 148 2.98 3.47 3.97 4.46 4.96 5.46 5.95 Frequency (GHz) -- n=3 n=4 ~C> 8~~~~n=5 n =6 n=7,+^"~~~~~'|~~~ | + | at a=0. 02286m. d=0. 1015m. =r 1. 0 r~~~~~~~~~~~~,~ ~~-l — ~7Er=7.O 1 00 1.20 1.40 1.60 1.80 0.00 0 C) INormalized Frequency (FN = ka/7-) 11.00 1.20 1.10 1.60 1.80 2.00 2.20 2.140 FIG 3-14(a): RESONANCE OF THE CAVITY (Digital Plot).

C> ", 98 2.37 2.77 3.17 3.56 3.96 4.35 4.75 Frequency (GHz).. n=3 n=4 n=5 n6 n7 o n=6 C1 n=7 o | l l | a=0. 02286m. d=O. 1015m. /r= 1. 0 er =11.0 0I co bJ 0 C0 o Normalized Frequency (FN = ka/7)'1.00 1.0 1. 40 1.60 1.80 2.00 2.20 2.40 FIG. 3-14(b): RESONANCE OF THE CAVITY (Digital Plot).

o Co CD 0 ----- t - ---- --- -4-. - -4 ----- -4 —- ------ + - 1 69 2.03 2.37 2.71 3.05 3.39 3.73 4.07 Frequency (GHz) o n=3 n=4 n=5 o11=3 114 1 n15 n=6 n=7 n=8 a=0. 02286m. d=0. 1015m. C' Ad II Ad I Ic II II Er=15. 0 CD 1.0 1.20 1.40 1.60 1.80 2.00 2.20 2.40 0 Co Co C' CI 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 FIG. 3-14(c): RESONANCE OF THE CAVITY (Digital Plot).

c) o'2 0.25 2i0.32 2i0.39 210.46 210.52 210.59 210.66 210.73 Frequency (GHz) f x 102 ~ | \ a=0. 02286m. X / \ d=0. 031m. CM(vj"N~~~~~~~~ / \ Mr=1.0 r= 15. -0 ~/ \ n=l c0 C o 0 —\ o\ g / Normalized Frequency FN x 102 124.10 124.14 124.18 124.22 124.26 124.30 124.34 124.38 FIG. 3-15(a): AN EXPANDED DIGITAL PLOT FOR THE RESONANCE PEAKS.

cl 0 ^ —---- 1 —-— + —-------— ~ —---------- I -+ --— 4 —---— + —----— 4 —---- -— + — 3)1.57 301.66 301.74 301.82 301.91 301.99 302.08 302.15 Frequency (GHz) f x 102 0 / \ a=0. 0\- a=O. 02286m. d=0.031m.../ \ r= 1.0 r=15.0 o / n=2 c oo: 0 C1. 1 0 / Co! Normalized Frequency FN x 102 - -- ---- - ------- - --— I — I 178.00 178.05 178.10 178.15 178.20 178.25 178.30 178.35 FIG. 3-15(b): AN EXPANDED DIGITAL PLOT FOR THE RESONANCE PEAKS.

C) C) - i- -— + — -— + —F-+- -+ — - -— 4 —----.. —— +- - --- - -4 -- - 1q4. 15 204.24 204.32 204.4 1 204.49 2 201.58 201.66 204.75 Frequency (GHz) f x10 o Co a=0. 02286m. b=0. 0106m. d=0. 1015m. o /r \ =1.0 -- ~ / ~\ Er=15. 0 n=3 bD CM o / \ 0 \ C) CMU o Normalized Frequency (FN= ka/T7) - ___________FN x 102'120.50 120.55 120.60 120.65 120.70 120.75 120.80 120.85 FIG. 3-16(a): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

C) Co??7.70 227.77 227.84 227.91 227.97 228.04 228.11 228.18 Frequency (GHz) f x 102 o a=0. 02286m.."' b=0.0106m. d=0. 1015m. r/=1. 0 r =15.0 n=4 o s\ o.3 C). / Normalized Frequency (FN=ka/7r) =~irI ~ I I _+ + _ +FN x 102 cu. - i —----- 4 --'4-I ---- -—. - --.... —- 4 -. -------- 134.40 134.44 134.48 134.52 134.56 134.60 134.64 134.68 FIG. 3-16(b): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

0o Co i4h. 81 254.89 254.98 255.06 2.15 255.23 255.32 255. 40 Frequency (GHz) fxl02 0C I~U~~~~~~~~~~t,"~ \ ~a=0. 02286m. b=0. 0106m. d=0. 1015m. / \r=1.0 C/ \ E=515. 0 _ i-/ \ n5 i'r 0 \ ) O o Normalized Frequency (FN= ka/ir) ^/O~~~~ ~~~~FN x 10 (1J 1 4 - 4_ | __ — 4 4 A 4 150.40 150. 45) 150.50 150.55 150.60 150.65 150.70 150.75 FIG. 3-16(c): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

C) 0 - - - ---------- ----- --- -- -------- - --- --- "2 4.16 284.54 284.63 284.71 284.80 284.88 284.97 285.05 Frequency (GHz) f x 102 a=0.02286m. o b=0.0106m. d=0. 1015m. 1.0 Er 15.0 C)^~ / \ ~~~~n=6 Cn~~~~~C 0 C I 0) CD C) 0 \ 0 /Normalized Frequency C0 /FN x 102'167.90 167.95 168.00 168.05 168.10 168.15 168.20 168.25 FIG. 3-16(d): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

0 -— t —--- ---- 4 -4 i-t — --- -4 4 4 —— t- -4 4 4 —C -~-. — 5.97 316.04 316.11 316.17 316.24 316.31 316.38 316.I Frequency (GHz) f x 102 0 0 a=0. 02286m. b=0. 0106m. CDP/~~ \ d=0. 1015m. I \ er=15.0 Cq ~~~~~~~~~~~~~~~~~~n=7 ~~~~ br" 0 U, CL ~~~~~~~~~~~Normaliz-,al Frequency FN x 102 M- / V fi~~ —t- /-t- \-f —-3- -~ —------ ~- ---— t- -t —— t- o Normalized Frequency FN x io2'leI.50 1854 185.58 186.62 186.66 186.70 186.74 18.78 FIG. 3-16(e): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

0 -. -+ — f —— _ —-f — — I+- ----- --- --, — —, 318.814 348.91 348.97 349.04 349.11 344918 349.24 349.31 Frequency (GHz) fx 102 Co a=0. 02286m. 0~"~~~~~~~~~~~~~~~ ~~b=0. 0106m. d=0. 1015m. A=1.0 r ^' — ^E 15. 0 o Cr^l5*O ~ol~~~~~ X \ ~~~~~~~~n=8 00 ~ 0 bfD 0 O ~~~~~~~~~Normalized Frequency FNx io2 a*~ ~ ~~~~~~~~~ -- / 4 ~I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ " 1 "/ Co 0 / ~~~~~~~Normalized Frequency FNx102'205.90 205.9'4 205.98 206.02 206.06 206.10 206.1m 206.18 FIG. 3-16(t): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

CD c::. c- 1. I — A —--— f- - -— 4 — -— + —--- — +t — — + —— f- ----- I I-I —----— ~~ —--- 3 2.72 382.79 382.86 382.93 382.99 383.06 383.13 383.20 Frequency (GHz) f x 102 C1 o) a=0. 02286m. b=0. 0106m. d=0. 1015m. r = 1. 0 o E =15.0 0 r n=9 CTI 0 0 0 C) U-. 0\ CA 09 0 Normalized Frequency FN x 102 -4 i 4 —---- -+ -t —~ —-- — 4- - 1 - -- - -4- - - - f- - 4 -4 --'225.90 225.94 225.98 226.02 226.06 2S6.10 226.14 226.18 FIG. 3-16(g): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

Example No. 5. For a= 5t, b =2", x = 3/2 ", r = 6. 63 and ~r = 12. 6, fc = 129.2246 MHz. Using Eq. (3. 11) the resonance frequencies are: n = 1 fl = 449.715114 MHz n = 2 f2 = 871.135869 MHz n = 3 f3= 1.298692 GHz 3. 6 Evaluation of Coupling From Sections 3. 3 and 3. 5 the coupling of two loaded guides opening into a perfectly conducting infinite ground plane can be obtained. From Eq. (2. 99) in Chapter I which is: 2 2 2 2 2 1 [cosh a, 9I cos -1 I+sinh ag Isin 3gQ y(O)| ig lgIg g Z sinh 2 lg2 cos 2l +cosh 2a sin I lg J 1Zlg 12 Lsmh ~lg lg 9l " J the power received is proportional to this factor. Therefore by multiplication of this equation by an appropriate factor and adding to the coupling from aperture to aperture in dB the total coupling can be obtained. Also the resonance frequency and the theoretical bandwidth can be obtained. The effect on the resonance frequencies due to the material perturbation and metal probe perturbation might be taken into consideration as in Example No. 2 120

Chapter IV EXPERIMENTAL INVESTIGATION 4. 1 General Discussion All the experimental work was performed in an anechoic chamber. The chamber dimensions were 50' x 30' x 15', with an aluminum ground plane, 12' square by 1/8" thick mounted in the center of one of the 30' walls (see Fig. 4-1). The nine individual sections of the ground plane were joined by 2" wide aluminum tape to form an electrically continuous surface at microwave frequencies. The center section has a removable 2'x3' section for the test antennas. Within the 2'x3' part there was a removable circular disk of 14" diameter. Inside the disk there was a rectangular hole of dimensions 10" x 3" in which the slots could be mounted. Also, there was an additional fixture to make the X-band slots to fit in the mounting as shown in Fig. 4-2. The ceiling and the floor were covered with B. F. Goodrich HV-4, vinyl-covered, four-inch hairflex microwave absorber. The rear 30' walls and the 50' walls are covered with VHP-18, pyramidalabsorber. The center of the rear wall was covered with VHP-26 absorber. The chamber had been lined with interlocking aluminum foil sheets to provide a known uniform termination for the absorbing material and to shield the room from external signals. The frequency range of the chamber is 500 MHz - 50 GHz and has a nominal reflection coefficient, for normal incidence, of -70 dB for X-band and higher frequencies,tapering off exponentially to -40 dB for lower frequencies. The accuracy of coupling measurements was approximately within the range of + 1 dB (i. e., for the swept frequency use). For the case where the swept frequency generator was not used, both transmitter and receiver were tuned by means of a double-stub tuner, and the accuracy was a little less than the former (i. e., within + 2 dB ). 121

"~~~~~~~~~~&:"~~~:~j*:@* ~~ ~s-~z:.. 424K''K'4"'~~~~~~~~~~~~~~~~~' ~;:s''4444~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~''::=: K" 444....~~~~~~~~,, ~~~~,**,~~~94//'*~~~~~'~4w 94/2 ~ ~::: K> 42'> 4K'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'d~":; 9 /'' K #t K+ 4' A 4~~~~~~~~~~~~~~~~~~~~~~O~:LSi./> K'.7 {~~~~~~~~~~~~~~~~~~~//4 t4 /4 7 4~~~~~~~~~~~~~~~~~ 4~~~~~~44444K' >4>A4'2 44..,.~~~~~~~~~~~~~~~~~~~~~~;;;~:::-:::: >494444""~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:j~-:::::::: ~~~~~~~~~~~~ 4<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: KK'4K44K"K'K'K'KKKKK>KK'K44'K'%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~h:::-~i;"-:::'#''4K4K4 B~~~~~~~~~~~444>7 122~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'_'::::::

T:EST ANTENNA MOUT. (Comparison soale is in inhes. ) 123

-;~-~~~- sj:i S~xZk: iili- RsXB —LJ i O':iae:.1,-:~~~ e:~i~a:-z:::ni:Ds9i5li~~:ij ac.::~a,:;~-,:j~:~C~~IOE:$gi:.: —~:- ~l p:_,aIi-::::::::r:::: ij::::::-:::::::-~ Bi:::::~:-:-:::::::: -x:m~~~w,;i~:~:.: a::s:al:i:t-:'::::::::::::::::::i-_:ai~fRPil4-I: —''ifi\i::-::~;:'::::::i~::: —_i I; —:::::: d-i B)i,:'-di-iP-r::::-:-::::::: ::-::-::-: —-:-::;,:-r;:_-i rl: I 9:i-:r:::::i:-:;:-:::-:::I:::-::::::-:rl: —:: —:: I:::: _:'ia: —: I:i::_:::::-::;:.;:j-~::::;-:- -:-:-.:;::-:_:_-:;::::i:::::: ::ii:-:l:b bi_;i:i;i:::i-i _i-:-::-:::,.:::::::-:.: -:-i:-.-:: B1 "-ia;isk-~:;bi b::: a,,,,,i-,,, i~ "1ID I lii a x ". ~c — C:i:3 n:j~:s~,;1~""j'"' r-r.;:,:::::-a?: ~:r iBE-" is i~~::~~ ~:i~~ Z D -~::: -:-,,,,.~/i:-.:.:...j,,,,,,,,,,, F]EQ. 42(1: AgIhbNGEMENTS Fgt LePDED XHND IAVE6ES. (Csunprison nc3le is i irnes. ) %24

."VIG 42(c- RRNGEMENTS FOR STANDA~RD S-~BAND'WAVEGUIDES. oparison scale sinchesb 125

4. 2 Coupling Measurements 4. 2. 1 Swept Frequency Most of the experiments in this section were made in the S-band frequency range, or, more specifically, the measurements were taken between 2. 0 and 4. 0 GHz. Also some experiments were carried out for swept frequencies in the X-band range of 8. 2 to 12. 4 GHz. For the X-band measurements, it was found that in feeding a loaded guide with X-band signal there is clear indication that there is more than the dominant mode propagating. This was observed from the shape of the radiation patterns. This was to be expected since the X-band range of frequencies is far above the cutoff frequency of the loaded guide. At an early stage of the experimental work a dielectric powder was used to load the X-band waveguide. It was then realized that it was very sensitive to any movements. By moving the transmitting antenna from one position to another, it seemed that the granules of the dielectric filling moved around. For this reason it was difficult to get a repeatibility in the graphs. A substitute to avoid this problem was a solid dielectric material which is machined to fit the size of the X-band waveguide. Emerson and Cuming's Stycast High K solid dielectric was used. The manufacturer gives a loss tangent for this material of 0.0008. Different values of the dielectric constant for the material were used such as K' = 5, 7, 11 and 15. The material was machined to fit the cavity and also a hole was drilled for the probe of the feed as shown in Fig. 4-3(a). Two different waveguide lengths were used, d=0. 031m and d=0. 1015m as in Fig. 4-3(b) to demonstrate the effect of the cavity length on the resonant frequencies. Also some measurements were performed for the cases of standard waveguides at Ku-, X- and S-band frequency ranges. It is interesting to note that the coupling obtained experimentally for these cases is within a fraction of one dB from the corresponding coupling obtained from the analytical formulations; 126

~i:~::s-g,'~~ _ nN:i:5":'"-:-;pi'dl;~Q -i~ s:::::,::;,: -3:s9i: ~~ —- s;c;;aa=I*"L~-ln-s- —-L:~531 %pi.:.a-~;-~a.~i;~-.';i Izi:ljI-i —~i'; 4!:"i; " U " iw i a -~. a"p x=: Ff i:i: 5- -"" i Y -Y9:s -": n ~ L h :*s ~z I a gL k "*B I " D """) Rls-b FH, 4 —3ga: TgE E8jbCE 9 QnN P$8 taEECTXG FHLls 27

128

A ~~&A,,.,,,,,t' ~ 7,,7<,.,,,., K ~ ~ 4947. ~~4 /~~r'4747'~ ~ ~ ~ ~~~~~~~~~~~~~~~~c 1<4*"'r "7,~

compare Figs. (4-4, 3-7), (4-5a, 3-8) and (4-5a, 3-9). Figure 4-4 is the coupling versus swept frequency between 2. 00 GHz and 4. 00 GHz for a pair of S-band waveguides spaced 0. 075m apart, center-to-center. The upper curve of the graph is a measure of the power level from the transmitter obtained by connecting the two coaxial lines; in a way it is a measure of power level. The lower part is the coupling level with the two apertures flush mounted 0. 075m apart and in the ground plane of the anechoic chamber. Notice that the coupling follows approximately the 6 dB/octave trend as frequency is varied. Also when the coupling was taken versus spacing for a fixed frequency it showed 6dB/octave of distance confirning the analytical results. Figure 4-6 shows the coupling versus frequency for an X-band waveguide cavity of length 0. 031m loaded with Stycast High K solid dielectric material machined to fit. In this particular case the material dielectric constant was 11. 0. The level curve represents the level of the transmitted power through the coaxial lines from the transmitter to the receiver, as indicated in the experimental set-up shown in Fig. 4-14, with a 25 dB attenuator added to the circuit. Note that the coupling resonant frequencies are approximately the same as those predicted in the analytical work. The small error may be due to the measurements of the guide length which might not be very exact. Also, errors in the measurement equipment may exist. It is believed that the split at the top of the resonance is due to the fact that the transmitter and receiver cavities do not have exactly the same length. Therefore, these two peaks represent the resonant frequency for each individual cavity. For that reason plus the occurrence of multiple reflections inside the cavities, the 3 dB or half-power bandwidth is wider than was expected theoretically. Figures 4-7 and 4-8 are the coupling and gain respectively versus swept frequency for a cavity of loaded X-band waveguide with dielectric constant of 15. 0 and the dimensions of the cavity 0. 02286 x 0. 01016 x 0. 1015 meters. The gain curve was taken with the set-up for far-field pattern measurements as shown in Fig. 4-9. The standard antenna which was used on the boom was 130

Coaxial power level or 0 dB level -10 _ -20 Frequency (GHz) I,....* — I _, 9I., A, A1 a _ 2.5 3.0 3.5 4.0 FIG. 4-4: STANDARD S-BAND WAVEGUIDE COUPLING VS FREQUENCY. x1=0. 075m. 131

-25 -30 0 -35 _________..__________ 8 9 10 11 12 Frequency (GHz) (a) -30 dB (Calibration) *-30 ~-35 13 14 15 16 17 18 Frequency (GHz) (b) FIG. 4-5: E-PLANE COUPLING VS FREQUENCY FOR TWO SLOTS SPACED 11.4 CM. (a) Standard X-band Rectangular Waveguide. (b) Standard Ku-band Rectangular Waveguide. 132

2.0 2.5 3.0 3.5 4.0 Frequency (GHz) n=2 -25 dB level -25 -35 n=1 bIf 0 -45 -55 FIG. 4-6: COUPLING VS FREQUENCY FOR LOADED X-BAND WAVEGUIDES, Ir=1.0, er= ll., d=0. 031m., x 1=0. 03m, a=0. 02286m., b=0. 01016m. 133

2.5 3.0 3.5 4.0 -15.... | Frequency (GHz) 9 n=8 t=7 n=6 -25 t S -35 bn 0 -45 FIG 4-7: COUPLING VS FREQUENCY FOR LOADED X-BAND WAVEGUIDES, /,r=l. 0, Er=15. 0, d=0.1015m., xl=0. 013m., a=0. 02286m., b=0. 01016m. 134

2.0 2.5 3.0 3.5 4.0,,,.. -, I - -- I. -- I Frequency (GHz) Supply power level through coaxial lines -35 3r=. r=5. 0- d=. 1015 m. ^~~~~~~~3

FIG 4-9CHAMBERLD PATTEN MEASUREMENTS. 136

an AEL No. ARN101A (1.0-11.0 GHz) log-periodic type. The boom was fixed in its position and the frequency was swept between 2. 00 - 4. 00 GHz. Note that the coupling resonances approximately coincide with those resonances of the gain of the antenna. Also, the dynamic variation of the coupling without matching is about 35 dB and the peaks are lower as the frequency approaches the cut-off frequency which is due to the mismatch at the feed. Similarly the coupling and gain graphs versus swept frequency were obtained for dielectric constants of 11. 0 and 7. 0 which are shown in Figs. 4-10 - 4-13. 4. 2. 2 Coupling Versus Frequency with Point by Point Tuning There was no simple way to match the input port of the transmitting antenna and output port of the receiving antenna with swept frequency. A circuit had been constructed to take measurements while the frequency is varied point by point; a double stub tuner was used at both the transmitting and receiving ends as shown in Fig. 4-14. For each frequency the stubs were adjusted for maximum power received. The experiments in this part were also conducted on the 0. 1015m. and 0. 031m. cavities. Also different values of dielectric constants were used as in the swept frequency. In Figs. 4-15 - 4-17 the zero dB power level represents the normalized coaxial power level. The line representing zero power level is not shown. It represents the transmitted power minus the power lost in the circuit from the transmitter to the receiver when the coaxial lines are connected as shown in Figure 4-14. The upper curve is the power level as in the coaxial case but it also includes the antennas with the two apertures connected together thus forming a cavity. The surface on the aperture was machined to be smooth enough such that the mating surfaces would not act as a discontinuity. Also the mating periphery was covered with foil tape such as to prevent power leakage and to form an electrically continuous surface between the guides. If this is done perfectly then the arrangement is like a cavity filled with dielectric material and there is an input at one end and output at the other end. 137

2.0 2.5 3.0 3.5 4.0 Frequency (GHz) n=7 -25 dB level n=5 n=6 -25 n=4 I n=3 -45 -55 FIG. 4-10: COUPLING VS FREQUENCY FOR LOADED CAVITY, r=1.0, er-ll. 0, d=0. 1015m., xl=0.013m.,a=0.02286m., b=0. 01016m. 138

2. 0 2.5 33.0 3.5 4.0 Frequency (GHz) Supply power level through the coaxial line -25 139 -35 P4 -45 FIG. 4-11: GAIN VS FREQUENCY FOR LOADED CAVITY, Par=1. 0, Cr=11.0, d=0. 1015m.

2.0 2.5 3.0 3.5 4.0 Frequency (GHz) n=5 -35 n=2 FIG. 4-12: COUPLING VS FREQUENCY FOR LOADED WAVEGUIDES, ur=l. O, Er=7.0, d=0. 1015m., xl=0. 013m. 140

2.0 2.5 3.0 3.5 4.0 -25 i Supply power level through coaxial lines. -35 i -45 FIG. 4-13: GAIN VS FREQUENCY FOR LOADED CAVITY, ur=l. 0, er=7. 0 and d=0. 1015m. 141

Alfred Sweep Attenuator requency Oscillator AT104 Meter (1- TransModel Empire4Ghz) mitter 650 Devices H-P Mlodel 536_ RG-9 Coaxial Cable Coaxial Line (Reference) RG-9 Double Stub Tuner Wide Range Re- Crystal c,, ~~~~Crystal | ceiving System Detector R De~tector }. S-A Model 402 P Receiver HP Model (Scientific-Atlanta 423 A FIG. 4-14: EXPERIMENTAL SET-UP FOR COUPLING WITH POINT BY POINT TUNING.

-10 n=2 T R -~20 ~ Kn=1 J >C 0a=O. 02286m. b=O. 01016m. d=O. 031m. - - / ^ x1=0. 03m. 0r.Lrl. 0 I C X V /a L r=l. 0' -30 -40. \/ ~ ~ Frequency (GHz) 2.5 3.0 3.5 4.0 FIG. 4-15: COUPLING VS FREQUENCY WITH TUNING BOTH TRANSMITTER AND RECEIVER AT EACH POINT. (x) Coupling with x1=0. 03m., ( ) Power Level through Cavity formed from Transmitter and Receiver.

-5 -10 -1n=05 a=O. 02286m.,n=7 L b=O. 01016m. -20 d=O. 1015mI 30 -40 Frequency (GHz) 2.5 3.0 3.5 4.0 FIG. 4-16(a): COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling with x=0. 013m., () Power Level through Cavity formed from Transmitter and Receiver.

-10 -20 2.5 3.0 3.5 4.0 n=2 0 -30 x a=O. 02286m. b=0. 01016m. d=0. 1015m. Xl=0. 013m. =l1. 0 -40 Frequency (GHz) 2.5 3.013 3. 5 4.0 FIG. 4-16(b): COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling vs Frequency with x1i0. 013m., (o) Power Level through Transmitter and Receiver forming a Cavity.

-10 n=5 ]Ert A /A- \1-~a=O. 02286m. n=3 nb=O. 01016m. l2.5Frequencd=0. 1015m. x1=0.013m. -202.5 3.0 Er=7. 0 rC n=l P L t T a R c Ca -30C, X X X -40 Frequency (GHz) 2. 5 3.0 3.5 4..0 FIG. 4-16(c): COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling with x1 (spacing) = 0. 013m., (.) Power Level through Transmitter and Receiver connected as a Cavity.

-10 a = 0. 02286m. b = 0. 01016m. ll n=5 d= 0. 1015m. n-40: x 0. X03m. - 2.50 n=3.0,r3.5- 14.0 with x(separation) = 0. 03m. () Power Level through Transmitter and7. 0 Receiver con=2 -30 x -40 X X Frequency (GHz) 2.5 3.0 3.5 4.0 FIG. 4-17: COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling vs Frequency with xl(separation) = 0. 03m., (-) Power Level through Transmitter and Receiver connected as a Cavity.

Therefore the curve in comparison with the zero dB level represents the power lost in the transmitting and receiving cavities. It is to be noted that the level of the power lost shown by these curves is a little high because of the discontinuity at the mating surfaces in spite of all the precautions which were taken. The difference is probably within the range of 1 - 2 dB. The lower curves of the graphs represent the level of coupling for various separations, cavity lengths, and dielectric constants. It should be observed that the differences between these graphs and the corresponding graphs with swept frequency are: i) the level of coupling is higher in the case of tuning, ii) the dynamic range of variations is less than before and it is around 20 dB, iii) the resonance points are about the same with or without tuning. 4. 3 The Far-Field Patterns The arrangement in Fig. 4-9 was used to measure the E-plane pattern which is with 0=0~; 0 varies approximately from -90~ to +90~. The O-plane is perpendicular to the plane of the aperture at y = a/2 and parallel to the x-axis (see Fig. 2-1). The reference antenna is AEL No. ARN101A (1 - 11 GHz) logperiodic and moves in a radius of 5'6". The H-plane patterns were also measured. For this case 0=90~ and the reference antenna moves in a plane perpendicular to the aperture at x= b/2. The variation of 0 is between -90~ and +90~. Different values of frequencies were considered and both E- and H-plane patterns were drawn. A representative graph is shown in Figs. 4-18(a) and (b). There was no noticeable variation in the shape of the curves with different values of frequency in the S-band range except for the level of the gain. The variation of the gain with frequency was shown in Figs. 4-8, 4-11 and 4-13. The differences between these patterns and those of a free-space S-band slot in a ground plane are: 148

, tt irj.....- I. i @10 f 1 * 4F i +, _iTi Hl I I j I 4. 14i- iiii i4[. 4! -i 4 i 11...... [' r!''!''l t- - t; 1:_i:-:iL li:~ L I i-::i-: I I "1 -90 -72 -36 0 36 72 90 0 (degrees) FIG. 4-18(a): E-PLANE RADIATION PATTERN OF A LOADED X-BAND WAVEGUIDE. f=3. 495 GHz, er=15. 0, ur=l. O 149

it:l l I iI i j i, W 1, I,,,.... I!' I;' I I ~ 1 I I,' I: r I I 10 4 _.... l il_ _ _ - I X-BAND WAVEG.IDE. f3.495 GHz, Er=15. 0 r=l. 0 X-BAND WAVEGUIDE. f=3.495 GHz, r. 5, 150

i) the E-plane pattern is almost a circular pattern, more so than for the corresponding unloaded case, ii) the H-plane pattern has the sidelobes somewhat higher in level than the corresponding unloaded case, iii) the gain is lower than in the unloaded case due to the losses in the dielectric and the mismatch at the aperture. 151

Chapter V SUMMARY AND CONCLUSIONS The preceding chapters describe a theoretical, numerical and experimental investigation of the coupling between slot antennas with the broadsides parallel; the slots were loaded or unloaded and placed in an infinite conducting ground plane. The two slots were backed with a material loaded cavity filled with homogeneous isotropic material. The investigation does not claim credit for development of new antennas since both loaded and unloaded slots have been introduced before in several papers and textbooks as mentioned in the Introduction. Instead use is made of existing antenna types and an investigation is made of the coupling for both cases. The results of the two cases are then compared. The investigation does not claim an advantage of loaded slots over unloaded ones or vice versa, since each has its own merits and drawbacks. Therefore it is up to the designer of the system, after taking into consideration the characteristics of each, to decide which would be the best choice for his system. For example, if a system has limited space and can tolerate the loss of some of the power it might be a worthwhile compromise to choose the loaded slots. As was seen in the investigation it was possible to use a standard X-band waveguided loaded with dielectric material for S-band frequency operation. It was also noted that the radiation patterns for this arrangement were rather similar to the ones obtained from the standard S-band slots. A serious drawback of the dielectric loaded waveguides is the frequency sensitivity as shown in both coupling and gain curves versus frequency. No experimental work has been done to obtain the coupling versus frequency for ferrite loaded slots but it is expected that the resonance phenomena would be less pronounced in this case. The coupling between the apertures with material loading did not follow the usual trend for the unloaded case which is a 152

drop of approximately 6 dB/octave of frequency. For the loaded slot case the behavior of the coupling between the apertures depended mainly on the parameter of the loading material. It was rather interesting to see that the coupling versus frequency was almost constant with frequency for low values of dielectric constant. For the higher dielectric constants the coupling changed more drastically. It varied from minimum to maximum as shown in Figs. 3-7 to 3-10 (Ch. nI). The variation of coupling with spacing followed the 6 dB/octave of distance characteristic. The phenomena of resonance was investigated and a method to predict the resonant frequencies and the half power bandwidth was obtained. A good agreement between experimental and theoretical results was obtained. It was noted that resonant frequencies occurred when the cavity length was such that = n (kg/2 ) where n is an integer. It is interesting to note that if it is possible to maintain Xk constant or approximately so corresponding to its value at the resonance peak, then it would be possible to have a wider bandwidth and reduce the frequency sensitivity. To look into the requirement to do so, consider Eq. (3. 10) (Ch. III) which is: 0.15 f = G Hz, where a is in meters, c a/^ a also f2 or 0. 3 X 0 3 - in meters, P/rCrf2 (0.15 )2 where f is in GHz and a is in meters. Then for Xg to be constant irrespective of f implies that UrCrf2-( 0 1 ) = constant. 153

If the material loading is to be taken specifically to be dielectric or ur=l then This indicates that one way to have a wider bandwidth is by using a material with a dielectric constant inversely proportional to the square of the frequency over a certain band of frequency range. The existence or the physical realizibility of such material needs further investigation. Another outcome of the investigation was that, as mentioned in the previous chapters, the resonant frequencies depend on several factors which are; 1) the length of the cavity, 2) the broadside length of the slots, and 3) the material paremeters. Therefore if two slots are spaced close to each other on a ground plane the mutual coupling can be controlled. For example, in order to decouple the two slots, the parameters affecting the resonance frequency could be chosen in such a way that the resonant frequency of one antenna coincides with the antiresonant frequency of the other antenna. For this case each antenna is acting as a frequency filtering device to the power which is intercepted from the other antenna. It is very interesting to note that the spacing of the resonance peaks is not uniform over the frequency scale. It is smaller nearest the cutoff frequency and becomes wider away from it. This was observed both experimentally (Fig. 4-7 and 4-16) and from digital plots of the resonance points ( Fig. 3-14(a)+(c) ). This can be explained by considering Eq. (3. 11) which is Squaring both sides there results: 2 2 0.15 rn2 1 154 r E~ ~ ~ ~ ~ ~ ~ ~~~~~~(.1 r+-

or f2 res n2 1 0.15 t Q2 a2 or f2 res n2 0.15 2 ( D I n = 1. (5.1) a -Er l/ a Equation (5. 1) can be put in the form 2 f2 2 res n 2 i2 - = 1, (5.2) A A2 1 2 where 0. 15 A: 15 (5.3) a 4Ar^ A2 = /a, (5.4) with a and x in meters. Equation (5. 2) is a hyperbola as shown in Fig. 5-1. From the figure it can be seen that the resonance peaks are not uniformly spaced but become more nearly uniform at higher frequencies or at the asymptotic part of the curve. Also, from Eq. (3. 10) (p. 75) it can be seen that A represents the cutoff frequency for the specific parameters Cr., e, a of the loaded cavity. 5. 1 Areas for Future Investigation 1) Experimental investigation of coupling for ferrite loaded slots. 2) Investigation or search for material to widen the bandwidth. 3) Study dielectric loaded slot array. 4) Study ferrite loaded slot array. 5) Design wide frequency band tuning arrangement for the feed. 155

A2 Asymptote n = - fres A2 n %24 __ f 6_ f6res FIG. 5-1: EXPLANATION OF NON-UNIFORMITY OF SEPARATION OF THE RESONANCE PEAKS. A FIG.~~~~~s \-:EPAAINO O-NFRIYO EAAINO THE_ REOAC PEAKS.N

REFERENCES 1. Adams, A. T. (March 1964), "The Rectangular Cavity Slot Antenna with Homogeneous Isotropic Loading, " The University of Michigan Cooley Electronics Laboratory Technical Report No. 147 (5549-7-T). 2. Lyon, J. A. M., R. M. Kalafus, Y-K Kwon, C. J. Digenis, M. A. H. Ibrahim and C-C Chen (April 1966), "Derivation of Aerospace Antenna CouplingFactor Interference Prediction Techniques: Final Report, " The University of Michigan Radiation Laboratory Technical Report 6633-1-F, AD 483051. 3. Ramo, S., J. R. Whinnery and T. Van Duzer (1965), Fields and Waves in Communication Electronics, John Wiley and Sons, New York. 4. Von Hippel, A. (1959), Dielectrics and Waves, John Wiley and Sons, New York. 5. Lewin, L. (1951), Advanced Theory of Waveguides, Iliffe and Sons, Ltd., London. 6. Levis, C.A. (August 1956), "Variational Calculations of Impedance Parameters of Coupled Antennas, " The Ohio State University Report 667-16. 7. Kaplan, W. (1959), Advanced Calculus, Addison-Wesley Publishers. 8. Harrington, R. F. (1961), Time Harmonic Electromagnetic Fields, McGraw-Hill Book Company, New York. 9. -- (June 1968), "Techniques for Integrating Solid-State Circuitry into Antennas, " The Ohio State University Electroscience Laboratory Report 2142-16. 19. Jennetti, A. G. (November 1965), "A 1 Gc Cavity-Backed Slot Antenna Field, " The Ohio State University, Contract AF33(657)-10386. 11. Svoboda, D. E. (September 1966), "Correcting for the Pattern Distortion Effects of Mutual Coupling in Antenna Arrays, " The Ohio State University Report 2142-6. 12. Richmond, J. H. (November 1961), "A Reaction Theorem and its Applications to Antenna Impedance Calculations," IRE Trans., AP-9, pp. 515-520. 43. Galejs, J. (February 1965), "Self and Mutual Admittances of Waveguides Radiating into Plasma Layers, " Radio Science, 69D, pp. 179-189. 14. Mikenas, V. A. and P. E. Mayes (July 1967), "Analysis of a Log-periodic Cavity-slot Antenna Using Three Port Network, " University of Illinois Technical Report AFAL-TR-67-109, Dept. of Electrical Engineering. 15. Rumsey, V. H. (June 1954), "Reaction Concept in Electromagnetic Theory,' Physical Review, 94, No. 6, pp. 1483-1491. 157

16. Wu, C. P. (March 1969), "Numerical Solutions for the Coupling Between Waveguides in Finite Array, " Radio Science, 4, No. 3, pp. 245-254. 17. Archer, D. H. and G. S. Hardie (December 1963), "Investigation of Coupling Between Closely Spaced Antennas and Development of High Insulation Techniques, " Raytheon Space and Information Systems Division, Santa Barbara, California 18. Swift, C. T. and D. M. Hatcher (April 1968), "The Input Admittance of a Rectangular Aperture Antenna Loaded with a Dielectric Plug, " NASA Technical Note D-4430. 19. Cockrell, C. R. (October 1968), "Higher-order Mode Effects on the Aperture Admittance of a Rectangular Waveguide Covered with Dielectric and Plasma Slabs," NASA Technical Note No. D-4774. 20. -- (November 1964), "Techniques for Integration of Active Elements into Antennas and Antenna Structure" Ohio State University Electrical Engineering Department Interim Engineering Report 1566-15. 21. -- (September 1967), "Techniques for Integrating Solid State Circuitry into Antennas, " The Ohio State University Department of Electrical Engineering Interim Technical Report 2142-12. 158

APPENDIX A THE REDUCTION OF QUADRUPLE INTEGRALS TO DOUBLE INTEGRALS For the numerical computations to reasonable in cost and programming for the computer calculations, it was necessary to reduce the quadruple integrals to double integrals. This was performed by means of coordinate transformations. It consisted of a rotation of axis, magnification and translations. Consider the general coordinate transformation; 1 = r7 cos 0 +y sin 0 1 = -rl sin 0 + y cos 0 If we choose 0 to be 45~, then 1 = - (-r + y) 1 2- + Take A2 = p1 2 = 1 (Magnification) T2 = 71 Then 2 = (7 + y) T2 = ( -rl + y) Performing translation along P2 such that the origin would be at the center of the rectangle as shown in Fig. 2-5 (Ch. I). Therefore = -2 -a or p + a = ~2 = r + y (A. 1) 159

Let 7=2 or = y - r7 (A. 2) Due to the magnification 1 dr)dy = 2 d/d dy Therefore the integration: fa a (a-)1 (a-+)) f f(y, r)d dy= f(, 4d- fr f( y)d 7dy (A. 3)'0'"0 (a-7) - a (a+y) If f(p, y) is even in ju and y then: f(-,M, Y):=f(, -7) =f(, 7Y) Therefore a a a a-y fI f(y, r)drldy=2 f f(u, y)ddy Similarly b b b (b-co) 1 0 (bi) ff f(, x)ddxT =' I f(-, v)dvdvr+ f f(a, zdvdc (A. 4) Wu<) *'0 -(b-ua) b T4-or) also if f(cr, v) is even in cr and v then (A. 4) reduces to b b bb-c ~~0bf(, z')d~dxt=2 f()dvdo In general 160

b aa 1 b a (a-) (b-c) f(x,y,), r)ddrndydxt=, I f(y,, A, z)dvd/dody <O )<) 00 <0 -(ra-y) -(b-co) a O (a-r" (b-h) +4 t-( a-) ( b) f(v, /, a, y)dz34dody -0 *b -(a-) -(b+-) + ~L b (a+)) (b -r) 4 fr f(v, PL, CT, jdv,?dod 4-a'O -(a+y) (b —) 1 0 0 ~(a+y) (b-ib) +4 I f f ( f(v, IA, o, y)dvd/Adody. -a (a+7) (b+o) (A. 5) In Chapter II f(y,, o, v) is an even function in y,,, v. Therefore: b aa b a b a-Ty (b-o) f rf6 ff(x,'y, n, )d, rdRdydx = f(7, r., cr, v)dv d doudy O^ )o "o o'o -(b-cr) a a.a-y (b+-) o+ bJ 0 (bJ ) f(7y,co, v)d7/ddC dy. (A6) 161 (b+ <) 161

APPENDIX B EXAMPLES OF COMPUTER OUTPUT DATA ON COUPLING B. 1 Coupling Versus Frequency B.1. 1 For an empty S-band waveguide for x1=0. 0341m., x1=0. 075m, A a, B - b, EPSR * er, FMUR s Pr, XI separation center-to-center, UC = coupling without interaction, and UCI = coupling with interaction. B. 1. la: A 0.71360000-01 8 a 0.346.36n00D-0n 0 0.00001')-01 EPSR = I.000000 n FMU a 1 *000000 NA 0 NI 3 0 3 NS - 30 NL 51) FOR XI a.3410000D-01 F-N UC tUCI 1. In -I l. 339^ -9.97155 1.on0 -1. 099 - 2.137R3. 300 -13.16771 1 3. 13351 1.00 -13.71613 *1A3.76355 1.500 -14.13574 -14.25094 1.60 -14.51635 - 14.6423 1.700 n 14 90229 -15.1055R I.) -r60 -15.31,60 -15.53R67 1 900 -15.7"103 -15.999A4.000n -16.30)64 -'165013 162

B. 1. lb: A =.71360000-01 R = 0.34036000-01 =- 0.50000000-01 EPSR 1.0000n0 FMUR - 1.000000 NATI 30 NRI = 30 NS 30 NL = 50 FOR XI 1 0.7500000D-01 FN UC UCI 0131 D9. 0r)0) 15 - 114^13 -512 735 I.On - 17 165t1 -17.074q6 I - 1 19?79Tq =q~R.O?q.^nn -1. 7?16- 1.~60'33 1 500 - 9 133 I - 9.09728 I.6no - 19 54507 - 19.541 1i. 70 -1991373 -19.96331 1.0n -20n.9591 - O.37?n 1.QO -P 02 7Ct9 -n.,77q57. 000 -P21.1621q -21 20163 B. 1. lc: A = 07136000D-01 B = 0.34036000-01 t 0O5000000O-01 EPSR = 1 000000 FMUR s 1.000000 NAI = 30 NRI= 30 NS 30 NL = 50 FOR Xl * O.2075000 FN UC UCI OR I)F 1.100 -q3.5791 -23.26242 1.?nn -P5. 34572 -5 32054 1.300 -26 33617 -26.3435R.400 -26.95072 -26.96499 500 -27. 40R1I -27 3961.1 1.600 -27. R1R2 -27.79544 1.700 -2R 17953 -28.19244 1. rOO -2. 56F64 -2R 56822 1.900 -28 98639 -28 97585 2.000 -29.44464 -29.44963 163

B. 1.2 For an empty X-band waveguide with spacing of xl=0. 013m., xl= 0.02286m., 0. 114m. B. 1.2a: A = 0.2e6n0 )-n F3- 0o.1016000no-01o 0.50000Ononnn)c;R - 1.000o000 FMUtR = 1*000000 NAI = 30 NFI = 30 NS = 30 NLt 50 FOR X1 0.13000001)-01 FN IJC JC 1.100 - 1.67733 -I 1.7S54 1?0 - i 3. -Rno - I 13.74 57 1.300 -14.45179 -14.64503 1.,00 -14.94471 -15.17159 1.50n -15. 930n - 5..3526 1 00 - 1 5.5 605 -1 5.2277.700 - 15s.637 -16.07R59 I. RO -16.16473 -16.33n55. 900 - 6.49024 - 6.59960 2.000 -1 6. 5484 - 6.90338 B. 1.2b: A = 0.2* q6000D01 R = 0.10160000-01 0 = 0.5000000D-01 EP.R = 1. 000000 FMUR = 1.000000 NAI = 30 N91 = 30 NS = 30 NL = 50 FOR XI = O.22860000-01 FN L.C UCI r13 DR 1.100 -15. 1675 -15.25937 1.200 -17.416429 -17.12500.0nn -1. n707 -17.96615.400n - 1.59560 -1q.47595 1.5n0 -1.96A40 -1R.R6764 I. Ar)o -4 9.27 11 6 - 19.22250 1.7n0 - 9.55607 -19.*56723 1.q00 -19.94716 - 9.90658 1.900 -20.16037 -20.24134 2.000 -20.50626 -20.57900 164

B. 1. 2c: A 0.2289600())-0-)1 8 0 10160000 —01 Q = O,'SOOOOnnO)- SP s- 1.00n00nnnFMI = 1.000000 NAIT = 0' NR I = 30 NS = 30,NL 5 FO XI = 0.1140000 F'N IJC UCI )B3 1) H 10nn -9?21674 -?q.03(O I.0.) -30. 254R4 -30 *6302 30nOq 1 1R.. -'31 18221 1.A0 -31. 740R -31 74599 1 500 -32. 1719 -32.12186 1 600 -32 44.31 -32. 44q5.700n -3?.73573 -32.73111 1.09 -33. n133 -33.03565 1.r90 - 33 34674 -33 34273 2 000 -33.69305 -33 69674 B. 1. 2d: Standard Ku-band waveguide coupling versus frequency output. A = 0.1579SRO0-01n 0.79994000-02 0 = n.5000000D-01 EPSS = 1.00000n FMIJUR 1.000.000. NAI = 30 NFI = 30 NS 5 30 NL - SO FO0 Xi - 0.1140000 FN IJUC lJCI O)R DR 1.o00 -32.996R2 -3.. 99475 I 30r)d -341. 02'749 - 34. 02494. 4nn -.34. 69 629 -34. 69953 1*500 -S3.l212i -35.2139R I.An00 -35.67041 -.35.66919 I1.700 -36.11913 -36.11730 1 900 -36.59391 -36.59572 I.9no -37.054 -37.09514?2r.0 -37.63593 -37.63455. 100 -38.*4509 -389.24580 165

B. 1.3 For a standard X-band waveguide filled with dielectric material. These parameters are common for the three tables which follow: a=O. 02286m., b=0. 01016m.,,r= 1. 0, separation center to center = xl=0. 013m. B. 1.3a: cy = 7.0 FN UGC ICI OR OB 1.100 -19.92142 -19.96417 I.?n0 -P?. 74857 -22.75982 1.300 -24.04925 -24.04954 1.400 -24.6473 -24 61905 1.500 -24. 7698 -24.77725 I.600 -24.70965 -24.69682 1 700 -24.49765 -24. 4R233 1.RO0 -24.21448 -24.19,717 1 900 -23.9813 -23. 7930 2 000 -23 57103 -23.55122 2 100 -23.24631 -23.22612 2.200 -22.931'5 -22.91165 B 1.3b: er= 11.0 FN UC UCI ) B 01 1.100 -19.R3601 -19.91287 1.200 -23. 0505 -23 23635 1.300 -25. 1408 -25.20028 1.400 -26.47726 -26.48592 1.500 -27.30095 -27.30501 1.600 r27.76787 -27.76875 1.700 -27.96353 -27.96200 1.800 -27.96154 -27.95808 I.900.-27.82349 -27.81837 2.000 -27.59736 -27.59080 2.100 -27.31824 -27.31042 2.200 -27.01053 -;7.00160 166

B.1.3c: ~r=15.0 FN lJC UG I I 100 -19 42214 -19.5154%. 200 -22.7672 -22 91717 1.300 -25.02914 -25.05214 1.400 -26. 5214 -26. 6652 1 500 -27.77097 -27.7RP05 1.600 -2. 64666 -28 65255 1~ 700 - 9 rt61 5,' -'9. ^2649 1,P00 -29.64q76 -29. 5036 1.900 -29 R4291 -29. 4302 2.000 -29. *q097 -29 R7991 2.100 -29 79956 -29.79730 2.200 -29.63168 -29 62845 B. 1.3d: A 22q6nnnr)-01 R 0 101 60000-01 Q = 0,50000000-01 F.PSR 20. o F00000M IMIJ 1.Onno00 NAT. 30n NRI = 30 NS = 30 NL = 50 FOR XI = 0*1300000n-01 FN.JC IJC I OR O) R 1.100 -19.31 11 -19 13669 1.200 -22.47551 -2P 52.64 I..n -0.6 3^q9 -24. 6171 1./40 -26. 9247 -26.3107q.500 -27.60q13 -27 620? 1. 00 -2?. A750 -2R. 6954 1.700 -?9.56q57 -29.57504 1.R00 — 30.2699 -30.27354 1.900 -30.79920 -30.0227 2.00 - 31.16974 -31. 17161 167

B. 1. 3e: A = -n.?o^600q-0e)-l = 01 O. 60ln)0-nl o = o.50n00000l)-0 no - n? r0r) n 1) oiM(JI o- 0 n so nonIO- o i 2S000000( FMUR 1.000000 NAI = 30 NI = 30 NiS 0 30 NL = 50 Fni0 X I -. 130n 0D)-01 FN IUC, UCT 1. l n - 1.50556 -11.20670 1* PO -1 3.1*SO4 -12*99226 1,o0 -10. 7 7? - - 3.6?3qR I.4 00 -13.99R55 -13.7707 1.500 -13.96r525 -13.97640 1.h*O -13.93971 -140l1147 1*700 -13. R555 -14.022R5 I.*00 -13.q2937 -14.02949, 1.900 -13.7R094 -14 039R0.n000 -13.750r1 -14.05700 B. 1.3f: A = O.q2RqAOOo-0)1 = 010160000-01 r 0.50000000-01 EPSR = 3.000000 KMUR 1.0000000 NA[ = 30 N1I = 30 NS = 30 NL =' 50 FOR XI 1 0.13000000-01 FN!JC i.iC 1) 13 0 P3, *. 00 -15.16019 -b*.03304 1.0nn -1 6.1772 -16.72254 1. 00 -17.3177? -17.23417 1o400 -17.4155? -17.34109 1. 5nO - 1 7.34165 - 1 7.7 1 I* AOO -17.19165 - 17 14257 1.700 -17.011 3 -16.9R099 I R00 -! 69,41.2 - 16.81 70 C.900 - 16. 6432R - 6.66336.on0r -16.4760? -16.52637 168

B._1. 3g: A= 0.?R60002nE)-01 8 = 0.10160000-01 n O.5o00o00D-Ol 4.00000O O MUR - 1 000000 NAI: 30,NI)I I 30 NS'=S 30 NL = 50 FOR XI = r013000000-01 FN.)I, IJ( 1)1 1.) F3 1.100n -7.4919? - 7.,44991.o0n - 19.34451 -19.3016 1.300 -1.Q1739 - 9.7RqR4 I.on) -?).01R99 -19.97.31 1.500 -19.91176 -19.q6559S I.00 -1 9.706R -19. 66379. 700 -19.46463 -19.42311 1.00 - 9.20q11 -19.17239 1.900 -1R.95446 - 1.92724 2.n00 -18.71?12? 18.6961A' B. 1. 3h: A 0.qR60nn0)-n R a 0.10160rn-01 Q = 0.5000o000D-01 S —,, = 5,n0000onr) " sMI I OcnnlOc) NAI.30 N:tI 0n NS F 30 NL = 50:'0R X I = 01300000o-01 FN UC UC, 100 -18.8711 - 1887?19 1.200 -1. 0/60 -21.03;350 1.3no -21.812q6 -21.79115 1.4n -. n0r0A0 -.1.'9q-29 1. 00 -R 1.934/78 -21.905 1 I.Aro -?.7?967 -21 69765 I.70 -.4 61 -1 -.42830 00) -2 I.1662 -2.13683 I.o -2. 75001 - 20. 3-4239. n0 -?0.5q375 -20.556q9 o. n0 -Po. 3054 -20.2670 2.an0 -n. 05) 019 -0n.03519 2~.3 - 9.q 10 -19.80O00 2. 1')O - 9.5q93 - 19.59373 2.500 -1 9.3366 - 19. 40432.,6nn - o. 0671 - 19..3530 p.7r00 -19.04364 -19.0595 P.3^o - l.q9901 - I.95541 2.9n0 -1*R77234' - 1.84271 3.000 - 186631 6 -1874688 169

B. 2 Coupling Versus Spacing In the following these notations are common: A a a, B * b, EPSR a Er, FMUR - r, GB = N1 and N2 where N1=G/Yo and N2 =B/Yo for the normalized frequency specified FN. Also, R= N3 and N4, where N3 is the real part of the reflection coefficient and N4 is the imaginary part of the reflection coefficient. Also, L indicates the separation between the two slots center-to-center in wavelengths. X1 indicates the equivalent in meters of L for the specific case aL hand. 170

.4 C 00 C C! I c -' — ct. ~' - r-. CV -C C'C. C c C'n'IV C, C". n r C c c C r c C; r C e n c C C vo Oj' " s-' *r %r C,,,' C C - r c in c F'. — t C. c c o C _, ~ C ~ ~. o.. O ~..... ~. O O...... 4.. C C c C\1 C —. c C P*r c cv,, J N Nc,, Lf M-l -cn,''- c, I r' c- c' CC'C.C, C C CC r c C CC C~ -. n - r c c r C C'.) c -X~, g T~ C C C; n C %'' q Ln \C r - W C. N CO) C — <, —. C --,. Cv C, un Cv', C-,4;:- C, \Co C,. V - ro o C C r I I i I i I i i I i i IC i i I i IC i i i I I I I ~ - o,-.-I C C'0 C ~e C,C *V ( C C C C C C C C C C C C C C C C C C C C CC C C C C CI C C'.' C'C C,C C C C C C, C,'C C C ~zC C C- C, 1"- ^ CC-'Cc,, CCC C C0 C' C fC C C' ~ " ".' - C CC C, C C C C C IC. C C — rC C C C C C. C uCC-CC C c rw y c < c c' c <, r" C:r ",. O- C r c:- _' t.C...' 0,' ". % - 0 0 0 0 0 0 0.,*,S 0. 0 r 0 T 0 ar ^ 0 - 0Ce CI C 1 * a- - - -.. - v- - -. -' - -.......... C -- C r r -. CC C C C C C CC C C- C C C c C C C C cCC C C c'- 0' II-c r -------- *v. r 1 - CccccC C CCC C cC C C C C C CCCC ccCC *C C c C ccC- C. C c C * C CCCCCCCCCCCC C CCCCCCCCCCCC C CC.. CCcc C Ccccc CC c c Ccc C C C c C C C CCCC. C 0 CC C C'Cccc C C C c C C cc.) C C C C C CC"'. C CC C 0,- o,<' rnr ac - ---- - 1.- c - c.. - - - - - - - n m * - - - - - -C.c' e -c t -... _ _ _ _. _. _. _. _ C_ _ c" q L.* r-. {:.c'***. — C* Z C CCCCC

B. 2.2 The parameters are as in Section B. 1.1. A = 0.2 RoS.60000-01 3 = 0.10160000-01 Q 0.50000r(lO-01 R'R = Io11.00 FMUR = 1. 00000 N AI nR N3 I = 30 NS = 30 NI = 50 FOR FN = 1.500000 G. 0n.30823470-01 -0.3090832 R 0.7810194 0.5343343 L XI uCl Il, 0. 10 0 r ). 1010907n01)- -3174 -3 *: 74057: 083 0.2000000 n.2021P1ID-O1 -31.24739 -31.24590 0. JOt))3)0)) 0). 30327;0-0 1 -33. 75363,33. 75455 0. no0or000r) 0. 40436290-01 -35;71 362 -'. S 71 77 0.5000c.000. 5054536)-01 -37. 37070 -37.37194 n., O non0 0l. S665^430-r1 -3. 793^A -3R.79316 0.7000000 0.70763510-01 -^0.03263 -4n.n017g.3000000 Or. 0725RO0)01 -41.1' 665 -41 ~ 2163 0.9n0nn00000 0.tnq00165-01 -42.10409 -i10 1.000000. 1010907 -42.99637 -42.98679 I. 1 n0'O 0.111199R.-43.7R97' -43.7.q97F l.oinnnO 01.?13099 -44.V?6r -44.52659.300000 0. 1 31179 - -/5.0075 -45 *.P073 1.400000 0. 415 70 -45. 3965 -45. 3977.500000 0- 151 6361 -46.4 O955 -46.42974 1600000 06. 117z52 -46.9s~45 -46.99?/46 l.7C10000 O.171q5a2 -7.50 65 -47.50251 i. qnr00000 0. l19633 -47.9937q -47.,9936^ 1.o00000n0.19o07?4 -4R. 45q7 -43.45393. 000000 0.20 1R4 - 4 o9053 -49. 900064?. 00000 0.11P905 -.31 -4099-9.32101 3.100000 0.3133912 -52. 6622 -5. 6R6P3 4.100000 0.4144720 -55.10325 -55.10o826 5. I00000 0.5155627 -57.00095 -57.00095 6. 00000. 6166534 -5. 55449 - 5.5544R 7. 00000 0.7177441 -59. 72~03 -59. R7203.1i00000 0.R81q349 -61*01590 -61.01590 9. 0000 0.9199256 -62.02657 -62.02659 10.10000 1.021016 -4"e-*-85 -62.93185 172

UNCLASSIFIE D Security Classification DOCUMET CONT ROL DATA R & D (Security classification of title, body of abstract rnd indexinrg lnnotation must be entered when tile overall report Is cflssllled) I. ORIGINATING.ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UN C LA S S IF I E D Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 ____ 3. REPORT TITLE 300 MHz Array Using Physically Small Slot Antenna Elements 4. DESCRIPTIVE NOTES (Type of report and Inclusive dates) Second Interim Report, Technical 5. AU THOR(S) (First name, middle i'i tlal. laa t namo) Medhat A. H. Ibrahim and John A. M. Lyon 6. REPORT DATE 7,,. TOTAL NO. OF PAGES 7b. NO. OF REFS August 1969 172 27 8a. CONTRACT OR GRANT NO. 9u. ORIGINATOR'S REPORT NUMBE.R(S) F33615-68-C-1381 1 — T 1770-2-T b. PROJECT NO. 6278 c- Task No. 627801 h. c TlHER REPO RT NO(S) (Anry other,numbers that may be as s lined this report) d. AFAL-TR'-69-299 10. DISTRIBUTI-ON STATEMENT This document is subject to special export control and each transmittal to foreign governme nts or foreign nationals may' be made only with prior approval of AFAL (AVWE), Wright-Patterson Air Force Base, Ohio, 45433 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Avionics Laboratory Air Force Systems Command Wright-Patterson Air Force Base, Ohio 13. ABSTRACT The coupling between two loaded rectangular waveguides opening in an infinite conducting ground plane has been investigated. The slots are equal in size and placed so as to have the broadsides parallel. The analysis is made in two steps: 1) the coupling between the apertures is derived, and 2) the effect cfthe backing cavities on the coupling is determined. The reaction concept, a form of the "variational technique" was used in the analysis. This method helped to simplify the mathematical work and provided very reasonable results. Substantial experimental work on the behavior of coupling versus frequency and coupling versus slot separation was performed. In order to simplify the experimental investigation the standard empty X-band waveguide (0. 9"x0. 4") was filled with dielectric loading of Emerson and Cuming Stycast Hi-K dielectric material. Materials having dielectric constants of 5, 7,11 and 15 were used. Most of the measurements for the loaded case were in the S-band range of frequency (2. 6 -3. 95 GHz). The choices of the S-band frequency range and the dielectric constant wer made to permit use of the anechoic chamber in the first instance with reasonable accuracy and small waveguide in the second instance. An extensive comparison of coupling for the case of standard waveguides with and without loading has been made and explanations are given for the observed differences. An iteration procedure has been used to determine the effect of the electromagnetic interaction between the slots. Detailed computer programs were developed for the analytical expressions. Numerical results for many important cases were then obtained. Methods and results can readily be exended ~to other frequencies and other slot sizes. DD FO RM 1473 UNCLASSIFIED So curit I',sIiic('hon -

Security Cla.ssification | ~~~~14.~~K Y LWDIN'K A LINK B LINK C K. WO DS. ______ ROLE WT ROLE W T ROLE 7WT Antennas Antenna Arrays Electromagnetic Coupling Coupling Reduction Dielectric Loaded Antennas I^ l'lr ltv ('!.ts, fi,.,,s ~

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