*, ENGN t) K s UMR0005 I _ 0 IOn Technical Report No. 147 the Rectangular Cavity Slot Antenna With Homogenous Isotropic Loading A. T. Adams COOLEY ELECTRONIICS LABORATORY Department of Electrical Engineering The University of Michigan United States Air Force Air Force Systems Command Aeronautical Systems Division Contract No. AF 33(857)-10607 Wright-Patterson Air Force Base, Ohio March,1964

Technical Report No. 147 5549-7-T THE RECTANGULAR CAVITY SLOT ANTENNA WITH HOMOGENEOUS ISOTROPIC LOADING \ by;,<\ A. T. Adams Approved by X t John A. M. Lyon COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan March 1964 THE UNIVERSITY OF MICHIGAN ENGINEERING LIBRARY

ACKNOWLEDGMENTS The author wishes to acknowledge the valuable help and encouragement of his thesis chairman, Professor John A. M. Lyon, as well as the assistance of the committee members. This report is based upon work performed on Contract No. AF 33(657)-10607. Those who assisted in the performance of the work on this project were D. K. Adams, R. M. Kalafus, J. C. Palais, W. B. Ribbens and A. I. Simanyi.

TABLE OF CONTENTS Page LIST OF TABLES iii LIST OF ILLUSTRATIONS v LIST OF SYMBOLS xi ABSTRACT xv CHAPTER 1. INTRODUCTION1 1.1 Historical Background 5 1. 2 Review of the Literature 7 1. 3 Statement of the Problem 8 2. RADIATION FROM A WAVEGUIDE FILLED WITH ISOTROPIC FERRITE MATERIAL 12 2.1 Introduction 12 2. 2 General Field Solutions 13 2.2. 1 Region No. 1 - Waveguide Fields 13 2. 2. 2 Region No. 2 - Fields in Half Space 16 2. 3 Derivation of Variational Expression for Admittance 17 2. 4 Evaluation of the Variational Expression for Admittance 23 2. 4. 1 Reduction from Quadruple Integral to Double Integral 23 2. 4. 2 Elimination of Singularity in the Normalized Admittance Function 26 2. 4. 3 Normalized Admittance for Large Values of iLr and er 41 3. THE COMPOUND IRIS 47 3.1 Introduction 47 3. 2 Derivation of the Variational Expression for Admittance 47 3. 3 The Compound Iris 49 3.4 Special Cases —Inductive and Capacitive Irises 55 3. 4. 1 Inductive Iris 55 3. 4. 2 Capacitive Iris 56 4. HIGHER ORDER MODES 76 4. 1 Introduction 76 4. 2 Higher Order Mode Coefficients 76 4. 3 Reduction of Quadruple Integral Forms 83 4. 4 Effect of Forbidden Modes 92 5. GENERALIZATION OF THE VARIATIONAL ANALYSIS 97 5.1 Other LSE Modes Incident 97 5. 2 Below Cutoff Considerations 102 5. 3 Superposition of LSEX and LSEy Solutions 105 5. 4 The Dual Problem 107 5. 5 The Vector Problem 108 5.6 Reaction Methods 111

TABLE OF CONTENTS (Cont.) Page 6. EXPERIMENTAL RESULTS 113 6. 1 Verification of Variational Results 113 6. 2 Measurements of Aperture Field 117 6. 3 Experiments with Dielectric and Ferrite Loaded Cavity Slot Antennas 126 6.3. 1 Introduction 126 6. 3. 2 Resonant Frequency 128 6.3.3 Bandwidth 131 6.3.4 Efficiency 135 6. 3. 5 Beam Patterns 140 6. 3. 6 The Tuned Slot Antenna 144 6. 3. 7 Transmission Line Tuning Effects 144 7. CONCLUSIONS AND RECOMMENDATIONS 145 7. 1 Optimization of Antenna Characteristics 145 7.2 Conclusions 153 7.3 Recommendations 158 APPENDIX A. COORDINATE TRANSFORMATION 161 B. EFFICIENCY OF A RECTANGULAR CAVITY SLOT RADIATOR 164 C. PERTURBATIONAL CALCULATIONS 177 D. NOTE ON MATHEMATICAL STEPS REQUIRING FURTHER JUSTIFICATION 180 REFERENCES 181 DISTRIBUTION LIST 186 ii

LIST OF TABLES Table Page 1 Higher order mode calculations. 91 2 Performance comparison of rectangular cavity-slot antennas. 126 3 Characteristics of ferrite and dielectric material. 127 4 Comparison of theoretical and experimental data. 134 iii

LIST OF ILLUSTRATIONS Figure Title Page l(a) Resonant dipole. 2 l(b) Electrically small dipole 2 l(c) Antenna equivalent circuit. 2 l(d) Spherical line mode characteristic impedances. 3 l(e) Loaded antennas. 4 l(f) Tuning effects. 4 l(g) A rectangular cavity slot antenna. 4 2 Material loaded rectangular waveguide radiator with aperture plane iris. 9 3(a) Material loaded rectangular waveguide radiator. 10 3(b) Regions of a boundary value problem. 11 4 Coordinate transformation. 26 5(a) Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff), 1r = Er = 1.0. 31 5(b) Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff), 1r = r = 1. 5. 32 5(c) Aperture admittance of a loaded waveguide radiator (FN = 1. 0 at cutoff), A r = r = 2. 0. 33 5(d) Aperture admittance of a loaded waveguide radiator (FN = 1. 0 at cutoff),'r = r = 2. 5. 34 5(e) Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff), Ir = Er = 3. 0. 35 5(f) Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff), Lr = = 4. 0 36 5(g) Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff),;r = Er = 5. 0. 37 5(h) Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff), p1r =e =6.0. 38 5(i) Aperture admittance of a loaded waveguide radiator (FN = 1. 0 at cutoff), j = er = 8. 0. 39 v

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 5(j) Aperture admittance of a loaded waveguide radiator (FN = 1. 0 at cutoff), Mr = er = 10.0. 40 6(a) F1( 1). 44 6(b) Comparison of asymptotic expression (koa < i<) for G/Y with computer data. 45 6(c) Comparison of asymptotic expression (koa << r) for B/Yo with computer data. 45 7(a) Compound iris in the aperture plane of a material loaded waveguide radiator. 48 7(b) Compound irises. 52 8(a) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), Mr = r = 1. 0. 58 8(b) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), jr =E r = 1. 5. 59 8(c) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), jr = 2. 0. 60 8(d) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), Mr = Er = 2. 5. 61 8(e) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), r = Er = 3. 0. 62 8(f) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), pr = r = 4. 0. 63 8(g) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), Mjr = r = 5. 0. 64 8(h) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff), Mr = er = 6. 0. 65 9(a) Aperture admittance above (2, 0) mode cutoff (F = 2. 0 - 3. 0), (FN = 1. 0 at cutoff), r Er = = 1. 66 9(b) Aperture admittance above (2, 0) mode cutoff (FN = 2.0 - 3.0), (FN = 1.0 at cutoff), Mr = r = 1.5. 67 9(c) Aperture admittance above (2, 0) mode cutoff (FN = 2.0 - 3.0), (FN = 1.0 at cutoff), Mr =er =2. 0. 68 9(d) Aperture admittance above (2, 0) mode cutoff (FN = 2. 0 - 3. 0), (FN = 1. 0 at cutoff), Mr r = 2. 5. 69 9(e) Aperture admittance above (2, 0) mode cutoff (FN = 2. 0 - 3. 0), (FN = 1. 0 at cutoff), Mr = r = 3. 0. 70 vi

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 9(f) Aperture admittance above (2, 0) mode cutoff (FN = 71 2.0 - 3.0), (FN = 1.0 at cutoff), p1r =r =4.0. 9(g) Aperture admittance above (2, 0) mode cutoff (F = 72 2. 0 - 3. 0), (FN = 1. 0 at cutoff), 1r = e = 5 0. 9(h) Aperture admittance above (2, 0) mode cutoff (FN = 73 2.0 - 3.0), (FN = 1.0 at cutoff), ri =er =6. 0 9(i) Aperture admittance above (2, 0) mode cutoff (FN = 74 2. 0 - 3. 0), (FN = 1.0 at cutoff), r = er = 8. 0 9(j) Aperture admittance above (2,0) mode cutoff (FN = 2.0 - 3. 0), (FN = 1.0 at cutoff), r = r = 10. 0. 75 10 Equivalent representations. 10l ll(a) Dielectric loaded waveguide radiator, (a) side view. 114 1l(b) Dielectric loaded waveguide radiator, (b) front view. 114 12 Slotted dielectric and wedges. 114 13 Experimental arrangement. 115 14 Geometry of the dielectric insert. 116 15(a) Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator. Suscep- 118 tance- r = 10, r = 1. 0, b/a =.444. 15(b) Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator. Conductance- er = 10, r = 1. 0, b/a =.444. 119 16 Apparatus for measurement of the admittance of a loaded waveguide with a compound iris in the aperture plane. 120 17(a) Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator with a capacitive or inductive iris in the aperture plane. Capacitive iris, frequency = 6 kMcs. 120 17(b) Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator with a capacitive or inductive iris in the aperture plane. Inductive iris, frequency = 6 kMcs. 120 18 Aperture coordinates for measurement of aperture field. 122 19 Apparatus for measurement of aperture field. 122 20(a) Measurement of aperture field. T bar feed - horizontal cuts, frequency - 324 Mcs (ferrite powder loaded). 123 vii

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 20(b) Measurement of aperture field. T bar feed - vertical cuts, frequency - 324 Mcs (ferrite powder loaded). 123 20(c) Measurement of aperture field. Cylindrical probe feed - horizontal cuts, frequency - 324 Mcs (ferrite powder loaded). 124 20(d) Measurement of aperture field. Cylindrical probe feed - vertical cuts, frequency - 324 Mcs (ferrite powder loaded). 124 20(e) Measurement of aperture field. Cylindrical probe feed - horizontal cuts, frequency - 735 Mcs (air loaded). 125 20(f) Measurement of aperture field. Cylindrical probe feed - vertical cuts, frequency - 735 Mcs (air loaded). 125 21 Loaded cavity antennas. 127 22 Simplified equivalent circuit of a cavity slot antenna. 128 23 Bandwidth factor. 132 24(a) Equipment for efficiency measurements —antenna and flange. 137 24(b) Equipment for efficiency measurements —aluminum "hats". 137 25 Efficiency of the ferrite powder loaded cavity antenna shown in Fig. 21. 138 26(a) Rectangular slot beam patterns. Coordinate system. 141 26(b) Rectangular slot beam patterns. E plane pattern. 141 26(c) Rectangular slot beam patterns. H plane pattern. 142 27 Comparison of experimental and theoretical beam patterns 143 28 Optimization of IT I2 for fixed p re product. 146 29 ITI2. 148 30 Choice of pr for maximum ITI2. 149 31 Bandwidth of a rectangular cavity antenna (fixed size and frequency —variable. rer product). 150 32 Bandwidth of a rectangular cavity antenna (fixed ka/rT — variable electrical size). 151 viii

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 33 A ferrite loaded scimitar antenna. 159 34 A model for efficiency calculations. 165

x

LIST OF SYMBOLS Symbol Description Reference II Hertzian electric potential function Eq. 2. 1 n* Hertzian magnetic potential function Eq. 2.1 4i(xy) potential function Eq. 2.6 k material wave number Eq. 2.1 k free space wave number Eq. 2. 12 r propagation constant Eq. 2.8 mn A mode coefficients Eq. 2.8 mnn B mode coefficients Eq. 2.8 mn R dominant mode reflection coefficient for Ex Eq. 2.10 F electric vector potential Eq. 2. 14 pu relative permeability Eq. 2. 19 er relative permittivity Eq. 2.20 emn number Eq. 2.26 mn GI(xY,,7 7) kernel function Eq. 2.30 D integral function Eq. 2.35 Y/Yo normalized admittance function Eq. 2.33 G/Yo normalized conductance function Eq. 2.33 B/Yo normalized susceptance function Eq. 2.33 x,y, z field point coordinates Eq. 2. 12,? 7,z source point coordinates Eq. 2. 12 A, jP, 6, v transformed coordinates Eq. 2.47 C1' D1 free space cutoff functions Eq.' 2.49 r, 0 polar coordinates Eq. 2.51 0 1 angle of the diagonal of the integration rectangle Eq. 2. 53 xi

LIST OF SYMBOLS (Cont.) Symbol Description Reference x 1' 61 small rectangle dimensions Eq. 2.53 Q relative size of small rectangle Eq. 2. 54 f,g,d angle functions Eq. 2.57 a,b rectangular waveguide dimensions Fig. 3(a) F1(0 1) configuration function Eq. 2.65 a, a2,bl, b2, dld2 iris dimensions Fig. 7(a) Y1 Y2 y, I- compound iris admittance functions Eq. 3. 5 o o y (dl, d2) aY normalized aperture admittance for rectanguo lar waveguide of dimensions d1, d2 Eq. 3.10 Cfghl quadruple integral functions Eq. 4. 5 Dmn linear equation coefficients Eq. 4. 12 Ymn/Yo mode admittance functions Eq. 4.20 A determinant of coefficients Eq. 4.22 C m Dm free space cutoff functions Eq. 4.26 Rfgfg self reflection coefficient for (f, g) mode Eq. 5. 1 Dfghl linear equation coefficients Sec. 4.4 Rr Ri real and imaginary parts of the dominant mode reflection coefficient Eq. 5. 12 Rfgfg(r) real part of self reflection coefficient for fgf gor (f,g) mode Eq. 5. 20 Rfgfg(i) imaginary part of self reflection coefficient for (f, g) mode Eq. 5. 20 G2(x y, y, r) kernel functions Eq. 5.30 G3(xy,,77) kernel functions Eq. 5.31 G4(x,y,, r7) kernel functions Eq. 5.32 Rm reflection coefficient —H — dual problem Eq. 5.24 Re reflection coefficient —Ey — dual problem Eq. 5.26 V. mode voltages Eq. 5.33 xii

LIST OF SYMBOLS (Cont.) Symbol Description Reference Yi mode admittances Eq. 5.33 Xo free space wavelength Eq. 6. 1 Bp probe susceptance Fig. 22 B' transformed probe susceptance Fig. 22 transformed probe conductanceFig. 22 G' transformed probe conductance Fig. 22 p B' transformed short susceptance Fig. 22 Ba aperture susceptance Fig. 22 Ga aperture conductance Fig. 22 ITI2 squared magnitude of the transmission coefficient Eq. 7. 1 K conductance-susceptance ratio Eq. 7.2 BI iris susceptance Eq. 7.3 PL time average power dissipated Eq. B. 3 Pr time average power radiated Eq. B. 3 E6', B", l, l" material constants Eq. B. 3 IRI magnitude of reflection coefficient Eq. B. 4 8r argument of reflection coefficient Eq. B. 4 10 real part of propagation constant Eq. B. 4 P 10 imaginary part of propagation constant Eq. B. 4 IAI 2 field constant for Region No. 2 Eq. B. 5 ~~~* ~indicates complex conjugate except when used in II* Eq. B. 10 Rs surface resistivity Eq. B.11 xiii

xiv

ABSTRACT The rectangular cavity slot antenna, with dielectric or magnetic loading, and with a compound iris in the aperture plane, is analyzed utilizing variational techniques to derive accurate design guides for efficiency, bandwidth, and resonant frequency. The problem is broken up into several steps. First, the admittance of a loaded semi-infinite waveguide opening out into an infinite ground plane is analyzed using variational techniques. The normalized admittance is computed for a variety of values of A r' Er b/a, and frequency. Next, the effect of the compound iris is analyzed. A stationary expression is obtained for the admittance of a loaded semi-infinite waveguide opening out into an infinite ground plane, with a compound iris in the aperture plane. This expression is evaluated using two different approximations of aperture field. It is shown, and verified by experiment, that the errors incurred are not serious. Finally, the effect of the feed and the finite length of the cavity are taken into account by, respectively, perturbation techniques and network transformations. The effect of higher order modes is analyzed and shown to be minor. The results are generalized to take into account any incident higher mode. Experimental results include: (1) Experimental verification of variational data. (2) Measurements of aperture field. (3) Comparison of theoretical and experimental values of bandwidth, efficiency, and resonant frequency, and beanf pattern for several models loaded with ferrite powder, solid ferrite, and solid dielectric. (4) Measurement of the effect of applied d-c magnetic field. It is shown that a solid ferrite model with a bandwidth of 2%o can be tuned over a greater than 30% bandwidth. General conclusions concerning the use of material loaded antennas are discussed. In all cases discussed, the admittance is proportional to J.r times an integral which depends only on the product pjr r. For a given size reduction, or pr er product, an optimum value xv

of I r, which can be predicted, exists. The optimum value may improve bandwidth by a factor as large as 2:1 over purely dielectric loading. Similarly, for equal loss tangents, efficiency is also improved. The analysis provides an accurate method of predicting the characteristics of loaded cavity antennas and of choosing a material for optimum antenna characteristics. The primary contributions are: 1. The derivation of accurate simplified design techniques for rectangular cavity slot antennas and the optimization of antenna characteristics. 2. Two contributions to mathematical techniques for variational calculations. (a) Discussion and example of the use of two forms of electric field in a single variational expression. (b) A proof that, in the problem at hand, the forbidden modes do not contribute to the admittance calculation. 3. General conclusions concerning the limitations and potentialities of loaded antennas. 4. Practical realization of several loaded cavity antennas of reduced size. 5. Two contributions to measurement techniques including (a) the measurement of aperture admittance using a wedged transition and (b) a proposed new method of measuring dielectric constant. xvi

CHAPTER 1 INTRODUCTION The idea of loading an antenna with dielectric or magnetic material has a certain natural appeal, yet relatively few loaded antennas are in use today. This condition arises not because loaded antennas lack useful applications, but because accurate design procedures are not available. There are many practical difficulties in the experimental use of material loading. Without accurate design guides, chances of arriving at a suitable design are not very good. In this study, the general subject of electrically small antennas will be treated. Particular emphasis will be placed upon obtaining practical antennas from the analysis of a class of electrically small dielectric or ferrite loaded structures. General conclusions concerning the potentialities and limitations of loaded antennas will be discussed. There are several intuitive ideas about electrically small antennas that have proved useful in this study. For example, small antennas may be regarded as less "sensitive" than large antennas, in that the smaller antenna extracts less power from an incident wave. However, upon comparing the half-wave resonant dipole with the infinitesimal dipole [Fig. l(a) and (b)], one finds that they have essentially the same directivity. They differ primarily in their radiation impedance. The half-wave dipole has a resonant impedance of 73 ohms and a fairly low Q. The small dipole looks like a small resistor and a small capacitor in series, so it requires external tuning to yield its full available gain. When tuned, however, the small dipole has a relatively high Q. Therefore, one of the main differences between a resonant dipole and a small dipole is bandwidth. Moreover, conductor loss is far more significant for the small dipole. The ratio RL/RR [Fig. l(c)] becomes large for small antennas, resulting in low efficiency. The reason for this is that RR depends critically on the electrical size of the antenna (for a small dipole RR is proportional to the square of its electrical length kT), whereas RL depends primarily on the skin depth and the current distribution, both of which change relatively slowly with frequency (skin depth is roughly proportional to the -1/2 power of frequency). Thus a second import

vI lF Fig. l(a). Resonant dipole. Fig. l(b). Electrically small dipole. RL RL= LOSS RESISTANCE Rr = RADIATION RESISTANCE Rr Rr EFF Rr RL+Rr Xa Fig. 1(c). Antenna equivalent circuit. ant difference between a resonant dipole and a small dipole is efficiency. Some of these distinctions may be made in an alternate manner in terms of equivalent spherical transmission lines. 1 2 3 The field solution to any antenna problem can be expressed in terms of a complete orthogonal set of spherical line modes, each with a characteristic impedance defined in terms of the ratio of transverse electric and magnetic fields. In spherical line modes, a phenomenon similar to a cutoff effect exists, in that the ratio of the imaginary to the real part of the characteristic impedance approaches large values as the wave is confined to restricted dimensions, that is, if we consider an orthogonal set of spherical line modes with a common origin, the ratio of the imaginary to the real part of the characteristic impedance approaches infinity as r, the distance from the origin, approaches zero. This effect is shown in Fig. l(d) (extracted from Ref. 4) which shows a plot of the real (Rn) and the imaginary (Xn) parts of the characteristic impedance of the TMon spherical line modes as a function of the electrical distance (kr) from the origin. Since the electrically small dipole pattern is identical to that of the lowest order (n = 1) radiating 2

3.5 n=l n=2 n=3 n=4 3.0o - I - XY XY R|I R2 R3 R4 Rn r =.16 2.5 Xn 2.0 -xn 1.5 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 kr Fig. l(d). Spherical line mode characteristic impedances. spherical line mode, we may consider that coupling from the dipole to the higher order spherical line modes is negligible; thus the equivalent circuit of the small dipole may be represented in terms of a junction between a TEM spherical transmission line (the dipole) and the lowest order radiating spherical transmission line. The reflection coefficient at such a junction depends on (1) the ratio X1/R1 and (2) the characteristic impedance of the TEM transmission line. The effect of the ratio X1/R1 represents the basic limitations of electrically small antennas. Any attempt to launch a spherical line mode in a restricted space (kr small) automatically leads to a large X1/R1 ratio. The effect of the characteristic impedance of the TEM transmission line is one over which we have some degree of control and this control is used to optimize the reflection coefficient in a similar problem (Section 7. 1). Thus a basic limitation of an electrically small antenna is noted in that it must convert energy to a highly reactive free space spherical line mode. Figure l(e) shows three types of antennas with, respectively, material, inductive and capacitive loading. All three types have proven useful in improving antenna characteristics. It is important at this point to distinguish between circuit and field effects. If these three types of loading merely add series or parallel reactance to the circuit, they are of no more use than the antenna of Fig. 1(f) with a tuner inserted in the transmission line. This suggests a critical test for any electrically small loaded antenna, namely, a comparison of efficiency and bandwidth with the loading removed and a tuner inserted in the input line. 3

A'r er Fig. l(e). Loaded antennas. TUNER Fig. l(f). Tuning effects. Coaxial InputRectangulr - Rectangular Cavity Aperture / _ Plane / iris Ground Plane Fig. 1(g). A rectangular cavity slot antenna. Some of these effects are brought out more clearly in a study of the loaded antenna of Fig. l(g), which shows a probe-fed loaded rectangular waveguide opening out into a ground plane. The reflection coefficient at the aperture can be analyzed in some detail and its effect on the antenna bandwidth and efficiency predicted. This antenna has the advantages of (1) flush mounting and (2) a single-lobed radiation pattern. It can be shown, for this antenna, that loading leads to a definite advantage in the reduction of aperture reflection coefficient. 4

1. 1 Historical Background The literature on electrically small antennas is extensive. Interest has centered around two antenna characteristics, namely, directivity and size. Interest in increasing the directivity of antennas of a given size has led to discussions of "supergain" antennas in the literature. Several writers showed that an arbitrarily large directivity could be obtained with sources confined to a finite area or volume. This was puzzling to many, since antenna designers had realized for some time that some sort of natural limitation on antenna directivity existed. For example, experience had shown that, for a given antenna size, directivity exceeding that of a uniformly illuminated antenna was difficult to achieve. 6 7 Schelkunoff and W. W. Hansen were among those who first realized that arbitrarily large directivity could be obtained in theory. Bouwkamp and de Bruijn proved that the directivity of a linear current distribution of fixed length can be made arbitrarily large by proper adjustment of the amplitude and phase of the current. H. J. Riblet9 proved the analogous theorem for a two-dimensional current distribution and noted the discrepancy between theory and experiment. T. T. Taylor showed that the behavior of the solution of the wave equation in spherical coordinates indicated severe practical difficulties in realizing high directivity with antennas confined to a small volume. He noted that an extremely sharp increase in reactive energy occurs as size is reduced below that of a critical "cutoff sphere. " This effect corresponds to the rapid change of Xn/Rn noted before in Fig. l(d). Associated with an antenna of high directivity will be a spherical line mode of large n and a cutoff sphere of large kr. Taylor also pointed out that such "superdirective" antennas require extremely rapid space variations of source amplitude and phase. The discussion and analyses1 2 stimulated by these papers established the fact that supergain antennas are theoretically realizable, in that it is always possible to specify the source distribution necessary to produce the required far field pattern. This is evident 13 from the Fourier transform relationship between aperture field and far field. Attempts to realize supergain antennas have encountered the following difficulties: 1. The realization of the required rapid phase and amplitude variations of the source is difficult, particularly when the associated mechanical tolerances are considered. 2. Even if the proper source distribution can be realized, the severe restric5

tions noted by Taylor lead to high Q, narrow bandwidth, and low efficiency. Despite these severe difficulties, it has proven useful in some cases to accept some of these limitations in return for modest improvements in gain. An example is found in Hansen14 Woodyard arrays. The history of supergain antennas may be described as an attempt to obtain high directivity with an antenna of moderate size. Alternatively, it may be described as the attempt to reduce the size of a high gain antenna while maintaining the directivity constant. A related problem is that of reducing the size of a low gain antenna while maintaining the directivity constant. While the same basic restrictions apply, they are less severe when only modest gain is required. The curve for the lowest order spherical mode changes [Fig. l(d)]; less rapidly than do those of the higher order modes; a size reduction of a given fraction results in a smaller ratio of imaginary to real parts of the impedance for a lower order mode. Thus the bandwidth and efficiency restrictions are less severe for the lower order modes. Several studies of the basic limitations of electrically small antennas have bn m. 1, 2,3,10,15-17 been made. Early in the history of microwave antennas, experiments were made with dielectric and magnetic loaded antennas in the hope of improving antenna performance or reducing 18 antenna size. Many of these experiments yielded negative results; only a few were successful. Two notable successes were the "loopstick" antenna19 and dielectric rod antennas. 20 The loopstick antenna had increased "pickup" compared with a loop of the same dimensions without a ferrite rod, if, in both cases a capacitor in a tuned circuit connected between the antenna and receiver was adjusted for maximum receiver output. The dielectric rod antenna had a small cross-section area compared with a horn of the same gain. It became evident that some of the simpler intuitive ideas were not adequate for the explanation of the effects associated with material loading. For instance, simple concepts associated with the shrinkage of wavelength X = X / y/)E were inadequate, as were attempts to explain the impedance behavior simply in terms of increasing series inductance by a factor of ir or capacitance by a factor of Er or in terms of a simple loaded quarter wave 21 transformer between the input transmission line and free space. Similarly, simple concepts associated with low frequency behavior may not be applicable to high frequency effects 6

(Chapter 7). As a further example, one unfamiliar with the limitations of electrically small antennas might assume that the use of material with pr = e would provide a reflectionless transition to free space. This is not the case; the reflection coefficient depends primarily on the free space electrical size of the antenna. The explanation of the behavior of the dielectric rod required the solution of the wave equation in cylindrical coordinates. Explanations of the behavior of the loopstick antenna have utilized the solution of the wave equation in cylindrical, spherical, and prolate spheroidal coordinates. There has developed a divergence of opinion on the usefulness of material loading, stemming largely from encouraging mathematics, generally discouraging experiment, and the failure of loaded antenna problems to yield to simple analysis or intuition. 1. 2 Review of the Literature Since the early days of radar, a number of theoretical studies of loaded antennas 22-29 have been made utilizing classical methods 29 to treat separable geometries, such as the 23-27 22 9 23 loop antenna loaded with a ferrite sphere, 23-27cylinder, or spheroid, the biconical antenna loaded with a ferrite sphere24 29 and the small dipole loaded with a ferrite sphere. 28 These studies have certain characteristics in common. They all show resonant increases in radiation resistance, indicating the possibility of increased efficiency. Because of the difficulties inherent in dealing with Bessel and Legendre functions of complex arguments, most of these results have not been extended to take efficiency and bandwidth for lossy media into account. In addition, any antenna whose dimensions are infinitesimal at any point24' 28 29 is not suited to efficiency analysis because of infinite energy storage, which results in zero efficiency for any finite a, e", or pI". Thus, the classical studies have aroused considerable interest in loading of antennas. However, because of their highly idealized nature, and the complexity of the mathematics, it has not been possible to answer in detail these basic questions: (1) For a given metal conductivity, what material characteristics are necessary to improve efficiency? (2) Can bandwidth be increased? (3) What types of trade-off are possible between bandwidth, size, and efficiency? Variational techniques have been used to analyze a number of antennas, both 7

30-39 30 loaded and unloaded. 39 Lewin30 has derived a stationary expression for the impedance of a waveguide (unloaded) radiating out into free space and has plotted curves for G and B as a function of waveguide dimensions. Many of the mathematical techniques used in this thesis 36 are similar to those used throughout Lewin's book. Cohen, Crowley, and Levis6 have solved the same problem, taking dielectric loading into account, and have published data on admittance for several values of b/a. The impedance of an unloaded rectangular waveguide 40 41 opening out into an infinite ground plane has been studied by several other methods. 42 In Volume 10 of the Rad. Lab. Series, the impedance of a waveguide opening out into a 31 restricted space is analyzed for several different geometries. Levine and Papas have analyzed the admittance of an unloaded coaxial line opening out into an infinite ground plane 33 34 (circular diffraction antenna). Galejs3334 has made a theoretical analysis of the dielectric loaded annular slot antenna utilizing variational techniques. He has also analyzed the impedance properties of a dielectric loaded rectangular cavity antenna feeding a thin slot in 36 an infinite ground plane. M. H. Cohen studied transient phenomena to analyze the Q of cavity antennas and compared theoretical and experimental data on cavity Q for a square waveguide cavity. A number of analyses of loaded cavity antennas have been made utilizing 43 other techniques. Counter has analyzed a loaded cavity by considering the cavity to be 44 half of a closed (parent) cavity, with fields identical in the two cases. Tector has made an experimental investigation of air loaded cavity antennas. Recently, integral transform 45-48 and Wiener-Hopf techniques have been applied to loaded and unloaded antennas; much of this literature might be included under the subject of "Surface Wave Antennas. " Under the impetus of these theoretical approaches, several types of dielectric 49-58 and ferrite loaded antennas have evolved. 58 The use of materials with relative permeability and relative permittivity greater than one provides an additional degree of freedom in the parameters r, e r. It would indeed be surprising if optimum results were always obtained with Lr = er = 1. However, the use of material loading is limited. No panacea is provided for small antenna problems. 1. 3 Statement of the Problem A particular type of material loaded antenna, shown in Fig. l(g) is analyzed in this paper. The antenna is a rectangular cavity, fed by any one of a number of coaxial probe 8

r Aperture —----- Plane Iris Fig. 2. Material loaded rectangular waveguide radiator with aperture plane iris. arrangements, and opening out into a ground plane. A compound iris is located in the aperture plane. This antenna is studied by means of the model shown in Fig. 2. This model consists of a semi-infinite waveguide loaded with material of characteristics Ar, Er opening out into an infinite ground plane, with a compound iris in the aperture plane. The problem is further simplified by analyzing the antenna of Fig. 3(a) (no iris in the aperture plane) and then taking the compound iris into account with a separate calculation. The input admittance is analyzed by variational techniques. The finite length of the antenna shown in Fig. 1 is taken into account by simple transmission-line calculations. The effects of the particular type of feed is taken into account by perturbation calculations. The finite size of the ground plane has little effect on the admittance, bandwidth, efficiency, and resonant frequency, although its effect on the E plane beam pattern is significant. Predictions of bandwidth, efficiency, impedance, resonant frequency, and beam pattern show good agreement with experimental data. In Chapter 2, a stationary expression for the normalized aperture admittance of the antenna of Fig. 3(a) is obtained. The dominant mode aperture field approximation is made and the expression is reduced to a double integral form suitable for computation. This expression is evaluated by computer for a variety of values of pr, Er, b/a (ratio of waveguide dimensions), and frequency, with increments small enough to allow accurate interpolation. Approximate expressions are obtained for large values of A rer. In Chapter 3, the compound iris is analyzed by variational techniques. The compound iris problem is split into two problems, one of which is solved in the literature and the other of which can be 9

Fig. 3(a). Material loaded rectangular waveguide radiator. related to the problem already solved in Chapter 2. The solution of the problem of Chapter 2 below dominant mode cutoff is required in some cases. In Chapter 4, the effect of higher order modes is analyzed. A general expression for the aperture field, involving an infinite number of higher order modes with unknown coefficients, is substituted into the variational expression. Using the condition for stationarity, an infinite number of simultaneous linear equations in the unknown mode coefficients is obtained. An expression for the admittance in terms of a linear combination of the unknown mode coefficients is obtained. The infinite set of equations is truncated for certain specific choices of higher order modes and computation of the mode coefficients and the correction term for admittance is carried out. Certain modes are forbidden by symmetry considerations and it is shown that these modes have no effect on the variational expression for admittance. In Chapter 5, certain generalizations of the method are considered. Among these are: (1) other modes incident (2) other cylindrical geometries (3) below cutoff effects. It is noted that variational expressions can be obtained for any mode incident. These expressions all have a form which consists of /1r times a double integral which is a function only of 10

REGION #1 REGION #2 Fig. 3(b). Regions of a boundary value problem. pr e In Chapter 6, experimental results are described. The variational admittance data of the antenna of Fig. 3(a) are verified experimentally for several values of r. The admittance of the antenna of Fig. 2 is verified experimentally for both inductive and capacitive irises. Simplified methods for predicting bandwidth, efficiency, resonant frequency, and beam pattern are described and compared with experimental results for several dielectric and ferrite loaded models. Aperture field measurements of several ferrite loaded waveguide models are described. The results of the analysis and experiments indicate that gross assumptions may be made concerning aperture field, feed admittance, feed location, modes present in the cavity, and the application of transmission-line theory without serious inaccuracy in the results. Chapter 7 contains general conclusions concerning the limitations and potentialities of material loaded antennas. 11

CHAPTER 2 RADIATION FROM A WAVEGUIDE FILLED WITH ISOTROPIC FERRITE MATERIAL 2. 1 Introduction This chapter outlines the first steps in the solution by a variational technique of the radiating characteristics of a material loaded antenna. The physical configuration is shown in Fig. l(g). The antenna consists of a rectangular loaded cavity open on one end, fed by a coaxial probe arrangement, and opening out into a ground plane, with a compound iris located in the aperture plane. The problem can be simplified to the antenna of Fig. 2 by (a) perturbational techniques taking the feed configuration into account and (b) network transformations to take into account the finite size of the cavity. The antenna of Fig. 2 consists of a semi-infinite waveguide with a dominant mode wave incident from the left and a compound iris located in the aperture plane. The problem can be further simplified to the antenna of Fig. 3(a). In Chapter 3, it will be shown that corrections can be made to the solution of the antenna of Fig. 3(a) to take into account the compound iris in the aperture plane. Consider the antenna of Fig. 3(a). The rectangular waveguide is semi-infinite and extends from z = -oo to z = 0. The waveguide is completely filled with isotropic, homogeneous material of characteristics r, e r The assumption that the material is isotropic and linear is equivalent in most ferrite materials to assumptions that (a) the applied d-c magnetic field is zero, and (b) the signal is small compared with the saturation magnetization. The flange is assumed to be infinite in x and y directions. It will be shown later that the finite size of the flange has only a minor effect on impedance properties, resonant frequency, bandwidth, and efficiency, although its effect on beam pattern may be significant. It is not necessary to assume that the frequency of excitation and the size of the waveguide are chosen so that all waveguide modes except the dominant (TE10) mode are cut off, although the network transformations introduced later are more reliable in this region. 12

Assume a dominant mode wave incident from the left. It will be shown later that the analysis can be generalized to take into account any incident mode. Information sought includes the complex dominant mode reflection coefficient and the aperture fields as a function of p r, r' and frequency. The far field beam pattern, the bandwidth, the efficiency, the resonant frequency, and the maximum effective aperture of the finite radiator of Fig. 1 are also of interest. Assuming that there is no Ey, the fields can be derived from Hertzian vectors II* of magnetic type with a single component in the y direction. The problem is split up into two regions [Fig. 3(b)], Region No. 1 including the semi-infinite waveguide and Region No. 2 including the half space z > 0. In each region a general solution is constructed. 2. 2 General Field Solutions 2.2. 1 Region No. 1 —Waveguide Fields. The fields in the waveguide satisfy the wave equation for the Hertzian potentials: (V2+k2) II (x,y,z) = 0 (2. 1) (V2+ k2) II* (x,y,z) = 0 where the electric and magnetic fields are: E = V xVxII- jwa V x II* (2. 2) H = jwc V x II+ Vx V xII* If we expand the above expressions in rectangular coordinates, it becomes apparent that, if II* = II* = II = 0, (2.3) X Z E will be zero and all of the other components will in general be nonzero. Similarly, if Y 1n = nz = n* = 0 then H will be zero and all other components will in general be nonzero. x z y This corresponds to the fact that the total field can be broken up into LSE (longitudinal section electric) modes and LSH (longitudinal section magnetic) modes derived from, respectively n* and H vectors. Similarly the total field could be broken up into LSE and LSH modes modes derived from Hertzian magnetic and electric vectors in the z direction, or into TM and TE riv frm H13rtzian magntic and electric vectors in the z direction

In our problem, since there is no Ey, the fields can be derived from a Hertzian magnetic vector II*. The fields are then y E = - jwA VxII* y y (2.4) H = V x V x II* y - y Solving for II*: y (+ k2) II y = 0 (2.5) where II* = 4(x,y) e+rF y (V2 + k2) (x, y) e z a +2 + L + (k+21 (x,y) = 0 [ ax2 ay2 J (VT +k') 4(x,y) = 0 where a2 a2?= a + ax ax2 ay2 (V2+ kc) X(x) Y(y) = 0 T C x Y x + = -k2 x y c - 2 Yk X + kz = _ Y = k2 x c y y X+(k2 -k2)X = = X+k X c y x 14

Y + k2 Y = 0 y cos k X cos ky Y(x,y) = (2.6) sin k X sin k y (2. x y Applying boundary conditions to 4/(x,y): 4i(x,y) = {sin kyy} {cos kxx} (2.7) oo oo A r z mn m Cos nfrx mn H1 = L rF sin a s b e m=O n=O mn oo oo B -r z mn m n x m n + m sin m cos e (2.8) m=O n=O mn We have chosen the mode coefficients to be A /r for convenience. mn mn r2 = (k )2 - k2, r = j mn mn mn mnm k = my a n7r x b k2 = k2 + 2 n2r2 c x y a2 b2 k2 = o2 IE m2 2 n2 n2 22 r2 = + ni t E mn 2 b2 with the assumption of a single dominant mode wave incident from the left, this becomes: oo A r z B -rz 1 mn s. mn 7ry nnx mn 10 T.ry 10 zI1 E rmn sin cos b e +- r sin-e (2.9) m=0 n=O mn 10 a 15

we make the following choices to simplify the form of E l: B = - 1- (implies a choice of the magnitude of the incident wave) 10 W/10 A — R 10 410 R = reflection coefficient for dominant mode E. x with these definitions: - zrlOZ rcZ - oc' A r z] (e -R e sin ry mn m cos nx mn 1 1 0 A sina+ s sina cos e (2.10) [L10 nm=l n=O mn E2 = +Rne R 10 sin -7TY xl m a oc o r z - 2 x0A sin my cos - e (2.11) mn a b m=i n=O where the prime on the double summation indicates that the 1, 0 term has already been removed. 2. 2. 2 Region No. 2 —Fields in Half Space. The Hertzian potential in Region No. 2 may be expressed as: a b -jkr H ^ (x, = -Y,7) -f f^ f(, In) d5 dij (2. 12) 2 2jo - o 00 r where r = /(x-)z + (y(y-)2+ z2, the distance between source and field point, and f(f,q) is an arbitrary source function. It can be shown59' 60 that: f(f,77) = E x(f,r) (2. 13) where Ex(5,?) = E x(U,,0) An alternate method of deriving the potential function is as follows: The aperture fields can 16

be replaced with a magnetic current sheet M = E x n backed by a conductor. 61 -s - Then the method of images can be used to replace the original problem with a magnetic current sheet M = 2E x n. In this case our only sources are magnetic sources, -- s so that the fields can be derived from an electric vector potential F. 62 E = - V xF (2. 14) I a b -jkr F = f f M e ds oo + 47 I - r 0 0 a b -jkr = f f 2E x n dx dy (2. 15) o o /1 a b e-j kr A F = - f f E dxdy y - -(5,2 x r a b -jkr F = -2r E r dx dy (2. 16) Y 2 2T J X r F =F = (2. 17) The F functions used here are related by a constant to the II* functions, just as the vector potential A is related by a constant to the vector potential II. / I a b -jly A E = +Vx ( aJ Ex dx dy) (2. 18) This definition of the electric field agrees with that derived from the Hertzian potential function II. It is interesting to note here that the planar geometry greatly simplifies the analysis. If the geometry of the plate was such that image method could not be used, then the representation of the fields would involve the Green's function for that geometry instead of the free space Green's function used here. 2. 3 Derivation of Variational Expression for Admittance In this section the E and H fields are equated at the boundary between Regions x y 1 and 2, the unknown mode coefficients A are eliminated, and an integral equation for the m 17n 17

aperture field Ex(x,y) is obtained. Then a variational expression for the normalized aperture admittance Y/Yo = (1-R)/(1+R) is obtained from the integral equation. Boundary Conditions We have already included the boundary condition that tangential E fields go to zero on the plate in our expression for II* [Eq. (2.12)]. a* aII i * az a- 2 (2. 19) az az r all* all* 1 2 1 (2.20) ax ax E r/(2 rr a2 II a2 n* yix = ayx2 (2.21) ayax ayax ( 2 ) II = + k II (2. 22) \ay2 ay2 0 2 a2 (.;-n*) ay = 0 (2.23) ay ax where rL, E r are relative permeability and relative permittivity. e = E e r o = /rl/lo We wish to form the quantity (1-R/l+R). Differentiation with respect to z yields a 1+R term and differentiation with respect to x or y yields a 1- R term. We will use boundary conditions (2. 19) and (2. 22). Ex = Ex2 (at z = 0): Ex(x,y) = (1+R) sin - m Ansin cos 1 (2. 24) 18 m 18

Multiplying Eq. (2. 24) by sin 7ry/a and integrating over the aperture: (1+R) = ab j Ex(, s) sin a d5 dr (2.25) o o multiplying Eq. (2. 24) by sin m'y cos and integratingover theaperture a b ntegrating over the aperture: ab Jf Ex(Q,7O) sin "mrI cos nr d da7 A ab (2. 26) a b dmd'A 4E'n o o m'n' where E mn = 2 (n =0) mn 2 = 1 (no0) switching from primed to unprimed notation: - 4E a b An = am b S Ex(,,) sin m7 cos - d dij (2.27) m' d dn (2.a27) 0 0 Hyl Hy2atz 0: j [( a2) a "'lO + (J +(k2 ~) f' sin cos a2- r~ a rio ay2 m= n=l mn (:y )1a b /,2 \-jk r 2-1 o[J E (2 k0') d5 d] (2.28) 0 L- 019 r

Substituting for A from (2. 27): mn o r f)f E Q"W 2 -kord sin - (1-R) F10 [J E( mn a r m1 n=0 abrrmn [ Ex(O' O sin M7T 7 nff 5 d~ ( a- + sin Cn7 cos d + k sin (2. 29) a C jb d a b Interchanging the order of summation and integration and transposing: a b a2 - Jkor ( a2 \ oo c, 4e _ e 2 mn m 7M.mfy ni7x ff E ( 7) - + ) )+ +k2) S S abk i sin cossinmrcoSn7X d 00 L\ay' \ m=1 n= r mn G (x,y,,) =- sin m ^10 (2. 30) 2oo~~ r S Sa EX(, 7) G (xIyj r11) d5 d = -sin (1R)r~ (2. 31) Multiplying by E (x,y) and integrating over the aperture: a b a b Ar f J S f Ex(x,Y)Ex(,)77)G1(x,yI, 7)dxdyd4d?7 1 -- R= - (2. 32) ab r10 J' Ex(xy) sin aY dx dy 0o Dividing (2. 32) by (2.) 25): a b a b -R-abur f' f f Ex(x,y))G(XY,,7 ) EX(_,7B) dxdy dd _ -1- R _ r 0o o o o jB 1+R = 10 b2 Y Y Y IS' Ex(xy)sin a dx dy Lo o (2. 33) 20

This is the final variational expression for the input admittance. Upon setting, =,O, 63 e =, this expression agrees with Lewin. It will now be shown that this expression is stationary. Proof of Stationarity: Elimination of R from Eqs. (2. 25) and (2. 31): a b (-R) f Ex(,1) Gl(x,y,,7) dd7 = - sin y (- 10 o o r r \ o o = - sin FTY ( a J a E(7,1) sin 7 d dq + 2 (2.34) ab 2 1'0 y D (y / D fS Ex(,)Gl(XYt,)dR d7 = - d0 sin - (1 -ab (2.35) oo r \ / where a b a b D = Jf Ex(,I7) sin a dd?1 = S J' Ex(x,y) sin - TY dx dy 00 a 00x a This is an integral equation for Ex(, 77). Consider small variations 6E(, 77) from the coro x rect value as determined by (2. 35). Taking the first variation of (2. 33) gives: x a abr 10 a b (u) dem -nm i nom 1 f Ex(xy) sin dx dy a b a b o o o o (num) f f f f E(77) Ex(x,y) G1(xy,7) dxdyd d7 0000 ~~~abab 21 a b a b

a b \ 6(denom) = 2 f f Ex(x,y) sin - dx dy / a b \ ( J f 0Ex(x,y)sin'Ty dx dy \o o Symmetry of G 1(x, Y,, t): -jk r Gi(x,y,,?J) = (ay2 +ko) er a 2+ 2\ c 00 r mn M7 n7 m7ry nx + m + k z m. si cos sin " 1L cos ay2 m=l1 n=O ablrr sin a G1 is symmetrical in x, 5 and y,7: Gl(x,y,,77) = G1(, 7 x,y) because -jkor 00 4E e d v m' mn.s n m ry n7x r and m nO abMlcosrn sa b a b m=1 n=O r mn are symmetrical and differentiation twice with respect to y introduces either constant multipliers or multipliers of form (y-r_)2. Therefore a b a b 6(num) = 2 f f f f E.x(,7) 5Ex(x,y) G1(x,y,a,i7) dxdyd5 dt7 0000 2r10D4 L0o o o o a b a b - 2D J' f f f Ex( ) E x() E(x,y) Gl(x,y,,7) dxdydt d) o o o o (ab 6E(,) sin 1T71 di di)] (2.37) 22

Evaluating the individual terms from (2. 35): fS E x(,)G1(x,y,,71)d d d = sin a) 1 - ) (2.38) Multiplying by 6Ex(x, y) 2D2 and integrating over the aperture: x a b a b 2D2 J f f f Ex(0,7) 6Ex(x,y) Gl(x,y,,r) dxdyd5 d7 o o o o 4D2 2r10 1D a b - r r abK ) J' S n f sin ay 6Ex(x,y) dx dy (2.39) /r \ b o o Multiplying (2. 35) by E(x, y), integrating over the aperture, and multiplying by a b 2D f f 6Ex(5,1) sin 7 d5 dr: o o 00 aba b a b 2D J J' f J' Ex(,I1) Es(x,y) Gi(x,y,,7/) dxdyd~ dr f f Ex( r7) sin 17d d d) o o o o \oo / -4r1D 2 / \ ab l —-r 1 - D J' f 6Ex(,) sin 1T7 d d dr (2.40) Ar a 0 \ ~ a Substitution of (2. 39) and (2. 40) into (2. 37): 5y = 0 (2.41) Therefore Eq. (2. 33) is stationary with respect to small variations of Ex(x,y) about its correct value. A variational expression for the input admittance of a material loaded rectangular waveguide opening out into an infinite ground plane has been obtained. The next step is the evaluation of this expression for a given aperture field. 2. 4 Evaluation of Variational Expression for Admittance 2. 4. 1 Reduction From Quadruple Integral to Double Integral. In this section the variational expression for admittance will be evaluated assuming that Ex = sin 7. This 23

assumption will be justified in later sections both by experimental measurements of aperture field and by the results of higher order approximations. Substituting E = sin Ty into (2. 33) gives: x a aa b a b -ab r f f f f Ex(X,y) Ex(,71) Gl(x,y,,T7) dxdyd5 dl Y G jB o o o o - -- + y (2. 33) o o o r a b 2 2r10 Sf Ex(x,y) sin y dx dy 10 x a LO 0 J -2tr a b a b f= J' f sin y sin 1 G (x,y,t1) dxdyd~ d77 (2.42) 10 ab a a 1o -2,u a b a b a2 \/ -= r' J J sin Y sin (2 + ko) e dx dy d 0o 0 0 0 / / ( ay (2. 43) Replacing a2/ ay2 with - a82 / ay a? and integrating twice by parts with respect to y and 77 gives: u = sin 7y dV = ( - a a dV y a 27( r / -jkr du = sin d cos dy V = a a- a a a77 2 7Tr b b a a 2/ -jk r = dx S d S d J sin sin Z aae )d ( 0 0 0 0 0 b b ( - e0 f dx f d4 f dq J sin 7-7 sin a "y _ dy o 0 o o 2 a a-jkor 0r i 2 f a ed (T sin 5 cs cos -17- dy -f dx 2drdr a a o b b-jk r Irr- a e. J'dx f df dy f sin cos a de (20 44) a a a?) 2vr 24

u = sin 7 cos y dV = d a a'a 27Tr -jkor e du = cos c cos = eV = a a a 2 rr ~b b a -- a -jkr - f dx f d d e sin Cos - f os cos y e d _ 2__r a a a a a 2_r o o 0 0o 2 a b a b -jkor a- J Sfff cos 2 cos -] e dxdyd~ dd (2.45) a oa 2ooo 10 0 0 a(0a (2. 46) At this point we note that r is a function only of the differences of variables (y-77) and (x- ). If, therefore, we make a transformation to sum and difference variables, we will be able to integrate over the sum variables, leaving a double integral in the difference variables. This is an essential step because the quadruple integral is not practical for numerical computation. Now introduce the change of variables: y-7 = X x- = a y+77 = a+jl x+ = b+v (2.47) This transformation consists of a 450 rotation of coordinates, a translation, and a linear magnification of f2 (see Appendix A). The transformation is shown in Fig. 4.....fd-jb +o aos d Y - =1b Sd _ S e do S d + k + cos 2lr~ f a2 d e a 250 0 0aa Yo ob oT2/X2 + (~2 o o 25

/ a,/I. // N /\ a,, \. ".1,y..''-, xN Fig. 4. Coordinate transformation. * r (b - ) [(a-) - _ ) I a + a ( 2 a) a]si 2 e dX " G jB (2.48) o o This expression also agrees with Lewin after setting i = py, e = e. The integrand has a singularity at the origin. The contribution of the area near the origin is significant, as shown by a computer printout, so it cannot be neglected. The next section will be concerned with the evaluation of the contribution of the small area near the origin. The double integral expression (2. 48) would be suitable for numerical integration were it not for the singularity at the origin. 2. 4. 2 Elimination of Singularity in the Normalized Admittance Function. Separating (2. 48) into its real and imaginary parts gives: - 2pi (k~a2' -^ a br ~ C ~.- sink 2 + a ~r2 G = r S - (b-a) 7T(a-X)COST+aDsin --- du dada T b ~ ~ b1 " (2. 49) 26 26

2M (k2a2- i2) a bF C 1 cosk X+a2 o2 - = ~ - J' S (b-a) 7T(a-X) cos a D sin da d 1722 b 2 2 2 ~0 L J (2.50) where C = k2a2 + 2 D = k 2a2 - 2 1 0 transforming to polar coordinates: X = r cos 0 a = r sin 0 G 21tr(koa ) 1 asec a snk rdrd G) + a 1sin 7 rr cos 0 sin k r dr d = Y'r(110f j { Rse08 (b-r sin ) if(a-r csios S ) cos k dr d O J Dl 2 aa b.,/k2a2- 0 1 a/2 becscl ar cos 0 + f 72/2bc 0 (b-r s +sin r cos a k r dr d (2.5 larity at k = sn (a-r cos cosmplicated. Th double inte gration in and r has the advantage of having no singularities. However, if we split the 7a a b O 772b2 csc0 r cos 10 ~a/2 bcsc (b-r sin0) 7(a-r cos +) cos a a-sin~ s cos k r dr do 0 k 1 aa] 0 (2.52) The transformation from rectangular to polar coordinates eliminates the singularity in the double integration in x and y occurring at the origin. The integration in r can now be carried out to give a single integral in 0. However, this single integral has a singularity at k a0 = ff cos 0. Moreover, the formulation is quite complicated. The double integration in 0 and r has the advantage of having no singularities. However, if we split the rectangular area into sections of equal increments in r and 0, the irregular shaped areas remaining require weighting factors which vary with the ratio b/a. Thus the polar coordinate forms do not appear to have significant advantage over the rectangular form. Accordingly, 27

both the rectangular and the polar forms have been used. The integral has been split up into two parts. For the first, a small rectangular area including the origin, approximations are made for small X/a, a/b and the integration is carried out explicitly using the polar forms. For the remaining area, the integration is evaluated by computer using the rectangular form. The two contributions are then added. The evaluation of the integral for a small area including the origin proceeds as follows: Evaluation of G: C1 2 22 (k2a- ) 1 Xsec0 aG 2Mr(koa o ~ 1 cos s in a )sin k r dr do C1 1T/2 C ac1csC a a in dd + f; / (b-rsin ) (a-rcos )cos ros + sin (r c si n k r dr do 9Y o o cos 8 (2. 53) where = tan1 b/a A 1 = Qa, a1 = Qb. In the computations for Fig. 5, Q =. 05.. 1 and a1 are the dimensions of the small rectangular area including the origin. Let c rTr cos 0\ 1 2r2 cos2 0 cos ----- = 1 - -- - a /2a2 sin k r = k r o o 28

Integration yields: C C1 Ya E 2 G 21Lr(k a2- l 2ka) b Q2 (for small Q) (2. 55) This represents the contribution to the total value of G of the small rectangular area includb g 2b +1 -lb- Q2 ( + - ( — + k)+.06Q -.616Q 31r(k O a3 ) [b d1) bd) 2 2_ d 2 a _12)](2. 5) 29 b 1 b2 1 a (k - )(ka27T+ k2a) + b u a (d —-- D + ad1 bQd (for sma ll Q) (2.55) ~ ~~~B2,(inkCg the (forigin. small Q) (2. 57) D0 Vk^2-2 LL a (- 1 Tr~~a2-^ L j (/C1 1)~~~~~2

where f = logtan 1 (a2 4 1 g = log tan -2 d = + 1 = sec 01 This represents the contribution to the total value of B of the small rectangular area including the origin. A computer program was prepared to evaluate and combine the contributions to B and G of the large and small areas. Results are shown in Figs. 5(a)-(j). This represents the basic aperture admittance data from which later calculations of bandwidth, efficiency, and resonant frequency will be made. Each figure represents a given choice of r., e. The data is plotted versus normalized frequency FN (FN = ka/7T, FN = 1 at cutoff). As expected, the ratio B/G increases rapidly with the /irEr product, indicating a rapidly increasing reflection coefficient. Near cutoff, both B/Yo and G/Yo change rapidly with frequency because of the factor 1/v/k2 a2 - i2 in Eqs. (2. 49), (2. 50). The behavior of B/Y with frequency is approximately that of a series resonant L-C circuit resonant at cutoff frequency (FN = 1). The data of Fig. 5(a) agrees with Lewin's data for the unloaded waveguide radiator. 65 As a check on the computation, two different increments were used in the integration and their results compared. The maximum error in the imaginary part of the admittance was. 7 percent. The integration in G involves positive and negative parts which nearly cancel as the / rEr product becomes large. Correspondingly the error in the real part of the admittance increased to a maximum of 2. 5 percent at Mr = r = 10. It is.estimated that the computation is accurate to within a few percent. Normalization of the Data: For all the data presented, lr is- equal to E. However, each figure with a given Atr and Er may be expanded to give a family of figures with the same pLre product but 30

.5,4.3 - | --- --- - 1.5 Yo, I11 I I FN - Fig. 5. Aperture admittance of a loaded waveguide radiator (F N =1.1 Oat cutoff). (a)| / I r'=. 0. 31

.3.2.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1~ ~ ~~.8 o'~Pt~tttttt tFN m e.7 B G Yo... ----— ttW~lr~~f~H tHff Yo -2.3 =23.C~fb~Ft~bM~m+WHM-Wfmt~............. -5. -.6....... -7..; O 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN Fig. 5. Aperture admittance of a loaded waveguide radiator (FN = 1. at cutoff). (b)M =E =1.5. 32

.2 -2 t ~~:-.4:.'.'.~:::::::;:::.. -I.O -1.4 -1.6 i.I 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN Fig. 5. Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff). (c).r = er = 2. 0. 33

.2 -.4 -.8 -.10 t12 B' Yo -.14......... -,18;20 -,22 r.24 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1,9 2,0 FN -O Fig. 5. Aperture admittance of a loaded waveguide radiator (FN =1. 0 at cutoff). (d) i =r = 2. 5. 34

O i..i. i o 4 -I. 2 1. 1 i 1.6 2 t, l,. j................c..l)...... - --- -2.0 -2.8 3.2 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN Fig. 5. Aperture admittance of a loaded waveguide radiator (FN = 1. 0Oat cutoff). (e).r = r = 3.0. 35

C-2. 0 0 15I I ------ II.' I I.,I1 I,':llllllll I; 11 I I ItiS-ll|!1 ItI I 1f11 -|11 1 N1. I.1 II1 -2.0..|'' |'.. d I N N' 15 -1.0, 1 1 M I. \...... I I.I I I I | I N I Ml,. ll',E I I'llllll'1 I II I MI II llll' ll 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN - Fig. 5i. Aperture admittance of a loaded waveguide radiator...ilt 1111IIII I (FN =. Oat cutoff). (f) I Ir TJI 4. 0. 36

-I.0.20 -2.0,15 -3.0 t.. - ~~~37~~~~~~~10 -4.0 5.0

-1.0 -2.0.20 -3.0 -4.0 -5.0. 15 -6.0 BN G Yo Yo -8..01 FN Fig. 5. Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff). (h) r = er = 6.0. 38

-2 -3 -4 t. 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN G -7.05.04..................0 2.01 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN Fig. 5. Aperture admittance of a loaded waveguide radiator (FN = 1. at cutoff). (i) /'r r =e =8. 0 39

-2 -3.04 -5 -6.03 Yo -7 -8 yoY.02 -10 -13 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN Fig. 5. Aperture admittance of a loaded waveguide radiator (FN = 1.0 at cutoff). (j) r = Er = 10. 0. 40

different ri /e ratios. This additional data is obtained by multiplying the real and imaginary parts of the admittance by tLr(actual)/ ur(figure). For instance, Fig. 5(e) can be converted to (jr = 4. 5, e = 2) by multiplying B and G by 4. 5/3. 0. The basic data shown in Fig. 5 is checked (a) by a direct measurement of admittance and (b) indirectly by measurements of resonant frequency. The experimental checks are described in Chapter 6.* 2. 4. 3 Normalized Admittance for Small Apertures. For electrically small apertures (koa small), or, alternatively, for moderate values of ka and large values of fr Er (such as in Fig. 5(j)), the results for the real part of the impedance decrease in accuracy due to the fact that the integration involves negative and positive parts which nearly cancel. This section will remedy this inaccuracy by integrating the expression for admittance using approximation for cos kor and sin kor. In addition, the asymptotic expressions derived will provide some insight into the behavior of the admittance functions. The approximations: sinX = X-X3/3! (2. 58) cosX = 1 -X2/2! where X = k /X2 + a2 were used. These small angle approximations can be made for large /ir and r because the aperture is now small in terms of air wavelengths. Evaluation of G: 21i(k 2 2 a2-2) k a b F2C)C k2(X2+ a (2) f J (b-r) (a-X) cos/ 3/2 (for sin-ma (2 61) Y - a D a11 6 a2b 2 k 2 0 0 L 1 JL j a b ~k'- f/k(2. 59) Integration gives: (k2a2 r 11 (k b)f3 [(may dominat 2br 0(k b) +I1 0 1+) + 32+ 2- 4)1 3) -.21T2 0 3 k2a2 2 \ 367v2D 1 (2.60) G 4 ka 3 b 1 4 1/2 E 3/2 (for small k b) (2. 61) 0 kga2-72r r *It should be noted that experimental verification is limited to a narrow range of values. It has been suggested that, at points where I1- RI << 1, the higher order fields may dominate, thereby invalidating the assumption of dominant mode aperture field. 41

Evaluation of B: Cl c1 2 2 a2 _ 2) aa b 2 B 2r(o 0 f f (b-(r) (a-) cos a sin do d 0 abir/ka2-12 o o /+ (2. 62) Transform to polar coordinates: A = r cos 8 a = r sin 0 C1 2 (k2a2 -T2) z 01 asecO a /aD (__ rB 2 br 0 k 0 ) (ar c r cos rcos0 a + J f (b-rs sin ) ina-r cos 0) cos s + sin dr63) 01 = tan 1 b/a Integration gives: a2b = k2 a2 {2 (tan2.+ J /(b-tanr sein ) a-r cos(0) co n ) secs r2 + )r cos r ) where 0= tan-1 b/a Integration gives: a r ( ( 1 C1 a- ) / 2 D log tan + + 2 + D (1 - see 0) 0 T/k2 a,2- ff2 + J' a+ta 0 ee -co. b ctn 0 2 + sec 0 + C os - ctn 0 a tan sin ctn 0)] ( C1) [ (s b ctn 0)1 d} (2.64) a~r a D a ~1~ ~~O (2.64a C For smallk a, D1 -— 1 42

B _ /r_ I BY _ F1(_ F (0) (2.65) 0 k2a2 -2 1 1 where 01 = tan1 b/a F () = 4 log tan ( I + ) ( - 1) 7r2 11 7[iT/ + j' -tan 0 sec 0 [ - cos - 01 b a ffb ctn 0 a,rb ctn 0 1,~ + j sec 0 cos a - tan 0 sin d 01 + 2 2 sec 0 1 - cos n dO (2. 66) For small k a: o B ir a F 1 G - 4 b ka 3 (2.67) Figure 6(a) shows the approximate form of F (0I), which was extrapolated from the computer data for Fig. 5(j) (jr = e = 10). A check with additional computer data for Ar = e = 100 shows good agreement (within 2 percent) with the curve for F1(01). Figures 6(b) and 6(c) compare computer data for admittance with the asymptotic forms (Eqs. 2. 60 and 2. 65) for G/Y and B/Y0. Figure 6(b) shows that the accuracy of the approximate expression (2. 60) increases rapidly with increasing pIrer. At ilrEr = 6. 25, the error is approximately 7 percent for the particular case shown in Fig. 6(b). Further checks on the conductance G/Yo for other values of b/a between. 1 and. 9, and values of ka/7r between 1. 1 and 2. 0 show a maximum error less than 10 percent over these ranges. Figure 6(c) shows the accuracy of the asymptotic expression (2. 65) for B/Yo. This expression is somewhat less accurate than (2. 60); the error does not approach 10 per43

.6F (e,).4.3.2 / 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 1.0 4 Fig. 6(a). F1(1 ). cent until /r1e E 36. The form of the asymptotic expressions for B/Yo and G/Yo allows us to draw some conclusions regarding basic limitations of this type of antenna. For instance, the curve for F1(01) has roughly the shape of K(b/a) /2, where K is a constant. When this type of dependence is assumed, Eqs. 2. 60, 2. 65, and 2. 67 become: G K1 I-r kaa b -KM ~ /ka\1 (k ~3b^) (2.68) o k2a2- B K2/rK,r b 1/2 A ~B ~ ~ ~ ~ K~~ 2 (bl/2~ ~(2. 69) 0o k2 a2 2 B I! 1/2 I G K3 () (2.70) B/ G is a basic quantity, sometimes referred to as the Q of an antenna, in analogy to the Q of a tuned circuit. As noted before, B/ G is closely related to the inverse of the fractional bandwidth. B/G is proportional to (a/b)/2; thus reducing b by a factor of 4 will reduce band44

.5,, -^ — CCOMPUTED DATA (FROM FIG. 5).25 5 ASYMPTOTIC FORM ( EQ. 2.60) 2.065.05 ka 1.5.04. 7T.03 b/a =.4.025.02.015.02 I-lr r Fig. 6(b). Comparison of asymptotic expression (koa << a) for G/Yo with computer data. 0 t1 -2 -3 1- -.COMPUTED DATA, ( FROM FIG. 5 ) 3 — /, ASYMPTOTIC FORM ( EQ. 2.65) =: 4 5<-4 46^ 7-809 06 -5- b/a =.4 COMPARISON OF ASYMPTOTIC EXPRESSION (k a << 7r) FOR B/Yo WITH COMPUTER DATA 45

width by a factor of about 2. This indicates that the price paid for b-dimension size reduction is not too severe, in contrast with a-dimension size reduction. It is further noted that the ratio B/G depends only on the free space electrical size of the antenna, and is unaffected by the material used. However, as noted before, the magnitudes of B/Yo and G/Yo are directly proportional to r. A more complete discussion of these concepts and their relation to the general limitations is carried out in Chapter 7. 46

CHAPTER 3 THE COMPOUND IRIS 3. 1 Introduction This chapter is concerned with the effect on the radiating properties of an inductive, capacitive, or compound iris located in the aperture plane of the ferrite loaded waveguide radiator. The configuration is shown in Fig. 7(a) for the general case of the compound iris. For the inductive iris b1 = 0, b2 = b. For the capacitive iris a1 = 0, a2 = a. Figure 3(b) shows the division of the problem into two regions, Region No. 1 (waveguide) and Region No. 2 (half-space). A variational expression for the normalized admittance is developed and shown to be stationary with respect to small variations of the aperture field about the correct value as determined by the defining integral equation. The development proceeds in a manner similar to that of Chapter 2 for the admittance of the loaded open-ended waveguide. Once the variational expression is derived, it is noted that the admittance expression is not amenable to simple evaluation, even with a simple assumption of aperture field, due to the fact that the part of the admittance associated with the second term of G1(x,y, ),77) involves a slowly converging infinite series involving sine and cosine terms not orthogonal over the aperture. In order to evaluate the expression for the normalized aperture admittance, two different assumptions of trial field are used. The use of two different assumptions of trial field makes it possible to reduce the admittance to two terms, one of which is related to the admittance of the same iris in an infinite waveguide, and the other of which is related to the admittance of a loaded waveguide of reduced dimensions radiating out into half-space. Stationarity is destroyed for the imaginary part of the admittance. That accurate results are still to be expected can be shown by appealing to related variational expressions. 3. 2 Derivation of Variational Expression for Admittance The representations of the fields in Region No. 1 (waveguide) and Region No. 2 (half-space) are the same as those of Chapter 2. 47

Fig. 7(a). Compound iris in the aperture plane of a material-loaded waveguide radiator. 0 _ _ e10 cOc oc A r z [(e 1ZRe 1- mn mvy nvx mn II sin._ __sin__ cos n e "-ioli ri a r a - *1 -jw (e eE10 a m=l n=O mn a (3. 1) a b -jkr 2 =Jf E x( 7) r d dr d (3.2) 2 27jwlo o o0 0 The only differences are (1) the coefficients An have different values and (2) Ex(,77) = 0 mn x on iris. A development parallel to that of Section 2. 3 yields a final variational expression: 48

a2 b2 2 b2 J f f f Ex(x,y) Ex(,7) Gl(X,y,I,1) dxdy d do 1-R Y abler al bl a1 b 1+R Yo 2r10 a2b 2 If E(x,y) sin y d ddy (3.3) La b a and a defining integral equation: a2 b2 f f E(, 7)sin n d~ dn a2 b2 -21r ai b f J Ex(,C1) G1(X,y,, ) d dr = - sin \ - a b'' r (3.4) Using the defining integral equation above, and a development parallel to that of Section 2. 3, it may be shown that Eq. (3. 3) is stationary with respect to small variations about its correct value. Equation (3. 3) involves, because of the second term of G1, a slowly converging infinite series consisting of sin m- and cos -. - terms which are not orthogonal over the a b aperture. Thus the integrals associated with these terms will not drop out as was the case in Chapter 2. The following sections are concerned with the simplifications of Eq. (3. 3) to a form suitable for computation. 3. 3 The Compound Iris a2 b2 a2 b2 J j S Ex(X,y) Ex(5,r) G1(X,y, I,7) dxdy d dd Y ab/lr aI b1 a1 bl Y1 Y2 Yo 210 br 2 Yo Yo f J E(x,y) sin ray dx dy al b 49 49

J' jI f J' E(x, y) Ex(, 1) i L cos - s mn sn m cos - dx dy d d Y1 abr a b. a1 b. m= n r mn brmn O 10 a2 b2 S Jo Ex(xy, y)sin dx dy] a b a Y m-1 r J' Ex(xy) sin m cos- n dxdy 1 r-10 m= n=O __ mn x __ a ~Yo" r"~ 2 2 12 L b Ex(x,y) sin dx dy l b1 (3. 5) This expression has the same form as one-half of the admittance of the compound iris in an infinite waveguide loaded with material of characteristics 1r, E. 66 In particular, x if the aperture field Ex(x,y) is identical in the two cases (infinite waveguide and semiinfinite waveguide with infinite ground plane), then Y1 reduces to one-half of the admittance of the compound iris in an infinite waveguide. The admittance of the compound iris in an 67 infinite waveguide has been calculated by Lewin. The addition of material with characteristics A r, er does not change the normalized admittance if frequency is changed (or if the linear dimensions are reduced by / rT) so as to maintain f/f. Y /Y will now be evaluated for the compound iris: 2 b2 b2 2 -jkor j f f f E (xy) E (5,) +- k2 1 dxdy di Y2 abM a 2 b a bo x x ay2 ~ 2Tr c1yd Yo 2r10 a2 b2 2 Jf f Ex(x,y) sin dx dy (3. al b x a (3.6) At this point we assume that E (x,y) is separable = E (x) E (y). 50

Integrating twice by parts with respect to y and r gives: a2 b b2 -jkr b2 a2 b2 -Jkor Y S Ex(x,y) Ex(,) e dxdydC dt - f f J f E,(x) E x() El(y) El() e dxdyd^ d Y2 ab a L al b 2 al bl al bl al bl 2 i- rio a b 2(3. 7) 0 2 2 f r Ex(x,y) sin a dx d a bx a where aE (Y) E'(y) E' (n) = ay Ln y' y- =- - b a2 - a = dl, r' =r (3.8) 7' = 7 - a = a - bl b2 b= d2 d d d d - -jkr ~Y2 abler o1 2 1,,, a2 2 2e Yo 2 r10 r dl d2 (y+a1 2 1T(y' +a 1 Lf 1f E (x',y') sin a dx'y'J (3.9) This form bears some similarity to the dominant mode approximation to the aperture admittance of a loaded waveguide of dimensions d1 and d2 radiating into half space [see Eq. (2. 33)]. Let YA(dl, d2)/Yo be the normalized aperture admittance of a rectangular waveguide with broad dimension dl, narrow dimension d2, material loading r1, r, radiating' 2 r r into half space. With the assumption of dominant mode aperture field: d1 d2 d Id2 a e -jkor YA(dld12) d12di f Jr f Jo E (xy) Ex( 71) L + k 2 er dxdyd d7 Y r - d2 2 j kd-v L Jo Ex(x,y) sin dx dy] (3. 10) 51

L____-___J _ ___(2 r- ---— 1 r4) i a,!, - a,(4) aH H I I _ [ Fig. 7(b). Compound irises. In certain cases, it may be reasonable to assume that the aperture fields for YA(di, d2)/Yo and Y2/Yo are identical. This would be a reasonable assumption for the cases shown in Fig. 7(b), which include all symmetric irises as well as irises asymmetric in x, and multiple irises each one of which is symmetric in y. Cases where the iris is asymmetric in y would have to be treated separately with particular assumptions of aperture field in Eq. (3. 7). r d2 -2 Y2 d 2 f f (xy) sin Y2 a'b.k d 1 o o (3. 11) YA(dl d d2) d 2ddd2 r 1 (ya) 2 f2 fka S J Ex(x',y') sin a dx' dy' Eo o xy a Let Ex(x,y) = sin 1 (y-a1) (3.12) in the aperture of the compound iris. Fir2 /a^ / b\ ^A//2^ / d^ N2 2 2 2 2 b\ 7k 2 dl Y2 = 16 \ (a) ( d(2) ( 1a2) csc a YA(d, d2) 2 L d1 \ 2/ a/272\ a/ -I (3. 13) Thus Y2 can be expressed as the product of an algebraic factor which is a function of the iris dimensions times YA(dl, d2), which is the normalized aperture admittance of a rectangular 52

waveguide with broad dimension dl, narrow dimension d2, ferrite loading Mr,er radiating into half space [see Fig. 3(a)]. The real and imaginary parts of YA(dl, d2)/Yo approach infinity as kdl -- 7 because of the factor 1//k2 d 2 in YA(dld d2)/Yo The factor k/ cancels the poles at kd1 = T and produces poles in the real and imaginary parts of Y2/Yo at ka = X (the cutoff of the actual waveguide used) as would be expected. Thus, the admittance of the compound iris has been reduced to two terms Y1/Yo and Y2/Yo. Y1/Yo can be related to the admittance of the same compound iris in an infinite waveguide and Y2/Yo can be related to the admittance of a waveguide of reduced dimensions dl, d2, radiating into half-space. However, in order to evaluate the separate expressions, it may be necessary to use two different approximations to the aperture field because aperture field assumptions convenient for the evaluation of Y /Yo may not be convenient for the evaluation of Y2/Yo. This, of course, destroys the stationarity, since Y1/Yo and Y2/Yo are not individually stationary. Y1/Yo is stationary with respect to small variations about the value as defined by the integral equation for the iris in the infinite waveguide, but is not stationary with respect to small variations about the true value as defined by the integral equation (3. 12) for the iris in the waveguide radiating into half-space. This implies that results derived from different assumptions may not be as accurate as results derived from a single assumption of aperture field, although different assumptions of aperture field are sometimes used in variational expressions without comment. Consider the approximation: Y 1(E 1) Y (E 2) Y = Y I + Y (E1(X, Y) 7 F2(x,y)) 0 0 0 At first glance,such an approximation (using two different values of aperture field) might appear to lack most of the advantages of a variational expression since the proof of stantionarity is no longer valid. However, it may be shown that most of the advantages of the variational method are retained because of the stationarity of Y1/Yo with respect to the defining integral equation for the infinite waveguide case. Consider the Inductive Iris: 53

A convenient assumption for the evaluation of Y2/Y is E = sin -T (y- al) which 2 0 X d1 1 reduces Y2/Yo to the waveguide half- space problem without an iris in the aperture plane. Now consider the expression for Y1/Yo, which is identical in form to 1/2 the admittance of the same iris in an infinite waveguide, with, however, different defining integral equations for, respectively, the infinite waveguide case and the problem under consideration. If we use the same assumption Ex = sin - (y -al) in Y1/Yo, we would expect good results because of stationarity. It is not convenient to use this approximation, but it may be argued that the results, if we used E = sin -T (y - al) would be close to those obtained with Schwinger's6 x 1 approximation, which is often used to evaluate the admittance of the inductive iris. The primary reason for using the Schwinger transformation is to obtain an infinite series which is orthogonal over the aperture and which is summable. Typical results show a difference of only a few percent between upper and lower bounds. Because of the stationary property of the admittance of the inductive iris in the infinite waveguide, we would expect that the assumption Ex = sin (y - a) would also give very good answers for the admittance of the inductive iris in the infinite waveguide. It is not convenient to use this assumption in the expression for Y1/Yo but it may be argued that the results for Y1/Yo would be close to those obtained with Schwinger approximation. Comparing the result of using a single approximation Ex2 = sin (y- al in both Y1/Yo and Y2/Yo with the result of using Ex2 in Y2/Yo and Exl (Schwinger's approximation) in Y1/Yo, we see that Y2/Yo is identical and Y1/Yo is very close, in the two cases. Thus we have argued that the individual terms Y1/Yo and Y2/Yo are accurately evaluated. Similar arguments would apply to the capacitive and inductive irises. It should be noted that the above limitations apply only to the imaginary part of the impedance Y/Y = Y1/Yo + Y2/Yo. Since 6y = 0, the real and imaginary parts of the admittance must separately be stationary (6G = 6B = 0). If (a) Ex(xy) is real and (b) higher 54

order modes are cut off, then Y1/Yo will be purely imaginary. The only contribution to the real part of the admittance comes then from Y2/Yo. As long as only one assumption is used for Ex(x, y) in Y2/Yo, the stationary property of the real part is preserved. 3. 4 Special Cases —Inductive and Capacitive Irises Consider the special cases of the inductive and the capacitive iris. These special cases introduce a higher degree of symmetry in the infinite waveguide -case than they do in the problem under study. Therefore, the forms for Y1/Yo do not reduce to the corresponding infinite waveguide admittance forms without further assumptions. 3. 4. 1 Inductive Iris. a2 b a2 b Jf f f Ex(x,y) Ex(), ) G1(X,y,,T;) dxdydf d;n y ablr a1 o a ~1 O Y1 Y Y 2 Y+ Y o a2 b ~ o o [ 2f Ex(x,y) sin Ty dx dy al o x a (3.14) Let us assume that E(x, y) is independent of x. This is a reasonable assumption since the physical configuration of the inductive obstacle does not vary with x. Ex(x,y) = Ex(y) (3.15) Ex(, 7) = Ex(n) a2 a2 o IF mo s mymin f 2f Ey) Ex(7) r sin mL sin a dy dr7 Y a1 a1 m=2 (6 a a) I- = I1 1 (3. 16) f Es(y) sin y dy x a La1 The equation above is similar to the normalized admittance of an inductive iris in 71 72 an infinite waveguide. 2 If the aperture electric field E (y) is assumed to be the same as x in the waveguide case, then Y1/Yo is equal to one-half the normalized admittance of the inductive iris in the waveguide case. 55

The expression for Y2/Yo reduces immediately to 2 Y2 162 \I - 1 ( YA(d l b) (3. 17) o L 16 Jk2aT2 cos 2a 1 / 3. 4. 2 Capacitive Iris. a b2 a b2 ff f Ex(x,y) Ex(Q,1) G1(X,y,,77) dxdy d dr y abMr o b o b1 Y1 Y2 Y0 210 r b2 2Y f f Ex(x,y) sin'T dx dy o b x a (3.18) assume that Ex(x,y) is separable = Ex(x) Ex(y) (2 m2-I2 Y1 abu mn 2 a a Y abji m- k2'mv S) S Ex(Y) Ex(77) sin m7Y sin dm71 ddy Y' 210 mbln 00 br2 2 2^ ^ ^ d5 a bI b b 2 a 2 Ex(y) sin Y f Ex(x) dx 0 x a b 1 Let E(y) = sin - (3. 19) Then the integration in y and f] yields a contribution only if m = 1. emn =1 for all terms. 1 2 2 n 2 r10 n r1 J' f Ex(x) Ex()cos cos n dx d Y, 1 n=1 1n b b x x y 1 n=1 i 1 1 (3.20) o b2 E56(x) dx 56

This term is equal to one-half the normalized admittance of a capacitive iris in an infinite waveguide if the fields Ex(x) are identical in the two cases. The expression for Y2/Yo reduces immediately to: Y2 b YA(a, d2) (3.21) YO ~d Y (3. 21) where YA( )/Yo is as defined in Section 3. 3. The theoretical formulations (3. 17) and (3. 21) of this chapter for the admittance of a loaded waveguide radiator with an iris in the aperture plane are compared with experimental results in Chapter 6. Figures 8 and 9 include additional data on the admittance of a loaded rectangular radiator [Fig. 3(a)] including (a) data below (1, O0) mode cutoff necessary for the evaluation of (3. 17), and (b) data above (2, 0) mode cutoff. 57

b/a =.81.8 20 L \q L I ^ T ^ - __B 2-k2a2 \ 1.6 1.8 b/a =.8 1. 1.6 -h \ \. \ \\.- - 1.2 \ \\ / 2 1.4 -..N 1.2 \'2\.6 9- "x \ \\ 1. 0,, \ \\ 1.2 \ - 0.8\ 0.8 _1.0 - \ \\ - -8 \ \0 0.4 / -0.4 0.6 02 0.2-,;/' - 0.2 0.4_\ 0.0 0. 2.2 0.0.2.4.6.8 1.0 1.2 FN Fig. 8(a) Aperture admittance of a loaded waveguide radiator below cutoff (FN =1. 0 at cutoff). (a) /ir = E6 =1. 0. 58

blba=. 8 — 0__ _/yG 2-k-a2, | 0 Id _ yO -0 | 2.6 1.0 O.6 \ |"1 as 4 2.4.9\ \\ b/la=. 8.4 a \ / - 2,, 0.8- \ \ N, x ~\ -2.0 0.7- \ \.6 (F 11. tctf)8 =e A.6 _\\ 5 \ \\ -1.8,Ij~ - w\\ // ciO _~.6-\ \\ 044- \ - 1 0.5i. \ r\' 0.4 0' -1.2 0.3- 3 -.2 1.0 0.2 0.8 0.1 0.6 0.0 0.0.2.4.6.8 1.0 1.2 FN Fig. 8(b) Aperture admittance of a loaded waveguide radiator below cutoff (FN =1. 0 at cutoff). (b) /i = Er =1. 5. 59

t b/a.8 -~- ^^w A A —- o —-- B /'2_k2 2 I bla = 8. I5 -\ bla-.8 < WY -_ G /\~-k~a _ 3. 0 0 4 _.6~~~~3..40-'.4 \\.6 \. 13. F b2g. 8\ o\ o ~30- \ Z6'h, ~~.3~0 4~~626 V \*4 - 2.4'.20 —-, 2 " 22 ^ / / x2. \N \.2 1- 2. 8 - 1. 6.00 I I A 1.4 0.0.2.4.6.8 1.0 1.2 FN Fig. 8(c) Aperture admittance of a loaded waveguide radiator below cutoff (FN =1. 0 at cutoff). (c) j, = Er = 2. 0. 60

b/a=.8 T~ __ B-4 - \YO.40. \\ \ 40 (:\ \ -_3.8 \ \ \ bla=.8 0- was 3. 6 4 \ \ I' \ — \ X6.30 \ cmx 3 3.4 6\ ^ ^ /r i3\2 -.20 - 3.0 2.8 2 62.6.10 - 2 - 2.4 2.2.00 I 2.0 0.0.2.4.6.8 1.0 1.2 FN Fig. 8(d) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff). (d) r: =e r = 2.5. 61

b/a. 8 -- o —-— G /r2-k2 a2 Yo —, — 0 —-- -/T.20 ".6 5.6 bla =. 8.15F -5.2 FN N,.1u = oe 6. 43 4 - ^4. 4 0,S ^ - 3.6.05.2 Y/ / -/ 3.2 24.00o0II. 0.0.2.4.6.8 1.0 1.2 FN Fig. 8(e) Aperture admittance of a loaded waveguide radiator below cutoff (F =1.0 at cutoff). (e) /r = cr = 3. 0. 62

— ~ — ---- G7r2"k2 a2 - 0- - r2-k2a2 7.6 bla. 8.20'f- 7. 2.6 - 6.8 -6.4.15 6.0.4 bla.8 cm 0l|.10 / 6 -5.2 6o|.4 00.4.2 o05 6.-2.2 0 at 4.0 3. 6.00 3.2 0.0.2.4.6.8 1.0 1.2 FN Fig. 8(f) Aperture admittance of a loaded waveguide radiator below cutoff (FN =1.0 at cutoff). (f) +r =r 4.)0. 63

--— "" —- _' / l —- ayoB7b/a2k-8 *92.10 < —-'... b_ b0/a= 8 -.88 09. _ _6 s —.84 08- 0_ / -.80.6 -6.76.06 - 4 o 05 -.68 0 2 4 6 8 10 FN Fig. 8(g) Aperture admittance of a loaded waveguide radiator below cutoff (FN = 1. 0 at cutoff). (g) r Er =5 64 ~ ~ ~~64.03.2 —.60.56 01 -. 2 0o0.2.4 06.8 1.0 l 2 FN Fig. 3(g) Aperture admittance of a loaded waveguide radiator below cutoff (FN 1.0 at cutoff). (g) /rl i r- = 5.-0.

O m mI-_- -. 10o8 b/a =.8 o_ 10.4.6.10 o --— o.., 10.0 "" o _. 2k2a2 O~ 9.6.09 - B i 9.2 x.06.6_8 0v/ o A ( 4g *05 7 8&0 z'I 04 7.6.4 * 03.0 - / -7.2 0.0.2.4 6.8 1.0 2 FN Fig. 8(h) Aperture admittance of a loaded waveguide radiator below cutoff (FN =1. 0 at cutoff). (h) r =r = 86.0. 65

G 0 ~ Yo ~yo bla= _-.-4 1.1.45 C- -.90.45..90.40 X J>< / -.80.35 - X,' O.70 B.30 -.60 G Yo Yo o00.25 -.50.4.20 -.40.15 X 1.30.10-<. -.8 20.05 ~ —---- - *10.05.10 0.0 I I I I I \I 0.0 2.0 2.2 2.4 2.6 2.8 3.0 3,2 FN Fig. 9(a) Aperture admittance above (2, 0) mode cutoff (FN = 2. 0 - 3. 0) (FN =1. 0 at cutoff). (a) I-r = r = 1. 0. 66

-0-o-O GYo O ---- 50 b/a =.6 1.0.50 1.0.45 - 0_A c.2.90.40 0 70.40 / 0.80.35 70 B.30 / -.60 G Yo / Yo.25-.c 0-. 2.50 020- ~ 6 -.40.15 -^ -^ — e.30.10 - 20 8.05 -.10 0.0 I_ I I I I I 0.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(b) Aperture admittance above (2, 0) mode cutoff (F = 2. 0 - 3. 0) (FN =1. 0 at cutoff). (b) Ir = er =1. 5. 67

G 1.10 YO.50 - - B Y1.00.45 - -.90 bla=,,, S/ 1.8.40 - --.80.35 0-.4 60 0*1""' o70 B.30 - 000 z/o.4 -.60 G Yo Y.25 Ole.50.20.; 40.15 -,.30.30.10 sop.20.2.05.10 o0.0 I 0.0I I I 200 2.2 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(c) Aperture admittance above (2, 0) mode cutoff (F = 2. 0 - 3. 0) (FN = 1. 0 at cutoff). (c) 4r = e r = 2. 0. 68

.25 G bla'.4 - -" O -- y —.20 0- ye.15 -.90.8 10.80.05 070 B.206-9 ^^J ~.60 0 -.05.- -- 50.~~~~~10 ~.40~ 2.0 202 2.4 2.6 2.8 3.0 3.2 ~~~.6 ~~~69 0.0 2.0 202 2.4 2.6 2.8 3.0 3.2 69

b/a-.4 02 G.6 — o yoo /8 B Yo.8 0.0 ",6 -.10 / / -^^.40 ~// / / -.20 / 4 3 Q 1 d,'4 -.3 0 2.2 2.4 2.6 2.8 3.0 3.2 F-.40(F =1. at cutoff) (e) = 3. 0. 0107.I 0.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(e) Aperture admittance above (2, 0) mode cutoff(F = 2. 0- 9. 0) 70

-.10 -- - G b/a =2 2 -o B 1.4 -.20 - o.6.40 -.30 - -.40 - - 2..50 /.30 B -.60/ G ~~yo /" 7yo -h-0'Yo -/80 - / I I________ I I 0000.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(f) Aperture admittance above (2, 0) mode cutoff (F = 2. 0 - 3. 0) (F =1.0 atutoff). (f) = =4.0. 71

- 30 bla;.2 - -. — -401.2 -.10~-.6.40 ~~~~-.90~. 72" ~~~~.20 -.70 - / / // 0.30,ya Coo 4; -1.2 - /.20 -1.1t- /J^0 01-0 -1.3 ~ ^ __o__ 02 | I-t_______________________0.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(g) Aperture admittance above (2, 0) mode cutoff (F = 2. 0 - 3. 0) (FN = 1. 0 at cutoff), (g) r =. = 5. 0. 72

b/a -.2 -.50 G_ B -.60 Yo.20 -.70 / /.6.8 -/.80 -.90 - / 15 -B -1.0-o / / 1.6 - Yo 9Yo -1. 1/ / / 00dP.10 -1.2' oAl4 -1.3- A0 -1.5- " -1.6 r r~~~0.0 2.0 2.2 2o4 2.6 2.8 3.0 3o2 FN Fig. 9(h) Aperture admittance above (2, 0) mode cutoff (F: = 2. 0 - 3. 0) (FN =1. 0 at cutoff). (h) r = E = 6. 0.

b/a.2.80 — o — -10->.O -:8 |.1o -1.2 X.09 -1.4.08 -1.6 / /.6 /.07 -1. 8 B / 06 -2.0 -3 // ^ ^ - 0.05 -2.2, /O,/ // F/ 9 p -.04 -2.4 ^/ /,,, -.03 -2.6- c'o.,~. 2 -2.8.01 I. __________I I.._ I I..... 0.0 2.0 202 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(i) Aperture admittance above (2, 0) mode cutoff (FbT = 2. 0 - 3. 0) (F = 1.0 at cutoff). (i) IJt. = E= r.0. 74

-1.0 __ G bla -.8 B o -.11 -1.2 -.10 -1.4 -.09 -1.6.4.2 - 08 -1.8.07 -2.0 -.8 / B 0 -l.06 G Yo -2.2 Yo -l Ole 6.05 -2.4 -/ /^/ - o.04 -2.6 - - 4 04 -2.8 -2.8 - j/ vr~^ ^.022 -3.0 i^^~goo-".01 -3.2 2,0 2.2 2.4 2.6 2.8 3.0 3.2 FN Fig. 9(j) Aperture admittance above (2, 0) mode cutoff (F T = 2. 0 - 3. 0) (FN= 1. 0 at cutoff). (j) ir = = 10. 0. 75

CHAPTER 4 HIGHER ORDER MODES 4. 1 Introduction In Chapter 2, the first (dominant mode) approximation was manipulated into a form from which computation could be made. In this chapter an infinite sum of modes with unknown coefficients is substituted for the unknown aperture field Ex(x,y). The condition of stationarity is used to derive an infinite set of equations in the unknown coefficients, and a corresponding infinite series for the admittance Y/Y in terms of the unknown coefficients Amn and known quadruple integral functions Cfhl. The quadruple integral functions Cfgmn mn fghV are reduced to double integral form suitable for computation. In the process, it is shown that Cfghl is zero if either (f + h) or (g + 1) is odd. As a result, it can be shown that certain modes, namely those modes forbidden by symmetry, do not affect the admittance calculation. Finally, computation of the higher order corrections to the admittance for the (3, 0), (1, 2) and (3, 2) modes yield corrections of the order of 2 percent. 4. 2 Higher Order Mode Coefficients In this section, the aperture field Ex(x,y) is expressed as an infinite sum of modes with unknown coefficients. The condition of stationarity applied to the input admittance Y/Y0 yields an infinite set of simultaneous linear equations in the unknown mode coefficients. Substituting this expression back in the admittance formulation yields an equation for the admittance in terms of the unknown coefficients. In Chapter 2, the following expression for the normalized aperture admittance was derived: a b a b -abUr f f f f Ex(X,y) Ex(f,?) G (x,y,n,7) dxdyd dr Y G jB 00.o (4. 1) -Y d- Y 0o o 0 o 0 a b 2F10 S Ex(x,y) sin dx dy 76

Let Ex(x,y) = sin l + z Z An sin cos (4. 2) m=1 n=O Consider the integral in the denominator: a b a J Ex(X,y) sin 7fy dxdy = a 00 Y G jB -2ur -:= - + - 0 [0 + Q + 0 + (4. 3) where a b a b ( = f f f f siny sin -G l(x, y,,) dxdyd d7d 0000 a b a b oo oo {2 =1 1 f f sin a~ A sin cos G( x Y7) a msn a b cos-G(y,S,) o o o o m=1 n=O dx dy d5 do7 abab 00 00 ~O = JJJ si Amn fsin n Zrm sin cos G1(xy,,77) a mn a-b 1 o 0 0 0 m=1 n=O dx dy d dr a b a b o0 oo, m 00 0 - ff f f L ZAmn sin cos 3 mn a o o o o m=1 n=O m=l n=O Am sin cosn G1(x,y,' 7) dxdy d d77 ( = C1010 (4.4) where a b a b 2 -jkr Cfghp = Jsin f Ycosgx sinh os (ay+ ko) edydd 77a b a b 2r dxdyd~do 77

Chffg ngff (4. 5) O + Q are identical since Gl(x,y,,,1) is symmetrical in (y, q) and (x, ). 00 oo + @ = 2 Z ~ Ah C 10h (changing subscripts from m, n to h, ). h=l l=0 (4.6) 00 00, A2 0 00 00 (4.7) ) r f=1 g=0 rfg a2 / f=l g=0 h=l g=0 - C + ab Z A2 ~Afg fk2 - Y o - rab 1010 4+ 2 f= AhClOh f-1 g fg fg a- 2 00 00, 00 00, + z E z I A (4. 8) f =1 g= h1 O Afg AhCfgh (4.8) The property of stationarity implies aY/Amn = 0.,3 (2 h=1 Z hCh a aAmn f=1 g=O h=l1 A= fgAh Cfgh 2 f =1 g=O AfgCfgmn (4. 10) because of symmetry of Cfg^h (CfgMh = C Mfg) aA r ab 10mn + 2Amn 4 Ir a2 ) fL g^o fg fgmn(4. 11) 00 00, Clmn + AmnDmn + C ~g AfgCfgmn 0 (4. 12) n l h=l =0 f f=l=1 g=0 where 78 78

D = () (k a2-m ) mn 4 u r r mn This represents an infinite set of simultaneous linear equations in the unknown coefficients A. The solution for any specific A is therefore the quotient of two infinite determinants. mn mn The above condition on the unknown coefficients contains terms similar to those in the expression for Y above. Changing subscripts from mn to hf, multiplying by Ahe and summing over hf, we obtain cc oc, o 00 c, 00 o00 cc c r Ah l~C!0h + Ah2 Dh + A C 2 Afg CAhfgh =0 h=l h =0 10M h=l O=0 h" hQ f=l g=0 h=l1 =0 fg h= fgh (4. 13) Substitution of the above equation into Eq. (4. 8) simplifies the expression for Y/Yo' ~ F= _ G jB Yo r10ab 1010 h+ 1 AhC10h Y + Y 2Mr o0 o0, = L X A C (4.14) F710ab L 1010 + Z mnlOmnj (4.14) ab 1010 tm=1 n=0 mn 10m This formula represents an equation for Y/Y in terms of the unknown coefficients Amn and the known functions C10mn. Now the coefficients must be determined by the conditions placed 1Omn' on them by the equation o o, C10n mn mn + m n AfgCfgmn ~ m=l n=0 Expanding: [SEE EQUATION ON PAGES 80-81] Some of the terms in the expansion are zero. From symmetry considerations, it can be seen that: A2n A4n = A6n (4.16) Am= 3A m5= *A = (4.17) An = 0 (4. 18) 79

A11(D11+Cllll) + A12C1211 +... +AlnClnll +A20C2011 +A21C2111 +. +A2nC2nll AllC1112 + A12(D12+C1212) +.. + Alnln2 + A20C2012 + A21C2112 +... + A2nC2nl2 AlCllln + A12C 12n + * * + Aln(Dln+Clnln) + A20201n + A21C211n + * + A2nC2nln AC +AC +... +AC +A (D+C +A AllC1120 + A12C1220 + + AlnC20 +A20(D20+C0 2 0) + A21C2120 +. + A2nC2n20 AlC1121 +A12C1221 + + Alnn21+A20C2021 +A21(D21+C2121) + + A2n 2n21 A llCl 2n+ A12c 22n + * A Aln ln2n + A20202n + A21 212n + + A2n(D2n+ 2n2n) +A C +. +A C +A C +A 2n0nnO AllClln + A12C12nO +. AlnClnnO + A20C20nO +A21C21n +. * + A2nC2nnO All cln + A12C12nl + + A lnClnn + A20C20nl + A21C21nl +' + 2n 2nnl A llCnn + Anc lnnn 20 20nn 21 2nn + + 2nc2nnn 80

+... +... + AA l ll + + A nnl= n0 + +. + Ann012 + Annl2 +. + A nnC nn 012 +... +... +A C +A C +. +A C =-C + nO n + n2* + Ann nn ln2 02 +... AA + + Ann +Anl 2 + + AnnCnn20= +,.. + * * + An0Cn021 + AnlCnl21 +.. * + AnnCnn2~ nO...n n. nun nnACn COin +..+... +A C + +A... +A C C nO n020 nl n120 nn nn2n +... +...+A C + C + A C 0 nnOln0 n 21 nn nn20 +.. +.. A C +A+ C+... + A C = 0 + + + AnO(DnO+cnOnO) +AnlCnlnO + + A nn nno -ClOnO +... +...+ A C +A(Dnl+Cn ) +... +A C = -Cl nO nOn nOnO nl nAn nn nnnO lOnO +...+... +A Cn+nn An C n n -C1nn (4.15) + + A n n + Onn + An1Cn ln n + + Ann(D Onn (415) 81

The method of solution of these equations involves truncation of the infinite matrix. The solution will be accurate only insofar as the terms included represent the dominant terms in the aperture field. Several specific truncations will now be carried out 1. Two Modes —(1, 0) and (m, n) C +A D +A C = 0 lOmn mn mn mnmnmn C 10mn Amn C -+D (4. 19) mnmn mn Y 2 r c l0mn 10 mn 2 C~~Lr~ y 2 y (4.20) Y ~ r 10ab 1010 0 C +D ~ Y y Y y o 10 L mnmn mnj o o Y 21i C2 mn _ r 10mn Y r4 ab C +D (4.21) o 10 L mnmn mn 2. Three Modes —(1,0), (m,n), (m',n') Amn(C mmnm + Dmn) Amn(Cmnmn = - Cmn mn mnmn n Cm'n'mn + Dmn = - Cm'n'10 C m'nmnCm'n' 10 CmnlO(C m'n' m'n' +D.) Amn (4. 22) C n C - C (C +D ) A - mn m' n'lO mnmn mnA ~AMm I mnmn (4.23) where (C mnmn + D n)(C ) Cnm+ 2 mm8 m 2'n m'n'mn 82

2~ Y Y Y Y r A C 10 Y mn Ymfn 10ab (C1010 + AmnCmn m'n' m'n'10) = + + (4. 24) o o0 0 0 Thus Y/Yo has been reduced, in Eqs. (4. 21) and (4. 24), to expressions depending only on the quadruple integrals Cfgh2. 4. 3 Reduction of Quadruple Integral Forms The Cfghf terms will now be reduced to general double integral form. In the process, it will be shown that CmnlO is zero if m is even or n is odd, and that C mnm n is zero if (m-m') or (n-n') is odd. The calculations for the first form will be carried out in detail; the results for the other forms will be given without derivation. Finally some of the double integral forms are evaluated by computer. Calculations involving the (3, 0), (1, 2), and (3, 2) modes are carried out, yielding corrections of the order of 2 percent to the input admittance Y/Yo calculated assuming the dominant mode. a b a b a2 -jk0r Cmnmn = J J' j sinm7Y os mx sin m cos f - a e dxdyd d7t ~mnm nn 00a b a b o ay T7, 2vr mnmn 4a' o o o o (4.25) a ba m 1nmn' 4a2 J' Jo [cosj (nx -n') + cos (nxn [Dmm, cos a (my - m'r7) - Cmm, cos a (my + m )] -jkor r e2 dx dyd dr (4. 26) where C = k2 a2+ mr2 m o D = k2 a2-n r2 m o r = (x-_)(y-_+2 = y _2 let 83

y-) = X x- = a y+r7 = j+a x+ = b+v In this case we do not have any symmetry in the integrand (Appendix A). f(X,a,P,,v) ~ f(~+X~,+~l,+v) cos (nxan'n) = cos (nn)b (nn' n )v + (n + n') a] b os cos a (mym'1)= cos -- [(m~m')a + (m~m'))L + (m+') m A] Cnnn= 1 Jff Jf cos 2 [(n-n')b + (n-n')v + (n+n')a] + cos b [(n+n')b mnn n 16a2 + (n+n')v + (n-n') a]} {Dmm cos [(m-m')a + (m-m')L + (m+m')x] - C, cos +m -a mm -2a -jko /a + X2 [(m+m')a + (m+m')j + (m-m')A] e dv du du dX 217 /a2 X2 where a b a-X b-a a o a-X b+u o b a+a b-a o o (a+X) (b+a) J'J J J J f J +j Jf f f + Jf f J + f J f f o o -(a-x) -(b-a) o -b -(a-XA) -(b+u) -a o -(a+X) -(b-a) -a -b -(a+X) -(b+a) (see Appendix A). a b o b Cmnmn, J G1(a) F1(X) du dX + f f G1(a) F2(X) du dX 4a7r2 0 -a o a o oo + f f G2(a) F1(X) d dX + f f G() F2() d dX d (4.27) o -b -a -b J where { 1 n r n'+ra.n(rvl / n') 1. n[rall Gl() = { n') [sin b cos(n-n')=7 - sin b n' cos(n+n') + sin ]} 84

G2( = { ( 1) [sin n cos(n-n') si n (n+n') T + sin F1( ) = { m [sin m cos(m-mr')- - i + [sin a cos(m+m')7T + sin ]} (mm mm maa m+m' a a F2() = Dmm' [sin r cosm-m - Ir - m; [si - cos((m-+m)') + sin mn If m-m' is even F1(.) = -F2(X) If m-m' is odd F1(X) = F2(A) If n-n' is even Gl(a) = -G2(a) If n-n' is odd G (o) = G2(C) Since all integrands are odd in a and X F a b a b Cnm'n' J Gl(a) F1(X) da dX - J G1(a) F2(X) d d 47T2 a o 0 o00 a b a b - S G2(a) F1(X) do dX + J f G2(a) F2(X) do dX oo o o If m-m' is odd: The first and second integrals cancel The third and fourth integrals cancel If n-n' is odd: The first and third integrals cancel The second and fourth integrals cancel 1. (m-m') or (n-n') is odd, and C' = if (4.28) mnm'n' = 2. mrm' and n/n' We have noted before that the only modes excited by the geometry of Fig. 3(a) are those with 85

m odd and n even. If (m-m') or (n-n') is odd, then either the (m,n) mode or the (m',n') mode must be a forbidden mode. If (m-m') and (n-n') are both even: ~ / 1 \4 _ r. n'o-. nfrf a = f f [n'y- sin n i mnmn k (n 2-n' m2)(m -m) 0 -jk r [mD sin m' rx-m'D 2 sin m7rx] e dx dy m'2 a m 2 a 2Tr (where 6 = b/a) (4. 29) 2 — b2-1 J' f In' sin n' iTy-n sin niTy] (n2-n2) (m-m') even for (-n') even rn'2 m2 x] e ddy As a result of the above calculations: 1. m is even or n is odd, and CmnIO = ~ if (4. 30) 2. m and nO 2 2 11 -jkax2+62y2 Cmni0 - () n,2- 1 sin nry[mD1 sin 7x - D 2 sin mirfx] e dx dy mnl0 a nr(m2-t) o 1 m2 abo cacuai (4. 31) 1. m is odd and n is even, and if 4 [2. ml1, ndO 86

The results of similar calculations yield: Za b ia b, I / 2 -jk or nmn= o sm cos bn- sin 1 cos -- dxdyd2 e ma b a b 0ayaI7 2vr dydd.(n-n') is odd, and Cmnmn, = 0 if (4. 32) m~0 b2 1 1 1 ~aC = a 3 n —f fS n' sin n'ry-n sin n7ry] mnmn' a m3 (n-n')(n+n') O O -jk a x + 6 y2 [m7r(I-x)D 2cos mrrx+C 2sin m7rx] e dxdy (4.33) m m Vx2+ 62y2 1. (n-n') is even, and if ( (4.34) 2. mAO, njn' Cmnm = 0 if m =0 (4. 35) Cn 10 = 0 if n is odd (4.36) 2 1 1~i 10 -jkoa /Vx2 + 62vy2 CmnO = -b- (1)- J f sin n7y[17Dl(l-x)cos 7x+C sin7x] e dx dy 0 o o Vx2+ 52y2 (4.37) if n is even a= a bS sin00 m7y n1 m'v1 nbr a(k a2b -jk 0r cos a -b- 2srr dx dy d d mnmtn = J J in cs si e02vr dx dyd o 1. (m-m') is odd, and C m = 0 if (4.38) mnm'n n | 2. n/o 2 1 1 ^Cm nn = nb vTm-r-m'f f [nlr(1-y)cos nry-sin nry] nzr (m+m')(m-m') o o 87

-jkoa x + 62y2 [mD,2 sin m'ix-m'D 2sin mrx] e dx dy (4.39) 22 X2 + 62y2 1. (m-m') is even, and if 2. n/O, mrm' COm'O = O if (m-m') is odd b2 2 1 1 = (-) 2 (m+mmm) sf f (1-y) [mD sin m'7x-m'D 2 sin mrxx] a) 2 (m+m')(m-m') o' m2 m jkoa x2+62y2 0 e _dx dy (4.40) rx2+ 62y2 1. (m-m') is even, and if f 2. m rnrnm' CmO10 = 0 if m is even (4.41) 2 2 1 -jkoa Vx2 + 2 y a a () r2 )J S (l-y)[mD sin rx - D 2 sin mx] e dx dy a. (m2- o m (M _1) (4.42) if m is odd a b ab -jkor mnn J S sin m cos b sin cos (2) aadxdyd de mnmy 2vr dxdy da dbr a b -jk 0 +fffJ s cos -sin mnu nx~ (k2) e + f s Sna T sin M cos s (k2) dxdy d du (4.43) 0 0 0 00 b2 1 1 1 = () 1 f f [ni(l-y) cos nry - sin nry] 2mn~3 o o 88

-jk a x2 + 62y2 rmr(l1-x)D 2 cos mIx + C 2 sin mvx] e dx dy m m 12 VX2 + 52 y if mtO, nfO (4.44) Cmnmn may have a finite value even though (m,n) is a forbidden mode. mnmn a b a b a2 -jkor CmOmO = J f sin my sina (_ a2 k2) e r dy dxdyd (4.45) C omo 0 0 a a \ m ay a1%) dx d cOmO 1a m o Is (l-y) [mr(l-x) D 2 COS mvx + C sin m1 mOmO a m-) 2 m m -jkoa x + S y2 e do dX (4. 46) Vx2+ 62y2 if m O. a b a b -jk m n limnn k n f f f sin d sin mrcos IT < x ady d2 de m~ O o o o o IcOm nIlim f f- amm'a a b a b m-0 0 0 0 a ) O O O d di} OICnm n< l i m a 2 - dx dy d5 d, k0 a 0 a m0 a a = 0 0 C0nm'n' 0 = CnOn' (4.47) We have shown that 89

1. (m-m') or (n-n') is odd, and C,= 0 if mnm n' 2. ms m' and n, n' 1. (n-n') is odd, and C = 0 if mnmn' 2. mO/0 C - 0 Onmn' 1. (m-m') is odd, and C = 0 if mnm'n 2. n4O COmO = 0 if m-m' is odd C / 0 for (m 0) even if m,n is a forbidden mode. mnmn Therefore Cmnmn = 0 if (m-m') or (n-n') is odd (4.48) Using the general formulas (4. 28 through 4. 46) developed above: 2 1 1 1 -jkoa Vx+ 62y2 C303 = (-) -- fS (1-y)[3ir(1-x) D cos 3Tx + C9 sin 37x] edxdy 330302 o o x2+6 (4.49) 2 () 1 [ -jkoa x+6 2y2 C3010 = () J' f (1-y)[3D1 sin rx-D9 sin 37rx] edxdy (4.50) 4va2 0 0 Vx2 + 62 y2 2b \ 1.1 -jkoa Vx2 + 62y C121 =- (-)( 3f J sin 2ry[tDDl(l-x)cos ix+C1 sin x] e dxdy \21 /o + Vx 62y (4. 51) 90

=b2 1 1 C122= (-) - f J' [2r(l-y) cos 2y - sin 27y] [rDl(l-x) cos 1x + C1 sin rx] a 43 o -jkoa Vx+ 62y2 e -dx dy (4. 52) rx2 + 62 y2 b2 1 1 C3232 = (-a) 1 f f [2r(1-y) cos 2iry-sin 27y] [3T(1-x)D9 cos 37rx+C9 sin 37x] 3123 0o -jkoa Vx2+ 2 y2 e- -dx- dyx (4. 53) VX22+ 62y2 1 1 -jko Vx2 + 32 y C3210 = - (-~ ) f sin 27ry[3D1 sin irx - Dg sin 3rx] e dx dy 3210 a 8"3 0 0 1 9 Vx2 + 62y2 (4. 54) The above formulas (4. 49 - 4. 54) were used to calculate by computer the higher order corrections to the admittance for the specific case (ILr = 1, er = 10, b/a =.5, FN = 1. 5). Results are shown in Table 1. The largest correction is 2. 1 percent of the corresponding dominant mode term. Dominant Mode Approximation Y 0 = 041 - j.31 o Higher Order Mode Correction Using Two Modes 12 (1,0) and(1,2): =.000116 - j. 00602 30 (1,0) and (3,0): y = -.000628 + j. 00636 Y 32 (1,0) and (3,2): y =.0000145- j.000135 Table 1. Higher order mode calculations. 91

4. 4 Effect of Forbidden Modes It is shown in this section that forbidden modes (m even or n odd) do not affect the calculation of admittance and that the Rayleigh-Ritz procedure yields coefficient values of zero for all forbidden modes. Consider the variational expression for the normalized admittance with the addition of one higher mode: Y _ 2r C mn1 0 10 mn (4. I Y (4.20) o 0 10 L'10 C mnmn mn Y0o o where Ymn 2_ r F C mn Y rab C + D o 10 L mnmn mn_ for m even or n odd, the higher order correction Ymn/Yo will be zero if the denominator is nonzero. Im C < 0 (ka < Tr) mnmn b mn(k2 a2 - m2 T2) There are five ranges of interest: D 2 nM~2 ~ 0 (~ k 7a2 272 n2l 292 mn a 2' 2Im C m n> 0 ()2 < k2a2 < 2 2) shown in Chapter 5 m2 2 2 2 mnmn a b_ 1r- a b2 m2irm2 22 nv2 ImC < 0 (k2a2 > +n mnmn a2 b2 There are five ranges of interest: 92

0 0 S S o o U U + + o + 8 Q 8 5 tt + t 0S S Q ~: Ct + C +T + SC 0 44-4 Cd a) n d n E n E ^' " 3' - - - tet c^ 0 0 A N 4 0 4S S I'-4-3 A 8 + v E i E V A: A g V I f A Ia A V bD -~ C sd C) 0dc D c> cq Sa)~S S Vg- A - 4 - > - Cj C., CM 93 E ~ E co E u Cq 5:)4 E 93cm

Therefore the denominator is nonzero, and Amn, Ymn/Yo are all zero for m even or n odd. In fact, the analysis can be generalized. Consider a finite or infinite set of allowed modes (aa'), (bb'),..., (ff') and one forbidden mode (m'n') with mode coefficients Aaa' Abb... *Aff Am'n' In addition let: /f = m\ Dfgmn Cfgmn +Dmn = Cgmn (otherwise) fgmn Equation (4. 15) reduces to o9 o9 1 Afg Dfgmn = CmnlO55) 0 Expanding Eq. (4. 55) yields, in matrix form: D D D D A? C aa'aa' bb'aa' ff' aat Dmn'aa t aa' Oaat Daa'bb' Dbb'bb'.. Dff'bb' Dm'n'bb' Abb' lObb' (4. 56) aa'ff' Dbb'ff * * * ff' ff' m' nff' ff ff!Daa'mtn' Dbbtm'n'. * Dff'm'n' Dm'n m'mn _ ClOm'nm where: D D D C D =D D =0 aamn = Dbbmn, = Dff',mn, = 1Omtn mTnaat m n'bb' = mnff Thus all the coefficients Dgmn in the bottom row and in the last column of the square matrix fgmn are zero, with the exception of Dm tn mn. In addition the term C10mn in the last row of the column matrix on the right hand side of (4. 56) is zero. 94

Using Cramer's rule to solve (4. 56) Daaa' bbaa'... Dff' aa' C10aa' Daa'bb Dbb'bb'... ffbb' ClObbt A = 5.. (47) Daaff' Dbb'ff'... ff ff C10ff 0 0 0 where A = the determinant of the square matrix in (4. 56). Expanding the determinant aa'aa' bbaa... ff' aa' DA='nmn Daabb Dbbbb Dffbb =D'm' -f (4. 58) minmn ~aa'bb' bb'bb' ffbm n mf bb-f Daa'ff' Dbb'ff, * Dff ff' A D A (4. 59) D D D m'n tn'mn' a-f D aa bbaa' aa bb'aa aa Dm n' mn Daa'bb' Dbb'bb'... DObb' A _, = _ _ Daa'ff' Dbb'ff' DlOff' (4. 60 A (4.60 ff' D A m'n'm'n' a-f The coefficient Aff, has the same value as would be obtained from the Rayleigh-Ritz procedure if the forbidden mode (m'n') were not present. The coefficient Amn, of the forbidden mode is zero except, perhaps, at zeros of the determinant A af But, if the determinant A were zero, this would imply that the set of simultaneous linear equations for the allowed a-f modes were linearly dependent and thus had no unique solution. Since the mode coefficient A m, is zero and C10mn is also zero, the term Amn, mn contributed to the dominant mode admittance by the forbidden mode is zero. 95

It is possible to generalize further. Consider an infinite set of allowed modes and an infinite set of forbidden modes. The matrix of the coefficients may be partitioned into two submatrices, one containing only allowed mode terms and the other containing only forbidden mode terms. rA;I allowed n ] lOmm (4. 61) where: A = allowed mode coefficients allowed A.d = forbidden mode coefficients forbidden A = submatrix of Dfgh terms associated with the allowed modes F = submatrix of Dfgh terms associated with the forbidden modes. Now it is possible to write separate matrix equations for the allowed and the forbidden modes. The allowed mode coefficients (A allowed) will be independent of the forbidden modes and the forbidden mode coefficients will all be zero if the matrix equations for the forbidden mode coefficients are linearly independent. Several facts concerning the higher order mode coefficients and their use in the Rayleigh-Ritz procedure have been pointed out. (a) If a forbidden mode with an undetermined coefficient is added to a finite or infinite set of allowed modes whose coefficients have already been determined by the RayleighRitz procedure: (1) The coefficient of the forbidden mode as determined by the Rayleigh-Ritz procedure is zero. (2) The dominant mode admittance is unchanged from its value as determined from the set of allowed modes. It is evident that, in the limit, as the number of modes included approaches a complete orthogonal set, the mode coefficients will approach their actual values and the mode coefficients for the forbidden modes will all approach zero. However, it is not evident, without the above derivation, that the Rayleigh-Ritz process yields a zero value for the forbidden mode coefficients at every step of the process, whether a finite or infinite number of modes is considered. 96

CHAPTER 5 GENERALIZATION OF THE VARIATIONAL ANALYSIS 5. 1 Other LSE Modes Incident Consider the configuration of Fig. 3(a) with an arbitrary one of the higher modes incident from the left. Fields in Waveguide: -1r(IigzR e~ gZ n1* _ = e fgfgeS ) sin fRY cos gcos x W1 1 fg a b oo o A z v v mn. m y n7gx mn sin cos + L L r- sn a cos b e (5.1) m=1 n=0 mn Fields in Half-Space: a b jk0r 2 2vIif fo Ex(5,) r dd (5. 2) 2 = jw- -2 o o o (5.2)r where Rfgf = self-reflection coefficient for (f,g) mode and where the prime on the double summation indicates that the (f, g term has been removed. xl = Ex2t z = 0: g(x m~My n7T (l+Rgfg) sn cos gX - Amn sin cos n-x = E (x,y) fgfg a b m=l n=O a 0ax multiplying by sin faY cos gbx and integrating over the aperture: (1+Rfgfg) = ab f f Ex(x,y) sin cos d dy 97

m'lry cosfn'?x Multiplying by sin m cos where either (m' W f) or (n' jg) and integrating over the a b aperture: 4e a b A = -mn n r x s my nTx Ann ab a d S Ex(x,y) sin m c dx dy (5.4) o o H =H at z = 0: yl y2 +) k2) [-(lg 1 ain cos + n nx 2E f ) CTsin aY cos gX mn sin n7TX o oo, 4e a b A 2 \ j y abr J S Ex(-b 7) sin a cos b -2 + ko. fg / bm=l n=O a mn a b -jk r )r 1 - (a2 \ e0 0io L 0 \ay2 0 1j J Ex Gly,,z) L2 sin f- cos (5 oc co 4 a b C- ( xm rn f i E s in j ff / n7T- + k 2 )= (\m= y y ml n= mn o o ay sin Ma. n7TX d d7] ab(lmRg nrg. moy n7rx sn ax cos b sin c cos (55 a b (1 JR r2 x ~a b a b fgfg fo = - ~ ~ ~ ~f___ b — o f___ g7- (5. 6) s m fg.. J Eysinacos dxdy o o r fg where a/ 2 2\ 5 -jk r a cc 4 2~ e2 2 mn 2vrk+ + k2 Z 27Tr 2 Li abL rF L ay ay m=n sin m cos nr sin nvx 1 a b a b multiplying by Ex(x,y) and integrating over the aperture: ab ab I(1 R-R r2 f f f f Ex(x,y) E(5, n) Gl(XY,,77) dxdy d dnd (1 Rfgfg)C) fo o o _ ___ =_ _ _ _ _ _ _ _ _ - 0 0 0 0 (5.6) 1r rfg b firy grx f f Ex(x,y) sin a COS bg x 98

1-R fr ab J Jf Ex(x,y) Ex(,r,) Gl(x,y,f,1) dxdyd5 d77 Y G f (5) 1o r 2fg fg [rf J sIn f7ry g rax LO O Ex(x,y) = sin y cos gbx x a b ab ab /2 2\ 0 ffn - _\ gX sin f fr gT 1 + k ed dy d Yfg 2 F b 0 00 s a b a b a2 27 dxdyddr7 if^ (m \y O),,/o o -jf 9 r fg f gff i yx fV ^ g 229 Y fg a b b in f'y Co gX- sin f1r7 gCo a +k 2 abOOOO a b a b a 0 fo o Fg4g a 2bn7r3 o m m2 -jk o Vx 6y e —------ dxg dyd -kx + 2y2 if (m 0) (n 0) 99

Cff~m =a m z m m g = r a2 b2 r 2 fo (f2 - k2) The expression for the input admittance of the higher modes is similar to that of the dominant mode. The dependence on IAr er is similar. In all cases, the normalized input admittance is equal to 41r times an integral which is a function only of the product j rer. The form of the admittance function (Eq. 5. 8) brings out several interesting points concerning the LSE (longitudinal section electric) modes. Consider the pole-zero pattern of the admittance functions. If g f 0, the real and imaginary parts of the admittance have zeros at cutoff for the (f, g) mode and poles at cutoff for the (f, o) mode. If g = 0, the real and imaginary parts of the admittance have poles at cutoff for the (f, o) mode. If g = 0, the real and imaginary parts of the admittance have poles at cutoff for the (f, o) mode. This type of behavior is characteristic of LSE modes. Consider an (LSE)y type mode. ~1' z n* = -1 sin miry nc x e mn y -jwiIr a b mn ~r z -. mry nfx mn E = (+) sin -- cos -n- e E =0 y /nr\. m7ry. n71x mn E = j j sin m sin nx e Ez =b 2 ~r z 1 nm7r mry si nrx mn H - + co s sin -- e x jwjur ab a b mn (k2 m2 T z - 2 ~r z H =-i a sin may cosnirx mn y jcowra a b mn 100

mfr ~ r z H = + C c os cos e z j]wp a b The only two components which contribute to power flow in the longitudinal (z) direction are E and H. These field components are related by the factor r2 / jo r X y mo mn r2 H = mo E (5.9) y jWjir x mn Thus for (LSE) modes, there are two critical frequencies (besides w =0) at which the ratio of E to H has a pole or zero, namely (m,n) cutoff and (m,o) cutoff. The fact that there is x y a pole at (m, o) cutoff is interesting because this does not occur for the ordinary TM and TE modes. This behavior is related to the fact that the waveguide is restricted in the x and y but not in the z direction. The poles and zeros in the ratio of H to E are identical to those y x occurring in the admittance functions Yfgfg. The behavior of TE and LSE modes at the critical frequencies is compared below: TE Modes (m#O) f - oo f -fc(m,n) f -fc(m,o) f -0 E - oo E - const E - const E -> X X X X E - oo E - const E - const E - 0 Y Y Y Y H oo H 0 H - const H - const x x x x H - oo H - 0 H - const H -- const Y Y Y Y H -_ const H' const H -_ const H - const z z z z LSE Modes (m 0) f -- o f - fc(m,n) f fc(m, ) f -- 0 E -* const E -k const E -- const E -- const x x x x E -- const E - const E -> const E -- const z z z z H -> -0 H - o H - const H - oo X X X X H - const H - oo H -- 0 H — o Y Y Y Y H -- 0 H -- const H --- const H - oo z z z z 101

We note that the behavior of the LSE and TE modes as f -- 0 is similar. The magnetic fields are predominant and the excitation of an LSE or TE mode in the aperture from a probe source can be considered as a problem of simple magnetic coupling. At infinity, the fields are predominantly transverse for both the LSE and TE modes. At (m,n) cutoff, the electric fields are predominant for the TE modes and the magnetic fields are predominant for the LSE modes. 5. 2 Below Cutoff Considerations We have limited the range of experimental verification of our results to a frequency range where only the dominant mode propagates. There are several practical reasons for this: 1. Standing wave patterns are hopelessly confused when more than one mode propagates, although this could possibly be remedied by a mode filter. 2. Special techniques would be required to accurately determine below cutoff behavior. Below cutoff the standing wave pattern can be represented as: sin'e ry 10z 2T10z E = sin- e 1(1+A e ) (5.10) x a mn where A is the complex aperture reflection coefficient for the (m,n) mode. mn -r 10 The rapid increase of the e term away from the aperture (for z negative) rl0z and the rapid decay of the A e term tends to mask the effects of A mn mn However, the theoretical results for the dominant mode reflection coefficient are valid outside the frequency range where only the dominant mode propagates as long as the assumption of dominant mode aperture field is valid. In most cases, this assumption would appear to be a valid one. The results of some below cutoff calculations of normalized aperture admittance are shown in Fig. 8(a)-(h). These calculations are useful for calculating the admittance of a loaded waveguide with an iris in the aperture plane as noted in Chapter 3. It is noted that, in the curves of Fig. 8, the imaginary part of the admittance goes to zero at zero frequency and that the real part of the admittance approaches a finite value. This can be explained by the following argument: Consider incident and reflected dominant mode waves in a rectangular waveguide opening out into an infinite ground plane at z = 0 (Fig. 3) 102

Incident: E sin y- e x a 4 2 1 - -k2 a2 - r R I F10z aH _ sin ye 10 (5. 11) y joW a Reflected: +F10 z f a2 ab(1 - I (ka > fr) (5.14) T = R sin tt e pwhere R is the dominant x a mode aperture reflection coefficient R Y2R = R r - jR y aco R [(1_-2) - B2] B = — 2Ri(5.16) The total power radiated is ab P = 1/2 Ref f E H* dxdy r [(1 + G)2 + B2] (1 + Rr)2 + Ri2 103 a ab -C1 - (R.) (ka < 7t) (5. 13) /2 k a ab - R12) (ka > i) (5.14) Thus the total power radiated below cutoff depends only on the imaginary part of the reflection coefficient. Y 1-R G B Y- 1+R = Y +J Y +2B [(1-Rr2) - i 2 R. G = (5. 15) 1 [(1+G)2+ B2] (1+Rr)2+R.2 = [(1-G2)-B2] B (5. 16) r [(1+G)2+B2] (1+Rr)2+R.2 103

If B goes to zero, then R. and the radiated power also go to zero. Thus the roles of G and B are interchanged below cutoff, the power radiated being proportional to B rather than G. In particular, G may go to zero without the radiated power going to zero. In order for the power radiated to be positive, R. must be positive and B must be positive. Similarly, if we consider incident and reflected waves of the higher order modes: P a4fg W g [fgfg(i) (k < f/a) (5. 17) 4Efg U 1A g R fgfg(i)]< < fg fgl... Ef 1 y- [1- IRfgfgI2] (ka> a-2 + %2 ) (5. 19) where fgfg fgfg(r) j fgfg(i) fgfg(r) >. (k > f2 2/a2 + g2 r2/b2) (5. 20) This last condition can be derived from the equivalent circuit representation. Since the net radiated power will always be greater than zero except at (f, g) cutoff and (f, 0) cutoff, the following conditions on the quadruple integral forms Cfgfg may be stated: Below (f, 0) cutoff Im Cfgfg < 0 (5.21) Between (f, 0) cutoff and (f, g) cutoff Im Cfgfg 0 (5.22) Above (f,g) cutoff Im Cgg < 0 (5.23) where I Cfgfg is equal to the imaginary part of Cfgfg The conditions (5. 20) through (5. 23) have been used in Chapter 4 to show that the denominator of the mode coefficients is nonzero. 104

5. 3 Superposition of LSE and LSE Solutions x y 73 Stratton and others have shown that an arbitrary field in a homogeneous source free region can be expressed as the sum of three vector potential functions. Applying this to the I and II Hertzian potential functions leads to the well known fact that an arbitrary field can be expressed in terms of a Hn and a H. vector function where i = x, y, or z, yielding, respectively, LSE and LSH modes, LSE and LSH modes, and TM and TE modes in recx x y y tangular waveguide. It is possible to show that an arbitrary field can be expressed in terms of other combinations of H and II* vectors for both rectangular waveguides and for other cylindrical geometries. Consider an infinite rectangular waveguide (Fig. 10) excited by sources to the right of z =0, which give rise to an infinite number of, for instance, TM and TE modes with arbitrary coefficients. Utilizing equivalence principles, the effect of the sources may be taken into account by introducing new sources at z =0. Two alternative 74 representations are:7. an electric current J = fHtangential backed by a magnetic conductor. 2. a magnetic current M s = Etangential x backed by an electric conductor. Using image theory, these representations may be replaced with 1. an infinite array of electric current sources in the plane z = 0. 2. an infinite array of magnetic current sources in the plane z = 0. The problem is now reduced to a homogeneous boundary value problem. The II and Il* Hertzian potentials, or the A and F vector potentials, may be found by direct integration. -jkr A 14 SSff Je dv sources 1 M -jkr F 1 SSS M e dv sources where r = distance from source to field point The result is: 1. nI and ny vectors —resulting from the electric current representation, or 2. nx* and y * vectors —resulting from the magnetic current representation. 105

(_____/I Sources (a) Sources (b) i Z O _____________ JK /JY (c) (3@ t Magnetic wall Electric walls My <X (d) _ Electric walls Fig. 10. Equivalent representations. Thus an arbitrary field in rectangular waveguide can be expressed in terms of (1) transverse II vectors or (2) transverse II* vectors. Increasing the number of sides of the waveguide does not invalidate the results, since all of the images are transverse magnetic or electric currents in the z =0 plane. Thus the results apply to a polygonal waveguide, and, in the limit, to an arbitrary cylindrical waveguide. However, if we attempt to extend the results to nonuniform waveguides, the images will include longitudinal as well as transverse n and II components. An immediate result is that the waveguide problem of Fig. 3(a) can be generalized to take into account any mode incident including LSH as well as LSE modes. The incident (m, n) mode may be broken up into a linear combination of a H*(m, n) mode and a i*x(m, n) 106

mode and the problem may be solved as a superposition of two problems of the type investigated in Chapters 1-4. As a matter of fact, there are fifteen different possible combinations of T and/or * T* vector potential functions which can be used as complete representations of the field in a source free rectangular waveguide. There are six basic mode types generated by 7x' y', 7z' 1Tx, ry, z*' vector potential functions. If we consider the TM-TE combination as a basic x y z complete orthogonal set, it can be shown that a combination of any two of the six mode types will also be a complete orthogonal set. This can be accomplished by actually solving for the mode coefficients of each combination in terms of the TM-TE coefficients to show that the representations are completely equivalent. 5. 4 The Dual Problem The problem dual to that of Fig. 3(a) yields some information of interest concerning the effects of permeability and permittivity on input admittance. Consider the geometry of Fig. 3(a) with the metal walls replaced with magnetic walls, and with a dominant mode wave incident from the left. The solution to Eq. 2. 6 of Chapter 2, appropriate to the magnetic walls, is: 4/ = sin ky cos k x The field representations in Region No. 1 (waveguide) and Region No. 2 (half-space) are: er10z riz \ r 1 \1 m e / sin ry A v A r z n- = -:- ------ - ----- ^- - — J- + mn M m7Ty n7Tx mn -ly = rI m sin y cos nx e mn (5.24) where: R is the dominant mode reflection coefficient for H. m x Region 2 a b -jk0r 2y - 2 je f f Hx( 7) e r- d7 (5.25) Using an analysis parallel to that of Chapter 2: 107

a b a b 1-R abe fS S S Hx(xy) Glm(x,y, 7) HX('1) dxdyd dr Rm 2r O a b 2 (5.26) m 210 af f H (x,y) sin -Y dx dy] 1o o where 9a 2\ v 0 mn m7ry n7Tx m n.abr - a bs a b m=l n=0 r mn For a dominant mode approximation to the aperture field, (5. 26) reduces to an expression directly proportional to. Consider R, the dominant mode reflection coefficient for Ey: R = -R e m 1-R 1+R m e 1+R 1-R m e Thus the electric field reflection coefficient is directly proportional to'ur for the electric wall problem and inversely proportional to er for the magnetic wall problem. 5. 5 The Vector Problem A variational expression for the vector problem will now be derived for the problem of Fig. 3(a). Methods used will be similar to those of Chapter 2 except (1) two simultaneous integral equations in E x(x, y) and E y(x, y) will be used and (2) an additional step is required to produce a variational expression. Consider the problem of Fig. 3a. A variational expression will be found for the input admittance. Methods used will be similar to those of Chapter 2 except (1) two simultaneous integral equations in E x(xy) and E y(xy) will be found and (2) an additional step is required to produce a stationary expression. The representations of the fields in Region No. 1 (waveguide) and Region No. 2 (half-space) are as follows: 108

Region No. 1 r 10z r10 z 00 o, A r z i* I e - R e sin — y-y +' mn * m~ry n~rx mn s -r ci r "ly = u -J ~/ --- I -- smZ ^ T- aC S LSE Modes ly j=W/1 r0 a1 m=1 n=O mn a Y co os B I z 11* =- cos - sin.... ___ - I mL n n n Ly SE Modes x j' Lm=l n=2 mn j j For the allowed modes, m is odd and n is even. Region No. 2 a b -jkor 2y 2jWJo SS Ex(f f 7 r d' dr -jk r (5. 28) a b o 2x 2-jwi f f -E Ey( ) d de7 2x 2 -7T JW 1o 0 0 r Equating Ex components at z = 0 yields an equation for Amn. Equating Ey components at z=0 yields an equation for Bmn. Equating Hy components and substituting for A and B yields the first integral equation. Proceeding as in Chapter 2 (multiplying the integral equation by Ex(xy) and integrating over the aperture) yields an expression for R which is not stationary. It can be made stationary by adding a second integral equation to the numeration of 1R' The second integral equation is obtained by equating Hx components at z =0, sub1+R L X stituting for Am and Bn, multiplying by E (x, y) and integrating over the aperture. The resultant integral equation has a left hand side equal to zero. This equation is added to the numerator (quadruple integral) of the previously obtained expression for 1-R. The result is 1+ R (SEE NEXT PAGE FOR EQUATION 5.29) where: Gl(x,y, ~, 7) is defined as in Chapter 2 [Eq. (2. 30)]. 109

0 0 i m Xs + i S 0 1 I 0 X - P So > > c~~~ ~~X

/ 2 X' Y,5,e-jk-r r G,m=l) ( ) - + n sin r cos c sin G2(,YI177) (xay 27Tr Z Z abr s a c b ina csby (5.30) F(L 2 (e]-jkor) a2 ) 0 4 G (XY -) sn Cos sn cos b (5. 31) many2of the problems 2ne d te var00 00 4 L x,+ ax) ) + m= n=2 m7ry. n7Tx m7rJ. nrx (5. 32) sin sin cos sin (5. This is a stationary expression for the input admittance. Reducing the ranges of integration appropriately will yield a stationary expression for the waveguide with compound iris. It is interesting to note that Eq. (5. 29) reduces to Eq. (2. 33) if E (x,y) is set equal to zero. information concerning the dependence of the admittance on jr and r. Harrington75 has developed a general expression for the input admittance of a junction of two waveguides of arbitrary cross section. If the waveguide on the right approaches half-space as a special case, Eq. 8-128 of Ref. 75 becomes (using the notation of Harrington fS ExH. ds + ZZ V.2Y 1- r Y = aperture (5 33) 1+Fr Y+ V 2 Y o o where: A V.2 Y. indicates a sum over all modes except the mode incident from the left, whose mode admittance is Y and whose reflection coefficient is r. 111

If there is no iris in the aperture then the assumption of incident mode field in the aperture will drop out, by orthogonality, the terms of the infinite sum L E Vi2 Yi, leaving only the integral over the aperture, which depends on the free space size of the aperture, the form of the aperture field E, and the Green's function relationship between E and H. The dependence of Y/Yo on r and er derives solely, then, from the factor 1/Yo-, the modal impedance of the incident mode. The dependence of Y/Yo on pr and Er is then as follows: Incident Mode Dependence of Y/Yo on Ar and er TE or LSE Mr TM or LSH 1/er TEM r r These three types of dependence give exactly the same type of behavior, in the following respect: Consider a material whose rer product is fixed. As ir and e are varied, Mr, 1/er, and Vl.r/er will all change by exactly the same factor. The process of optimizing the Mpr for a given,rEr product is discussed in Chapter 7. 112

CHAPTER 6 EXPERIMENTAL RESULTS 6. 1 Verification of Variational Results In general, variational analyses have yielded accurate results, especially in interior problems. However, there are several considerations which make it desirable to obtain an accurate experimental verification of the results of Chapters 2 and 3. 1. Upper and lower bounds are not obtainable. 2. Field approximations may be less accurate than those used in similar interior problems 3. In some cases, poor results have been obtained in exterior scattering problems. 76 In addition, discrepancies have been noted in the literature in similar problems. 4. In the case of the iris loaded radiator, two different approximations to aperture field have been used. To the author's knowledge, this type of calculation has not been checked experimentally. An experimental check was made on the theoretical values (Fig. 5) of aperture admittance for the loaded waveguide radiator. Checks were made using Emerson and Cumming Stycast Hi-K with = 9. 68. The dielectric material was machined to fit into an x-band waveguide of dimensions. 400 in. x.900 in. The waveguide was then mounted in a 2 ft. x 3 ft. ground plane which was mounted in a larger ground plane at one end of an anechoic chamber. Equipment used in the experiment is shown in Figs. 11 and 12. A sketch of the experimental arrangement in the anechoic chamber is shown in Fig. 13. Light absorbing pads were mounted around the 2 ft. x 3 ft. ground plane to simulate, as nearly as possible, an infinite ground plane. A slot was machined in the dielectric to allow a probe to sample the field. Impedance was obtained by conventional slotted-line methods using an x-band slotted line and probe carriage. The geometry of the slotted dielectric insert is shown in Fig. 14. Because 113

Fig. 11. Dielectric loaded waveguide radiator. Fig. 11. Dielectric loaded waveguide radiator. (a) Side view. (b) Front view. Fig. 12. Slotted dielectric and wedges. 114 Rfi~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i:aii'iiii~''i~ii~:~ii~iii~il: i~~~~~~~~~~~~~~~~~~~~~-i~i~iaii-~::-i:::- ii-iii|rii-:: ~:i~~~i:::~~:i::i'::-:-:::::l_:-:i:-:~:::~~~~~~~~~~::-:: i~~~li~~i-:::,:l~~~~a::-ia::r:::i-::;::::-~~~~~~~i:::i~~~l-:-i~:i~_-:i:;:_:i::i:- is:::::-i~~~~~~~~~~i~si:-::.1:lii::iuiii iii::i~~~~~~~~i~~i~~i~i-:ii:-:j:.~ ~ ~ il:::ii:-i:~iliii: —:-::_lii-i:i':'-::':':: _iiiii —i-iii —i_:iii'''''-::: _iiii~-~iiij~iii~~: —-~-::~-~ii::-:::-:-:::. il —: —-::: F_:~:~-:::-..- ~::i g.::::-:::'''-;''- 12. —-iii-i'i Slotted dielectric and w edges.~l- i':ilii~: —is:_ ii::-iij:':'-:-: —i::-:::::1 1 4:,~-~

LOSSY MATERIAL COVERING GROUND PLANE' 2'x3' GROUND PLANE / WAVEGUIDE I/ /// 14/ / 1' L —---------- - 30,' -- Fig. 13. Experimental arrangement. of the slot, the geometry differs from the theoretical model. Tapered dielectric wedges (Fig. 12) were used in the slot to provide a measurement of the impedance of a geometry [Fig. 11(b)] more nearly corresponding to that of the theoretical calculations. Impedance was measured both with and without the wedges inserted. The tapered wedge constitutes a gradual transition from the rectangular dielectric cross section to the slotted dielectric cross section. The length of the taper is 1-1/2 in., or about a wavelength at FN = 1. 5. The effect of the taper section on the impedance measurement, which depends on the shunt susceptance at the ends of the taper and the transformation through the nonuniform taper section, is not precisely known. However, the admittance is uniquely determined by the value of the VSWR and the angle of the reflection coefficient. The taper introduces little reflection at frequencies where the taper length is appreciable compared to the guide wavelength. Near cutoff, the reflection introduced by the taper may become significant. At frequencies slightly above cutoff, the VSWR should correspond to that of the theoretical model. Similarly, the measured angle of the reflection coefficient depends only on the electrical length of the taper, if shunt susceptance at the ends of the taper is ignored. The electrical length of the taper is taken into account by a measurement of minimum position with a short on the waveguide. 115

SURFACE A /- / / /.125 SURFACE B /400 — L.' — " —.900 Fig. 14. Geometry of the dielectric insert. Thus, the data with wedge represents a measurement of the admittance of a geometry identical to the theoretical model, with some inaccuracy expected near cutoff because of shunt susceptance and reflection introduced by the taper. The data without wedge represents a measurement of the admittance of a geometry differing from the theoretical model, with accuracy expected over the entire range of frequencies tested. The presence of the slot in the dielectric modifies the cutoff wavelength as well as the equivalent circuit of the discontinuity. Unfortunately, the effect of the slot cannot be taken into account by perturbation formulas. The boundary conditions on E at the sides of x the slot (surface A of Fig. 14) require continuity of Ex, while the boundary conditions at the bottom of the slot (surface B of Fig. 19) require continuity of e E. According to perturbation formulas, these two conditions would tend to change A in different directions. However, we know that guide wavelength has approximately the form: o 1 Ag - - _ _ 1 where eeff = effective dielectric constant of the dielectric-guide and air-slot configuration. Two measurements of guide wavelength determine the two unknowns e ef and f. Let;~g l \2 2 = ( g1 /(Og2 A (6. 1) 116

Then f 1-A (6.2) c 1 A f2 f2 2 Using this method, f was calculated using several measurements of guide wavelength and c averaging the resultant values of f. This value of f was used in plotting the data without wedges. The cutoff frequency of the guide when completely loaded was used in plotting the data with the wedges. This cutoff frequency depends only on e r er was measured by completely filling the slotted line with dielectric material and measuring g in the cutoff region bounded by the dielectric and the tapered metal slot of the slotted line. In order to obtain accurate measurements of VSWR, a dial gauge was attached to the probe carriage and the double-minimum method was used. This improved the accuracy of the measurement of G somewhat, although the correction to G due to attenuation in the dielectric was as high as 50 percent in some cases. As a result, the G data are somewhat less accurate than the B data. Results are shown in Fig. 15. The data with wedge agrees within a few percent with the theoretical data over most of the range of the data, with considerable divergence near cutoff as expected. Experimental measurements of the aperture admittance of a loaded waveguide radiator with an iris in the aperture plane were made. er = 3.9 was used as the dielectric material. Figure 16 shows the experimental arrangement. All measurements were made with wedges. Figure 17 shows the results. Theoretical values were determined using the 40 formulas of Chapter 3 and using The Waveguide Handbook to determine the susceptance of the inductive and capacitive irises in an infinite waveguide. The experimental results of this section indicate accuracies comparable to those obtained in variational calculations for interior problems. 6. 2 Measurements of Aperture Field Measurements of the aperture field of a rectangular waveguide mounted in a ground plane were made for two different feed arrangements and both air and ferrite powder 117

1I. 1.0 09 ~ \ XQ~ Experimental without wedge 08 -\ 0 Experimental with wedge -— Theoretical 0.7 0.6 ~B |~ \~ ~Theoretical Yo 5 03.02 0,1 0.1 o I I I I I I I 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 FN Fig. 15. Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator. (a) Susceptance - Er = 10, t r = 1. 0, b/a =.444. 118

a Experimental - wedge E Experimental - no wedge ---- Theoretical Theoretical.05,04 | G.03 Yo.02, I l I l I I l' l. l 1.0 1.1 L12 1.3 1.4 1.5 1.6 1.7 1.8 1.9 FN Fig. 15. Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator. (b) Conductance - r = 10, r = 1. 0, b/a =.444. 119

Fig. 16. Apparatus for measurement of the admittance of a loaded waveguide with a compound iris in the aperture plane..7 --.6 - (a).5- EXPERIMENTAL --- THEORETICAL B4Y.3 CAPACITIVE IRIS.2.3.4.5.6 b7.8.9 d2/b.3.4 5. 6 d/ 7.8.9 0 I I I I -2 -, ( (b) -4 - B -.6- o INDUCTIVE IRIS o 8 - -1.0 -1.2 / -1.4 -1.6 Fig. 17. Comparison of theoretical and experimental data for the loaded rectangular waveguide radiator with a capacitive or inductive iris in the paerture plane. 120

loading. The magnetic field H normal to the aperture plane was measured, using a shielded z loop fed by a miniature rigid coaxial cable. The coaxial cable lies along the ground plane and is perpendicular to the electric field in the aperture. A two dimensional positioning apparatus with micrometer drive was used to position the probe in the aperture. The complete field cannot be determined from H but the measurements give us z some idea of the mode configuration. All of the TE modes could be determined from H z measurements. The size of the rectangular aperture was 12 in. x 13 in. The waveguide was 4 in. deep and was fed either by a cylindrical probe or a T-bar each located about 2-1/2 in. from the aperture. Figure 18 shows the coordinate system used and the actual cuts taken. Figure 19 shows the measurement system. The waveguide was mounted in a 3 ft. x 3 ft. ground plane. Measurements of ground plane currents indicated rapid decay away from the aperture and thus justified the use of a finite ground plane. Figures 20(a) and (b) show the results. The fields are, in general, very similar to those that would result from a single dominant mode wave, although the presence of some sin 3y component is indicated by the steepness of the horizontal curve near x =0 and the a presence of some cos b component is indicated by the depression of the vertical curves near the center of the waveguide. There is not much difference between the cylindrical probe and T-bar feed arrangements. The T-bar horizontal measurements are more symmetric; this is reasonable since the T-bar is fixed but the probe may tend to slant. This same reason explains the unequal peaks in the horizontal cylindrical probe measurements. In the vertical measurements, there is little difference between the probe and T-bar feeds. The deviations from dominant mode fields appear to be greater for the air loaded than for the material loaded case. All of the curves fall rapidly at positions close to the waveguide walls. This is caused by the finite size of the loop. The average magnetic field threading the loop is measured rather than the magnetic field, exactly at the coordinates. Thus, the power received by the antenna starts decreasing as soon as any part of the loop passes outside the aperture. The loop diameter is approximately 1/2 in. Measurements were also made of the aperture field of an x-band (. 400 in. x. 900 121

VERTICAL CUTS x=.300 200 4.100 7900 9.800 1( 700 7 -.-+- - - - - - - -..- -t ---- I I I I | I! I. | — "'..-... —.-.-... —-+ —-----—.!. —4 -------- -- y= 2.000 3 -.-+ --- --— _' -— ~ —--'-4 —--- y= 1.500 HORIZONTAL,| _. t —-a+- t @ —- +. ------ y= I0I, CUTS X |-~- ~.... +-.............-..... —-- --- y=.500oo I1 I I I, I,!' - 12_- 1, I 2-.1 Fig. 18. Aperture coordinates for measurement of aperture field. Probe Ground Plane Plexiglass Cover Positioning Apparatus Aperture Fig. 19. Apparatus for measurement of aperture field. 122

1.0 L*.90 x.80 i x a. *70 _ ^ x E 6'B. f.60 o- ~ E.50 ~2x.40 _::0 x o y=0.5.30 -A y=1.0 /G i o0 y=1.5.20 - 0 8 o y=2.0.10 0.0 2.0 4.0 6.0 8.0 10.0 12.0 x Fig. 20. Measurement of aperture field. (a) T-bar feed - horizontal cuts, frequency - 324 Mcs (ferrite powder loaded). 1.00 //.95 - -,\. 7//.65 ~ x. 95.60 0 1 2 3 Fig. 20. Measurement of aperture field. (b) T-bar feed - vertical cuts, frequency - 324 Mcs (ferrite powder loaded). 123

1.0 -.90.80 8 00.*70 - 8 ^y. g.20^ t d ~ y 2.0.160'. 50 A io E' s.40 - I N a~f ~B~o y'0.5:.30 - c y-l.0 L1 S A y'1.5.20 - ~ y-2.0 ata^^~'y ^2.5.10 -9 0.0 2.0 4.0 6.0 8.0 10.0 12.0 x Fig. 20. Measurement of aperture field. (c) Cylindrical probe feed - horizontal cuts, frequency - 324 Mcs (ferrite powder loaded)..01/ x,, x.3., \,.65 0.0 1.0 2.0 3.0 Fig. 20. Measurement of aperture field. (d) Cylindrical probe feed - vertical cuts, frequency - 324 Mcs (ferrite powder loaded). 124

1.0.90.80.70,-, a.60 X 4 88.N x.30 - x o y=0.5 &.1 a y-2.0.20 - x A y 2.l5 0.0 2.0 4.0 6.0 8.0 10.0 12.0 x Fig. 20. Measurement of aperture field. (e) Cylindrical probe feed - horizontal cuts, frequency - 735 Mcs (air loaded). 1.0.0 - fNX myX2.25 6<4. I I2 \ _ I = 4II. 0 0. 0 1 2 3 y Fig. 20. Measurement of aperture field. (f) Cylindrical probe feed - vertical cuts, frequency - 735 Mcs (air loaded). 125

in.) waveguide mounted in a ground plane and loaded with dielectric material (r = 9. 68, 12. 18). Results were similar to those shown. Depression of the vertical curves was not noted because of the large size of the loop relative to the. 400 in. dimension of the waveguide. 6. 3 Experiments with Dielectric and Ferrite Loaded Cavity Slot Antennas 6. 3. 1 Introduction. A number of rectangular cavity slot antennas were constructed, using powdered ferrite, solid ferrite, and solid dielectric material. Experimental measurements of resonant frequency, efficiency bandwidth and beam pattern were performed and compared with theoretical predictions. Figure 21 shows three of these models, all operating at roughly the same frequency. Table No. 2 indicates the characteristics of these antennas, showing the size and weight reduction accomplished through the use of material loading as well as the price paid in terms of reduced bandwidth and efficiency. Operating Size Volume Bandwidth With Flange Bandwidth - No Flange Efficiency Directivity Weight Frequency VSWR = 3.0 VSWR = 6.0 VSWR = 3.0 VSWR = 6.0 Air Loaded 30" x 7 " x 10" 2250 cu. in. 64 Mc 90 Mc 90 % 5.8 25-3/4 300 Antenna pounds Ferrite Powder 12" x 3" x 9" 144 cu. in. 22 Mc 50 Mc 20 Mc 36 Mc 65 % 5.0 16-3/4 320 Loaded Antenna pounds Solid Ferrite 5" x 2" x 12" 15 cu. in. 19 Mc 34 Mc 19 Mc 32 Mc 30 % 5.0 3.6 352 Loaded Antenna pounds Table 2. Performance comparison of rectangular cavity-slot antennas. Later sections of this chapter describe methods of predicting antenna characteristics using the basic variational data, each section being devoted to a single characteristic. Comparisons are made between theory and experiment. The characteristic of the materials used are shown in Table 3. The characteristics of the solid ferrite material was measured by the manufacturer using disk samples and standard slotted line measurement techniques. The listed characteristics of these measurements were also checked at The University of Michigan Solid State Laboratory. The listed characteristics of the dielectric are also those of the manufacturer. The characteristics of the ferrite powder have been approximately 126

Fig. 21. Loaded cavity antennas. Ferrite Powder'e' = 8.3 AI'; 2.2 Q = A'/" = 30 at 300 Mc Solid Ferrite A = 6.63' = 12.6 Q = i'/L" = 30 at 300 Mc Solid Dielectric' = 10 Q = e'/e", Q > 1000 Table 3. Characteristics of ferrite and dielectric material. The ri e product was measured by loading a slotted slab line with ferrite powder material and measuring the distance between minimum positions in the air directly above the powder. The relative permeability was measured by standard techniques, but 127

Short Probe Aperture BP + GP BA X GA e — d2' —' dl ----- Transforming to the aperture: BS BP GP BA BA *Admittance of short At resonance: GP =GA Bs + Bp + BA = 0 Fig. 22. Simplified equivalent circuit of a cavity slot antenna. since the thickness of the sample could not be determined accurately, the data must be regarded as approximate. 6. 3. 2 Resonant Frequency. Utilizing the variational data and a simplified equivalent circuit (Fig. 22), it is possible to predict very accurately the resonant frequency of a ferrite material loaded rectangular cavity slot antenna. The resonant frequency, or the frequency at which the input reflection coefficient is minimum, is calculated simply by calculating the frequency at which the short susceptance transformed to the aperture plane (B?) is the negative of the aperture susceptance (BA). The probe admittance is taken into account by perturbation calculations (Appendix C). The short position required for resonance at a given frequency is calculated simply by reading off a Smith chart, the electrical distance (d res/X) from the complex conjugate of the aperture admittance to a short circuit. Similarly, for a given length, the frequency of resonance can be calculated either by trial and error, or by noting the frequency 128

at which the curves for dre /A and dact/X cross, where dat/Ag is defined as the actual res/ g act/ g d act/ g electrical length of the cavity. Two examples are shown below: Example 1 —Solid dielectric loaded cavity = 1 = 10 r r b = 3 in. a = 11-3/4 in. d = 3-3/4 in. cutoff frequency = 159 Mcs From computed data: FN B/Yo G/Y dres N __o 0__o res/ g 1.95 -.125.0279.2700 2.00 -.115.0290.2685 By trial and error the frequency at which d is the proper length for resonance is 313 Mcs. At 313 Mcs: \ = 35.35 cm d =.2695 X g g Perturbation correction: (Appendix C) Probe size = 1/4 in. radius x 1 in. length =. 049 cu. in. abd = 132 cu. in. o~ -4= = 2.43 x 10 The predicted resonant frequency is 313 Mcs. The actual resonant frequency is 316 Mcs. The error for this example is less than 1 percent. 129

Example 2 —Solid ferrite loaded cavity r = 6.63 Er = 12.6 b = 2 in. a = 5 in. d = 1-1/2 in. cutoff frequency = 129. 5 Mcs Interpolating from the data of Fig. 9: FN G/Y B/Y d /X d /X N /o /Yo res g act g 2.6. 025 -1.111.384.361 2.8. 025 -1.011.376.392 Curves for dres/Xg and dct/Xg cross at FN = 2.715 - 351 Mcs Perturbation Correction: Probe size = 1/4 in. radius x 9/16 in. length =. 0276 cu. in. Size of Teflon plug below probe = 1/4 in. radius x 1-7/16 in. length =. 0705 cu. in. - = -.794 C abd = 15 cu. in. w-w 0 = +.213% 0 Predicted resonant frequency is 351. 7 Mcs. Actual resonant frequency is 347 Mcs. The error for this example is 1.4 percent. It is interesting to note that the variational data are 130

valid at FN = 2. 6, which is well above cutoff frequency for the (2,0) mode. The accuracy of these calculations suggests a new method of measuring dielectric constant, namely completely loading a rectangular cavity which is open on one end and utilizing variational data and a measurement of resonant frequency to determine dielectric constant. This method could also be used to determine either 11 or Er if one were known. Some of the advantages of this method would be: 1. The resonance is sharp, especially for electrically small cavities. 2. Measurement would not depend critically on wall losses; movable walls could be used for different size samples. 3. Sample preparation would involve only drilling a cylindrical hole in a rectangular block. The theoretical data on admittance and perturbation effects could be refined further, using more accurate calculations and more extensive higher order mode data. Accuracies on the order of 1 percent or better for the determination of dielectric constant should be obtainable. 6. 3. 3 Bandwidth. For purposes of bandwidth calculation, several assumptions are made concerning the equivalent circuit of Fig. 22. 1. B = 0. The probe introduces no reactance into the cavity and does not, therefore, affect the bandwidth or efficiency of the antenna. The reactance of the probe is taken into account by perturbation calculations in the prediction of resonant frequency. 2. G' = GA. The probe conductance is taken into account by assuming that, p A over the frequency range of operation, the probe conductance, when transformed by transmission line transformations to the aperture, is equal to the aperture conductance. 3. The use of transmission line calculations involves the assumption that higher order modes can be neglected. There are two sources of higher order modes in the cavity, namely, the probe and the aperture. In some cases, the distance from probe to aperture or probe to short is small in terms of guide wavelength, on the order of Xg/8. The equivalent circuit calculoations and transformations are theoretically valid only if all obstacles, and the point of observation of the impedance, are all separated by large distances. The bandwidth is calculated using simple Smith chart techniques. Referring to Fig. 22, since G' = GA, the total real part of the admittance at the aperture is 2GA. The 131

6.05.0- Q 4.0 - 2.0 1.0 1.2 4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 FN Fig. 23. Bandwidth factor. total susceptance at the aperture (B' + BA) is zero at resonance. The half power points, s A then, occur when B' + B = 2G. The behavior with frequency of B' and B is known. As S A A s A the frequency moves off resonance, the change of B' with frequency can be calculated by s transforming the short to the aperture. The behavior of BA with frequency is known from the basic variational data (Fig. 5). Thus, the half power bandwidth can be calculated by trial and error. A single trial and linear correction were found to be accurate. The single trial might consist, for instance, of assuming that the changes in BA and B' were equal, A p calculating the frequency at which the change in B' is equal to GA, and then finding BA from p A A the variational data. It should be noted that it is necessary to use a bandwidth conversion factor in the trial and error process to convert from a percentage change in X to a percentage change in g frequency. This bandwidth ratio is equal to: 1 dXg / dXo X df df g o which reduces to FN2/FN - 1. This bandwidth ratio is plotted in Fig. 23. It would be possible, of course, to transform all the admittances to the probe position and to assume that G was 132

constant, and to use the off resonance values of GA rather than the value at resonance. It was found that this did not make an appreciable difference in the results. The efficiency, of course, effects the bandwidth. This can be taken into account by assuming an additional lumped conductance at the aperture. This new conductance G will be equal to GA(PL/PR) where PL/PR is determined from efficiency calculations as described in the next section. The effect of this additional conductance will be to increase the bandwidth by the factor (1/efficiency). Table 4 compares theoretical and experimental bandwidth data. RESONANT FREQUENCY Theoretical Experimental Solid Dielectric Cavity 313 Me 316 Mc Solid Ferrite Cavity 351.7 Mc 347 Mc Maximum Error = 1. 3 % HALF-POWER BANDWIDTH Frequency (Mc) Theoretical Experimental. Solid Dielectric Cavity 316 4.9 % 5.7 % Solid Ferrite Cavity 352 8. 0% 9.7 % Powder Ferrite Cavity 320 12.6 % 15.6 % Maximum Error = 19 % EFFICIENCY Frequency Experimental (Mc) Theoretical Experimental Method Powder Ferrite Cavity 315 62. 6% 66.5 % R. T. + G. C. Solid Ferrite Cavity 352 21 % 30.0 % * G.C. Table 4. Comparison of theoretical and experimental data. Only an approximate efficiency measurement was obtainable. R. T. - Reflection techniques. G. C. - Gain comparison. 134

6. 3. 4 Efficiency. A method of efficiency calculation is outlined in Appendix B. This method assumes a dominant mode only in the cavity, and utilizes the aperture admittance data and the material constants to calculate the efficiency. The analysis for one of the simpler cases (probe near aperture, low losses, magnetic losses only), was used and theoretical data for efficiency were calculated and compared with experimental results. From Appendix B (Section B. 3, Eq. B. 28) Magnetic volume losses only: Efficiency = PL 1 + R' i A a2 (2d)+ (k2 a2 - 22) sin 210d (6 P, L./k2 2_ 2 PL =__________K a 7 - -(6. 3) PR (1- IRI2) k2a- where d = length of cavity R = complex aperture reflection coefficient /k2 a2 7T2 10 = a Using the above formula, theoretical efficiency data was calculated for several cases. d was calculated using theoretical aperture admittance data. An example of efficiency calculation is shown below. Example No. 1 - Efficiency of the powder ferrite loaded cavity antenna shown in Fig. 21. /rer 8.3 r = 2.2 "/' =.0290 b = 3 in. a = 11.75 in. Frequency = 315 Mcs cs F = 1.805 = ka/v Cutoff Frequency = 174. 5 Mcs N 135

Interpolating from the data of Fig. 5: = -.268 G. 0746 Y 0 1- IRI2 =.244 From a Smith chart plot of B/Y and G/Yo: 2 10d = 210~ d/a =.388 PL/P =. 595 Efficiency = 62. 6 percent The measured efficiency was 66. 5 percent. The measurements are described below. Efficiency Measurements: The efficiency of the ferrite powder loaded antenna shown in Fig. 21 was meas70-72 ured using reflection techniques. The method is analogous to that described by Deschamps73 for transmission-line components. In the transmission-line method, the sample to be evaluated is placed in a transmission line. As a short circuit is moved on the output end, the impedance is measured on the input end. The locus of the impedance values is a circle on the Smith chart and is thus determined by three points. The efficiency is determined by four measurements, i. e., the three measurements using a short to determine the Smith chart circle and a measurement of impedance with the output terminated in its characteristic impedance. Similarly, in the application of this method to antennas, the short circuit consists of a metal enclosure which reflects all of the energy back toward the input. In this method, at least three different sizes of enclosures are required to determine the circle on the Smith chart. The tests described were made in an anechoic chamber. A flange was mounted on the rectangular cavity slot antenna and the flange was mounted in a large ground plane at 136

Fig. 24. Equipment for efficiency measurements. (a) Antenna and flange. Fig. 24. Equipment for efficiency measurements. (b) Aluminum "hats". one end of the anechoic chamber. Four aluminum "hats" of different sizes were alternately placed over the aperture and bolted to the flange. The antenna, flange, and "hats" are shown in Fig. 24(a) and (b). The Smith chart plot of efficiency is shown in Fig. 25. The figure shows efficiency calculated directly from the data and a corrected efficiency, which takes cable loss into account. The efficiency, as defined in Ref. 70 is equal to 137

0~!~~~~~~~~~~~~~~~~ OA~~~~~~~~~~~~~~~O) C'0 2 0S/ olxc~~~ a CO~ UCAEr- OPON (A0 4, x r~~~~~~ EFFCIECY 61 6 E F ICINC Ya 6.3 Fig. 25. Effciency of te ferrite, owder loade I, O:'ait ntna hwni Fg 21

2D1D2 (D1 + D2) ( - D2/R2) where D1 = longest distance from Za to the edge of the circle. D2 = shortest distance from Za to the edge of the circle. D3 = distance from Za to center of the Smith chart. R = radius of the Smith chart. Za = impedance of the antenna in free space with an infinite ground plane. For the measurement of Za, light lossy pads were placed on the ground plane and a large (10 ft. x 10 ft. ) array of lossy pyramids was placed in front of the antenna in order to reduce reflections as much as possible, thereby simulating an infinite ground plane. This method of efficiency measurement is generally regarded as a reliable one. However, a few limitations should be mentioned. The analogy to transmission-line methods is strictly correct only if a single radiating mode is excited. This assumption is realistic for electrically-small antennas but would not be expected to hold for electrically-large antennas. The method has sometimes been justified by the fact that the data points do, as predicted, lie on a circle. This is not a proof of the accuracy of the method; even if a perfect circle occurs, the location of Z can significantly affect the result. a It was desirable to have some check of the results by another method. Accordingly, the gain of the ferrite loaded antenna was compared to the gain of a larger air loaded antenna (Fig. 21) at the same frequency. The directivity of each antenna was calculated approximately from the E and H plane patterns. Relative efficiency was calculated from the following formulas. (gain)1 = (directivity)1 (efficiency)1 (gain)2 = (directivity)2 (efficiency)2 (6. 5) (efficiency)2 (gain)2 (directivity)1 (efficiency) (gain) (directivity)2 139

The result was a relative efficiency of about 75 percent, as compared with an absolute efficiency of 66. 5 as measured by reflection techniques. This data indicates good agreement since the efficiency of an air loaded waveguide radiator should be on the order of 90 percent. There are many sources of error in the gain comparison method and, as a result, it can be regarded as only a rough check on the reflection techniques. Table 4 shows a comparison of theoretical and experimental efficiency data. It was not possible to obtain a measurement of efficiency by reflection techniques for the solid ferrite antenna of Fig. 21 because only a small part of the Smith chart circle was defined. 6. 3. 5 Beam Patterns. The far field beam patterns of a loaded rectangular waveguide opening out into an infinite ground plane were calculated, assuming dominant mode fields. Figure 26(a) shows the coordinate system used. The E plane and H plane patterns are: E Plane (0 = i/2) k b sin 0 2 1^1" -( 2Fsin (kb s in IEoI 2 = [ sin ] (6. 6) k b sin 0 H Plane (0 = 0) k a sin 2 -E = (: 2 ) c(6.7) [2 _ (koa sin 0)2] The fields are normalized so that the. maximum squared magnitude is unity. It is noted that the E plane pattern depends only on the free space electrical size of b and that the H plane pattern depends only on the free space electrical size of a. Figures 26(b) and 26(c) are plots from a computer program for Eqs. 6. 6 and 6. 7 showing the theoretical E and H plane patterns as functions of, respectively, k b and k a. As the size of the aperture becomes electrically small, the pattern approaches that of a magnetic dipole oriented parallel to the broad wall of the waveguide. This agrees with the image theory concepts used in Chapter 2. Figure 27 compares experimental beam patterns of a 140

Fig. 26. Rectangular slot beam patterns. (a) Coordinate system. 1.0 lEe\2 0.5 \ y=8 0.4 y=ID 0.3 0~2 y=l.4 10 20 30 40 50 60 70 80 90 e Fig. 26. Rectangular slot beam patterns. (b) E plane pattern. 141

1 0 0.9 008 0.7x =.25 50 \ X 1.50 0.4- xx51.75 10 20 30 40 50 60 70 80 90 0.3 - 0.2 x =ka/ir 0.1 -.00 10 20 30 40 50 60 70 80 90 Fig. 26. Rectangular slot beam patterns. (c) H plane pattern. solid ferrite loaded rectangular cavity slot antenna mounted in a 3 ft. x 3 ft. ground plane with the corresponding theoretical patterns. Beam patterns are also shown for the same antenna without a flange. It is noted that the H plane patterns (with flange) show good agreement with the theoretical data, whereas the E plane patterns do not. These data show the same trends as that reported by Butson and Thompson for electrically larger air loaded waveguide radiators with finite flanges. More accurate H plane results could probably be obtained using their approximate methods. It is noted further that the effect of physical asymmetries is more pronounced (note H plane pattern —no flange) for electrically small antennas without flanges than is the case either for small antennas with flanges or large antennas without flanges. These characteristics have been noted in other experiments by the author. 142

--- THEORETICAL, INFINITE GROUND PLANE ----- EXPERIMENTAL, NO FLANGE - EXPERIMENTAL, WITH FLANGE 1.0.,9 \ H PLANE.7 2 /.2 __ / \ - 4/ i - \.9 -'1\ E PLANE, -3 ( DERE,.4 \ Ii / Fig. 27. Comparison of experimental and theoretical.1-.1^ v43 0 -180 -90 0 90 180.99.PLA EDEGREES.*4~~~~~~ ~143 Fig. 27 Copriso o -f e xermnta n dtericl

6. 3. 6 Magnetically Tuned Antennas. Preliminary experimental data were obtained on the effects of magnetically tuning a ferrite loaded cavity antenna. The antenna size was 2" by 5" by 2k" (same as the small antenna of figure —except that the length is 2-" instead of 1"). The resonant frequency with no applied magnetic field was 243 Mcs. Applying a magnetic field parallel to the "b" dimension, with the pole pieces located at the center of the top and bottom walls, it was possible to change the resonant frequency from 243 to 320 Mcs. Magnetic field was approximately 400 gauss at 260 Mcs; 1000 gauss at 290 Mcs. The antenna was fairly well matched at each frequency. VSWR at resonant frequency varied from 1. 50 to 1. 80. The bandwidth for a VSWR of 3. 0 was about 5 Mcs at all frequencies. The probe depth and probe position were not changed throughout the tests. Thus the device is automatically matched as the applied field and resonant frequency are changed. Efficiency is estimated at about 15 percent. 6. 3. 7 Transmission Line Tuning Effect. A critical experiment was performed to determine if comparable performance could be obtained with the ferrite material removed and a tuner inserted in the transmission line [Fig. l(e)]. If this were possible then the ferrite loading would offer no advantage. The ferrite powder material was removed from an antenna of size 12" by 3" by 4". The probe was placed as close to the aperture as possible (i"), adjusted for optimum length, and a double stub tuner was inserted in the coaxial transmission line feed. Bandwidth was reduced by a factor greater than ten. Efficiency was also reduced by a factor greater than ten, judging from a comparison of field strength on the peak of the beam pattern. 144

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 7. 1 Optimization of Antenna Characteristics Several processes of optimization of bandwidth and efficiency are considered. The first three processes discussed are: 1. For a given rer product and given size of aperture, what is the optimum M r? 2. For a given size of aperture, what is the optimum /,rer product? 3. For a given "a" dimension, what is the optimum "b" dimension? Only process (1) has a true optimum. Processes (2) and (3) have monotonic types of behavior which must be weighed qualitatively against other antenna characteristics. Consider process (1). What are some of the factors that affect bandwidth and efficiency? Certainly the squared magnitude of the transmission coefficient (IT 2 ), which is equal to 1 - IR 12, is of basic importance. IT 12 represents the fraction of the power transmitted for a wave incident from -oc. iT 12 enters directly into the equations for efficiency [Eq. 6. 3]. The maximum value of ITI 2 can be calculated. 1-R Y +R -= = G + jB i+R Y 0 IT|Z = 1- JR12 = 4G (7.1) (1+G)2 + B2 let K = G/B. For a given physical size of aperture, given / rer, and given frequency, K will be constant since both G and B are proportional to / r. IT 12 4KB (1 + KB)2 + B2 Differentiating with respect to B: d(T)} 1 d(T) = 0 at B = dB N 1+ K2 145

The maximum value of T 2 is 2 K \ 1 + K2 1 + K2 i +K 2 1 + K2 The maximum occurs at that value of pr which makes B = V1/(1+K2). Figure 28 shows a plot of bandwidth and 1 - JR [2 as a function of 1r for the following given characteristics = 9 r rr b/a =.3 FN = 1. 8,4.3 ITi.2'.1 12 3 4 5 6 7 8 9 Pr Half-Power 10 Bandwidth 8 % 6 4 2 1 2 3 4 5 6 7 8 9 Or Fig. 28. Optimization of I T 2 for fixed /rcr product. 146

It is noted that the maximum value of IT 12 occurs at pr = 6. 8 as predicted by Eq. 7. 2. However, the maximum value of bandwidth occurs for pr = 5. 0. The reason for this is that, as jLr increases, the electrical length of the cavity increases. This effect tends to decrease bandwidth and leads to a bandwidth maximum at a value of Ar slightly less than the,r value for maximum IT 2. This same discussion would also apply to the efficiency characteristics, since the increase in cavity length will increase losses. It is interesting to note that, according to Fig. 28, the maximum bandwidth is almost double that for the purely dielectric case (ir = 1, r = 9). This is supported by the experimental data in Table 4 for the dielectric cavity and the powdered ferrite cavity. These antennas are exactly the same size and operate at almost exactly the same frequency. Figure 29 summarizes the information on IT 12. It is a plot of IT 2 vs. B, with K as a parameter. Every plot of I T 2 vs. Ar (such as that of Fig. 28) will lie along one of the constant K curves. For a given aperture size, i re product, and frequency, K can be determined from the variational data. Then, as /r varies the constant K curve describes the variation of IT 12 and B. Thus the behavior of IT [2 as a function of /r can be determined immediately from the curves of Fig. 29. Figure 30 shows a plot of optimum /1r as a function of the product /re for ka/7r = 1. 9. It is noted that the optimum value of ur decreases with increasing r' Consider optimization process (2). For a given size of aperture, what is the optimum rer product? A check of the computed data for G and B reveals that, for a given aperture size and frequency, different values of Ir e yield almost exactly the same values of K. This agrees with the asymptotic expression for 1/K or B/G (Eq. 2. 67) B -7T a F(1) G 4 b 3 This expression is independent of the product and depends only on the free space elecThis expression is independent of the j er product and depends only on the free space electrical size of the aperture. This implies that a further optimization as in process (1) would yield identical values of IT 12. However, as noted before, the bandwidth will depend not only on iTi2, but also on the rate of change with frequency of the aperture admittance parameters B', BA andGA. 147

I0 5 2.5 1.5 Y L0.25 _.2 Fig. 29. IT I2vs. b/y..15 1.01.015.02 025.03.04 05.06 07 ".1.15.2 25.35".4.5 1.0'lf2 Fig. 29. TI2 vs. b/y. 148

BA and GA are proportional to 1/ F2 - 1 and change very rapidly with frequency near cutoff. The rate of change of B' with frequency is proportional to the bandwidth s ratio plotted in Fig. 23. Thus, all parameters change very rapidly near cutoff and bandwidth improves monotonically as cutoff frequency is decreased by increasing the rer product. Figure 31 shows a plot of bandwidth vs. i. er product for a fixed aperture size and operating frequency. For all points on the curve, Ar is chosen for optimum IT 12. The curve can be closely approximated by a curve of form: Bandwidth = K/Bandwidth factor of Fig. 23, (where K is a constant). It is noted that significant improvement in bandwidth can be obtained by using material with a high jirr product. Figure 32 shows a plot of bandwidth vs. free space electrical size for a given ka/ir. This curve shows the basic relationship between bandwidth and size reduction. The bandwidth decreases rapidly with size reduction as noted in Chapter 1 in connection with the spherical line analogy. The curve is plotted assuming 100 percent efficiency; actual bandwidths would be greater than those plotted by the factor ( -cic). The data can be closely k a 3 efficiency' approximated by a curve of form: Bandwidth = K ( —) where K is a constant. This agrees with Eq. 2. 67. Consider optimization process (3). For a given "a" dimension, what is the optimum "b" dimension? From Eq. 2. 70, it is noted that K has roughly the dependence (b/a) /2 In addition the maximum value of TI 2 approaches 2K (for small K). Thus IT[2 10.0 \ 9.0 b/a =. 8.0 --—, 7.0 6.0 - 5.0 ba Z 4.0 - 3.0 2.0 - =1.9 7' 1. I 0 10 20 30 40 50 60 70 80 90 100 /rEr Fig. 30. Choice of lr for maximum ITI 2. 149

1 -:12:: 8 6 a_ 4 / i 2 0 1.0 1.2 1.4 1.6 1.8 2.0 22 24 ka 2 ------ Mathematical curve (Bandwidth = K/Bandwidth factor of Fig.23) K=17.7 ------ Bandwidth (Calculated assuming Lr is chosen for 2 optimum I|T,b/a = 4, koo= 6 ) Fig. 31. Bandwidth of a rectangular cavity antenna (fixed size and frequency —variable rer product). which we have found to be roughly proportional to bandwidth and efficiency, has the approximate dependence (b/a),/2 indicating that the price paid for reduced "b" dimension is not very severe. Several other types of optimization processes are considered below. 4. Optimization u ui 1l ur a given /rEr product, given frequency, and a given physical configuration including an iris in the aperture plane. Y = GA + A + j BI proportional to /ur independent of /ur 150

-0-Bandwidth (Calculated assuming /r is chosen for k2 _ I g optimum IT, b/a =.4, - =1. ) ---— Mathematical curve (Bandwidth= K(koa)3) K=300 16 14 12 10 / / Icn I~~~j~ / 4 n// - 6- // C / r 4- / 0.1.2.3.4 k0a 27r Fig. 32. Bandwidth of a rectangular cavity slot antenna (fixed ka/ir and variable electrical size). where B, = iris susceptance = 1/2 susceptance of the same iris in an infinite waveguide. 4GA ITI2 = (7. 4) (1 + GA)2 + (BA + BI)2 151

let K = GA/BA Then IT 2 has a maximum of K2( + B2) 2 i1 + K2 (7.5) K2(1 + B2) 1 + 2(1 + B/K) 1 N 1+K2 At the maximum: 1 + B2 IB2 2 BA = ~~ X +B2 (small k) (7.6) 1 + K2 Only one of the two possibilities (~) corresponds to a realizable BA. Several optimization processes have been considered. Conditions for optimum T1 2 have been derived under a number of different conditions. Because of the effect of the increase with pr of the electrical length of the cavity, the optimum efficiency and bandwidth values will occur at slightly lower values of j,r An additional effect which is not included in the optimization process is that of the relative frequency sensitivity of some of the more complicated geometries such as that of the compound iris. Such factors will enter into the optimization process and effect the calculation of optimum bandwidth conditions. The analysis of Chapter 3 and the curves of Figs. 5, 8, 9 provide the necessary tools to carry out a more exact numerical optimization process for any given geometry. The approximate optimal condition derived in the section could serve as a starting point for such an analysis. The optimization process (1), utilizing Fig. 29, can be applied to any other type of loaded slot antenna which satisfies the conditions that that aperture field may be represented approximately by the form of the incident mode. All that would be necessary would be to plot new variational data similar to that of Fig. 5, appropriate to the given geometry. Then, once K was determined from the variational data, the curves of Fig. 29 could be used to determine the optimum choice of ur. 152

7. 2 Conclusions This section will cover conclusions other than those concerned with optimization. 7. 2. 1 Basic Limitations of Electrically Small Antennas. The basic limitations on electrically small antennas, as discussed by Chu,2 Wheeler, 16 and Harrington, pose severe restrictions on antennas small compared with a free space wavelength. There is no implication in this thesis that these basic limitations can be overcome. As a matter of fact, the analysis and experiment tend to emphasize in many different ways the severity of these restrictions. The factor K, which determines the applicable curve of Fig. 29, is determined solely by the free-space electrical size of the antenna. Thus, the maximum IT 2 obtainable depends on the free-space electrical size of the antenna. Whether the maximum is actually attained depends on a proper choice of material. The data of Table 4 and Fig. 32 also show the reduction in bandwidth and efficiency accompanying a reduction in size. 7. 2. 2 Properties of Loaded Rectangular Cavity Antennas. The effort has been made, working within the basic limitations of electrically small antennas, to (a) accurately predict antenna characteristics and (b) to provide such optimization as may be possible. It has been shown that characteristics such as (1) resonant frequency (2) bandwidth, and (3) efficiency can be accurately predicted for the rectangular cavity slot antennas, using data obtained from variational methods in conjunction with simplified Smith chart techniques. The normalized aperture admittance has been formulated for a number of similar problems including (a) dominant mode incident-scalar problem, (b) other modes incidentscalar problem, (c) dominant mode incident-scalar dual problem, (d) dominant mode incidentvector problem. In all cases it has been possible to show that, if the aperture electric field has approximately the form of the incident mode, then the normalized aperture admittance is proportional to /r. The presence of an iris in the aperture complicates the dependence somewhat, introducing a lumped reactance which is one half of the reactance of the same iris in an infinite waveguide and which is independent of 1r in addition to the complex normalized aperture admittance which is proportional to r' 153

As a result of the dependence on Ar,' it is possible to optimize the bandwidth and efficiency characteristics by choosing material constants for maximum IT|2, the squared magnitude of the transmission coefficient. 7. 2. 3 General Properties of Loaded Slot Antennas. Many of the characteristics of the rectangular cavity slot antenna may be carried over to a more general class of antennas, namely the class of loaded cavity slot antennas. The loaded cavity slot antenna is defined here as a section of uniform transmission line feeding a ground plane. The ground plane may be infinite, finite, and of any shape. A special case would be a loaded waveguide radiator with no ground plane attached, such as the antennas of Fig. 21. We can say, in general, that if two conditions are satisfied, the admittance functions will be directly proportional to /Jr 1/E r or V7I r/ and that the ratio of B/G will be independent of relative permeability or permittivity. The two conditions are (1) the form of the aperture electric field must be roughly that of the incident field and (2) there must be no aperture plane iris. If either of these conditions are not satisfied, then the admittance will contain additional terms which will be independent of /r and er. An example is the iris problem of Chapter 3, which yields an additional reactance term BI which is independent of L r and e. Similarly, if the aperture field is radically different in form from that of the incident wave, the infinite sum, Vi Y. 75 V2 Y 0 0 will yield an additional admittance term which, for a given )rEr product, will be independent of [', because of the factor (Y./Yo). This additional term will be reactive if the mode (i) is cut off and resistive if the mode (i) is propagating. As a result, for a given physical configuration, given frequency, and given IrE r product, an optimum choice of materials is always possible for the general class of antennas referred to as cavity slot antennas. The discussion and analysis has been limited to the case of a single incident mode. If a number of different modes are incident upon the aperture, each mode will have a different admittance and a different optimum. The over-all optimum may depend, for instance, on the relative strengths of the different modes. The experimental data seems to indicate 154

that, in the cases observed, the effect of higher modes is negligible, even in cases where several modes may propagate. One reason for this is certainly that the excitation is such as to launch predominantly the TE10 mode. The generality of the conclusions suggests applications to other types of antennas. We have discussed a rectangular waveguide with a unidirectional radiation pattern. The conclusions also apply to a waveguide radiator whose cross section consists of a doubly connected region, such as a coaxial waveguide. One form of this antenna is the annular slot antenna, 33 which is in common use. The analysis implies that the characteristics of the annular slot antenna could be improved through an optimum choice of material constants. Variational 31 data in the literature3 could be readily adapted to such an antenna. There are some significant differences between the rectangular cavity slot antenna and the annular slot antenna which would perhaps, lead to different applications. The beam pattern of the rectangular cavity slot antenna is unidirectional, whereas the pattern of the annular slot is doughnutshaped. A combination of these two patterns would give complete half plane coverage. 7. 2. 4 Variational Calculations. Several points concerning the mathematical techniques of variational calculations are noted. The use of variational calculations may be separated into three parts: (a) The formulation of the variational expression and proof of stationarity. (b) The reduction of the variational form to a form suitable for computation. (c) The computation of results. Numerous examples of variational expressions are available in the literature. Familiarity with similar forms often makes the process of deriving a variational expression and proving stationarity fairly simple. The computation of results, although it is an essential part of the process and requires a thorough process of checking, is straightforward. Part (b), the reduction to a form suitable for computation, represents an important part of the calculations for those problems studied in this thesis. It is in this area that two contributions to mathematical techniques are made. They are concerned with: (1) The use of two expressions for electric field in a single variational calculation. (2) The effect of forbidden modes on the Rayleigh-Ritz procedure. 155

In addition, several interesting points are made concerning: (3) The effect of assumptions concerning the scalar nature of a problem, and (4) The evaluation of higher mode coefficients. In Chapter 3, a variational expression was derived for the admittance, of a loaded rectangular waveguide radiator with an aperture plane iris. The expression consisted of two terms. It was not convenient to use the same form of the electric field in both expressions. A form E1(xy) was used in one term and a form E2(xy) was used in the other. This invalidated the proof of stationarity. Nevertheless, it was argued, by appealing to related variational expressions that each of the individual terms would be accurately evaluated. Experimental data from Chapter 6 indicates that accurate results are obtained in a particular problem (a loaded waveguide radiator with capacitive and inductive aperture plane irises). It is shown in Section 5. 4 that the forbidden modes have no effect on the RayleighRitz process. The forbidden mode coefficients are zero and the contribution of the forbidden modes to the admittance is zero. This result is proven for a specific problem (the Spaded rectangular waveguide radiator). This fact is primarily of mathematical interest, since one would probably exclude forbidden modes from a Rayleigh-Ritz process as a matter of course. If this fact is true in general and if, in addition, its inverse is true, namely that a zero coefficient implies that a mode is forbidden, then this fact could be used as a test for forbidden modes. It has been shown in Chapter 5, in two particular examples, that the assumption of a scalar problem (Ey = 0) before forming a variational problem is no more serious than the same assumption after forming a variational expression for the vector problem. Very few vector problems have been treated in the literature. One reason for this is that the complexity of a vector problem is greater than that of a similar scalar problem by at least a factor of four. However, we need only a crude assumption for aperture field. In many vector problems a scalar field assumption is justified. It has been noted, in connection with the evaluation of higher mode coefficients of the particular problem under study, that the quadruple integrals can be reduced to double 156

integral form and evaluated by methods similar to those used for the evaluation of the dominant mode approximation. Gross approximations have been made in the analysis in order to apply the results of the problem of Fig. 7(a) to the configuration of Fig. l(g). In view of these gross assumptions, the accuracy of the results is surprising. 7.2.5 Experimental Techniques. Several brief conclusions may be made concerning experimental techniques. The use of wedged transition from the rectangular dielectric block to the slotted dielectric block has apparently yielded accurate admittance measurements. This conclusion must be hedged slightly since the agreement of the wedged data and the theoretical data has been used to conclude both (a) that the theoretical data is accurate and (b) the wedged transition admittance measurement method is accurate at frequencies slightly above cutoff. A new method of measuring dielectric constant has been proposed (Section 6. 3. 2). This method has not been thoroughly investigated, but it is believed that it may offer some advantages over conventional techniques in (a) measurement accuracy, (b) sample preparation and (c) rapidity of measurement. 7. 2.6 Very Small Antennas. For very small antennas (antennas which are small compared with a material wavelength Xo/ iTrer ), certain conclusions can be drawn. For a very small loop antenna, the magnetic field is proportional to 1/r2 whereas the electric field is proportional to 1/r3. Therefore energy storage is primarily in the magnetic field and the effects of dielectric loading are negligible. Similarly, for the very small electric dipole, the effects of magnetic loading are negligible. It has been noted that increasing er decreases the radiation resistance of the very small loaded dipole, whereas increasing 1r of the very small loop increases the radiation resistance of the loop. This may be explained in terms of the fact that induced polarization currents -p of the dipole oppose the currents on the conductor whereas the induced magnetization currents A x M of the loop aid the currents on the conductor. For antennas large compared with a material wavelength, phase differences between induced currents and driving currents must be taken into account. As a result the above conclusions are invalid for antennas large compared with a material wavelength. 157

7.2.7 Summary. What has been accomplished through the use of ferrite loading? The use of ferrite in a rectangular cavity slot antenna may be described in simple terms as an effective method of synthesizing the simusoidal aperture distribution, and one which permits accurate prediction of antenna characteristics. In addition, the use of material loading also reduces the reflection coefficient. An additional advantage in ferrite materials is that they may be tuned to change resonant frequency (Section 6. 3. 6). However, it should be emphasized that there may be other methods of accomplishing the same thing without ferrite loading. It is certainly not claimed that the results obtained cannot be obtained by any other method. It is claimed that the results show one method of obtaining significant size reduction at moderate expense in terms of bandwidth and efficiency. The contributions of this thesis are outlined below. 1. The derivation of accurate simplified design techniques for rectangular cavity slot antennas and the optimization of antenna characteristics. 2. Two contributions to mathematical techniques for variational calculations. (a) Discussion and example of two forms of electric field in a single variational expression. (b) A proof that, in the problem at hand; the forbidden modes do not contribute to the admittance calculation. 3. General conclusions concerning the limitation and potentialities of loaded antennas. 4. Practical realization of several loaded cavity antennas of reduced size. 5. Two contributions to measurement techniques including: (a) The measurement of aperture admittance using a wedged transition. (b) A proposed new method of measuring dielectric constant. 7. 3 Recommendations for Future Work 1. The accuracy of the bandwidth prediction could be improved by a complete solution of the coaxial probe to waveguide junction. The equivalent circuit could then be used in the same type of Smith chart calculation. It is also possible that a more sophisticated model of the junction, short of a complete solution, would improve the accuracy of bandwidth prediction. 158

2. The application of ferrite loading to traveling wave types of antennas shows some promise. Results of experiments on a loaded scimitar antenna are shown in Fig. 31. The scimitar has essentially a high pass characteristic. The low frequency cutoff is lowered by a factor of two or three in frequency. Bandwidth is essentially unaffected and efficiency is 80 percent. Similar results have been noted for tests on loaded spiral antennas. 15 1-2A IR CASE 12 LOADED CASE 9 - BALSA Iu WOOD-1 \ J*^%%%^% "7-" 736" FERRITE [ I\ POWDER LI t \ I SCIMITAR 55" — O:6Y 61 \ IGROUND PLANE >/' I \ I 36 1 3100 300 600 900 1200 1500 1800 2100 FREQUENCY (MC) - Fig. 33. A ferrite loaded scimitar antenna. 3. The preliminary experiments performed on the magnetically tuned slot antenna show promise. It is interesting to note that what has been accomplished is not only a change in the cavity resonance, but essentially automatic matching since it was not necessary to mechanically readjust the probe depth at each frequency. The bandwidth is very narrow, but this may have some inherent advantages in high rejection for off-resonant frequencies. 4. Directive antennas. Material loading may have some application to directive antennas. Certainly the decreased wavelength XA/ ViJL e may help to realize the required 159

supergain distribution; some progress has already been reported in increasing the gain of helices by dielectric loading. 82 Of course, the basic limitations of supergain antennas cannot be overcome. As an example consider a horn antenna of a given size. Material loading may facilitate the launching of the higher modes necessary for higher gain. If higher gain is attained, it must be accompanied by narrow bandwidth and low efficiency. The beam pattern of the horn antenna depends only on the aperture field. If material is added and the aperture field distribution is not changed, then the directivity will be exactly the same. 160

APPENDIX A COORDINATE TRANSFORMATION In Chapters 2-4, the following coordinate transformation is used extensively. y-?7 = X x- = a y+r7 = a+j, x+~ = v+b The purpose in transforming to sum and difference coordinates is to allow reduction of the quadruple integrals to double integrals. e kr/r cannot be integrated but the other terms of the C nm,n, coefficients arising in Chapters 2 and 4 can be integrated. The transformation reduces e kr/r to a function of two rather than four variables. This transformation consists of a 45~ rotation of coordinates, a reduction of scale, and a translation. Consider the general coordinate rotation:.u" = 77 sin 0 +y cos 0 X" = -7] cos 0 + y sin 0 let 0 = 45~ J'I" = 1/f2 [y+r7] A" = i/f2 [y-l7] Now perform the transformation /it = f2 li t reduction of scale A.' = f2 A" A' = +r7 A' = y-r? 161

Now perform a translation along the ji' axis 1 = I'-a A = A' This translation locates our coordinate system at the center of the area of integration and makes more readily apparent any symmetries which may exist. These operations are shown in Fig. 4. Consider an integration over the square area of Fig. 4. a a ff f(y,.) dy dr o o dy drj = d,. dX (because of the reduction of scale) a a a a-A o a+X f f f(yI) dy d = f( ) d d + f(j,X) diu dX (A. 1) o o o -(a-x) -a -(a+A) If f(,,AX) is even in t and A: f(-,XA) = f(,,-X) = f(i,X) then a a a a-X f f f(y,) = 2 f f f(iX) dA L dXd (A. 2) 00 00 The integration over the square area reduces to four equal integrations over a triangular area. Similarly, for the other half of the transformation b b b b-a o b+o f f f(5,x) da dx = f f(oa,v) d da + f(a,v)dvdo (A. 3) o o o -(b-a) -b -(b+a) b b-a = 2 f f f(a, v) dv da[if f(a, v) is even in a and v] (A. 4) o o 162

Similarly a b a b a b a-A b-a f f fS f(x,y,,rl)dxdydd d71 4 f f f(v,,oa,A) dMddadA o ooo o o -(a-A) -(b-cr) a o a-A b+a + f f f f f(v,,,,,A) dv dju dadA o -b -(a-A) -(b+a) o b a+A b-( (A. 5) + 4 fS f f f (v,,,a,A) dvd/ dadAd -ao -(a+X) -(b-a) o o a+A b+ac + f f f f f(v,/,(a,A)dvdjidadXA -a -b -(a+A) -(b+a) aa - b-a a a-A b a o = f f f (v,,a,AX)dvdidadA + f f f f f(v,,a,AX)dvdidadX o o o -(b-a) o -b o -(b+a) (A. 6) [if f(v, Mt,(a,A) is even in j and A] a b a-A b-a o b a+A b-a = f f f f f(v, C, a,, )dvdidadA + f f f f f(v, ji,u, A)dv dL dadA o o -(a-A) o -a o -(a+A) o (A. 7) [if f(v,,u, a, A) is even in v and (a] a b a-A b-Ca = 4 f f f f f(v,,, o,A ) dv dL da dA (A. 8) 000 0 if f(v, /, a, A) is even in v,, C,A X. 163

APPENDIX B EFFICIENCY OF A RECTANGULAR CAVITY SLOT RADIATOR B. 1 General Formulation The efficiency of a rectangular cavity slot radiator has been analyzed, assuming an attenuated dominant mode only in the cavity and utilizing the reflection coefficient at the aperture which has been evaluated by variational techniques in Chapter 2. An electric current sheet source exciting the dominant mode is located at the probe position [Fig. 33(a) and (b)]. The source may also be assumed to be magnetic current, in which case the complex constant A, which determines the fields in Region No. 2, will have a different value. The electric current source gives rise to a discontinuity in H whereas the magnetic current source produces a discontinuity in Ex. The radiated power is calculated by integrating Poynting's vector across the aperture. The formulation takes into account magnetic and electric volume losses and wall losses with certain simplifications resulting in special cases. The fields are represented in terms of a Hertzian vector potential of magnetic type in Regions No. 1 and No. 2 -ir10z r10z] * -1 [e 1-R e Ay ( 11 -A +e 1 i. A2) 2 JW y 0 A sin - y (B.2) J = Jo sin 7Y *x electric current s 0 a or M M0 sin y * y magnetic current 164

MATERIAL )E d2 - di (a) V/I/. REGION /2 // REGION #///I Z~ (bJs (b) Fig. 34. Model for efficiency calculations. where R is the complex dominant mode reflection coefficient evaluated in Chapter two. Efficiency a b time average power radiated = P = Re f E H * dx dy o (evaluated at z = 0) 165

time average power dissipation in the cavity = PL = fffS WIE12 dv + SSff wl"lHI2 dv cavity cavity + Rs fff IH2 da walls where E and H are rms values, E = cE - jE" =' - j., R = surface resistivity P efficiency = (B. 3) Pr +PL PL r L L 1+ - r since we are concerned with the ratio PL/Pr, it is immaterial whether we use peak or rms values in the above equations. Solving for the Fields in Regions No. 1 and No. 2 12 S2 Y -2za 10 2 10z (B. 4)l IEx,2 = sin2 [e 10 + RI2 e +10 21RI cos (210z + 0) (B.4) 2 -2a 10 z 10 IR (2 + je +JR 112e e - 2 IRI cos (210z + 0)] Region Hz- 2 = 1az cos2 y Hz 1 - cos a -2a z 2a 10 z [ 10 e + 2 10 +2 IRI cos (2 oz + or) 166

E 2 IAr sin2 " e 10 + e 0 - 2 cos 210oz (B. 5) X2 a 10 y2 Wk2 a2 _ r 2 a e + e + 2 cos 203z c Region L J No. 2 H 2 I1 2 NW2 Cos 2 7TY z2 awli a -2a 10z' 2a 0z e 10 + e 10- 2 cos 2 z' where: Subscript one vectors are zero in region two and vice versa. Applying boundary conditions at the source: Electric Current Source: El] = Ex2] z = -d 2 z10 = -d2 167

A e 10 Re 10 +ro10d2 - r1od2 e -e +2a 10d1 -2a10 d e + IR2 e + 2 IRl cos (2103dl - er) IAI2 =2 2 -2a 10(B. 6) +2v 10d2 -2 0d2 e + e - 2 IRI cos (2 1d2) cosh 2a 10d + cos (2 10d - 0 r) R = 1(B.7) cosh 2a1 d - cos (2 3d) J d dL = d =d The value J0 may be derived by applying boundary conditions on tangential H. Radiated Power: Radiated power = Re fff (Ex H) z dxdy (B.8) aperture Y E = (1+R) sin — Y (B.9) A a H * = (k a: a)(- T 2 1 (1-R) siny Radiated power = Re [(l+R)(l -R*) (k a2 -7)*( +1 ) 1 ]ab (B. 10) where: * indicates complex, conjugate except when used with II. rl0 [ /_ [ 2a2 (wIa)* = (ka) A[] [L] R = complex aperture reflection coefficient ka = k)a /(E Ic - iE+ ) = I /r = r r 168

E = e' - jE" r r r = Mr/ o e = E r o It should be noted that the proper definition of the square roots must be chosen so that all the quantities have the correct physical significance. r10 is in the first quadrant, (woLa)* and ka are in the fourth quadrant for a time dependence ewt. P Efficiency = r P + PL r L where PL = We" fff IExI2 dv + w a" v fff [IHy2 + IHI2 dv (B. 11) cavity cavity + Rs ff IHtangl2 da walls The following integration formulas are useful: -d [e-2a 10z 2a ]01 f [e2101 + RI2 e 0 +2 R cos (210z + 0) dz 2a [R12 e 0 + (1- IR2 (B.12) 2a 10 RI [sin(2 10d- 8)+ sin0 r] 10 1 r -d -2a 1z 2a 1z S [e 10 +e 10 +2 cos 2/310z dz 0 ~ a10 169 ~ae -e ~ sin 210d (B. 13) 2a10d 2a 10d -sinh 20 10d sin 20 10d a10 / 10 169

Magnetic Volume Losses Wc" Sff [IH y2 + IHy22 + IHz,12 + IHz2] dv cavity + ( + ( + (B. 14) Cl"la = kal * | | I'Iol 1 Ika2 - 2 Iw \a1 = |kal I1 | k2a2- L 7T21 k E/LII b O Ikal 2 {2c [0 IR2 e 10dl + e 10 1 (- 1 R12)] - RI [sin (210d1- r)+ sin o]} IAI2 Ik2a-2a - 1_1-1T2 I / 1' I- bI +s sinin 210d2l VaI = M 2 1 10 lO f (- IVkIr l b f[-|R| 2 e 110 k10al 2 -2a10dl + e2a10dl (1 1R2)] O II?~~~-IRI +e - ( = Ikal 2 - 2a 1 + 1 [sin (2 0d - ) + sin Or IAl 2 1'1 I I "I b + sinh2a10d sin 2 d ^(~I kal 2 a 10 J 170

Electric Volume Losses Ce"a = ka|. I1/ 1 |Ie WE" fff [ IEx12 + IE x2 ] dv 1'1 1 [A~ -'b 1 2a1u0d 1 2aidl R dI1 = Ikal I/II | 20 10 [I2 e e10+e (1 IR2) 2RI ~ IRI sin (230d - 9 ) + sin 0 + [sin 10 1 r r + A2 [ sinh 10d1 -s 210 d1 (B. 15) al0 ~10 resistivity Losses Rs wf IHHtangI2 wall = 2Rs ff l [I Hy2 +lHy212 +lHz112 +IHz212] dydz s top wall L- J + 2R s ff lHz, l2 +,H z2 ] 2dxdy side wall back wall [~ + ~ + ~ + ~]+ [ + ]+[ (B. 16) aRsIk2a2_ 7T21 I f E [I -2a d 2a d _lk a _____ ___ [RI2 1 e 10 1 (1- R12) - R [sin (2 d - 0 ) sin } |A|2 aRslk2a2 - 21 I |I + sinh 2a d2 sin 210d2 Ik2a21 10 10 171

() 2 aRsl { I RI2 e +e( 72 aR I- | r 1 r -2a 10 dS 2a 10 d1 ~(e1 =22 t 12ao - IRI2 e +e (1 - IRl2)] Ik2a L +e'Ii 10 [sin (210dl sin r]} A2 IA12 a RSi I I + sinh 2ad2 sin 2 O d I k2 a21 a 10 10 2T2 bR S I I -2a d 2a d i |k j LR12 e 101 +e 10 1J +in [sin (2 o 10d1 - 0 ) + sin ]} IAow 222 bRfl 21- + sinhd2a sin 20 d u = Ik^2 I "10 2 10 2 Ikal B. 2 Efficiency Using High Q Material Certain simplifications to the above formulation can be made in some cases. If a 1012 as10d2 the loss tangent of the material is small enough so that e, e are nearly unity, then the cavity fields can be represented by unattenuated exponentials for purposes of calculating losses in the cavity. The power radiated can be calculated simply by multiplying the power flow in the incident wave by (1 - 1|R2) The problem will be reformulated using these assumptions. It should be noted that the assumption of unattenuated exponentials will hold if the loss tangent is sufficiently low even in cases where a low efficiency may result due to the high reflection coefficient. In Sections B. 2 and B. 3, tL, E, k are assumed to be real as used in the equations. For purposes of calculation, /i may be replaced by I I, c by e', and k by wc/ T7 in Eqs. B. 17 through B. 30. |IE |2 = sin2 I 1 +1112 + 2 R cos (210z + ) (B. 17) 172

|H 12 (k2-a2-_ ) e sin2 _yy I k2a2 u a 1 + IR12 - 2 |R| cos (2310z + r) Region 2,R, cos(2fi10z+ 0r)] No. 1 H 2 = Cs 7cos2_y zl k 2 ^ a 1 + R12 +2 IRI cos (23 10z + 0r) IEx212 = 21|A2 (1 - cos 2/10z') sin2 7TY (B. 18) IH12 = 21A12 (k a2 2 ) - (1+ cos 2310 ) sin2 egion y2 k2a2 10z a No. 2 IHz 21 = 21AI 2 2 (1 - cos 2 10z) coS2 a k2a2'1 Radiated Power Radiated power = 1 - IRI2] T = PR (B. 19) ka 2-= PR Electric Volume Losses W" It Jff [IEX1 + IEx21 ] dv ka E { ab (1 +|R|12) IRI + sin 0 + sin 0 L, 01 r r L +,Aad2 sin 2 10d2 + jA1{2 kaV a - I ~ (B. 20) 2L/k2 a 2- 7T2 Magnetic Volume Losses WII' IH1 f Hy1l+ IH ly2 + IH112 + IH z12 dv = + ( + 3 + ( (B. 21) 173

~ (( ka - t2 (+ JI { R + II [sin (2 1 d- 0 ) + sin 0 ka 1L Vl+ ) a 1/k2 a2 _ T 2 2,A2 k2 a2-T_ 2 /2 sin 2 " d2 ~ = 1A2 k akaI ab + 2j_1ad2 } OL si213d2 s in 0d2 } ka g +1 a- 8 + 2 2 7T: ab 12" JR Surface Resistivity Losses 2R fL [Y12 +lIHy212 +lIHzl I2 +H z212] dydz 111121 ]Hy2]z dx dy + 2R sid [ewl [IHZ12] idx dwz +s sback wall dxdy = ( + ~ + ~ + ( + + + (B.22) =R (k2 a2 - 12) e a2 f(1+RI 1 - IRI [sin (2 d- -O)+sinO } kaa2 a s 1 rlJJ $_ = 21A12 R (ka2- 7r2) e/ + 22 2} __) = 2|A|k 1T e a2/k a{ 2 2 _ 22- _ 174 e} =R~ -ad aRI [ (1 + sIn =a + s2d1 - I )s G =21A1 R { s kT - a2;- ___ ____ 2k2a2-2 a2]_)2 JA 2f 2 ab si n {210d22 k2a2 aa k,/'k2 a - ( d_]

_() = 41 Al2 R 2 Eab d4 ssin 23 10d2 (~) = 4A — ab - - — } k2 a 2 a 2 = 21AI2 (k2a2- ).. ab k2a2 A B. 3 Efficiency with Probe Near Aperture If the probe is located near the aperture, the problem can be simplified to a single region. Let II* 1 ( -1 ) y = 3 — 1 j )cos z10' sin 7Y pi j10 a Exl2 = sin2 Zl? sin2 xY (B. 23) IH 2 =2 (k2a2_ ) _ ( ) cos2 z sin2 IHzI2 = 7 2 sin2 1OZ' cos2 k a2 12 12= 2 c\/ a Radiated power = (1- IR2) Re f f E H* dx dy (1 - IR2) P 00x y 8 ka V A r (B. 24) Electric Volume Losses fffs2 2 a sin2 2 20 102 CO" ff sin2 z sin dv = w [ d (B. 25) cavity aL 22 Magnetic Volume Losses 82sin 2 d t J [IH 12 + IHZ12] dv = W a b e. ka2 ( )+ 10 (k2a2- 272)] w~( Scav ity 1 a, 8Jk k2A2 a ka2 -k a (B. 26) 175

Resistive Wall Losses R ff H | 12 A Rs a2 e 2d 2a2 + 2b 2 all walls ta dA k2a 2'4 a a +in 10d (k2a2272 - 2T12 b + 2b (k2a2- r2)] (B.27) a a a For Magnetic Volume Losses Only: [, ( )+ sin 2310d 2 at a -2 )(k2 - 2 72)...... = u" = --------- vka~r (B. 28) Pr (1- IRl 2)k2a2For Electric Volume Losses Only: k2a2 [ 2d sin210d ] a J L^ ( ) E'ka (B.29) Pr (1-RIS2) J/kFor Resistive Wall Losses Only: =.^^s/F ~ f^2^ 2).^ ^a2.2712 ka 2b0d k2 _ sh210d 2a 2 ).00531 Rl rr ) [ a ~ 1r2 + (ka 2ir2 2i2 — V ^r L a-\ a a PL r Vl/ka2 a2-_ 3 r ka (1 - IR12) k/2i (B. 30) 2b (k2a2- 2) a J In Eqs. B. 17 through B. 30,u may be replaced with,u', e with E' and k with wvAT for purposes of calculation. 176

APPENDIX C PERTURBATIONAL CALCULATIONS The effect of a coaxial feed structure on the resonant frequency of a loaded rectangular cavity slot antenna can be calculated utilizing perturbational formulations. A simplified approach which treats the antenna as a resonant cavity is discussed. Perturbation techniques for calculating the percentage shift in resonant frequencies of cavities due to small changes in material loading or deformation of cavity walls are well known. 83, 84 The loaded rectangular cavity slot antenna may be considered as a resonant cavity since the magnitude of the aperture reflection coefficient is nearly unity for electrically small antennas. A more accurate method would be to assume a two region model with a current sheet source as in Appendix B, but this degree of sophistication is unnecessary since the shift in resonant frequency is very small in most cases. Figure 34 shows the model employed, which is a rectangular cavity resonant in the lowest (TE 101) mode with a perturbation caused either by the insertion of a cylindrical metal probe or by the material perturbation of a cylindrical section of material. The shift in resonant frequency may be calculated from the following formulas. 85 Metal Probe _o - 1fff (IHO2-E1E12) dv o'- Probe_ (C. 1) fff (IHoI2+ elEol2) dv cavity Material Perturbation W co fff (AIEol,2+ AlHo 0) dv o cavity (C. 2) ~O S fff (IHol2+ e Eol0) dv cavity 177

where Eo and Ho are the unperturbed fields, AE,A4 are the changes in material constants, AE, A/ are positive for an increase in material constants, AC, AJL are negative for a decrease in material constants. IEo12 = IEx2 = sin2 7 sin2 ay IHo12 = iHy2 + IH Z Hy2 = (k2a2 _ X2) ( )cos2 E sn2 k2a2 a ]H z_2 2r2 7Z 2 7TY IH Z12 = (-) sin2 Cos k2a2 d a Assuming that the fields are constant over the region of perturbation, these formulas reduce to: Metal Probe WW - [ 2V 2v7d 2 i7Td ________ _ 1V 1^ ____ __2 cos 2Cos2(C. 3) wo abd cos d k 2 s2 abd 2 where V is the volume of the probe. Material Perturbation WW 2V i k rd 7rd1 - - abd [(ka2T2) ( -) cos2 d + e sin2 d (C.4) If the perturbation consists of the removal of a cylindrical section of material, leaving an 178

air space: - abd L kd ) f d where V is the volume of the air space. 2v 7T2 Vid 1 - cos2 d (for large AirEr) abd k2 a2 d2 2V abd (dl d2) It is interesting to note that the effects of a probe and an air space of equal size cancel, the probe decreasing the resonant frequency by 2V/abd and the air space increasing the resonant frequency by the same amount. 179

APPENDIX D NOTE ON MATHEMATICAL STEPS REQUIRING FURTHER JUSTIFICATION Throughout Chapters 2, 3, and 4 certain mathematical steps have been made which cannot be justified in terms of the Riemann integral formulation. Among these are: (a) interchange of the order of integration (b) interchange of the order of integration and differentiation (c) integration by parts. -jkor Because of the singularity e /r occurring in the integrands, these steps cannot be justified by theorems of Riemann integration which require continuity of the integrand. However, the singularity is absolutely integrable, which m'eans that we could discard a small area about the origin and justify the steps by theorems of Riemann integration. It would be preferable to formulate the problem in terms of the Lebesgue integral, for which some of the steps could be justified immediately. For example, Fubini's theorem permits the interchange of the order of integration if the integrand is absolutely integrable. Singularities of this type often occur in boundary value problems, because of the idealized nature of the sources assumed. In Chapter 2, the singularity occurs because of the idealized magnetic current sources, which after imaging, consist of a thin current sheet in free space. If the problem were formulated ini terms of distributed volume sources similar to any realizable source system, the singularities would not occur. 180

REFERENCES 1. N. Marcuvitz, The Waveguide Handbook, Vol. 10 of The Radiation Laboratory Series, McGraw-Hill Book Co., N. Y., 1950., p. 47. 2. L. J. Chu, "Physical Limitations of Omni-Directional Antennas, " J. Appl. Phys., 19, Dec. 1948, pp. 1163-1175. 3. R. F. Harrington, Time Harmonic Electromagnetic Fields, McGraw-Hill, N. Y., 1961. 4. J. A. M. Lyon, A. T. Adams, and R. M. Kalafus, "Analysis of the Miniaturization of Resonant and Non-Resonant Antennas Utilizing High "Q" Materials, " Transactions NEC, 1962. 5. Proceedings of the Symposium on Microminiaturization of Electronic Assemblies, Hayden Book Company, Inc., N. Y., 1958. 6. S. A. Schelkunoff, "A Mathematical Theory of Linear Arrays, " Bell System Tech. J. 22, Jan 1943, pp. 80-107. 7. W. W. Hansen, Notes on Microwaves, MIT Radiation Laboratory Report T-2. 8. JC. J. Bouwkamp and N. G. de Bruijn, The Problem of Optimum Antenna Current Distribution, Phillips Research Reports, Vol. 1, 1946, pp. 135-158. 9. H. J. Riblet, "Note on the Maximum Directivity of an Antenna, " IRE Proceedings, May 1948, pp. 620-623. 10. T. T. Taylor, "A Discussion of the Maximum Directivity of an Antenna, " IRE Proceedings, Sept. 1948, pp. 1134-1135. 11. S. Ramo and J. R. Whinnery, Fields and Waves in Modern Radio, John Wiley and Sons, N. Y., 1953. 12. S. A. Schelkunoff and Harold Friis, Antennas - Theory and Practice, John Wiley and Sons, N. Y., 1952. 13. J. D. Kraus, Antennas, McGraw-Hill, N. Y., 1950, p. 348. 14. W. W. Hansen and J. R. Woodyard, "A New Principle in Directional Antenna Design," IRE Proceedings, Vol. 26, March 1938, pp. 333-345. 15. L. B. Felsen and N. Marcuvitz, "Slot Coupling of Rectangular and Spherical Waveguides, " J. Appl. Phys., June 1953, pp. 755-770. 16. H. A. Wheeler, "Fundamental Limitations of Small Antennas, " IRE Proceedings, Dec. 1947, pp. 1479-1484. 17. M. H. Cohen, "On the Bandwidth of Cavity Antennas, " J. Appl. Phys., May 1954, pp. 582-587. 18. R. H. Barfield and R. E. Burgess, "Small Aerials in Dielectric Media, " Wireless Engineer, Aug. 1948, pp. 246-253. 19. W. J. Polydoroff, High Frequency Magnetic Materials - Their Characteristics and Principal Applications, John Wiley and Sons, N. Y., 1960. 181

REFERENCES (Continued) 20. D. G. Kiely, Dielectric Aerials, Methuen and Co., Ltd., London, 1953. 21. R. C. Hansen, "Two Famous Inventions, " Microwave J., Aug. 1961, p. 15. 22. M. Islam, "A Theoretical Treatment of Low-Frequency Loop Antennas with Permeable Cores, " IRE Trans. PGAP, March 1963, pp. 162-169. 23. R. E. Burgess, "Iron-Cored Loop Receiving Aerial, " Wireless Engineer, June 1946, pp. 172-178. 24. J. Herman, "Thin Wire Loop and Thin Biconical Antennas in Finite Media," Diamond Ord. Fuze Lab. Tech. Rept. No. TR-462, May 1957. 25. 0. R. Cruzan, "Radiation Properties of a Thin Wire Loop Antenna Embedded in a Spherical Medium, " IRE Trans. PGAP, Oct 1959, pp. 345-352. 26. Saburo Adachi, "Impedance Characteristics of a Uniform Current Loop Having a Spherical Core, " Ohio State Univ. Res. Foun. Rept. No. 662-26, April 1959. 27. C. T. Tai, "Radiation From a Uniform Circular Loop Antenna in the Presence of a Sphere, " Stanford Res. Inst. Tech. Rept. No. 32, 1952. 28. D. M. Grimes, "Miniaturized Resonant Antenna Using Ferrites," J. Appl. Phys., Vol. 29, No. 3, March 1958, p. 401. 29. C. Polk, "Resonance and Supergain Effects in Small Ferromagnetically or Dielectrically Loaded Biconical Antennas, " IRE Trans. PGAP, Dec 1959. 30. L. Lewin, Advanced Theory of Waveguides, Iliffe and Sons, Ltd., London, 1951. 31. H. Levine and C. H. Papas, "Theory of the Circular Diffraction Antenna," J. Appl. Phys., Jan 1951, pp. 29-43. 32. C. T. Tai, "Application of a Variational Principle to Biconical Antennas," J. Appl. Phys., Nov 1949, pp. 1076-1084. 33. J. W. Galejs and T. W. Thompson, "Admittance of a Cavity-Backed Annular Slot Antenna," IRE Trans. PGAP, Nov 1962, pp. 671-678. 34. J. W. Galejs, "Admittance of a Rectangular Slot Which is Backed by a Rectangular Cavity, " IRE Trans. PGAP, March 1963, pp. 119-126. 35. J. E. Storer, Variational Solution to the Problem of the Symmetrical Cylindrical Antenna, Cruft Lab. Rept. TR 101, Harvard Univ., 1952. 36. M. H. Cohen, T. H. Crowley, and C. A. Levis, The Aperture Admittance of a Rectangular Waveguide Radiating Into Half Space, Ohio State Univ. Res. Foun. Rept. 339-22 of Nov 1951. 37. V. H. Rumsey, "Traveling Wave Slot Antennas, " J. Appl. Phys., Nov 1953, pp. 1358-1365. 38. M. H. Cohen, "The Normal Modes of Cavity Antennas, " Doctoral Dissertation, Ohio State Univ., 1952. 39. C. A. Levis, "Variational Calculations of the Impedance Parameters of Coupled Antennas, " Doctoral Dissertation, Ohio State Univ., 1956. 182

REFERENCES (Continued) 40. J. S. S. Kerr, "The Radiation Impedance of a Flanged Rectangular Waveguide," Doctoral Dissertation, Univ. of Ill., 1951. 41. C. C. Nash, "The Input Impedance of a Rectangular Aperture Antenna, " Doctoral Dissertation, Univ. of Ill., 1949. 42. N. Marcuvitz, op. cit., p. 184. 43. V. Counter, Miniature Cavity Antennas, Microwave Laboratory, Stanford Univ., Rept. No. 105, Jan 1950. 44. R. J. Tector, The Cavity Backed Slot Antenna, Electrical Engineering Research Lab., University of Illinois, Technical Report. No 26. 45. H. M. Barlow and J. Brown, "Radio Surface Waves, " Oxford Univ. Press, 1962. 46. H. Levine and J. Schwinger, "On the Radiation of Sound from an Unflanged Circular Pipe," Phys. Rev., Feb 1948, pp. 383-406. 47. B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon Press, Inc., N. Y., 1953. 48. L. A. Vajnshtehn, Propagation in Semi-Infinite Waveguides, Six papers translated by J. Schmoys, New York Univ., Inst. of Math. Sci., Rept. No. EM-63. 49. F. P. Brownell and D. E. Kendall, "Miniaturized Cavity Fed Slot Antennas," 1960 WESCON Records, Part 1, pp. 158-166. 50. F. E. Butterfield, "Dielectric Sheet Radiators, " IRE Trans. PGAP, Oct 1954, pp. 152-158. 51. C. W. Morrow and J. L. Moore, "High-Dielectric Rod Antenna Arrays for UHF," Electronics, Feb 1960, pp. 60-62. 52. K. E. Niebuhr and E. H. Scheibe, Surface Wave Dielectric Disc Antenna, Univ. of Wisc. Internal Rept., Contract No. DA-36-039- sc-71158. 53. W. Rotman, Metal Clad, Progressive-Phase, Dielectric Antennas, Air Force Cambridge Research Center, Rept. No. E 5081, Nov 1951. 54. R. W. Wagner, The Dielectric Plate Antenna, USASRDL Rept. No. 2240, Fort Monmouth, New Jersey. 55. K. G. Balmain, The Radiation Pattern of a Dipole on a Finite Dielectric Sheet, Elec. Eng. Res. Lab., Univ. of Ill., Tech. Rept. No. 41, Aug 1959. 56. J. W. Eberle, C. A. Levis, and D. McCoy, "The Flared Slot: A Moderately Directive Flush Mounted Broad Band Antenna, " IRE Trans. PGAP, Sept 1960, pp. 461-468. 57. B. T. Stephenson and C. H. Walter, "Endfire Slot Antennas, " IRE Trans. PGAP, Oct 1954, pp. 152-158. 58. Special Issue on Electromagnetic Waves in the Earth, IRE Trans. PGAP, May 1963. 183

REFERENCES (Continued) 59. L. Lewin, op. cit., p. 123. 60. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle, Oxford Clarendon Press, 1939, p. 157. 61. R. F. Harrington, op. cit. 62. Ibid., p. 98. 63. L. Lewin, op. cit., p. 124. 64. Ibid., p. 126. 65. Ibid., p. 126. 66. Ibid., p. 90, Eq. (4. 39). 67. Ibid., p. 96, Eq. (4. 52). 68. Ibid., p. 91. 69. D. S. Saxon, Notes on Lectures by Julian Schwinger, "Discontinuities in Waveguides. 70. R. F. Harrington, op. cit., Ch. 8. 71. R. E. Collin, Field Theory of Guided Waves, McGraw-Hill Book Co., N. Y., 1960, p. 349. 72. Ibid., p. 351. 73. J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Co., N. Y., 1941, p. 393. 74. R. F. Harrington, op. cit., p. 108. 75. Ibid., p. 76. C. L. Dolph, "A Saddle Point Characterization of the Schwinger Stationary Points in Exterior Scattering Problems, " J. Soc. Ind. Appl. Math., Sept 1957, pp. 89-104. 77. Interim Engineering Report of Research on Miniature Zero Drag Broadband Tunable Cavity Antennas - 1 July to 30 Sept., 1953, Univ. of Oklahoma Res. Inst., Contract No. AF 33(038)- 10405. 78. C. F. Arantz, Measurement of Efficiency of Small Linearly Polarized Antennas by Reflection Coefficients, Aerial Reconnaisance Lab., Wright Air Development Center, Wright Patterson AFB, Ohio, March 1958. 79. D. E. Royal, Efficiency Measurements Using Reflection Techniques, RamoWooldridge, March 1959. 80. G. A. Deschamps, "Determination of Reflection Coefficients and Insertion Loss of a Wave-Guide Junction, " JAP, 24, No. 8, Aug 1953, p. 1046. 81. P. C. Butson and G. T. Thompson, "The Effect of Flanges on the Radiation Patterns of Waveguides and Sectoral Horns, " IEE Proceedings, July 1959, pp. 422-426. 184

REFERENCES (Continued) 82. V. P. Shestopalov, A. A. Bulgakov and B. M. Bulgakov, "Theoretical and Experimental Investigations of Helix-Dielectric Aerials, " Radiotekhnika i elektronika, Vol. 6, No. 7, 1961, pp. 1136-1145. 83. R. F. Harrington, op. cit., Chap. 7. 84. A. D. Berk, "Variational Principles for Electromagnetic Resonators and Waveguides," IRE Trans. PGMIT, April 1956, pp. 104-110. 85. R. F. Harrington, op. cit., p. 319. 185

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