mszmm a; mama Loam 3101 Mama arm A WRFBWRE mm fiwsis far fits Degree 2:? mm ‘ mmm mm mm mamas m 1921 * This is to certify that the thesis entitled Investigation of Impedance Loaded Slot Antennas and a Short-Backfire Antenna presented by Tsing-Zone Hsieh has been accepted towards fulfillment of the requirements for Ph. D. degreein EleC. Engr. Calm P. Wat Major mate“ 0 Date August 3, 1971 0-7639 ABSTRACT INVESTIGATION OF IMPEDANCE LOADED SLOT ANTENNAS AND A SHORT-BACKFIRE ANTENNA BY Tsing-zone Hsieh The circuit and radiation properties of impedance loaded (rectangular and annular) slot antennas and a short-backfire antenna are investigated in this thesis. In Part Ithe loaded slot antennas are studied, while the short-backfire antenna is investigated in Part II. A simple, flush-mounted antenna consists of a slot cut in a ground plane and excited by a potential maintained between its edges at a point along its axis. The circuit and radiation properties of electrically small slot antennas doubly loaded by lumped impedances connected between their edges are investigated in this research. Loaded rectangular and annular slots are studied both analytically and experimentally. An electrically small slot is difficult to excite efficiently due to its small input resistance and relatively large input reactance; its directivity is also relatively poor. These circuit and radiation characteristics can be improved by choosing an Optimum doubled loading to appropriately modify the electric field distribution in the aperture of the slot. In the theoretical investigations, integral equations for the electric field distributions excited in the impedance loaded slots by a 6-function current source are formulated and solved numerically. In terms of these slot fields, the input impedances and the radiation fields of the loaded slot antennas are calculated. These analytical results are confirmed qualitatively by those of complementary expe rimentaly studies. It is demonstrated that the electrically small slot can be forced into a near antiresonant condition by the selection of an optimum purely reactive loading. A large input resistance is subsequently obtained. This optimum reactance is found to be usually capacitive while the TSING- ZONE HSIE H antenna input impedance is inductive. Low loss capacitive loading and tuning impedances can thus be utilized to implement a high efficiency, small antenna. A second optimum reactance loading leads to radiation field patterns which can be highly directive in the case of the rectangular slot or greatly modified for the annular slot radiator. A numerical-physical optics method is applied to study the circuit (impedance) and radiation characteristics of the short-backfire antenna. This radiator, developed through extensive experimentation by AFCRL, consists of a dipole exciter located between a large rimmed reflector and a small secondary reflector. It has wide band- width and high directivity comparable to sophisticated reflector antennas. In the numerical—physical optics method, the following steps are followed: (1) a set of coupled integral equations for the currents excited in the dipole and on the surface of the secondary reflector are formulated and solved numerically, assuming for this step that the large reflector is infinite; (2) the surface currents of the large reflector are approximated by a truncated form of those calculated for the infinite conducting sheet; (3) the radiation field maintained by the currents of steps (1) and (Z) is calculated; and (4) a diffracted field correction can be made to account for the finite dimensions of the large reflector and its rim. This method has the advantage, relative to earlier studies, that it can successfully predict the antenna's circuit characteristics. Excellent results are obtained for both square and circular geometries. Comparison is made with experimental measurements made by AFCRL. INVESTIGATION OF IMPEDANCE LOADED SLOT ANTENNAS AND A SHORT-BACKFIRE ANTENNA BY Tsing-zone Hsieh A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1971 ACKNOWLEDGMENT The author wishes to express his sincere appreciation to his major professor, Dr. D. P. Nyquist, for his guidance and assistance throughout this study. His active participation in the project and his willingness to discuss problems as they arose made working with him a rewarding experience. Thanks also go to Dr. K. M. Chen for his many suggestions and technical advice on problems encountered in the research. He wishes to thank the other members of his guidance committee for their time and interest in this study: Dr. B. Ho, Dr. J. Asmussen, Jr. , Dr. P. D. Fisher and Dr. J. H. Hetherington. The research reported in this thesis was supported by the Air Force Combridge Research Laboratory under Contract No. Fl9(628)-70-C-OO72. Finally, the author. thanks his wife, Lian-wu, for the under- standing and encouragement that only a wife can give. ii TABLE OF CONTENTS Acknowledgment ........................................ List of Figures ......................................... List of Tables .......................................... PART 1. Investigation of Impedance Loaded Slot Antennas 1 Introduction .................................... 2 The Loaded Rectangular Slot: An Electrically Short Antenna with Enhanced Radiation or Improved Directivity ............................ 2. 1. Introduction .............................. 2. 2. Physical Structure of the Loaded Rectangular Slot ........................... 2. 3. Integral Equations for the Electric Field Distribution in the Slot ................ Z. 4. Reduction to a Quasi-One-Dimensional Hallen- Type Equation ...................... Z. 5. Numerical Solution of the Integral Equation . . . 2. 5. l. Trigonometric Series Solution ....... 2. 5. 2. Solution by Pulse-Function Expansion ......................... 2. 6. Radiation Field of the Loaded Short Slot Antenna .............................. Z. 7. A Short Slot Antenna with Enhanced Radiation: Numerical Results .............. 2. 8. A Short Slot Antenna with Improved Directivity: Numerical Results ............. 3 The Loaded Annular Slot: An Efficient Electrically Small Antenna ....................... 3. 1. Introductory Remarks ...................... 3. 2. Physical Structure of the Loaded Annular Slot .............................. 3. 3. Integral Equations for the Electric Field Distribution in the Slot ..................... 3. 4. Reduction to a Quasi-One-Dimensional Integral Equation .......................... 3. 5. Numerical Solution of the Integral Equation. . . Page ii xi 12 l7 17 Z3 Z8 30 41 46 46 47 47 54 58 3. 6. Radiation Field of the Loaded Annular Slot Antenna ...................... 67 3. 7. Numerical Results ........................ 7O 4 Experimental Investigation of Loaded Slot Antennas .................................. 84 4. 1. Introductory Remarks ..................... 84 4. 2. Anechoic Chamber and Experimental Setup . . 84 4. 3. Rectangular Slot Measurements ............ 91 4. 4. Annular Slot Measurements ................ 95 5 Summary and Conclusions ....................... 101 References ............................................ 103 Appendix A ............................................ 105 Appendix B ............................................ 107 Appendix C ............................................ 117 PART II. The Short-Backfire Antenna: A Numerical— Physical Optics Study of its Characteristics 1 Introduction ................................... 133 2 Calculation of the Induced Currents .............. 137 Z. 1. Introductory Remarks ..................... 137 2. 2. Physical Structure of the Short-Backfire Antenna .................................. 137 2. 3. Integral Equations for the Induced Currents. . 140 Z. 4. Simplification of the Integral Equations ...... 146 2 5 Numerical Solution of the Integral Equations . 151 3 Radiation Field of the Simplified Short-Backfire Antenna Model ................... 159 3. 1. Introductory Remarks ..................... 159 3. Z. Induced Currents Excited on the Large Reflector ................................ 159 3. Z. 1. Current Excited on Image Plane by Dipole ......................... 161 3. Z. 2. Current Excited on Image Plane by Small Reflector ................ 162 3. 2. 3. Total Image Plane Surface Current . . 163 3. 3. Radiation Field Calculation ................ 163 3. 3. 1. Radiation Field Maintained by Truncated Image Plane (Large Reflector) Surface Currents ........ 164 3. 3. 2. Radiation Field Maintained by the Small Reflector Surface Currents. . . . 165 iv 3. 3. 3. Radiation Field Maintained by Dipole Exciter Current ............. 3. 3. 4. Total Radiation Field Maintained by the Induced Currents ............ 4 Numerical Results .............................. 4. 1. Introductory Remarks ...................... 4. 2. Induced Currents on the Antenna Structure . . . 4. 3. Input Impedance to the Primary Radiator ..... 4. 4. Radiation Fields Maintained by the Induced Currents .......................... 5 Summary and Conclusions ....................... References ............................................. Appendix A ............................................. Appendix B ............................................. Appendix C . . . . ......................................... 165 166 170 170 172 176 179 187 190 191 194 212 .10 .11 .12 .1a .1b Physical structure of loaded annular alot antenna . . . . . ......................... Integration subdivisions for solution of the integral equation ..... . ................ Geometry for radiation field calculation ..... Comparison of slot field distributions from quasi-one-dimensional theory with Storer's current distributions for a thin-wire complementary loop (unloaded case) ........ Comparison of slot impedances from quasi- one-dimensional theory to Store r's input impedances for a complementary loop antenna (unloaded case) ................... Typical slot voltage distributions for a loaded annular slot with flobzo. 5 (¢0=1r, 52:10. '1) . . . . Typical slot voltage distributions for a loaded annular slot with pobzo. 25 (¢o=1r, {2:10. 7) . . . Input impedance, reflection coefficient, and standing wave ratio (for RC=5052) as functions of loading reactance for a slot with fiobzo. 5. . Input impedance, reflection coefficient, and standing wave ratio (for RC=SOQ) as functions of loading reactance for a slot with Bobzo. 25. Optimum reactance loadings to obtain anti- resonance or Rin:50 ohms as functions of pob ................................ Optimum loading reactances to obtain anti- resonance or Rin:5o ohms as a function of (b0 for cases of fiob=0. 25, fiob=0. 5 .......... Radiation patterns of loaded and unloaded annular slots of fiob=0. 25 in their three principal planes . . ....... . ................ Experimental model of rectangular slot; loadings, coaxial feeder, and slot field probe indicated ........................... Slot antenna backing cavity and instruments in experimental setup ..................... Page 48 63 63 71 72 74 75 76 79 80 81 85 85 4.6 4.7 4.8 4.9 4.10 Experimental model of annular slot; loading, coaxial feeder, and slot field probe indicated . . . . . ...................... Slot antenna backing cavity and instruments in experimental setup ..................... Anechoic chamber and block diagram of experimental setup ........................ Rectangular slot with its coaxial feeder, loadings and probe ........................ Annular slot with its coaxial feeder, loading and probe ......................... Physical structure of loaded slot antenna and its measured field distributions for various loadings .......................... Input impedance, relative amplitude of radiation field, and feeder line SWR for short slot antenna loaded by various shunt reactances .......................... Typical measured slot field distributions for a loaded annular slot with pob=0. 5 ...... Input impedance, relative amplitude of radiation field, and feeder line SWR as functions of loading reactance for an annular slot with fiob=O.5 ................................. Typical measured slot field distributions for a loaded annular slot with (Bob-=0. 22 ..... Input impedance, relative amplitude of radiation field, and feeder line SWR as functions of loading reactance for an annular slot with Bob=0.22 ................................ viii 87 87 88 89 89 92 93 96 97 98 100 PART II. 2.1a 2. lb 2. 2a The Short-Backfire Antenna: A Numerical- Physical Optics Study of its Characteristics Typical physical model of a short—backfire antenna ................................. Typical physical model of a rimless short-backfire antenna ................... Geometry of the rimless short- backfire antenna ......................... Current and voltage due to image theory Geometry for large reflector induced current calculation ...................... Geometry for radiation field calculation . . . . Amplitude and phase distributions of current Iz(z) in the dipole exciter ......... Amplitude and phase distributions of surface current Kz(x, z) excited on the small reflector ................................ Amplitude and phase distributions of surface current Kz(x, z) excited on the large reflector .......................... Input impedance to a backfire antenna with a dipole exciter for various exciter electrical half-lengths h/AO .............. Frequency dependence of the input impedance to a short-backfire antenna with a dipole exciter ........................... E-plane (y-z plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach ....... FI-plane (x-y plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach ....... E-plane (y-z) plane radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach for various small reflector radii b ............ ix 138 138 139 139 160 160 173 174 175 177 178 180 181 182 4. 4. 10 11 H-plane (x-y plane) radiation patterns of a short-backfire antenna obtained by the numerical-phys ical optics approach for various small reflector radii b ........... 183 E-plane (y-z plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach for various excitation frequencies ............ 185 H-plane (x-y plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach for various excitation frequencies ............ 186 LIST OF TA BLES Page Part 1. Investigation of Impedance Loaded Slot Antennas 2. 1 Efficiency study for the rectangular slot with h=0.1>\0, d:0. 7h ......................... 42 3. 1 Efficiency study for an annular slot with electrical circumference gob-:0. 25, ¢O=180 degrees .............................. 83 PART I INVESTIGATION OF IMPEDANCE LOADED SLOT ANTENNAS CHAPTER 1 INTRODUCTION A theoretical and experimental investigation on the circuit and radiation properties of impedance loaded, electrically small slot antennas is performed. The slot radiator consists of a narrow slot cut in a thin conducting ground plane of large lateral extent and excited by a potential maintained between its edges. Such slot antennas are popular for aerospace applications due to their inherent adaptibility to flush mounting. The loaded slot antennas investigated here utilize a double impedance loading, shunt connected between the edges of the slot at appropriate locations to modify their circuit and radiation characteristics. Electrically small slot radiators are difficult to excite efficiently due to their inherently small input resistances and relatively large input reactances. Their radiation patterns are characterized by relatively low directivities. It is demonstrated by both theory and experiment that an appropriate optimum double reactance loading can be utilized to implement a modified electric field distribution in the aperture of the slot antenna. By appropriately modifying the slot field distribution, the slot antenna can be forced into an antiresonant condition or its radiation field patterns can be drastically changed. In the former case, a large increase in the antenna's input resistance, and consequently its radiated power, can be achieved, while in the later case an improvement in its directivity can be implemented. Narrow, loaded slots of both rectangular and annular shape are considered. A short, narrow rectangular slot antenna has poor directivity and, due to its small input resistance and large inductive input 1, 2, 3 reactance its radiated power is relatively small. A narrow, shunt loaded slot radiator is the complementary antenna to a thin, series loaded cylindrical dipole. It is well known that a conventional, electrically short linear antenna is characterized by small radiated l power (input resistance) and low directivity.4 By a double impedance loading technique, Lin5 demonstrated that significant improvements in its radiated power or directivity could be achieved. Since the series loading of the dipole can be adjusted to control its current distribution,5 then according to Babinet's principle for complementary slot and wire antennas, the electric field distribution in the complementary slot can be similarly controlled by adjustment of its shunt loading. A quasi-one-dimensional Hallen-type integral equation for the electric field (or voltage) distribution in the loaded rectangular slot is developed. A numerical solution of this integral equation is obtained by expanding the slot field distribution in series of either trigonometric7 or pulse8 functions and subsequently point matching the integral equation to convert it to a system of linear algebraic equations for the series coefficients. The system of algebraic equations is solved by matrix inversion, and the series are summed numerically on a high speed digital computer. The input impedance and radiation field pattern of the loaded slot are subsequently calculated in terms of its voltage distribution. This study demonstrates, both theoretically and experimentally that if the electric field distribution in a short slot antenna is appro- priately modified by a double reactance loading, its radiated power or its directivity can be significantly increased. Since the optimum loading reactances are found to be capacitive, it should be possible to realize practical, low loss implementations of these loadings. A simple, practical flush-mounted antenna consists of a narrow annular slot cut in a conducting ground plane and excited by a potential difference maintained between its edges at a point along its circum- ference. An electrically small annular slot is difficult to excite efficiently due to its very small input resistance and large capacitive input reactance.9’ 10’ 11 It is demonstrated that the small annular slot can be forced into a near antiresonant condition by shunt connecting an appropriately located, double reactance loading between its edges. A large input resistance is subsequently obtained. The shunt loaded annular slot radiator is the complementary antenna to a series impedance loaded, circular wire loop antenna. It has been demonstrated in the later case13’ 14 that the series loading can be adjusted to implement an appropriate modification of the current distribution excited on the loop antenna. Babinet's principle6 for the complementary circular loop and annular slot radiators again suggests that the electric field distribution in the aperture of the slot can be similarly modified by adjustment of its shunt impedance loading. A quasi-one-dimensional integral equation for the electric field (or voltage) distribution excited in the loaded annular slot by a 6-function current source is formulated and solved numerically by a Fourier series method.12 In terms of this slot field, the input impedance and the radiation field of the antenna are calculated. It is found that the impedance loading can be quite effective for the purpose of implementing a modified slot voltage distribution. Appropriately chosen loading reactances lead to either the increased input resistance (enhanced radiated power) alluded to above or a drastic change in the radiation pattern of the slot antenna (although its directivity is not increased). The optimum load reactances are found to be capacitive and the slot input impedance to be inductive. Low—loss, capacitive loading and tuning impedances can therefore be utilized to implement a high efficiency, electrically small antenna. The slot field distribution, input impedance and radiation field are qualitatively confirmed by experimental measurements. Chapters 2 and 3 present the analytical formulations, solutions, and numerical results for the investigations of shunt loaded rectangular and annular slot antennas, respectively. The results of an experimental study on these antennas are presented in Chapter 4. A brief summary as well as a statement of conclusions is included in Chapter 5. CHAPTER 2 THE LOADED RECTANGULAR SLOT: AN ELECTRICALLY SHORT ANTENNA WITH ENCHANCED RADIATION OR IMPROVED DIRECTIVITY Z. 1. Introduction In this chapter the loaded rectangular slot antenna is considered. Specifically, the feasibility of enhancing the radiated power or improving the directivity of an electrically short rectangular slot is studied. A short antenna has poor directivity and, due to its low input resistance, its radiated power and efficiency are relatively small. A narrow, shunt-loaded slot radiator is the complementary antenna to a thin, series loaded strip dipole (or cylindrical dipole of equivalent radius).6 It has been demonstrated by Harrison15 and by Lin5 that the current distribution in a cylindrical dipole can be controlled by an appro- priately adjusted double impedance loading. According to Booker's extension of Babinet's principle for complementary antennas,6 the electric field distribution in the complementary slot can be controlled by adjustment of its double shunt loading. The general problem of determining the electric field (voltage) distribution in the loaded slot, its input impedance, its radiation field pattern and its total radiated power is considered in detail. It is assumed that the slot is driven by a 6-function current source and doubly loaded by lumped impedances. An integral equation for the electric field distribution excited in the loaded slot is formulated and is solved numerically by applying the method of moments. This method consists basically of expanding the unknown slot voltage distribution in a series of either trigonometric or pulse functions and point matching the integral equation to determine the coefficients in the expansion. The details of the formulation, corresponding to the achieve- ment of enhanced radiation or high directivity, are presented in sections 2. 3 through 2. 6, while the numerical results are collected 4 in sections 2. 7 and 2. 8. It is demonstrated that the nearly uniform slot field distribution implemented by an appropriate optimum capacitive loading leads to significantly enhanced radiated power. A second optimum capacitive loading, which produces an appropriate phase reversal in the slot field distribution, is shown to lead to a radiation pattern of high directivity. 2. 2. Physical Structure of the Loaded Rectangular Slot The basic structure of the doubly loaded slot antenna config- uration to be studied is indicated in Figure 2. 1. A rectangular coordinate system, with its origin at the center of the antenna aperture is chosen to describe the free-space region. A rectangular slot occupying the region (Z) S h , (y) S a is cut in an infinitesimally thin, perfectly conducting ground plane of infinite extent. Excitation is provided at the center of the slot by a 6-function current source 10 of angular frequency w , and a double shunt impedance loading ZL is connected between the slot edges at z = id. 5 is the surface of the entire y-z plane at x = 0, SS is the slot aperture surface, and S - SS is the surface comprised of the infinite, perfectly conducting screen excluding the slot. Two induced currents Id are excited in the loading impedances at z : id in directions opposite to that of the exciting current as indicated. The dimensions of interest for an electrically short slot are 11/).0 = pOh/zns 0.2 with h>> a and a/xO IBOa/Z'rr < < 1, where X0 is the free-space wavelength, and Bo : 21r/XO is the corresponding wave numbe r. 2. 3. Integral Equations for the Electric Field Distribution in the Slot 16,17,18, 19' 20 that a solution to Maxwell's It is well known equations for the EM field EEG"), BR?) at any point in (otherwise unbounded) free space can be expressed in terms of its values on a closed surface S which encloses all the electric sources which maintain the field as $2 = 2£n(4h/a) thin, conducting ground plane / / .7 :6 / / ZL /E:L % / IE: // __ z:6 Z‘: -d+6 ZL Id zz-d-é // ZI-h Figure 2. 1. Physical structure of loaded rectangular slot antenna. O +[QxE('f')] xVGO(‘£~’,'r")}ds' (2 1) w» A -> —>—> A —-> —>—> H(r) = -§ {[n xH(r')] x VGO(r,r') -j(,)eo[an(r')] G0(r, r') s + [41- “fi(}")] VGO(I~’,?')} dS' (2. 2) where GOG": .15) is the Green's function for unbounded free space 1's l'r'-‘r"l ') = e 0 (2.3) 4n)¥;?}' CG? 0 ! and Q is the outward directed unit normal vector to S. Jacksonl has demonstrated that when the closed surface S assumes the form of an infinitesimally thin (plane) pancake shaped surface of infinite lateral extent which separates two half-space regions, then the EM field in either half-space can be expressed in terms of the fields which are maintained on its plane boundary as _p 1‘ ET!) = -S [th(?‘)] x VG( 3-") dS' (2.4) S —>—> A —> —.-> A —>-> -—>—> H(r) : -S {-jw(0[n xE(r')] G(r, r') + [n- H(r')]VG(r, r')}dS' S (2.5) A where n is the unit normal vector to the half-space boundary S which is directed into the field region of interest, and G(?,?') is the free- space Green's function for a half space _,_, e-J'Bolr-r'l G(r.r') = __,_, ~ (2-6) 271’] r-r'l The latter expressions can also be developed through the introduction of equivalent electric and magnetic surface current sources to represent the effects of the electric and magnetic fields maintained on the closed surface S. 19’ 20’ 21 When S degenerates to a plane surface of infinite lateral extent, then image methods can be applied to show that the EM field at any point in space can be expressed in terms of the equivalent magnetic currents (and charges) alone. This procedure again leads to equations (2. 4) through (2- 6) abUVC- Due to the assumed idealized physical structure of the loaded rectangular slot radiator as indicated in Figure 2. 1, equations (2. 4) through (2. 6) are an appropriate set of basic equations for the calculation of the EM field maintained at any point in space by the aperture field of the slot. These basic equations are subsequently applied to formulate an integral equation for the electric field distribution in the loaded slot. A numerical solution to this integral equation leads to an accurate knowledge of the slot field. In terms of this field distribution, the input impedance to the antenna is obtained and the radiation field is calculated based on equations (2. 4) and (2. 6). The integral equation for the electric field distribution in the rectangular slot is based upon the boundary condition for the tangential components of magnetic field at its aperture. Let HUN-r') be the (2)6?) be that in the region x < 0, then the boundary condition for the tangential magnetic field at any point in the region x > 0 and H magnetic field at the slot aperture requires that A n x [-—l-'I(l)(0Jr ,y, z) - fi‘2’0 is readily related to the electric field E5") and the magnetic field H(F‘) in the slot aperture by equation (2. 5) above, and is given by 'fi‘l’tr’) = -S {-jweo[?1x‘1:’(‘r")]G(?,?')+ [3' fi]VG 0, and G(?,?') is the half-Space Green's function _' __ -j(30R(I-',?') G(r, r') : G(x,y, z,x',y',z') : e 2nR(?, :71) where R = (1.2331) and R(_1r,—1"') : [RI 2 "112?: I : , 2 , 2 2 1/2 , , [(x-x ) + (y-y ) + (z-z') ] . The boundary conditions at the surface S in the x = 0 plane are in the aperture surface SS of the slot 13(9) :1 o /\ -—> -> n x E(r') : 0 A in the surface S - SS outside the slot. n ° H(r') = 0 The integration over surface S in equation (2. 7) therefore reduces to an integration over the aperture surface S8 of the slot as .fi(?')]vc(?,}")}ds'. 131mm = ‘) {-jw€O[/rl x 1565)] GE?) + [ ‘Ss (2.11) 10 To satisfy boundary conditions (2. 8), only the tangential components 139%?) of the magnetic field are required. These are obtained from equation (2. 11) as —>(1)—> _. . A E." —>—>' A —>—>' —>-r>t ' Ht (r) _ - {-Jw€o[nx (r )] G(r,r ) + [n° H(r )] VtG(r,r )}dS S S (2.12) where Vt : In the slot aperture surface SS, n = x and dS' 2 dy' dz' such that A A y 8/ 8y + z a/Bz is the transverse gradient operator. A Efr") = ESG") = x Bis/u z') +9Ej-+ ' up» n- H(r')=/)\c- J V'xE(r') (2.14) (UFO Substituting equations (2. 13) and (2. 14) into equation (2. 12) yields ->(l)-> . /\ /\ s A S /\ s —>—» Ht (r): - s {-Jw60X x[xEx(y', z') + yEy(y', z') + zEZ(y',z')] G(r, r' S + (3,1 94v xfis 0 Upon the introduction of the following notation for the field at the x = 0+ plane 1 A /\ H: )(0+,y, z) = Hél)(y, z) = yH;l)(y, z') + zH(Zl)(y, z) and with further simplification, the following result is obtained ->1 , A H: )(Y.Z) = -S {-Jw€O[/2Es(y', z') - yE?(y',z')]G(y, z,y',z') Y 2 S S + J [ a E (Y! zl)__E (yr 21)] wpo W ' 8' ’ /\ a A a ' ' [Ya—y + Z 5;] G(y, Z.y ,z')} dy'dz' (2.16) where G(y, z,y', z') : G(0,y,z, 0,y'z'). 11 Equation (2. l6) expresses the tangential magnetic field + maintained in the slot aperture at x 2: 0 in terms of its electric field distribution. The y and z components of the magnetic field vector in the aperture are easily found to be Hm, Y E (y', z') - 73—3—7 Es(y', z')] v.2) = y -% G(y, z,y', z') + piEjy'. Z')G(y. z,y', 2'1} dy'dz' (2.17) 1 ° 8 8 H(Z )(Y!Z) : Biz; SS {‘[W E:(Y',Zl) -851 E:(Y',Z')] S 8 2 s 5; G(y, Z,y', Z!) + fioEyhfl, Z')G(y, z,yl, zl) }dy'dz' (2.18) where L0 = «((10760 is the intrinsic impedance of free-space. The development of equations (2. 17) and (2. 18) above has assumed that the slot radiated into the half-space x > 0. For the (2 ) magnetic field H (r) at any point F in the half-space x < 0, the derivation will be exactly the same, except that in the later case A A ->(2) ”(2) n = - x. Hence the scalar components of Ht (0-,y, z) : Ht (y. Z) in the slot aperture at x : 0- are obtained as H(Z:)(Y!z) Tg‘SS {[8—- HE1Y,7')-—az—Ey( (Y'1Z'11 8 2 s E G(y, 2.3", 2') + fioEz(yl’ Z')G(y, z,yl’ z')} dy'dz' (2.19) H(2 ) 8 s (v.2): 41‘s:- [-— E (y'.z 2') ‘30 go ‘ 8y. Z ] a l l 2 S | ' I ' ' l 5-; G(y,z,y.z)+(30Ey(y,z )G(y,z,y,z )}dy dz (2. 20) 12 If the impressed currents of equation (2. 9) and expressions (2. 17) through (2. 20) are used to satisfy the boundary conditions described by equation (2. 8), then the following pair of integral equations are obtained a S a S 1 I i 1 1 SS {‘[W EZ(Y': 2') ‘ 5-27 Ey(Y 1 2)]82 G(Y,Z.y 1 Z 1 S 2 + Bo ij'. mew, z,y', z')} .1,» dz' 3'13 1; = - 729—0 {105(2) - Id[6(z-d) + 6(z+d)]} (2.21) 8 S 1 1 a S 1 1 8 1 1 5‘8 {[W EZ(Y ,Z ) - WEY(Y ,Z )] '—a-_Y_ G(y, Z,Y ,Z) S 2 s 1 1 1 1 1 1- +(30Ez(y,z)G(y,z,y,z)} dydz _o. (2.22) Expressions (2. 21) and (2. 22) are a pair of simultaneous (coupled) integral equations for the unknown components E:(y, z) and E:(y, z) of the electric field distribution in the aperture of the slot. These equations will be simplified by a quasi-one-dimensional approximation in Section 2. 4 and solved numerically in Section 2. 5. 2. 4. Reduction to a Quasi-One-Dimensional Hallen-Type Equation In principle, the coupled integral equations (2. 21) and (2. 22) could be solved simultaneously by a numerical method to determine the unknown components E:(y, z) and E:(y, z) of the slot field distribution. However, this procedure would require excessive temporary storage and execution time when implementing the numerical solution on a digital computer. These difficulties can be avoided by ignoring the relatively small longitudinal electric field component in the slot. This approximation has been used by several investigatorsl’ 2’ 3’ 17’ 19 and has been considered to be legitimate . for a narrow slot. Assume the slot is narrow in the sense that h>> a and (303. < < 1, then approximately 13 S I I.’ Ez(y,z)- 0 8?— ES(Y1’ z') 4 0 y . . . approximations appropriate y z for the case of a narrow slot. H ( '.Z') =° 0 y Y 1 Therefore equation (2. 22) is satisfied identically and equation (2. 21) can be simplified as a S I I a 2 S I I I I I I 5.5 [dz' Ey(y ,z )52-+ [30 Ey(y ,z )]G(y,z,y ,z )dy dz 5 15040 - — T‘ {106(z) - Id[6(Z-d) + 6(z+d)]} - (2. 23) The induced currents Id through the loading impedances ZL at z = id can be expressed in terms of E:(y, z) at z : d as V(z:d) -l 5-61 s I : T— = -;—-— E (y, z:d) dy d L AL +a y 2 a s 3 Z—- gl E (y, sz) dy L ‘ 0 Y such that the integral equation becomes 8 s 8 2 15 [—327 Ey"’"‘"" a; + 13., ESW'J'H G(y.z.y'.z'>dy'dz' = - __2__ O z) - [(225111 [6(z-d) + 6(z+d)]} (2.24) L Upon integration by parts with respect to z', the first term on the left hand side of (2. 24) can be simplified as follows: 14 1 [3—27 Efr‘y"z"a—: + BiEfrW'm'nG(v.z.v'.z'1dv'dz' S _ 3 S 1 I3 2 S 1 1 1 1 1 1 _ SaS-‘h [WE-3y“, , z )Bz + BOEy(y ,z )] G(y,z,y ,z )dy dz a s 8 Z'Zh :1 [Ey> a and fioa < < 1, it is permissible to use a quasi-static field approximation for E:(y, z). The electric field in a slot of width 2a cut in a thin, conducting screen of infinite lateral extent has the approximate quasi—static forml’ 2’ 3 (obtained by a conformal mapping technique) V(Z) 1T /a2_y2' where V(z) is the voltage distribution along the slot. This expression (2.33) S E , z : Y(y ) is consistent with the definition for the voltage difference distribution between the edges of the slot. According to the usual definition for voltage difference an identity is obtained as follows: -a a V(z) = -S ES(y,z)dy .—. W795 ___dY_._ = V(z) a y T’ -a 2 2 a-Y Using this approximation for E:(y, z), equation (2. 32) becomes an integral equation for the voltage distribution along the slot of the form 5:: .1: “JLG (Y» Z»y', Z')dy'dZ' 140 = --—{C locosfi z+I Osinfiolzl - flgfl[sinfiolz-dl +sinfiolz+d(]} - (434) Noting that since E :(y, z)- — E :(y, -z) , then V(z) = V(-z) 17 such that the integral equation takes the form h a 1 1 jg .1 .1 V(z'1Kw’z’y’z) dY'dZ' = ' "'43 {CICOS 1302 O '3. 2 ,2 11' a -y + IC sin (30 (z) - 1232:2122 [sin (30(z+dl + sin (Bolz-dl]} (2.35) where K(y.Z.y'.Z') = G(y, z,y'.Z') + G(y, z,y', -z') (2.36) Equation (2. 35) is a quasi-one-dimensional, Hallen-type integral equation for the voltage distribution V(z) along the loaded rectangular slot. The singularities of the integrand on its left-hand side at y = :13 and (y, z) = (y', z') are both integrable, and can be handled by approximate analytical methods. Numerical solutions to equation (2. 35), which are implemented on a high-speed digital computer, are described in the next section. 2. 5. Numerical Solution of the Integral Equation In this section, the numerical solution of integral equation (2. 35) is discussed. The point matching method, a special case of the method of moments, 8 is applied to reduce the functional integral equation to an algebraic matrix equation. The voltage distribution V(z) is first expanded in a series of appropriate functions, after which the integral equation is subsequently point-matched to reduce it to a system of linear algebraic equations for the coefficients in the expansion. Numerical processes of integration and matrix inversion are applied to calculate the expansion coefficients; the series for V(z) is then summed numerically to reconstruct the voltage distribution. All of the numerical Operations are implemented on a high-speed digital computer (CDC 6500 system). Two alternative series expansions are investigated in connection with the solution. 2. 5. l. Trigonometric Series Solution It has been shown7 that for the Hallen-type equation (2. 35), 18 the following expansion in trigonometric functions is an efficient repre sentation for V( z) N V(z) = Z on sing—E (h-z) . (2.37) Substitution of this expansion for the approximate voltage distribution into the integral equation (2. 35) leads to the following re sult- I I Sh 5a Elan sin (h-z' )] K(y,z,y ,z) dy'dz' Z 2 o-a 1 ‘rr 8. - y jto N a sing—£(h-d) -——{Cl cosBz+I sinBIzI- n o o 0 Z 1 IL n21 [sin solz-dl + sin BOIerdII} . (2.38) Equation (2. 38) can be rewritten as {I :5: K9“ 2’ Y ’2) sin n" (h-z')dy'dz' 112011 “Al—7 25 Y 11.0 - 717— sin g—(h- -d)[sinfiO Iz- dI + sinBO Iz+dI]} L jgo jgo . . + Tcl cosBOzz-TIOSInBOIzI . (2.39) The N+l unknowns in this equation, C1 and N coefficients an, for n = l, 2, 3, . . . , N , can be determined by point matching equation (2. 39) at N+l equally spaced points along the center of the slot, say (y: 0, z=zJ.) for j: 1,2, 3,...,N+1. Def1neC1= an+1' then N+l . 1&0 . _ T I0 8111 BOIsz (2. 40) M p U L“lib E: 11 n=l where, for n=1,2,3,...,N and j:1,2,3,...,N+l .nn _, , , 51112-5 (h z)dydz — jé" sinn"(h-d)[sin(3 Iz-dI+sin(3 Iz+dI] (2 41) 42'; 2h 0 j 0 j ' ' ' For n=N+l and j = 1, 2, 3,. .. ,N+l, then Aj n has the value 140 Aj,n — T cos fiozj . (2. 42) The matrix form for equation (2. 40) is expressed as 1 '- F— D "- . ”I I111,1 A2,1 AN+1,1 Cl1 51“ Bozl A1,2 A2,2 AN+1,2 “2 . $1“8022 1&0 = - —— I 4 o A1,N+1 A2,N+l AN+1,N+1 ClN+1 51“ L3sZN+1 I. __ __ _I L __J (2. 43) This matrix equation can be inverted numerically to obtain the coefficients on once the matrix elements A. n have been calculated. The series expansion (2. 40) for V(z) is subsequently summed numerically to complete the solution for the slot voltage distribution. There is no simple analytical expression for the integrals which occur in the definition of Aj n , but they can be calculated numerically by various approximations. Define I1 and 12 as h a G(0,z.,y',z') n I (j,n) = 3 sin 7" (h-z')dy'dz' (2.44) 1 213 o -a. I 2 '2 1T a - y h a G(0,z.,y',-z') n I (j, n) = Y J sin -—"— (h-z')dy'dz' (2. 45) 2 t 2h 0 -a 2 2 wa-y' then A. 11 becomes (for nt N+1) D 20 it Aj’n = Il(j,n) + 12(3 ,n) - .470: [sing (h-d)] [sin ISOIzj-dI + sin (30):;ij . (2.46) Suppose the slot is divided into L x M rectangular subsections as shown in Figure 2. 2, where M = K(N+l) (K an odd integer) is the number of subdivisions in the z-direction and L is the number of subdivisions along the y-direction. K and L must be odd integers in order that the matching points be at the center of the integration subsections (see Figure 2. 2). VI , I = l, 2, . . . , L, and zm , m = l, 2, . . . , M , locate the center of a subdivision of area AS : AyAz = (2a/L)(h/M) defined by the intervals (Ay)I : yI - Ay/Z < y < y! + Ay/Z and (Az)m : zm - Az/Z < z < 2m + Az/Z. Integrals I1 and 12 can then be expressed as L M G(O,z.,y',z') z 1 1 J 1 ' (Ay') (Az') 2 2 I 21m:1 I m TT a — y' n“ _ 1 1 1 $111 75 (h z )dy dz (2.47) L M G(O, z.,y',-z') 129’“) 2 E E I; I J , 1' 1 1 1 =1 m=l AV )1 (921m 2 ,2 11 a - y sin g (h-z')(1z'dy' (2.48) In order to facilitate integrating the singularities which occur in the Green's function for (y'xO, z'zzj) and at the slot edges y': :ta, the subsectional integrals are approximated for three different cases. (1) Green's function singularities (case of I =0, mzj) at (y':0, z':z,) In this case the integrand, excluding the singular Green's function, can be regarded as constant and equal to its value at the Center of the appropriate subsection. The Green's function singularity Can then be integrated analytically as shown in Appendix A leading to t he approximate expre s 5 ions AzM 21 (AY)(L_1)/'2 IAYCI) (IAYIL-IVZ z:h 1 I 1.1 1 1_ —zM 777:177__ —zM_l 777377_ 777n77_ _I—7_7‘—I_I—7 OA(yo,zm)m:1,2,...,M A (yo. 2 m ) (7+1) mzl, 2, . . . , M are the matching points for trigono- metric series solution. AZ3 A22 Azl Figure 2.. 2. 777W77: 777377 777377: 777377_ 777W77_ 777i77_ 777W77_ 777W7ji 777U7|A 777377# 777377_ 777377 777377— Y_(L-1)/2 3113’s y1 V(L_ are the matching points for pulse function solution. Integration subdivisions and matching points for numerical solution of the integral equation. 22 G(O z ,y , z ) n11 S 5 sin 2h (h-z')dy'dz' (Az')rn(Ay')I 2 2 na-y' . 1 A A Z 8111 9% (h-zj)x E {2Azfn[Z—;—f+ (25%) +1] fi 2 4 1+ A A + 1 . + ZAy lnI A3,;AZ) I - jBOAyAz} (2. 49) G(O,z.,y ,-z') 5 S 3 sin 1.5% (h-z')dy'dz' I I (Az )m (Ay )1 1T Iaz _ y,2' '2 1 'n nw(h-z)xG(0z O-z)A Az (2 50) ._ j S]. ‘27; j , j, , j y . . (2) Edge singularities (case of i :1, L) at y' 2 :ha In this case the integrand, excluding the NIaZ - y'2 factor, can be regarded as constant and equal to its value at the center of the apprOpriate subsection. The square-root singularity can then be integrated analytically, leading to the approximate expressions S 5* G(O, zj,y' , 2') sin (h- z ') dy'dz' (Az 1m (Ay 1, w Ia 2 T 2 71" G(O, zj,yp, zm) ,1, r111 _, . _ slnzfi-(h /.m) x H Al. _1 1',,+Ay/2 -1 1,43172 ' ..___ - ' _.__._._._ 2. 51) {sm ( a ) s1n ( a )} ( G(0, z. ,y' , - z') n S y 3 sin 2% (h-z')dy'dz' (AZ' )rn (AY'), 2 2 11a -y' ______a )- sin"1 (y! ; AY/2>} . (2.52) 23 (3) No singularities in integrand (all cases not included above) In this case the integrand can be regarded as constant and equal to its value at the center of the appropriate subsection, leading to the approximate expressions G(O, z., y', z') I S 3 sin a}; (h-z')dy'dz' (Az')m (A3“)! ,I 2 12 ira-y . n-rr G(O’ 2j’ y! ’ zm) = sm 2h (h-zm) x AyAz (2. 53) S S G(O’zj'y"-z') ' n" (h )d d 8111 -z' y' z' (Az')m (Ay')I 11 I32 _ y,2 2E G(01 zjvyls'z ) m w Ia2_ yz f 2 . n11 _ _ sm 211 (h zm) x AyAz . (2. 54) If L and M are sufficiently large, then approximations (2. 44) l and I2 when the summations indicated in equations (2. 47) and (2. 48) are carried out. through (2. 54) are valid and lead to values for I Finally, the matrix elements A. n are calculated according to equation (2. 46). The coefficients in the series expansion for the slot voltage distribution are subsequently obtained by matrix inversion as The series (2. 37) is summed numerically to evaluate the voltage distribution V(z) in the loaded slot. A computer program was developed to carry out the numerical calculations outlined above on a CDC 6500 computer system, and a listing of the program is included in Appendix B. Typical elapsed central processor time accumulated during execution of the program is 30 seconds. A discussion of the numerical results is included in Sections (2. 7) and (2. 8). 2. 5. 2. Solution by Pulse-Function Expansion Suppose the aperture is partitioned into L x M rectangular subsections as demonstrated in Figure 2. 2. M is any integer while 24 L must be an odd integer. The area of each partition is AS = AyAz :(Za/L)(h/M)- Yl91:11210"1L9 and zmsm:1129"‘!M’ locate the centers of subsections defined by the intervals (Ay)I y, - Ay/2 < y < yI + Ay/Z and (Az)rn:z1m - Az/Z < z < zm + Az/Z. Define the set of pulse function fn(z) as 1 forz in (Az)n f (z) = (Z. 55) n 0 . . . for z not in (Az)n. and let the slot voltage distribution be represented by V(Z) = Z (Inf (Z) - (2-56) n21 Substituting expansion (2. 56) into integral equation (2. 35), and point matching the equation at the set of points (0, Zj)’ j = 1, 2, 3, . . . , M, which locate the center of subsections defined by (Ay)0 and (Az)j , the functional integral equation is reduced to the matrix equation M 1'1; . ___o . 2 £j,nan +j3OTrC1 cos Bozj — 4 IO Sin Bozj , n=l j:1,2,3,...M (2.57) where a K(O,zj,y',z') f. = dy'dz I I . 7—5 a (AZ )n 11’ a2 - y"Z j30nen(d) .. T [Sin polzj-dl + 8111 pOIzj+dII (2.58) and 0 for x< zn-Az/Zoer zn+Az/2 €n(X) = 1 for zn-Az/ZS x - sin ( )I a a . 2 27 -j[30\Izj+zn) + Y! + e fl Az I 2 2 (zj+zn) + yI a a [sin-1(w) - sin-l1} (2.63) (3) No singularities in integrand (all cases not included above) In this case, the integrand can be regarded as constant and equal to its value at the center of the appropriate subsection, leading to the appropriate expression 1 . 2 2 -Jfi0~/(zj+zn) +Y, 12(1.j.n)g AVAZ {e E nsIaZ - Z N/(z.+z )2 + y2 Y1 J n 12 . 2 2“ e-JfioJ(zj-Zn) + Y1 + , } . (2.64) 2 2 J(zj-zn) + yI If L and M are sufficiently large, then approximations (2. 62) through (2. 64) are valid and lead to values for 1,0, 11) when the summation indicated in equation (2. 59) is carried out. Finally, the matrix elements A., n are calculated in terms of the lj, obtained according to equation (2. 61). The coefficients in the series expansion (where aM=0) for the slot voltage distribution are subsequently obtained by matrix inversion as [a1=[A 1"1—'—91 sins?) ' J o o‘j The series (2. 56) is summed numerically to evaluate the voltage 28 distribution V(z) in the loaded slot. A computer program was developed to carry out the numerical calculations outlined above on a CDC 6500 computer system, and a listing of this program is included in Appendix C. Typical elapsed central processor time accumulated during execution of the program is 46 seconds. A discussion of the numerical results is included in Section 2. 7 and 2. 8. 2. 6. Radiation Field of the Loaded Short Slot Antenna Let the origin of spherical coordinates be located at the center of the slot, and let .1" be the position vector from the origin to any point in space, while .13" is the position vector locating any source point in the slot aperture (see Figure 2. 3). The electric field at any point in space is then calculated from, equation (2. 4) as EG’) = -5‘ [’i‘ixE(?')] xVG(?,?')ds' . (2.65) SS It is noted that VG(’i-','r") = 3% VR(?,?1) - (31:25) semi - (lJrijOR> C-jflzR % l l and subject to the usual approximations in the radiation zone (60R >> 1), with the additional restriction that Boa < < 1, then /\ -> -. -. - I ' -. z-ITR — 211’]: — ZTTI‘ r such that for points in the radiation zone _. .. 7.11301. _. 1 VG(r,r') s - jpo 327,?— e 1302 C059 ’1) (2.66) A A . For points in the half space x > 0, then 11 : x and at source pomts in the slot aperture 29 Figure 2. 3. Geometry for radiation field calculation. 30 S —>, :' A E(r) yEy (ZW such that the radiation field becomes ->r _* e-Jfior a h A s A jfioz'cose E (r) : jfso W5 5h z Ey(z') x r e dz'dy' —a - A A A Since z = rcos 8 - 0 sine , then the radiation field takes the final form dz' . (2. 67) . -jI3 r _’ _. /\ 3‘3 be 0 h . , Er(r) : s 02 811195 1225(2')eJBOZ C05 9 Tl'l‘ h y ejBOz'cos 0 : For a short slot with Bob < <1, 1 + jfioz' cos 0 - % (fioz')2 c0520 such that (2. 67) is approximated as —>r—> _ A. b e-j'30r E (r) — ¢JE0XZ —-r—— F(I30h,0) (2.68) where F(Boh, 8) = sin 0(1- K c0520) (2.69) and E0, K and E2 are defined as follows: E h I Es(z')dz' o -h y h _ 1 12 S 1 1 E2 - 2 Shmoz) Ey(z )dz K = EZ/EO The radiation function F(I30h, 9) depends upon the parameter K, which is in term dependent upon the distribution of the slot field (voltage) E:(z) . 2. 7. A Short Slot Antenna with Enhanced Radiation: Numerical Results In order to check the validity of the theoretical-nume rical solution and the computer program which was developed to implement it, the voltage distribution and the input impedances of the slot and its complementary dipole were calculated for an unloaded rectangular 31 slot antenna. This corresponds to the special case of ZL = - joo . The pulse function solution was utilized predominantly here (although the trigonometric series was also found to yield excellent results). The results are then compared with King's4 input impedances and current distributions for the complementary dipole antenna. Figure 2. 4 indicates the distribution of the normalized amplitude and the relative phase of the slot voltage and complementary dipole current for antennas with h/AO = 0. 125, 0. 25 and 0. 5 for a width specified by Q = 2 fn(4h/a) = 10. 0. Figure 2. 5 demonstrates the input impedance Zin of the rectangular sloZt as well as the impedance Zco of its complementary dipole (ZCO = (DO/4 Zin’ {,0 : 120 11). In each case, an excellent agreement is noted between the theoretical- numerical results for the slot radiator and King's results for the complementary dipole antenna. Numerical results obtained from the trigonometric series solution are also included in Figure 2. 5. It is found that they are almost identical to those for the pulse function solution. However, it was found that for certain loadings the trigonometric series solution is inaccurate, particularly for the case of a phase-reversed voltage distribution. Although, intensive studies of the later method have been carried out, the numerical results obtained by this method are omitted. The convergence of the pulse function solution is relatively fast for short slot antennas;a study on the convergence rate indicated that 40 matching points (or terms in the pulse-function expansion) are sufficient for all cases considered here. Figures 2.6 through 2. 11 present results based upon the pulse-function solution with 40 matching points for an antenna with Q = 7. 37 . Recall that the radiation field ER?) is expressed in equation (2. 68) as "1511?) J), E 1?— e-jfior F( h e) 7 J 0 A0 an B ’ A 2 relative amplitude of slot voltage distribution 100 degrees d1stribution phase 1n Figure 2. 4. 32 I T z , 3- / /iI///////{:/// V V + /// 7/ W/ // /, al/ In... L/ rectangular slot antenna complementary cylind rical dipole h/xoeos 9:10 __ quasi-one-dimensiona ‘ slot theory — ‘ O o.‘ complementary cylindrical dipole (King's result) .... o ‘ O l 1 1 l l 1 1 1 0.1 0.2 0.3 0.4 0.5 position z/AO along the slot LO—O-—O—o——O I s- ’ I C) .1. d)- -100 p Comparison of slot field distribution from quasi—one- dimensional theory with King's current distributions for a complementary cylindrical dipole (unloaded case). I Q t T a—A Kz T I + h / 1 . Z. +§ J— / 4 i;% in in _ / 1' // 7.- 7-1): 2a b-2a h __J cylindrical dipole strip dipole SIOt antenna (human/a) = 7 52:21n(4h/a)=7.37 800 L- _ A, ’s‘ — ‘ j -. I i — ‘1. ' 1 , x 7: ‘ K ' / \P’ ‘ N-H -— ’ \ ‘ I \\ \o\ I I ,, / (-1 \ ,X3 2 ’ ' / \ \ “I I 'o 100: ’A I ‘ / \ a) __ I / - \ 2‘ _ / \ ' A / j; : . ‘ /I/* \"' 1° /A o 9 (+) | o ’ 9-1 _ E , / \ l )5 I /(+) .. 1 / ’ 0‘ ' \o‘» ' ' O / \ I I /A "’ 1 E 10)— / $7, 'I o / ' l \ I R. for slot c1, ' in E II(+) ‘ I ____ x. for slot 8 _ ' I in g) - I, I ' ' - - — Rin of complementary to L- / I -—--— X. of complementary 8 r— , I In 8101; 8 — l . A . . I Rin for slot (tr1gonometr1e 8 I I series solution) 31 I A X. for slot (trigonometric a"; I 1 series solution) I-a 1* .I O cylindrical dipole R. I: I in : I 0 cylindrical dipole Xin 0. 5 J 1 l L l L 0.0 0.1 O. 2 0.3 0.4 0.5 slot half-length in wavelengths h/Ao Figure 2- 5- Comparison of slot impedance from quasi-one-dimensional theory to King's input impedances for a complementary cylindrical antenna (unloaded case). 34 whe re 1") 11 h s 5 E (z')dz' o -h y E _15h 'ZES'd' 2—7_h(BOZ) y(z) Z Since [Bob < < 1 for a short slot antenna, then E2 is normally very small compared with E0, provided that E0 is not forced to approach zero by reversing the phase of the voltage distribution along the slot. Thus IEr(?)I is proportional to E0, and E0 is the area under the distribution plot for the slot field (proportional to the area below the voltage distribution curve). The total power radiated by the antenna is /\ r prad _ 2g (r.<§>)ds s O n/Z 1T 2 LI 5 rZIErIZsinB d6 ds gO"Tl'/2 O (I) where is the time-average Poynting vector, (r, 0, <11) are the variables of a spherical coordinate system with origin at the center of the antenna, and S0 is the closed surface consisting of the y-z plane and a hemisphere of infinite radius in the half space x > 0. Since I E; I is proportional to E0 for a short slot antenna, then the radiated power is proportional to E: for such a short slot. The mechanism for the enhanced radiation is demonstrated by Figure 2. 6. This figure indicates the slot voltage distributions corresponding to various purely reactive loading impedances for an antenna with d = 0. 7 h and h = 0. 1 A0. These results were obtained L : - j 100, -j 116, and - j 150, the amplitude of the voltage distribution is nearly from equation (2. 57). For reactance loadings of Z constant (with uniform phase) between the antenna input terminals and the location of the loading. These distributions are to be compared with the essentially triangular voltage distribution for the unloaded slot. Evidently an enhancement of the radiated power is implied since the area under the distribution plot is significantly increased for the reactance loaded slot. 35 f=600MHz $2: 2£n(4h/a): 7. 37 ZL 2a: 0. 02AO ’///////////////////////////////////////////)V////// I) 772///////////////////////// /////:I/////1V/ // \\ \\\\ ..e d20.7h N7 h:0.1AO .._, H O 0- -j65.5(phase) +100 1:: .3 ED. E .20. —50 “o 0 3° 33 :0. 8 0 no > a) +1 '0 30' O 1: (D 0H ‘H 0) ° 3 g0. 11; is 50 E . m 0 .20. ‘5 B “o 100 0. 0 1 1 1 ‘1- ’ ‘1' ’ 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0 position z/h along the slot Figure 2. 6. Typical slot voltage distributions for a loaded rectangular slot with h:0.1>\0 and d:0.7h . 36 Figures 2. 7 and 2. 8 present the input impedance, reflection coefficient, and standing wave ratio (for an RC = 5052 transmission line exciting the slot) as functions of loading reactance for a slot antenna with h = O. 1 A0 and d = 0. 7 h. Figure 2. 8 indicates the results obtained when the input reactance to the slot is tuned to zero, while Figure 2. 7 gives results for the case when the input is not tuned. For a capacitive loading of Z = - j 116 ohms, an antiresonant slot impedance is obtained asLshown in Figures 2. 7 and 2. 8; the corresponding slot voltage distribution is indicated in Figure 2. 6. A capacitive loading of ZL = - j 120 ohms leads to an input impedance with a resistive component nearly equal to 50 ohms. If the inductive input reactance is tuned to zero by a low loss series capacitor, the short slot is essentially matched to the RC = 50 ohms feeder line. A significant enhancement of the power radiated by the short slot can consequently be achieved (relative to that of the unloaded slot). The optimum reactance loadings to obtain an antiresonant input impedance or one with Rin = 50 ohms are presented in Figure 2. 9 as a function of h/kO for the case of d = 0. 7 h. It is found that the optimum loading reactances are decreasing functions of antenna length for a fixed loading position. The optimum loading reactances to obtain slot impedances which are antiresonant or have Rin = 50 ohms are presented in Figure 2. 10 as a function of leading position. It is noted that the optimum loading reactance is an increasing function of leading position as the impedance is moved toward the extremities of the slot aperture. It is found that the input power, supplied through a 50 ohm transmission system, to a slot antenna which is doubly loaded to implement a 50 ohm input resistance (with the input reactance tuned to zero), is always greater than that supplied to an unloaded slot antenna with a tuning network at its input terminals. If the loading impedances or input tuning network are not ideal, and consequently dissipate a certain fraction of power supplied by the transmission system, the efficiency of the antenna should be considered. It is assumed that the input reactance to the optimum loaded slot is tuned to zero by a series impedance with a quality factor of Q = Qt while for the loading Q = QL' The efficiency of the slot antenna with an optimum double loading is calculated as 37 Z. ZL ‘ m ZL ///////ir////////////// //1/1//// / 10$ iEy L' //////t///////// ////7d////// //// h ——*‘ 21n(4h/a)=sz=7. 37 h:0.1)\0 d:0.7h 2. =R. +jX. 1n 1n 1n P = reflection coefficient 8 : standing wave ratio 500 F +— F (+) “\ l/j 10°C ’/--- _ / l reflection coefficient |I"| input impedance (ohms) or SWR 10L: r—n C 1: 0.5 b 4 Ar 0.0 o -40 -80 -120 -160 00 loading re actance (ohms) Figure 2. 7. Input impedance, reflection coefficient and standing wave ratio (no input tuning and Rc =5052) as functions of loading reactance for a slot with h— - 0. 1x0 and d— - 0. 7h. input impedance (ohms) or SWR ZL /// \ 38 47;; 0 Z L /////}///////////////// ;/// l — 2a 52 = 2 1 n (4h/a)=7. 37 I G) r77777fu/f/ /|/////// ////f//7 / h20.l>\ d.:0.7h o z. = R. + j x. 1n 1n 1n P: reflection coefficient S : slanting wave ratio 500 '- 10 L— : l - h l—- +3 C1 .._ .2 U 55‘ *— G.) o o c: o 1 - '3 — U ._ cu — C: __ o o. 5 l 1 o ‘* O -40 -8O -lZO -l60 co loading reactance (ohms) Figure 2. 8. Input impedance, reflection coefficient and standing wave ratio with input reactance tuned to zero (RC = 5052) as functions of loading reactance for a slot with h = 0. 1X0 and d = 0. 7h. 39 ZL ZL ////////////////,V/////////////////////////;/;////// / /////////////// //////// ////////////////f//// M ‘I‘—— d z :jX *"——h d:0.7h 922£n(4h/a):7.37 -500 " F' / / A )— /// U) a _ , / ‘8 x” v / /’ r—l // >4 100 _ / : , ’ -—- // c: +- I ’ (U _ / I 3 I 8 h / . . . . ,4 _ / for 1mproved d1rect1v1ty OD / c: / "S / for antiresonance o H E for R 2509 5 -10 —- in .§ - a _ o _. -5 '- _1 1 1 1 1 1 0.025 0.05 0.075 0.1 0.125 0.15 slot half-length in wavelengths h/kO Figure 2. 9. Optimum reactance loadings to obtain antiresonance, Rin = 50 ohms, or improved directivity as functions of slot half-length in wavelengths h/kO . (ohms) optimum loading reactance XL 40 ZL ZL //Z///////fl////////////// ///////////////////flfl / Io Q1 E y _ %/////////////////////// (///// //////7/A V/////// /. % -¢——d——>4 —<—-—h—————>¢ Z :jX L L Q=2£n(2h/a):7.37 h:O. l X o -180 (L -l60 -140 for R. 25052 1n —120 -100 '- -80 — for antiresonance -60 "' -50 J l l 1 0.5 0.6 0.7 0.8 0.9 loading position d/h Figure 2. 10. Optimum loading reactance to obtain antiresonance or Rin = 5082 as a function of loading position d for case of h = 0. 110. 41 1(2) Rin - 2 I§[RL]0 E ff = 2 P x 100% e I (R. + R ) 0 1n t where Rin = input resistance of the slot (2' 50 ohms) Rt = Xin/Qt = resistance of the input tuning impedance f—1 RL]op = [XL]op/QL = resistance of the loading impedances IO = input current at the center of the slot Id = current through loading impedances at z = id . For the unloaded slot antenna, a transmission line section with short- circuited single stub tuner is used to implement a tuning network which transforms the input impedance of the slot antenna to match the 50 ohm system. Conventional transmission line theory is applied to calculate the efficiency of this network (vp = 1. 97x108 m/sec and a = 0. 019 neper/m assumed for the line)- Table 2. 1 indicates the efficiencies of both loaded and unloaded slot antennas for various QL and Qt' 2. 8. A Short Slot Antenna with Improved Directivity: Numerical Results Recall that the radiation function Fmoh, 0 ) is expressed in equation (2. 69) as F(fioh,0) = sin 0(1- K c0520) (2. 69) where K : EZ/Eo and h E S Es(z') dz' Y O E -h h l ,2 s , , 2 ZShmOZ) Ey(z)dz Normally Fmoh, 0) 2' sine for the electrically short unloaded slot, since Eo >> E2 when Boh < < 1. However, when the phase of E: is reversed by selection of an appropriate opimum reactance loading, the integral for E0 can become very small (of the order of E2). The value L : jx The directivity of the slot radiator is the ratio of its maximum of K can consequently be adjusted by an appropriate choice of Z L' radiation intensity to its ave rage radiation intensity, i. e. , 42 QL ZL Z1n Qt effic1ency 0%) 100 94.79 100 1.18-j118 51.44+j282.3 1000 99.45 10000 99.95 100 94.09 1000 0.116-j116 50.13+j315 1000 99.38 10000 99.94 100 93.88 10000 0.0116-j116 48.45+j316.1 1000 99.35 10000 99.93 stub . tuner m 0.249+1203 020.019 17.77 neper/rn Table 2. 1. Efficiency study for the rectangular slot with = 0.1x ,(12 0.7h4 o 43 (dp/d9)max D(directivity) = 1 (2- 70) 71-17 rad where Prad is the total radiated power, and the power radiated per unit 3 olid angle is (113 .2A —- r _ . < > Z . . . . . m r r 8 rad1at1on 1nten81ty The time-average Poyntings vector is -> -> —> A =é Re( xH*)= —%— [Erlzr 0 <1) and the total radiated power is therefore (where S0 was described in Section 2. 7) Prad = 25 (?-<—§r>)ds S O w/Z n =_1_ (‘ r2 [Erlzsin0d0d¢. (2.71) Q0 -1T/2‘“0 ¢ The directivity D can then be expressed as 2 4w[r lEglzlmax _ 'rr/Z 1T 2S 5 r2 lErlzsin Gdedq) -v/2 0 ¢ 2 l" : 2112,: 1m... SW [Eilzsine d9 0 The radiation field E; can be expressed as r . 2 [13¢] = CF(soh,e) = c $1n0(1- K cos 9) where C is defined by rib-jar blEl C=jonE : o 44 Carrying out the integrations leads to the following expression for directivity 105[ sinZ 0(l—K cosZO)2] max 70 - 28K + 6KZ This result relates D to K = EZ/EO; K can be controlled by adjustment of the reactive loading Z = j XL (for a given X K can be calculated in terms of the slot voltagLe distribution). L Normalized plots of the radiation field patterns, as well as a tabulation of the corresponding values for K, the beamwidth, the directivity, and the direction of maximum radiation for the antenna L 2 j X L = j 65. 5 ohms (K =1. 725) the directivity is significantly increased relative to that for the unloaded are indicated in Figure 2. 11 for various reactance loadings Z L' It is clear that for a loading of Z slot (K = 0. 036). The voltage distribution corresponding to the ZL = - j 65. 5 ohms optimum loading (relative amplitude and phase) is presented in Figure 2. 6. Figure 2. 7 shows that the input resistance to a loaded slot with high directivity (ZL = - j 65. 5) is very small (as is typical of such super-directive antennas). Physically, the later result is readily explained; the radiated power is essentially proportional to E3), and E0 is very small due to the phase reversal of E3. The efficiency of a highly directive loaded slot is thus relatively small and its Q is very high. The optimum reactance loading to achieve high directivity is indicated in Figure 2. 9 as a function of h/XO for the case of d = 0. 7 h. 45 om: doughnuts Emu “2m 32m- 0» popmoa Eoumzmonmmm mgoucm ”631:9? mo mnuofimm 36G nofiflpmm pomp 933 m unmanEfi moonmop 5 m oncm umfioa ova on: om: 2L 0mg QNH o: ooa om cm 05 op om 0v om .HH.N wusmwm m.mofi-u 8H1“ mew-” N N 1 _ ~ _ _ _ fl p131; uoyi'etpex }0 epninduxe oezuewzou _ q d _ A _ can coo em.“ omo.o coda- ovv coo v~.v ~n.~ m.moh- ovm omm Ho.v pa.» mom- can ooa Nm.a omo.o a ”we fiBBESm o o M AN CHAPTER 3 THE LOADED ANNULAR SLOT: AN EFFICIENT ELECTRICALLY SMALL ANTENNA 3. 1. Introductory Remarks An electrically small, annular slot antenna is difficult to excite efficiently due to its very small input resistance and large capacitive 9,10 input reactance. Extending the concepts developed in Chapter 2, it is demonstrated here that by connecting a loading impedance between the narrow sides of the annular slot radiator, it can be forced into an antiresonant condition. A large input resistance and an inductive input reactance which can be tuned by a low-loss series capacitor are subsequently obtained. The general problem of determining the electric field (voltage) distribution in the loaded annular slot, its input impedance, its radiation field pattern and its total radiated power is considered in detail. An integral equation for the electric field distribution excited in the loaded slot by a 6-function current source is formulated and solved numerically by a Fourier series method.12 The details of the formulation, corresponding to the determination of the field distribution in the loaded annular slot as well as its input impedance and radiation field, are presented in sections 3. 3 through 3. 6, while the numerical results are collected in section 3. 7. It is observed that the maximum of the slot voltage distribution V(ct) occurs at 4) = 17 for the unloaded slot, while this maximum can be shifted to ct) = 0 through the choice of an optimum reactance loading. This condition results in a significant increase of the input resistance to the small, loaded slot radiator as it approaches an antiresonant condition. If the loading reactance is adjusted to reverse the phase of the slot field distribution, the radiation pattern of the annular slot antenna can be greatly modified. 46 47 3. 2. Physical Structure of the Loaded Annular Slot The basic structure of the loaded annular slot antenna to be studied is shown in Figure 3. 1. The electrically small, narrow annular slot of mean radius b and width 2a is assumed to be cut in an infinitesimally thin, perfectly conducting ground plane of infinite lateral extent. A cylindrical coordinate system, with origin at the center of the radiator in the z = 0 plane, is chosen to describe the free-space regions on either side of the ground plane. Excitation is provided by a 5-function current source I0 at (I) = 0, and a double, lumped, shunt impedance loading ZL is connected at c); = :1:ch between the slot edges. S is the surface of the entire x-y plane at z = 0, SS is the slot aperture surface, and S - SS is the surface comprised of the infinite, perfectly conducting ground screen excluding the slot. Two induced currents IL are excited in the loading impedances at <1) 2 :hcbo in a direction opposite to that of the exciting current as indicated. 3. 3. Integral Equations for the Electric Field Distribution in the Slot As indicated in section 2. 3 of Chapter 2, the basic equations for the calculation d the EM field maintained at any point .1" in space by the aperture fields of the slot are equations (2. 4) through (2. 6)18 m?) = -( [91 x‘E'('i-")] x VG(’r’,?')ds' (3.1) is ->-> /\ —> —> -> mm=- {-um[anuMGum' S O +[Qo'i’1('r")]vc(?, 177)} dS' (3.2) -Jsol?-'r"l m??):e g, (in 217' r-r'l Where 3 is the unit normal vector to the half-space boundary S , which is directed into the field region of interest, and G(?,?') is the free- space Green's function for the half-space. The integral equation for the electric field distribution excited in the annular slot is based upon the boundary condition for the tangential component of magnetic field at its aperture. Let HUN-r.) be the magnetic field at any point in the region z > 0, and HRH?) be that in 48 thin, conducting ground plane \ $2 : an(4:b ) Figure 3. 1. Physical structure of loaded annular slot antenna. 49 the region z < 0, then the boundary condition for the tangential magnetic field at the slot aperture requires that 1) QXI‘fi‘ (r.¢.o+)-fi‘2’) flew») (3.4) A where n = z is the unit normal vector pointing from region (2) into region (1), and Ke(r,¢) is the impressed electric surface current maintained through the exciting element and the loading impedances. The scalar component equations from expressions (3. 4) are - prl)(r,¢, 0+) + Hsz)(r,¢, 0’) = K:(r,¢) (3.5) (1) + (Z) - _ e Hr (r,¢,0 )- Hr (r,¢,0 )_ K¢(r,¢) It is assumed that K8 has only a r-component which is approximately independent of the transverse coordinate r in the aperture of the narrow, loaded slot such that -IO/ZbA¢ for [(1] < Ac); e . e . lim Kr(r,¢) - K1443) ‘ ZA¢-'0 IL/ZbAq; for l¢—¢O( 0 is readily related to the electric field E(?‘) and the magnetic field Han) in the slot aperture by equation (3. 2) above, and is given by fi‘l’m . -5 {-jwofixmancwfi) S + [91- '1—‘1(?*)] VG(?,?')} dS' (3.7) A A where n = z is the unit normal vector pointing out of the slot into the half-space z > 0, and G(?, 1°") is the half— space Green's function 50 w —> -jfiOR(-;’ .1?!) r') = G(r.¢.Z.r'.¢'.Z') = e 211m}? 3") RG30 = [r‘2 + r"2 - er' cos (¢-¢') + (z-z')2] 1/2 The boundary conditions at the surface S in the z = 0 plane are A —> -> n x E(r') :t 0 A at the aperture surface S of the slot " s n H(r '):t 0 A -> -+ n x E(r') = 0 A _‘ _. at the conducting surface S- 55 outside n ° H(r') = 0 the slot. The integration over surface S in equation (3. 7) therefore reduces to an integration over the aperture surface 85 of the slot as H(l)(r) : - {-Jwgo[an(?')]G(r,r' S8 A —>—> —>—> + [n' H(r')]VG(r,r')}dS' . (3.8) To satisfy boundary conditions (3. 5), only the tangential “(1) ( components Ht r) of the magnetic field are required. These are obtained from equation (3. 8) as o s A —> +[n-H(r')]VtG(r,r')}dS' (39) where _/\a Ala Vt”??? ¢Fa ac). is the transverse gradient operator. In the slot aperture surface S /\ A n = z, dS' = r' dr'dct' and I 1555) = 1535*) = Q) E:(r',¢') + $1125 I I AI S I I ¢(r ,¢)+ z Ez(r ,¢) (3.10) 51 suchA that /r\1xE(r) ¢'Ers(r,¢') - r' E:(r, 41'.) It is noted that 8': ¢cos(¢- ¢')+rsin(¢- (b') and [r'-[rcos (<19 ¢')-¢ sin(¢- 43') and therefore ’1‘. x E(?‘) = $[cos(¢-¢') E:(r',¢') + sin («p-0m: (r'. cw} + /1\'[sin(¢-¢')E:(r',¢') - cos (¢-¢')E:(r',¢')] . (3.11) From the Maxwell equation V x E 2 - jg) (LO-H A.»-—> _/\ J ' ' n H(r') — z w—TO V x E(r) -;#—O{-r—r§'r—r[r E¢(1":¢)]—r—$r Er(r,¢>)} (3.12) Substituting equations (3. 11) and (3. 12) into equation (3. 9) yields 'fiéllfi) = -55 {-11.60% (cos(¢-¢'>E:(r"¢')+ sin(¢-¢')E:(r'.¢')) S A - 1 S 1 1 I S I l ”*1 + r (s1n(¢-¢ my .42 )- cos<<1>-<)> )E¢(r ,1. ))] G(r.r) 1 l 8 011 O[Fa—Sr(r'E:(r',¢'))-FW E:(r'.¢')] A A a I —>—> [r' 0?? +1; %]G(r,r')} dS' A .55 {'jWG'Olrd) (COS(¢-¢')E:(r',¢')+ Sin(¢-¢')E:(r',¢')) s A ~ I S 1 l S 1 + r(S1n(¢-¢)}:r(r.¢)- cos(¢>- ¢'I")E¢( ¢'))]G( . ) <1> + /1\' (c::(¢ -¢') 5E1,- + 3:11- sin(¢-¢')§¢—,) ] G(?,?')} r'dr'dct' . (3.13) Let G(?,?') = G(r,¢, 0, r',¢', 0) = G(r,¢, r',¢') in equation (3. 13), then this equation expresses the tangential magnetic field maintained in 52 + . . . . . the slot aperture at z = 0 in terms of its electric field d1str1bution. The r and 4) components of the magnetic field vector in the aperture are easily found to be H(rl)(r,¢) = -55 {-jweo[sin(¢-¢')E:(r'.¢') S - cos(¢-¢')E:(r'»¢')] GU14): r',4)') j 1 8 r' , , l 8 Es , -JH—[F§;Tar( Effl(r¢))-FTE rI’( '4)” [cos(q)- W') + sin(c))- ct') )% 3:7 ] G(r,¢,r',¢')}r'dr'd¢' (3.14) H(l)(1‘ <13) ‘ -§ {-' [C08(<1>- ')ES(1" (V) d) ’ - S JOJEO ¢ r , S + sin(¢-¢')E: (1". ¢')]G(1‘: <1)» 1", CV) j l 3 S g y 1 8 rl I -w—)L;[?r—0-I_‘T(r'E¢(r’¢))- ETWEf.” 93)] [cos(<1>-<)>');lr 5’3. - sin(¢- c) $171600. r'.<)>')}r'dr'd¢'. (3.15) The development of equations (3. l4) and (3. 15) above has assumed that the slot radiated into the half space 2 > 0 . For the magnetic field Haw-1") at any point if in the half space 2 < O, the derivation will be A A exactly the same, except that in the later case n = - z. Hence the H(Z) scalar components of Héz)(r,¢, 0-)- — (r,¢) in the slot aperture at z = O- are obtained as H(Z)) 5591-wa o-[cosw 4,431?!” 4” H¢ + sin(4)-4)')E;(1".<1>')]G(r .¢.r'.¢') j l 8 a 1 - (1)7;[173r-rh 153:)(1‘14) ))- QMWESU ¢' )1 [com-cw) fi— 5%: - sin(¢-¢' 5?;— Jc-% a)» 133'” 1"” l [C05 99'4”) Fr % - Sin(4)-4)') B—ir]G(r’ (1), r', (131)} r'dr'd<1)' 1 27, {106W - IL[6<¢-¢O> + 6(¢+¢O)]} (3.18) $53 {-Jweo [8111019 -r¢')E 9".(I' ¢')-COS(¢ 49' ):(:(r 1" ¢' )] GI‘(,¢,I":¢') J l 8 r1 1 l 1 I -T['r_"5?r(r Eih‘ 4)))-.1782—7 ES(I' 43'” [cos(4)- -4)' )5?— + sin(4)-4)' —l-, —%]G (r, 4), r', 4)')} r'dr'd<1)' : 0. r (3.19) Expressions (3. l8) and (3. 19) are a pair of simultaneous (coupled) integral equations for the unknown components E:(r, 4)) and E:(r, 4)) of the electric field distribution in the aperture of the slot. These equations will be simplified by a quasi-one-dimensional approximation in section 3. 4 and solved numerically in section 3. 5. 54 3. 4. Reduction to a Quasi-One-Dimensional Integral Equation In principle, the coupled integral equations (3. 18) and (3. 19) could be solved simultaneously by a numerical method to determine the :(r, 4)) of the slot field distribution. However, this procedure would require excessive temporary storage unknown components E:(r,4)) and E and execution time when implementing the numerical solution on a digital computer. These difficulties can be avoided by ignoring the relatively small circumferential electric field component in the slot. This approximation has been used by several inve stigators,1’ 2’ 3’ 17’ 19 and has been considered to be legitimate for a narrow slot. Assume the slot is narrow in the sense that b >>a and 130a < < 1, then approximately E r', I __°_ 0 1 ¢( ¢) Hr(r',4)') 2' 0 1 approximations appropriate for the case of a narrow 52)—,[r'E¢(r',¢1)] =' 0 ) SIOt' 1‘ Therefore equation (3. 19) is satisfied identically and equation (3. 18) can be simplified as SS {-jweo cos<¢-¢') Ef');1— 3:7 G(r,<1),r',¢)') - sin(¢-<)>') a—E, G(r.<)>. r'.¢')] } r'dr'd¢' = 71,—, {Io 60>) - IL[6(¢-¢o) + 6<¢+¢O>H - (3. 20> The induced currents IL through the loading impedances Z at 4) = i4)o can be expressed in terms of E:(r,4)) at 4) = 4)O as L V(¢=¢O) l b'a S( I = ——z——— = - — E r, 4)=4) ) dr L L ZL b+a 1' 0 l 5b+a s = — E (r,4)=4) )dr ZL b-a r 0 such that the integral equation becomes 55 S {-jweo COS(¢-¢') E:(r',¢') G(r.¢.r'.¢') S - l 8 + i. .5 5%, ES (r ' ¢')[cos<¢-¢')-r—r 5;, G(r.¢.r',¢>'> sin(¢-¢') air, G(r <1> r' <)>')] } r'dr'd<)>' 1 V=<<1> 4’0 ) - 72-5 {105(11)- ZL [6(<)>- "01¢ >+ 61¢+<1>0)]} (3. 21) It can be verified easily that a (:0 <1 r' 011 = 160 ¢ r' 04 (3 22) 6? 9 9 9 4).:11' 8r} 9 9 9 ¢':‘W . a G(r 4. r' ¢')l = Low 4) r' 41') (3 23) 8—4)T ' ’ ' 4)'=1r 84)t ’ ’ ' 4)'=-1T ' and by symmetry E:(r', 4)'=1r) : E:(r',4)'=-1T)- Using the above relations and upon integration by parts with respect to 4)‘ , the second term on the left-hand side of equation (3. 21) can be simplified as follows-. . 1 SS 6%:— F %rE :(r"¢ )[COS(¢' 9") —1r 5%;- G(r,¢, rl,¢l) O S - 3111(4)’ 99') )Ea—r G(r, 4) r' (15' )] r'dr'dq)‘ b+a, j 2 fix??? r' <3 ) b-a Ito—7 .1. [cos(4)-4)') 8%G(r»¢,r':¢') rI Sin(4)- 4)‘ )— B—r' G(r,<1),r',<1)')] r'd4)'dr' b+a ' 1 a = b-a m)? r'dr'{E:(r'.¢')[cos(<1>-¢')F5$T G(r.¢.r'.¢'> 4": 11 s 8 sin (<1)-49%?r G(r,4>.r'.4>')]| -5 Er(r',¢') 5713-.- 4)'=-1r -1T [com-0);)- 5%.- G(r. ¢. r'.¢') - sin(¢-¢') a—fr 00. <1). r'. <1> ')]d<)>'} 56 b+a. 11’ L l S ' ' a : -5‘b-a S17 ”“0 F Er” ’4) )W [COS(¢-¢') Flt-3851’ G(r7¢9 r'9¢') - sin(¢-¢') 5% G(r.<:>, r'.¢')] r'dcb'dr' Equation (3. 21) can subsequently be rewritten as “73:55 {Bi cos<¢-<:>') E:(r',¢') G(r. 4» r', '> S + _r Eru' ¢') 53,—. [cos(<:>-<>') ;1r 5% G(r,¢,r',¢') - sm(¢-¢') 5% G(r,d>, r',¢')] } r'dr'd¢' 1 V(¢=¢O) = 73 {10 M) - ‘7?"— [6<¢-<1>0) + 6<¢+¢O)]} (3.22) For a narrow slot having b >>a and Boa < <1, it is permissible to use a quasi—static field approximation for E:(r, <1)) . The electric field in a slot of width 2a cut in a thin, conducting screen of infinite lateral extent has the approximate quasi-static forml’ 2’ 3 Bin-249) =' VW’ = VW) am (3. 23) 1‘-’\/a2 - (b-r')2‘ where for sim plicity l nJaz - (b-r')2, is defined to represent the square-root edge singularity, and V(cb) is f(r') : the voltage distribution along the slot. This definition is consistent with the definition for the voltage difference distribution between the edges of the slot. According to the usual definition for voltage difference an identity is obtained as follows: 57 b-a b+a w ) = - ES(r.¢)dr = W” dr dr = V(¢). ¢ b+ 1‘ 1* b- 2 Z a 3 Ja - (b-r) Using this approximation for E:(r, 4)) , equation (3. 22) becomes an integral equation for the voltage distribution along the slot of the form ‘1“ lwlma V<¢')£ w” -‘n’ b-a 0 + 331-5847 [cos(¢-¢>') ?IT 35:1- G(r,¢, 1'" (V) - sin(¢-¢') 3:7 G(r,¢, r',¢')] } r'dr'dd)‘ 1 V(¢=¢O) = “2'3 {10 5(4)) ‘ T [5(¢‘¢0) + 5(¢+¢O)l} (3. 24) Equation (3. 24) can finally be rewritten as 17 ijO V(¢=¢O) S" V(¢')K(¢-¢')d¢' = - Tb {10 5W) - —-——ZL [5(¢-¢0)+ 6(¢+¢0)] } (3. 25) where the Kernel K(¢-¢>') of the integral equation is defined as b+a 2 K(¢-¢') =S f(r') {BO COS(¢-¢') G(rflb, r'.¢') b—a + ;1r 5%[cosw-w 3% 8%.- G(r.¢.r'.¢'> - sin(¢-¢') 33;, G(r,¢,r',¢')] } r'dr' . (3.26) This is a quasi-one-dimensional integral equation for the voltage distribution V(¢) along the loaded annular slot. A numerical solution to equation (3. 24), which is implemented on a high- speed digital computer, is described in the next section. 58 3. 5. Numerical Solution of the Integral Equation A Fourier series solution to the integral equation for the voltage distribution V(¢) along the loaded annular slot is discussed in this section. A solution of integral equation (3. 25) can be obtained in the form of an exponential Fourier series.12 This solution is obtained by expanding both the kernel and the unknown voltage distribution in exponential series. A numerical integration process is applied to calculate the Fourier coefficients; the series for V(¢) is then summed numerically to reconstruct the voltage distribution. All of the numerical operations are implemented on a high-speed digital computer (CDC 65 00). Let the kernel of the integral equation and the unknown voltage distribution be expanded in exponential Fourier series as (I) K(¢-¢') = z anejn(¢‘¢') for 05(¢-¢')s 2n (3.27) nz-m _ jm¢ W4” ‘ Z Vme for 0 54) 5 Zn (3.28) m:-CD where the Fourier coefficients on and VIn are to be evaluated. The On are given by 1 " -'n(<)>-<1>') on = 2—11 5‘ K(¢-q>')e 3 do' (3.29) ‘TT and are evaluated by numerical integration of the kernel function which is defined by equation (3. 26). It is found, by substituting the series (3. 2'7) and (3. 28) into integral equation (3. 25) and exploiting the orthogonality of the exponential functions, that the Vm can then be related to the known an coefficients. Upon substituting expansions (3. 27) and (3. 28) into integral equation (3. 25), and applying the orthogonality relation 11' . , S e3(m‘n)¢ dqv = 211' a; (3.30) -'n’ where 621 is the Kroneker delta function, the left-hand side of equation 59 (3. 25) can be expressed as 11’ 00 “IT . , S V(¢')K(¢-¢')d¢' = 2 an) e3“‘¢’¢ ’ V<¢')d¢' 'TT ‘TT nz-m a) 1T . , 00 . , : 2 (1 S eJn(¢'¢) 2 V eJm¢ (143' n m nZ-(D '1T m:-@ = 2n 2 a V ejn‘l’ . (3.31) n n nz-(D If both sides of equation (3. 25) are multiplied by eflmd) (after making use of relation (3. 31)), and integrated with respect to 4) from -w to w. the integral equation reduces to 2 _ jpro 1 "in“? W“) 411' Vmam - ' "25- [IO ‘ 7]: (V((bo) ‘3 O + V(-¢O) e 0)] such that (4))-Lo 1 _.m¢ m4) vm = ———2—— {lo 7— [V(<)>O) e J C>+ V(-<(>O)eJ 0]} j8b1r a. L m (duo ZV(¢O) : ——2—__{IO-—z—cosm¢o} (3-32) j8b1r am L where the later step follows since by the symmetry V(¢O) : V(-¢O) . By inserting relationship (3. 32) into the Fourier series (3. 28), V(cb) can be expressed in terms of the am coefficients such that v(¢) = Z vm e3m¢ mz—CD °° wu 2V(<)> ) . = Z 0 I - —.Z—o cos m (b } ede) mz-oo j8b1r a o L o (3. 33) 60 V(¢ ) 18 evaluated by requiring solution (3. 33) to be satisfied at 4) 4) which leads to 2V(¢O ) ejmcbo V(¢0) = m2%:[10 cos m. (1)0 ] —a——-—- j81rob m °° jn¢0 e (3. 34) LOP-010 .——.—<: -—.—— j81r b nz-OD n where constant k is defined by k = 00 w” Jmcpo o e 1 + cos m4) _——— 0 (1 m 3417 ZLb m:_m Upon the application of (3. 34) to eliminate V(cp ) in equation (3. 33), the final solution for the voltage distribution in the loaded annular slot takes the form 00 . Jn¢o -jw)10 Io { V(¢) - 2 1+ ———2-—— cos m¢o 4n bZL an am 811' b mz-OO n:-00 (3. 35) The coefficients am occurring in solution (3. 35) must be They are obtained determined from their definition in equation (3 27) -' - I 3mm) (1) ) and integrated 1f equation (3. 27) is multiplied on both sides by e with respect to ¢' from -1r to TI’, the result is Tr . , an1 = 7% 5‘ kq¢-¢')e'3”“¢‘¢ )d¢' (3.36) "TT Let ¢-¢' = , then a/a¢' = - a/ao and R = Jrz + r'2 — 2 rr' cos(¢-¢') = N/rz + r'2 - 2 rr' cos (I: such that with definition (3. 26) for K(¢-¢') the an are given by 61 l 17 b+a 2 an : E; 5‘ Sb f(r') {[30 cos(¢-¢')G(r,¢, r',¢') -1r -a + plr 5% [cos(¢-¢')-r1-. £1" G(r,.r'.<)>') - sin<<1>—¢') 3?: G(r.<)>. r'.<)>')] } e'jn‘¢'¢" r'dr'd<)>' 1 'rr b+a 2 212—175‘ Sb f(r){|30cosG(r,r,) -17 -a 1 8 1 8 , + F 56 [COS Q ?" ‘a—Q G(r9 r 9 Q) + sm 5" G(r, r’, )] } e-Jnc15 r'dr'dCP . (3. 37) Once the coefficients an have been calculated, the Fourier series expression (3. 35) for V(¢) is subsequently summed numerically to complete the solution for the slot voltage distribution. It should be noted that r remains as a parameter in the expression for the am . In all subsequent calculations, r = b is used so the integral equation is satisfied along the mean radius of the slot. The re is no simple analytical expression for the integrals which occur in the definition (3. 37) of an , but they can be calculated numerically by various approximations. Define three new quantities as follows: 1 Sir b+a pie A n —2 S 211' --1r b-a JaZ _ (b-r')2 -'n . J e".ll3oR I I 1 coséb W— rdrdCD (3. 38) 1 ‘7 b+a (Jim 3 1 a e-BOR , —- , EEO”??? ‘21—" 1dr” Bn : 2 217 M" b-a «[az - (b-r')2 (3.39) 1r b+a -jn -j[30R C = —1 S S e 8 [sin 8' e ]dr'd n 2 ? 84> 8r 717R 211' -1r b-a L2 _ (b_r.) (3. 40) 62 then a = A + B + C n n n n Upon integration by parts with respect to (b twice, Bn can be simplified as follows: B _ 3” _:S‘:L:e QCOSQ J9— e-JBOR dr'd n rJ: 2' 8CD ZTTR a - (b- r'z) : "LS: S:+:e-¢(jncos<1>+sin<1>) EgiZOR dr'dCI> r'Z’Ja - (b- r')7 (3.41) Similarly, after integration by parts with respect to CD, Cn can be simplified as b+a . n . e-jBOR ' = —— -:5_b a 2 24 31nd) 17 ZwR dr d . f:J - (b- r' O) Q) (3. 42) Noting that the odd parts of the integrands in (3. 40) - (3. 42) will integrate to zero, then An, Bn and Cn can be simplified further as follows: b+a eO—jfi R A = T‘EZS: 5;) C03 “‘1’ com r'dr'd<1> (3. 43) n an 2 7 717R - (b- r '2) _ b+a (n cos (D cos n - sin sin n) e-JfiOR , B _ dr d n 2 ? ZTTR r'Nla - (b-r') (3.44) n " b+a sin n sin (D 8 e—jBOR C : dr'dCP n 2 Z __ b-a f2 2' ar' ZwR " T’ a - (b-r') (3. 45) Suppose the slot is subdivided into IxJ subsections shown in Figure 3. 2, where I denotes the number of subsections in the (I) direction, and J is the number in the r-direction. i , i = l, 2, . . . , I rj , j = 1, 2, . . . , J, locate the center of a subdivision of area 63 integral equation is satisfied _______ along the center-line , , ’ ~ c of the slot I ’ ‘ \ Ill-o-II |----l (Ar)l (Ar)2 (Ar)J 1' r2 Figure 3. 2. Integration subdivisions for solution of the integral equation. l / F i " / "‘0 / /' /6 / / R/: / I / 2a / ¢'., /___\ I / / X Figure 3. 3. Geometry for radiation field calculation. 64 As 5 b A<1> Ar = b(Z'rr/I)(Za/J) defined by intervals (A <1)i = <1>i - AQ/Z 5 Q 5 Qi + AQ/Z and (Ar)j : rJ. - Ar/Z S r S rj + Ar/Z. Integrals An, Bn and Cn can then be expressed as [32 I . A B02 “2 Z S ‘3‘ cos nQ cos bQfi 62.131512 r'dr'dQ n . (A <12). (Ar) Ja 2__ 2 TI i: 13:1 (3.46) I -'(3 R -n ncos nQ cosQ-sianian eJO , Bn : v 71TR dr dé —T_T_Zi:1j=l (A)i (Aflj 'J 2 - (b-I“)Z (3.47) I J - C - n 2 Z S S SinQ sin nQ 8 e-JfioR dr'dQ n — ——z 7 31" ZTTR 211' izljzl (AQ) (Ar)j N[3:12_(b_r,)2 (3.48) which lead finally to the Fourier coefficients an I An + Bn + Cn In order to facilitate integrating the singularities which occur in the Green's function for (Q = 0, r' = b) and at the slot edges (r' = b-a, r' = b+a), the subsectional integrals are approximated for three different cases. (I) Green's function singularities (case of Q = 0, r' = b) In this case the integrand, excluding the singular Green's function, can be regarded as constant and equal to its value at the center of the appropriate subsection. The Green's function singularity can then be integrated analytically as shown in Appendix A, leading to the approximate expressions 5‘ 5‘ cos nQ cos Q‘ G(b, r', Q) r'dr'dQ (AQ)i (Ar)j a2 -<1<>-r')Z cos nQi cos i rJ. AQ Z a {Jfio rj Ar AQ + [21"]. A@ m(—E— (erAcp) 1+~[I‘J.ZAQ/Ar) +1 +——J_——z._A +l>+2ArIn( rjAJQ/Ar )]} (3. 49) 65 S S n cos nQ cos Q - sin nQ sinQ G(b, r', Q) dr'dQ (A)i (Ar)j 1...];2 - (b-r')Z n rjAQ rjAQ2 fl : a—F {-Jfiorj AFAQ +21‘j AQIn[—E—+ (T) + 1] 1+ '\/(rj A<1>/Ar)‘2 +1 + ZAr fn[ rj AQ7Zr (3. 50) sin nQ sinQ 8 , , 1 J «[21 - (b-r') sin Qi sin nQi ._ I _ a [G(bi r 9 Q) ©Z¢i r':b+ Ar/Z _ t G(b,r ,<1>) QzQi ] A<1> . (3.51) r’zb-Ar/Z (2) Edge singularities (case of r' = b-a or r' = b+a) In this case the integrand, excluding the IN? - (b-r')Z1 term, can be regarded as constant and equal to its value at the center of the appropriate subsection. The square root singularity can then be integrated analytically, leading to the approximate expressions 5‘ 5‘ cos nQ cos Q ‘ G(b. r', (1))r'dr'dQ (Aq’h (Ar)j £2 - (b-I")Z :° cos nQ. cos Q. G(b, 1". <1)) . 1 1 1‘ :r. J Q=Q. 1 b-r.+Ar/2 -l b-r.-Ar/2 r. A)dr'Q (A<1>)i (Ar)j £2_ (b_r,)2 - sin nQ sin Q 8 G(b r' Q) - 1 i w ’ ’ 1"”) =i b-r.+Ar/2 b-r.-Ar/2 [sin'1( J )- sin'1( J )1 a a (3.54) (3) No singularities in integrand (all cases not included above) In this case, the integrand can be regarded as constant and equal to its value at the center of the appropriate subsection, leading to the approximate expressions S S cos nQ cos Q fl G(b, r', Q) r'dr'dQ (A<1>)i (Ar)j Jaz _ (b_r,)2 cos nQi cos Qi -_°- G(b,r', cp) r,:r rj ArAQ (3.55) ’Jaz - (b-rj)—Z' J Q=Q. 1 j r"\/a2 - (b-r')Z (n cos nQ. cos Q. - sin Q. sin nQ.) :- 1 1 1 1 G(b, rj, Qi) ArAQ fl rj J32 - (b-rj)Z 5‘ S n cos Q cos nQ - sianin nQ G(b, r', (1)) dr'dQ (A<1>)i (Ar)j (3. 56) 67 5 5‘ sin nQ sin Qfi 36:3 G(b,r',Q) dr'dQ (A<1>)i (Ar)j L23 (b_r,)2 sin nQ. sin Q. 8 : 1 1 “—‘r G(b,r',Q) , . ArAQ —‘ 8r r :r l 2 ' Z J a - (b—r) (1):qu (3.57) If I and J are sufficiently large, then approximations (3. 52) through (3. 57) are valid and lead to an evaluation of An, Bn and Cn when the summations indicated in equations (3. 46) through (3. 48) are carried out. The series (3. 35) is summed numerically to evaluate the voltage distribution V(cp) in the loaded annular slot. A computer program was developed to carry out the numerical calculations outlined above on a CDC 6500 computer system, and a listing of the program is included in Appendix C. Typical elapsed central processor time accumulated during execution of the program is 30 seconds. A discussion of the numerical results is included in section (3. 7). 3. 6. Radiation Field of the Loaded Annular Slot Antenna Let the origin of spherical coordinates be located at the center of the annular slot cut in an infinite ground plane which occupies the x-y plane, and let Y be the position vector from the origin to any point in Space while 1?: is the position vector locating any source point in the slot aperture (see Figure 3. 3). The electric field at any point in space is then calculated from equation (3. l) as E(?) = -5‘ [fixfifi'w] x VG('r’,'£~") dS' . (3.58) s S Recall from equation (2. 66) that for points in the radiation zone __,_, 'jpor ' ”if VG(r.r') = 4130 ?'7??— eJB°r r9 - (3.59) /\ /\ For points in the half space z > 0, then n = z and at source —> /\ points in the slot aperture E(r') = r Er(r', (6') such that the radiation 68 field becomes _’ ' ejflor b+a 2w . . Era“) — ——2————b‘§ 305‘ [(2 x r' x r] ]Er(r',¢>')eJB0 r'dcp'dr' From the vector relations (9 3:15)}: i}: [9‘ sine sin(<(>-c(>') + 6 c059 sin(¢-¢')+ $cos(¢-¢')] x r /\ /\ 9 cos(¢-c)>') - (p cosG sin(<(>-¢') (3. 61) and Q- 3'" = r' sine cos(¢-¢') (3.62) then the radiation field takes the final form E14?) : e-jfior b+a ba5~:rr[ /\ jBO T_ 0 cos(¢- cb') - q) cosO sin(¢>- (6') )] ejfi Or'sirflcos(¢- 43') Er(r ' <1>') r'dcp'dr' . (3.63) For a narrow annular slot is can be assumed Er(r', (6') £2 Er(¢') and r' = b, such that [9 cos(<)>-<)>') - $ cose sine-19)] I‘ -> . jBOabe-Jfior 5‘2“. /\ r _ TTI‘ O Er(¢,) ejfiob sine cos(¢-¢') d¢' . (3. 64) Radiation factors to describe the 6 and ((9 components of the radiation zone electric field are subsequently defined as Zn . . ' F9 (9 ,(b) : S COS(¢-4)') Ell-(4") eJfiOb Slne COS((1)-4)) O (143' (3.65) 217 . . , F¢(9,¢) : 5‘ cost) sin(¢-¢') Er(¢>') eJBOb smG COS(¢-¢ ) dcb' o (3.66) 69 For an unloaded annular slot with (Bob < < l, Er(¢) 2' EC) is essentially constant, so for this case F9 (6,4)) = waiob E0 Sine F (9, ) =' O 4) C) In the case of a loaded annular slot, Er(¢) can be a strongly varying function. Assuming an electrically small slot with (30b < < 1 , the exponential term can be expanded in a power series. Retaining only the leading two terms of this series leads to 211 . . ' F9(9.¢) = S Erw) COS(¢-¢') eJfiobsme ”SW"? ) dq)‘ O 211 = y Er(¢')COS(¢-¢'){1+jfiob51n9 C°S(¢'¢') O l 22 -2fiob sinZB cosZ(¢-¢') + . . . } dcb' ZTT 2 :' cos¢5 Er(¢') cosq)‘ dd)‘ + jflob sine cos ¢ 0 211 2 S Er(¢')cos (13' d4)‘ O 211 + Jfiob sinO sinzog Er(¢>') sinzo' dd)‘ (3. 67) o 211 . . , S COSO sin(<()-¢') Er(¢l) erobSII’le COS(¢-(l) )d¢' O F (9.¢) 211 :5 cost) sin(¢-¢') E ((6') {l + jfi b sine cos(¢-¢') ° 2 2 r 0 60b 2 2 - —2—— sin 9 cos (¢-¢') + . . . } do' II- 211 sinog cosd)‘ Er(¢')d¢' + jfiob sin¢cos¢ sine l, O 211 2 S Er(¢')cos 113' d¢' + jfiob sin¢cos¢ sine o 211 2 l E W) sin ¢' d¢'. (3. 68) . o r 70 If three constant coefficients are defined as E o 211 5 Er(¢') COS¢' d¢' O m u 211 2 l jfiObS Er(¢')cos cp' d<(>' O 211 2 E2 = jfiobS Er(¢') sin 43' d¢' 0 then the radiation factors (3. 67) and (3. 68) can be rewritten as Fe(9,¢) 2 E0 cosq) + sine (E cosz¢+ E sinzcb) (3. 69) l 2 (9,4)) [EO sin¢+ (E - E )sin¢cos¢ sinG] cost) . F6 1 2 (3. 70) By appropriately adjusting the impedance loading, the distribution of Er(c(>) can be modified in such a way as to adjust the relative magnitudes of E0, E1 1 and E2 for a small, unloaded slot, but if the phase of Er(<)>) is reversed, , and E2. For example, E0 is very small compared to E then EO can be made to dominate over E1 and E2. The radiation field pattern of the loaded annular slot can therefore be significantly modified. 3. 7. Numerical Results In order to check the validity of the Fourier series solution and the computer program which was developed to implement it, the voltage distributions and the input impedances of the annular slot and its complementary loop we re calculated for an unloaded slot antenna. The results are compared with Storer's12 input impedances and current distributions for the complementary loop antenna. Figure 3. 4 indicates the distributions of the normalized amplitude and the relative phase of the slot voltage and complementary loop current for antennas with (30b = 0. 1, 0. 5 and l. 0 for a width specified by Q = 2 ln(411b/a) = 10. 7 . Figure 3. 5 demonstrates the input impedance Zin of the annular slot (52 = 2 £n(411b/a) = 10. 7) as well as the input impedance Zco of its complementary loop (52 = 2 £n(211b/a) : 11, and Z = LZ/4 Z. , l; = 120 Tr) as a function of the antenna's electrical co 0 in o 2a complementary loop antenna 0 .0 CD \0 T T .9 \1 I O 0‘ 17 $2 : 10.7 .0 .1; I quasi-one -dimensional relative amplitude of slot voltage distribution 0 UJ l . slot theory 0. 3n- . o g ‘ complementary thin-wire 100p 0- 2* O (Storer's theory) 0. 1‘ 0. 0 J l l l 1 l l l l O 20 4O 6O 80 100 120 140 160 180 100 angle 11> in degrees phase distribution in degree 5 Figure 3. 4. Comparison of slot field distributions from quasi-one- dimensional theory with Storer's current distributions for a thin-wire complementary loop (unloaded case). complementary loop antenna {3 : 21119212) 3 z 11 (<1) 2a - 9+ Z. 1000 )— 1“ 9' K (ohms) I \ 1n 1 100 t TTWI’ I I Rin for slot X. for slot n ---- 1 _ -._ Rin of complementary loop _.._ X. of complementary 1n loop Storer's loop Rin O Storer's 100p Xin resistive or reactive component of input impedance Z. 1 J 1 J 0.0 0.2 0.4 0.6 0.8 1.0 electrical circumference (30b : (211b)/>\0 Figure 3. 5. Comparison of slot impedances from quasi-one-dimensional theory to Storer's input impedances for a complementary 100p antenna (unloaded case). 73 circumference fiob. In each case, an excellent agreement is noted between the theoretical results of the Fourier series solution for the slot radiator and Storer's results for the complementary loop antenna. The convergence of the Fourier series solution is relatively fast for an electrically small annular slot; a numerical study on the convergence rate indicated that retention of the leading 10 terms in the series is sufficient for all cases considered here. Figures 3. 6 through 3. 13 present results based upon the 10 term Fourier series for slot antennas with a width specified by $2 = 10. 7 . The mechanism through which the resistive component of the input impedance to the slot is increased is demonstrated by Figures 3. 6 and 3. 7. These figures indicate the slot voltage distributions corresponding to various purely reactive loading impedances located at 4:0 = 11 for antennas with (Bob = O. 25 or O. 5. The slot voltage distributions are modified such that the maximum of the slot voltage is shifted from (1) = 1800 to the excitation point at (p = 00. This shift in the maximum voltage location is implemented through the choice of an optimum reactance loading, and an input impedance with a greatly increased resistive component is subsequently obtained. As indicated in Figure 3. 7, the slot voltage distribution has a phase reversal at d) = 900 when the loading consists of an impedance ZL = - j 35 located at ¢o = 11. The radiation pattern of the loaded slot antenna will change significantly subject to this condition of phase reversal as mentioned in Section 3. 6. The radiation patterns in the three principal planes of loaded and unloaded annular slots having Bob = 0. 25 are presented in Figure 3. 12. It is observed that in the x-z plane of 4) = O0 or 1800, the radiation pattern is modified from an almost sinusoidal pattern (for Z = 00) to an essentially omidirectional one L (for Z = - j 35). In the y-z plane of ¢ = 900 or 2700, the radiation L pattern is changed from nearly sinusoidal (for ZL = 00) to essentially cosinsoidal (for Z = - j 35). Finally in the x-y plane of 6 = 900, the L radiation pattern is modified from an essentially omidirectional pattern (for Z = 3'00) to a nearly cosinsoidal one (for Z = - j35). L Figure 3. 8 presents the input impedance, reflection coefficient L and standing wave ratio (for an RC = 50 ohm transmission line exciting the slot) as functions of loading reactance for an annular slot with Bob 2 0. 5. The input reactance to the slot can be tuned to zero by resonating 74 (30b = 0.5 9 210.7 1. o ZL: co 0 9_ (unloaded 9 00 .0 \1 P 01 I litude of slot voltage distribution 0 o In. o l l relat1ve amp 0 o -- N t» l l l \ z :-j50 - 2:2 ZL _] 0 /_ZL: '35 0.0 l J l 1 1 l I L 0 20 4O 60 80 100 120 140 160 180 angle ((1 in degrees Figure 3. 6. Typical slot voltage distributions for a loaded annular slot with (3016 = 0.5 (60: 11, 12:10.7). 75 (30b 0.25 52 : 10.7 O \0 O G) 9 K) O 0‘ P m l :‘2 ZLJS e of slot voltage distribution pDhtud 53 w 4: N] l I I :3 U1 relative am 0 N l N t“ H L—I on .0 p—o l O. O l l J l l l L L l 0 20 4O 6O 80 100 120 140 160 180 angle 4: in degrees Figure 3. 7. Typical slot voltage distributions for a loaded annular slot with pobz0.25 (4)0211, @210. 7). input impedance (ohms) or SWR 1000 500 100 50 76 (3b:0.5, 52:10.7 0 Z. : R. + jX. 1n 1n 1n (Fl, 81) : reflection coefficient and SWR with no input tuning (F2, 82) : reflection coefficient and SWR (for R = 5011) with . c 1nput reactance tuned to zero X. 1n ‘ \ \ \ \ \ \ \ \ \ \ \ \ \ o reflection coefficient If) l" 10 —- _ 1 _ 5 F'— 1 1 1 4 0- 0 O -10 - 20 -30 -4O 00 loading reactance (ohms) Figure 3. 8. Input impedance, reflection coefficient, and standing wave ratio (for RC = 50 52) as functions of loading reactance for a slot with (30b : 0. 5. 77 its input impedance with a lumped, series reactance at the input terminals; the input reflection coefficient and SWR for both the tuned and untuned cases are indicated in the figure. For a capacitive loading of ZL = - j 5 ohms, an antiresonant slot impedance is obtained as shown in the figure; the corresponding slot voltage distribution is indicated in Figure 3. 6. Figure 3. 9 indicates the input impedance, reflection coefficient and SWR (for an RC = 50 ohms transmission line exciting the slot) as functions of loading reactance for an annular slot with 60b = O. 25. The input reflection coefficient and SWR for cases where the input reactance to the slot is both tuned and untuned to zero by a series reactance at the input terminals are presented in the figure. For an inductive loading of ZL = j75 ohms, an antiresonant slot impedance, as shown in Figure 3. 9, is obtained; the corresponding voltage distribution is indicated in Figure 3. 7. A purely reactive, capacitive impedance of Z = - j25 loaded L at 4’0 = 1800 in a slot with electrical circumference (30b 2 O. 5, or an inductive loading of Z = j70 located at (to : 1800 in a slot with Bob = O. 25, leads to alri input impedance with a resistive component nearly equal to 50 ohms. If the associated inductive input reactance is tuned to zero by a low-loss series capacitor, the loaded annular slot can be essentially matched to the Rc = 50 ohms feeder line. A significant enhancement of the power radiated by the slot can subsequently be achieved (relative to that radiated by the unloaded slot). The optimum reactance loadings to obtain an antiresonant input impedance or one with Rin = 50 ohms are presented in Figure 3. 10 as functions of the electrical circumference (30b of the annular slot. It is found that the optimum loadings are decreasing functions of antenna circumference for a fixed loading position. The optimum loading reactances to obtain slot impedances which are antiresonant or have Rin = 50 ohms are presented in Figure 3. 11 as a function of loading position for a fixed antenna circumference specified by [job = 0. 25 or O. 5. It is noted that the optimum loading reactance is a decreasing function of loading position as the impedance is moved toward the location (to = 180 degrees. It is found that the input power, supplied through a 50 ohm transmission system, to a slot antenna which is doubly loaded to 78 (30b : 0. 25, $2 = 10. 7 (Fl, SI) 2 reflection coefficient and SWR with no : . , . . in Rin + inn ' 1nput tun1ng 1 (F2, 52) : reflection coefficient ’ ‘ and SWR with input ,' \ reactance tuned to I ‘ zero \ 1000 — 500 '- “a“ m 100 - r ’ ’ ‘ 1. N o E .r: 50 "' S 3 2 R. R. in in 2 0 0 $3 «1 '3 I 2" _ I: 1“l E. a 10 _ L . 1. 0 33 1:: l" ‘H .._. 2 "H v 8 U 5 “' 1: .2 Z ‘8 o 8.1 0 ( .. 1‘ 1 l l I l W O. 0 0 25 50 75 100 co loading reactance (ohms) Figure 3. 9. Input impedance, reflection coefficient, and standing wave ratio (for RC = 50 Q) as functions of loading reactance for a slot with (Bob : O. 25. 6- 250 *— ZL = RL + jXL I": Q : lO 7 E ,1: 0 V 200—- ,_l X 0 0 £1 .‘3 o 8 150 *- 1. co .5 "U .23 Optimum reactance to E 100 / obtain antiresonance L :3 E '3 CL / O optimum reactance to obtain R. = 50 ohms 1n 50 b 0 1 4 1 0. 0 0.1 0. 2 0. 3 0. 4 \0. 5 electrical circumference (30b : (ZTTb)/)\O - 50 1—- Figure 3. 10. Optimum reactance loadings to obtain antiresonance or R. = 50 ohms as functions of (3 b. 1n 0 8O $2 : 10. 7 L : RL + JXL 100 r A 90 _ U) E 3 80 - .4 )— >< 7O / g for R. : 50 ohms Bob 2 0- 25 s 60 r m a) +9 f5 Q, 50 " 1. E” 40 — '6 8 for antiresonance .._. 30 h g for R = 50 ohms 20 - in f: *5. o 10 - 0 1 1 1 1 L 2} ' 100 120 140 160 \180 - 10 — loading position (1)0 (degrees [30b : 0. 5 - 20 ”' - 3O .— Figure 3. ll. Optimum loading reactances to obtain antiresonance or Ri = 50 ohms as a function of (to for cases of (30b : O. 25. (308 = 0. 5. 81 z () x—z plane of ¢=00. 180° L x 0 : 90° 2 e = 90° 6““9 l. O ZL: -j35 4» 0o 8 y-z plane of A JIErWH Z + lEr(9)l 2 o o 6 ¢ <1): 90 , 270 0-6 ZL - co 0. 4 ¢ = 270° 10. ¢ = 900 :Y O 6 = O0 9 = O 9 4y 9 4» l. 0 /_ ZL : co 0. r 8 x-y plane of [It-33(9)) )1 0. 4 0 2 . 0 (i _x 6 = 180° «1 = 0° Figure 3. 12. Radiation patterns of loaded and unloaded annular slots of (Bob = O. 25 in their three principal planes. 82 implement a 50 ohm input resistance (with the input reactance tuned to zero), is always greater than that supplied to an unloaded slot antenna, with a tuning network at its input terminals. If the loading impedances or input tuning network are not ideal, and consequently dissipate a certain fraction of power supplied by the transmission system, the efficiency of the antenna should be considered. It is assumed that the input reactance to the optimum loaded slot is tuned to zero by a series impedance with a quality factor of Q = Qt while for the loading Q = QL' The efficiency of the slot antenna with an optimum double loading is calculated as 1 . - 2 I [R ] Eeff _ o 21n L L op x100 % I0(Rin + Rt) where Rin 2 input resistance of the slot (5 50 ohms) Rt = Xin/Qt = resistance of the input tuning impedance L input current at the driving point of the slot [RLlop = [XL]op/Q = resistance of the loading impedances I 0 IL I current through loading impedance at (to = 11. For the unloaded slot antenna, a transmission line section with short- circuited single stub tuner is used to implement a tuning network which transforms the input impedance of the slot antenna to match the 50 ohm system. Conventional transmission line theory is applied to calculate the efficiency of this network (vP : 1. 97x10 m/sec and a = O. 019 neper/m assumed for the line). Table 3. 1 indicates the efficiencies of both loaded and unloaded annular slot antennas for various Q and Q with electrical circumference 8 b : 0. 25. L t o 83 QL ZL Z1n Qt efficiency (‘70) 100 22. 9 100 l. 24+j124 47. 3+j748 1000 26.18 10000 26. 55 100 52. 37 500 0. 27+j135 53.1+j1227 1000 63. 01 10000 64. 32 100 61. 00 1000 0.135+j135 43. 4+j1228 1000 76.11 10000 78. O4 stub . tuner 00 0. 279-j77 (1:0. 019 11. O9 ) neper/m Table 3. l. Efficiency study for an annular slot with electrical circumference (30b 2 O. 25, (to = 180 degrees. CHAPTER 4 EXPERIMENTAL INVESTIGATION OF LOADED SLOT ANTENNAS 4. 1. Introductory Remarks In this chapter, the results of an experimental investigation on the loaded rectangular and annular slot radiators are presented and correlated with the theoretical-nume rical solution described in Chapters 2 and 3. Typical slot field distributions, input impedances, relative amplitudes of radiation fields, and feeder standing wave ratios we re measured for antennas consisting of a rectangular slot with electrical half-length h = 0. l )‘o loaded at d = 0. 7h and annular slots with electrical circumferences of (30b : 0. 22 and 0. 5 loaded at (to = 180 degrees. Since it was necessary to back the slots by a cavity in order to prevent radiation outside the anechoic chamber, the experimental loaded slot antennas radiated only into a half space. Since the experimental study deals with electrically small slot antennas, energy stored in the backing cavity and dissipation due to the backing cavity and the coaxial feeder line strongly influence the measured results. The results of the experimental investigation are therefore not quantitatively comparable with those of the theoretical-numerical solutions. However, a strong qualitative agreement is found between the results of these experimental and analytical studie s. 4. 2. Anechoic Chamber and Experimental Setup The experimental arrangement consists basically of an anechoic chamber constructed with an aluminum image or ground plane forming one of its walls. Experimental measurements are made upon slot antennas which are cut in the ground plane and which subsequently radiate into the chamber. The purpose of this anechoic chamber is to simulate a free half-space environment. A photograph of a typical anechoic chamber and experimental set up for the loaded rectangular slot is shown in Figure 4. l. A similar photograph of a loaded annular 84 85 Figure 4. la. lixpi-rimvntal llMHlL‘l of rectangular slot: loadings. (manual l|‘l'(l('r. and slot field prolm imilratt-(l. I I ..__ O _ . . .._. . -. l 7:5" ;. y ,2". ’-. .0. ,\ nag ° ' , . . OI lgwll) g .".,,..;‘((‘ <. ,., 86 slot is shown in Figure 4. 2; the details of its arrangement are shown in Figures 4. 3 and 4. 4. Figure 4. 3 indicates the complete experimental set up including the anechoic chamber and the connection of all the instrumentation which was used, while Figure 4. 4 shows the detailed construction of the rectangular and annular slots (including the schemes used to excite the slot, implement its loading, and probe the field in its aperture). In order to prevent radiation outside the chamber, the slots were backed by an adjustable cavity structure as indicated. The anechoic chamber (dimensions of 8 ft. wide, 6 ft. high, and 6 ft. long) was constructed entirely from appropriately covered wooden frames. Its interior was completely enclosed by an aluminum ground plane on one wall and B. F. Goodrich type VHP-8 microwave absorbers covering the remaining three walls as well as the floor and ceiling. Except for the details of their geometrical structures, the experimental models of the rectangular and annular slot antennas we re constructed similarly. Consequently, the basic experimental setup as well as the details of its operation are identical for the two slot radiator geometries. The discussion which follows is initially focused upon details and measurements for the loaded rectangular slot, while a corresponding discussion on the annular slot is presented at the conclusion of this section. A short, rectangular slot antenna of half-length h = 5 cm (h/k0 = 0.1 at 600 MHz) having 2 h/b = 10. O was cut into a brass plate of thickness t = 0. 475 b. The brass plate was mounted on the large aluminum plane (linear dimensions of 5 X0 per side) which forms one wall of the anechoic chamber. An excitation frequency of 600 MHz was used for all experimental measurements on the rectangular slot. The slot was center-driven by a coaxial feeder as shown in Figures 4. 3 and 4. 4. This coaxial line (RG58), which excites the antenna, has a characteristic impedance of 5052 and an outer conductor diameter of 0. 195 in. Provisions were provided for connecting lumped load impedances between the narrow sides of the slot at any points along its axis. A small coaxial line (RG 196) probe, with characteristic impedance of 5052 and outer conductor diameter of 0. 076 in. , was installed (along with a positioning assembly) to facilitate measurement of the electric field distribution in the slot. The loading impedances 87 Figure 4. 22. Experimental model of annular slot; loading. coaxial feeder. and slot field probe llldlfattd. Fluilz‘n- 4.21). Slut autmiiia l1.v kll'lu 1:111), am: lllFll‘lilHt'IllS m, < \.;,)<-I‘1>-)-<:)l.il setup. 88 "F 8 ft. .._) microwave . anech01c absorber |___’.__(, chamber dipole receiving antenna * :3 \0 V aluminum movable /gr0und plane probe slot amplitude ' / g detector l I —J VSWR 50 Q slotted indicator line section _ ' , R. F. ad ustable TE _ .VdS.WR d mplitude méde backing 10 Osc1llator in icator etector cavity 1 kHz SOS-2 amplitude termina- modulation tion Figure 4. 3. Anechoic chamber and block diagram of experimental set-up. 89 /' /’ / I o E / coaxial : / coaxial probe : //’ feeder ‘ froniRWFX source to detector Figure 4. 4a. Rectangular slot with its coaxial feeder, loadings and probe. loading hnpedance /' coaxial orobe to ,/ detector coaxial feeder / fron111.F\ source Figure 4. 4b. Annular slot with its coaxial feeder, loading and probe. 90 were implemented by short sections of open- or short-circuited coaxial line (RG 58 or RC 196) connected between the edges of the slot at appropriate locations. The slot antenna was backed by an adjustable, short-circuited TE10 the 400-800 MHz band. The physical dimensions of the waveguide are mode waveguide cavity designed for dominant mode operation in 24 inches for the tunable length with a cross section 4 inches high and 14. 75 inches wide; it was constructed of brass with a thickness of 0. 125 in. A movable short-circuiting plate was provided to facilitate tuning the TE mode cavity to resonance or antiresonance. The cavity protrudes outiiode the anechoic chamber and prevents radiation in that region, such that the slot radiates effectively into a half— space. In order to avoid excessive energy storage and dissipation in the backing cavity, its length was adjusted to produce an antiresonant condition. A small, movable receiving dipole was provided to monitor the radiation field maintained by the loaded slot, and is located a radial distance of 3 ft. 4 in. away from its center. The length of the coaxial line feeder was adjusted to k/Z such that its input impedance (measured by conventional slotted line techniques) was equal to that of the slot antenna. A block diagram. of the arrangement of experimental equipment is presented in Figure 4. 3. For the experiment on the annular slot, the experimental arrange- ment is the same as that for the rectangular slot; only the subsection of the ground plane in which the slot is cut is modified. An annular slot with an ave rage radius of 4. 5 cm and a width of 1 cm is cut into an aluminum plate of thickness 0. 188 in. which is subsequently mounted on the large ground plane. To obtain an electrical circumference of (30b 2 0. 25 (for which thorough numerical results were presented in Chapter 3), a frequency of 280 MHz should be used. Due to the presence of strong harmonic frequencies in the output of the only available R. F. signal generator covering this frequency range, it was necessary to utilize a different signal source at a frequency of 250 MHz. Consequently, a slot with Bob = O. 22 was investigated experimentally. For the case of a slot with electrical circumference (30b 2 0. 5 (extensive numerical results again collected in Chapter 3), an excitation frequency of 560 MHz is utilized. 91 4. 3. Rectangular Slot Measurements Measured slot field distributions, corresponding to various capacitive, double loading impedances located at d = 0. 7 h, are indicated in Figure 4. 5. The distribution of the field in the aperture of the unloaded slot (ZL = - jco ) is the expected linear distribution (corresponding to the current in a complementary short dipole). For impedance loadings on =-j l60andZ L L uniform. These loadings will lead to a radiated power which is enhanced = - j 120, the slot field becomes increasingly relative to that for the unloaded case. When the capacitive load impedance is decreased in magnitude to the order of ZL = - j 47. 5 to ZL = - j 40, a phase reversal occurs in the slot field distribution as shown in Figure 4. 5. This later phase reversal can lead to an improve- ment in the directivity of the slot radiator. In Figure 4. 6, the input impedance to the loaded slot antenna, the relative amplitude of its radiation field in dB, and the standing wave ration on its coaxial line feeder are indicated as the load impedance is varied between Z = - j 40 and Z L L impedances were implemented with open-circuited sections of RC 58 : - j 00 . The capacitive load coaxial cable of appropriately adjusted length (their impedances were measured at 600 MHz using a slotted line). Energy stored in the backing cavity influences the measured input impedances such that they cannot be compared meaningfully with those based on the theoretical-numerical solutions. For example, the measured input impedance to the unloaded slot is Zin : 2. 14 + j 28. 5 compared to the theoretically calculated impedance of Zin : O. 249 + j 120. The theoretical input resistance should be greater than the experimental one by a factor of two since the slot was assumed to radiate into a full space in the analytical study while it radiates into a half-space in the experiment. The large measured value of input resistance can be attributed to dissipation in the coaxial line feeder and backing cavity (which can be very significant in this case because the radiation resistance of the slot is so small). In spite of the discrepancies in the quantitative details, the qualitative nature of the results presented in the cases of Figure 4. 5 closely resemble those predicted theoretically in Figure 2. 6. As the load impedance is decreased in magnitude from Z : - joo L toward ZL = - j40. 0, Figure 4. 6 indicates that the input resistance to g— z:h : d ZL -— z=d \ \ \ z + ‘1- antiresonant v ' ,J z=0 TElo mode 0 _ rectangular cavity ——. ._ b d 1 fl - feeder line \ Z —z:-h L ‘ \ \ adjustable ._ 2: -d : Sl'lOI‘t. strip dipole ¥ slot antenna and Clrcmt ground plane ’ J‘/— 2L: —j120 \\. \ 2L: -j160 ’ ‘\‘ - -100. 0 7 \ x U) V Q) o: >~ "' -50.0 3 8 <22 s .8 73 f: mv>~ 0 ~ 0. 0 “J 8 2?. 3 .8 '3. i. 0 § 1'. g 0. '- 50. O o .3 5" 2 ZL: -j40 \ g g 0. 2" . o a 7 10.0 0. 1- 2L: -j40(phase) - 100.0 loading position 0. 0 I I T l l T l I F 0.0 0.2 0.4 0.6 0.8 1.0 position z/h along the slot Figure 4. 5. Physical structure of loaded slot antenna and its measured field distributions for various loadings. 93 (swqo) 30112101231 pus aoueisisai indui .moocmuomou ocsnm magnum; >3 popmofi essence no? unocm you mgm on: nopoou pcm .30: coflmwpmu Go opsufidgs 9,330." docmpoaem 39.: .o .v 0.":th Amczpmofi 0: .fi:ofio-¢oaov W AM 3:20. oocmuomou poo“ ucsnm ofifioddmo mm .. ow—u ow? com- co- CM- 0 o .o «r r r _ _ _ _ F o . 3:33 I m 2 L mucoumfimou “:95 l\ v o .3. 1 23$ on?" 1 m 0>m3 9:23: I m .hm J I / \ 1 Nu I \ I I \ I I O ’0 m .No \ I l l / o~ \ , I . l VN 0 mo / \ Amfino. 1. VA monsoon." \’ . on: m S L \ / 6 a , r 3 7 I / +’ \ o .ooT I a, T Nm EB 83:88 I\/ . . \ . 1 3 £033 on \ . u m. N: E u n ..\. A. o {6 18m , o .2 u {an o . O 08.2.. v\ A o u {a 13 (8P) m~u N“ (814181 lapse; sun [syncs no one; 3A2M fiugpueas (39) pm; 1101121va :0 691111th 811112191 94 the slot as well as the relative amplitude of its radiation field increase rapidly while the feeder line SWR decreases. For an optimum capacitive loading of [ZL]op : - j 120 ([ ZL]op = - j 120 theoretically), the radiated power is enhanced by 10 dB relative to the unloaded slot and the input resistance has increased to nearly 50 ohms. Since the input resistance is drastically increased, the enhanced radiation is not simply a tuning effect. If the inductive input reactance were tuned to zero by a low-loss series capacitor, a further increase in the radiated power would of course occur, and the short slot antenna would be nearly matched to the Rc = 5052 feeder line. The electric field distribution in the slot is nearly uniform for the [Z 1) = - j 120 loading as shown in Figure 4. 5. L10 A corresponding optimum load impedance for the complementary dipole . d 18 [ZL]op = 7;:/4 ZL : j 296, which yields a nearly uniform current distribution along the dipole according to the theory by Lin.5 As Z is further decreased in magnitude, the input resistance to the slot alrJid its radiated power fall off rapidly. An important practical point is that the optimum load reactance and the necessary input tuning reactance are both capacitive such that low-loss implementations of these reactances can readily be realized. When Z is further reduced in magnitude to Z L L (ZL = - j 65. 5 theoretically) a phase reversal in the distribution of = -j45.0ohms electrical field in the slot aperture occurs as indicated in Figure 4. 5. As discussed in Section 2. 6, the phase reversal can lead to a radiation pattern with significantly improved directivity. Figure 4. 6 indicates that the input resistance to a loaded slot with high directivity (for Z = - j 45. O) is very small (as is typical of super-directive antennas). LSince the radiated power is proportional to E: , as defined in equation (2. 69) of Chapter 2, and E0 is small due to the phase reversal of E5, it is evident that the input resistance should be very small in this case. The efficiency of such a highly directive, loaded slot is therefore relatively small. The radiation patterns of a short slot antenna appropriately loaded to achieve high directivity we re not measured in detail due to the inadequacy of the available anechoic chamber at 600 MHz. However, the general shape of the patterns and their beamwidth were confirmed experimentally. 95 4. 4. Annular Slot Measurements Measured slot field distributions are indicated in Figures 4. 7 and II 4. 9 for annular slot antennas with electrical circumferences of Bob 0. 5 and 0. 22, respectively, for various purely reactive loading impedances. The loading is located at (to = 1r. It is observed that the maximum amplitude of the slot field distribution occurs at d) = 17 for the unloaded slot (ZL = - j 00), while this maximum can be shifted to <1) = 0 through the selection of an optimum reactance loading. This shifting of the voltage maximum to the input terminals results in a significant increase of the input resistance to the small, loaded, annular slot radiator. For the case of an annular slot with [30b 2 O. 22, a loading impedance of Z = - j 20 leads to a phase reversal in the slot field distribution as shown in Figure 4. 9. This phase reversal leads to the drastic change in the radiation field pattern described in Section 3. 6. In Figure 4. 8, measured results for the input impedance to a slot antenna having [30b = O. 5, the relative amplitude of its radiation field in dB, and the standing wave ratio on its coaxial line feeder are indicated as the load impedance is varied between Z : - j 5 and ZL = - jab . The capacitive load impedances were inldplemented with open-circuited sections of RC 196 (50 Q) coaxial line of appropriately adjusted length. The measured input impedance to the unloaded slot is Zin = 3. 5 + j5. 5 compared to the theoretically calculated value of Zin = 2. 18 + j4. 92. As discussed thoroughly in Section 4. 3, the discrepancy between measurement and theory can be attributed to energy storage in the backing cavity as well as dissipation in the cavity and feeder. As the load impedance is decreased in magnitude from Z = - joo L toward Z = - j 10. 0, Figure 4. 8 indicates that the input resistance to the slot iii-creases rapidly. For an optimum capacitive loading of [ZL]op = - j 42. 5 ([ZL]op = - j 25 theoretically), the radiated power in enhanced by l. 5 dB relative to that of the unloaded slot, and the input resistance has increased to nearly 50 ohms. If the inductive input reactance were tuned to zero by a low-loss series capacitor, a further increase in the radiated power would of course occur, and the small annular slot antenna would be nearly matched to the RC = 50 Q feeder line. The slot field distribution corresponding to the Z = — j45 loading L is shown in Figure 4. 7. 96 relative amplitude of slot voltage distribution Figure 4. 7. zL : joo ‘ zL — -J 5 2L : -j15 ZL : -j 45 I l I J L l l | l 20 4O 6O 80 100 120 140 160 180 angle (I) in degrees Typical measured slot field distributions for a loaded annular slot with [30b : O. 5. 97 (swqo) aoueio'eal pue aoueqsisal indui .m .onnon no?» no? .3353 cm u0w mocmoomou mcfipmofi mo mcoflocsw mm mam on: nopoom paw 63$ coflmfipmu mo mpsuflaam 9,320.“ .oocmpoacfi 39: 38:3 oocmoomou mcfipmoH 8 cm: mvu own mm: om: mm: ON: mas o o )x _ _ _ _ _ _ _ 2 $833 I , oocgmmmou 3&5 , om J 23m. 0:2 1 ’ o>m>2 wcflucmum \ l I \l l a \\ 'Il Ill-l .I III- I \/\ \ OOH In/ I: /’II .II.‘ lulu 1 / 53 63:38... L / Ema 53.2me // 0.: T / l 1 / / + I I I I l I ’ Amegov ooamnomou 395 |\\. // CON l / l /\ .w a. .35 O (HMS) (HP) N u-¢ v on w on Na m- A: 0 ON m VN 0 mm o NM NH (HMS) lapse; {eixeoo no one; aAem Buipueis (gp) ptai} uopeipex }O apmudwe angels; 1. O. O. Q .g. 0. :3 .o 'E.‘ 40-; O. 0H 'o 3.1. m 0. 4.) H o > u 0. o H U) '«H O O. o "o :3 $9 :51 O. E“ a: a) 0. > "-1 4—3 «s '5 o 34 Figure 98 -- Zz-jZO '0 lllllllll O 20 4O 6O 80 100 120 140 160 180 angle 4) in degrees 4. 9. Typical measured slot field distributions for a loaded annular slot with Bob 2 O. 22. 99 In Figure 4. 10, measured results for the input impedance to a slot antenna having [30b 2 O. 22, the relative amplitude of its radiation field in dB, and the standing wave ratio on its coaxial line feeder are L = j 10 and ZL = jco . The inductive load impedances were implemented by short- indicated as the load impedance is varied between Z circuited sections of 150 ohm two-wire transmission line of appropriately adjusted length. The measured input impedance to the unloaded slot is Zin = 3. 27 - j47. 5 compared to the theoretical value of Zin = O. 278 - j77. 3 (the discrepancies can again be explained as in Section 4. 3). L = j 10 toward ZL = j40, Figure 4. 10 indicates that the input resistance to the slot increases rapidly. For an optimum loading of [ZL]op = j 25 ([ ZL]op =j 8O theoretically), the radiated power is enhanced by 2. 5 dB As the load impedance is increased in magnitude from Z relative to the unloaded slot and the input resistance has increased to nearly 50 ohms. Since the input resistance is drastically increased, the enhanced radiation is not simply a tuning effect. If the inductive input reactance were tuned to zero by a low-loss series capacitor, a further increase in the radiated power would occur, and the annular slot would be nearly matched to the RC = 50 ohm feeder line. 100 (swqo) 93119331291 pue aoueisisax indut 0 Ln OOH 02 com .NN .o u non Lisp “2on 53353 cm Mom oocmuomou mcfipmofi mo mcoflocsm mm mam we: umpoow pam .30: coflmfipmu mo opgflafls “wk/Sofie.” .oocmpomew ”.59: .3 .4 €ng Amanov oocmuommu wcwpmofi ON. 00 om 0% cm om OH 0 $8.23 oocmumfimou 39: fl 1 .I/ / AmCEov ooCmuomou / / “sac“ I / l / l / / II / II I. / EB @3338... / I 3...: Sufism... / 1 \ “‘I " ,"'.III II V\..A\ \\.. ./ / 1 I/ \ I 1| I ..L .I I II 339 032 98.5 mnwpamum (HMS) O OH ma ON mm om mm ow (SIP) om- mm: OH: OH (gp) p191; uoii'erp'ex jo apquwe BA‘QBIBJ (HMS) lapse} Gun [eixeoo uo one: SA‘BAA Butpueis CHAPTER 5 SUMMARY AND C ONCLUSIONS A theoretical-numerical solution for the aperture field distribution and the input impedance of a short, loaded rectangular slot antenna was presented in Chapter 2. These analytical results were obtained by first developing the appropriate integral equation for the voltage distribution in the slot and subsequently solving it numerically. It has been demonstrated that if the electric field distribution in the electrically short slot antenna is appropriately modified by a double reactance loading, its radiated power (input resistance) or its directivity can be significantly increased. These phenomena result from the implementation of a slot voltage distribution which exhibits, respectively, either a nearly uniform amplitude or a phase reversal. Since the optimum loading reactances are capacitive, it should be feasible to implement a low-loss optimum impedance loading. The efficiency of the doubly loaded slot with enhanced radiation has been shown to greatly exceed that obtainable with conventional input tuning networks. A highly directive slot antenna can be achieved only through the sacrifice of radiated power and efficiency. An appropriate integral equation for the aperture field distribution in an electrically small, loaded annular slot antenna was developed in Chapter 3. This equation was solved numerically by a Fourier series method to determine the voltage distribution in the annular slot as well as its input impedance; its radiation field was formulated in terms of this voltage distribution. It has been demonstrated that the maximum of the slot voltage distribution can be shifted from 4) = 1800 (diametrically opposite the driving point) to <1) = O0 (at the input terminals) through the choice of an optimum reactance loading; this results in a significant increase of the input resistance to the electrically small, annular slot radiator. In effect, the antenna can be forced into a near-antiresonant condition. The efficiency of this loaded annular slot with enhanced radiation again exceeds that which can be achieved by conventional input tuning schemes. Appropriately chosen loading reactances can also lead to a phase reversal in the slot voltage distribution; this later aperture 101 102 field modification results in a drastic change in the radiation field pattern of the slot antenna. The equipment arrangement and results for the experimental investigation of loaded rectangular and annular slot antennas have been described in Chapter 4. Due to loss in the coaxial feeder and energy storage and dissipation in the backing cavity, only a qualitative agree- ment has been found between the experimental results and those of the theoretical-numerical solutions. This qualitative comparison between theory and experiment confirms, however, all of the essential features of the slot voltage distributions and input impedances predicted analytically. 10. ll. 12. 13. REFERENCES Galejs, J. , ”Excitation of slots in a conducting screen above a lossy dielectric half space, " IEEE Trans. Antennas and Propagation AP-lO , 436-443 (July 1962). Galejs, J. , "Admittance of a rectangular slot which is backed by a rectangular cavity, ” IEEE Trans. Antennas and Propagation AP-ll, 119-126 (March 1963). Galejs, J. , "Hallen's method in the problem of a cavity backed rectangular slot antenna, " Radio Science 67D, 237-244 (March- April 1963). King, R. W. P. , The Theory of Linear Antenna, Harvard University Press, Cambridge, Massachusetts (1956). Lin, C. J. , D. P. Nyquist and K. M. Chen, ”Short cylindrical antennas with enhanced radiation or high directivity, " IEEE Trans. Antennas and Propagation AP-18, 576-580 (July 1970). Booker, H. G. , "Slot ae rials and their relation to complementary wire aerials (Babinet's Principle), ” J. Inst. Elec. Engr. (London) 33, 620-626 (1946). Neff, H. P. , Jr. , C. A. Siller and J. D. Tillman, "A simple approximation to the current on the surface of an isolated thin cylindrical cente r-fed antenna of arbitrary length, " IEEE Trans. Antennas and Propagation AP-l8, 399-400 (May 1970). Harrington, R. F. , Field Computation by Moment Methods, The Macmillan Company, New York (1968). Galejs, J. and T. W. Thompson, "Admittance of a cavity-backed annular slot antenna, " IRE Trans. Antennas and Propagation AP-lO, 671-678 (November 1962). Cumming, W. A. , "Design data of annular slot antennas, " IRE Trans. Antennas and Propagation AP-6, 210-211 (April 1958). Wait, J. R. , "A low-frequency annular-slot antenna, " Radio Science 62, 59-64 (January 1958). Storer, J. E. , ”Impedance of thin-wire loop antennas, " AIEE Trans. 15, 606-619 (November 1956). Iizuka, K. , "Circular loop antenna multiloaded with positive and negative resistors, " IEEE Trans. Antennas and Propagation AP-13, 7-20 (January 1965). 103 14. 15. 16. 17. 18. 19. 20. 21. 104 Nyquist, D. P.,and K. M. Chen, Quarterly Report No. 12 for AFCRL Contract AF l9(628)-5732, Oct. -Dec. 1968, pp. 5-6. Harrison, C. W. , Jr. , "Monopole with inductive loading, " IRE Trans. Antennas and Propagation AP-ll, 394-400 (July 1963). Levine, H. , and J. Schwinger, "On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen," Comm. Pure and Appl. Math. 2, 355-391 (1950). Silver, S. , Microwave Antenna Theory and Design, Dover Publications, Inc. , New70fl<(1965). Jackson, J. D. , Classical Electrodynamics, John Wiley and Sons, Inc. , New York (1967). Collin, R. E. , and F. J. Zucker, Antenna Theory, McGraw-Hill Book Company, New York (1969). Fradin, A. Z. , Microwave Antennas, Pergamon Press, London (1961). Harrington, R. F. , Time-Harmonic Electromagnetic Fields, McGraw-Hill Book Company, New York (1961). APPENDIX A APPROXIMATE EVALUATION OF A DEFINITE INTEGRAL The double integral SZ+AZ/Z y+Ay/2 e-jBOR I : S ___.___ z-Az/Z y-Ay/Z where R = [(y-y')Z + (z-z')2] l/Z dy'dz' (1) , and Ay, Az are small quantities (i. e. , BoAy < < l and BOAz < < l), is improper since its integrand has a singularity when y'=y and z'zz. This integral can be evaluated analytically by the procedure outlined below. Since [30R < < l in this case, by expanding the exponential into a power series and retaining only its leading two terms, equation (1) can be approximated as SZ+AZ/2 SerAy/Z l - jBOR R dy'dz' z-A z/Z y- Ay/Z .._: 52+Az/2 Sy+Ay/Z dy'dz' - jBOAyAz . (Z) z-Az/Z y-Ay/Z "R If the change of variables u = y-y' and v = z-z' is made, equation (2) can be rewritten as SAz/Z SAY/2112 du dv -Az/Z Ay/Z 2+ 2 V - jBOAyAz Az/Z Ay/Z 45“ S du dv 'jBOAYAz' (3) 2 u+v 105 106 u=Ay/2 5...). 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The representative short-backfire antenna described above possesses a radiation pattern with inajor-lobe beam widths of approximately 32° in the E-plane and 38° in the H-plane; its sidelobe and backlobc levels are at least 25dB 133 134 below the peak intensity of the major lobe. Although the short-backfire antenna has received extensive experi- mental investigation, no accurate theory for its circuit and radiation characteristics has been developed and the basic principle of its oper- ation is currently not well understood. An early study by Chen et. a1. 4 demonstrated that the major lobe of its radiation pattern could be well approximated by assuming the short-backfire antenna to consist of an open- cavity radiator which radiates from a cosinusoidal field distribution in its circular aperture (this cosinusoidal distribution was based upon experimental near-zone field measurements). Later research by Hong et. a1. 5 showed that improved radiation patterns, which, except for the sidelobe structure, compared well with experimental results, can be obtained by using a waveguide excitation approach to determine the amplitudes of the various modes which can contribute to the aperture field of the Open cavity. Nielsen and Pontoppidan6 performed an analysis of the short-backfire antenna according to the following outline: (l) the rim of the large reflector is ignored completely; (2) for the purpose of determining the currents in the remaining components of the structure, the large reflector is assumed infinite in extent; (3) the small reflector is replaced by a tapered dipole array; (4) all dipole currents are as- sumed to be sinusoidal and their amplitudes are determined based on the "induced e. m. f. " method; (5) the large reflector currents are ap- proximated as a truncation of those excited on the infinite image plane by the system of dipole currents; and (6) the radiation fields main- tained by all the induced currents in the system are calculated. The inherent inaccuracies associated with neglect of the rim, replacement of the small reflector by a dipole array, and use of the induced e. m. f. method lead to significant inaccuracies in the predicted major-lobe beamwidth and an inability to accurately ascertain the antenna's input impedance. A NUMERICAL-PHYSICAL OPTICS method? is applied in this research to analytically predict the characteristics of a short-backfire antenna with greater accuracy. In this approach, the EM boundary value problem is formulated theoretically and solved numerically to determine the induced currents excited on the conducting surfaces of various components of the radiator. 135 The analytical scheme of the numerical-physical Optics method can be outlined as follows: (1) For the purpose of calculating the induced currents on the re- maining components of the short-backfire antenna structure, the pre— sence of its rim is neglected. (2) A coupled set of integral equations for the induced currents excited in the cylindrical dipole and on the surface of the small reflector are formulated based on the boundary condition that the tangential elec- tric field vanish at all conducting surfaces. (3) In calculating the currents induced in the dipole and on the small reflector, the method of images is applied by approximating the large reflector as an infinite conducting sheet. (4) The currents excited in the dipole primary radiator and on the surface of the small reflector are expanded in appropriate (efficient, rapidly converging) series. The integral equations are subsequently point-matched to reduce them to a set of simultaneaous, linear algebraic equations for the unknown coefficients of the series. These algebraic equations are solved by matrix inversion, and the series for the induced currents are subsequently summed using numerical methods and a high- speed digital computer. This completes the determination of the induced currents in the dipole exciter and on the small reflector. (5) The surface current excited on the large reflector is approxi - mated by a truncated form of that current induced on an infinite plate by the known currents in the dipole and on the small reflector ( as well as by their images). (6) The radiation field of a rimless short-backfire antenna model is calculated by superposing the radiation fields maintained by the cur- rents in the dipole and on the surfaces of the small and large reflectors. (7) The input impedance to the rimless short-backfire antenna model is calculated in terms of the input current to the dipole exciter. (8) The computer program for the numerical solution is written in such a manner that it can handle reflector geometries consisting of arbitrary plane shapes. (9) The effect of the large reflector rim upon the radiation patterns of a short-backfire antenna can be accounted for by applying the geometri- cal theory of diffraction to calculate a correction term based on the field diffracted by the rim edge. 136 The application of this numerical-physical optics method (integral equation approach) to the study of a short-backfire antenna permits an accurate determination of the radiator's input impedance. Accurate radiation fields are obtained, since the amplitudes and phases of the induced currents excited in the dipole and on the surface of the small and large reflectors are carefully determined, and the radiation fields maintained by these currents are calculated directly. Chapter 2 presents the integral equation method for the determination of the induced currents excited in the cylindrical dipole primary radiator and on the surface of the small reflector. A calculation of the surface current excited on the large reflector as well as the radiation field maintained by all the induced currents in the system (except those on the large reflector rim) is presented in Chapter 3. All of the numerical results are calculated in Chapter 4. A summary as well as a discussion of conclusions is included in Chapter 5.. CHAPTER 2 CALCULATION OF THE INDUCED CURRENTS 2. 1 Introductory Remarks A simplified, rimless model of the short-backfire antenna, as illustrated in Figure 2. lb, is investigated theoretically in this chapter. The rim of the large reflector (present on the short-backfire antenna) is neglected in this approximate study. The EM boundary value problem is formulated theoretically and solved numerically to deter- mine the currents induced on the conducting surface of the rimless antenna model. In calculating the current in the dipole exciter and the induced cur- rent on the small circular reflector, the method of images is applied by approximating the large reflector as an infinite conducting sheet. A set of simultaneous integral equations for these unknown currents are formulated based on the boundary condition that the tangential elec- tric field must vanish at each of the conducting antenna surfaces. The currents in the dipole primary radiator and on the surface of the small reflector are subsequently expanded in trigonometric series. The in- tegral equations are then point matched to reduce them to a set of simultaneous, linear algebraic equations. This system is solved nu- merically by matrix inversion, and the trigonometric series are sub- sequently summed numerically to complete the solution for the induced currents. A high-speed digital computer is applied to implement the nume rical calculations . 2. 2 Physical Structure of the Short Backfire Antenna A photograph of a typical short-backfire antenna is presented in Figure 2. la. A similiar photo of an approximate rimless model for this backfire is shown in Figure 2. 11). A sketch of the approximate rimless antenna model is indicated in Figure 2. 2 a. This radiator consists of a small circular reflector R of radius 1) and a large circu- la r reflector S of radiusw arranged parallel to one another with 137 138 Figure 2. 1a Typical physical model of a short-backfire antenna. Figure 2. 11) Typical physical model of a rimless short-backfire antenna. 139 .- c .q d dipole exciter L .1 circular reflector w ‘ w=1.o>. \ o y h:d~:0.25ko b:0.2)\ o c:0.5k o 9: Zln (—-Z—r-‘-)210. a typical dimensions Figure 2. 2a Geometry of the rimless short-backfire antenna. small reflector dipole u :5 I ll 1 u l image elements actual elements infinite, plane, conducting reflector (image plane) Figure 2. 2b. Current and voltage due to image theory. 140 separation c. The planes of both reflectors are transverse to the longitudinal antenna axis. The dipole primary radiator D is centered along the antenna axis between the large and small reflectors at a dis- tance d from the large reflector. The center of the large circular reflector is placed at the origin of a Cartesian coordinate system whose y-axis lies along the short- backfire antenna axis. The dipole primary radiator is parallel to the z axis with half-length h and radius a, and its center is located at y = (1 along the positive y-axis. The small circular reflector is paral- lel to the x - 2 plane with its center along the positive y-axis at y = c. 2. 3 Integral Equations for the Induced Currents In practical short-backfire antenna configurations , the radius w of the large reflector is related to that of the small reflector b as well as to the other appropriate antenna dimensions approximately as wé 4b é 4c é 8d. For the purpose of determining the currents excited in the dipole or on the surface of small reflector, therefore, a negligible error is intro- duced by allowing the large reflector radius to become unbounded such that w ->°°. By assuming the large reflector is infinite in extent, the usual image technique is applied for the determination of the currents induced in the primary dipole radiator and on the surface of the small reflector (as well as their images). Figure 2. 2 b indicates the dipole and small reflector as well as their images. In calculating the radi- ation field of the antenna, the large reflector currents are truncated by using the actual radius of this reflector. The unknown currents to be determined are Iz (z) = axial current in dipole exciter —> Kr(x, z) A Q er (x, z) + z Krz (x, z) : surface current on small reflector R. The pertinent boundary conditions on the tangential electric field at the conducting surfaces of the antenna are Ecz (z) = E: (z) . . . at surface of cylindrical dipole exciter D 141 Erz (x, z) E (x, z) I'X H O H O at surface of small reflector R where the tangential field components maintained by the currents in the system are Ecz(z) at the dipole surface and Erx(x’z ), E z(x, 7.) a. the surface of the small reflector. The impressed electric field E:(z) maintained by the potential difference V0 across a gap of width 26 at the center of the dipole is V 1 for | z) < 6 _° 26 E:(Z)- - 0 for |z| > 6 ll impressed electric field at cylinder surface due to V0 across gap of width 26. The induced field at any point if in Space maintained by all the currents in the antenna system is given by -’ E(r)_-ve(?)-joii(r) 113 MR?) 1 -> e O - - VS p(r') dV' 471cc V R(r , r') 1'qu _. _.' e-JBOR(F.I") --—4TT—. V J(I‘ ) _’ _* dV' . R(r,r') (2.1) Applying the equation of continuity V. Jr (r') : -jwp(r ‘), equation (2. 1) becomes 11 -j ..1 1'13 R(?.?') E(r):——§2 VS V'.J(r')e O dV' 471136 V R(?,}") _’_’ -j(3 R(r,:') +5: Sler') e O- _. dV' R(r,r') (2.3) where 80 : Zir/ko is the free-space wavenumber, g0 : ‘J 11 7c is the o , g o I r - r 'l is the distance .-p D between the field point at r and a source point at r ' _, ~> free-space wave impedance, R(r , r ') -- , and V is the vol- . -> ume including all the currents in the antenna system. J (r ') dV' in the 142 present case consists of the current 912(2')dz' in the dipole and the surface current K rd: ')ds' : [Qer(x', z') + QKrz(x', z')] dS' on the small reflector as well as the effects of their images. Currents 212 and K r are imaged into the y = 0 plane to replace the effects of K s on the large (infinite) reflector and concurrently satisfy the boundary con- ditions at its surface. Thus the induced field E (I?) can be expressed finally as —> -p h _. *‘fl -. R' —> —>' E (r)=-j_h VB 8' I(z,)ejpiRcE’r)_eJpogcfsrfldz' 411130 "h 3z 2 R (1. ,rl) RI (1., r') c c -> —> I _. .5 —> -> 'jfloR (r, r ') 'jPORr (r: r ') +SR V'.Kr(r') 8 gr» - e”, »' dS' Rr(r,r') Rr(r,r) h . “‘5 "’| -. ' -> —>' + B: ’25 Iz(z') e-JfioRcu’ r )_ e 3130162”. 1‘ ) dz' —h RC (2:, ;|) Ré(;’ rt) er’, F') R}(?, r') (2.3) where R and R are the distances from current elements on the cylinderrand recflector, respectively, to the field point, while R; and R' are the corresponding distances from the imaged current ele- ment: to the field point. R designates the surface of the small reflec- tor. Applying equation (2. 3) to evaluate the electric field at the con- ducting surfaces of the simplified short-backfire antenna model leads to the following results: 143 (i) Tangential electric field at surface of dipole exciter: Ecz(z) jgo 8 h 8 , e-JpoRcc(z' z') e-JpoRcc(z’ z') , E(‘z(z) : - 41113 8z 7)? Iz(z) - dz 0 -h R (z,z') R' (z,z') cc cc 8 I ___a_, I +SR (W er(x,z')+ 3z' Krz(x'z')) _’ I _' I I I e JfiORCI‘(z,X',Z) e JfiORCI'(z’ x’z) — dx'dz' I I I I Rcr(z,x,z) Rcr(z,x,z') h -jB R (z z') 1's R' (z z') + 2 1 ( , e 0 cc ’ _ e 0 CC ’ d , ‘30 -h z z) , , , z RCC(Z’ Z ) RCC(Z)Z) ... R I I _' l . ' +5 K (X' 2') 8 J60 Cr(z, X 9 Z) e JfioRcr(Z, X , Z ) R rx ’ ' ' ' dxldzt Rcr(Z.X . z) R'Cr(z,x', z') I (z. 4) The various distances are defined as 1/2 1/2 RCC(z, z') :. [ (z-z')Z + a2] R'CC (z, z') : [ (z-z')2 + 4d2] 1/2 1/2 R (z,x', z') : [ x'2 + (c-d)Z + (z-z')2] R'C (z.X'. 2') = [ X'2 + (c+d)2 + (z-z')z] 144 (ii) Tangential electric field at surface of small reflector: Erx(x’ z) E (X,z):—.j-§o- L[Sh a 1(z1) I'X Z 41rf30 3): -h 32' e-jg3Rm (X, Z, Z') e-j%Ri'C(xv Z, Z') — dz' ch(x, z, z') Rf'c (x, z, 2') +5 8 I a I R (FEES): (x',z)+—a? Kama“) e'jfi)Rrr (X, Z, X', Z.) e’iji'r (X, Z, X's 2') '- dx'dz' R (X. Z. X’. 2') R' (X. Z. X'. Z') 1']? rr 2 -jQ)Rrr(xv Z, X', Z') ’jfi R. (X, Z, X', Z.) + B. l. K... > er(x, 2). By assuming K (x, z) é O, the coupled integral equations can be simplified rx to the form h a , 8 2 ‘ t ' . $.11 [‘TZ. Iz(z ) 82 + (30 Iz(z):l (3CC (z, z) dz 8 , , 8 2 , , t +SR Bz' Krz(x’z) 8z +{soKrz(x’z) Gcr(z,x',z') dS' (2.10) o z r __€’__. . . 3 +5R[8z'Krz(X’z) Bz The coupled integral equations (2. 10) and (2. 11) will be simpli— h S [-—58z—, Iz(z')—§a;- + [32 I (z') Gtc (x,z,z') dz' 2 + (30 Krz(x" Z') (3:1. (x, z, x', z') dS'=0. (2.11) fied further to facilitate the trigonometric series solution. In doing so, the following symmetry properties and definitions will be used: (1) Symmetries of the induced currents excited on the dipole primary radiator and the reflector: I(z') =1 (‘2') z z Krz(x', z') : Krz(-x', z') : Krz(-X', -Z') = Krz(x" ‘2') 148 (2) Symmetries of the induced current derivitives: 3 I _ 3 I 82' 12(2 ) - - 82' Iz(-z) 2 2 8 Z lz(z') ; 8 2 Iz(-Z') 82' 82' _Q__ I I __ a I I __ a I I 82' Krz(x ’ z ) - az' Krz -X ’ z ) — az' Krz(X ’ -z) -._ a K (XI I) 32' rz - ’ -z 2 8 2 Kr (x', z') : 8 2 K -x', z') : 8 2 Kr (x', -z') 82' z 82' rz az' z Boundary equation for edge of the small reflector: - ’Jbz - x2 for a circular reflector zmax (x) : f(x) - (3) Integrating by parts and using the above properties, the first in- tegral in equation (2. 10) becomes h E) 8 t I , 5:1] 8z' 12(2') 82 Gcc(z’z) dz Z'Zh z': _ t 3 I t . 3 I - - Gcc (z, z') 8z' Iz(z) ' - Gcc (z, z) 82' Iz(z) ' z - z :-h h t 82 +5 G (z,z') ZI (2') dz' -h cc 82' z 8 z':h h 82 = -G (2.2') . I (Z') +5 G (z,z') zI (Z') (12' cc dz 2 z':0 0 cc 82' z (2.12) 149 and 2. h t 2 Sh I I I __ I I I poS—h1z(Z)Gcc(z’ z ) dz - (30 O Iz(z ) Gcc (z, z) dz (2.13) where G (2 z') -.—. Gt (2 z') + Gt (z -z') cc ’ cc ’ cc ’ Proceeding similiarly with the second integral leads to SR _§-z—'Krz(x"z') a: Gdr (z,x',z') dS' b (X') :S 5‘ aa' Kr (x',z') 88 Gtr(z,x',z') dz'dx' -b -f(x') z z z c b t 8 z':f(x') :5 - G r(Z,X:Z') a I Kr (X'sz') dX' -b C Z z z'z-f(x') b (x') t 8 +3 Gcr (z,x',z') 2 K (x',z') dz'dx' -b -f(x') az' ‘2 8 z':f(x') : - G ‘r(z,x', z') W Kr (x' , z') dx' 0 C 2 z':0 (X') I I I I ' I + 0 0 Gcr(z,x,z) 3sz Krz(x,z) dz dx (2.14) and Z (X') I) K (x',z') G (z,x',z') dz'dx' (2.15) o O 0 rz cr 150 where I I- t I I t I I Gcr(z,x,z)—Gcr(z,x,z)+Gcr(z,-x,z) + Gt (z, -x', -z') + C:t (z,x', -z') cr cr Equation (2. 10) therefore takes the final form h 2 zl: S GCC(z, z') 3 2 + B: Iz(z') - Gcc(z’ z') 8 Iz(z') 0 82' 82' z': 82 2 I I I I I I + SR Gcr(z’ x , z) 2 + [30 Krz(x , 2) dz dx l 82' b 8 z':f(x') 341$ e - G (z,x',z')—7K (x',z') dx' = 0 E (z) cr az rz z 0 z'zO o (2.16) where R1 is the first quadrant of the small reflector. Similarly, equation (2. 11) can be rewritten as h 2 S G (X) z) Z') 62 + [32 I (Z') dZ' O rc 82' o z z':h - G C(X. 2. Z') a . 12(2') ' z :0 82 2 +5 G (x,z,x',z') + (3 K (x',z') dz'dx' R rr 2 o rz l 82' Sb z':f(x') - G (X. Z.X'.Z') —— K (X'.Z') ._ o a ' rz z':0 dx‘ 0 (2.17) whe re I_ t I t _I Grc(x,z,z ) _ Grc (x,z,z)+ Grc (x,z, z) t G (x, z,x', z') : G rr r t (X, Z,X', Z') + G (X, Z, -x"z') 1' rr + __J +6 (X,Z,-X,-Z) G (x,z,x, 7) 151 Numerical solutions to the coupled integral equations (2. 16) and (2. 17), which are implemented on a high-speed digital computer, are described in the next section. 2. 5 Numerical Solution of the Integral Equations The unknown induced currents are expanded in series as follows L 12(2) : 2 A2 F2 (7*) (2.18) =1 M N Krz(x, z) : Z Z anHmn(x, z) (2.19) mzl n:l where FI(Z) and Hmn(x, z) are apprOpriate expansion functions such that the series for Iz(z) and Krz(x’ z) are efficient and converge in a relatively few terms. It is shown later that excellent results are obtained by cho- m-l . _ . 1 1T _ Sing Fl(z) - Sin _-2h (h-z) and Hmn(x, z) _ x sin fig; E(x) - 2 These functions have been found to lead to efficient series expansions by earlier investigators8. The series expansions for the unknown currents are substituted into integral equations (2. 16) and (2. 17). The L + MN unknown coefficients are subsequently evaluated by point matching equa- tion , (2. 16) at L points along the primary dipole exciter in the interval 05 zS h and by point matching equation (2. 17) at MN points on the surface R1 in the first quadrant of the small reflector. Integral equations (2.16) and (2. 17) are thus converted to a system of linear algebraic equations for the unknown expansion coefficients. These equations are solved on a digital computer using the numerical processes of matrix inversion and numerical integration. The series are summed numerically to com- plete the solution for the induced currents. Assume the dipole primary radiator is partitioned into R sub- sections with zr, r:1, 2, 3, . . . , R, located at the center of each sub- section, and define the following notation for the interval spanned by the rth subsection; (A z)r: zr -- A z/Z 5 z 5 2r + A z/2. Suppose the small reflector is approximated by a piecewise rectangular geometry and sub- divided into ST rectangular subsections, where T is the number of sub- divisions along the x-direction and S is the number along the z-direction. x‘, t l, 2, . ., 'l'. and z , s- 1, 2, ..., S locate the center ofa s 152 subdivision of area AS 2 AxAz : (b/T) (f(xt) /S) defined by the intervals (Ax)t : xt - Ax/Z s x S xt+Ax/2 and (A2)S : zs- Az/Z S 2 5- 28 + A 2/2. Substituting expansions (2. l8) and (2. 19) into integral equations (2. 16) and (2. 17) leads to the following expressions: L h a2 2 I I I Z A! 50 GCC (z. z ) [82.2 + (30] Fl(z) dz [=1 8 z'zh - I __ I Gcc(z' 2) 82' F1(2) ' + z :0 M N b f(x') 3 z , ' ' ' 'I' Z Z an SO SO Gcr(Z, X‘, z') 2 ‘1' fl Hmn(x , Z ) dz dX m:l n=l b 2':f(x') 34116 e .1 G (z.x'.z'> . H (x',z') dx = °E (z) 0 cr 82 mn 2‘20 {,0 z (2.20) L h 32 2 Z A! Grc(x,z,z) ,2+Bo Fl(z)dz O 82 1:1 3 z'zh I I ' C'rc(x’ z, z) 82' 171(2) I z _0 M N b f(x') +2 2 B G (x,z,x',z') mn O O mzl n=l 2 [a 2 +(32 H (x‘,z') dz'dx' 8 , 0 mn z b 8 z'—f(x') -S G (x,z,x',z') —,— H (x',z') dx' :0 0 rr 82 mn 2,20 (2.21) Equation (2. 20) is point matched at locations 2i (i: l, 2, . . ., L) while equation (2. 21) is point matched at locations (xj, zk) (j : 1, 2, 3, .. . , M; kzl, Z, 3, . . . , N), and with the following definitions l I lu¥ I‘ll 153 I 1 -Sh G ( ') 82 + 2 F (2') dz' I(zi’ )‘ 0 cc ‘1" 82,2 ‘30 I (2.22) . 3 z'=h I (z., 1) = - G (2., z') -——,- F (2') 2 1 cc 1 82 l z‘=0 (2.23) f(x') 62 2 I3 (Zi, m, n) = So So Grc (zi’xi, zI) [37+ $0] Hmnbc" zl)dzIde (2.24) 8 z'=f(x') 14(21, m, n) = -S Gcr(2i,x', 2') —8-;,- Hmn(x', 2') dx' 0 2'20 h (2.25) I(x l)-S G (x z ') 82 + 2 F(2')d' 5 j: 2k: " 0 re j’ k, 2 32,2 50 I z (2.26) 3 z'=h I (x., z ,1): - G (x., z , z') , F1 (2') 6 J k rc J k 32 ztzo (2.27) b f(x') 2 u I—ggm W 7 xj, 2k, m, n - 0 0 rr xj, 2k, x ,2 82.2 Bo H n(x',z') dz'dx' (2.28) I I- S G ( z x' ') 3 8 (Xj’ zk’ m' n ‘ ' 0 rr "j’ k’ 'z 82' 2'=f(x') H (x',z‘) mn 2'20 (2-291 then equations (2. 20) and (2. 21) become L 2 A! [11(2i, 1)+ 12(2i, l]+ Z ZNBm 1:1 m=l n=l 13(21, m, n) + 14(zi, m, n)] j4nao z—go E2359 forizl, 2, 3, ...L (230) 154 L M Z A! 15(xj, 2k, 1) +16 (xj, 2k, 1) +2 [:1 m: 17(xj,2k, m,n)+ 18 (xj,2k,m,n)] = 0 for) 1,2, 3,...,M k:l’2,3.ooo.N. (2.31) N 2 B mn ln=l 11 These two expressions form a set of L 4‘ MN simultaneous, linear alge- braic equations. The L + MN unknown coefficients A! and an of the series for the induced currents are obtained by solving this system of simultaneous equations numerically by matrix inversion. The integrals defined in (2. 22) through (2. 29) can be approxi- mated as follows and subsequently evaluated numerically: h I (2 l)-S G (2 2') 82 + ['32 F (2') dz' 1 i’ - cc 1’ 2 o 1 0 82' R 82 2 _ I I I - Z S Gcc(zi’ 2) 82.2 + (30 F1 (2 ) dz r=l (A2)r R 2 éZ G (2 2 ) ——-2-8 + B2 F (2') AZ cc i' r 82' o 1 z'-2 r11 ' r r i - A2 JAzZ 2 Z —- + T-I- a + ——7—8 + [32 F (2') Zln --Z 82' o I 2'=2. a . e"2j‘30d 1: ' “304” ' TAZ + See “Y “Q A” (2.32) 2'=h I ...L I 12(zio I ) = - GCC(zi' 2) azl Fl (2) 21:0 (2. 33) 155 32' 0 T S .2 Z 5 5 G 2...... Cl' 1 t: 1 8:1 (Ax)t (A2)s b f(x') 2 1 (21. m, n) =5 S Gcr(zi’ x', 2')[ 8 2 +£3ch Hmn'x" 2') dz'dx' 82 2 —_T + (30 Hmn(x',2') dz'dx' T s 2 1 3 Z AxAz -- Z Z Gcr'zi'xt’zs)l: 3 '2 + [30 Hmn(x" 2') ztzz t: 1 z S b 3 2':f(x') 14(21, m, n) - 0 Gcr(zi’ x', z') 72,— H n (x', 2') dx' T 8 2':f(x') : - Z S G (z.,x',z') , H (x',z') dx' cr 1 82 mn 2,:0 l ZI— .. G r(2 ,xt,f(xt) 82' Hmn(xt’ 2') L“ ) Ax 2‘ "t (2.35) h 2 I(x z I):S G (x z 2') 8 +le F (2') dz' 5 j’ k’ 0 rc j’ k' 82,2 0 1 S ~25 G(xzz') 32+2F(')d' _. rc J" k' 82.2 (30 l 2 2 5:1 (A2)S S 2 22 G (x z 2) —-§——+(32 F (2') AZ rc j’ k’ s 82'2 o l 2,22 3:1 5 (2.36) 156 I __3__ I 16(xj,zk’l):-Grc(xj,z’Z) BZ'F1(Z) " z'=0 (2.37) b f(x') I x.,2,m,n : G x.,2,x',z' 7 large : \ Rl I reflector I _. l S I rl r : I \N‘ I I I : \\\ R3 I I \\ :1 9‘ I \3. ' dipole ' small \\ reflector ' \ \ I \ I \ I \ \ I ‘ s] Figure 3. 2 Geometry for radiation field calculation. 161 3. 2. 1 Current Excited on Image Plane by Dipole As shown in Figure 3. 1, R1 is the distance between an arbi- trary source point along the dipole axis and a field point at any point in space. Thus the vector potential at any point in space maintained by the dipole current can be obtained easily as A (x, y, z) = ’2 It}: Sh Iz(2') s—J-fi—(PE-l— dz' R1 (3.1) where u o is the free-space permeability and 1/2 R1: X2 + (y-d)2 + (z’z')z : Rl(x9 Y: Z, Z') By exploiting the symmetry of the current induced in the dipole, Iz(z') -.-. IZ(-z'), the scalar component of equation (3. 1) can be rewritten as (.1 L . '10 :T Z Iz(zl)kl(x,y,z,2l)Az 1:1 (3.2) where L is the number of partitions of the dipole, A2 : h/L, and -' I -_ - ' k (x y z z.) = e JBOR1(X.Y.2, z) + e 130R1(x,y,z, 2) 1 , , ’ Rl(x, y, z, z') R1(x’ Y, Z, -Z') . (3. 3) Applying the definition of the vector potential function A F1 : l VXX ”o (3.41 and the current excited on the image plane by the dipole can be obtained as 162 —> A —§ Ksl(x,z):an-I y:0 :‘J—QX(VXA) y=0 H'o 1A A 8 A8 :— xx-—-A - —-A (Loy (8y 2 Yax 2) y:0 "‘o y y=0 L 4 A l 1( )ilu A _-z 4“ 22! 3y lx,y,z,zl -0 2 . [:1 y“ (3.5) 3. 2. 2 Current Excited on Image Plane by Small Reflector As shown in Figure 3. 1, R2 is the distance between an arbitrary source point on the small reflector and a field point at any point in space. Thus the vector potential at any point in space can be obtained easily as H -J'I3 R 4° S K (x',z') e R° 7‘ dS' " R ”z 2 (3.6) A(x,y,z)=9 where R is the surface of the small reflector and 1/2 R2 = [(x-x')2 + (y-c)Z + (z-z')2] : R2(x, y, 2, x', 2') By applying the symmetry prOperties of Krz’ equation (3. 6) can be re— written as “ b f(x') K0935 21:9 4: S S Krz(x',z')k2(x,y,z,x',2') dz'dx' O 0 M N LA P-o K ( z)k(x 2x Z')AxAz ”Z41: rzxm’n Z’Y"m'n m=l n=1 (3. 7) where M is the number of partitions of the small reflector in the x-direc- tion, N is the number in the 2-direction, Ax = b/M, A2 : f(xm)/N, and 163 e-jfioszc, y, 2, x', z') + e-jfioR2(x, y, 2, x', -2') RZIX. y. z, X'. 2') Rabi. y. z, X'. 4') I I _ kZ(X.y.2.X.2)- _’ _ I l _’ _ _ I + e JBoR2(x'Y’z' X.2) + e JflORZ(X.y.2. 2". 2) R2(xn Y. Z, -X', Z') Rz(xo Y. z! -X., '2') (3. 8) The current excited on the image plane by the small reflector is therefore obtained as M N E(XZ)-gl 2: 2 K( I-a—k(x ) AA S 2 ’ " ' 4" rz xmp Zn 8y 2 , y. 2, Xm, Zn -0 x y m=l n:l ' (3o 9) 3. 2. 3 Total Image Plane Surface Current The surface current excited on the image plane by the dipole and small reflector is the summation of the component currents in equa- tions (3. 5) and (3. 9). Since the images of the dipole exciter and the small reflector excite exactly the same current on the image plane as their real counterparts, then the total induced current is L -> A 1 8 KS (x,z) : -z 2" Z 12(zl)—8y k1(x, y, 2, z‘) -0 A2 1:1 Y- M N 8 + Z Z Krz(xms Zn) .5;- k2(x0 Y. Z, xm! Zn) _0 AKA z m=l n21 Y‘ (3.10) 3. 3 Radiation Field Calculation The total radiation field is found by superposition of the fields maintained by all of the currents in the simplified short-backfire antenna model: the dipole exciter current and the surface current distributions on the small and large reflectors. In order to numerically calculate the radiation field in terms of the various current distributions, the dipole exciter is again partitioned into a number of subsections, and the two reflectors are partitioned into a number of rectangular subareas. Over each partition the induced current is assumed to be approximately 164 constant and equal to its value at the center of that partition. 3. 3. 1 Radiation Field Maintained by Truncated Image Plane (Large Reflector) Surface Currents: The vector potential at any point in space maintained by the truncated image plane with its center situated at the origin of a spher- ical coordinate system, as shown in Figure 3. 2, is given by ~13 R .1 u A(r,6,¢): 4: S 2K8z(x',z') i—lf—J- dz 'de (3 11) s 1 . where S is the surface of the truncated image plane, and R118 the distance between a source point on the large reflector and the observation point. By applying the conventional far-zone approximations R1 5 r . . . for amplitude terms Rlér-Q. F'2r-(x'coscbsin8 +z'c038) forphase terms equation (3. 11) becomes -j(3 r-(x'cos ¢sin8 +Z'COSBJ e O dx'dz' . _, u A(r.3.¢)=-4—:—SS QKSZW'J') r (3.12) . . . A A A . Usmg the coordinate transformation 2 : rcosO -8 Sine , the vector po- tential at any point in space maintained by the large reflector currents takes the final form -JBor X(I,9,¢)Z'S—_ jfio(x'cos (I) sine +z'cos 9 ) 4'II'r S ’rcosGK (33,2'18 S 82 - 3 sine K (x' z') ej(30(x'cos¢sin8 +z'cose) dx'dz' , sz ’ (3.13) From the vector potential definition B : V x-A , the magnetic field maintained at any point in space by the large reflectoris given by 165 B+ z'cosfl) R rz ' dx'dz' h . , . . +5 I (z')eJBo(z cosO+d Sine sm¢) dz' . -h z (3. 20) The E-plane (y-z plane of <1) : TT/Z) radiation factor is defined as FE(9 , w/Z) : sine S Ks (x', z') ejfioz'cose dx'dz' S 2 +5 K (X,,z,)ejflo(z'cose+c sine) dx'dz' R rz h +5 12(2.) ejpo(z'cose+d sine) dz, -h ° I ' I : Zsine {S KsZ(x" Zn) (eJfioz cose+ e-Jfioz c089) dx‘dz' S l . . . , _. , +5 Krzh“, z') eJBoc Sine (eroz cosB + e Jfioz c059) dx'dz' R l h . . . , _. , +5 Iz(z') eJBod Sine (83502 cosG+ e Jfloz c056) dz' 0 I J i Z sine Z 2 K (x,, z.) (e‘l‘sozj(:()se+e‘lfioC 81116) Ax.Az. . 32 1 J 1 J 1: l jzl K L jB c sine (jB z cosB -j[3 z c051) + e 0 Z Z Krz(xk'zl) e o l + e o I AxkAzl k=1 [:1 K + eJBOd Sine Z I (z ) (ejpozmcose + e-JBozmcosB) A2 2 m m me] (3. 21) 168 where S1 is the first quadrant of the large reflector and R1 is the first quadrant of the small reflector; I is the number of partitions of the large reflector in the x-direction, J is the number in the z-direc- tion, Axi = w/I, Azj : f(xi)/J; and K is the number of partitions of the small reflector in the x-direction, L is the number in the k = b/K, A2! = f(xk)/L; and M is the number of par- titions of the dipole, Azm = h/M. The H-plane (x-y plane of 9 211/2) radiation factor is defined similarly as z - dire ction, A x F H,(1r/Z (b) :SS Ks z(,x' z') ejpo x C084, dx'dz' + SRK (x', z') ejBo(x'cos+ c sintb) dx'dz' rz h +3 I (z') ejfiod sincb dz' -h z = 2 S K‘z(x', z') (ejfiox'ms‘l’ +e'jfiox'°°s¢) dx'dz' s 1 + SR Know. 2') emoc 3i” (ejflox'C08¢+ e-jpox'cos¢)dx.dz. 1 ~ h +3 I (z') erosimb dz' 0 z I J ' x. scb -' x.COS¢) Z Z Z Ksz(xi' zj) (eJfio 1C0 + e 3‘30 1 AxiA zj i=1 j=l . K L +8Jp°csm¢ Z Z Krz("k’ 21) k=l [=1 (ejfio xkcos¢+e ~jfl oxkcoscb) AxkAz1 + ejfio d sin¢ 21:11 :11 (3. 22) 169 A computer program for calculating the surface current on the large reflector induced by the currents of the primary dipole exciter and the small reflector was develoPed. This program subsequently determines the E-plane and H-plane radiation patterns of the sim- plified short-backfire antenna model in terms of the known currents on each of its components. A listing of this program is included in Appendix C. A complete discussion of the numerical results is included in Chapter 4. CHAPTER 4 NUMERICAL RESULTS 4. 1 Introductory Remarks A theoretical-numerical solution for the currents excited on a short-backfire antenna was carried out in Chapter 2 by expanding the induced currents Iz(z) in the dipole and Krz(x’ 2) on the surface of the small reflector in efficient series as Iz(z): Z A! sin 21h" (h-z) M N -1 Krz(x, z) :2 Z anxm sin g—f—I-(x) EX)- %. m=l n=l Based on these induced currents, the surface current excited on the large reflector as well as the radiation field maintained by all of the induced currents on the simplified short-backfire antenna model were calculated as described in Chapter 3. In the present chapter, numerical results for the current distributions excited in the dipole primary radiator as well as on the small and large reflectors, the in- put impedance to the dipole exciter, and the radiation fields of the short-backfire antenna are presented. An antenna having a square reflector geometry was initially studied to simplify the numerical work. The computer program deveIOped to implement the numerical solution was later generalized to handle arbitrary reflector geome- tries with a boundary contour described by 2m“: f(x). A circular short-backfire antenna structure was subsequently considered. In the investigation of an antenna with square reflector geome- tries, two numerical solutions are studied in which the small reflec- tor current is approximated by the double series truncated after 9(M: 3, N23) and 15(M=5, N:3) terms, respectively. Although the 15 I70 li'llluiilj‘l‘lullil 171 term current series is undoubtedly the more accurate of the two, the 9 term solution is found to result in radiation patterns which agree more closely with those patterns measured for a circular short- backfire antenna. This result can be explained as follows. The 15 term series can successfully predict the (mathematical) edge sin- gularity in the surface current excited on the small square reflector, while the 9 term series does not. Since the edge singularity is much less pronounced with a circular reflector geometry, the relatively smooth current obtained with the 9 term series provides better agree- ment with the measured results. Only the results obtained with the 9 term series for the current on the small, square reflector are dis- cussed in this chapter; the trigonometric series for the induced cur- rent Iz(z) in the dipole exciter is truncated after 5 terms (L:5). A detailed study of the circuit and radiation properties of the simplified, rimless short-backfire antenna model with circular re- flector geometry is made in this chapter. In obtaining these numer- ical results, a five-term series for the dipole exciter current (L:5) was utilized, while the series for the surface current on the small, circular reflector was truncated after twenty terms (M25, N=4). The results predict radiation patterns in very close agreement with those of the of the AFCRL experimental measurements. It is found that the numer- ical -physical Optics method can successfully predict both the radia- tion field patterns of a short-backfire antenna and its behavior as a circuit element (input impedance). The typical dimensions of Ehrenspeck's experimentally opti- mized short-backfire antenna model are length of dipole primary radiator : 2h = 0. 5 X0 distance from dipole to large reflector : d = 0. 25 x0 distance between reflector plates : c = 0. 5 x0 diameter of large reflector : w = l. 0 X0 diameter of small reflector : b = O. 2 k0 dipole antenna thickness parameter : $2 = 2 1n (Zh/a) = 10. 0 where a is the dipole radius and X0 is the free-space wavelength. The experiment was conducted at a frequency of 3. 0 GHz. These parame- ter values were adopted initially (and varied later) to obtain the nu- merical results of tie numerical-physical optics method presented 172 in the remainder of this chapter. 4. Z Induced Currents on the Antenna Structure Figure 4. 1 indicates the amplitude and phase distributions of the current Iz(z) induced in the dipole exciter. It is observed that this current distribution excited in the primary radiator is compara- ble to the essentially sinusoidal distribution of the current in an iso- lated, resonant length, cylindrical antenna. The total change in the phase of the current along the dipole is limited to about 8 degrees. The input current at the center of the dipole exciter is obtained as IO : 7. 38 x 10-3e-j75° 9 when the slice voltage generator is assumed to maintain a one volt potential difference (Vo = 1.0 eJ 0) across its gap. The amplitude and phase distributions of the surface current Krz(x, z) excited on the small reflector are demonstrated in Figure 4. Z. In this figure, the currents Krz(x:0’ z) and Krz(x, z:0) along diameters of the reflector at x=0 and z=0 are shown. The variation of the current amplitude along the z-direction is essentially sinu- soidal, and behaves similarly to the current along a cylindrical di- pole as might be expected. The current distribution along the x-direction decays relatively rapidly in amplitude. This result is not unexpected since if the reflector is visualized as a closely Spaced array of dipoles parallel to the z-axis, then the dipoles not close to the z-axis have lengths which are smaller than their resonant length and which become shorter as x is increased. A rapid decay of the induced currents excited at the centers (along the x-axis) of these array elements is therefore expected for increasing values of x. The phases of the surface currents along both directions arefissen- J ) excites the dipole primary radiator, the surface current excited at "l j990 l e tially constant. When a one volt slice generator (V() = l . 0e the center of the small reflector is 7. 0 x 10 , which is approximately 180° out of phase with that of the current at the center of the dipole. Figure 4. 3 shows the amplitude and phase distributions of the surface current Ksz(x, z) excited on the large reflector. Again, the distributions of Ksz(x: 0,2) and Ksz(x, 2:0) along reflector diameters at x=0 and 220 are indicated. This current is determined in terms II P N U1 >’ O -100 / .1 -75 z o \I I phase of Iz(z) .0 c» I IIZ(Z) relative amplitude of I (z) 9 o 4; m I I .0 w l . O Iz(z:O) : 7. 38x 10‘3’ea'175°9 0. 0 l I l l l l l l i 0 0.0 0.2 0.4 0.6 0.8 1.0 position z/h along dipole exciter Figure 4. l Amplitude and phase distributions of current Iz(z) in the dipole exciter. phase of Iz(z) in degrees Small Re fle cto r R Z O ‘3 O 0‘ relative amplitude of K (x, z) 53 U1 0.0 Figure 4. 2 174 phase of Kz(x, z:0) _, ’ .- ————-c---‘ J l l l i L l 0.2 0.4 0.6 0.8 1.0 position x/b or z/b along small reflector Amplitude and phase distributions of surface current Kz(x, z) excited on the small reflector. 150 100 50 -50 100 phase of Kz(x, z) in degrees 175 Large Reflector S o Kz(x:0, 2:0) 2 9.8 x 10'2 ”2‘8 1.0 .. 150 0.9 -— _. 0.8 - - 100 - ‘ =- A § phase of K (X, z=0) N Q 2 350 7 - e \ 3 MN \ \ phase of Kz(x:0,z) _ \ _ “50.6 § \ 50 \ 3 \‘ \ 30.5 - \ \ _. :3 \ z? \‘ \ “0.4 ~|KZ(X.z=0)| \ \ _) 0 Q) \ > \ .3 \ \ £0 3 " \ \ _( o \\ H :0 IKZ(X , z)| \\ O. 2 '_ \\ -4 -50 \\ \\ 0.1 L- _ 0.0 J 1 1 J 1 1 1 l -100 0.0 0.2 0.4 0.6 0.8 1.0 position x/b or z/b along large reflector Figure 4. 3 Amplitude and phase distributions of surface current Kz(x, z) excited on the large reflector. phase of Kz(x, z) in degrees 176 of the near-zone electromagnetic field maintained by the currents in the dipole exciter and on the small reflector (and their images) as described in Chapter 3. It is demonstrated by this figure that the currents along the x- and z-directions both decay very rapidly in am- plitude as the edge of the reflector is approached while their phases change significantly. The amplitude of the surface current near the reflector edge is only approximately 1% of its value at the reflector's center. It is therefore quite an accurate approach to assume the large reflector is infinite for the purpose of calculating the dipole and small reflector currents, and then truncate the image plane to its actual size when calculating the radiation fields. The surface 0 current excited at the center of the large reflector is 9. 8 x lO-Zejgz' 8 and is approximately 180 degrees out of phase with the current at the center of the dipole (nearly in phase with the small reflector currents). 4. 3 Input Impedance to the Primary Radiator The input impedance Zin to the short-backfire antenna is deter- mined in terms of the input current to the dipole exciter. Input im- pedance to a short—backfire antenna having a dipole primary radiator with various electrical half-lengths h/Xo are presented in Figure 4. 4. It is found that the resonant half length of the dipole primary radiator in a backfire antenna is h : O. 212 X0 (at which point Rin is approx- imately 25. 0 ohms), which is somewhat shorter than that of an iso- lated cylindrical dipole antenna (for which h : 0. 234 X0 at resonance). The resistive and reactive components of the input impedance to the dipole exciter behave in a manner which is qualitatively similar to the impedance variation of an isolated cylindrical antenna as h/k0 is varied. For the optimized AFCRL dimensions determined experi- mentally by Ehrenspeck, the short-backfire antenna input imped- ance is 42. 01 + jl44. 0. The frequency dependence of the input impedance to a short- backfire antenna with a dipole exciter is indicated in Figure 4. 5. In these results, the frequency is varied between 2. 0 and 4. 0 GHz while the antenna dimensions are fixed at values which are optimum at a center frequency of 3. 0 GHz. It is found that the antenna's input impedance is nearly 50 ohms at a center frequency of approximately 177 "" d dipole exciter I— _l. circular h b reflector T— — I - y w L 700 ’- P IS? 600 - in : Rin+ inn 45 h: 0.212). “"L‘Ho V _ o I: _ N“ = resonant half-length b ' 0' 2 )‘o ‘H 500 ,_ for which Zin= (22 + J0) d = 0. 25 X o o 5 t— C : 0. She g / o. Q=Zm(2ha)=10.0 g 400 ‘ o o o _ .3 Xin \ g 300 r o H H) ‘4 r— o o .2 200 - R. z; x. m \ '8 in e _ .. ( ) 100 )- 0 l V i 1 1 0.15 0.20 0.25 0.30 0.35 electrical half-length h/ko of exciting dipole Figure 4, 4 Input impedance to a backfire antenna with a dipole exciter for various exciter electrical half-lengths h/xo. 400 300 (ohms) In 200 100 resistive or reactive component of Z. -100 -200 Figure 4. 5 178 dipole excite r circular h reflector T— _ w Dimensions at Center Frequency of 3.0 GHZ w: 1.0)\ b l 3.0 3.5 4.0 excitation frequency (GI-Iz) Frequency dependence of the input impedance to a short-backfire antenna with a dipole excite r. ‘1) 179 f : 3 GHz. Again, the behavior of the input impedance as a function of frequency is qualitatively similar to that of an isolated cylindrical dipole. The results described above confirm an important experimen- tal observation regarding the circuit preperties of a short-backfire antenna. It is found that the input impedance to the antenna is deter- mined primarily by the characteristics of the primary radiator and only secondarily by the remainder of the antenna structure. A broadband short-backfire antenna, for which the desirable radiation field patterns remain essentially unmodified, can therefore be im- plemented by selecting a primary radiator whose pattern is essen- tially dipolar but for which the impedance is relatively frequency independent 4. 4 Radiation Fields Maintained by the Induced Currents The E -plane (y-z plane) and Iii. -plane (x-y plane) radiation pat- terns, which were calculated for simplified short-backfire antenna models with both circular and square reflector geometries according to the eXpressions developed in Chapter 3, are presented in Figures 4. 6 and 4. 7 for the indicated dimensions. These radiation patterns are compared with those obtained experimentally for an optimized short-backfire antenna (Ehrenspeck's experiment at AFCRL3) and very satisfactory results, even for the backlobe level, are obtained. A twenty-term series was used to calculate the current excited on the small circular reflector (leading to an input impedance of Zin : 42. 01 + jl44. 0 ohms), while a nine-term series was utilized for the case of a square reflector (Zin = 39. 8 + jl42. 1 ohms) as discussed in Section 4. I. Since the nine-term series is inadequate to represent the mathematical edge singularity of the current on the square reflec- tor, it should be expected to act similiarly to a circular reflector. This observation is confirmed by the numerical results of Figures 4. 6 and 4.7, which show that both the radiation fields and the imped- ances obtained for the two geometries are very similar. In Figures 4. 8 and 4. 9 the F: -plane (y-z plane) and if -plane (x-y plane) radiation patterns of the simplified, circular short-backfire antenna model, obtained by the numerical-physical optics approach, z 180 P—C —- —-( d dipole exciter circular _ reflector I Fi—‘l \ :1.0>.0 y w h:d=0.25)\ o b:0.2)\ o c:0.5>\ o 52 : 21n(2h/a) 2: 10. 0 AFCRL sho rt-backfire antenna Numerical solution with 20 term current series for CIRCULAR reflector (Zin = 42. 01 +j 144. 0 ohms) Numerical solution with 9 term current series for SQUARE reflector (Zin = 39. 8 + j 142. 1 ohms) Correction for field diffracted by rim edge added to (---—); the major lobe is not changed relative amplitude of radiation field in dB -35 F- -40 1 1 1 L 1 1 1 1 1 0 20 40 60 80 /100 120 140 160 180 angle from axis in E-plane in degrees Figure 4. 6 E-plane (y—z plane) radiation patterns of a short-backfire antenna obtained by the nume rical-physical optics approach. 181 'P— C 4 d ipole exciter 1— .1 circular ‘ h b - reflector T— I y w y o h : d = 0. 2.5 X o b = 0. Z X o c : 0. 5 X o 52 = 21n(2h/a) = 10.0 A FC RL sho rt -backfi re antenna ...... Numerical solution with 20 term current series for CIRCULAR reflector (Zin : 42. 01 + j 144. 0 ohms) Numerical solution with 9 term current series for SQUARE reflector (Zin = 39. 8 + j 142. 1 ohms) Correction for field diffracted by rim edge added to (----); the major lobe is not changed \ \J -35_ \ / \l -40 1 1 1 1 1 L 1 1 1 0 20 40 60 80 100 120 140 160 180 angle from axis in fi-plane in degrees relative amplitude of radiation field in dB Figure 4. 7 FI- plane (x-y plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach. 182 4:4: dipole exciter I LI. 1 circular reflector T— W \ , E-Plane \ w = 1' 0 )‘o Y b = d z o. 25 x O 0 s c : 0.5 ho %\ QzZln(2h/a)=10.0 -5 __ \ .\ \\ -10 _ \ \\ \ \\ -15 _ "a b = 0.30 x Q \ \ 6b = 0.25 x: .5 \\\ // . = o 210 E ‘ ‘ ’/ H 5 H \ ,’—\ /I E3 ‘2 ~ \\ \ I \. ‘ 1’ 3 1 / '1 .._. \ ‘ 'U _ I 3 3O "’ \ \ ' I" I ’b -_- 0.15 x0 1.8 \ \l I \\ / q, -35 __ \\ /’ \ I "o \ I» I :5 \ :2 ‘1 I \ / g. -40 _ \ ‘1' (U \ A\ ‘y \ 3 -45 __ ‘ \ .5 \ \. I ‘* -50 1 1 1 1 1 L 1\ ,1 4 0 20 4O 60 80 100 120 140 160 180 angle from axis in E -plane in degrees Figure 4. 8 E -plane (y-z plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical Optics approach for various small reflector radii b. 183 c 17 dipole exciter LI. 1 circular reflector I urn c:0.5)\o Q=2!n(2h/a)=l0.0 b=0.3X o -10 m -15 'U .5 T, -20 ...4 O a E -25 35 '3 -30 H ‘H O o -35 'U 3 "a3. 40 E «I Q) 3 -45 ._ ‘13 H O H _50 1 l l l I l l l J 0 20 40 60 80 100 120 140 I60 180 angle from axis in II -plane in degrees Figure 4, 9 II -p1ane (x-y plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical Optics approach for various small reflector radii b. 184 for various small reflector radii "b" are demonstrated. It is evi- dent that the optimal radius for the small reflector is b = 0. 2 X0; this theoretical conclusion is in agreement with the Optimum reflec- tor radius determined experimentally by Ehrenspeck3. The frequency dependence of the 1:; -plane (y-z plane) and PI - plane (x-y plane) radiation patterns determined by the numerical- physical optics method for the simplified, circular short-backfire antenna model are indicated in Figures 4. IO and 4. 11. It is ob- served that the most desirable field patterns, with minimum beam- width and sidelobe levels, are obtained at the 3. 0 GHz design center frequency used by Ehrenspeckz. The 5 ..plane patterns are rela- tively broadband while the PI -plane patterns degrade more rapidly as the excitation frequency deviates from 3. 0 GHz; the later sensi- tivity is due to the fact that the directivity of the patterns in the PI - plane depends critically upon the relative amplitudes and phases of the currents excited in the various antenna components. The results of the numerical-physical optics solution for a simplified, circular, short-backfire antenna model, as indicated by the radiation patterns in Figures 4. 6 through 4. ll, predict a very close agreement between the theoretical major lobe shape (and beam- width) and the AFCRL experiment. Likewise, the E ~plane sidelobe structure and the backlobe are quite accurately predicted. It appears that the only major shortcoming of this theory involves its prediction of the relatively large —15dB P—I -plane sidelobe, which is not observed experimentally. This sidelobe evidently arises due to the neglect of the large reflector rim in the simplified model of the short-back- fire antenna. With the induced currents excited in the dipole and on the surfaces of the small and large reflectors known, however, a diffracted field correction, based on the geometrical theory of diffrac- tion, can be made to account for the presence of the rim. The calcu- lation of an expression for this correction term and the computation of numerical results have been carpleted in a complementary but sep- arate investigation. This diffracted field correction leads to greatly improved radiation patterns in which the major modification is a 5 dB reduction in the I? -plane sidelobe level. Numerical results for the total radiation field patterns, including the effect of the diffracted fields are indicated in Figures 4. 6 and 4. 7. ~10 ~15 ~20 ~25 ~30 ~35 ~40 ~45 relative amplitude of radiation field in dB ~50 185 dipole exciter ! ! h circular -——.-y reflector r- I i; x Y w=l.0)\ o \ h=d=0.25)\ Y X 0 b:0.2 o c=0.5)\ s‘ o \ f:2GHz 9:21;, 2h)=10.0 \ a \ f=4GHz \ \\ “f23GHZ \\ I ‘ \ \ /l /\ I“ I’- I J ‘ ’ \a ’ \ \x I a l l l l l l l J A 20 40 60 80 100 120 I40 160 180 Figure 4, 10 E ~plane (y-z plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach for various excitation frequencies. 186 1’. ‘\ dipole exciter LI. 1 J}. _ h Circular reflector —-———.y 1*“1 \ W: 1.0 K0 \ h 2 d = 0.25 X y O b: 0.2 X o c = 0.5 RC 0 \\-~ 52: 21n(——azah 210.0 -5 _. \ ‘\ /‘=4GHZ \ \ \ \ -10 _ CD -15 )— \ \ '° \ I: \ .,4 \ \ E -20 1— ‘ \ é) \ /’\ C ‘ I .2 -25 1— ‘ / i; v '6 f: 3GHz «1 ~30 1.. L4 “-0 o .8 ~35 ... 3 g” -40 _ a: o '5 '45 I— .‘3 o “ _50 1 1 1 1 1 L 1 1 1 O 20 40 60 80 100 120 140 160 180 angle from axis in I; -plane in degrees Figure 4. 11 PI -plane (x-y plane) radiation patterns of a short-backfire antenna obtained by the numerical-physical optics approach for various excitation frequencies. CHAPTER 5 SUMMAR Y A ND CONCLUSIONS A numerical-physical optics method (integral equation approach) is applied in a theoretical investigation of the circuit and radiation properties of a short-backfire antenna. The currents excited in the dipole primary radiator and on the surfaces of the small and large reflectors of the backfire antenna are determined through this theo~ retical-numerical solution, in terms of which the radiation field main- tained by the antenna as well as its input impedance are subsequently evaluated. These analytical results are found to compare favorably with the experimental observations of earlier investigators. The EM boundary value problem is formulated theoretically, in terms of a coupled set of integral equations for the unknown currents induced in the dipole exciter and on the surface of the small reflec- tor, and solved numerically in Chapter 2. In this approximate study, the rim of the large reflector is neglected. For the purpose of deter- mining the currents excited in the dipole or on the surface of the small reflector, the large reflector is approximated as an infinite conducting sheet such that the method of images can be applied. A numerical solution is implemented by eXpanding the unknown induced currents in efficient trigonometric series and point matching the inte- gral equations to evaluate their coefficients. Chapter 3 presents the calculation of the total radiation field maintained by the currents in the dipole and on the small and large reflectors of the simplified, rimless short-backfire antenna model. The current excited on the large reflector is first approximated by a truncated form of that current induced on an infinite ground plane by the near-zone field of the currents in the dipole and on the small re- flector (as well as their images). The total radiation field of the sim— plified antenna model is then determined by superposing the radiation 187 188 fields maintained by the known induced currents in the dipole primary radiator and on the surfaces of the small and large reflectors. All of the numerical results calculated by the numerical- physical optics method are collected in Chapter 4. These results include the amplitude and phase distributions of the currents excited in the dipole primary radiator and on the surfaces of the small and large reflectors as well as the E-plane and H-plane radiation fields maintained by a short-backfire antenna and its input impedance. The theoretical radiation patterns are compared with those measured experimentally. It is found from the numerical results that the surface current on the large reflector decays very rapidly as its edge is approached. This result consequently justifies the approximate technique used in Chapter 2 to calculate the currents in the dipole exciter and on the small reflector. The nume rical-physical optics method, which is based on an integral equation technique, has the advantage that it can accurately predict the circuit characteristics (input impedance) of a short- backfire antenna; prior investigations have been relatively approximate in this regard. It is found that the input impedance to a short-backfire antenna depends upon the exciter length and its excitation frequency in essentially the same manner as does that of an isolated cylindrical dipole. This analytical result agrees with the experimental observation that the circuit characteristics of a short-backfire antenna are primarily dependent upon the configuration of the primary radiator and only secondarily upon the details of the remaining antenna structure. A broadband primary radiator might therefore be adopted to implement a short-backfire antenna which is relatively frequency independent. It is found that the predicted radiation field patterns maintained by the induced currents in the dipole and on the surface of the small and large reflectors (calculated by the numerical-physical optics method) of the simplified, rimless short-backfire antenna model agree very favorably with those measured experimentally. The theoretical and experimental results for the shape of the major lobe, the backlobe level, and the E—plane sidelobe structure compare very closely. The only major shortcoming of this theory involves its prediction of the 189 relatively large -15dB H-plane sidelobe, which is not observed experimentally (-25 dB level). This sidelobe evidently arises due to the neglect of the large reflector rim in the simplified model of the short-backfire antenna. It has been found that this sidelobe is greatly reduced when the field diffracted by the rim edge is included. It is furthermore found that the numerical-physical optics solution correctly predicts an optimum small reflector radius of b = 0. 2X0 to achieve minimum beamwidth (maximum directivity) with a minimum sidelobe level; this agrees exactly with the optimum radius determined experimentally. Finally, the theoretical-mime rical solution indicates a moderately broad frequency bandwidth for the radiation patterns, which again conforms with experimental observations. In conclusion, the numerical-physical optics method appears to provide an accurate means to analytically investigate both the circuit and the radiation properties of a short-backfire antenna. It leads not only to radiation patterns which compare well with experi- mental measurements but also to an accurate prediction of the input impedance to the antenna. R (‘1'0 rcn (‘0 s (l) H. W. Ehrenspeck, "The backfire antenna: new results, " Proc. IEEE _5_3, 639-641 (June, 1965). (2) H. W. Ehrenspeck, "The backfire antenna, a new type of direc— tional line source, ” Proc. IRE :18, 109-110 (January, 1960). (3) H. W. Ehrenspeck, "The short-backfire antenna, " Proc. IEEE _5_3, 1138-1140 (August, 1965). (4) K. M. Chen, D. P. Nyquist and J. L. Lin, "Radiation fields of the short-backfire antenna, " IEEE Trans. Ant. and Prep. AP-l6, 596-597 (September, 1967). (5) M. H. Hong, D. P. Nyquist and K. M. Chen, "Radiation fields of Open-cavity radiators and backfire antennas, " IEEE Trans. Ant. and Prep AP-18, 813-815 (November, 1970). (6) E. D. Nielsen and K. PontOppidan, "Backfire antenna with dipole elements, " IEEE Trans. Ant. and Prop. AP-l8, 367-375 (May, 1970). (7) T. Z. Ilsish, D. P. Nyquist and K. M. Chen, "The short-backfire antenna: a numerical-physical optics study of its characteristics, The 1971 USNC/URSI-IEEE Spring Meeting, Washington, D. c. (8) H. P. Neff Jr. , C. A. Siller and J. D. Tillman, "A simple approx— imation to the current on the surface of an isolated thin cylin- drical center-feed dipole antenna of arbitrary length, " IEEE Trans. Ant. and Prep.AP-18, 399-400 (May, 1970). (9) T. Z. Hsieh, D. P. Nyquist and K. M. Chen, ”The short-backfire antenna- A numerical-physical optics study of its characteristics, ’ Scientific Report No. 2 for Contract No. Fl9(628)-70~C-0072 with Air Force Cambridge Research laboratories, July 1971. 190 APPENDIX A APPROXIMATE EVALUATION OF THE DEFINITE INTEGRALS The double integral z+Az/2 x+Ax/Z dB R 11 = e R0 1 dx'dz' z-Az/Z x-Ax/Z 1 (1) 2 1/2 where Rl : [(x-x') + (z-z') ] , and Ax, A2 are small quantities (i. e. , (SOAX < < 1 and BOAz < < 1), is improper since its integrand has a singularity when x' :x and z' : z. This integral can be evalu— ated analytically by the procedure outlined below. Since (30R < < l in this case, by expanding the exponential 1 into a power series and retaining only its leading two terms, equation (1) can be approximated as Z+Az/Z x+Ax/2 115 S 1' JpoRl dx'dz' z-Az/Z x-Ax/Z R1 z+Az/Z x+Ax/Z ..- —R1—- dx'dz' - ijAxAz z-Az/Z x-Ax/Z 1 (2) If the change of variables uzx-x' and v:z-z' is made, equation (2) can be rewritten as Az/Z Ax/Z dudv 1- S -A z/z -Ax/Z u2W2 -J°(3 AXA z 0 Az/Z Ax/Z II A dud" — jpoA xAz O 0 N] u2+vz (3) 191 192 AZ/Z Ax/Z Az/Z uzAx/Z dudv S 1 ( / 2 z) = n u+ u + v dv 0 0 Z Z O u + v u = 0 Az/Z Ax Z : In —— ‘1' AX (1" SO 2v /1 +sz Az/Z = S sin h.l g—vx dv O (4) Let Ax/Zv : y, then dv = szdy/Ax = -Axdy/Zy2 and equation (4) becomes Az/Z Ax/Z Ax/Az dudv : ”A? 5 12 sin h-ly dy 0 0 Z 2 on y u + v Ax/Az =_Ax --l—sinh'ly-znl”‘j12+l 7- Y Y m 2 _ Az Ax Ax - 2 In __Az + «K—J) +1 2 Ax Az ,/ A2) + Z In Ax Ax +1 (5) The final approximation for integral I1 is therefore of the form 2 2 , Ax /Ax Az ’(Az) Il - 2A2 In _A2 + (A—z) +1 + ZAyln _Ax + _Ax +1 - jfioAxA z. (6) 193 The inte g ral +A 2 . z Z/ e-JBORZ I2 = R dz' z-Az/Z 2 (7) 2 2 1/2 where R2 = [(z-z‘) + a ] ,and a and Az are small quantities (i. e. , Boa < < l and (30A 2 < < 1), has a sharp peak in its integrand when zzz'. This integral can be approximated analytically by the procedure outlined below. Since fiORZ < < 1 in this case, by expanding the exponential into a power series and retaining only its leading two terms, Equation (7) can be approximated as z+A z/Z . 1 . 1 : dz' - 35 Az z—Az/Z R2 0 ° (8) If the change of variable x:z-z' is made, Equation (8) can be rewritten as Az/Z 1 -= d" jp AZ 2 7 - o -Az/Z ’xZJraZ Az/Z : 2 d" -jp AZ 0 2 2 0 x +a Az/Z ’2 2 . : Zln (x+ x +a -JBOAz O 2 2 I Zln Az[2+;/Az /4+a -J'(30Az (9) The final approximation for integral I2 is therefore of the form 2 2 1 :. “n Az/Z+;[Az/4+a - jBOAz (10) A PPE NDIX B COMPUTER PROGRAM TO IMPLEMENT NUMERICAL SOLUTION FOR THE INDUCED CURRENTS (CHAPTER 2) 194 oILUZthF .afinum 195 ucudn WPZUOO .duruuxu QOFOMkai uOhumJuml l‘.‘ D .UOkouuum oflCIUUJtDQ JJ<. oUUkHUXU HOLUuLUgT x; 11 J4cf: 2(bew0HCIx HIP he FZ~CQ CZ~IUF<§12X DO ZCHFLhmelx UIF PC UFZ~OG UUULCW QC CJU—ulx Z. :Ckuunhwf Jndzw wit CCU mYLfikhkcco uC .OZIu 2» Ctkficxu UJCOHC HIP no «DefeclmDafdc .Ckbh. 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