IY L, Wm“? mm!“ I“. \m uem' rm...“- L _.____A . . ‘.. ~ ‘- ,,_ \.. u . ~ . '~ “-. ’ . x . .- .‘ ,, ;.- . ‘4 \o‘, A ‘. .\‘_ L“ ‘. - . - .‘ ‘ a 0-“--_.. ‘C. . V . A§‘._;‘ . ‘ . -‘ . \ “' . - ‘~ ~ .‘A- . ._“\E '._ .~ -‘ 5.‘.. I.. a - .'\ A .. .. ‘vn n .V ‘A. 5’ ~- \. t .- 9 ‘§ . .._ s § . v ‘ ‘\ Q ’. .': -‘. u - . . ,v. ¢_.- ‘. . O ‘§ u ~§ .\ - - .- ‘a ' O ABSTRACT A THEORETICAL AND EXPERIMENTAL ANALYSIS OF ANTENNAS IN BOUNDED MEDIA WITH APPLICATION TO ELECTROMAGNETIC PROBING By Carl Edward Grove A theoretical and experimental investigation is made of coupled linear antennas in rectangular waveguide filled with a homogeneous, linear, isotropic medium. It is assumed that the antennas do not span the waveguide, the waveguide has arbitrary terminations and the medium is known in terms of its macroscopic electrical parameters. The integral equations for the unknown antenna currents are found by applying modal analysis and the Lorentz reciprocity relation to the electric field boundary condi- tion on the antenna. First a delta function source model is assumed with both the thin—wire and thin-strip approximations made for the antenna. The integral equations are then solved numerically by the method of moments and the thin-strip approximation is shown to exhibit improved convergence properties over the conventional thin-wire approximation. The modal series in this method of moments solution are related to sampling theory for estimating optimum numerical truncation points. Besides the delta function model, the source is also modeled as a TEM mode coaxial aperture (for a single antenna only) in a parallel development. The coaxial contribution Carl Edward Grove to the fields in the waveguide is found as Bessel function series and the numerical solution is used in comparing with the delta- gap model infinite input susceptance. An experimental system is described for verifying the theory at L-band frequencies for semi- infinite waveguide and for the cavity configuration with air and dissipative media. Although verification is obtained by systematically varying all possible experimental parameters, emphasis is placed on the single antenna in a cavity. From the extensive data it is found that new and unique antenna current distributions exist which relate primarily to cavity resonance coupling (and not simply to the typical antenna height resonance). Finally, experi- mental results for distilled water, salt water solution and a sample of sewage treatment plant effluent give support to the fundamental proposition of this research, namely: that the antenna— waveguide-media system can be effectively used as a tool for environmental media monitoring. A THEORETICAL AND EXPERIMENTAL ANALYSIS OF ANTENNAS IN BOUNDED MEDIA WITH APPLICATION TO ELECTROMAGNETIC PROBING By Carl Edward Grove A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Electrical Engineering and Systems Science 1975 TO BUD ii as ,3 ..;- I n .v A- oh. I .s u» . . a b s .\. '0 Q Q . a. v. ..c a» at 3 Oh .. c u . a guy . . .v. T . .. . . In. 4. (II\ .n o n. y.— .s. v I . ‘ . .. 0.. ~\.. n.. .u. 9‘: . .I A... -\u «x A t. g . . . q. i ‘W . en. .1 ’1 v. .w. . .. ‘F. -. :— an. .r. ACKNOWLEDGMENTS From the beginning of my graduate education to the con- clusion of my research I have continually needed encouragement, guidance, criticism, and advice. For this and for the lasting in- fluence it has had on my personal and technical development, I will be forever grateful for the wise and willing counsel of Dr. David Fisher. From conception to completion the work for this disserta- tion has required not only my own devotion but also that of an experienced advisor. I am thankful for the time and effort of Dr. Marvin Siegel who ably served in this capacity. Also deserving of much credit for this research is Dr. Dennis Nyquist. His availability and patience for my endless stream of questions, as well as his knowledge of both the theoretical and experimental aspects of this problem, have become a benchmark for my own development. I aunalso indebted to Dr. Ken-Mm Chen and Dr. Robert Ball vflwse time Spent on the advisory committee helped form both the rmrpose and perSpective of this dissertation. Finally, I am anxious to recognize my wife, Carole, who has not only sustained me spiritually and physically through this endeavor, but whose hours of graphing data and copying manuscript have been invaluable in the compiling of this work. iii . . .. - T. g. I. I3 .e. v . n . . c . a . . .. 5 r.\ . a .h. c.. A L ..‘ cl. ~ 5 nu. < .‘4 pi. ‘. . .u. . a o I u n c . . ~. a . . a . s . a a a . s q s VI. » u ‘ s VI. .cJ ..u -.u sod niJ a . A. y ‘ ~p. . 44 -~ Q! ..\ q \- . . . .\ Qtv .1. add 844 ~n~ II. III. OUTLINE PREFACE INTRODUCTION 1.1 Background 1.2 Discussion of the Problem and Related Works 1.3 Method of Investigation THEORETICAL DEVELOPMENT 2.1 The Model 2.2 Boundary Conditions; Slice Generator Approximation 2.3 Electric Field in the Waveguide 2.4 Integral Equations; Reduction to Hallen- Type Equations 2.5 Cylindrical Antenna Approximations 2.5.1 Thin-Wire Approximation 2.5.2 Strip Approximations 2.6 Coaxial Gap Solution 2.7 Conclusion; Comparison with Literature NUMERICAL METHODS 3.1 Matrix Equation Representation 3.1.1 Method of Subsections Mathod of Point Matching Galerkin's Method Matrix Equations for the Strip and Coaxial Gap Solutions 3.1.5 Numerical Determination of Charge Distribution 3.2 Convergence Properties 3.3 Comparison of the Solutions wwu F‘F‘F‘ bWN EXPERIMENTAL INVESTIGATIONS 4.1 Description of Apparatus 4.2 Experimental Mathods iv Page \DJ—‘N l3 13 17 20 34 38 39 4O 48 61 66 67 69 7O 72 76 82 83 100 109 110 128 Page V. NUMERICAL AND EXPERIMENTAL RESULTS 140 5.1 Correlation of Theoretical and Experimental Input Admittance 143 5.1.1 Infinite Driving Point Susceptance 143 5.1.2 Terminal Zone Correction Network 146 5.2 Single Antenna 156 5.2.1 Waveguide with Matched Termination 157 5.2.2 Air Filled Cavity; Current Distribu— tion 172 5.2.3 Air Filled Cavity; Charge Distribution 215 5.2.4 Air Filled Cavity; Input Admittance 220 5.2.5 Cavity Filled with Dissipative Media; Current Distribution 252 5.2.6 Cavity Filled with Dissipative Media; Input Admittance 267 5.3 Coupled Antennas 275 5.3.1 Waveguide with Matched Terminations 278 5.3.2 Air Filled Cavity; Current Distribu— tion 282 5.3.3 Air Filled Cavity; Admittance 294 5.3.4 Cavity Filled with Dissipative Media; Current and Admittance 299 5.4 Special Applications to Sewage Plant Effluent 306 v1. CONCLUSION; RECOMMENDATIONS 329 BIBLIOGRAPHY 337 GENERAL REFERENCES 340 APPENDICES I DESCRIPTION AND DEFINITION OF NOTATION 342 II EVALUATION OF DEFINITE INTEGRALS 349 o O. A C 4.- 4. ; ~.-- Lu - .‘u ' . . .‘ no. O'Ia C .t .0- O I. . “°>-~ .n-o .. C. : ‘.-- ‘- - ,.v. I . .. . -. ‘14 .‘H- I. . I ' a .u,.‘ ..- .ts- "- ."'- on ‘ -’ -. \ -~ ‘ a... ,. . - 'r;"}r~ .-.__ I...w . . -- P 2 u . ' '0‘. 5 .-:"v-.. M53- .24.\ v . ‘- 3 : I . b I‘- -- . .. ..," r... ‘U "U . A ’4 .k -- a _. -_ . | 6.. . v.‘ '-' . ‘;:' Q ‘ .“. d . -- . . ‘x. . ."5 5.. , I P ‘\ 'O' § ‘- V .. ' a. . .V‘ . . ‘ “‘ . a . l~ ‘ J ‘d‘, s u c A'. ’ y “9‘. - . . .‘. . r ’ ~ a.- ‘ ~I V .- e... 6 I. § - . ... O ..“ ”"4: ' - s‘-- 9 .- ‘t. .~ (i .. I.. ' . ‘ I ‘4 :6.- ~“ . t "-. T. A“ n. ‘ r-_ r" ‘ b. .. A- . ~‘I F‘s‘ .-‘ 9 ¢_. ‘ M. 'O . .‘aL .9. D 0". a f “-1 ;._. .‘ _.. L‘. Table 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 LIST OF TABLES Input Admittance as a Function of in a Semi-Infinite Waveguide (dl 1.05 Ghz) Input Admittance as a Function of in a Semi-Infinite Waveguide (dl 1.3 Ghz) Input Admittance as a Function of tion in a Semi-Infinite Waveguide 1.05 Ghz) Resonances of Rectangular Cavity (a b = 8.255 cm, L = 60.96 cm) Resonances of Rectangular Cavity (a b = 8.255 cm, L = 60.96 cm) Input Admittance as a Function of Near a Cavity Resonance (a = 16.51 cm, d Input Admittance as a Function of Near a Cavity Resonance (a = 22.6 Input Admittance as a Function of 0.6 - 2.2 Ghz (a = 16.51 cm, (11 = Input Admittance as a Function of 0.85 - 1.9 Ghz (a = 22.6 cm, (11 = Input Admittance as a Function of 0.6 - 2.2 Ghz (a = 16.51 cm, (11 = Input Admittance as a Function of Location (a = 16.51 cm, 1.2 Ghz) Input Admittance as a Function of Location (a = 22.6 cm, 1.2 Ghz) Input Admittance as a Function of Location (a = 16.51 cm, 1.53 Ghz) Input Admittance as a Function of Location (a = 16.51 cm, 1.4 Ghz) vi Antenna Height = 2.5318, Antenna Height = 0.2511g, Antenna Loca- (h = 6 cm, 16.51 cm, 22.6 cm, Frequency 1 = 0.51) Frequency cm, (11 = 0.5L) Frequency 0.5L) Frequency 0.5L) Frequency 0.291) Antenna Antenna Antenna Antenna Page 169 170 171 177 180 223 225 227 229 231 236 238 240 242 o u i. I u‘. . A - ‘ ‘5. - 8 . . .o g I ,b. . 6 v__ . a... I - u . o c . v - . ‘ - . An..~ _. . ...,.‘ -. , O ‘ - - .u. - O V -- O c “‘h-s 3 I~~..,‘ ..-_:‘. V .._ _ a... -6..~“ cu . uv;- .4" “~"»-I .. . ;‘~..' ‘._‘I ~ ‘.I~ - .. Table 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 Input Admittance as a Function of Antenna 0.238 , 1.05 Ghz) Admittance as a Function of Antenna 0.133 , 1.1 Ghz, a = 16.51 cm) Admittance as a Function of Antenna 0.133 , 1.1 Ghz, a = 22.6 cm) Admittance as a Function of Antenna 0.133 , 1.2 Ghz) Admittance as a Function of Antenna 0.133 , 1.3 Ghz) Admittance as a Function of Antenna 0.500 , 1.53 Ghz) Height Height Height Height Height Height Admittance in Highly Dissipative Media Comparison of Theory with Experimental Literature Input Admittance in Distilled Water at Different Frequencies Input Admittance Locations Input Admittance Heights Input Admittance at 350 MHz Input Admittance at 1.0 GHz Admittance for Coupled Antennas in Semi- Infinite Waveguide Input Admittance for Coupled Antennas as Function of Separation (f = 1.3 GHz) Input Function of h Input Function of h Admittance for Coupled Antennas as 1 (1.2 GHz) Admittance for Coupled Antennas as 1 (1.3 GHz) vii in Distilled Water at Different in Distilled Water for Different for Distilled and Salt Water for Distilled and Salt Water Page 246 247 248 249 250 251 269 270 271 272 273 274 281 295 296 297 . .. 1 . c. ' u. . “I 4 ,' .,_ 4 I o- t .. D I o. u ,‘ u. ‘ . '0 . v. '- C . .a x... a u o.‘ -v- - no , O .... a. . I]: § - 1.- . .~—. ’~-. PM‘ f...t ‘-.t ‘05. 1.... th..: - .- a”: .' o " \ 5 ..oc . ‘IV .1. -o .5 —_. d -.o 1“ .r, Table 5.31 5.32 5.33 5.34 5.35 5.36 5.37 5.38 5.39 Input Admittance for Coupled Antennas as a Function of h2 (1.3 Ghz) Input Admittance for Coupled Antennas in Dissipative Media at 350 MHz Input Admittance for Coupled Antennas in Dissipative Media at 1.0 GHz Analysis of Sewage Sample Admittance for Single Antenna in Sewage (350 MHz) Admittance as a Function of Height of Single Antenna in Sewage (1.0 Ghz) Admittance as Function of Location for Single Antenna in Sewage (1.0 Ghz) Admittance as Function of Height of Single Antenna in Sewage (1.2 Ghz) Theoretical Admittance for Coupled Antennas in Sewage at 350 MHz viii Page 298 304 306 313 320 321 322 323 326 I w. .P. De L“ a. s . v1. or . s .. . A a . o > . . _ Q 9 s 3.. a o . . . . . "I. . . q _ ‘us sv. - 0. n s O“ ‘§ #3 .JJ: 55 0. 5.. € .. . . 2. a“: ... «u. at V. Q O >\ b a 0‘” ha 9“ ’k .r. .\. a. .u. .u. .3 3. 4.. . s ... n- a. ... a. .. . . u . . re a V. M. .... .u. M”. T . l. . . p c . C ..I p . u. . n c :- It. m o. .u o. u . Figure 1.1 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6 3.7(a) 3.7(b) 3.7(c) 3.7(d) 3.7(e) 4.1 LIST OF FIGURES Cutaway View of Antenna-Waveguide System Physical Structure and Geometry for the Antennas in the waveguide Geometry for Slice Generator Approximation Geometry for Application of the Lorentz Reciprocity Relation Coaxial Gap Excitation Geometry for Finding Gap Induced Fields Using the Lorentz Reciprocity Relation Pulse Function for Expansion of Antenna Current Comparison of Galerkin's Method and Point Matching Comparison of Off-Diagonal to Diagonal Elements of [S] Matrix Convergence as a Function of m Convergence as a Function of n for the Thin Wire Approximation Convergence as a Function of n for the Transverse Strip Approximation Comparison of Various Approximations Comparison of Various Approximations Comparison of Various Approximations Comparison of Various Approximations Comparison of Various Approximations Rectangular waveguide Section in Cavity Configura- tion with Antenna Measurement System in Place ix Page 14 18 23 49 53 68 75 88 90 94 97 102 103 104 105 106 112 _.-.~ . ’b “ .‘\ Figure 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 End View of Rectangular Waveguide Section with Antenna Impedance Sensing Block Slotted Antenna with Current and Charge Probes Cutaway View of Slotted Antenna with Probes Second Antenna(s) for Coupled Antenna Measurements Total Experimental Set-Up for Current Distribution Measurements Representative Measurements of Conductivity Glass Tubing Properties Representative Measurements of Dielectric Constant Effect of Charge Probe on Current Measurements Effect of Probe Orientation on Current MEasurements Effect of Infinite Susceptance on Numerical Solution Lumped Equivalent Terminal Zone Networks Functional Behavior of Terminal Zone Correction Network Correction of Measured Data via Terminal Zone Network Current Distribution for Antenna in Semi- Infinite Waveguide (h - 5 cm, f = 1.05 Ghz) Current Distribution for Antenna in Semi- Infinite Waveguide (h = 5 cm, f = 1.3 Ghz) Theoretical Current Distribution for Semi- Infinite Waveguide Configuration with Antenna Location as the Parameter Current Distribution as a Function of Fre— quency Near a Cavity Resonance (a = 16.51 cm, cl1 = 0.51.) Page 112 115 117 117 118 118 122 125 127 131 133 145 148 148 152 159 161 164 178 I .go I,._{"... c u r.- . 3‘33: 4 C ' a... "t... . Vbb§ ‘ a n I? - ‘0' v -a IV.” . hict...‘ . O ' 1 A - "o ‘0. o ‘ O 7’- do..... - . .-.. - l -"‘~l5.... - D O r In. 5-5.-..‘ . ‘~Ao. ~P . "*lso... . vv;,' b...‘.‘~ ~ ‘ ‘--0. -__ U. 1 ““c-.. . v..-. p ".‘*‘c~ ~ .. . ' I;;..~. ..- ““kwg .. a v. I--‘_. . .‘F v . “~... . v .,. _ - —" "lfi‘A“ U u I..‘_. - us“.‘ ‘ p ' ‘5. I -t . .5. x : . ..'. - E‘I‘I" .~ ~ '- . I. N.‘ I .tw‘ ,‘ A ., v“‘.’:n. - H‘ .- I» Q _ ' qt‘in‘ I v' 1 fl. "9.. . v"-.... ‘\ J “ '. fl . - . I, . J '4 A IF . -.;’N. . 1.“: ”A I. A . “ = ! ‘J.§ Q V. .0 “.:‘r;::‘ A A.“ ._ t. " "I a 3 1‘ , . J.J ‘ u. I.‘Edr:‘..‘ u \.‘ .- .;--.; a.-“:‘ ‘t I. - i a J-_ ‘ S‘. J Figure 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 Current Distribution as a Function of Frequency Near a Cavity Resonance (a = 27.6 cm, d1 = 0.5L) Current Distribution as a Function of Frequency 0.6 - 2.2 Ghz (a = 16.51 cm, d1 = 0.5L) Current Distribution as a Function of Frequency 1.1 - 1.9 Ghz (a = 22.6 cm, d1 = 0.56) Current Distribution as a Function of Frequency 0.6 - 2.2 Ghz (a = 16.51 cm, d1 = 0.2906) Current Distribution as a Function of Antenna Location (1.2 Ghz) Current Distribution as a Function of Antenna Location (1.4 Ghz) Current Distribution as a Function of Antenna Location (1.53 Ghz) Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.05 Ghz) Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.1 Ghz) Current Distribution as a Function of Antenna Height (a = 22.6 cm, 1.1 Ghz) Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.2 Ghz) Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.3 Ghz) Current Distribution as a Function of Antenna Height (a - 16.51 cm, 1.53 Ghz) Charge Distribution as a Function of Frequency ((11 = 0.5 , h = 6.25 cm) Charge Distribution as a Function of Frequency (d1 = 0.29 , h = 6.25 cm) Theoretical Input Admittance (Susceptance) as a Function of Frequency Near a Cavity Resonance (a = 16.51 cm, (11 = 0.56) Theoretical Input Admittance (Susceptance) as a Function of Frequency Near a Cavity Resonance (a a 22.6 cm, d1 = 0.5L) xi Page 181 183 186 188 197 199 201 206 207 208 210 212 214 217 218 224 226 .-': in. .. or.__".,. . \ ' Au».n>bo ,ron . a. .- <0- . r r“1 susq‘“ ’ ..I v-I~';".- . ..-..\ .t. .u» . ‘0 ‘OI-O ' r r . .“-.".. . H .- l ‘ I-\':‘DO. p .h.‘ .u- .«5‘. - ".: .. ’.r v . .s-ugs... .H ‘ .-~-'-o.- ‘ v u 1.. bush-u..- o |.O..-- p a 5L..-.... _ "' .- ‘ .-' ' I. P '5 .“ “‘-h~~A. O ...'F‘- o a . I‘I-v..t... fl '1. , .- ‘ h-. .._~ . p '0“ ..u .”_‘.‘.‘\ o 1' -n.._ "‘ a ‘h-I‘..o.. u I.. — ’;‘V;..- ..o ...._. “ . h- '0 . “o "";»o o ¥¢C.,“‘ . v ‘- - .1"; -- u‘““ w- ' ‘O. I.. a F a V .§‘§“-. .. . -.‘ -v y- . visa...“ .~ In ., u. ‘ .r';hb Ult|~.\‘-. .C. .. . .._' ’.-. - U“ P ‘QI. ~ Q .. ...-' I‘..: ”u. §~ .‘ v.- I n“ . 'v . Ba.‘v". \ u‘ v ‘ I 4.: ‘9. ”a. P "In. A“".~ '. P Figure 5.26 5.27 5.28 5.29 5.30 5.31 5.32 5.33 5.34 5.35 5.36 5.37 5.38 5.39 5.40 Theoretical Input Admittance as a Function of Frequency 0.3 - 2.2 Ghz (a = 16.51 cm, d1 = 0.5L) Theoretical Input Admittance as a Function of Frequency 0.85 - 1.9 Ghz (a = 22.6 cm, d1 = 0.5L) Theoretical Input Admittance as a Function of Frequency 0.6 — 2.2 Ghz (a = 16.51 cm, (11 = 0.29L) Theoretical Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.2 Ghz) Theoretical Input Admittance as a Function of Antenna Location (a = 22.6 cm, 1.2 Ghz) Theoretical Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.53 Ghz) Theoretical Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.4 Ghz) Current Distribution in Highly Dissipative Media Comparison of Theory with Experimental Literature Current Distribution in Distilled Water at Different Frequencies Current Distribution in Distilled Water at Different Locations Current Distribution in Distilled Water for Different Heights Current Distribution for Distilled and Salt Water at 350 MHz Current Distribution fpr Distilled and Salt Water at 1.0 Ghz Driven Antenna Current for Coupled Antennas in Semi—Infinite Waveguide (f = 1.05; 1.3 Ghz) Driven Antenna Current for Coupled Antennas as a Function of Separation (f = 1.3 Ghz) Driven Antenna Current for Coupled Antennas as a Function of hl (1.2 Ghz) Driven Antenna Current for Coupled Antennas as a Function of h (1.3 Ghz) 1 xii Page 228 230 233 237 239 241 243 256 259 261 262 264 266 280 285 288 290 . . \ y... ‘1 '4 '. t'..‘~u. . .. . .-.. ‘. \- n- ...-.4 vo‘qo . u .- 5...-..5 . ~Vu-c-O 0 .— U'OOs-As -. \ fin ~ . A . '.'—. ‘ ’ ' VJ.....~ .-.."- . r . ..-... - ‘I . “ ‘ \u‘ . “‘ ".‘L 0. s an a N "no . 9...:H‘ .I . 4. .I t c- 0. ~ -. d.‘;v.-‘ rue.“ ‘. n .2 _ I... 5.....1‘ 'u .. 3 §‘.' ., u- ‘u. V . H“‘-l~ . I .‘n (L. ~""~ .‘I...‘ u . s v.-_. 4. .9:( \ P- ._". ‘ V's; F . .. .5 \‘AH \ .. .~ Figure Page 5.43 Driven Antenna Current for Coupled Antennas as a Function of h2 (1.3 Ghz) 292 5.44 Driven Antenna Current for Coupled Antennas in Dissipative Media (350 MHz) 302 5.45 Driven Antenna Current for Coupled Antennas in Dissipative Media (1.0 Ghz) 303 5.46 Current for Single Antenna in Sewage (350 MHz) 314 5.47 Current as Function of Height for Single Antenna in Sewage (1.0 Ghz) 315 5.48 Current as Function of Location for Single Antenna in Sewage (1.0 Ghz) 317 5.49 Current as Function of Height for Single Antenna in Sewage (1.2 Ghz) 318 5.50 Current for Single Antenna in Sewage at 1.6 Ghz 319 5.51 Theoretical Driven Antenna Current for Coupled Antennas in Sewage (350 MHz, h1 = 5 cm) 324 5.52 Experimental Driven Antenna Current for Coupled Antennas in Sewage (350 MHz, h1 = 6 cm) 325 5.53 Received Voltage versus Separation for Coupled Antennas in Cavity Filled with Dissipative Media 327 11.1 Transformation to Polar Coordinates for Cap Regions 357 xiii \“ l-“ . .. “mtg. uh. :‘I'o. 1. v5. ..v. .p A ~ 5.. .o . .. . . . 6 ~ .2 .V. . C. r . - . s a 5 p k 0 L ‘ n s . ... vs 6. o ‘5 5L. 3.. .5. >. «He . u u . .P. ... .s. 1 p“ L C ... . 3‘ r». A . m. u s .A. n A n s c. u u c I . :- . ...: .. ‘. s. O. .... a. I. ..I.. .I a _ . a .u a an. t . 5‘. .3 o . .u~ .1. b a as O a . n . .. . 5 I. . r6 . I c n .n . V a :- 6 u .v. a n . ... .P- n: . . s . :u u! .4.- VK .~~ . .n . i \ -\. ,. a . . o - - . t ..w .3. n\. . n . o . .1. . I . ~ 0 . . . . a . . . ... .... .. . ..i ...; . . .. .. . .0. v. x ‘h. a . . . I. . ... ... . . -.. “A PREFACE In the formative stages this research was aimed at a direct extension of the recent developments of antenna theory in dissipative media. The result was to be a comprehensive theoretical and experimental investigation into the area of electromagnetic probing with direct application to ecological media. It was hoped that extensive experimental work on clean and polluted air, and fresh water, and water containing organic and inorganic matter would pro- vide the foundation for effective use of antennas as environmental measuring tools. However, as was so accurately stated by George Bernard Shaw, "Science is always wrong. It never solves a problem without creating ten more." Thus it was that as the research developed, it became obvious that a well defined, closed, bounded, and portable system was needed to meet the requirements of .applicability. The final system chosen was that of coupled antennas in a rectangular section of waveguide or rectangular cavity. This system satisfied the desire to exploit the correlation between anntenna characteristics and properties of a medium, as well as (offering an attractive theoretical, experimental and practical solu- tion to the problem. In order to use this system, however, an curtensive amount of effort was needed to formulate the general tfunoretical problem. Therefore, this report primarily addresses itsealf to the develOpment of this theory and the subsequent xiv In _ .' ufi‘ .W " . , ....— -' C ..,, .rs " ,D - ' 5h ..~ .uot -'~"‘ \ \ ...... 9-. . 'Q l. ' ‘-‘ .. he .. -0 Q. ~ ..." .. ...._ u- .. —— ... _ u. . .- ‘ ~- " .-. . - experimental verification of the results. Because of the lack of adequate literature on this problem, this develOpment in and of itself provides a significant contribution to the field of electro— magnetic theory. Although some representative cases of Specific dissipative media are considered briefly, time has not permitted the type of comprehensive study first sought. This type of study is still badly needed, and it is hoped that the results presented in this report may be used as the basis for such an effort in the future. C. Edward Grove Holt, Michigan December 22, 1974 CHAPTER 1 INTRODUCTION When the cry went up to"Save the Environment: there was a call to the sciences to measure, monitor and quantify what man had produced. In the field of electromagnetic theory, this stimulated renewed interest in investigating the possibilities of using electromagnetic theory to develop new information gathering and diagnostic tools for use by the environmentalists, ecologists, biologists and chemists. The approach has been to apply fundamental field principles and well understood microwave systems to obtain meaningful relationships between measurable electromagnetic phenomena and the environmental quantities of interest. With this approach the electromagnetic engineer has been able to obtain useful, reliable and even pertinent data concerning the environment; the most dramatic example is in the rapidly expanding area of remote sensing. The possibility of useful results has also been found in the area of zintenna and probe theory. By applying the fundamental principles of electromagnetic wave propagation and the theory of antenna inter- zaction with.a general medium, unique relationships have been found adrich demonstrate the feasibility of using antennas as measuring tcxols. The research described in this report addresses itself to t1": further development of the theory of antennas as diagnostic tools by investigating a microwave system which can be directly applied to prcflalems of environmental interest. 1 6.,- .... .--... ..- gun U s o no. . I . I I .o . ... ... .A- ‘1 A . s . ~ F— V .~ 1 .r. .r. .. v. .... .. . s s 2. . L. It _. . .. .. . ... .s. . o a h . :. C« .3. .3 u . L. 9. o u _ ufi .t b s . ‘ ... .u. n . m s as 3 1| t . A n A 0‘.‘ ‘3. n s o .. . — ... a u . o. g s s 9!. u. 3.. v. . a. ;. n. ,h. ... ... .- L . at. nu . . s ... .ux a. o s... . § \ . I ~ .. u 1 t . . I . c . ‘~ - , . 0 . a o n .n . ... u u A . a . . .u o . u s . . . .\. ... u u | . a. . . nua an. . - P\e aJ . .. . ... u . . . v n . .- \ . .. . .4 n\. . c 3%. . a . .. .. . . . . . . . .. ..... . . . . ‘0‘ wk .K- .. .\.o~ \. ‘L . o \ .b‘.\ 1.1 Backggound. Basic investigation into fundamental probe and antenna theory has previously been done for the purpose of analyzing and predicting both the qualitative and quantitative composition of the earth and the ionOSphere (see Grant and West (1965) and Ament (1964)*). Extensive theoretical analyses of electromagnetic propagation in conducting media have been carried out (see Brekhovskikh (1960) and wait (1962)) which are applicable to antenna-media interaction in in- duction studies. Further theoretical and experimental work on probes in various types of conducting and stratified media has in- dicated the ability to obtain meaningful correlation between ueasurements and media characteristics (see Ament (1964), Iizuka (1963, 1964, 1965), Iizuka and King (1962), and Biggs (1968)). Experimental investigations attempting to verify the validity of several of the approximate formulas for the electromagnetic field of infinitesimal dipoles in a dissipative half-Space include Saran and Held (1960) for fresh water, Kraichman (1960) for a tank containing salt solution and Blair (1963) for scale model tank measurements. Experimental checks have been made 1_ situ for ocean and fresh water environments which verify a general theory (see Siegel (1970)). Studies involving water and dissipative media must necessarily involve electrochemical considerations. water alone involves complicated electromagnetic and chemical interactions as well as The references by author and date in this section are given in the general references following the bibliography. N- ' - -- 5». I “an .9 ... u... _ '9"- r ‘\b.- I' . ~ .‘fliv1gu . . I"'~s.‘. ... - l ‘ l .- . I‘o ...; ‘h _ "" 5n»- .. ‘ 0. " I. ..._\ "t‘ :n J , , r-Izw . 9 .§ ‘ .....- -. ~ t. . I . .v .-.: -. u... ...“. \t 0 ., , . - '0. o \_k “c H. -A‘l “" ...‘9- us. ‘) -.' n ' ‘5. “'s.‘ ...P. u“- .‘:I.;~‘ . v 4 . ... g ‘. “- '. . ‘ “A ”an ..."F “.v 9.. ':"n .-. ... q ‘ ‘ a i _ a .H ‘ C...o - -~Q I . . ..-““p§£ - I.‘ I, ‘7 a ' | ... , N ‘u ‘l "51 m‘ a .'&5 . r. ' \ ._- = O ; ‘I 5 ‘I " ‘n- I ...-“ . " Q t C v V s I 1 :- ..'.._ fl: . “- C ,. \ o! '\‘. '- ..o . . ‘.-—v Ra" I .‘ u 4‘” . n.‘ ‘ d . fl 0 " 7 '1 L. a l ‘g F . 1,. a .. ,. ‘ “. diffusion and electrical potentials (see Golterman (1969)) which may be useful in predicting and interpreting various eXperimental phenomena. Existing methods of electrochemical analysis include the conductometric, electrOphoretic, electrolytic, polentiometric and polarographic methods (see Muller (1956)). Of these methods the last three involve electron exchanges at an electrode which is directly applicable to antenna probing investigations. Another area of importance in electrochemistry is the method of high-fre- quency conductance (see Blaedel (1950) and Johnson and Enke (1970)). This method directly utilizes the microwave frequencies to monitor chemical changes in solutions below the range where Spectroscopy is useful and would be useful in identifying Specific phenomena relative to antenna probing in water solutions. Electromagnetic measurements also have application in.ai£ quality studies (see Sauter and Chilton [3]). The major areas of concern are particulate matter (presently detected optically in the visible and infrared region), irritants and oxidants such as 802, N02 and ozone compounds, and systemic poisons such as CO and H28 (presently measured by chemical reaction in solution). Measure- ments of these pollutants could best be performed in situ near the source (smoke stacks, etc.) where concentrations are on the order of five parts per thousand. Thus far, the little application of tnicrowave analysis that has been done shows promise particularly for compounds which have molecular absorbtion Spectra in the lower frequencies. Antenna impedance, current, and charge measurements correlated to the macroscopic properties of air in the presence of various pollutants of controlled concentrations could be useful ad. - v .‘u .4 u I ‘ O xx. _ . t ~ ~ g 5 m: .. . l . r m: ‘ . Out ~.. . . ...- .-. .\. . ~ . . .. . 1.. o. O u g“ c a nu ., g H..- ..— .P. .. . ‘- ., u: .3 s ; a .x. .F. .. a v. u .b» g s s .. . 4 d- in monitoring the output from industrial exhausts. This would thus involve the same instrumentation as in water studies and the general theory could be applied to predict and interpret the antenna measure- ments made. With the above mentioned areas of concern providing the back- ground and impetus, the electromagnetic engineer is capable of extending theoretical analyses of antenna measurements to the area of ecological monitoring. The research described in this report draws on this background in an effort to extend the use of antennas as diagnostic probes. 1.2 Discussion of the Problem and Related Works. The prime motivation for this research is in the recent extension of the theory of antennas in the presence of arbitrary, general, dissipative media. This theory relates the terminal characteristics; e.g., impedance or admittance parameters, and the current and charge distributions on single or coupled antennas to the constituitive electrical parameters; i.e., the electrical conductivity, permittivity and permeability of the medium surround- ing the antenna. Much of this theory for infinite (or semi-infinite) ‘media is presented in the book by King and Harrison [18]. Much of the important preliminary work is found in the works of King and Iizuka [12, 13, 14, 19] and of King, Scott, and Mishra [21, 27, 35, 36]. In particular, King and Scott ([21] and [35]) have formulated and tested the theory of the single antenna while Mishra, King and Scott ([27] and [36]) have treated the coupled antenna case. These latest works clearly demonstrate the possibility of rekujng the characteristics of an antenna to the electrical pro- mnties of a medium. Extensions of these theories to the direct zunflication of antenna probing techniques is most recently reported tw'Rogers and Peden [30] in their studies of deep polar ice. Perhaps the most obvious drawback in all the above mentioned vaelopments on antennas in arbitrary media is the need for an (at kmst effectively) infinite or semi-infinite environment. This is, cf course, very attractive theoretically and can be closely approximated experimentally, but is unfortunately restrictive when itis desired to apply the antenna probing technique to a generally lawlized environment with boundaries that cannot be easily char- acterized or modeled. The most attractive alternative to this pnflflem.is to consider the antenna and media of interest in well «mined, fixed surroundings. Thus, it is this consideration that firms the starting point for the research presented here. By approaching the antenna probing problem primarily from an applications vieWpoint, the selection of an appropriate effective system was the first consideration. The system chosen to meet the requirements of theoretical, experimental and practical viability is that of an array of antennas in a section of rectangular wave- guide as depicted in Figure 1.1. This system has the potential of being portable and easily adaptable for use as a continuous flow- through monitor or discrete sampling system for application to the study of fluids. It is experimentally convenient since the waveguide and antenna size can be chosen physically large for easy construction while the results remain valid for any size system by prOper scaling. The theoretical analysis of this system is also 833$ opmdwocwmgu madofiumm mo 33> >m3m§0 3 M mb 0: c 1. I a . _ r. . uh .1. a. . . OI. ah. Pl 6 .r. s s . a a o . .... 95 . a .. c c L J... .0 a o . .6 v . . .3 —.. .5. .3 r . .1 .v. C. .C a. ... .~ . k. . ..-. q: ... «l. 5.5 .. . a. .x -Q A. ”I 1 . u s a s... . . =- u‘. u a .. a .. a .. .. r. u.» . . v a o . ... . «a. .o . s . .. e . ..w. .. vflk . ..u .I . \Iv .I . Q l 6 x Q o o n m1“ uni . We a . A\_ v.1 .5... H o “no. .U\.. nus '1 ..w . .. fia .u... . 9- . I s V. on ... vs D. vs. ‘C .b. .4. I. . 3 v._ .... .6 .u. .. a. . ... .u. V. . .1 .C . n.‘ v . .u . o ..u . ._ a . . use .n u o s . \ .... a .. is. . a duo. - . n a\. . a “ ... . s a . n p.” . vi .. .u. ... a. . o . J» u ‘4 rs. .. 1.. 10.1 .. an 1a . . . . .. . . . . h... .n t t c x... u\ . . . . . u a Q \ .ua.‘ . . .. u. fit. 6 . . . H's. - \. .~a\ \. ~ . . ... . x , . . atuzctive because the theories of both the waveguide and the zuuenna are well understood and the basic techniques used to solve aufinproblem separately can be applied to yield a tractable solu- tion for the combination. Unfortunately, an adequate general theory fin an array of antennas in a rectangular waveguide is not avail- abhain the literature. Therefore, it is to the development of Hus theory that the research reported here is primarily directed. To approach the theoretical problem an understanding of the mflntion sought must be made clear. Since it is desired to expand mIthe previously mentioned conclusions of antennas as diagnostic mxmes, the system chosen must be approached as an antenna problem. flmt is, the antenna characteristics are to be viewed as a function ofthe waveguide boundaries and the electrical parameters of the medhnn filling the waveguide. ‘When approached in this manner, the (ksired result is seen to be the solution of the antenna terminal duracteristics and antenna current and charge distribution in terms ofthe waveguide dimensions and terminations, and the conductivity, [Ermittivity and permeability of the medium. The fundamental solution for the antenna characteristics hithe current distribution on the antenna. From this the charge (fistribution is readily obtained by application of the continuity equation; the terminal characteristics (e.g. self and mutual admittances) are available from the known (Specified) antenna driving voltages and the derived value of input current by Ohm's law. Therefore, the theoretical problem is basically to develop expressions for the current distributions on the antennas. It is this require- ment that makes the present literature on the theory of antennas in waveguides inadequate for this problem. . 1‘ ...e L: .H A p b" .\o O. .. —. ~_i .P. .. v. . 'u c. .— .. an is . ... no — y- “...-- v . .A a. ah .\~ P. o . 5. ad. .\. . g D u at. .V. ... . I I I .— ‘0‘ v . QN‘ ... p . a . v . s.» .q . Lu 3,. «\J \u .. I» .. ..P. u u o . s . bL n. P ..-. s c. .u‘ a o .. a t . I... h .4 . l 1],. fr.— ~ t C; a‘ . A w‘ I n s a: u . ... a. . . h x a. . . . . u s . r» . 54: v F5 p . ...uw H»... . 1 A ‘ o s s a. fit t ..IN I I I) l v... ....n: In the literature available on the theory of antennas in waveguides, consideration is either given to very Specialized cases of the problem considered here or assumptions are made on the antenna, mveguide, current distribution or medium which are unacceptable for the solution sought here. Thus it is, in the book by Lewin [24] where only a single antenna is considered and the antenna current distribution has an assumed approximate form. Similarly in the paper by Eisenhart and Khan [7], only a single antenna is handled. Their paper also considers only antennas which Span the waveguide (i.e., posts). (It may be noted that this condition leads to a simpler theoretical solution for the antenna current than does the case of an antenna which does not Span the guide.) Rao [29] has adequately solved the case of coupled antennas but again the antennas Span the waveguide (which is taken to be a parallel plate Waveguide rather than a rectangular waveguide). Finally, in the monograph by Lewin [25], again only the case of a post is con- sidered. It is characteristic of much of the above literature that the theories are developed from the vieWpoint of the waveguide tenninals rather than from the antenna or post terminals; i.e. , they are regarded as waveguide problems rather than antenna prob- lems. This restricts the application of these works to the prob- lem undertaken in this research. It goes without saying, however, that each of the above articles is helpful in providing verifica- tion of the general theory through Special cases as well as being of assistance through various aSpectS of the development. D a . v . u- .5 ... t . .... .1 . . s c . . . ... . n .H. a 9 . u. s .t u . ~u C. .5. . . .u» . . . C . o o nu!» p» H .v- . ”L. n“ a. D... . I. -\ o u». ‘5 .\. u. of.‘ .s .. .k l L. t. h . I A . a l I. I n‘. .Q. a Ia.\ 35 .6 AM“ he. "P; v. c .fi5 5 a A‘§ PM» vs. a ‘ K O § ‘ “H a ._ Q . 2» Q. .f. on . . . s e .s . r .. a... 1. u . C. vs ... .. o .. ‘ . as .w. at h cc ‘. .2 a. a. I t . «4 a. u . c. u . a s on O.» I o 3» a .. . L U .C . A: ‘n n . F. a a o ... -.. on ... 3: ts .. x s." .. .... y. L. a» .‘u. u S .c b . u A ‘d a: i s .. .t a . 3. u V n . .3 .. . ‘ .p. s . . 3‘ .~ 5L .1 .. T. ...]. I g. p. :s as . . .pu ....- . . ... . ‘5 c a .4 as .. a an~ t ... .‘N. . Mu. ..a, .4." ..t. .w I has” .93 —- FV. v-uu >- ~ ~ na~ .... .u Ms. . .. . u . ~ . \.... .... o \ ... . . L ~ ~\» . . uh . .... \.. .2... N). .-c\. ...l. v- u... 1.3 Method of Investigation It has been established that the antenna-waveguide-medium System is to be approached from the antenna vieWpoint and that the fundamental solution is the antenna current distribution. This implies the derivation of an integral equation (or set of integral equations for coupled antennas) for a suitable theoretical model of the antenna. In Chapter 2 these integral equations are developed based on the electric field boundary condition at the antenna sur- face. The electric field is found as a superposition of fields in- duced by the unknown currents on each antenna. These unknown fields are first expressed as modal expansions in terms of the rectangular waveguide modes. The modal expansion coefficients are then found as functions of the unknown current distributions on each antenna by application of the Lorentz reciprocity theorem in the waveguide volume. The resulting induced electric field involves integrals of the unknown current volume density functions. The antenna excitation is first modeled as a slice generator or delta-gap source and applica- tion of the electric field boundary condition results in the desired integral equations for the unknown current distributions. These equations are then reduced from the delta function form to Hallen- type equations. The unknown volume current density functions are finally converted to one dimensional current distribution functions by two different approximations on the cylindrical antennas: the thin-wire approximation and the infinitely-thin strip approximation. The resulting one dimensional set of coupled integral equations is then in suitable form for numerical solution. i ‘ p: " I—o’ 7" . .-. o~ . ”"3" ‘5 at." 0 v4.- ..- ... ~. ‘p ... \ l U ‘ ‘- '».AL . . V ‘ ‘ . ""‘.l bp- Os .1 "‘ ....n ..l .1 .I x t. ""5. . Q. ‘ \ ....Jl..“. ‘- . ‘ I '.~ ; ‘0. ,‘ -;_- ..... ,i h- .‘. . - ~ .'--' .4 ...e .. -. ' O '- “ 5"; .. ‘ ‘- '. ‘5. . *--, . .. in; .. _-.— -...... N _ .“ ‘I... .‘a‘;~--. .. _‘ ‘ ‘y ‘---..‘ ..\ . ... . .‘ - ‘~\ ‘ ‘ ‘ .. q”. ' -‘ ‘ I ._ . u o t‘ . “ . ._ -‘ .\\.‘.‘S . . . '5 P!_ ‘0‘ kl.: ‘. l':“:.-. ‘SNRE H...- r- -. ‘\ . N . . i'§.3 ‘ .""‘ Q's-I" ...; ‘l ""‘25‘ -,J . r ‘1‘- C . . : ._ ‘-.; $_’ .§ . ‘.“y “ . . . \‘ ..~ ._ ' .. .‘- \- ~\-‘-‘.c‘ ‘ . ,‘ ~ 5 o u 9‘ ‘U. _ .- ‘ u'L Q. ..E . “‘ ‘>“L-‘u- ‘\‘..: ‘ -i =. I 5‘ ‘L. ‘ ‘.‘u fi" — so- ‘1 K4 1-T~s= ‘ m“ A 10 To improve the approximate antenna model consider first, the problem is approached again as described above but with the antenna excitation modeled as the physical coaxial gap instead of as a Slice generator. Because of the increased complexity this is only done for a Single antenna and the cylindrical antenna is handled only with the thin-wire approximation. ‘No reduction to a Hallen- type equation is necessary and the integral equations are directly solvable by numerical methods. The numerical methods for the solution of the integral equa- tions developed in Chapter 2 are treated in Chapter 3. Discretized equations in matrix form are found by the moment method using the method of subsections and point matching. Comparison is then made between this point matching solution and the results of applying Galerkin's method. No Significant improvement in the solution is found, thus supporting use of the Simpler point matching results. The solutions of each approximate form of the integral equations developed in Chapter 2 are also compared with reSpect to their convergence properties. It is shown that the thin Strip approximation exhibits improved convergence over the thin-wire approximation. The delta-gap and coaxial gap solutions are found to give comparable results for the current distributions. Also, criteria are developed for preper truncation of the two infinite series modal summations which improve the numerical evaluation process. In Chapter 4 the apparatus for experimental investigation of the antenna-waveguide~media system is described. Attention is paid to the details of construction and quirks of operation for the components used to measure current distribution and antenna input a .. 1 - . .3 v. .w. .n o v. \ . v. D; .0 a '0 o v P‘ - n... .t .a o y 9 u -\.~ .u. . .v‘!‘ n! u ' Q .\. .ua .I 5?; J» v» c u u w . a ...u. C a 4.- F‘. I" l ‘1‘ I Q‘. l I I. I n” .... .. . . . ...n . . . u .o v~ w. a “A ..x Q Q .. .a" )5 ‘9“ ’ , a“. --. ... .Ps .o A L .. t s a . ... an. nu. n- .\J n a. 11‘.» . h .t . . s-s- ‘ O 4 .PK . . ... . . x . n~ ... .C \ b\ D a n u v. t a ~. ~ o .4... .fl. I.‘ ~\- fl . an. .5 .‘u ... ..... we. . . ..I. u 1 V a . . . P: ~c ah; ha ... Q» . .u~ .. Q .... .. a a 2.. |. h ‘ n n o s... ‘ ~ ... a. t; ... I ... .... .. a n. a u... "L . . .n. ll admittance (the impedance sensing block). Data is presented which illustrates several sources of experimental error. Methods are also described for the independent determination of dielectric con- stant and (low-frequency) conductivity of the dissipative media con- sidered. The presentation of theoretical results and the comparison with experimental data is contained in Chapter 5. The problems of correlating theoretical and experimental input admittance are treated in general in Section 5.1. Evidence of the infinite Susceptance introduced by the delta gap theory and the problems associated with applying conventional terminal zone correction networks are described. New ways of approaching the determination of a correction network for the case of the antenna in a bounded region are presented along with typical results for several physical and electrical configura- tions. The experimental admittance results given in the following sections, however, are not corrected for terminal zone effects to avoid obscuring the original data. The results for current distribution and input admittance are primarily divided into the cases of the Single antenna and of two coupled antennas with the Single antenna in an air filled cavity receiving the most emphasis. Comparison of the results predicted by the theory and experimental work available in the literature for highly dissipative media is also included. Results of experimental measurements on distilled water and one salt solution are compared with the theory for a limited range of parameters. Finally, the results of investigations on one sample of effluent from a sewage treatment plant is included in Section 5.6. 12 In general the results Show excellent agreement between theory and experiment for the current distributions over all variations of the physical and electrical configurations. Discrepancies between input admittance values exist, but, in general, are expected. The overall agreement substantiates the ability of the theory to predict the measurable data over the range of parameters of interest. Of particular interest are the current distribution and input admittance found for the Single antenna in the air filled cavity. Variations in current and admittance are strongly coupled to the resonant frequencies of the cavity with the current distribution showing dramatic variation from that eXpected in an unbounded (free Space) region. It is anticipated that advantage can be taken of this unique feature for implementation of the antenna-cavity-media system as a disgnostic tool. In Chapter 6 the results are summarized and conclusions drawn on the applicability of this system for monitoring the electrical characteristics of media over the lower microwave fre- , quencies. Indication is also made of areas of this research which Show promise for further investigation. Recommendations are made for expanding, continuing, and complementing the findings reported here along with possible areas for improving the analysis. .u. .. b .. — .1 .o . s . ‘ a . q a. . var: r e ~. .\~ 5 I a . a a.» s . HA. ...}. a ¢.~ - .5». .s . I ... v. .4 . a t ..s A . . .. ... . a h l-% a. 4‘I‘A - t . . .s . .. .... 2. ,. . ... .. g 0PM .5. ..- a... a. 2 E ,v; E ‘ O ork I I ~ .- ..a 5.... e I~< ..44 A . Sc 5 9” .. 0“ 9.. .. s e .-.- 'u.. o; C b ... u . .p. ... .... . a . ii. 5 a.“ . h D: g. Q r. N.- ..V i ax .. CHAPTER 2 THEORETICAL DEVELOPMENT 2.1 The Model The system under investigation is that of N linear antennas in a rectangular waveguide. A solution for the antenna characteristic (in particular, the current distributions) is sought for the boundary value problem with an arbitrary medium filling the waveguide. This chapter contains the details of the theoretical development of the necessary integral equations and the reduction of the integral equa- tions to one dimensional forms Suitable for numerical solution. The physical structure and geometry of the general problem are shown in Figure 2.1. The antennas form a linear array with co- planar axes located in the center of the broad wall of the waveguide and parallel to the y axis. They are cylindrical antennas formed by extending the center conductor of coaxial transmission lines beyond the inside surface of the waveguide wall and are capped with hemiSpherical Surfaces. A rectangular coordinate system is used with the origin located in one corner of the waveguide such that the center of one antenna is at z = O. The waveguide has interior transverse dimensions a and b in the 2 and 9 directions, reSpectively, and longitudinal dimension d1 + d2 in the a direction. For generality, the waveguide is terminated at the planes 2 = -d1 and z = +d2 in arbitrary complex reflection 13 l4 035?:me on» E 925353. 9% n3 .rnuogooo n5... oufiogum 303.3% H .N MMDUHR |:—,o ...- , ‘. , . s. L ’-- .... "u"“' ... ‘ IITLTEC' "w a' ._ :.. h“"‘~.-.s .‘ a... . 'l ‘ , .s- :lt 3 fl. .- a C I .v. p;‘ “5 ‘a'! t- 1 QI.‘C K n I ’ ‘; ' .§ ..: -. ‘.' 311 i ‘ J. -~ ?.'!.~, 1‘ A k-rfnu. ‘ l 1'. gt»... A..’:P -‘~_I a , .‘r 'r-‘ “~"M.. <.' a “ . D .. .- '\.J\ 4: a‘. I): a‘.‘ v 1 ...~ ' D ‘t ‘— 15 coefficients F11. and FZL’ reSpectively. The medium filling the waveguide is characterized by the constituitive electrical para- meters: namely, permittivity, ergo; conductivity, 0; and per— meability, u. For this development, the conductivity and permittivity are combined into an effective complex permittivity as AS stated in Chapter 1, the most fundamental results to be obtained are the current distributions on the N antennas for known excitations or loads at the antenna terminals. These antenna currents are a direct consequence of the electric field boundary condition on the antenna surface and thus it is on this boundary condition that the development is based. The interrelationships of the other field components in the waveguide are based on the boundary conditions on the interior waveguide walls. That these boundary conditions are met is tacitly implied in the modal expansion technique (only fields satisfying these boundary condi- tions are permitted). Therefore, explicit application of the electric field boundary condition on the antenna surfaces serves to sat: isfy all boundary conditions and results in expressions for the unknown current distributions. A knowledge of the assumptions and approximations made in this development serves to define the degree of generality and the 1' ° Hints of applicability of the results. Therefore, the various ass - Umpt lons and approximations are listed here for reference. The det aIlls and their implications will be described more fully as they .... .v. Av; .. c. 16 are invoked. 1) 2) 3) 4) 5) 6) 7) 8) 9) The medium filling the waveguide is assumed to be linear, isotropic, and homogeneous with complex permittivity, e, and permeability, u. The waveguide terminations are assumed to be independent of waveguide mode and produce no mode coupling. All conductors are assumed to be perfectly conducting. The antenna terminals are handled in two ways: (a) primarily, the coaxial gap in the waveguide wall is removed and the slice generator or delta-gap approxima- tion is used. (b) For the Special case of a single antenna, the theory is developed for the coaxial gap excitation assuming a coaxial TEM field in the gap (Section 2.6). In no case does the antenna Span the guide; i.e., they do not touch the wall opposite from their terminals (h < b). The antenna radii are assumed to be electrically Small and not comparable with the waveguide dimensions (r8 < k0, ra << a). Each antenna radius and height is assumed to satisfy r8 < h. The unknown antenna currents are approximated as having only a y or axial component. The cylindrical antennas are approximated in three ways: (a) thin-wire approximation (a line current is assumed to flow on the axis). (b) transverse strip approximation (the cylinder is repre- sented by an infinitely thin strip lying in the (x,y) ;--.. ...“. g tu-L I. .“I . ---E \ L I .J‘ { “C b a. v.‘ V . ‘. nib. '. ‘0‘, ‘~.: r'. I: -. x .. ‘s l_q ‘a: A " (c .v u.‘ . . 2’ ,, - n ‘\‘:S . ’ .Q ‘ ls, ‘ ...3 l .1 5'25 ‘ ‘n9'N - .5... ‘. VA. ‘,- . \ u at ‘ I 9 V" ..- . l7 waveguide plane) . (c) longitudinal strip approximation (same as (b) lying in the (y,z) waveguide plane). 2.2 Bounda_r1 Conditions; Slice Generator Approximation To solve the general problem of N antennas, the slice generator or delta-gap approximation is made for each antenna terminal. For this approximation, the physical coaxial gaps are replaced by conducting Surfaces and a time harmonic (eJ ) voltage of radian frequency u.) is maintained across an infinitesimal gap between the waveguide wall and the antenna. The Specific geometry is shown in Figure 2.2. Each antenna has radius ra and is centrally located in the broad wall at x = a/2. The general antenna is located at longitudinal position 2 = z,; is excited by a voltage Vj’ referenced to the waveguide wall; has total height y = hj; and carries an unknown current Jyj(x,y,z) in the 9 direction. The assumption here that the currents have only a y com- ponent implies that the sufficient boundary condition on the antenna Surfaces is the continuity of the 3: component of the electric field. The N boundary conditions for all N antennas are then: (2.1) i e E x, ,Z =E .X, ,Z . -1,2,000,N yj( y ) yJ( y ) J for locations (x,y,z) on each antenna Surface. The impressed electric field (superscript e) is the field maintained directly by eXterna 1 sources; i.e., all ideal generators and the fields set up by internal system losses (imperfect conductors). The induced field (SuperScript i) is the "radiation" field which results (via 18 (1) (2) (j) (N) I l | n y :hj -——(-\ | | m I rlL h o o o o o o FZL I : Jvl t JYZ 1 VJ 1 JYN T I I l [ V1.-—V.2. ij VNf IL: I l I I l I Z=-d 2:0 2:22 z=zj zzzN z=dZ 1' e/ a (1) (2) (j) (N) FIGURE 2. 2 Geometry for Slice Generator Approximation O ..___. x:0 x.3 _2 x:a l9 Maxwell's equations) from the external source fields. Since it is assumed that the antennas are perfect conductors, the impressed electric field is identically zero along the antenna surface. (This assumption can easily be relaxed by approximating a finite con- ductivity with a surface impedence as noted in Section 11.4 of King [17].) The impressed electric field in the infinitesimal gap at y = O is non-zero and may be handled mathematically by employing the Dirac delta function (see Appendix I, page 273 of Papoulis [28]). This yields the desired prOperties of an infinite electric field for the zero width gap which when integrated across the gap gives the finite applied voltage. Some care must be exercised, however, in using the delta function to represent the infinitesimal gap electric field when only the range y 2 O is considered. In other words, the delta function centered at y = O, 6(y) , does not appear appropriate. The most unambiguous approach is to consider first a small but non-zero gap width and then apply the solution in the limit as the width approaches zero (see Section 2.4) . By assuming the non-zero, but vanishing small, gap width to be '1' and the voltage applied across this gap to be V, a proper expression for the gap electric field is E(y) = -V 6(y - T/Z) The negative line integral of this electric field from the lower to the higher potential (+ 3: direction, Figure 2.2) then yields the applied voltage as it must: T "J E(y)dy = V 6(y - T/2)dy = V - O 0" 3—1 a». I) if It" at 20 Note that had the delta function centered at y = O, 6(y) , been used, the integration from the lower limit of y = O (the boundary of the valid range of y) would have lead to an ambiguity in evaluating the integral; i.e., the integration would not have been formally "across" the delta function. Using this infinitesimal gap width 'r for each of the N antennas, the impressed electric field for the j-th antenna then becomes: e E (x, ,z = -V. 6 - T/Z N y ) J (y ) O_<.y:~:hj (2.2) j=l,2,...,N (x,z) on j-th antenna Surface With the impressed electric field Specified, it now remains to determine the induced electric field for application of the boundary condition, Equation (2.1) . 2.3 Electric Field in the Waveguide The field maintained by any general current in the waveguide Inay be expressed as a superposition integral of the current and a dyadic Green's function (see Chapter 2 of Collin [2 1). The dyadic Green's function is the solution to the inhomogeneous vector Helmholtz equation in the presence of a unit intensity, point localized, current source and is subject to the boundary conditions that the electric field to be derived from the Green's function has Zero tangential component on the interior waveguide walls. Stated mathemat ically, the induced field for a general current 36:.) may be written as 21 E(?) = jmu £ 3(?') - E(?\?')dv where G is the solution to 5 = -fi 5(? - ?') = 2 VXVXG-ko under mixed Neumann and Dirichlet boundary conditions. Here, u is a unit dyadic, 6(13 - f') is the three dimensional Dirac delta 2 2 function, and k0 = w us is the propagation constant of the medium filling the waveguide. In the above, as well as throughout this development, the primed coordinates locate Source points and the unprimed coordinates locate field points. In order to apply the boundary conditions of Equation (2.1), it is necessary to determine a formulation for the y-component of induced electric field in the waveguide, E;, with the N unknown antenna currents considered as sources. Since the antenna currents are assumed to have only y-components and since only the y-component of electric field is required, the vector superposition integral for the Green's function technique becomes a scalar integral and only the W—component of the dyadic Green's function iS required. By expressing the total electric field as a linear superposition of the fields maintained by each antenna current independently, the follow- ing expression may be written down: ' N 1 E (X, = z J . I, v, 'G ’ , u, I,zl dvl 2.3 y 37.2) i=1ii'y1(x y 2)yy(xy Z\x y ) ( ) 1 Where the factor jw has now been included in the unknown function G yy’ Aga in, the primed coordinates locate source points (in this CaSe . on the i-th antenna); whereas, the unprimed coordinates locate 22 field points (on the j-th antenna). .Application of Equation (2.3) now requires detenmination of the appropriate Green's function for the interior of the waveguide. However, rather than seek the Green's function Specifically, the induced electric field will be found directly by another method, and, based on Equation (2.3), the Green's function will be identified. This approach avoids the possibility of misusing a Green's function derived for free space in the assumed general medium filling the waveguide which can have non-zero con- ductivity. (The problems related to applying the free Space Green's function in a current carrying region are discussed in a paper by Van Bladel [39].) The method used to determine the general induced electric field is to apply the modal analysis technique to an appropriate fonmflation of the Lorentz reciprocity relation. This development ammlthe notation used are similar to that in Section 4.10 of Collin [ 3] or Section 5.6 of Collin [:2]. The Lorentz reciprocity relation fcm electric current sources in a volume V enclosed by a surface S is £021 x H2 - E2 x fil)-n d8 = t£03241 - E1°J2)dV (2.4) fl = outward normal to S where (iEl, fii) and (fig, fiz) are independent solutions to Maxwell's equatir>r153 with externally applied source currents 3i and 32, respectiitxrely. Application of Equation (2.4) will be made in a volume V i (corltaining the unknown current on the i-th antenna) within the w aveguide as shown in Figure 2.3. In Equation (2-4) the ChOice '1"?! El 3'... 23 14:9 1. Region AL, Volume Vi Region B; g .1 g 1.) s / 1' l A E" + | 0.__” V ! n ! CUEV : l 4 t . 1 L a; a: I'21. . 5| '0' | I + ' ' - l 13va I T Jy,1(v') l DUEV : . "__’ I 0 ¢————- | L I 3‘ Vi l. l l I I z = ZA 2 = z' z = zB c s. = waveguide cross section FIGURE 2. 3 Geometry for Application of the Lorentz Reciprocity Relation 24 J2(X.y,2) = 0 is made such that the correSponding fields (E2, 11.2) are any fields which can exist in a source free region in the waveguide; the particular choice for (E2, R2) will be made later. The current 3.1 in Equation (2.4) is taken to be the unknown current distribution on the i-th antenna: _,= IU'A J1 in0<,y ,2)y , such that the correSponding fields (E1, H1) are the desired induced fields in the waveguide. The solutions to Maxwell's equations in a source free (externally applied) region inside the waveguide are the complete Set of orthogonal waveguide modes. Employing the notation used by COllin [ 2, 3 ], it is expedient to separate the transverse (x,y) dependence of the eigenmode fields, and to jwt and longitudinal (z) Suppress the assumed harmonic time dependence factor (e ). The Waveguide fields are expressed as +‘Y Z +Y Z -'|" -0 V A V E\,(x,y.2) - et,v(x,y)e i;z ez V(x,y)e ; z 1 2 4+ _. Y .. Y H" (x,y,z) = i h (x,y)e v + 2 h I(x,y)e V V t,\) Z,\) where the transverse (t) and longitudinal (2) vector field com- POnents are also distinguished. The upper case letters (E,H) are L386 to represent the total x,y, and 2 functional dependence; whereas, the lower case letters (eu v’ h 'v) are used to represent : U: 25 only the Spatial dependence of the field on the two transverse plane coordinates x and y. (The duplicate use of the lower case e as the exponential function should be unambiguous since this will always have an exponent associated with it; whereas, the field will always have a subscript indicating the component and/or the mode index.) The upper series of Signs in the above expressions (E , H+) refers to fields propagating in the positive '2 direction (e-Yz) , and the lower series (E-, H—) refers to propagation in the negative (e-Wz) "z direction The indices for the eigenmodes are repre- sented by the single subscript v as a short-hand notation implying all combinations of indices (n ,m) for TE and/or TM modes. When the distinction between modes is necessary, fields for TB modes are denoted by a superscript h and fields for TM modes by a super- script e. The mode function solutions for the fields in the waveguide inVOlve an arbitrary amplitude constant which may be evaluated by assuming a value for the amount of power carried by each mode in the waveguide. By Specifying the power in each mode, the mode fields beCome normalized and form an orthonormal set of basis functions for the interior waveguide region. The choice made for the normaliza- t ion condition is arbitrary as long as it is used consistently through- OUt the development. In this development it is convenient to use the following normalization condition: 1/2 v=\)' where c.S. denotes a waveguide cross section. The normalized L 26 modes for this development are given in Equations (2.5) (see, for example, Section 3.6 of Collin [3] for their derivation). for TE modes: h ex’v(X.y) h e yWhey) h hx , V(x J) h hy,v(X.y) Vvtuere c 5‘ c 0 +1 6 6 Z k v on om _y 2 ab k Zhe e k -——-—-—— -§-sin(k x)cos(k y) X Y cos(k x)sin(k y) c x y 2 ab k c (2.5a) (x,y) (x,y) -£-COS(k X)COS(k Y) X Y v 2 z: kx - ab k; COS(kxX)Sln(kyy) 2 2‘: k - —1-Sin(k x)cos(k y) ab kc x y 2 23) kc + ab q; Sin(kxx)31n(kyy) (2.5b) 1 e ' *e (X,) ze y,v y v 1 e -—'e (x,y) ze x,v v 12 we ’ w ’ Yv 27 arm eon All other notations are described in Appendix I. The complete expressions for either a TE or a TM mode field (in a source free region) propagating in either the i 2 direction can now be expressed as + z -d' .. Yv - — + e EV (et,v — z ez,v) :Y z (2.6) -—7+ —-+ A V H— =(+ h + 2 h )e .v " t,v z,v where, for example, H = a e + A e . etav X)V y ya), Because the fields of Equation (2.6) form a complete ortho- normal set of basis functions over the interior waveguide region, any waveguide field can be expanded in an infinite series of these functions with unknown expansion coefficients. Thus, for example, any electric field propagating in the a direction may be expressed as l:3'(x,y.2) = 3"va - Referring now to Figure (2.3) , the general fields outside the volume of interest are thus: 28 in region A: A vv vv v S _, - 4’- z 2A = + HA ZAvHv BvHv v in regionB: ”E =2CE++DE- B v vv __. r a” ZZZB HE vv+D~H II M O The expansion coefficients can be interrelated via the waveguide termination reflection coefficients as Bv = Av Flv D = v C I'Z‘v Where -2-Y d = ‘ v 1 Fly II'll. e — d 1" = I‘ e ZYV 2 2'\) 2L Therefore, the general fields become -o 4t = A E + E EA 2, v'( ' I'1v' v') V z s ZA (2.7a) —. _. —-rl- HA — Ev A\)'(H\)' + I'lv'Hv') and 29 1 4+ 4— B .Cv'(Ev' + I'Zv'Ev') v z 2 2B (2-7b) ['11 ll b1 4 “B -»+ q— 23’. CM“... + rum,» The coefficients Av and Cv are now determined from the reciprocity relation, Equation (2.4), in terms of the unknown current source in(x',y',z'). In Equation (2.4), let "’ I u I _a t v I J1(X 3y ,2) y in(X ay ,2 )2 II C O O z S 2 E uat o 2. . C . E ua 2. . To determine the coefficients Av’ let 32(x'9y'32') = 0 and choose %=fl+%vv H2=ifj+r2Vfiv Where this choice of linear combination of mode fields is a solu- tion to Maxwell's eQuations in a source free region. Using the above, the reciprocity relation becomes: 4+ —+- - -o A c i AEEA x (HV + I‘ZVHV) - (E: + I‘Zvi’v) x HA]-(-z)dS + d a+ a- 4+ - a A _ + {3333 x (Hv + I‘ZVHv) - (EV + r‘zvfiv) x HB].z dS — C = IKE: + II'2vgv) '3’ indv' ’ V. 1 30 since the surface integrals are identically zero on the perfectly conducting waveguide walls (e.g., (EA X {112/n = (a X FIB-Ill) and h x E. E 0 on the waveguide walls). This equation may be simplified by performing the following steps: (a) Substitute Equations (2.7) into the reciprocity relation (b) apply orthogonality; e.g., J‘fi xfi,)-d'§=o for v#v' (3.8 V V (c) expand cross products and cancel appropriate terms (d) expand mode fields according to Equation (2.6) (e) collect and cancel appropriate terms (f) invoke the assumed normalization condition; i.e., J‘G’ xfi')-e dS =1/2 v v Thus, after Substituting and applying orthogonality, there results, ---I.‘ 4+ —-9— — J“ A[(E +I‘1E)X(I_{"-+1"2H) c .S.A v v V V V V V 4+ -o- - - (Ev + PZva) x (fiv + Puffy-la dS + x cw: m x 1': m c .S.B 2v v -'+ —0- - A _ - (Ev + I‘ZVEv) x (if: + szfivaz dS - -'+ -o- = + .15 (IV! é. (Ev I'Zva) y in i The integrand in the second integral is identically zero (hence the choice for (EZ’fiZ) to solve for Av) . When the cross Products in the first integral are expanded, the terms with the Single coefficient F1 or 1‘2 cancel. When the remaining fields v v 31 are expanded according to Equations (2.6) , the eXponential z dependence factors are seen to cancel. Since the integrand involves the scalar product with a, only the a-component of the cross pro- ducts contribute to the integral so that the terms involving a ezw and E h have no contribution. With the above the equation 9 becomes: _ —9 X3 - -o —9 . = 2A I [(et,v t,v) I‘l\)I"2\)(et,\) X ht,v)] 2 d3 _'+- «- = + -“ dv' J'(E\) FZvEV) nyi i Invoking the assumed normalization then yields the desired reSult: l ...sz sz Av = mgley’V0‘ ,Y )[e + sze ]in(x :y :2 )dV (2-8) ’ 9 i h which is different for TE or TM modes accordingly as e:\J and ey'v a re different . To determine the coefficients Cv’ choose 31 and (£1.31) as above and again let 32 = 0. but: now choose a = fi’4+ r fi+ . V Following steps (a) through (f) above, the result for Co is: C 1 +sz' -sz = l u u t l u é'ey \)(x ,y )[e + F1 Ve 1Jy1(x .y .2 )dV (2-9) 1 V P]- 9Vr2 ’V -1 3. 32 whicki is again TE and TM mode dependent. It should be observed that, in the course of the simplifica- tion (step (d) above), the expansion coefficients lose any dependence on the particular choice of waveguide cross-section planes at 2A and 28. This choice may thus be taken as close to the i-th antenna as desired. These eXpansion coefficients may now be substituted into the general eXpansions for the fields in a waveguide, Equations (2.7a,b), to yield eXpressionS for the fields induced in the wave- guide by the unknown current on the i-th antenna. At this point, however, it is most expeditious to recognize that only the y-com- ponent of the induced electric field is required for application of the boundary conditions, Equation (2.1). From Equations (2.7) then, +y z -y z v v + F1,ve ) z 5 GA ’ ’ i 2'1 Ey,A(x y 2) )3 Avey’v(x y) (e ( 0a) V -sz +sz 2. 2 Cvey V(x,y)(e + P2 ve ) z 2 2B ( 10b) 9 3 Ey ,B (X :y :2) Upon substitution of Av and Cv from Equations (2.8) and (2.9), it may be seen (after some algebra) that Equations (2.10a) and (2.10b) can be combined into one equation (unconditional in z) by appropriate use of absolute values in the eXponential terms. Thus, the field induced by the i-th antenna current is: E10! Z)‘ _ ____L—_ J ' ' ' -Y\)\z-zv\ + y ,Y, (i) - 5 P1 vrz v_1 ' yi(x ,Y :2 )ey,v(x:Y) (6 , , 1 -vv(z+z') +Y (z+z') +yv\z-z" +1“ e +r e V +I‘ I“ e )dv' (2.11) 1,V 2,\) 1,\) 2,\) 33 Equation (2.11) may now be eXpanded by applying Equation (2.5) for ey v' The distinction must here be made between TE and TM modes. ’ The integrand contains the term e e x', ' e x = e ,v( y ) y.v( .y) y h h xv, 'ee X, +8 xv, I e y ,v( y ) y.v( y) y,\)( y ) yN(x.y) Using Equations (2.5), this becomes 2 2 2 Z: k 22 e neom kx [ 2_y +- O ]Sin(k x)sin(k x')cos(k y)cos(k y') ab kc 2 ab kc x x y Y which reduces to 3 e 2 2 _o—In _ ' ' ' ' [abwe Yv (ko ky)]Sin(kxx)31n(kxx )cos(kyy)cos(kyy ) Further, the order of volume integration and modal Summation in Equation (2.11) may be interchanged. Finally, the total y-com- ponent of induced electric field in the waveguide is found by a linear superposition of the fields from each of the N antennas. The complete result is: . N e 1 = I I v 0m 2_ 2 . . u Ey(x,y,z) 1:1 £'in(x ,y ,z )E abwevv (ko ky)81n(kxx)31n(kxx ) i l I I cos(kyy)cos(kyy )fv(z\z )dV (2.12) where -y \z-z" -v (z+z') +y (z+z') +y \z-z' e v + F e V + F e V + F F2 e V f =2z ” wafianxwmuxw yy n m 0 y X x cos(kYY)cos(kyy )fnm(z\z ) (2.14) such that Z i E 2 a = J " '3 'G a ) ': ': 'dV'. 2°15 y(xy2) 121$ yi(x y 2)yy(xy2\x y 2) ( ) i Equations (2.12) through (2.15) now completely define the y-component of the induced electric field in the waveguide in terms of the N unknown antenna current distributions. The equations necessary to solve for the current distributions are obtained in the next section. 2.4 Integral Equations; Reduction to Hallen-Type Equations. Having thus far obtained expressions for the impressed and induced electric fields in the waveguide, it now remains to apply the boundary conditions on each antenna surface and solve the result- ing integral equations for the unknown current. A straightforward solution is not attempted, however, for two reasons. First, the fact that the antennas do not Span the waveguide implies that one set of boundary conditions does not apply across the entire wave- guide cross-section. This makes an analytical solution unwieldy, and thus a numerical solution is sought. A second reason, with this in mind, is that the assumed mathematical delta function excitation cannot be implemented numerically. Therefore, the in- tegral equations are eXpanded into equivalent integro-differential equations which are separable. These integro-differential equations are readily reduced to Hallen-type equations by solving the in- dicated differential equation. The result is then solved numerically \ in the next chapter. g 35 The boundary conditions on the surfaces of the N antennas are given in Equation (2.1): i e E .(x, z =13 (x, .2) j =1.2.-.-.N (21) Y] y’ ) yj y Substitution of Equations (2.2) and (2.15) for the impressed and induced electric fields yields N 2 J x', ',z' G x, ,z x', ',z' dV'= -V.6 - T/Z £' yi( y ) W( Y I y ) J (y ) i=1 (2.16) ...j=1,2,...,N OSyShj (x,y) on j-th antenna surface where ny is defined in Equation (2.14). Equation (2.16) is the desired system of N coupled integral equations in the unknown currents for the assumed slice generator approximation. To convert these pure integral equations tointegro-differential equations, the following observation is made: 2 (k2 + a-§-)cos(k y) = (k2 - k2)cos(k y) (2.17) 0 3y Y 0 Y y where ky = mn/b. The right hand side of Equation (2.17) is recognized as a factor in the Green's function, Equation (2.14), and the equi- valent differential operator is substituted. Since the derivative with respect to y is no longer a function of mode index m (ky) and is also independent of the volume integration over (primed) source variables, it may be taken outside the integral and summa- tion so that the integral equations (2.16) become: 36 2 N 2 (k + 5-5) Z J i(x',y',z')G1 (x,y,z‘x',y',z')dV' = -V,6(y-T/2) O 8}, i=1 ly yy J i (2.18a) ... j =1’2,...,N OSyShj (x,z) on j-th antenna surface where the new "Green's function" or kernel is: 1 1 60m G (x, .z X'. '.2' = Z 2 yy y \ y ) . . | l EEZEEf-Sin(kxx)31n(kxx )cos(kyy)cos(kyy ) n m nm f (z‘z') nm This is now a system of N coupled integro-differential equations. By letting N ...1 Bj(y) = 1:1 J'in(x .y .2 )ny(x.y.2\x'.y',2')dV' . (2 19) 1 Equations (2.18a) become: 2 (k: + a--§-)Bj(y) = -Vj6(y - T/Z) j = 1,2,...,N (2.18b) By The solution to these differential equations is (a similar equation is solved in Section 3.2 of King and Harrison [ j) V Bj(y) = cj cos(koy) - 21].: sin(ko\y ~ 'r/2\) (2.20) where the symmetry property B = B - j(y) j( y) has been invoked (as seen from the cosine dependence of Giy and hence Bj(y)). Equating the right hand sides of Equations (2.19) and (2.20) yields pure integral equations for the unknown currents: 37 N V 1 I = - _i_ - - E I in nydV Cj cos(k0y) 2k Sin(ko\y 7/2‘) i-l V o ... j = 1,2,...,N o s y s h (x,z) on j-th antenna surface These integral equations have been derived for the slice generator approximation with a finite gap width T. As stated in the dis- cussion leading up to Equation (2.2), the zero width gap, which is desired, is found by taking the limit as T approaches zero. It is necessary to delay this limiting process until this point so that the ambiguous nature of a delta function centered at y = O is avoided in the solution of the differential equations (2.18b). Performing the limit as T approaches zero is straightforward. Also, since the integral equations are valid only for y 2 0, the absolute value may be removed to give: N V 1 j , Z J . x', ',z' G x, ,z x', ',z' dV' = C, cos k - Sin(k i=1 ' y1( y ) W( Y \ y ) J ( 0y) Zko DY) i (2.21) j=1flwnfi Osyshj (x.z) on j—th antenna surface Equation (2.21) is the desired system of N coupled Hallen-type integral equations in 2N unknowns; namely, N current density functions and N new unknowns Cj which result from the solution of the differential equations. There are also available N new boundary conditions which render the equation solvable. These result from the fact that the antenna current is zero at the end of each antenna; i.e., 38 in(x,hi,z) E 0 1 = 1,2,...,N (2.22) (x,z) on i-th antenna surface . It should be emphasized that the vanishing of the current will only be true for antennas for which hi < b, and the foregoing reduction to Hallen-type equations would be pointless for the prob- lem of posts which span the waveguide. 2.5 Cylindrical Antenna Approximations. A rigorous solution for the antenna current, under the assumption of a perfectly conducting antenna, would yield a cylindrical sheet of surface current flowing on the surface of the antenna. The complexity involved, eSpecially numerically, in obtaining an accurate description of the current in three dimensions is obvious. In this thesis, therefore, approximations are made for the cylindrical antenna to simplify the numerical solution in rectangular coordinates. First, the thin-wire approximation is employed as in standard linear antenna theory [17]. It is found, however, that this approximation yields a divergent series in the numerical solution (see Chapter 3) and is not useable. Therefore, two other approaches are taken in which the cylinder is approximated by an equivalent strip. This approach is similar to that of Eisenhart and Khan ['7] for a post Spanning a waveguide, except that a more realistic representation is made of the strip surface current density. In this section, the three approximations are described and the integral equations corresponding to each approximation are obtained. 39 2.5.1 Thin-Wire Aggroximation The simplest approximation, and the one most often used in antenna theory, is the thin wire approximation. This approximation assumes that the antenna radius is small compared to the wavelength; i.e., ra << K- Under this condition the current on the antenna is approximated as a line current flowing on the antenna axis. For the case in hand, the thin-wire approximation implies in(x',y',z') = Ii(y')6(x' - a/2)6(z' - zi) 0 s y' shi The boundary condition on tangential electric field on the antenna is still maintained at the cylindrical Surface. However, for simplicity of numerical implementation, the boundary condition is only satisfied along lines on the antenna surface. The lines chosen are defined by: x = a/2 + r - a O s y s h. J z = z . .1 Under this approximation, the integral equation (2.21) becomes (using superscript tw for thin-wire) N hi v t j . 2 j‘ 11(y')Gi¥(Y\Y')dy' = 0]. c08(koy) - 2k sm(koy) i=1 y'=0 o (2.23a) Ogysh, j=1,2,...,N J where the triple volume integration was reduced to the line integral by straightforward evaluation of the x and 2 integration over the Specified delta functions. The resulting kernel for the integral 40 equations (thin-wire approximation) is thus ° ” j e tw , _ ' Gij(Y\Y ) - E 2 abwe cos(era)cos(kyy)cos(kyy )fnm(zj\zi) (2.23b) m Ynm o d where ‘Y \z -z.\ -v (z.+z.) +y (z,+z,) +y \z.-z“ e V j 1 +_ ive v J 1 + sze v J 1 + Flnfzve v J 1 f (z. z.) = - V 3‘ 1 rmrzv 1 (2.23c) and v represents both n and m. The appropriate boundary con- ditions from Equation (2.22) are Ii(hi) E 0 i = l,2,...,N (2.23d) Equation (2.23a) is now a system of N coupled integral equations in one dimension which is suitable for numerical solution. The details of the solution are discussed further in Chapter 3. 2.5.2 Strip Approximations A second approximation made for the cylindrical antenna is the strip approximation. Several references are available stating the equivalence between the strip and the cylinder through a con- formal mapping (in particular, see page 3-6 of Jasik [15] and section 1.7 of King [17]). For the infinitely thin strip, the strip width, w, is related to the cylinder radius by w = 4 r . (2.24) The total axial (y) current flowing on the strip must satisfy the relation 41 I(y') = g Jy(x'.y'.z')dS' where S' is the cross-section of the Strip. It will be assumed that the current is separable into x,y, and z dependences such that Jy(X'.y'.Z') = I(y')f(X'.Z') - This is only approximate for the antenna of finite length, but is a realistic approximation under the previous assumption of r8 < h. From the above, it is obvious that the current cross-sectional dependence must satisfy the relation g f(x',z')dx'dz' = 1 . (2.25) I Thus, by determining the functional dependence that satisfies Equa- tion (2.25), the problem is again reduced to a one dimensional one, as in the thin wire approximation. AS will be seen, the function f(x',y') is easily obtained and results in a straightforward re- duction of the problem. If this had not been the case, the (more rigorous) two dimensional problem could have been approached numerically. The degree of numerical complexity, however, would greatly increase and this in itself encourages the approximate approach. For the antenna in the waveguide, the lack of "rotational" symmetry with reSpect to the antenna axis requires that the orienta- tion of the strip be specified. This thesis considers two cases: i) the strip in a transverse or (x,y) plane, and ii) the strip in a longitudinal or (y,z) plane. 42 1) Transverse Strip Approximation For the strip oriented in a transverse waveguide plane, the current in the strip is a surface current and the cross-sectional dependence may be expressed in terms of the Dirac delta function as f(X'.y') = g(X')6(Z' - zi) . for the i-th antenna. The condition (2.25) then becomes a/2+w/2 f g(x')dx' = 1 (2.26) a/Z-w/2 for the antenna located at x = a/2 on the waveguide wall. An .appropriate guess for the transverse variation of current density, g(x'), is deduced through the following reasoning. The y-component of current on the strip is proportional to the §-component of magnetic field (by the boundary condition for tangential magnetic field) and thus g(x') will be proportional to the variation of Hx with x. This variation is obtained from the "physical dual" or dual complementary problem consisting of a slot in an infinite plane (see Chapter 13 of King and Harrison [18]). In the complementary slot, the dual quantity of interest is the fi-component of electric field, So that g(x') is proportional to the variation of Ex with x. The appropriate distribution to use for Ex in a Slot is stated in Galejs [,8] (derivable by a conformal mapping technique using a Schwartz transformation), the result being g(X') = 1 (2.27) “fix-92 - (x' - «vi/2)2 for the geometry of the case in hand. This is seen to satisfy Equathx1(2.26) and thus the condition (2.25). 43 It should be noted here that in the strip approximation made by Eisenhart and Khan ['7], the x distribution is chnsvn as constant across the strip. They subsequently find the resulting input impedance in error and consequently reduce the equivalent strip width to obtain agreement. Based on the foregoing arguments, one would expect the error in impedance to be caused by the choice of current distribution rather than the strip width. Of course, the equivalent strip width is truly an approximation for the cylinder in a waveguide since the arguments of complementarity only hold in an infinite unbounded system. However, this approximation is eXpected to be valid as long as the antenna radius is small compared with the waveguide dimensions. To proceed with the transverse strip approximation, the current distribution on the i-th strip will be taken to be 5(z' - zi) 'nq/(Zra)2 - (x' - a/Z)2 for 0 S y' s h. ; a/2 - 2r s x' s a/2 +12r 1 a a in(x'.y',z'> = Ii(y'> (2.28) The j-th antenna surface, along which the integral equations must be satisfied, is described by the line x = a/2 O Sy Shj (2.29) z = z That only a line is necessary, results from the assumption of an analytical form for the x distribution of current; this being 44 tantamount to satisfying the boundary condition across the entire strip width. Substitution of Equations (2.28) and (2.29) Into the integral equations (2.21) gives N 6(2' - 21) 1 z 1(y') v G (3/2:Y9z.\x',y',z')d\l' = i=1 '1 2 2‘ W J i n (Zra) - (x' — a/2) '31— O S y S hj = C cos(k y) - sin(k y) j 0 2kO 0 .j = l,2,...,N where G;y(x,y,z\x',y',z') is given in Equation (2.18). The volume integration may be performed successively, the 2' integration ' begin straightforward. Integration over the x variable requires evaluation of a/2+2ra sin(k x') I x 'dx' - 2 2 a/2 2ra \/{2ra) _ (x' _ a/2) By a change of variables, u = x' - a/2, this becomes cos(kx u) fig: du for odd n a‘/Q2raz ) O for even n or, by equation (23) in Galejs [8 j, {-1) n go(2era) odd n where 30 is the zeroth order Bessel Function. With this, the integral equations reduce to (using superscript st for transverse strip): 45 h N 1 st V. 1 t n = _ I - E 'f 11(y )Gij(y\y )dy Cj cos{koy) 2k 81n(koy) i-l y o (2 .303) OSyShj j=lflpnfi The kernel for the transverse strip approximation is j e om St 1:00 __ Gij(Y\Y ) 2 abweYnm 03d where fnm(zj‘zi) is given in Equation (2.23c). The system of EMS 30(2era)cos(kyy)cos(kyy')fnm(zj‘21) (2.30b) integral equations (2.30) is now in final form for the numerical evaluation to be done in Chapter 3. ii) Longitudinal Strip Approximation For the strip oriented in a longitudinal waveguide plane, the analysis proceeds as above. In this case, however, the current distribution on the i-th strip is, by analogy, in(X'.y'.Z') = I (y') (2.31) for 0 s y' 3 hi ; zi - 2ra s z' s 21 + 2ra The j-th antenna surface along which the integral equations are to be satisfied is, as before, the line described by x = a/2 O s y s h. J z = z The huegral equations (2.21) become 46 N . 2 P - 58 ‘3”) cl dV' = 1:16! 2 ' 2 yy J i n (2ra> - (z - 2i) (2.32) YJ_. O S y S hj = C. cos(k y) - sin(k y) J 0 2ko O j=1,2,...,N where G;y(x,y,z\x',y',z') is given in Equation (2.18). Again, the separable kernel allows the triple volume integral to be per- formed successively. Here, however, the x' integration is straight- forward and the 2' integration is the more complicated. Integra— tion over 2' requires evaluation of the following integral ' .- 1 . dz — Fv(zj‘zi) The details of this integration are given in Appendix II; the results are, from Equations (II.15a) and (II.16b), Fv(zj\zi) = 420(2Wra)f\,(zjlzi) i aé j (2.33a) Fv(zi\zi) =.Jo(2yvra)fv(zi‘zi) - {g(Zera) i = j (2.33b) where .JO(2yvra) is the Modified Bessel function of zeroth order and complex argument. 1%(2vvra) is the Modified Stuve function of zeroth order and complex argument. and fv(zj\zi) is given in Equation (2.23c). \ It is worthwhile to recall here that the subscripts (i,j) refertthhe Source point antenna and the field point antenna, -——r"‘ 47 respectively. Therefore, the condition i # j implies the use of the field excited by the current on one antenna to satisfy the boundary condition on a different antenna; whereas, the condition i = j implies that the source current and field matching points are on the same antenna. When the above integrations are applied to Equations (2.32) the resulting system of integral equations is (using St for longitudinal strip): N hj st v 2 I Ii(y')Gij(y\y')dy' = C cos(koy) - 2%_ sin(k y) i=1 y'=0 o o (2.34s) 0 S y S h j = l,2,...,N The kernel for the longitudinal strip approximation is j a s1. . = °° __ Gij (fly ) 2 abwe'Ynm 8M8 ' n cos(kyy)cos(kyy )an(zj\zi) (2.34b) odd with an(zj\zi) given in Equations (2.33). This system of equa- tions is now Suitable for numerical evaluation. To summarize Section 2.5, the three dimensional set of N coupled integral equations which were derived in Section 2.4 have been reduced to one dimensional equations via three approximate approaches: Equations (2.23) result from the thin-wire approxima- tion; Equations (2.30) result from the strip approximation applied in a transverse waveguide plane; Equations (2.34) result from the strip approximation applied in a longitudinal waveguide plane. It alsouay be recalled that all of the above expressions are based 48 on the delta-gap or slice generator approximation for the antenna terminals. In the next section, the slice generator approximation will be removed and the exact representation of the physical coaxial antenna terminals will be made. 2.6 Coaxial Gap Solution. In the foregoing sections of this chapter, the physical prob- lem of N antennas coupled to the interior of a waveguide through coaxial ports was analyzed using the slice generator approximation for the antenna terminals. It is possible, however, to approach the problem without this approximation by applying the techniques of the previous section. The complexity of this problem necessitated the consideration of the Special case of a single antenna and its coaxial gap- The Specific geometry for the coaxial gap solution is shown in Figure 2.4. AS can be seen, the problem is the same as that solved previously except for the addition of the annular region made by the coaxial transmission line in the waveguide wall. This annular region, Sg’ is defined by an inner radius ra and an outer radius rb, centered on the broad wall of the waveguide at x = a/2. A time , t harmonic (ejw ) voltage of magnitude V0 is maintained across the coaxial transmission line, referenced to the outer conductor and waveguide wall. The electric field that exists in the coaxial gap is assumed to be the fundamental TEM coaxial field; namely, E (r) = ° “r . (2.35) 49 A Y A x A z I -... | I | K'N , | I I‘ r1L h 2L I J ' ' Y(Y) | S l ...—g... ._-—. J—y:0 I V=O | +V z=-d 0 z:dZ —.X:O ._x_.§ _Z _x:a FIGURE 2. 4 Coaxial Gap Excitation 50 In other words, the higher order coaxial field modes, which are excited by the abrupt junction between the coaxial line and the wave- guide, have been ignored. As already stated, the approach to this problem is the same as in the slice generateo approximation; that is, integral equations are developed based on applying the boundary conditions; where the fields used in applying the boundary conditions are expressed as modal expansions via the Lorentz reciprocity relation. Boundarypconditions To begin, the boundary conditions to be satisfied for the continuity of tangential electric field are: as before, E;(x,y,z) = E:(x,y,z) ... on antenna surface (2.36a) and the new condition for tangential electric field in the coaxial gap: -§ X §1(x,y,z) = -y X Ee(x,y,z) ... in coaxial gap (2.36b) lgppressed fields The impressed electric fields, from Figure 2.4, are seen to 13(2: (Dr: the conducting antenna E:(x,y,z) E 0 (2-373) for 0 s y s h ; (x,z) on antenna surface and in the coaxial gap, 51 fife(x,y,?) = §g(X.y.2) (2.37b) for (x,y,z) on S 8 Induced fields The induced field required in Equation (2.36a) is the linear superposition of the field induced by the current on the antenna with the coaxial gap removed (replaced by conductor), and the field induced by the electric field in the gap with the antenna removed. The former is exactly as found in Section 2.3 for the case N = 1 and V = 0; whereas, the latter is the field Specified by (and resulting from) the boundary condition (2.36b). Using superscript J for the current induced field and Superscript G for the gap induced field, the total induced field becomes, by superposition, E;(x,y.z) = E:(X.y.Z) +E§f . (2.38) The current induced field, from Equation (2.15), is J E 3 3 = J '3 '3 ' G 3 3 '3 '3 ' dv' y(x Y Z) 4;. y(x y Z) ”(X y Z\X y Z) (2.39a) for 0 S y s h , (x,y) on the antenna surface Where 3 3 2 2 G = z z ——29— (k - k )Sin(k x)sin(k x') W abwev 0 y X x n nm I I cos(kyy)cos(kyy )fnm(z\z ) (2.39b) and, fnm(z\z') is given in Equation (2.13). 52 MM As before, the gap induced field is to be found as a modal expansion, where the coefficients are determined by application of the Lorentz reciprocity relation. In this case there are no physical current sources, so the reciprocity relation becomes: 3 (E1 x HZ - E'Z x fi1)-’n d8 = 0 (2.40) The problem is shown in Figure 2.5 (a) where S = (c.S.A) + (c.S.B) + Sg + (waveguide walls). The analysis proceeds in the same manner as in Section 2.3, but before beginning, some forethought is required. It should first be recalled, that the induced electric field is sought in order to satisfy the boundary condition (2.36a) on the antenna surface. In other words, the field must be evaluated on the inner circle of radius Ia in Figure 2.5 (a). It is also recalled, however, that the reciprocity relation only yields results which apply on or outside the chosen surface; i.e., the results are valid for the regions 2 s -rb and z 2 -l-rb for the problem of Figure 2.5 (a). Because (Bf the mixing of cartesian and cylindrical coordinates, the analytical evaluation of the integral over S is difficult unless Guare is taken in selecting the volume for the reciprocity relation. 455180, the total coaxial gap must obviously be accounted for in <313taining expressions for the induced field. For these reasons, it :15; best to consider the two independent problems shown in Figures :1..5 (b) and (c), which constitute the whole problem by superposition. '11: this way, the antenna surface (inner circle) is included on the 53 Region A Region B ""’"W H t" c.S.A c.S.B "1 N> C N N W hn—upu- ””1 F ..S" -—z‘ J'- l u 1 I I 13 3 2| 3 :3 L ‘0:- a? <2 :1: ES 2:“ '3 - - E c: “3' a s ‘4‘ U c .o ' O O ’ 0 § 0;. .640 .81) 'm. .640 e °| as d V .22 I I R | .1 1 ' .4 I I Z z13(1) z = z21(2) z = Z8(2) (b) (C) Independent Problems Giving the General Problem by Superposition FIGURE 2. 5 Geometry for Finding Gap Induced Fields Using the Lorentz Reciprocity Relation 54 surface enclosing the source for each case and the results can be used to satisfy the boundary condition (2.36a) . Considering first the case shown in Figure 2.5 (b), the modal expansions for the waveguide fields are: in Region A(l) , —o _ (1) —o- —'l' EA(1) ’ fAv (Ev+ F13) z s (c.s.A(l)) (2.4la) _ (1) n- 4+ ifA(l) - 5 Av (Hv + 1-.l\2Hv) and in Region B(1), _ (1) 4+ *- if13(1) ' 2 ex» (Ev + rszv) v z 2 (c.S.B(l)) (2.41b) —. _ (1) 4}- -+- H13(1) " gov (Hv + FZvHv) These fields are used as (fil’fil) in Equation (2.40) , while -o—o ,/ (E232) are chosen as fields for a source free region as in Section 2-3- The surface integral in Equation (2.40) is then iterated over the waveguide walls, the cross -sections, and Sg(l) . 0n the wave- guide walls, the tangential electric field is zero so that (E x H) -fi = (i) X E) H a: 0, and these integrals disappear. 0n the crosS-sections, the step8 (a) through (f) in Section 2‘3 (preceeding E qu""Iion (2.8)) are again employed to reduce and evaluate the in- t egra ls. Thus, for example, when solving for A31) , the choice is Inade -o -0 _ -"I" - 4+ -0- -o -a that (E2,H2) - (Ev -I- 1"2\)EV , Hv + PZ‘vHv) , and (E1,H1) are t ak‘eh from Equation (2.41). With this choice, the integral over c I.038 section B(l) is identically zero (as in Section 2.3) so that 55 J‘ (Elxfi’z-E‘Zxfil)-nds+o (11’ xii -E xH)-hdS c.s.£(l)+ 1 2 2 1 c.s.A(1) c.s.B(l) ( _ 1) _ o _ Av (1 Flvfzv)(2ios.(evxhv) 2dS) =A(1)(1 - r r) v 1v 2v In the remaining surface integral over Sg(l), the following observations are made: (E1 X H2)'h = (n x E1)'H2 . by vector identity = (h X E1)-H2 by choice of E} as induced field = (fi X Ee)-fiz by Equation (2.36b) = (a x fig) {1'2 by Equation (2.37b) and (E2 X H1)~n - (n X E2)'H1 by vector identity 0 H Since E2 is chosen as a linear combination of waveguide modes (for source free region) and h X EV = 0 for y = 0. Thus, (E xii -1'=.’ xfi')-fids= (yx§)-fi ds £g(1) 1 2 2 1 ig<1> g 1 A. fter combining the above steps, the reciprocity relation as stated in Iscluation (2.40) yields: (1) 1 A if - A =———— (y XE)-( +r fi)ds (2.423) v P1vr2v.1 {8(1) 3 V 2v v and ‘“ 56 (1) 1 A ‘2 ""' "+' C = ———-_—' (y X E )-(H + I‘ H )dS (2.42b) v rlvFZV 1 ig<1> g v 1v v Following the foregoing procedure for the problem of Figure 2.5 (c) yields (2) 1 A "2 “'1’ ~0— A =--—-— (y XE)'(H +1“ H)dS (2.433) 8 (2) 1 A -0 —r}- —o— c = -—-—-_ (y x E )‘(H + 1‘ H )dS (2.43b) where, for example, the following expansion was made §B(2) = E c32)(fi': + I‘ZVEQ z 2 (c.s.B(2)) (2.44) With the eXpansion coefficients determined in Equations (2-42) and (2.43) , in terms of the known gap field, the problem is essentially completed since substitution into the modal exl’élIlSions defines all the desired field quantities. The field which is sought for the boundary condition is seen to be the superposition of the electric fields on cross-section c.s.A(l) and on cross- section c .s .B (2) . Thus . «G _ ~ 4 _ _ E - EA(l) + EB(2) for z — zi (— 0) (2.45) S 0 1“(1e these are all that is required, further attention need only Au) (2) V e focused on the coefficients and C . v As can be seen, the coefficients are dependent upon whether TE 0r TM modes are considered. It is best to make the distinction at this time, such that, from Equation (2-418) : 57 —o e -*e— -*h+ EA(1)=§AW(E +F1VEV+)+zAh(E: +r1va) and from Equation (2.44) , _. -e- h —oh+ —h- B(2) — v v v Zva ) + E Cv(Ev + FZVEv) Since only the y-component of these fields is required for the boundary condition, the fields of interst are, e e +sz + -sz Ey.A(1) — E A\) ey,v(x,y) [e rue ] + +y z -y z Aeh V v + EA v (x,y)[e + The 1 (2.463) and -y z +y z e e v v E = C , + + Y,B(2) E v ey,v(x y>[e sze 1 h h 'sz +sz + 2 C\, ey’v(X.y)[e + sze ] (2.46b) v In Equations (2.46) the expressions given in Equation (2.6) have been used to separate the (x,y) and z dependences. Expressions for ey,v(x,y) are given in Equations (2.5). The coefficients in Equations (2.46) may be eXpressed (from Equations (2.42a) and (2.43b)) as e l e+ e- Av - ———F1v:2v'1 [draft 1‘“ 3(1)] (2.47a) h + - Av = 1‘1 1‘2 -1 [31(1)+F2v5}(1)3 (2'4”) Ce = ___1___ v FlvFZv 1 [3(2) + 1‘2 V3(2)] (2.47c) h 1 h- cv = __1" 12oF _1 [3(2)+ Bah”) (2.47d) 58 ;( )+ (8 h)+ h = (yx Eg) fi'h ds (2.48a) 3(1) (1) g e)+ (e )+ J; _£ MR h d3 (2.48b) 3( ) 3(2)® E8) j These eight integrals are evaluated in Appendix II. As can be seen from.Equations (11.31), (11.41), (11.43) and (11.44), the expressions 7 for the integrals are extremely lengthy. Subsequent substitution into the expressions for the coefficients, Equations (2.47), and then the expressions for the electric field, Equations (2.46), re- quires a great deal of algebra, albeit straightforward. The final result for the y-component of the gap induced electric field is, from EQuation (2.45) , n+1 f;-s --jQL--< 1)—§—. €°m (kx ) s cos(p¢)/p} P odd for z = 21 (= O) and n = odd (2.49) DBES(p) = gp(rb§) - yp(ra§) k 2 2 2 § = kx - Y, t = arctan(kx/jvv) Otl"er notation is described in Appendix I. 59 Now that the gap induced field is explicitly evaluated, the total problem undertaken in this section is complete. Substitu- t; ion of Equations (2.49) and (2.39) into Equation (2.38) gives the S‘r—component of induced electric field. The boundary condition (2.36a) wi th Equation (2.37a) then represents an expression which may be solved for the unknown current distribution. That is, J G Ey(x,y,z) + Ey(x,y,z) = 0 (2.50) is the desired integral equation for the coaxial gap problem. In the slice generator approximation, it was found convenient to reduce the integral equation to a Hallen type by creating an integro-differential equation. This reduction is not necessary here because all functions are non-singular (there is no delta function). In order to obtain a numerical solution, however, it is best to approximate the current distribution, as before, in order to reduce the problem to one dimension. It can be seen that the strip approximations made for the slice generator problem are not acceptable hare because the coaxial gap would also need to be transformed. Therefore, the most straightforward solution is found by making the thin-wire approximation. To apply the thin-wire approximation to the coaxial gap Problem, the current is again taken to be Jy(X',y',Z') = I(y')6(X' - a/2)6(z') O s y' s h “"’here the one antenna considered is taken at the origin (2' == 0) of coordinates. This approximation is now applied to the integral e3"ipression for the current induced field, E5, given in Equation pv— 6O (2 -39), and the x' and 2' integration performed. Next, the integral equation represented by Equation (2.50) is satisfied on the cylindrical antenna surface along lines defined by x = a/Z + r -a OSysh z=0. With these steps, the integral equation becomes (from Equation (2-50)): J G Ey(a/2 i_ra, y, 0) +-Ey(a/2 i ra, y, 0) — J _ _ G Ey(y) - Ey(Y) (2.51a) (2.51b) Where, from Equation (2.39) after integration, the current induced field is h J - I 2 I Ey(y) - I I(y )6 (y\y )dy 0 S y s h (2-52a) I yy y =0 With 3 e 2 2 221' E2; ' = 0m - k - -1 ny(y\y ) t2“. t2:“ mko y)( 1) cos(era)( ) odd U cos(kyy)cos(kyy )fnm(0\0) (2.52b) and: from Equation (2.49) , the gap induced field is ~EGQI2“—'1;7"r—"€vx‘30~°'(k r a)cos(k y) y( a) abm y I‘le‘:———v_—- Q x {(1 + F1V+r2+ 2)1'r DBES(0) + j4(1- -r r, ) 2: DBES(p)Cos(p¢)/p} odd (2.53) 13 61 Equation (2.51b) is now the integral equation to be solved for the unlcnown current distribution. It is in suitable one dimensional form for numerical evaluation, which is discussed in Chapter 3. 2 - 7 ConclusionLComparison with Literature. As was stated in Chapter 1, the problem approached in this research has not been solved in the published literature. Nonethe- less, there have been similar problems solved which represent important special cases of the theory presented here. Thus, by giving proper consideration to the approaches of other authors, it is possible to obtain support for this general theory through Cotnparison with their results. For example, in his book Advanced ILeog 9_f Waveguides (see Chapter 4 of [24]) Lewin approaches the tuned post (i.e. the post does not Span the waveguide) from the vieWpoint: of the waveguide and derives an integral expression in terms of the antenna current for the component of electric field in the waveguide which is parallel to the antenna or post. Since in deriving this expression Lewin has made assumptions equivalent t 0 the thin wire approximation considered in this report (see eXpression 4.3 of [24]), it is possible to compare Lewin's result to the expression for induced field obtained in Chapter 2: namely, Equation (2.12). Since in Equation (2.12) the thin wire approxima- tion has not been made, this must be done before comparing to I‘ewin's expression (4.3) . Thus using I I I = I I _ I _ I , in(x ,y ,z ) 1i(y )6(x a/2)5(z 21) for 0 S y s h in Equation (2.12) and integrating over the Specified delta functions, 62 the induced electric field in the waveguide is found. Also, since Lewin considers the Special case of an infinite waveguide, the expression (2.12) must further be specialized by considering zero re flect ion coefficients: Wl'ien this is done, Equation (2.12) becomes (with the additional Simplification of a single antenna, N = 1, located at 21 = O) ' ‘j 6 2 1 _ om _ 2 . . Ey(x,y,z) — Z a—bweY (ko ky)sm(kxx)Sin(kxa/2) n,m nm ‘Y Z h cos(kyy)e “m f 1(y')cos(ky')dy' . (2.54) y.=0 y This is seen to be identical to Lewin's expression (4.5) for the Case of a post centered in a broad wall of the waveguide (i.e. Stat d = a/2 in his reSult). (Note that Lewin has chosen to inter- change the roles of x and y coordinates and uses a different “Otation for Neumann's number, eom') This equivalence between the tWo theories lends support to the results found in this report. Further comparisons may be obtained from the developments of Eisenhart and Khan [7]. In their paper they considered the single post and approached it as an equivalent Strip. Their final expression for induced electric field cannot be directly compared to the general theory presented here because of its dependence upon the assumption of a current of uniform density across the strip, in contrast to the inverse square root edge Singularity form dQVeloped in this report (Equation 2.27). However, various aSpects of their development are comparable; in particular. their Green's 63 function and their consideration of a terminated waveguide. The creen's function is given in their expression (5) and is obtained from the work of Tai [38]. It is seen to be identical to the Green's f Lulction which is independently derived in this report, Equation (2 .14), except for the coefficient and the exclusion of the factor accounting for waveguide terminations (i.e., rlL = FZL = O in Equation (2.14)). The difference in coefficients results from the term jun. being included in Equation (2.14) (as described following Equation 2.3)). Removing this term from Equation (2.14) gives exactly the Green's function shown in (5) of Eisenhart and Khan [7]; namely , e 2 ‘Y \Z‘z'\ G (r r' = 2 -—91———— (k - k )e sin(k x)sin k x' yy 1 ) n,m absznm o y X ( X ) O cos(kyy)cos(kyy') . (2.55) To include the case of arbitrary waveguide terminations, Eisenhart and Khan have stated that this Green's function may be multiplied by a factor, denoted by '1', which takes the terminating boundary COnditions into account. This factor (from expression (30) of [7]) tray be written in the notation used in this report as l + + 1“1 ,nm I"2,nm + I«l,nmr'Z Lnm T = (2.56) 1 - I‘ I" l,nm 2,nm where F1 nm is defined in Appendix 1. This is the same as the fa(Itor fv(0\0) given in Equation (2.13) (except for a minus sign whitm is incorporated in the expression ny of Equation (2.14)). (Racall that v is used to represent both n and m.) In other 64 words, the factor 1’ of Eisenhart and Khan agrees with the results 0 f this report in the case of T1118 condition implies that the source point, z', and field point, z , are at the same location; i.e., on the post (taken as the origin of coordinates). This is, of course, all that is required for the Single post problem and thus supports the results of this report's general theory in the Special case. Since Eisenhart and Khan were only concerned with this Special case, there is no difficulty in the ir use of this factor '1'. However, it is extremely important tO realize that this factor '1' is not appropriate for arbitrary Source and field points, z' and z, in the waveguide. In this reSpect the Statements of Eisenhart and Khan must be modified, since they imply that the general Green's function for a waveguide with arbitrary terminations is obtainable by multiplying the Green's function for an infinite waveguide by this factor '1'. Since 1' is independent of coordinates, their implication is that the fields reflected from each termination experience the same amount of phase change regardless of the location of the field point. This is seen to be obviously incorrect when the field point is located arbitrarily with reSpect to the center of coordinates and the source point. A s innple discussion to extend the results of Eisenhart and Khan is not possible, but it is seen that the theory developed in this re- Port includes the proper phase shift factors in the term fv(z\z') ShO'wn in Equation (2.13). It may therefore be concluded that the Green's function for a waveguide with arbitrary terminations is £92. 65 d irectly obtainable from the infinite waveguide Green's function by a Simple multiplying factor. The proper "Green's function" is given in Equation (2.14). It bears repeating at this time that the term "Green's function" is used loosely in referring to ny of Equation (2.14) . AS the reader Should be well aware, the factor jw is included in ny and this must be considered before using ny in the sense of a general Green's function in another context. The theoretical develOpment for the problem of N coupled antennas in a rectangular waveguide has been presented in this chapter. Although it has been necessary for conciseness and clarity to omit much of the algebraic details, the resulting el-tp‘ressions have been shown to be valid in comparison with similar Specialized cases available in the literature. Further verifica- tion comes from the agreement among the reSultS from each of the approaches considered (see Section 3.3) and from the ability of the theory to predict published experimental results of other authors as well as the experimental results obtained in this research effort (See Chapter 5) . I “_ CHAPTER 3 NUMERICAL METHODS In Chapter 2, the integral equations for the current dis- tributions on N antennas in a rectangular waveguide were de- ve loped. Three different representations for the slice generator model and one representation for the coaxial gap model were obtained in approximate one dimensional forms suitable for solution by numerical methods. Although analytical solutions could be sought in the manner of Hallen or of King and Middleton (see King [17]) the results essential to this report can most readily be obtained numerically. As will be shown in this chapter, the solutions Obtained from the single antenna coaxial gap theory indicate the Validity of using the slice generator model for comparison with experimental results. Also discussed in this chapter are the reasons for selecting the transverse strip approximation in the slice generator model as the best approach. Section 3.1 contains the techniques used to obtain the numerical solution while Section 3.2 discusses the convergence pro- perties of each representation. The last section presents some representative numerical results which demonstrate the agreement betWeen the theories. The numerical results obtained in this report are computed using the Fortran Extended Language on a Control Data ' 6500 Computing System. 66 67 3 - ‘1 Matrix Equation Representation The integral equations obtained in Chapter 2 are solved numerically by the method of moments For the problems in hand, his is a two step process: first, the unknown Current is approximated by the method of subsections (see section 1 S of Harrington [11]); second the equations are satisfied discretely by the point matching method (see section 1-4 of Harrington [11]) The results of this procedure is a matrix equation in the unknown current distributions which is easily implemented for computer Solution. This process is the same regardless of the form of the Integral equation and thus is described in detail only for the thin- Wire case (the other approximations will be treated analogously) Equations (2.23) for the thin-wire approximation are re- peated here for reference N h' i V. f I. (y' )6: ‘Jf'my )dy' =c. cos(k y) -,_—1—sin(Ay) q q 2 i (AY)i = hi/Ki u. ' fl 1’pW) 1 -I.. *l I I Ia. (Ay)i-ev-{ : I l I I I I Y' ' I I I l l l l . yP=1 : yp-Z yp-l : yp l l I FIGURE 3. 1 69 3.1.1 Method of Subsections. The first step in converting the above system of equations (3.1) to a matrix equation is to expand the current in a sot of basis functions by the method of subsections. The ith antenna is subdivided into Ki equal subsections and the functional dependence of the current is specified in each subinterval by the basis functions. The set of basis functions used here are the pulse functions shown in Figure 3.1. The current is then approximated as K. i I ___ I Ii z ai’pui,p(y > (3.2) p=1 where the pulse functions ui p are 1 ' s ' < ' ' Yp_1 y yp ui,p(y ) — (3'3) 0 elsewhere and the subsections are defined by k4 II MAY)i (3.4) (Ay.) = h./K, 1. 1 1 as shown in Figure 3.1. The current is therefore assumed constant Over each interval and the unknown current distribution becomes a Set of K unknown amplitudes oz. 1,9. When the current of Equation (3.2) is Substituted into 1 Equation (3.1) the indicated integration becomes simply: ' i P cos(kyy')ui,p(y')dy' = cos(kyy')dy' (3.5) ‘<¢——45‘ ‘4“:‘4 I: I o p-l For conciseness, let .JI-n..__ 70 Y J. cos(kyy')dy' = (Tm)i,p (3-63) yp_1 where, using a trigonometric identity, 2 ii; Sin(ky(AY)i/2)C°S(ky(é)’)i(P"5)) m I o (T ) - (3.6b) m 1.9 - (Ay)i m = 0 where the subscripts indicate the p-th subsection on the i-th antenna and emphasize that the result is dependent only on mode index In (Ry). With the method of subsections then, the system of integral equations become a system of algebraic transcendental equations: K N j m m jeom E E or 2 2 ————cos(kr)f (z ‘2.)(T). cos(k y) = i=1 p=1 1,13“ m abweynm xa nm j i mi,p y odd V. = Cj cos(koy) - fil—Zko Sln(koy) (3.7) OSySh. j=1,2,...,N J The condition that the current be zero at the end of each antenna becomes the set of conditions ai,K a o 1 = l,2,...,N (3.8) 3.1 .2 Method of Point Matchirg This system of algebraic equations is now converted to a matrix equation by the method of point matching. Instead of satisfying the equations continuously in the variable y, an 71 approximate solution is found by matching the two sides of the equation at discrete points in the interval 0 s y s hj. This correSponds to satisfying the boundary condition at a discrete set of field points on the j-th antenna. In order to obtain a con- sistent set of equation, the number of matching points must equal the number of subsections on the same antenna. The midpoint of each subsection is chosen, so that for the j-th antenna, the matching points are (see Figure 3.1): = (q-%)(AY)j q = 1,2,---,Kj j = 1,2,---,N (3-9) yj.q with , = h (Ay)J j/1dy 2“. y=031“(ko>’)“j,q(y>dy (3.13) From the left-hand side of Equation (3.13) the integral is h yq k d = k d 3.14 g cos( yy)uj,q(y) y I c08( yy) y ( ) yq_1 In the notation of Equations (3.5) and (3.6) this may be eXpressed as J- 74 q I cos(kyy)dy = i‘ sin(ky(Ay)j/2)cos(ky(AY)j(q-%)) ... m # o y _1 y q (AY)j m = 0 Again this notation indicates the result (Tm) is for the q-th j’q subsection on the j-th antenna and is a function of the mode index m (Ry) only. The right-hand side of Equation (3.13) involves the integral h 2 —— ‘ -;’ y£OCOS(ROY)Uj’q(Y)dy k0 Sin(kO(AY)j/2)cos(ky(Ay)j(q 2)) 3. ) (T)j.q ( 1° which is dependent on the wavenumber of the medium (kc) and the q-th subsection on the j-th antenna. The last integral on the right-hand side is denoted as h . a. . . _ yio Sln(ROY)Uj’q(Y)dY k0 Sln(ko(Ay)j/2)Sin(ko(Ay)j(q %)) . 3.17 (T )j.q ( ) Substituting the results (Tm) , (T)j q’ and (T') given in .1"! 3"} (3.15), (3.16) and (3.17), respectively, back into the Equation (3.13) gives N i m m jeom . -———-——- s k r f . T . T . = 121 p21 al’p E 3 abweYhm CO ( X a) “m(zj\zl)( m)1,p( m)J,q odd L = c, T - T' , (3.18) J( )j.q 230 ( )J.q ¢(Y). degrees |I(y)\, ma/volt percent A . 40 20 1.6 1.2 0.8 0.4 75 . A Galerkin's Method (a) P H 0 point matching “ Absolute current per a a volt in polar form - g _ II I: a .. B _ a a a - o _ O O - é - 6 O I- e 4 l l J J 1 1 4 J 1 .2 . ‘ 4 y/h 6 8 l O I O " I D B u L :1 Z. .— 6 b (1 0° 3 l n a _ 0,, a=2b=l6.51 cm '3 a d = 8.08 cm I B 1 . . , . , 4 1. "43—4 f = 1:3 Ghz .2 .4 6 -8 1-0' h=5 cm K = 20 _ MMAX = NMAX = 34 __ (Galerkin) - (pt. match) a o r A - Galerkin X 100/" loo 0 L .2 .4 o 0 ° W n . .0 Al 4 L 0A I #17; fl 1 v if j ' l . . o o o o o o 6 y/h 8 1 0 Percent Difference - o L 0 FIGURE 3.2 Comparison of Galerkin's Method and Point Matching 76 Equation (3.18) is now the matrix equation desired for the unknown current distribution amplitudes ai,p which results from applying Galerkin's method. In order to investigate the degree of improvement in the numerical solution, several cases were solved both with the result given in Equation (3.11) from the point match- ing approach and Equation (3.18) from the Galerkin's method approach. Despite the differences in form between Equations (3.11) and (3.18), the results in all cases were practically identical. An example of the computed current distribution for a single antenna-waveguide configuration is shown in Figure (3.2a) with the percent difference shown in Figure (3.2b). (Note the highly expanded scale.) Obviously, for this problem the point matching method is as accurate as re- quired. Since point matching also offers the greatest intuitive understanding of the numerical process, it is used throughout the remainder of this report. 3.1.4 Matrix Equations for the Strip and Coaxial Gap Solutions. In the previous subsections the details of applying the method of moments to the thin-wire approximation integral equation are presented. The same methods may be applied in an analogous manner to the integral equations that have been developed for the other approximations to obtain matrix equations suitable for numerical solution. The similarity of forms between the various equations, and the fact that only basic trigonometric or delta func— tion integrals are involved makes it possible to write the results down by direct comparison with the steps of Sections 3.1.1 and 3.1.2. 77 For the transverse strip (superscript "st”) approximation from Equation (2.30); namely, N 1 V 2 J‘ I,(y')GS,'f(y\y')dy' =C.COS(1< y) - “Lsirwk y) i=1 y'=o i 1] J o 2ko o osysh. j=1,2,...,N J ‘9T11:h 90(2era)cos(kyy)cos(kyy')fnm(zj‘zi) St co co jeo G1j(y\y')= z Isa—Lb n m weYnm od d Isy'the method of subsections as in Section 3.1.1 this becomes i oomjg 2 a. Z Z 1 p=l 1’p n m abweYnm om - 90(2era)fnm(zj‘zi)(Tm)i,Pcos(kyy) — V. _ _ _L_ - . . = _ Cjcos(koy) 2RD Sin(koy) o S y s J J l,2,...,N (where the boundary condition (3.8) has been used). Now, by point tnatching as in Section 3.1.2, this becomes N Ki'l -V. 2 z (33‘) a, - C.cos(k y, ) = z—l sin(k y_ ) (3.19a) i=1 p___1 ji Pq 19p J O Jaq k0 O J9q (1:192! ,Kj J =192) 2N Where (58:) = E ; jeom ' (2k r )cos(k )f (z \z )(T ) (3 19b) ji pq ‘-—‘“’“'#o x a yyj,q nm j i m i,p ' n m abmeynm odd With (Tm)i,p and fnm(zj\zi) given in Equations (3.6) and (3.1a), reSpectively. 7' Elli ‘0 78 It should be noted that the only difference between the thin- Wire approximation (Equation (3.11)) and the transverse strip approximation above is the replacement of cos(era) by 510(2er3). This will be seen to be significant in relation to the convergence 0 f the series described in Section 3.2. The numerical solution of the matrix equation found above is, of course, straightforward t17'n.':ough the use of standard matrix manipulation techniques. For the longitudinal strip (superscript "31") approximation from Equation (2.34); namely, h N j 51 V, I I I = _ __l , . E f'_ 11(y )Gij(y\y )dy CjCOS(kOY) 2k Sln(koy) i-l y -o o OSyShj j=l,2,...,N with 631(ny') = 2 02° 1.0m cos(k y>cosF (z \z) 13 n m abu“an y y nm j 1 odd and Jocos(uY)/u} (3.22e) u odd fv(0\0) is given in Equation (3.1c), and v represents both n and m. The symbols used are defined following Equation (2.49) and in Appendix I. Combining the above results the matrix equation for the single antenna coaxial gap problem may be written as: K cg m m S a = Z cos(k p51 pq p E m Rhm yyq) q l,2,...,K (3.23) odd . Cg . d d 3.22 d .22 With Spq and Ehm efine in Equations ( c) an (3 e), reSpectively. In summary, any of the four matrix equations (3.11), (3.19), (3.20) and (3.23) may be used to calculate the current distribution on the antennas. The results will be a set of Ki complex amplitudes, ai,p’ for each of the i = l,2,...,N antennas (N = 1 for the co- axial gap problem Equation (3.23)). Although the amplitudes are assumed constant over each subsection, they are the correct amplitudes only at the center of each subsection (yi q) where point matching 9 was applied. The real and imaginary parts of the ai p correSpond , to the in-phase and quadrature components of current on the antenna (referenced to the driving voltage). The input currents are taken 'Eh‘ga’. c... 82 to be: the amplitudes calculated on the first subsections, ai,1° Fran these input currents and the known applied voltages, all Partinent characteristics of the system (as viewed from the antenna terminals) may be determined. Finally, it should be pointed out that the dependence of the current on the electrical parameters 0f the medium is contained explicitly in the matrix representation. In other words, the complex permittivity and the permeability must first be calculated for the Specified dielectric constant, con- ductivity, and relative permeability of the medium of interest. The numerical results are presented in Chapter 5. 3.1.5 Numerical Determination of Charge Distribution. Up to this point, all work has been aimed at determination of the antenna current distribution. Also of interest, however, is the distribution of charge on the antenna. It has not been necessary to be concerned with the charge density in the theoretical development since it can be determined from the current via the continuity equation. Although the charge density provides no independent information theoretically, the current and charge can be measured independently in an experiment. Therefore, it is desirable to calculate the charge distribution for use as an alternate experimental check. For a y component of current with only y dependence, the continuity equation is d 711.5113, jw Q(y) = o . (3.24) This gives the charge distribution, Q, referenced to the input 83 currernt. It is desired, however, to obtain the charge referenced, in phase, to the input voltage (denoted as 6). This is just ‘— d em = H mm = $7151)- - (3.25) Numerically, Equation (3.25) is evaluated by the method of finite differences; that is, charge at a point between two adjacent sub- sections on the antenna is approximately proportional to the slope of the (straight) line connecting the current amplitudes at the center of each subsection. Therefore, '_ (a - a. ) =.l 1:P+1 1’9 p = l,2,...,K (3-26) Q14) 00 (A301 1 Results obtained for the charge distribution are presented in Chapter 5. 3.2 Convergence Properties. The matrix elements for each of the representations found in Section 3.1 involve two infinite summations over the waveguide mode indices m and n. For numerical evaluation, these summa- tions are truncated at MMAX and NMAX, where these values are determined by the desired accuracy and the convergence properties of the series. In this section the nature of these series will be discussed and the criteria derived which determine the optimum values for MMAX and NMAX. Because of the complicated nature of the series and the coupling that exists between the effects of indices m and n, much of the analysis is necessarily qualitative. 84 Under: suitable approximations, however, the criteria for truncation are found and it is possible to compare the results to the work of other authors . The series involved in the matrix equation for the thin- Wire approximation is given in Equation (3.llb) and is repeated here for reference. je om -————- cos k r k , abweynm ( x a)COS( ny,q) o d (3.27) (T ) f (z z ) m i,p nm j‘ i 2 m It is simplest to first dispense with the reflection coefficient factor, fnm(zj‘zi) for large m and n. As given in Equation (3.1c) this factor is (again using v as a shorthand notation for both m and n) f z = vint matching (according to Equation (3.9)) - that is, the Spatial period of the field matching - and at m = MMAX the field has a s’13.9.tial period of twice this amount, then the criterion of Equation (3 .36) corresponds to the use of a "sampling" frequency which is twice that of the field being "sampled". For those readers acquainted with sampling theory this observation is seen to be equivalent to the fundamental theorem for sampling (Nyquist criterion) ‘IEEI. .III.‘ 90 E mo coHuocsm w mm mocomuo>coo o.H o.~ ¢.m mmonm £\Mn¢ n\xnm z xxxnm nxxn oq on om 0H d . . _ n q . .J _ _ _ a a _ _ _ _ q — .4 _ _ _ q . q q _ . _ . fl « a . .4 .. a. _ - . _ .. _ 1 - _ 4 - m. u a u a q _ 4 I. H H V II: Q U . DI. _ 4 l m u M . «\n u n . I _ 4 "Lugs Asm.mv cowumsvm wo meow Hmwuuma m \MV 1 .. . d a an Cam». I . “------Quw ... .. . . .. .--......--.-.. ---: MY D Nu . 1”! .b!l...culoooou.0uu .D DID-mom D D D D 0 lg unuuuuuoooua?.. .. . mono; _________ mamadgdm_....tacaccaa..__ 111 um dd (ms) 3 N p92119m10u ‘ y. 91 which states that the minimum sampling frequency must be twice that of the signal being sampled in order to accurately portray or re- construct the signal. This apparent connection between the method Of moments in modal analysis and sampling theory provides an inter- e S t ing point for further study for those familiar with sampling theory and techniques. For the moment, however, it serves to supply some helpful intuitive insight into the choice of truncation point. Further observations concerning the criterion given in EQuation (3.36) can be made by comparing this to the results obtained by other authors concerned with convergence of solutions in modal ana- lysis problems. For example, in the paper by Eisenhart and Khan [73 the convergence of an admittance function is studied for a gap (having an assumed constant impressed electric field across it) in a Post spanning a waveguide. They determined that convergence was rea.<:hed by taking the maximum mode field m = MMAX given by WAX=b/g , WQre in their case g is the gap width or interval over which the aneguide field must be matched to the impressed field. A QQ‘lrreSpondence can thus be drawn to the case studied here where the i"iterval over which the fields are matched is given by h/K. In th is context then the criterion of Eisenhart and Khan and the one tie\Jeloped here are seen to be equivalent. Another pertinent example in the literature may be found in the paper by Lee, et a1. [22] where mode matching across a Waveguide iris leads to similar convergence phenomenon. They determine that convergence is reached when the number of modes 92 considered in the incident waveguide (MMAX) is related to the number of modes chosen in the iris (say, K) by the expression veil-1e re b is the incident waveguide dimension and h is the iris d imension. This criterion is obviously the same as that chosen in Equation (3.36) if the antenna subsectioning is viewed as a modal expansion (which indeed it is since the current is expanded in a set of basis functions). A further point of interest which is observed by Lee, et a 1 - [22], is that the number of waveguide modes, MMAX, should be Se lected first and that the number of iris modes should then be determined by Equation (3.36); i.e. , S ince in their problem the solution sought diverged if MMAX (from Equation (3.36)) was greatly exceeded. It is certainly not obVious that this type of relative convergence would be expected, and it is not known if the solution of this report also exhibits th is phenomenon, but it is a point well taken that the criterion of Equation (3.36) should be satisfied as nearly as possible. Having studied the convergence of the series given in EcIliation (3.31) as a function of mode index m, it now remains to investigate the series as a function of mode index n to determine a suitable truncation point NMAX. In this light, Equation (3.31) may be written as CW (3 ) = c -——-—-———- (3.37) pp mn m It is obvious that m and n are not independent because of their relation to v . Assuming m and n are large enough to dominate mn 2 the term k and assuming waveguide dimensions given by a = 2b, 0 then / 2 2 v N n/a n + 4m mn With MMAX already determined above, Equation (3.37) becomes tw cos (n11 ra /a) m C . (Spp)MMAX,n MMAX 2 . (3 38) n + 40411311002 This may now be studied for the example case of Figure 3.4, where WAX = 10. If it is assumed, for the example, that ra = a/20, the result of summing terms of the form given in Equation (3.38) is as shown in Figure 3.5, curve (a). Obviously convergence is not reached even when summing terms as large as N = 100 and the reSult of premature truncation could be totally erroneous. Part of this apparent problem stems from the fact that m and n have been considered independently. In fact, if the convergence as a ELI‘laction of n is considered for the m-th terms of the series in with m is much less than n, then Equation (3.37) becomes tw cos (n11 ra /a) a: C .3 The convergence of these terms is shown in curve (b) of Figure 3.5. (Note: Equation (3.39) decays as (II)-1 while Equation 3.39 decays 2 2 - as (n + 4(mAX) ) 35). In this case convergence is reached to 3.0 2.0 “:3 23’ .H 1.0 H E 3—4 0 c: " o a as E s 513. =25 tel s: 41.0 .0 94 - AA _ / ‘\-—~ 10 or n > 1.6(a/r ) , x a a where the n.-35 dependence provides the required convergence. As 96 was done above in Equation (3.37), the n-dependence of the matrix elements for the transverse strip approximation may be obtained from Equation (3.19b). The (m,n)-th term in the double summation may be written as g (2k r ) (Sgt) =c —9-—"—"1— (3.40) pp Inn m Yum With the MAX given by Equation (3.36) this becomes st — }O(2era) S — C ( pp)mn MMAX V2 2 n + 4(MMAX) for a = 2b. The convergence of the sum of these terms is shown (3.41a) 1 in curve (a) of Figure (3.6) for the same parameters as studied in the example of Figure 3.4. Obviously the series is much more rapidly convergent than when the pure cosine term is present. The convergence is even more dramatic when the terms involving m < n are considered such that EQuation (3.40) becomes (2k r ) (Sgt) ec LL?— . (3.411)) pp mn m n The series of these terms is shown summed in curve (b) of Figure (3.6). As before, the true convergence of the series involving mode index n lies between the two extremes shown in Figure (3.6) depending on the relative size of index m (ky). As shown in Figure (3.6) the smallest value of n at which the series can safely be truncated is given by a NMAX — zra (3 .42) 97 coHumexouaad awwum wwum>mcme ago wow a mo coHuocsm m mm wocowwo>coo o.m 553m m m m m 2 “\mm u\mm u\m wN\m ow 0m 00 om oq om ON 0H _ a a a A \4 \4 fizz _ Aaaq.mv cowumsvm mo 55m Hmwuuma u Aav o>wso _ Amaq.mv coHumsvm mo 55m Hmwuwma u Amv m>wao _ I _ .Yo.H 3c _ u _ 1 OIOIOIOIO:O-OIO10 O.O|O-OIO:O-O -Ouo-O..O-OoAOIOIO-OIO-O\O-O‘MV\%©:O/Oblo \0 4\4/4 4 ‘QIQ / d W4\4I4/ 4 /d a\ . I .-..-----.4./.a/.. ---: -...q\a\.::..-- ..-vq-..---::..q.\-.:.: -:-a./..-: ...---.:\.--- - -.w GIA$|4\\ ./< \. 4 d _ I {4‘4 //4 \ . 4 l va /4\4 \ow _ l -.-.....+..--.--.- lo.N f 4 l // 4 \ 4 L a _ rm _ _ F um dd u pazlteuuou (:33) i1 :nal C‘ '1. .rfu.‘ ‘ “"3 .yovV‘ ‘ nl ‘ 3355.. 28 sq. I 4 u ‘c‘actcr: a at Asx. Mr... v‘l .UL “:5 r 5.; s x ‘ ._, 5C . ‘nr «up a ‘ a e t '\ .>\ is I. it ’8‘ 98 At this point errors less than 8% can be expected and with the additional convergence of the series for index m this error will be substantially reduced. Therefore, although the choice of an appropriate truncation point for the thin~wire approximation was difficult, the transverse strip approximation has lead to a satisfactory result given in Equation (3.42). As with the criterion developed for MMAX in Equation (3.36), some reflection on the implication of the criterion for NMAX offers intuitive appeal for its choice. For the y component of electric field considered, the NMAX-th mode exhibits one-half wavelength variation in the x direction in the distance a /NMAX = Zra Since this represents the diameter of the cylindrical antenna or the interval of field matching, this criterion can again be related to the Nuquist criterion for sampling as before. In general, of course, this connection with sampling theory is not definitive (as it now stands) but the insight is beneficial. The NMAX criterion of Equation (3.42) can also be compared to the results of other authors concerning similar problems in the literature. In particular, Eisenhart and Khan [7] in analyzing the thin post Spanning a rectangular waveguide determined that their finpedance function series would be satisfactorily convergent to 17° for a truncation point given by NMAX = a/2ra which is identical to Equation (3.42) above. In actual use they nature the c: 2C8 3D I 'n? 9 5 -1 Ed. ‘6' Lazas - c M" u I. ‘ ...3 5.“ any- bps ~- y g m. ..Ab» o than A vs.. ‘ r d . Q- C" ii I :Y Q A ta: qali .- '» .3 an( x :H' § iuu LI; a .3 r .L 3 r; f e ‘1\ ,5 .. . .13. .... ...: v . . . 99 decided that half of this value was sufficient due to the stationary nature of the impedance formulation with respect to approximations made on the currents. It is not apparent from the above discussion that a similar easing of the criterion given in Equation (3.42) is desirable and so this is not done here. The use of Equation (3.42) throughout the numerical solution is, however, substantiated through this comparison with the literature. Quantitative studies of the convergence of the other approximate forms for the integral equations, namely, the longitudinal strip approximation (Equation 3.20) and the coaxial gap solution (Equation 3.23) is not presented here. However, the results would be qualitatively comparable to the above discussion. By comparing Equation (3.20) with Equation (3.19), one sees that the longitudinal strip approximation exhibits the same dependence on mode index m as the transverse strip and a similarly convergent dependence on mode index n (i.e., both the ordinary Bessel function, ’0’ and the modified Bessel function, 4%, decay for increasing arguments). Also, by comparing Equation (3.23) with Equation (3.11), one sees that the coaxial gap solution should have convergence properties similar to the thin wire approximation. The comparison of the results for the four approximations is done in Section 3.3 where the criterion of Equation (3.36) and Equation (3.42) are used for all solutions. In summary, the discussion of the convergence properties of the various solutions has lead to the following criterion for truncating the numerical evaluation of the series in each formulation: ‘F I .ll ”45 N» 1 . r. van nJ. fly. a P\~ E «Jr I! O u! . v s v a nu. O 5 0|. ‘— Y .2 \Iv a a ». ~‘ 3 ‘i S .1 a 2. E a: ,7. II s i o v . 3.. :n n1... C; S a . O; y a r ,_ A... as». n. u .N a a» s e we ...J 2.. .u. L. .a. s a.“ a C B it a . . ~ .14 ‘Wo at ... s u - s . . . c s .s a .a .. e .... .3 a . ft we Liv \. «l A .3. Pt 5 a $ \ . .. ..: 100 bK mu: " h (3.43) NMAX = 59-— r a It has also been shown that the thin-wire approximation leads to a series which converges very slowly compared to the series for the strip approximation. This fact is the primary reason for not using the thin-wire approximation for determining the final results, and is an important point to investigate whenever the thin-wire approxima- tion is made for antennas in a modal analysis type solution. 3.3 Comparison of the Solutions. In this section the four approximate forms of the integral equation for the antenna current are compared. The examples shown are representative of the excellent agreement for the current dis- tributions among the four solutions deSpite the differences in functional form. Because of this basic agreement in current dis- tribution, it is concluded that the transverse strip approximation with its computationally simpler form and attractive convergence properties already discussed is the most desirable representation of the solution. The values computed for the antenna input admittance (note, the values are related to the overall scaling of the current distributions) show varying degrees of agreement among the four representations. The differences were found to depend upon the particular configuration and frequency studied with no significant pattern being detected. These descrepencies of admittance are included in the following examples where current distribution along 101 the antenna is presented in terms of milliamperes per volt of excitation and the results are ng£_normalized. The discussion involving attempts at absolute admittance correlation is presented in Chapter 5 where experimental results will also be included. Observations concerning the infinite Susceptance resulting from the delta function excitation will also be given in Section 5.1. The examples considered are shown in Figure 3.7(a) through Figure 3.7(e). They are all for the waveguide in the air filled cavity configuration (i.e., short circuit terminations). As can be seen the form of the current distributions are nearly identical in all cases except that of Figure 3.7(d) where the thin-wire approximation gives a current which is somewhat shifted toward the driving point. This is thought to be related to the convergence problems associated with this solution. No significance should be attached to the differences in the absolute magnitudes of the solutions shown. Even though the differences are large in some cases, this does not detract from the overall validity considering the dramatic dif- ferences in the mathematical forms and convergence prOperties of the four approximations. It can also be seen from Figure (3.7) that excellent agreement is obtained between the transverse and longitudinal strip approximations and that apparently no significant information is lost by choosing the transverse strip representation over the longitudinal strip for the overall problem solution. For different electrical and physical configurations, one approximation may be closer to the true currents and fields than the others. Thus, the results shown here indicate the range of variation the various approximations have on the solution. Since the purpose of 102 mm H V222 E o m u N '0 u mcowumeflxouaa< moowpm> mo acmwuwano é .IIIN H ”II. .I 0" 3:6 memo: OOH we 00 «0 NO _ _ _ . _ a _ . I ucmumcoo u com: u Axve 1 o '4 a '. mow T3580 4 I muw3-:w£u O 1 a mango mmum>mamuu D 1' D o l o o a o o o 4 o a1 D O o o o o o o o 4 o n I. 4 D D D D D 1 4 F 4 4 4 l 4 d 4 4 . . . a r muhfim<_ _ 0.... aqntosqa ‘JIOA/Bm ‘ \(&)1\ 103 mcojméflxouaaa. wsoflum> mo cemwumanu 3:6 mange is O.H w. o. G. N. _ a _ . _ _ _ a a oomuwcoo n cm. H Ahve 00 mm H x422 u g 1 4 o mafiaugnu O 1 ON u M 4 U 0 “Show Hmafionuwwcoa Q 1 4 0 “Show moses/mama... D I o . in o NP—O .qu “ W d O O l . N a? o 0 Eu mm m.» u p T 6 o o 41 O O O O . .. H do 0 0 Eu we 2 I p a E o 04 5% do n _ ."E , . IL. _ a F p _ _ a _ NH 0H annlosqe ‘JIOA/Ew ‘ \(A)I\ 104 ox Nam. u so m~.e New 3.4 So ms.ea so we.afi so Hm.ea u pm mcoHumEonuaaa. mooHum> mo comwumano 3:6 memo: ;\s o.H m. e. a. N. 4 . q . a a _ q T ucmomcoo n oom- u T38 a 4 O I o 1 o 4 o muwsacflsu 0 I o I. 4 now Hmemoo Q o o I 3.3m omum>mcmuu D I 4 o M o 4 o I S D O I. 4 o o H M I. 4 1 N a _o o H U r 4 H o o o - u e 4 o o o o H w T- 4 c O O O 1 t Q 0 I 0 o 4.. 4 sll Ne l.._ as _ m m m . _ _ . _ _ _ _ _ NH 0H om annIosqe ‘HOA/I‘?ul ‘ \(5)1\ r1 ....-iE «v.6. 105 mcoHumexouog mooHum> mo :OmHumano 9:6 .25: is O.H w. 0. 49. N. ucmumcoo u oom- u Axve .. 41 r muw31cwnu 0 .1w. M awuum Hmawvnuwwcoa 4 1 4 3.3m mmwm>mcmuu D D1 0 mm u 522 u N42: 1 o 1 04 ON u M 0 Eu wow. u Eu mmé u L O 4 m ~30 o.H u m I l 0.4 N EU w®.Nm " va I Q 4 D 1 H m 4 Eu wo.w u c o m. 4 o .4 4 4.44.4 o o anméaunuum I anon on o 1N.m o D o r o o o 1 N —H o o 0 ‘III mulllll' my - _ p . — b _ b P — 0.4 annIOSqe ‘JIOA/Em ‘ \(A)I\ — T —. ‘ I'll: l'li' _‘I|IIIII1|1 L mcowumEHxOMn—aaw moonm> mo somwpmaaou MW am u xazz_u xazz 1. ON u e ox wow. u Bo m~.e u a use 0.4 n u so ma.om u as u as Eu Hm.eH u LN u a Na — He _ Amos.m mesoHe OJ osmomcoo u oom+ u Abe 3.5 o mascara—Emu... D A: Mum Hmcflugwwcoa Q aqnlosqs ‘JIOA/Bm ‘\(K)I\ r;:u5 a va {BIL}? IE t1 r°r* ‘ § pier J ,. . ...”. .15: a ALP-Se .' .14 I ‘9i: 1 ‘. ,. C‘r‘pe IQ 107 Figure (3.7) is to show the comparable results obtained from the various approximations of the integral equations, a discussion of the nature of the current distributions shown is reserved until Chapter 5. One remark is appropriate, however, concerning the phase angle of the current. With the current separated into in-phase and quadrature components referenced to the driving voltage as I(y) = 1"(y) +J' I'(y) (3-44) where I(y) is in units of amperes per volt of excitation, the phase angle of the current is defined as $(y) = arctan(I'/I") (3.45) so that the current may be written in polar form I(y) = \I(y> \ejw) Since the configuration chosen for all curves in Figure (3.7) is the antenna in a (ideal) cavity, there is no in-phase component of current (no power can be radiated by the antenna). Therefore, the phase angle will be constant along the antenna and equal to +900 or -90°. This result is the same for all approximations and each curve. The conclusion which may be drawn from the above discussion is that all approximations lead to essentially the same form of current distribution. Therefore, the transverse strip approximation is primarily used throughout the remainder of this report to des- cribe the theoretical results. The coaxial gap solution, however, is not entirely abandoned at this point because of the different ofi V3 :38 Jim ‘ A s:I 108 value of input admittance it will yield owing to the difference in assumed form of excitation. When appropriate, the coaxial gap solution will be compared with the transverse strip approximation (with delta function excitation) and with the observed experimental values in Chapter 5. CHAPTER 4 EXPER IMENTAL INVES TIGAT IONS The theoretical and numerical developments presented in Chapters 2 and 3 provide the analysis necessary to predict the per- formance of the chosen rectangular waveguide and antenna system. In order to support this analysis, an experimental system was con- structed of sufficient flexibility to adequately test the theory. This chapter describes the eXperimental system and the methods used to obtain the data. In designing the experimental set-up, the choice was made to perform the experiments around the L-band frequencies; i.e., 0.9 to 2.4 GHz, for several reasons. First, the vector voltmeter which was chosen as the primary measuring tool had a maximum Operat- ing frequency of 2.4 GHz. Second, the physical size of the L-band waveguide (inside dimensions = 6.5" X 3.25") meant that the field probes could be large enough to easily construct and handle without greatly perturbing the fields. Finally with an eye toward direct application, it was noted that Sauter and Chilton [31] reported in their 1970 study that research emphasis is needed at the low end of the microwave spectrum for use in air quality monitoring. It goes without saying, however, that the choice made for experi- mentally testing the theory places no restriction on its application at other frequencies. 109 110 Other considerations in designing the experimental system included the decision to use at most two (2) antennas and to primarily use the waveguide with short circuit terminations. In designing the two antennas, only one was constructed to provide access for measurement of current and charge distribution and measure- ment of input impedance. This antenna was also constructed to be continuously variable in length. The second antenna was made in interchangeable discrete lengths and could be either driven or loaded as desired. The decision to use short circuit terminations on the waveguide (i.e. to primarily test the case of a rectangular cavity) was made not only because of the experimental simplicity, but also because this case represented the most critical condition for testing the theory. (Again, no restriction is placed on the validity of the theory.) The rectangular cavity is also attractive from the applications vieWpoint, since it offers a well-defined, closed system whose behavior is independent of the environment surrounding the apparatus. 4.1 Description of Apparatus. Historically, measurements of antenna characteristics has required somewhat elaborate systems of microwave components for accurate determination of both amplitude and phase of pertinent signals (see section 8.8 of King, et al. [20]), each component and interconnection increasing the possibility of error. At pre- sent, however, the state of the art in commercial vector voltmeters (VVM) has greatly reduced the need for discrete components, since the VVM performs both absolute magnitude and relative phase 111 measurements for two microwave signals over a broad frequency range. The eXperimental work done in this thesis takes advantage of this simplicity by using a VVM as the heart of the measurement system. This not only offers experimental advantages, but also makes the total system convenient for direct practical application. The vector voltmeter chosen for this work was the PRD 2020 (PRD Electronics, Inc.) primarily because of the useable frequency range of 1.5 MHz to 2.4 GHZ. This allows operation up to the L-band (approximately 0.9 to 1.8 GHz) which has both the eXperimental and practical advantages noted earlier. With the VVM selected, the remaining apparatus was designed to take maximum advantage of its use: the waveguide was constructed such that the dominant mode (TRIO) would propagate for the L—band frequencies, and an impedance sensing block (to be described later) was employed to simplify the impedance de— terminations. The rectangular waveguide, as shown in Figures 4.1 and 4.2, is made of one-half inch (top only) and one-quarter inch brass plate. The narrow inside dimension (theoretical ”a") is 3.25 inches, which is the standard for rectangular L-band waveguides. For experimental flexibility, the wide dimension (theoretical "b") was made adjustable in three steps of 6.5 inches, 7.7 inches, and 8.9 inches corresponding to a cut-off frequency for the dominant TE1,0 mode of 0.908 GHZ (L-band), 0.767 GHZ, and 0.664 GHZ. The waveguide length corresponding to the theoretical dimension d1 + d2 was set at two feet or approximately 0.6096 meters. The flanges on the wave- guide section were adaptable to the standard L-band flange. As shown in Figure 4.1, the usual configuration was to short circuit 112 FIGURE 4. 1 Rectangular Waveguide Section in Cavity Configuration with Antenna Measurement System in Place (Numbers described in text) FIGURE 4. 2 End View of Rectangular Waveguide Section with Antenna . \ L aka V 6 e d E 1 n C. n a «A. "A.— v. 1' ‘ "J 7'. MC» Abi . i-x- ~Avn Y "t 5‘ 1... L-.. Q O‘ ‘I‘ .... <5 pt .QL a“ Au. 1‘ .O I . .. Q F {...V s \.4 ..,',e «Ln n,» 4 113 the ends of the waveguide section with brass plates (indicated as number 1) to create a rectangular cavity. Access for the antennas was made via fifteen (15) tapped holes in the center of the broad wall. These were filled flush to the inside wall with threaded plugs when not in use. The hole spacing was 3.2 centimeters with the center hole (hole number 8) being equidistant from the end flanges. Although the holes restricted the flexibility of axial location, this was felt to be outweighed by the lack of internal field perturba— tion which a continuous slot would introduce (eSpecially at fre- quency where higher order modes could exist). Also shown in Figure 4.1 is the antenna measurement system placed in the center hole (hole number 8). This antenna feed system is made of air filled coaxial transmission line having 0.25 inches (O.D.) diameter brass tubing as a center conductor and 0.5625 inch (I.D.) diameter brass tubing as an outer conductor (standard General Radio rigid coax outer conductor tubing was used). This configuration yields a char- acteristic impedance of 48.66 ohms. The antenna protrudes into the waveguide as a continuous extension of the center conductor beyond the inside surface of the waveguide wall. Other parts of the antenna measurement system shown in Figure 4.1 are the sliding tee (number 3) and the impedance sensing block (number 4). The sliding tee is used to excite the antenna while allowing the center conductor to slide, thus allowing a con— tinuous variation of antenna height inside the waveguide. Above the sliding tee is an adjustable short circuited section of coaxial line which allows one degree of matching to increase power transfer from the generator to the antenna. (Obtaining maximum power transfer by 114 completely matching with a double stub is not done because of the increased complexity required.) The shorting plunger was taken from a General Radio adjustable stub. The impedance sensing block (number 4) is used to probe the voltage and current at the fixed physical location on the coaxial line, in this case 12.55 centi— meters (physically) from the end (the inside wall of the waveguide; i.e., the antenna terminal plane). The construction, operation and effectiveness of the impedance sensing block was also reported in Scott [35]. The idea is basically simple and its effectiveness is based on the use of a vector voltmeter. A sketch of the block and probes is shown in Figure 4.3. Sensing of the coaxial field is accomplished by a short electric probe and a small magnetic probe which protrude through the outer conductor of the coaxial line. The electric probe is sensitive to radial electric field and thus has a voltage induced across its terminals which is proportional to the voltage (amplitude and phase) at that location on the line. The magnetic probe is a small shielded, loaded 100p which is sensitive to circumferential magnetic field and thus has a voltage induced across its terminals which is proportional to the current (amplitude and phase) at that location on the line. The theory and properties of these probes is discussed in section 8.6 of King, et. a1. [20]. The probes are constructed of 0.023 inch (O.D.) solid copper jacketed coax (obtained from Uniform Tubes, Inc., College- ville, Pa.), and are of physical dimensions much smaller than a wave- length so that negligible line loading occurs. Determination of impedance using these probes is done by using the vector voltmeter to measure the absolute magnitude and relative phase between the 115 Q Laugw at Z” x 2” X lll Aluminum ,5 0 (a) Cutaway View of Half the Block Showing: (1) Coaxial Line, (2) Probe, (3) and (4) General Radio Connector 0. 023" dia. coax. ‘— Center conductor p—- solder filled ‘ I j 'I ll! . 1!! n— dia. brass “H I f‘g dia. brass ‘| With 31°” l (with center I hole) I I l GR- 874- c 174A ’ C able connector 3"". Current Probe Mount Voltage Probe Mount FIGURE 4. 3 Impedance Sensing Block 116 voltages induced in the probes. The ratio of these voltages is then proportional to the impedance at that point on the coaxial line. In order to obtain absolute impedance, it is necessary to compensate for the proportionality constants in amplitude and phase for each probe. Although this can be done analytically for the probes (for example as done by Scott [35]), it is much simpler and more reliable to obtain the proper compensation for the entire probing system experimentally. The method used to obtain the calibration constant for the impedance sensing block is described in the next section. As mentioned above, the antenna for the antenna measurement system is the extension of the center conductor of the coaxial air line beyond the inside surface of the waveguide wall. This antenna is constructed so that current and charge distribution measurements can be made, as shown in Figure 4.4. At the terminal plane the center conductor is supported by a one-eighth inch thick Styrofoam wafer as indicated by the number 1 in the figure. This is the only dielectric support necessary in the entire system because both the sliding tee and sliding short circuit act to center the conductor. The quarter inch diameter brass tubing is slotted axially to accept the probes for current and charge distribution measurements. The probes are mounted in the key of a hollow brass plug which is attached to a small stainless steel tubing push rod, both sliding freely in- side the antenna tube (see Figure 4.5). The charge probe (number 2 in Figure 4.4) is constructed as the extension of the center con- ductor ofuflniaturesolid dielectric coaxial line. Since this probe is mounted radially (normal to the antenna surface), the voltage induced in it by the electric field is prOportional to the charge 117 FIGURE 4. 4 Slotted Antenna with Current and Charge Probes to VVM (Kali—EEK. cl I a) miniature coax to b) stainless steel tubing push rod VVM c) brass tubing antenna/center conductor d) slot in antenna e) brass carrier plug f) key on plug FIGURE 4. 5 Cutaway View of Slotted Antenna with Probes 118 I? 0a 'll/ // / s ' ’ l/ ' . Ill/[I’léllllu l a? s I I [I I I ll 0 a . FIGURE 4. 6 Second Antenna(s) for Coupled Antenna Measurements FIGURE 4. 7 Total Experimental Set-Up for Current Distribution Measurements (Numbers are described in the text) 119 at that point on the antenna (by the boundary condition for normal electric field). The current probe (number 3 in Figure 4.4) is a shielded, loaded, circular loop which has a voltage induced in it by the circumferential (antenna) magnetic field. This voltage, by the boundary condition for tangential magnetic field, is propor- tional to the current at that point on the antenna. The minature coaxial lines (0.034 inch diameter, solid copper jacketed) of which the probes are made run continuously from the probes to the free end of the center conductor (beyond the short circuited section of line) where subminiature coaxial connectors are attached. At this point, the vector voltmeter is used to meaSure the voltage induced in the probes as they are moved axially along the antenna surface (see Section 4.2). When it is desired to experimentally test the case of two antennas in the waveguide, the antenna shown in Figure 4.6 is used. This antenna is connected directly to a General Radio type connector via a short section of coaxial air line. Five antennas were made in lengths ranging from one (1) to three (3) inches in half-inch steps. They are made of one quarter inch diameter solid brass rod and are threaded on the end for attaching to the coaxial line center conductor at the terminal plane. The total set up used to obtain experimental data is shown in Figure 4.7. The microwave generator used for L-band frequencies is the General Radio type 1218 B shown by arrow number 1 in the photo- graph. The generator is connected through a lOdb attenuator (to minimize loading effects) to a cavity type frequency meter indicated by arrow number 2. Although not visible in the photograph, this is 120 followed by a 2 GHz low pass filter (for work in the L-band) which drives the antenna measurement system and waveguide described in the preceding paragraphs. In the photograph of Figure 4.7, connections are made for current and charge distribution measurements on the antenna in the waveguide. The solid miniature coax and subminiature connectors at the top of the antenna measurement system (arrow number 5) are connected to flexible miniature coax which goes to the coaxial switch (arrow number 3). This switch is used to select either the current probe or charge probe for connection to the vector voltmeter. The vector voltmeter, indicated by arrow number 4, has the signal processing and metering unit with the controls and a sampling head which converts the two applied signals to an intermediate frequency. The sampling head is connected through a ten foot cable to the main unit. The reference channel input for the VVM is shown in the photo- graph connected (arbitrarily) to the voltage probe of the impedance sensing block. This offers a convenient reference for making relative phase measurements along the antenna. The experimental set-up shown in Figure 4.7 is used for measure- ments with the air filled or empty cavity. For measurements with water filling the cavity a wooden box large enough to hold the wave- guide section is lined with plastic sheeting and filled with the water. The water level is kept even with the top of the waveguide to minimize the forcing of water into the antenna driving line. Air is removed from the waveguide by venting through the antenna access holes. This method of studying fluids is of course cumbersome, but would not be necessary had the waveguide been constructed water-tight. This, however, would have meant either sacrificing experimental 121 flexibility, or increasing the complexity of the waveguide design, and neither alternative was acceptable. For practical application to fluid monitoring, the waveguide or cavity size would be chosen and constructed water tight, and ports could be cut and covered with fine wire screening to allow for filling or continuous flow studies. The screening would provide gross filtering while maintaining, at least approximately, the conducting surface at the cavity wall. In order to test the theory for the case of a dissipative medium the values of relative dielectric constant, er’ and con- ductivity, a, for the medium must be known. For water, a great deal of literature is available (e.g. Dorsey [5], Grant [10], Saxton [32, 33], and Lepley [23]) based primarily on the Debye equation. How- ever , it is best to determine the properties independently for the Specific water samples used. In this report the conductivity and permittivity are measured at low frequencies and high frequencies reSpectively, much as was done by Scott [35]. Measurement of conductivity is performed using the low fre- quency test cell shown in Figure 4.8. The test cell, a cylindrical capacitor, is filled with the sample and the impedance of the capacitor measured on a low frequency bridge (e.g. GR type 1606-A) at frequencies in the vicinity of 1 MHz. This impedance is a func- tion of the complex dielectric constant of the medium filling the capacitor as well as the physical dimensions. The impedance can be obtained theoretically by solving the problem via Maxwell's equa- tions (Scott, [35]). The same result, based on electrostatics with fringing fields neglected -- a good assumption with water having er >> 1 -- can be obtained from the formula for a capacitor filled 122 1 :5. 25 cm —-> m—rlzo. 318 cm brass rod 4; plexiglass brass pipe Z = R + jX K M/K‘2 - 4(Rw6 e )2 0' : 1’ O : E U/m 2R R r _ 1 Z _ K -— '2—1TIID(r—l-) -— 8 799 (a) Coaxial Capacitor Salinity Frequency Conductivity, cr , (NaCl) MHz millimhos /meter 0. 5 gm/liter 0. 5 97. 7 97. 5 96. 3 1. 0 gm/liter 1. 0 196. (b) Representative Measurements of Conductivity FIGURE 4. 8 123 with a pure dielectric medium (i.e., by calculating the "complex" capacitance for the complex dielectric constant case). Thus, from electrostatics, the capacitance of a coaxial capacitor of length L is = L c Mrs/ta) (4.1) where rb is the inner radius of the outer conductor and r8 is the radius of the inner conductor. For a complex dielectric constant e=ereO-Jo/w the corresponding "complex" capacitance is 6' = (1/K)(ereo - j o/w) (4.2) where K =0n(rb/ra)/2n L (4.3) is a constant dependent upon the physical dimensions of the capacitor. The impedance of this complex capacitance is then simply l/j (06 N II K/0'+ Jw ergo) =R+jX where R = Ko/(o2 + (m erso)2) X = -weK/(o2 + (w ereo)2) Solving for o in terms of er and R gives E ‘FE plant's. it; 124 A fl )2 2 + - = K JK 4(R u) ergo a 2R . (4.4) For the test cell used in this study the constant is (see Figure 4.8 for the dimensions): K 4 8.8 For all the media used in this study, the measured value of R from the low frequency bridge was less than 200 ohms. Therefore at fre- quencies below 1 MHz, the factor 2 2 4(R u) ergo) S 3.2 << K for an assumed er = 81, and the conductivity is approximately given by a = K/R (4.5) which is insensitive to errors in the assumed value of er. The approximation, Equation (4.5), is of course not used, but only serves to demonstrate that this method of determining conductivity is effective even without a precise knowledge of the dielectric constant. The results obtained for two concentrations of sodium chloride in distilled water are tabulated in Figure 4.8. Based on available literature (see Saxton [32]), it is assumed that these conductivities are constant up to the L-band frequencies of interest. The measurement of relative dielectric constant is accomplished in this study by the cavity perturbation method. Two cylindrical cavities were available as shown in Figure 4.9 (a) for absorbtion type measurements. Based upon perturbation theory for a cylindrical cavity excited in the TMOIO mode, it can be shown (see Section 6.8 125 BNC W 1‘] Connectors T I 0th Probes ----- I L .5 - . —fiL.t-1 I I -—-I““ 10. 16 cm I I I I I I 11.75 cm 1 “ """""" rr”"‘“" I 10. 251cm 11. 873 cm 1.1203 GHz 0. 9672 GHz Hole on Axis for Inserting Glass Tubing (a) Cylindrical Cavity ral /’ ra2 r = O. 0825 cm al 5 r = 0. 146 cm a2 r b 6 = 5.18 r2 air, Er : l k 0)) Glass Tubing Properties FIGURE 4. 9 126 of Johnson [16]) that for a dielectric rod (dielectric constant = e ) r located on the cavity axis the shift in resonant frequency, 5f, is 2 (e - 1)(r /r) = ‘ r a b 0.54 (4'6) "flO‘ H1 0 where rb is the radius of the cavity and r8 is the radius of the dielectric rod and f0 is the unperturbed cavity resonance. It is desired in this study to place a water sample held in small glass tubing on the axis of the cavities and thus the theory must be modified to account for the double dielectric (glass-water) rod. This can be done by following the method in section 6.8 of Johnson [16]. The result for tubing of inner radius r and outer radius al r32 made of material having a dielectric constant e 2 and filled r with a material having dielectric constant (see Figure 4.9 (b)) erl is 2 2 2 if (erl - 1)ral +’(er2 - 1)(raZ - ral) f = O 2 (4.7) 0.54 rb To use Equation (4.7), the eXperimental set up shown in Figure 4.10 (a) is used. First, the unperturbed resonant frequency of the cavity, f0, is measured. Then the empty glass tubing is placed on the axis of the cavity and the shift in resonant frequency 6fg (g for glass alone) is measured. Since in this case a = 1, r1 Equation (4.7) becomes 2 2 f -1 - 6 g = (erZ )(raZ ral) (4.8) 2’ o 0.54 rb which can be solved for grz. For the glass tubing used in this experiment the value = 5.2 was found. It is not necessary cr2 127 Sweep Generator Frequency Sample Detector Horizontal Vertical 50 S2 Term. Circuit Diagram for Permittivity Measurements (21) Salinity Frequency Dielectric Const. (NaCl) f , GHz 6 o r Pure water 0. 9672 83. 5 I. 1203 82. 9 0. 5 gm/liter 0. 9672 93. 4 l. 1203 85. 7 1. 0 gm/liter 0. 9672 99. 3 (b) Representative Measurements of Dielectric Constant FIGURE 4. 10 128 to solve Equation (4.8), however, since combining with Equation (4.7) yields: 2 5f - 6f - (erl - Dral 4 f f - 2 ' ( '9) Thus by measuring the resonant frequency shift for both glass and water, bfgw’ and for glass alone, ofg, the dielectric constant of the water (erl) is given by Equation (4.9). The results obtained for two concentrations of sodium chloride in distilled water are shown in Figure 4.10 (b). Since these results are obtained near 1 GHz, they are assumed constant over the entire L-band frequencies of interest in the remainder of this report. The appropriate values of permittivity as well as conductivity for each medium studied will be presented with the other experimental results in Chapter 5. 4.2 Experimental Methods. As stated in Section 4.1 the use of a vector voltmeter to make antenna measurements is a relatively new procedure. For this reason some time will be Spent in this section briefly describing the methods used to obtain the data. Also, by describing these pro- cedures, account may be taken of the possible introduction of error from the procedures alone, so that, if it is deemed necessary in the future, proper refinements may be made toward obtaining results with a high degree of precision. Even though the most precise methods are not used, it is found that the precision is more than adequate to verify the theory and to indicate the feasibility of application. Since the vector voltmeter reads directly the absolute magnitude and relative phase between two signals, most of the pro- 129 cedures necessary to obtain useable data center around selecting the preper signals and calibrating the system to convert the VVM readings from relative to absolute. As with all antenna measurement, the data for current and charge distribution need only be relative, in both amplitude and phase, to the input current or applied voltage at the antenna terminals. Therefore, no account need be taken of the phase shift or transmission line losses in the connecting lines between the antenna current and charge probes and the VVM (these being constant for a fixed frequency). A constant phase reference must be used, however, for each probe as it is moved axially along the antenna. In the set—up used here (as shown in Figure 4.7), the reference is taken to be either the voltage or current probe (whichever is larger) at the impedance block. This signal is main- tained at a constant level throughout the measurement to assure con- stant driving condition and its phase is, of course, independent of location of the probes on the antenna. By subtracting off the phase measured at the driving point from the phase at all other locations on the antenna the data is then referenced to the current at the input terminals. Regarding the current and charge probes on the antenna, the question may be raised as to the effect of having both probes on the antenna simultaneously. In answer, it is first realized that the current and charge distribution are inter-related via the con- tinuity equation and not independent. Thus, measuring both is a duplication of effort except that they may be used as checks for each other. In designing these probes, it was realized that the current distribution was to be the fundamental result and so this 130 probe was built and some representative current distribution meaSure- ments made. Then the charge probe was added and the measurements repeated. The results showed that the charge probe had no appreci- able effect on the data. This data is presented in Figure 4.11. As can be seen, only the data near the driving point is changed and since current probe measurements at this point are always unreliable due to unavoidable loading, nothing is lost. 0n the other hand, the current probe is expected to have an appreciable effect on the charge probe measurements owing to the relative size and proximity (no comparative study was made) and thus the charge probe data may be inaccurate. Therefore, the current probe is primarily used to obtain data and the charge probe is occasionally used as a check. One further consideration needs to be made concerning the current distribution measurements; namely, the orientation of the current sensing probe. Unlike antennas in an iniinite medium, the fields surrounding the antenna in the waveguide are not rotationally symmetric. Because of the physical size of the current probe (radius = 0.25 cm), it is sensitive to the magnetic field a short ways from the antenna surface and thus is excited by fields other than those at the antenna boundary. Even though this distance is much less than a wavelength, the fields are rapidly varying near the antenna, since besides satisfying the antenna boundary condition, the fields must also satisfy the waveguide boundary conditions. For example, consider the case of the current probe lying in the (y,z) plane; i.e., the longitudinal plane at the center of the waveguide. In this orientation the current loop is excited by magnetic fields which are a directed -- normal to the plane of the 1.0 I 131 0 without Q probe A with Q probe . ...... theory - (a) f = 1.2 Ghz FIGURE 4.11 Effect of Charge Probe on Current Measurements I— —. ...... theory .. 8 0 without Q probe . A with Q probe A F -4 I- . u: (13) f = 1.5 Ghz 132 100p. By looking at the distribution of the x-component of the waveguide mode magnetic field (see Equations (2.5a)), one sees that : I x e h 2 z___. a: sin n x a L x,nm ( TI / ) (:53 h which is a maximum at the center of the waveguide (x = a/2) for all n = odd modes (the only ones excited according to the theory). Thus, for example, near the end of the antenna, where the current is approaching zero, the magnetic field on the antenna surface must be small; yet a short distance away from the antenna this component of magnetic field must approach the maximum for the waveguide mode. These rapidly varying fields will obviously adversely affect any attempt to relate the induced probe voltage to the tangential magnetic field or current. On the other hand, consider the case of the current probe lying in the (x,y) or transverse waveguide plane. In this orientation the loop is excited by the E-component of magnetic field. The waveguide mode field, again from Equation (2.5a), is distributed as h h cc cos (n11 x/a) .7 2 ,nm ;. a e '3') E 0 (.... 2,nm Therefore, all modes of odd order n go through a minimum at the center of the waveguide (x = a/2). In the x,y plane, then, the current loop voltage is primarily induced by the magnetic field nearest the antenna surface and more nearly represents the current on the antenna. Figure 4.12 shows some representative results for the probe in the two orientations. In either orientation, the probe-size will introduce errors, but, based on the above arguments sand the results of Figure 4.12, these errors are expected to be 1.0 a = 22.6 cm .8 - -+-theory . b = 8.25 cm probe in trans- dl — d2 _ 30 48 cm ' verse plane h = 6.25 cm .6. probe in . f = 1.0 Ghz longitudinal lane m L ° P .4 - .2 . .2 .4 y/h' .6 .8 1.0 FIGURE 4.12 Effect of Probe Orientation on Current 1-0‘ , Measurements flrd. ~— .2 —q 4 .8 . - .—o theory (b) .. 0 probe in trans- ~ a = 2b = 16'51 cm verse plane d1 = 8.08 cm .6 - A probe in < (12 = 52.88 cm longitudinal h _ 7 I _ plane — cm I I f = 1 2 Ghz .4. .2 - 134 minimized for the second case. For this reason, all the experi- mental data presented in the next chapter is taken with the current probe oriented in a transverse waveguide plane. With the current and charge distribution data taken relative to the input or terminal values, the absolute antenna characteristics are determined from a knowledge of the absolute input admittance. This requires a calibration of the impedance sensing block and the interconnecting cables. Based on the theory of the probes used, the ratio of the current to voltage signals is proportional to the magnitude of the admittance, and the phase angle between these signals is within an additive constant of the phase of the admittance. Since it is nearly impossible to theoretically account for the losses and phase shift introduced by the probes and probe mounts, the system is most easily calibrated experimentally. Because the impedance sensing block is not located at the antenna terminals, it is also desireable to accurately determine the equivalent electrical length through which the impedance data must be transformed; this is dif- ferent from the physical length because of the dielectric support wafer and the physical size of the probes. Both the effective length and the calibration constant are determined in the following way. At a fixed frequency, a short circuit is attached at the antenna terminals (the end of the outer conductor), and the phase angle of the susceptance (j;jB) at the impedance measuring block determined (exact length is not required unless it is near a multiple of a quarter wavelength). Using the vector voltmeter phase offset and zero adjustments, the phase angle between the current and voltage 135 signals is set to the proper value (i 90°). This calibrates the phase angle measurements. The current and voltage signals are re- corded, and from transmission line theory, the following equation holds: K ”II = 2 tan (5;) (4.10) z TIIT c ’ where K2 is the unknown impedance amplitude calibration constant and L is the unknown length between the antenna terminals and the measuring point. ZC is the characteristic impedance and B is the free space prOpagation constant. The existence of the dielectric wafer is explicitly ignored in writing Equation (4.10) so that its effect can be included in the effective length and calibration con- stant. Next, the short is removed and a fixed length (say, d) of outer conductor is attached to the antenna terminals, thus extending the length of line along the antenna a fixed distance, d. (This is facilitated by having the threads on the antenna support plug com- patible with General Radio parts so that a G-R nut is used to secure the extension.) The short circuit is again attached and the above process repeated. The phase angle remains calibrated and the trans- mission line equation gives: \VZI Kz ‘72—‘- = 2C tan (6(4 + d)) , (4.11) where V2 and 12 are the new voltage and current readings from the impedance sensing block. Equations (4.10) and (4.11) are now two equations in the two unknowns K2 and L. Dividing the equa- tions eliminates the unknown K2 and gives a transcendental equation 136 to be solved for L, which is very easily and rapidly solved with a calculator since an approximate value of L is known (physical length). The resulting value of L then yields the calibration constant via Equation (4.10) or (4.11). The two step process des- cribed above now calibrates the impedance sensing block for obtaining absolute complex admittance data. That is, the VVM phase is left untouched and the data is taken for the desired eXperimental condi- tions. The measured voltage and current are multiplied by the calibration constant and the resulting complex admittance is trans- lated to the antenna terminals through the effective length L via the transmission line equation. Although this procedure is seemingly lengthy, the bulk of the work is handled easily by a computor or calculator and there are very few places for experimental error (in fact, fewer than in the slotted line method). It is also a char- acteristic of this Specific set-up that the Short circuit method is necessary. In a future arrangement it would be best to remove the length as an unknown by proper compensation of the support wafer. Then only the calibration constant need be determined and this is easily done by applying a matched load at the antenna terminals and taking one set of readings. One drawback of this impedance System is that it is inherently frequency dependent and thus must be calibrated for each frequency used. This information, however, can be determined independent of the experiments and stored for retrieval and use by a computing system. (This drawback, unfortunately, prohibits performing direct swept frequency experiments, but to my knowledge, this problem is shared with other techniques.) As an example of the frequency 137 dependence, it was found that over the frequency range of 1.0 GHz to 1.4 GHz, the effective length changed by no more than i;0.#% (around 12.7 cm) and was repeatable to within 1 0.1%. Owing to the rapid variation of the tangent function, however, no attempt was made to Standardize this variable, thus avoiding the introduction of unnecessary error. The calibration constant was much more fre- quency dependent, having more than 10% variation, repeatable to within 1.0%, over this frequency range. One further comment concerning the use of this impedance meaSuring system and its inherent errors in this application should be made. When the waveguide is in the cavity configuration, the antenna input admittance is very nearly a pure susceptance. (The theoretical lossless cavity results are, of course, exactly so.) This implies that the voltage standing wave ratio in the antenna driving coaxial line is infinite. As a result, for some conditions the voltage to the current ratio at the impedance measuring block terminal plane is nearly infinite or zero at a given frequency. This would then yield a Signal below the noise level of the VVM from either the current or voltage probe and no data could be taken. (The same problem could, of course, arise when calibrating the probe using a short circuit as a reference, as discussed previously.) Coupled with this problem is the nature of the transmission line equation for transforming impedance. Because the transformed impedance as a function of electrical length has a steep slope near the infinities, small errors in impedance measurement (or in length) can lead to large errors in transformed antenna input impedance near the frequencies where this tangential infinity occurs. For 138 example, if the antenna input impedance (or admittance) is near infinity, a Small error in the electrical length through which the measured ratio of \VI/III is transformed could lead to large errors in the transformed experimental data even though the raw experimental data was accurate. This source of error is unavoid- able, but it should be noted that this problem arises because the antenna impedance being measured is purely reactive. For antenna impedances which involve a comparable resistive component (a wave- guide with matched terminations or a cavity filled with a dissipative medium), the standing wave ratio in the line is not infinite and accurate results are obtainable at all frequencies. A potential source of error which arises in the measurement of phase should also be mentioned. It was found that the flexible coaxial line used to connect the impedance measuring block to the VVM introduced variations in the measured phase angle. AS the cable was moved, the phase angle was found to vary by less than i 2°. Thus as the location of the antenna in the waveguide was changed, there was a slight change in the phase angle reference from the value determined during calibration. At a fixed location, however, no variation was found. Thus, in the cavity configuration, the measured phase angle was found to be within j;2° of the 900 phase reference determined during calibration and this variation' was ignored. Although one may be concerned about the effect of wall losses (or cavity Q) causing an impedance which has a small resistive component, the experimental system used showed this to be a negligible effect within the accuracy described above. (In fact, the theoretical solution was altered to include the approximate 139 formulation of wall losses as described by Collin [2], and the phase angle of the antenna input impedance changed by less than 0.50 from pure reactive.) Therefore, the phase angle measurements are assumed to be accurate to better than 20 for this experimental system. CHAPTER 5 NUMERICAL AND EXPERIMENTAL RESULTS The previous chapters have dealt with the theoretical de- velopment, numerical solution, and the experimental techniques for the study of linear thin wire antennas in a rectangular waveguide. The characteristics of the antenna, namely, the current distribu- tion and input admittance, have been viewed as a function of the physical configuration and the electrical properties of the medium filling the waveguide. In order to fully and Optimally exploit the relationship which exists between the antenna characteristics and the properties of the medium, it is necessary to understand the operation of this antenna-waveguide-medium system as an in- dependent function of the physical configuration alone. This chapter, therefore, is devoted primarily to the establishing of this relationship through the comparison of theoretical and experimental results for various configurations. In order to obtain meaningful correlation between theory and experiment, how- ever, consideration must be given to the basic differences between the theoretical and experimental models. In particular, there exists well known (and extensively studied [17, 18, 20]) dis- crepencies in antenna input admittance. The problems associated with these discrepencies in relation to the antenna-waveguide system are discussed in Section 5.1. 140 141 To effectively understand the operation of the antenna- waveguide system, the correlation between experimental and theoretical results is divided into the case of the Single antenna (Section 5.2) and the case of two coupled antennas (Section 5.3). In each case the parameters of operating frequency, physical dimensions, and properties of the medium are varied in turn. Also, for each configuration the antenna current distribution and input admittance are treated separately. The current distribution is always presented in normalized form. The normalization point is taken as the point of maximum current. When this point is at the input, the normalization is done at the second numerical current solution point; i.e., 01,2 from the theory. This was necessary to avoid introducing dis- crepencies in the nature of the distributions which arise from the delta-gap solution, as described in Section 5.1.1. This also was close to the nearest point at which experimental data was able to be obtained thus facilitating the normalization of measured current. Where possible, the current on the first numerical subsection (i.e., ai,1) is also included in the figures. In general, only the delta gap theory with the transverse strip approximation is used and presented. However some theoretical results from the coaxial gap model are also included for comparison. The variation in the phase of the current is also pre- sented with the relative experimental data being normalized to the same second current point as discussed above. In the case of the cavity configuration, however, the phase of the current is 142 a constant 1L90° along the antenna with exception of possible 1800 phase reversals occurring at points of current zero crossings. Therefore, the cavity phase data is given as the constant theoretical value obtained. The experimental phase measurements were found to differ by no more than 30 from constant with the greatest variation occurring at the end of the antenna. This was therefore felt to be primarily caused by experimental errors resulting from variations in the flexible coaxial line leading to the probe as discussed at the end of Chapter 4. Additionally, Since the antenna surface impedance introduces losses along the antenna, some variation from the idealized theoretical results is expected. At the point of phase reversal the experimental data showed a somewhat gradual change, but because of the small magnitude of the current at this point (zeroI), the VVM was unable to measure the relative phase values. The reversal was very rapid, though, and occurred in a distance less than 0.05h (the antenna height) for this cavity case. The measured and calculated admittances for each configura- tion studied is presented following the current distributions. In the theory the admittance is taken to be the value of the current on the first Subsection (ai,1) for a one volt excitation. This is different from the actual input point by one-half the sub- section length (0.5 Ay) which is taken generally as 0.025h (20 seg- ments). Since the delta gap theory and coaxial gap theories differ most significantly in this value of input admittance, both results are presented where they have been obtained. The experimental data is given in unaltered form from the value obtained by trans- forming down the driving transmission line from the measuring 143 terminal plane. Discussion of the use of impedance matching net- works is given in Section 5.1.2 but, for reasons to be discussed, no matching is applied to the data. 5.1 Correlation of Theoretical and Experimental Input Admittance. In the theoretical model of the antenna developed in Chapter 2 the electromagnetic source exciting the antenna is approximated either as an ideal delta function generator or as a coaxial gap supporting a perfect TEM mode field. The experimental system described in Chapter 4, on the other hand, is obviously different than either approximation. This difference is known to result in disagreement between predicted and measured input admittance. Much effort has been Spent in trying to account and correct for this difference, eSpecially concerning the infinite susceptance predicted by the delta function generator (e.g., see Section 3.10 of King and Harrison [18] or Duncan [6]) and concerning the de- velopment of terminal zone correction networks (e.g., see Chapter 8 of King, Mack and Sandler [20]; Sections II.1 through 11.10 of King [17]; or Chang [1]). In the following paragraphs these areas of interest are discussed as they relate to the results obtained in this study. 5.1.1 Infinite Driving Point Susceptance. The delta function generator model for the antenna source leads to an infinite input Susceptance, since the zero width source region implies an infinite capacitance and, hence, an infinite driving point current. AS pointed out in Section 3.2 of King and 144 Harrison [18], this current singularity is logarithmic in nature and extends only for a very Short range along the antenna. An approximate order of magnitude value for this effective distance is given as -1/k r h e ° a (5.1) which for a one-quarter inch diameter antenna at 1.4 ghz gives (2 X 10-5)h. King and Harrison go on to point out that the re- moval of this infinity from the solution is essentially accomplished as long as the antenna current is found as an expansion of con- tinuous functions and terms of extremely high order are not in- cluded. Unfortunately, the solution for the current by the numerical methods of Chapter 3 is not obtained from continuous functions and, in fact, the step discontinuity presented by the pulse functions necessarily includes effects of infinitely high order. Therefore, the solutions obtained for the delta function generator model in this report must include the effect of this in- finite susceptance. It is very difficult to theoretically predict the contribution of this singularity, but its effect can be seen in the numerical results. An example of this is shown on a much expanded scale in Figure 5.1, where the smooth extrapolation of the calculated current amplitude at points along the antenna obviously does not coincide with the calculated value of the in- put current at the first point on the antenna. It is interesting to compare this result to the solution obtained with the approximate coaxial gap excitation. In this case there is no infinite input cl... ESE-r lie, 145 1 .o -— ”ow-MG" «... ”db-ii E-.ztt.fi Eyes“ A delta gap theory :3. ,. ...-9'5". 0 coaxial gap theory P ..9.’I .91.. ...-'7’ _ s-—— b ——-+ - ’ -o.5{, h (a) \II ' .8— norm. A I. .7- o [, ¢(y) = -90 = constant ..111..1-..|...11AJL 0 .1 .2 y/h .3 .4 Y 1 2 —-(3 _ ' L | a = 2b = 16.51 cm 1.1 - -0.5{, h h g 6 25 cm ' 42.4%“ f = 1.0 Ghz 1.0 - “at, I '3:., I I " "‘-A_~\ A delta gap theory (b) norm. __ "4:. .9 -..,\.g~.:\\‘ O coaxial gap theory 8 _ .X:.:\‘Q *- .--A-..::“Q o7 '—' ...A-.‘.“~u - "-4.: .6 " ¢(y) = +900 = constant % ‘/ /r 4 I n l 1 l 1 l 4 1 A J 1 l L I 3 X 1 2 y/h 3 4 FIGURE 5.1 Effect of Infinite Susceptance on Numerical Solution 146 susceptance and (see Figure 5.1) the calculated input current agrees well with the continuous extrapolated value. It may be noted that, as expected, the error in input current from this capacitance singularity is in the direction of increased positive susceptance (note the phase of the current referred to the driving voltage in each figure). Since it is difficult to theoretically predict and correct for this inherent error in driving point susceptance, and since any attempt to compensate for the effect by extrapolation would be imprecise, the numerical results in this chapter for the delta func- tion generator model always include this effect. 5.1.2 Terminal Zone Correction Network. One technique which may be applied to obtain satisfactory correlation between predicted and measured antenna input admittance involves the determination of a terminal zone correction network. The terminal zone network is a lumped element approximation which attempts to account for the coupling between the transmission line and the combination of the antenna and terminating structure. There has been considerable effort Spent in the past to obtain analytical expressions for these terminal zone effects (e.g., see King [17] and Wu [40]) with results which yield sufficiently accurate correction to the observed differences between measured and pre- dicted admittance. As pointed out by King, et. al. (Chapter 8 of 147 [20]), however, the simplest procedure for determining these net- works (once the theory has substantiated their existence) is by a direct comparison of the experimental and theoretical admittances. This reduces the problem to one of network analysis and the solu- tions are much more readily obtained. The lumped equivalent terminal zone network is shown in Figure 5.2 in two equivalent representations. The circuit relation- ships between calculated admittance (impedance), YL (ZL), and observed or apparent admittance (impedance), Ya (2a), are simply: for the circuit of Figure 5.2(a), _ 1 Ya — —-——ZL + ijT + ijT (5.2a) and for Figure 5.2(b), l = -—-----+ jML (5.2b) a YL + ijT T Z For application to an experimental system which has the antenna as the load on the coaxial transmission line, the physical Significance attached to the lumped capacitor is basically the effect of terminat- ing the transmission line and the effect of the terminating structure, whereas the lumped inductor includes the effect of the coupling between transmission line and antenna. To determine the appropriate. values of LT and CT from Equations (5.2) the measured and theoretical values of admittance (in general, complex) for one antenna configuration lead to two easily solved eQuations from the real and imaginary parts of Equations (5.2a or b). The computed LT and CT are then applied to data from similar configurations and are l. 1 148 5: LT 3. LT a c z z T L a CT YL % LT 3: LT (a) (b) FIGURE 5.2 Lumped Equivalent Terminal Zone Networks FIGURE 5.3 Functional Behavior of Terminal Zone Correction NetWOrk 149 assumed frequency independent. In general, changing the physical or electrical properties of the configuration is expected to change the values of LT and CT [40]. The antenna-waveguide system considered in this report offers some unique variations from the typical antenna driven over an infinite ground plane. In particular, since the corrective net- work elements are dependent upon the transmission line terminating structure, one expects the network to be different for each wave- guide configuration. Furthermore, since the waveguide fields (and hence the transmission line coupling to the waveguide) are highly frequency sensitive, the network is expected to be frequency dependent. For the special case of the rectangular cavity, which is extensively studied here, both of these effects become enhanced because of the existence of resonances. The one physical parameter which is not expected to cause drastic variation in the correction network is the antenna height (although for the cavity or antenna near resonance this could also prove an important influence on coupling effects). With these problems in mind, the task of de- termining a useful corrective network.appears formidable. Nonethe- less, a great deal more can be learned about the implication of the terminal zone network through initial attempts here to apply the theory as it now stands. The necessity of investigating a terminal zone network is based on discrepencies on the order of a factor of two between observed and predicted antenna input admittance. Although seem- ingly large, similar differences have been reported (see Section 3.11 of King and Harrison [18]) for which a corrective network 4' 150 provided satisfactory agreement. It is the purpose of this section to study the possibility of the network providing similar corrections and to give some representative results. Since a great deal of in- sight into a problem is gained by the preper approach, some time will be Spent discussing the methods used to obtain the corrective networks. Because of the complicated dependence of the network on the configuration studied in each case, the results presented in the later sections of this chapter are not corrected for terminal zone effects in order to avoid obfuscating the experimental data. To apply the Equations (5.2) for the case of the antenna in a rectangular cavity,* it is noted that the antenna input admittance is a pure susceptance, since there can be no real power radiated by the antenna for the assumed ideal case. This means that two sets of values for observed and calculated admittance must be used to determine LT and CT‘ Based on the arguments of the preceding paragraphs the best parameter to use as an independent variable in the sets of values is the antenna height. Using the network of Figure 5.1(a) as an example, Equation (5.23) becomes -1 B (h) =—-——-——+B (5.3a) h +'x a XL( ) L c where XL = T (5.3b) Bc = wCT The author is indebted to Dr. J.A. Resh, Department of Electrical Engineering and Systems Science, Michigan State university, for his helpful discussions concerning this problem. 151 If the apparent Susceptance is viewed as a function of the theoretical reactance, then Equation (5.3a) results in the curve shown in Figure 5.3. In view of this, by graphing a set of data as Ba versus XL (for fixed frequency) one can obtain approximate values for the terndnal zone inductor and capacitor. Solving for XL (LT) and Bc (C from Equation (5.3a) is seen to be equivalent to selecting T) the hyperbole which is a best fit to the data given and any curve fitting criterion can be used to optimize the results. AS an example of this process, the results from three com— binations of physical configuration and frequency are shown in Figure 5.4. The inductor and capacitor are determined by a least square minimization method via an interactive routine on a Hewlett Packard model 9821A calculator. Because of the nature of the data the problem was very nearly ill-conditioned, and it was found that a weighted Squared error criterion was often desirable. A.weight- ing factor equal to the inverse square of the slope of the Ba(h) versus XL(h) curve (i.e., (dBa/dXL)-2) was therefore used to reduce the effects of the critical asymptotic points. As is seen in Figure (5.4) the corrective network reduced discrepencies from the order of a factor of two to differences of 30% or less. Better agreement was not attained in every case primarily because of the possibility of experimental errors and the fact that the antenna height could be a factor in the coupling and perturba- tion of the waveguide modes and the transmission line fields. In- deed, as shown in Figure 5.4(c), when the antenna input admittance does not pass through a zero or an infinity as the height is varied, the corrective network is seen to give highly satisfactory results. input reactance, X, ohms 152 A theory 300 '" 0 measured 4' _ . corrected ,' ' measurement ' 200 -— 100 '- F 0 l U 1 _ 3 h/x L -100 — L. L mLT = 38.18 ohms -200 - LT = 5.5 x 10'9 henrys -3 I. L .... “£1, 3 ’2.08 X 1.0-12111110 u CT = -0.32 x 10 farad r...“ ‘11 3 without weighting factor a = 2b = 16.51 cm L = 60.96 dm d1 = .133; h = parameter f = 1.1 Ghz FIGURE 5.4(a) Correction of Measured Data via Terminal Zone Network 100 50 0 E ..: ‘50 O x. J U fi 3 o -100 ‘0 0.) H U 3 D. G -.-4 ~150 -200 153 F' P A theory r: 0 measured P O corrected P measurement F I I I I - .1 F r- - I - mL = 10.18 ohms T -9 h L = 1.35 x 10 henry k T -3 * wcT = -7.745 x 10 mho " _ -12 H— L ____.1 CT - -1.03 X 10 farad __ d u with weighting factor 1 ‘35 = 2b = 16.51 cm L = 60.96 cm d1 = .343 L h = parameter f = 1.2 Ghz FIGURE 5.4(b) Correct ion of Measured Data via Terminal Zone Network 154 80 F ” A theory 1- 0 measured 50 - . corrected 0 g _ measurement " E — ma 5 40 - as U r- O- a: U r— 0) :3 m l- U 3. 20 - C. 'H .- P- G I I r r I I I T I F I T T I—j ’ .1 .2 3 h/xo p_______. 4, .______.1 ll col. = 29.28 ohm "— —"25 T -9 1 g; LT = 4.25 x 10 henry a = 22.6 cm wcT = -0.l68 x 10'3 mho b = 8.255 cm CT = -0.0244 X 10.12 farad L = 60.96 cm without weighting factor (1 = 0.54, 1 h = parameter f = 1.1 Ghz FIGURE 5.4(c) Correction of Measured Data via Terminal Zone Network 155 A further consideration as a possible source of error is the varia- tion in accuracy of the numerical results. Since the finite series evaluation of the matrix elements is a function of the antenna height (see criteria for truncation in Equation (3.43)), the accuracy of the numerical results will be somewhat dependent on height. Also, for a fixed number of segments on the antenna (K = 20), the distance to the first matching point is a function of height. This will result in varying degrees of divergence of the input current (input admittance) resulting from the infinite delta- gap susceptance as discussed in Section 5.1.1. Each of these effects indicates that the problem of admittance correlation is by no means a simple one for this antenna-waveguide-medium system and that a correction network is expected to yield a variable range of agree- ment. Another observation from the results for the correction network shown in Figure 5.4 is the variation in values obtained for the inductor and capacitor. For comparison, it may be noted that for the same antenna above an infinite half plane the corrective network for a coaxial transmission line (see Section 11.38 of King [17]) consists only of a capacitor which, for the dimension used in this problem, has a value of -0.95 X 10_13 farads. This variation clearly indicates the dependence of the corrective network on the physical and electrical configuration. As the antenna is placed in different locations and as the frequency is changed, the electric and magnetic fields which will interact with the coaxial trans- mission line may be changed drastically. This will result in varying degrees of electric and magnetic perturbation in the terminal 156 zone and consequently require variation in the corrective network to compensate. More insight into the complicated nature of the variation in coupling which can occur as the configuration is changed can be best obtained from the collective results presented in the following sections. 5.2 Single Antenna. By first studying the case of the single antenna, a great deal of insight can be gained about the operation of the antenna in the waveguide. A common configuration of an antenna in a waveguide is the coaxial to rectangular waveguide coupler - the waveguide is shorted at one arm‘and the other is assumed matched to a guided system. This configuration is studied first with results compared to the usual assumptions made in the classical analysis of this waveguide system. In the cavity configuration, the current distribu- tion and input admittance are presented first as a function of frequency from below the first cavity resonance to about 2.0 Ghz. Correlation is made between the nature of the current and admittance, and the resonant frequencies. The results are then viewed as a function of location where the effects of coupling to the cavity modes are observed. And, finally, the antenna characteristics are presented as a function of antenna height which is seen to be the least significant factor of the three in determining current distribution .and input admittance. In the final section the results of applying the theory to dissipative media filling the waveguide or cavity are compared both to pertinent data available in the literature and to experiments performed here. 1 .. 157 5.2.1 Wavgggide with Matched Termination. The coaxial to waveguide coupler has been studied in many classiCal texts (see Collin [2, 3] and Lewin [24]) with the primary purpose of determining the input impedance to the antenna or probe. The treatment is basically similar in approach to that used in this report except the Current distribution on the antenna is taken to have an assumed form: I(y) = Iosin ko(h - y) 0 s y s h (5.4) that is, the approximate distribution expected on an antenna in unbounded space. The resulting input impedance is assumed to be fairly accurate because of the stationary nature of the impedance formulation and the form of the approximate impedance gives a useful starting point for design considerations. By using the assumed current of Equation 5.4 the input resistance to the antenna is found to be 22*1‘0 2 2 R ='-- - sin (5 d )tan (k h/2) (5.5) In 2 1,01 0 ab k 0 when TE1 0 mode propagation exists, where k0 is the free Space prepagat ion cons tant . This configuration of the semi-infinite waveguide has been studied with the theory developed in this report. Experimental measurements were made by filling one end of the waveguide section described in Chapter 4 with microwave absorber (visible in the right background of Figure 4.7), since a matched L-band load was not available. The other end of the waveguide was covered with the 158 brass shorting plate and the antenna was located at approximately one-quarter of a guided wavelength from this short for each operating frequency. A test was run to determine the degree of matching pro- vided by the microwave absorber by using a second short antenna (h = & inch, r8 = 1/8 inch) to probe the field between the driven antenna and the absorber. These measurements indicated an approximate standing wave ratio of VSWR,R$1.15 at 1.3 Ghz Since part of this mismatch could possibly be attributed to the perturbation caused by the probing antenna, this result indicates that a good match was obtained. Current Distributions In Figure 5.5 and 5.6 the theoretical and experimental current distributions for different antenna heights at 1.05 Ghz and 1.3 Ghz are compared. The distribution of the current is seen to show excellent agreement. The discrepancies in phase at 1.05 Ghz are most probably caused by decreased matching by the absorber, since this frequency is relatively near the cutoff frequency for the waveguide (fc = 0.909 Ghz) thus causing a long guided wavelength (kg = 57.1 cm). Also included in these figures is the shifted sinusoidal distribution of Equation (5.4). This distribution is seen to be fairly repre- sentative of the character of the current, but of course assumes that the current is everywhere in-phase with the driving voltage. The theoretical and experimental results, on the other hand, indicate that the in-phase and quadrature components of current are comparable. 159 \1\ norm. 4—0- theory A experiment H: II 2b = .5 16 1 C“ 14.48 cm 5 em = .175 10 b 0.2531g 1pm 1.05 Ghz FIGURE 5.5(a) \ Current Distribution for Antenna in Semi-Infinite waveguide (h = 5 cm, f = 1.05 Ghz) E n... Planking! IK- 1.0 \I\ norm. 160 -o-o- theory A experiment m ll HI I r 1 .05 Ghz 2b = 16.51 cm 14.48 cm 6 em = .21 )‘0 0.253718 LCD FIGURE 5.5(b) Current Distribution for Antenna in Semi-Infinite Waveguide (h = 6 cm, f = 1.05 Ghz) 161 |I\ __ norm . -o-o- theory A experiment a = 2b = 16.51 cm 8.08 cm 0 0.25171 = 1.3 Ghz g FIGURE 5.6(a) Current Distribution for Antenna in Semi-Infinite Waveguide (h = 5 cm, f = 1.3 Ghz) ERIE‘AIIIIII ill 1.0 m norm. 162 +0- theory A eXperiment = 2b = 16.51 cm 3-08 cm b 1pm = 6 em = 0.26 x0 0.251),g = 1.3 Ghz FIGURE 5.6(b) Current Distribution for Antenna in Semi Infinite Waveguide (h = 6 cm, f = 1.3 Ghz) 1.0 ';” —% n—- ——J m -— .1 norm. .5 ... —u theory F" A exper imen t a = 2b = 16.51 cm 8.08 cm. C!) h = 7 cm = 0.304 )‘0 b “— 0.251% "" t f = 1.3 Ghz FIGURE 5.6(c) Current Distribution for Antenna in Semi Infinite waveguide (h = 7 cm, f = 1.3 Ghz) 164 woumeouom ago no :ofiumoofl mocoucd rows coHumuowchoo mpaowo>m3 mugcwwcHuHEmm pow coHuonwpumfiQ ucmuuoo Hmowuouoose huomsu now Hmwxmoo ---- muooSu omm wuaoc .IIII ~.m mMDOHm .0 2o 84 u a a 3.0 u so 0 s .80 Hm.oa u LN m H 165 mucosu amw Hmwxmoo I--- Ae.usoov .huoosu now muamv n.m MMDUHR 166 If the distribution of the in-phase and quadrature components are derived from these figures, one finds that the in-phase component behaves very nearly the same as the shifted sine function while the quadrature component decays as an inverted sinusoid from the driving point to the end of the antenna. This is not shown here, since it will be most evident in the results for the cavity configuration in the following sections. In Figure 5.7 the effect of varying the antenna location is shown with comparison between the delta-gap and the coaxial gap theory. (No experiments were run for this case.) The current is seen to change from having a large in-phase component (small phase angle) at locations near the short to having a large quadrature component (large phase angle) as the location approaches one-half a guided wavelength from the short. Since a large in-phase component implies larger radiated than stored energy, one would expect the locations closer than xg/4 to be the better for consideration in a coaxial to waveguide coupler. In fact the result at d1 = 0.142 Kg having nearly zero phase angle at the input would provide the best conditions for radiating energy into the waveguide. Input Admittance The antenna input admittance for the cases considered above are given in Table 5.1 through Table 5.3. Also included is the resistance value computed from Equation (5.5). There is some difficulty in comparing resistance values, since errors in the imaginary part of the admittance from the theory (e.g. the delta gap infinite sus- ceptance noted in an earlier section)become combined with the real 167 part of the impedance upon inversion of Y. With this in mind, the theoretical input resistance compared with that computed from the approximate formula of Equation (5.5) show fairly good agreement, at least in overall behavior as a function of antenna height or location. Agreement at 1.05 Ghz in Table 5.1 between theoretical and experimental admittance is very good in view of the difficulties noted in Section 5.1. However, there is reason to suspect that at this frequency the experimenta1.'natched" boundary is failing and thus this agreement may be anomolous. In Table 5.2 at 1.3 Ghz, where the experimental set-up is expected to be fairly accurate, there is seen to be acceptable agree— ment in the real part of the admittance between the coaxial gap theory and the experiment. The delta gap theory, on the other hand, shows lack.of agreement in both parts of the admittance. This indicates that the coaxial gap provides a better model than the delta gap for this semi-infinite waveguide configuration. However, since both theories give comparable susceptance values, one sees that neither Inodel has adequately included the effect of stored energy near the antenna as compared with the experiment. This is the effect for which a corrective network of the type described in Section 5.1.2 is classically used as compensation for the approximations. Because of the scarcity of data, no network.was developed for this case. The results of varying the antenna location in Table 5.3 follow much the same agreement as noted above. However, the agree- ment in susceptance between the two theories becomes worse as the antenna is moved away from the point of maximum y-component of 168 electric field (d1 = 0.25 x8). Again this effect results from the different approximations on the antenna driving field and the way each approximation ignores effects of Stored energy. No attempt is made to discuss the comparison in impedance values given in the tables because of the effect of mixing of in- formation in the real and imaginary parts. That is, since the fundamental theoretical solution is the current for an assumed fixed applied voltage, the admittance results are the best means of correlating theoretical phenomenon to the assumptions made. 169 TABLE 5.1 Input Admittance as a Function of Antenna Height in a Semi-Infinite Waveguide (d1 - .253 kg’ 1.05 Ghz) bF. (11:0: a: = 2b = 16.51 cm Y, millimhos h 5 cm = .175),O 6 cm = .21),o 7 cm = .246),O Theoretical (Delta gap model) 2.6 + j3.94 2.44 + j3.02 2.0 +j2.56 Theoretical (Coaxial gap model) 3.92 + j4.25 3.73 + j3.06 3.03 + j2.56 Experimental 4.71 + j4.66 3.93 + j4.26 3 + j4 Input Impedance , zin’ ohms h 5 cm 6 cm 7 cm Theoretical (Delta gap model) 117 - j177 162 « jZOO 189 - j242 Theoretical (Coaxial gap model) 117 - j127 160 - j13l 192 - j162 Experimental 107 - j104 117 - 3127 119 - jl6O Approximate Rin Eqn. (5.5) 85.6 13.7 214.2 170 TABLE 5.2 Input Admittance as a Function of Antenna Height in a Semi-Infinite Waveguide (d1 = 0.251xg, 1.3 Ghz) d1 1‘ .. a = 2b = 16.51 cm Y, millimhos = .2 = . = . h 5 cm 17),O 6 cm 26x0 7 Cm 304),o Theoretical (Delta gap model) 4.69 + j4.73 4.0 + j3.ll 3.06 + j2.61 Theoretical (Coaxial gap model) 7.33 + j4.75 6.05 + j2.47 4.50 + j2.l6 Experimental 8.9 + j8.43 7.7 + j6.05 7.17 + j5.l Input Impedance , zin’ ohms 5 = .2 6 = .2 = . h cm 17),,o cm 6x0 7 cm 304),0 Theoretical (Delta gap model) 105.5 - j106.4 155.6 - 3121 189.2 - j161 Theoretical (Coaxial gap model) 96.1 - j62.2 142 - j57.7 180 - j86.7 Experimental 59.3 - j56.2 80 - j63 92.5 - j65.5 Approximate Rin Eqn (5.5) 68.42 118 206.6 171 on 2: - mam mam: - 5:” on: + :4. $2.1. so; wx «3. m 2: we: - 0.3 Na: - «.3 8.: + and 3;: + an; .6 8m. 9m: 3: - 9o: 93: .. m: 8;: + 36 2.: + S.~ mg :m. w Rs 8: - m2 8: - a: 8.: + 25 86?. a: .1 m2. w 24 o: - OS 8: - SN 8.: + 33 a: + an .1 8n. w 2 3.: + N: as: - SN 2:: - 26 3.2 + 23 a N3. Am.mv .com cam knowsfi anoosy mucosa mnoonh oUuEaxouooa new flowxmou now ouHon new amaxwoo new ouaon Hp a: .c: are . N 8:353 a use mo.~ u m noumaoumo a an s ... a .. soon; audafiufiu. £6 83 :5 e a 5 ovasmo>o3 ouacamcHuaamm m a: soauoooa mssoua¢ mo coauoaom o no mocuuua8p< unocu m. m mafia. 172 5.2.2. Air Filled Cavity; Current Distribution The antennadwaveguide~media system in the cavity configura- tion has been discussed previously as the most attractive choice for application to probing of the medium. It is also an interesting configuration theoretically because of the inter-relationship and superposition of the two resonant systems: the cavity and the antenna. Although use of an antenna to excite a cavity is common, very little has been done to determine the characteristics of the antenna in this configuration. The theoretical and numerical solution of this problem discussed in the previous chapter accurately predicts the antenna current distribution with excellent experi- mental agreement as will be seen in the results which follow. The complete understanding of the characteristics of this system is gained when the nature of the input admittance is considered. This is done in Section 5.2.4. The case considered is a cavity of fixed length (L = 24 inches), fixed height (h = 3.25 inches), and two widths (a = 6.5 inches, 8.5 inches). The antenna is centered on the broad wall (at x = a/2) and is moveable axially (z-direction) in discrete steps. The antenna height is also a parameter. The following results are presented first as a function of frequency, then as a function of antenna location, and finally as a function of antenna height, with the other parameters fixed in each case. Current Distribution with Frequency as the Parameter The dominant factor in determining the antenna current dis- tribution is the resonant nature of the cavity. As the frequency is varied, the existence of the cavity resonances results in a pattern 173 of current distributions which is dependent upon the proximity of the operating frequency to the antenna-cavity resonances. For the ideal empty cavity considered here with the narrowest width (a = 16.51 cm) the first several resonances up to about 2.0 Ghz are given in Table 5.4. The location of the antenna in the cavity determines which resonances are excited and thus is also important in describing the nature of the current distribution. When the antenna is centered longitudinally (equidistant from the waveguide short circuits at d1 = 0.5;), only the resonances having odd numbered longitudinal modes ("L" = l,3,5,...) are excited. The variation in current distribution in the vicinity of one of these resonances (f = 1.171 Ghz) is shown in Figure 5.8 for a fixed 1,0,3 antenna height. As the frequency is varied through this resonance the current distribution changes through a complete "cycle." Start- ing from essentially a triangular distribution well below the resonance (and well above the next lower resonance) the distribution becomes similar to a quarter-wave cosine distribution as the resonance is approached. At the resonance the distribution is like a half-wave sinusoid and as the frequency goes above resonance, this distribution shifts outward along the antenna until it again appears essentially triangular. This "cycle" is repeated through each cavity resonance which is excited. Since resonance is defined in terms of admittance or impedance at the input terminal plane (e.g., when the susceptance or reactance is zero), it is not evident from the current distribution alone when the resonant condition is achieved. This will be examined in detail in Section 5.2.4, but since the current is purely in phase -- 174 quadrature with the driving voltage, resonance may be expected to be the point at which the current is zero at the driving point. Thus resonance is seen to be just below 1.15 Ghz for the case shown in Figure 5.8. This is, of course, different than the ideal resonance at 1.171 Ghz, since the antenna perturbs the cavity and is a resonant system itself. For the cavity in a wider configuration (a = 22.6 cm) the ideal empty cavity resonances are given in Table 5.5. The current distributions at frequencies near the f1,0,3 resonance are shown in Figure 5.9. Again the same repetitive pattern of current dis- tributions is seen to exist as the frequency is changed from below to above the resonance which is excited, and again the resonant con- dition occurs below the ideal empty cavity resonance. In both of the above cases, the experimental current dis- tributions at frequencies where data was taken is seen to compare extremely well with the theoretical results. The slight differences *which are shown near the resonance points are readily accepted as the failure of the theoretical model to precisely describe the perturbation of the ideal cavity resonances by the presences of the antenna. Indeed, the model assumes a strip approximation for the antenna which will obviously perturb the cavity differently than the physical cylindrical antenna. Because of the rapidly changing nature of the current over a relatively narrow frequency range the agreement which is shown is very satisfactory. The current distributions for the entire frequency range between 0.6 and 2.0 Ghz is shown in Figures 5.10 and 5.11 for the above two cases of cavity width (a = 16.51 cm and a = 22.6 cm, 175 respectively). The agreement between theory and experiment is again excellent and the nature of the distributions is seen to be dependent upon the relative proximity of an excited cavity resonance. One discrepancy between theory and experiment occurred at 1.2 Ghz in the wide (a = 22.6 cm) configuration shown in Figure 5.11. The measured current distribution did not agree with the theoretical result at 1.2 Ghz but is shown to agree with the theory at 2.4 Ghz. This is a consequence of the experimental set-up alone. When making experimental measurements, the sliding short circuit above the tee in the antenna driving system (see Figure 4.1 and 4.7) was adjusted to give maximum signal from the current measuring probe near the antenna input plane. It was possible, however, to erroneously tune to the first harmonic of the source frequency if the full range of the sliding short was not investigated. This in turn de-tuned the fundamental frequency with the result being primarily dependent on the first harmonic frequency. It was possible to detect this error before making measurements by noting the phase angle between voltage and current probes at the impedance sensing block. Since the block was calibrated at 1.2 Ghz and a phase angle reference was obtained, the measured impedance should be very nearly 1:900 with respect to the reference. Since only in the most unlikely of circumstances is this phase angle reference the same for both the fundamental and the first harmonic, the measured impedance phase angle for the first harmonic is‘significantly different from the expected : 90°. Al- though this was overlooked in the single case of 1.2 Ghz in Figure 5.11, care was taken to insure tuning to the fundamental in all other cases. 176 Another indication of experimental difficulty regarding tuning to the first harmonic is the deviation of the measured current distribution from the repetitive pattern shown in Figures 5.8 and 5.9. That is, the current distribution at frequencies above the resonance determined by mode index "n" = 2 has an inflection point instead of the straight line or triangular char- acteristic evident at the lower frequencies. In the case of Figure 5.11 at 1.2 Ghz this inflection point was evident and thus experi- mental difficulty was suSpected. In Figure 5.12 the current distributions are shown as a function of frequency again for the case of the narrow cavity (a = 16.51 cm). However, the antenna is not centered longitudinally and thus couples to (or excites) all cavity resonance given in Table 5.4. In this case the repetitive pattern of the current distribu- tion occurs twice as often as the frequency is changed from below the first resonance to 2.2 Ghz. Agreement between theory and experiment is again very good with acceptable differences accountable by the approximations of the theoretical model as discussed previously. 177 TABLE 5.4 Resonances of Rectangular Cavity (a . 16.51 cm, b = 8.255 cm, 4, = 60.96 cm) n m L fres’ Ghz 1 0 l .941 l 0 2 1.033 1 0 3 1.171 1 0 4 1.339 1 0 5 1.529 1 0 6 1.734 1 0 7 1.947 1 1 1 2.046 1 0 8 2.168 h» 178 L a = 2b = 16.51 cm L = 60.96 cm (11 = d2 = 0.56 h = 6.25 cm (fres)1,0,3 = 1.171 Ghz -o—o—- theory A experiment 1.0 —Y I I I I I I ' 1.0 . 1.0 Ghz 111 ' \1| .5 - 5 )- ¢ = +900 0 J I J l l J l l I O 5 y/h 1 o FIGURE 5.8 Current Distribution as a Function of Frequency Near a Cavity Resonance (a = 16.51 cm , d1 = 0.5L) 179 1.0 1.15 Ghz 1.0 \I\ ' 1.0 FIGURE 5.8 (concluded) \I\ O 180 TABLE 5.5 Resonances of Rectangular Cavity (a = 22.6 cm, b = 8.255 cm, L = 60.96 cm) n m L fres’ Ghz 1 0 1 .708 1 0 2 .826 1 0 3 .992 1 0 4 1.187 2 0 1 1.350 1 0 5 1.397 2 0 2 1.416 2 0 3 1.519 1 0 6 1.618 2 0 4 1.652 2 0 5 1.809 1 0 7 1.845 1 l 1 1.950 2 0 6 1.985 1 1 2 1.996 3 0 1 2.006 ."v b I»\ 1’ , . l. h A. ..: 181 bw—dl o—d2——— L a = 22.6 cm b = 8.255 cm L = 60.96 cm (11 = d2 = 0.5L h = 6.25 cm (fres)1,0,3 = 0'992 Ghz 0 experiment 1.0 \I\ FIGURE 5.9 Current Distribution as a Function of Frequency Near a Cavity Resonance (a = 27.6 cm, d 1 = 0.5L) 182 1.0 Ghz 1.0.. 1.05 Ghz .5. L O --—¢=+90-¢- _ r l l l l 0 y/h 1 0 .0 FIGURE 5.9 (concluded) 183 a = 2b = 16.51 cm .— d1 d2—-. b L '= 60.96 cm d1 = (12 = 0.51, L h = 6.25 cm f = parameter H theory 1.0)— 0.6 Ghz 1. \Il ‘ r .5- F. L ¢ . +9o° 0 J i l L L I l L 5 y/h 1 o 1.0- 0.8 Ghz 1.0“— 0.85 Ghz ~ F \1\ . |I\. )- r- .5 1 .5 .. ‘ F 0 0 L n 1 n l L 1 I L .5 y/h 1.0 .5 y/h 1.0 FIGURE 5.10 Current Distribution as a Function of Frequency 0.6 — 2.2 Ghz (a = 16.51 cm , d1 = 0.54,) 184 I a = 2b = 16.51 cm 1*. d ‘._.. d2 __' b L = 60.96 Cm 1 - — L h = 6.25 cm f = parameter ._.- theory 0 experiment 1.0*- 1.3 Ghz 1.0 FIGURE 5.10 (continued) 185 y/h 1.0 .5 y/h 1.0 FIGURE 5.10 (concluded) 186 -0-0 theory .—— dl—bflv—dz —-.b L 0 experiment 1.1 Ghz 1 FIGURE 5.11 Current Distribution as a Function of Frequency, 1.1 - 1.9 Ghz .0 ‘1! ms‘mbo‘m 22.6 cm 8.255 cm 60.96 cm = d2 = 0.5L 6.25 cm parameter \ + o \ (2.4 Ghz /‘ o 0‘ as = -90 ) ‘K (see text) I. K 1 1 1 1 l L 1 1 0'5 5 y/h 1.0 (a = 22.6 cm , d1 = 0.54,) 187 FIGURE 5.11 (concluded) 188 b .- d1 e (12 ——— L a = 2b = 16.51 cm L = 60.96 cm (11 = 0.2901, h = 6.25 cm f = parameter 1 .0 1.0 \1\ - \1\ .5 ‘ .5 0 l L J 1 l L L 1 4 0 .5 y/h 1.0 FIGURE 5.12 Current Distribution as a Function of Frequency, 0.6 - 2.2 Ghz (a = 16.51 cm , d1 = 0.29041.) 189 1.0 \11 1.0 0.95 Ghz .5 y/h 1 FIGURE 5.12 (continued) .0 190 1.0 1.0 ‘4 _ 1.1 Ghz _ 1.15 Ghz \I\_ m .5- 5 ' +- ¢=+90o _ G 1 L 1 1 J 1 1 1 1 0 1 1 .5 . y/h 1.0 1. L 1.16 Ghz 1’0” 1.17 Ghz \11. m1 . m = -90° . I- b o 1 1 1 1 L 1 1 1 1 O .5 y/h 1.0 FIGURE 5.12 (continued) 191 1'0 1.25 Ghz \I\ ” .5 ’ 0 l 1.35 Ghz ¢ = +900 1 1 1 1 I 1 1 1 1 5 y/h l 0 1.0 1.37 Ghz . ' l1\ - FIGURE 5.12 . (continued) .5 - 192 FIGURE 5.12 (continued) 193 1.75 Ghz ) ¢ = +90‘ 5 y/h 1 0 1.9 Ghz ¢ = -900 l I I L L l l 5 y/h 1 0 FIGURE 5.12 (continued) 194 2b = 16.51 cm 60.96 cm = 0.290L = 6.25 = parameter 1 Harmon: FIGURE 5.12 (concluded) 195 Current Distribution as a Fonction of Antenna Location In the preceding paragraphs the antenna current distribution was seen to be strongly dependent upon the relationship between the operating frequency and the cavity resonant frequencies. The loca- tion of the antenna in the cavity determined the degree of coupling to each cavity resonance. In Figures 5.13 through 5.15 the current distributions on an antenna at various axial cavity locations is shown for fixed operating frequency and antenna height. As can be seen, the general nature of the distribution (i.e., triangular, quarter-wave cosinusoid, shifted sinusoid, etc.) is determined by which cavity resonance is excited for a given location. That is, for example, in Figure 5.13 at 1.2 Ghz, two basic distributions are exhibited which for convenience may be called "concave" (with a de- creasing slope along the antenna) and "convex" (with an increasing slope along the antenna). These distributions are qualitatively related to the degree of coupling to different cavity resonances for each location. That is, the resonance described by mode index "L" = 3 (f1,0,3 = 1.171 Ghz for the empty cavity) has zero y-com- ponent (parallel to the antenna) of electric field at the axial locations 2 = 0.333L and z = 0.667L. Thus at the locations d1 = 0.290L, d1 = 0.3434 and d = 0.396L there is reduced 1 coupling to this f1 0 3 resonance and the nature of the current depends on the coupling to the nearby "L' = even resonances f1 0 2 and f1 0 4. At the other locations considered in Figure 5.13, however, there is stronger coupling to the f1 0 3 resonance and 9 9 the current distribution depends on the proximity of the 1.2 Ghz Operating frequency to the system resonance. This is 1,0,3 196 obviously a simplified view but serves to describe general overall behavior. There are, of course, differences between all the dis- tributions which reflect the degree of coupling to each resonance and the relative nearness of the operating frequency to each resonant frequency. This qualitative description of the nature of the current distribution applies to the results shown in Figures 5.14 and 5.15 as well. In each case the coupling to the odd or even numbered "L" mode index resonances is determined by the antenna location and the current distribution can be identified with a different part of the "cycle" described at the beginning of this section. In Figure 5.15 the operating frequency is very nearly equal to the empty cavity resonance f1,0,5 = 1.529 Ghz. (The antenna- cavity system resonance is below this frequency.) In this case, the locations d1 = 0.185L and d = 0.396L do not couple to the 1 “L" = 5 resonance (which has Ey = 0 at z = 0.2; and z = 0.4L) resulting in different current distributions. Greater discrepancies between theory and experiment are seen in this case for basically two reasons: first, the differences between model and experiment and thus between theoretical system resonance and physical system resonance; and second, the large magnetic fields which exist in the cavity near resonance thus increasing the current probe errors (dis- cussed in relation to Figure 4.12). These probing errors are accountable for the apparent shift in zero-crossing of the experi- uental current distributions, since the shift is dependent upon location in qualitatively the same manner as the cavity magnetic field. 1.0 71 1‘0 N 1; 1 1.0 m ‘ 197 d1 = 0.2904, . A; 0 ¢ = +90 1 £3 . = 0.2384 1 o . ¢ 1: +90 A A .. £5 . £5 1 1 l L I 1 .5 y/h. 1.0 [:0 experiment ...... theory a = 2b = 16.51 cm L = 60.96 cm d1 = parameter h = 6.25 cm 0.25 x0 1.2 Ghz 1"! II FIGURE 5.13 Current Distribution as a Function of Antenna Location (1.2 Ghz) 198 1.0 .5 y/h 1.0 FIGURE 5.13 (concluded) 1'01- k," I 111 ’ 1.0 N .11 .’ 0x 199 A 0 experiment + theory dl I...—.+._ d2 _..| We. -.- 1 . L - a = 2b = 16.51 Cm L = 60.96 cm (11 = parameter h = 6.25 cm f = 1.4 Ghz FIGURE 5.14 Current Distribution as a Function of Antenna Location (1.4 Ghz) 200 FIGURE 5.14 (concluded) 1.0 t'vr‘; : -a ,, 1 A ‘ . A ' d - o 5 ' <> 0 1 — L o _ £5 = ~90 o . ¢ ‘ \Il ’ A <> . AA <> 1 ,5 _ d1 = 0.448: A 0 q ¢ = +90 ' <> (5 . £5 . o 1 1 1 1 l 1 L 1 1 .5 y/h 1.0 1.0 (— I1’ (I I." T 1.4 II 201 1.0 exPerment A :11 = 0.1334, - theory —‘_ E] experiment 0 d1 = 001354. ‘ ‘1‘ theory ‘- b ‘ d 1 . 4 F vl“ d2 -——.-| .5 #- ¢=+900 0 ¢ = _%01 d----fl---— I b F D a = 2b = 16.51 cm b ‘3 - 4. = 60.96 cm d = parameter r- A a q 1 15 h = 4 cm ' 0 x" f = 1.53 Ghz 0 —l 1' g {I 1 ‘ . y/h 1.0 FIGURE 5.15 Current Distribution as a Function of Antenna Location (1.53 Ghz) 202 1.0 |1| = 0.3964, +90o . . .5 —- _ o . o . o L A A 0 1 4 y/h 1.0 1'0 - 0 448 4A dl ' L ,, + .. o . d = 0.5 [1‘ 1 0!. 4 - 1 1. (b = _%o - .5 ' .. D U o . A . A .‘A. 0 . . . . y/h 1.0 FIGURE 5.15 (concluded) 203 Current Distribution with Antenna Height as the Parameter The final independent parameter considered in the antenna- cavity system with air as the medium is the antenna height. The height is, of course, of primary importance in studies involving antennas in unbounded Space but is found to be of secondary importance in this bounded medium configuration. The current distributions as a function of antenna height at various fixed frequencies and fixed axial locations in the cavity are shown in Figures 5.16 through 5.21. Agreement between theory and experiment is excellent in all cases except at 1.53 Ghz in Figure 5.21 which is near a cavity resonant frequency and suffers the theoretical and experimental errors already discussed in relation to Figure 5.15. In general, the antenna height is given only in physical centimeters because the current distribution is not dependent upon the height as much as on the cavity resonant characteristics. As can be seen in the figures, the general shape (concave, convex, quarter wave cosine, etc.) of the current distribution is not affected by antenna height but instead the height determines "how much" of the shape is included on the antenna. This is, of course, qualitatively similar to the unbounded case, but here the unusual changes in the nature of the current distribution as a function of cavity resonances predominates. As an example of the above discussion, consider Figure 5.16 where antenna heights are shown in electrical free Space wavelengths. Iflle nature of the current is obviously different than the analogous unbgundgd 3.... The interesting point in this figure is that the cxrrrent zero crossing point (or point of phase reversal) remains 204 physically at approximately the same location on the antenna: (at yiw 4 cm.) as the height is varied. This physical location falls very near the midpoint between top and bottom waveguide walls (i.e. at b/Z). This behavior then implies some type of coupling to a waveguide mode having mode index "m" = 1 rather than the dominant "m" = 0. Although this correlation is not precise, it implies that there is strong influence §£_this frequengy from these higher evanescent "m" = 1 modes. In Figures 5.17 and 5.18 the current distributions appear more like that expected in unboundedSpace but are still, of course, related to the cavity resonances and operating frequency. The effect of varying height is in general similar to that expected. Note, how- ever, that at 1.2 Ghz (Figure 5.19) the effect of varying antenna height is seen to have very little effect on the nature of the current distribution, while at 1.3 Ghz (Figure 5.20) the antenna height has more effect on the current than would be expected in unbounded space. Finally at 1.53 Ghz (figure 5.21) the theoretical current again appears relatively independent of antenna height. It is interesting to compare the results shown in Figures 5.21 and 5.16. In contrast to the current at 1.05 Ghz discussed above, the physical location (y, in centimeters) of the current zero crossing at 1.53 Ghz does not stay fixed as the height is varied. Instead, this phase reversal point moves so that it occurs at approximately the same normalized distance (y/h) along the antenna, and there seems to be very little correlation between evanescent cavity mode distribu- tions and the current distribution. 205 In general, the effect of varying the antenna height on the current distribution depends upon the operating frequency and near- ness of cavity resonances. There is apparently no unique relation- ship between antenna electrical length and current distribution; it is difficult to describe resonance or anti-resonance behaviour of the antenna from the height alone. 206 A 0 experiment -ih- | theory 2b = 16.51 cm 60.96 cm = 0.238L = parameter = 1.05 Ghz FIGURE 5.16 Current Distribution 1.0 '; as a Function of Antenna Height \, ‘ (a = 16.51 cm, “‘ 1 .05 Ghz) o h = 6 cm = 0-21A ll 0 h) .p U1 7 0 o D 9- II I 207 A 0 0 experiment 50-.- theory d gt... .2 -—«-. 1;: L = 2b = 16.51 cm = 60.96 cm = 0.133L = parameter = 1.1 Ghz 1 H1 3' 0. ¢- m FIGURE 5.17 Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.1 Ghz) 208 E.-.——-— d —————'l b L 22.6 cm 8.255 cm 60.96 cm = 0.133L parameter 1.1 Ghz HI 3‘ O- 6- U‘ m ll .... theory <> experiment 1.0 \I\ FIGURE 5.18 Current Distribution as a Function of Antenna Height (a = 22.6 cm, 1.1 Ghz) 1.0 :1! . 1.0 209 FIGURE 5.18 (concluded) 210 ‘11-. b L a = 2b = 16.51 cm L = 60.96 cm d1 = 0.133L h = parameter f = 1.2 Ghz -o—o- theory A experiment 1.0 I I U I ‘r 1 I v \Il 5 F 0 J 1 1 l J J 1 n n .5 y/h 1.0 FIGURE 5.19 Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.2 Ghz) 211 FIGURE 5.19 (concluded) 1.0 III b L. .5_ 212 :: d2 b d1 L a = 2b = 16.51 cm L = 60.96 cm d1 = 0.133L h = parameter f = 1.3 Ghz -o-+ theory 00 experiment 1.0 VJ-Vr ‘ r f u t I v v . Q h- 0 - 0 1 O O O . h = S cm \I‘ . ¢ = -900 b o d O 0 ° ’ h = 4 cm ‘ o = + .5 _ <0 90 O , . O L l L 1 l 1 J L L 5 y/h 1.0 FIGURE 5.20 Current Distribution as a Function of Antenna Height (a= 16.51 cm, 1.3 Ghz) \II 1.0 \I\ .7 . ~ ' 0 b 0 G . <> . P . o " <> ¢ = -900 d I ¢ = +900 1 1 1 l 1 1 1 1 y/h 1.0 O - <3 0 0 t1==8 cm 0 o o ' O O b. - _ O _ ¢ = -900 O’ P I 3! e 1 1 1 4 1 1 FIGURE 5.20 (concluded) 1.0 \II 1.0 U! 214 experiment theory = 3 cm :- ..v0 = 4 cm _ + I + ‘7 I- ). ¢ = -900 «II \ ///’/' - / \ . o O b ¢ - +90 <> 4 '- vv . - ..V. .’ V y/h 1.0 = 6 cm = 7 Cm . ¢ = -900 . V — o d V 9 _ 8 .. . - (P = +90 8 . .5 y/h 1.0 d *u‘— - l 1 §5 d2 l b in L fi 2b = 16.51 cm 60.96 cm = 0.5L = parameter 1.53 Ghz FIGURE 5.21 Current Distribution as a Function of Antenna Height (a = 16.51 cm, 1.53 Ghz) 215 5.2.3 Air Filled Cavit ; Charge Distribution The distribution of charge on the antenna is determined numerically as the derivative of the current based on the continuity equation (see Section 3.1.5). Experimental methods of measuring relative charge distribution are discussed in relation to Figure 4.5 and in Section 4.2. Since the current distribution is taken to be the fundamental result, very few experimental charge distributions are presented here. In Figures 5.22 and 5.23 the results for several different frequencies with two different physical configurations are shown. One experimental factor which is unique to this charge distribution data is that the dielectric support wafer at the antenna terminal plane was made of nylon (replaced by styrofoam for the current and admittance measurements). This results in large experi- mental charge at the input terminals because of the build up of charge on the dielectric. Therefore, the experimental charge measure- ments are not normalized to the theoretical point nearest the input plane but instead to the points y/h as 0.3 or y/h w 0.5 depending on which has the greatest magnitude of the charge. The normalization point chosen is shown in the figures. As can be seen, the overall agreement between theory and experiment is very good. Many of the differences are directly related to the errors found in the current distributions. One observation concerning the numerically determined charge distribution is the oscillation in magnitude which is seen, for example, in Figure 5.22 at 1.5 Ghz. This is a result of the truncation 216 of the series in the theoretical solution. When the series is trun- cated prematurely by just one or two modes, this oscillation appears. This observation was subsequently used as a qualitative measure of the appropriateness of the chosen truncation point and adjustments were made if the oscillations were large. It is perhaps interesting to note that these oscillations occur only toward the tip of the antenna -- the worse the truncation point, the closer the oscilla- tions came to the driving point. Another interesting point concerning the theoretical charge is the values predicted at the driving point (the first numerical subsection) and at the tip (the last subsection). These are in- cluded in Figures 5.22 and 5.23 where possible. The jump in magnitude at the driving point is a result of the anomolous rise in current predicted by the delta-gap model as discussed in Section 5.1.1. The rise in magnitude of charge at the tip has been observed by others (see King and Harrison [18]). It was not physically possible to experimentally verify this charge build up on the antenna tip because of the presence of the current probe (see Figure 4.5). 217 a = 2b = 16.51 cm b...— d1 h L= 60.96 cm d1 = 0.5L , h = 6.25 cm charge is shown as absolute in units of coulombs -°-0- theory 0 experiment Qnormalization point 0 O . 1.0 Ghz 1 .0 1.15 Ghz O 2.0 - i 2.0 — . O 1 f w , m -11T ' 415 X10 _ o (X10 ,. O()O 1.0 — ‘ 1.0 _ L r b 0 = _%o 1 b 0 1 1 1 1 l 1 1 1 1 0 5 y/h l 0 1.0 1 _ O l. 7 Ghz . 1.5 Ghz . W . 2.0 9 x10.11 ~ \Q\ ' L- -11_ X10 0 .5 r ' ~ 1.0 — P L b- h- t . 0 . FIGURE 5.22 Charge Distribution as a Function of Frequency (d1 = 0.5L , h = 6.25 cm) 218 a = 2b = 16.51 cm b -* (11 h L = 60.96 cm d1 = 0.29L , h = 6.25 cm charge is shown as absolute in units of coulombs —'-°- theory Oexperiment 0 normalization point 5.0 1— 1.0 Ghz 2.5 _O L- r- \Q1\1. o X10- L M L 070.0'66.0.3*" 00 1.0— o r 0.5” _ ¢ = -9o° O l l l l l l l l l 1 0 5 y/h 1 0 5'0 1.2 Ghz 10°C 1.3 Ghz 1- . ,— , m M ' X10-11“ ° -11_ q} X10 0 . 5.0- .. 0 COO . 1.0- o . co = -90 0 1 1 1* 1 l 1 1 1 1 5 y/h 1 0 FIGURE 5.23 Charge Distribution as a Function of Frequency (d1 = 0.29L , h = 6.25 cm) 219 10.0 M xlo'11 5.0 r- 1.4 Ghz 0 10.0 _ 1.6 Ghz \Q\ ' X10 0 _ 0000000 5.o- ° $>o() _ = -9o° o 1 l 1 1 1 1 5 y/h 2.5 L3 2.0 Ghz ‘ |Q| x10"11 . o . 2.5 101. x10 -11 1.5 Ghz charge is in units of coulombs FIGURE 5.23 (concluded) 220 5.2.4 Air Filled Cavity; Inppt Admittance The input admittance of a single antenna in an air filled cavity for all the cases considered in Section 5.2.2 is presented in this section. The experimental data is in unaltered and uncom— pensated form for comparison with the theoretically predicted values. The comparative tables are followed by correSponding curves showing the theoretical admittance for each case. Where possible, the dis- crete theoretical points are connected by approximate lines to in- dicate the qualitative dependence predicted. These dashed lines are meant only to aid in visualizing overall behavior so that various phenomena such as resonance and anti-resonance can be readily identified. Extrapolation of the discrete theoretical results to form these curves is based on the known or expected continuity of the solution and Such fundamental results as the Foster reactance theorem. Since the cases considered here are only for the cavity con- figuration and a single antenna, there is no theoretical mechanism for loss (all conductors are assumed perfect). Therefore, the input admittance will be purely susceptive. Although conduction losses are present in the experimental system, these were not detectable . 0 within the i;2 accuracy of the admittance measuring system used. Input Admittance with Frequency_as the Parameter As was stated in Section 5.2.2, the dominant factor in de- termining the behavior of the antenna-cavity-media system is the resonant nature of the cavity. The "cyclic" nature of the current distribution which is observed as the frequency is swept through a 221 cavity resonance is also seen in the input admittance. This data is rshown in Tables 5.6 and 5.7. In the correSponding Figures 5.24 and 5.25 the theoretical admittance is plotted along with the appropriate approximate current distributions. In these figures the "cycle" in the shape of the current distribution is seen to have a unique re- lationship to the resonances and anti-resonances of the system. If the system is visualized as an equivalent parallel (lossless) LC resonant network then resonance is defined as the point at which the susceptance is zero and anti-resonance as the point at which the sus- ceptance passes through infinity. This definition is chosen, since the system resonance falls very near the theoretical empty cavity resonance considered in each of the figures. As can be seen in Figures 5.24 and 5.25 the current distribution at anti-resonance is similar to a quarter wave cosinusoidal distribution, while at resonance it is similar to a half-wave sinusoid. The input admittance over the frequencies between 0.6 Ghz and 2.2 Ghz for a cavity of width a = 16.51 cm is shown in Table 5.8 and Figure 5.26. In Figure 5.26 the resonant nature of the admittance is evident. Comparison with the current distributions presented pre- viously in Figure 5.10 shows the correlation between the shape of the current and the cyclic nature of the admittance as depicted in the composite Figure 5.24. The theoretical empty cavity resonances from Table 5.4 are also included in Figure 5.26. Since the case shown is for a centered (d1 = 0.5;) antenna, only the odd "L" modes are excited and these are shown as solid triangles in the figure. The even "L" mode resonances are included as open triangles. 222 In Table 5.9 and Figure 5.27 similar results are presented for a centered antenna (d1 8 0.5;) but for a wider cavity (a = 22.6 cm). The same correlation between cavity resonances and zeros of the input admittance is shown. As in all the admittance data (see Section 5.1), the theoretical and experimental data show notable discrepancies in magnitude. However, the behavior as a function of frequency is qualitatively comparable. The input admittance with frequency as a parameter is lastly considered for the case of the antenna off-center (d1 = 0.29L) in Table 5.10 and Figure 5.28. Here the antenna couples to all cavity resonances and the cyclic nature of the impedance is seen to repeat for every "L" mode resonance. Again magnitude discrepancies between theory and experiment are found but in view of the rapidly varying nature of the admittance, the agreement in overall behavior is very satisfactory. 223 TABLE 5.6 Input Admittance as a Function of Frequency Near a Cavity Resonance (a = 16.51 cm , d1 = 0.54) b [::_d;::u}; ] a = 2b = 16.51 cm, L = 60.96 cm d = 0.5;, h 8 6.25 cm l-o-——L—~i 1 current distribution in Figure 5.8 Input Admittance, millimhos frequency, Ghz Theory Experiment 1.0 + j5.6 + 113 1.05 + 321.3 - 1.07 - j121.4 - 1.1 - j3.26 - j10.l 1.15 + j0.44 - 1.17 + j1.25 - 1.2 + j4.45 + j7.3 1.25 + j8.6 - input susceptance, B. millimhos 25 1- I, I I '0 ca’ ’ "‘ I ‘g 20 ' l '\. I X I ". I 41 I I 15 - / I I I I I Ii I 1 ’ a 10 - / a ,mt I, .‘1 3’ -._‘. '-. -' ‘ I a \ ’I’ 1 {I '1 I ‘ I ,-" ..\‘ S P I \ ‘-. f../ \" 11 , i g ” I 4' ’ o L A A J L l 1 1 L L 1 1 1 1.0 1.05 1.1 ,,” 1 15 ’I (’9 I I ’ ~~ s. -5 _ I I ' 0 I, '2, ' a I 3 I '1 -10 " 0' \ .' < :' I I -120 1.. ,‘ "1.. I 3 CD '1, ' 1 ’ FIGURE 5.24 224 I ‘. ..x \- ‘\ 4 'l K.“ I ,e x I I I I I I e ’ I I f, Ghz Theoretical Input Admittance (Susceptance) as a Frequency Near a Cavity Resonance (a - 16.51 cm, d1 = 0.54) theoretical cavity resonance (Table 5.4) See Table 5.6 for data and configura- tion Function of 225 TABLE 5.7 Input Admittance as a Function of Frequency Near a Cavity Resonance (a a 22.6 cm , d1 = 0.5L) E'd I ] a = 22.6 cm, b - 3.255 cm, I, = 60.96 cm b 1 h _ d1 = 0.5L , h 8 6.25 cm h—L—A current distribution in Figure 5.9 Input Admittance, millimhos frequency, Ghz Theory Experiment .85 + j8.45 A 4-jl9.4 .9 + j21.88 + leS .95 - j5.15 - j8.26 .975 + j.165 +-j.856 1.0 + j2.65 + j5.13 1.05 + j6.08 + j13.7 input susceptance, B, millimhos 226 25 20 15 10 5 0 See Table 5.7 theoretical for data and cavity -5 1— configuration resonance 1 (Table 5.5) L FIGURE 5.25 Theoretical Input Admittance (Susceptance) as a Function of Frequency Near a Cavity Resonance (a = 22.6 cm , d1 = 0.5L) 227 TABLE 5.8 Input Admittance as a Function of Frequency 0.6 - 2.2 Ghz (a = 16.51 cm, (11 = 0.5L) b -—-d1—][h 8 = 2b = 16.51 cm, L = 60.96 cm d 1 = 0.5L, h = 6.25 cm h—¢—-4 current distribution shown in Figure 5.10 resonances in Table 5.4 Input Admittance, millimhos frequency, Ghz Theory Experiment .6 + j4.73 - .7 + j7.5 + j18.9 .8 -+ j31.65 - .85 - j12.67 - .. see Table 5.6 ... 1.3 + j24.6 + j44 1.4 - j6.8 - j10.8 1.5 + j.77 + j2.79 1.6 + j18.7 + j28 1.7 - j6 - jl7.3 1.8 - j1.4 - j2.83 1.9 + j1.27 + j3.64 2.0 - 118.6 - 2.2 + j.187 - 228 .26 Sac n at .80 Hméa u my 55 ~.~ I m.o homosvmum mo cofiuogm m mm 3:33.55. usacH Hmofiuouoofi. o~.m 55min ~~ ‘_-_—-——.. ---_-- ------_-_-.._o_ ¢.m manna mmocmaommu G I. zufrmo ‘ IL m.o . _ . . I I 1.... .0 I xiw.. \\Q\ I. cemumuswamcoo new name you w.m magma mum ON: 0 "1' O OH ow om lo.» soqurnum ‘g ‘aoueqdeosns nndut 229 TABLE 5.9 Input Admittance as a Function of Frequency 0.85 - 1.9 Ghz (a = 22.6 cm , d .915“. I+——L—-4 a d 1 = 0.5L) = 22.6 cm, b = 8.255 cm, L = 60.96 cm 1 = 0.5L, h = 6.25 cm current distributions in Figure 5.11 Input Admittance, millimhos frequency, Ghz Theory Experiment 0.85 + j8.45 + jl9.4 0.90 + j21.88 + j105 0.95 - j5.15 - j8.26 0.975 + j0.l65 + j0.86 1.0 + j2.65 + j5.13 1.05 + j6.08 + j13.7 1.1 + j10.l7 - 1.2 + j427 - 1.3 - j5.52 - j10 1.4 + j3.22 +-j8.06 1.5 - j49.5 - j102.5 1.6 - j4.33 - j5.52 1.7 - j0.78 - jl.90 1.8 + j1.74 + j2.6l 1.9 - j2.41 - j5.07 390 u use m.H I mw.o xocwsvopm mo cowuocsm m mm oucmuqum< DDQCH Hmowumpomne m.m manms moocchmou G F .Auwtrmo 4 Maw IILWVCRPLFILVLGFIIJIEW 1Fn! “I 230 COmeustmzou mam moan now m.m manme mam H. e .8 9%" 3 m~.m mMDUHm OH 0N soqm1111m ‘g ‘aoueqdeosns undur Input Admittance as a Function of Frequency 0.6 - 2.2 Ghz (a = 16.51 Cm, d1 = 0.29L) h——L——'| 231 TABLE 5.10 a d1 = 2b = 16.51 cm, L = 60.96 cm = 0.29L, h = 6.25 cm current distributions in Figure 5.12 Input Admittance, millimhos frequency, Ghz Theory Experiment 0.60 + j4.72 - 0.70 + j6.70 + j18.0 0.80 + j26.8 - 0.90 - j2.14 - 0.95 -+ j2.95 - 1.00 - j2.90 - j3.05 1.05 + j3.24 - 1.10 + j5.98 + j12.9 1.15 + j57.0 - 1.16 - j7.22 - 1.17 + j1.7 - 1.20 + j9.7 + j20.4 1.25 + 3289 - 1.27 - j33.2 - 1.30 - j7.43 - j11.5 1.35 + j6.84 - 1.37 + j53 - (continued) 1 h—L——* 232 TABLE 5.10 (concluded) = 2b = 16.51 cm , L = 60.96 cm a d1 = 0.29L , h = 6.25 cm Input Admittance, millimhos frequency, Ghz Theory Experiment 1.4 - j9.42 - j16.9 1.5 + j0.78 + j3.14 1.52 + j2.75 - 1.6 + j35.5 + j41.4 1.65 - j8.35 - 1.7 - j0.92 - j1.05 1.75 + j7.28 - 1.8 - j16 - j69.0 1.9 - jl.8 — j5.05 1.93 - j1.l6 - 2.0 + jo.34 - 2.05 - j1.59 — 2.2 - j9l.0 233 20 0. Wm s AQmN.o u H c .80 Hm.oH u mv Maw N.N I 0.0 mocmsvam mo cowuocsm m mm oosmuuw8w< unacH Hmowuwuooza wN.m MMDUHm mam mumv How _ 0N1 _ _ _ ¢.m oHan N N “ moosmGOmmn \6 ~ _ N ~AUH>8 1 OH... . A a . a . s 1 . . \ ... f. .1 .. .0 e no «mm . L‘Wb W‘I p Tire . a F — a O o.H m. .. w .u .u o--- \ § \ s Q N Q 0\\ J . . a \x . .9 ... \\ I OH ~ \ s _ . 1 . a _ . — . _ H 4 ON “ . _ u _ - _ m . " cowumuswwmcoo O 1 o~.m manmfi mam soqm;111m ‘g ‘aoueqdaosns undui 234 Input Admittance with Antenna Location as the Parameter As the location of the antenna in the cavity is varied, the degree of coupling to cavity resonances differs. This (as shown in Section 5.2.2) affects the general nature of the current distribution. The input admittance also varies and the coupling effect is more obvious. For the same cases discussed in relation to the current dis- tribution the theoretical and experimental input admittance values are compared in Tables 5.11 through 5.14. As in other cases there is a sizeable difference in magnitude but both theory and experiment dis- play the same behavior as the location is varied. The theoretical values are plotted for each case considered in Figures 5.29 through 5.32. The lines drawn in the figures are only approximate to indicate the general behavior of the admittance. The indication that the sus- ceptance passes through infinity in some cases (e.g. Figure 5.29) is based upon the continuity and smoothness expected of the admittance or impedance. The shape of the admittance as a function of location in the cavity can be related easily to the distribution of fields in the cavity. In Figure 5.29, for example, the operating frequency of 1.2 Ghz is very near the ideal cavity resonance f1,0,3 = 1.171 Ghz. This resonance exhibits zeros in the y-component of electric field at locations z/L = 0.333 and z/L = 0.667. These locations also correSpond to points of maximum magnetic field. The input admittance shown in Figure 5.29 readily demonstrates the change from electric (capacitive) to magnetic (inductive) coupling as the location is varied. The magnitude of the admittance is related to the degree of coupling; the lower'values of admittance correSpond to the locations of maximum coupling. 235 The cavity fields (steady state) at frequencies off resonance are ideally zero for the case of the perfectly conducting boundaries considered here, but the dominant evanescent fringing fields are apparently related to the nearby resonances. The change from capacitive to inductive is not demonstrated in all cases considered, since the value of admittance depends on the electrical prOperties of the antenna as well as the cavity. However, in all the figures there is a direct correlation to the distribution of fields for the closest resonance. One observation that can be made from these results is that the sensitivity to variations in location is related to the nearness of the operating frequency to the resonant frequencies of the cavity- antenna system. Near a resonance (e.g., Figure 5.31) the admittance is relatively insensitive to location over portions of the cavity, whereas farther from a resonance (e.g., Figure 5.32) the value of in- put admittance varies widely with antenna location. These differences in sensitivity can be exploited to advantage in designing an antenna- cavity system as a tool. These relationships are also important for consideration of coupled antenna systems. 236 TABLE 5.11 Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.2 Ghz) -— d %h b a = 2b = 16°51 Cm , I; = 60°96 cm 1 d1 = parameter , h = 6.25 cm L I f = 1.2 Ghz current distributions are in Figure 5.13 Input Admittance, millimhos d1 Theory Experiment 0.133L +-j4.82 + j9.9 0.1854, + j4.24 + 17.94 0.2381, + j5.38 + 38.62 0.29OL + j9.7 + j20.4 0.343L - j96.1 - j125 0.396L + j10.75 + j20.2 0.448L + j5.15 + j9.1 0.500L + j4.46 + j7.3 input susceptance, B, millimhos 237 .— '1 : : E = I 20 >— : '| : | —J 1 ' ' | I | I ' I 1 l ‘ ' ' .1 * . ‘1 : ‘. I, “ ,' \ I ‘ o \ 10 r- )5 9‘ ’I q I' - \ I \\ I’ \ I \\ D’l \ ’I \h d, t ‘a-c’ \c>“‘=o-v”o “0’ d O : . c t { f I f —D .1 .2 .3 .4 .5 .6 .7 .8 .9 z/L r. 1 -100 — | ' ‘ ' a = 2b = 16.51 cm, L = 60.96 cm .' ----- . b d1 = parameter, h = 6.25 cm N) data in Table 5.11 FIGURE 5.29 Theoretical Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.2 Ghz) 238 TABLE 5.12 Input Admittance as a Function of Antenna Location (a = 22.6 cm , 1.2 Ghz) :22. = . = . “d1‘flh b a 6cm,b 8255cm,L 6096cm d = parameter, h = 6.25 cm 1 f = 1.2 Ghz Input Admittance, millimhos (11 Theory Experiment 0.133L + j3.65 + j7.25 0.185; + j4.45 + j9.8 0.238L + j27.7 + j74.0 0.29OL + j8.25 + jl6.5 0.343L + j3.92 + j8.34 0.396L + j3.73 + j7.75 0.448L + j5.48 + jll.5 0.500; + j427 - j357 input susceptance, B, millimhos 239 430 " T ‘3‘. b :' . I I I I l I 420 - I I — JV . ‘1/ I I ' I l I ' .. | I | - I ' ' I I ‘ ' 1 : 4 20 '_ : I I l I I l I II I l .’ ' i I \ 10 " ’ I 1 d I \ I ' \ I, O\ 1/ \\ ’0 ‘1 I \ I x I 0’ \b /O b\\ ’I \O ‘ ~o/ ~—o’ 0-’° “0’ 2 .4 .5 .6 .7 .8 .9 z/L L .1 l- d1 -I a = 22.6 cm, b = 8.255 cm, ---- --- b p L = 60.96 cm, (11 = parameter "2 h = 6.25 cm, f = 1.2 Ghz data in Table 5.12 Figure 5.30 Theoretical Input Admittance as a Function of Antenna Location (a = 22.6 cm , 1.2 Ghz) 240 TABLE 5.13 Input Admittance as a Function of Antenna Location (a = 16.51 cm , 1.53 Ghz) ‘_ d1]?! b a = 2b = 16.51 cm ,L = 60.96 cm d1 = parameter , h = 4.0 cm I*-——- L ——*l f = 1.53 Ghz current distributions in Figure 5.15 Input Admittance, millimhos d1 Theory Experiment 0.133L + j3.64 + j4.83 0.185L + j4.93 + j12.4 0.238L + j3.80 + j6.85 0.290L + j3.61 + j5.1 0.343;, + j3.66 + 36.4 0.396L + j10.58 + j27.8 0.448L + j3.71 + j6.07 0.500L + j3.61 + j5.13 241 12 r- - 8 * I E10 + II I? ~ ...-I L H I. H II I < ”a II I“ . 8 - ". . . ~ In : I I ‘ 4 a b ' “ " ‘I 8 6 " I I ‘ I “ ‘ '4 I: I I ' cu I I ' \ \ 48‘ I- o I \ I q 8 4 I, ‘s I, ‘\ I, II I \ " ‘3 ' a °‘-o—O’ °~-o——d °--o«—d ‘0' U] I- a U A a 2 - I: "4 .. 0 a - 4 4 4. i I I I —- .. .1 .2 .3 .4 .5 .6 .7 .8 .9 ‘ z/L lo- d1 -1 H a = 2b = 16.51 cm, L = 60.96 cm b (11 = parameter , h = 4.0 cm .__ I, __.|——z f=1.53chz II? data in Table 5.13 FIGURE 5.31 Theoretical Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.53 Ghz) 242 TABLE 5.14 Input Admittance as a Function of Antenna Location (a a 16.51 cm , 1.4 Ghz) 60.96 cm __ d1:[h a = 2b = 16.51 cm , 4, d h 6.25 cm = parameter , f = 1.4 Ghz current distributions in Figure 5.1% Input Admittance, millimhos d1 Theory Experiment 0.133L + j9.24 + j18 0.185L + j10.5 + j19.8 0.238L - j17.3 - j27.4 - j17.6 * 0.290L - j9.42 - j16.9 0.343L + j13.5 + j25.8 0.396; + j8.62 + j16.5 + j12.6 * 0.448L + j44.9 + j171 - j20.4 * 0.500; - j7.16 - j12.2 - j9.41 * * Coaxial-Gap theory input susceptance, B, millimhos 243 50 ' ' I- o o J | I I I I 40 l I ' _ I I ' I I 'I I/ ' ‘ ‘ ‘4? .1;' : ‘. _ I I - 20 I, ‘ I I I I I I I I I I ‘ ' I ' I . I O l I 0 I \ 5 ‘I ' I l, I I I I O I- I | I -I 1° 9 . 55 22 x I O r, . , } ¢4 4— I I -—-. .1 .2 .3 .4 .5 .6 .7 .8 .9 2H. Ex I .. '10 I— e?‘ I I P l I : I I I _ I I I I ' I ‘ I I 9 I -20 __ I I d |._d1_4 b a = 2b = 16.51 cm, L = 60.96 cm ?—-- ----. A d1 = parameter, h = 6.25 cm -—o-Z r~————- I, _____.4 f = 1.4 Ghz I data in Table 5.14 FIGURE 5.32 Theoretical Input Admittance as a Function of Antenna Location (a = 16.51 cm, 1.4 Ghz) 244 Input Admittance with Antenna Height as a Parameter The comparison between theoretical and experimental input admittance as a function of antenna height is given in Tables 5.15 through 5.20. (Because of the small amount of data in each case, the theoretical results have not been plotted.) As usual there is a lack of agreement in absolute magnitude of admittance but good agreement in behavior as the height is varied. Some attempts at correcting the magnitude discrepancies are given in Section 5.1. In Table 5.15 both the delta-gap and coaxial gap theories are included for comparison. The coaxial gap theory obviously re- presents the relatively constant nature of the experimental admittance better than the delta gap theory. The variation in the delta-gap theory results from the change in size of the sampling interval in the numerical solution. That is, as the height is increased, the sampling interval gets longer (Ay = h/K, K = constant). Attempts were made at maintaining a more constant sampling interval by changing the number of sections as the height changed, but this introduced variations in convergence properties which were less readily predicted and recognized. Nonetheless, the similarity in value of the coaxial gap and delta-gap theories is, of course, satisfactory. In general the greatest insight into the effect of varying height is gained by comparing the results for admittance with the current distribution. The occurrence of resonances and anti- resonances is then more obvious. It may be noted that there are conditions when the admittance is either insenSitive to changes in height or strongly a function of height. As with the sensitivity to location discussed previously this change of sensitivity can be used to advantage depending on the properties desired. 245 It is also interesting to note that in the case when the current distribution is independent of height (at 1.53 Ghz) the admittance increases with height (compare Table 5.20 and Figure 5.21), whereas when the input admittance is independent of height (at 1.05 Ghz) the current distribution changes to maintain a con- stant physical location for the zero crossing (see Table 5.15 and Figure 5.16). 246 TABLE 5.15 Input Admittance as a Function of Antenna Height (d = 0.238L , 1.05 Ghz) l u = 2 = .5 = . _._ d1 h b a b 16 1 cm , L 60 96 cm d1 = 0.238L , h = parameter I‘— L ——'l f = 1.05 Ghz current distributions in Figure 5.16 Input Admittance, millbmhos h, cm Delta-Gap Theory Coaxial-Gap Theory EXperiment 4 +-j3.56 + j3.63 +-j5.73 5 + j3.56 + j3.68 +-j5.73 6 + j3.33 + j3.03 + j5.73 7 + j3.18 + j3.68 +—j5.73 247 TABLE 5.16 Input Admittance as a Function of Antenna Height (d l = 0.133L, 1.1 Ghz, a = 16.51 cm) “cl—4L1 l k—L—ol a = 2b = 16.51 cm , L = 60.96 cm b d1 = 0.133L , h = parameter f = 1.1 Ghz current distributions in Figure 5.17 Input Admittance, millimhos h, cm Theory Experiment 4 + j8.61 + j21 S + j22.2 - j116 6 — jl7.6 - 119 6.25 - le - j15 7 - j3.95 - j5.4 248 TABLE 5.17 Input Admittance as a Function of Antenna Height (d1 = 0.133L, 1.1 Ghz, a = 22.6 cm) ‘] J a = 22.6 cm, b = 8.255 cm, L = 60.96 cm _ d1 = 0.133L , h = parameter I L I f = 1.1 Ghz current distributions in Figure 5.18 Input Admittance, millhmhos h, cm Theory Experiment 4 + j7.4l + j15.5 5 + jll.7 + j29 6 + j53.7 + j740 6.25 -j3660 - 7 - j14.3 - j24.9 8 - j4.l3 - j7.38 249 TABLE 5.18 Input Admittance as a Function of Antenna Height ((11 8 d1 = 0.133L , 1.26112) = 2b = 16.51 cm, L = 60.96 cm = 0.133; , h = parameter f = 1.2 Ghz current distributions in Figure 5.19 Input Admittance, millimhos h, cm Theory Experiment 4 + jS.ll + j9.35 S + j4.93 + j9.80 6 + j4.84 + j9.90 6.25 + j4.82 +-j9.90 7 + j4.77 + j9.75 250 TABLE 5.19 Input Admittance as a Function of Antenna Height (d1 = 0.133; , 1.3 Ghz) a = 2b = 16.51 cm , L = 60.96 cm b_-d1 lib b d1 = 0.133; , h = parameter [<— L ———~| £=1.3 Ghz current distributions in Figure 5.20 Input Admittance, millimhos h , cm Theory Experiment 4 + jl72 - j152 5 - j5.50 - j7.88 6 - jl.30 - jl.22 7 + j0.20 + jl.87 8 + jl.l8 + j3.36 251 TABLE 5.20 Input Admittance as a Function of Antenna Height (d1 ._ q}. I-_——I.——.I 1 = 0.500L , 1.53 Ghz) a 2b = 16.51 cm , L = 60.96 cm d = 0.500L f = 1.53 Ghz , h = parameter current distributions in Figure 5.21 Input Admittance, millimhos h, cm Theory Experiment 3 + j2.90 + j4.69 4 + j3.02 + j5.14 6 +-j3.48 +-j6.8 7 + j3.83 - 252 5.2.5 Cavity Filled with Dissipative Media; Current Distribution To study the effect of dissipative media on the theory as well as the experimental system, three cases are considered. First, the theory is tested against some special cases of measured current dis- tributions available in the literature. Second, the theory is compared with experimental measurements taken on distilled water. The third case involves experimental measurements in sodium chloride solution. For comparing the results of the theory with experimental data published in the literature, it is necessary to use results for highly dissipative media. If the losses are high enough, the cavity or waveguide walls are essentially shielded from the antenna and measurements taken in semi-infinite media will be comparable to the theoretical results for the waveguide. In the report published by Scott [35] such highly dissipative data are available. By proper choice of Operating frequency, conductivity and permittivity the attenuation characteristics of the media studied by Scott can be duplicated in the theory presented here. The comparison of the measurements by Scott to the theory is shown in Figure 5.33 for three values of dissipation and is seen to be excellent. The theoretical curves were all obtained at 1.2 Ghz with appropriate choices of con— ductivity and relative permittivity. The comparison of the theoretical and experimental results for distilled water are shown in Figures 5.34 through 5.36. The value used in the theory for relative permittivity of 83.5 is that determined near 1.0 Ghz by the cavity perturbation method described in Section 4.1. The value of conductivity is approximate because 253 measurement of conductivity is only done at low frequency. Since the conductivity of distilled water is a strong function of fre— quency, the low frequency data (not presented in this report) are not useable. The value of 0.2 mhos per meter was therefore taken as approximate from the literature. (See, for example, Lepley [23] for typical values of water conductivity as a function of frequency.) The theoretical curves shown in Figures 5.34 through 5.36 are there- fore only for reference and should not be taken as absolute measures of the experimental results. In Figure 5.34 the current distribution is shown for three frequencies. A constant value of conductivity was used for reference in the theoretical results at all frequencies even though the exper- imental data clearly indicates an increasing conductivity. Both the delta-gap and coaxial-gap theories are presented in this figure to show the effect of conductivity on the different antenna excitation models. Both theories indicate approximately the same effective wave— length and attenuation of the current along the antenna (as expected) but a phase displacement is evident in the coaxial model. This dif— ference is only evident at 1.2 Ghz in the phase data, however. The phase of the current distribution (i.e., maxima and minima) of the experimental data appears to roughly fall between the delta—gap and coaxial-gap theories. The effect of changing the antenna location in the cavity (Figure 5.35) is seen to be nearly negligible. This indicates that the losses for these frequencies are sufficiently high to mask the presence of some of the conducting boundaries, if not all. 254 The study of experimental results in a sodium chloride solution is shown in Figures 5.37 and 5.38. Again the theory is only used at a constant conductivity of 0.2 mhos/meter for reference. The effect of lower frequency is obvious at 350 MHz in Figure 5.37 where the distilled water gave much less attenuation than the salt water. This change in conductivity changes the effective dielectric con— stant and hence the effective wavelength, thus accounting for the phase shift between distilled water and salt water experimental data. The effect of changing location is much more pronounced at this low frequency for the distilled water case indicating the effect of the cavity boundaries. At the higher frequency (1.0 Ghz) in Figure 5.38 the con- ductivity of distilled water and the salt solution are much more comparable. This is the well-known dipolar relaxation effect [23] of the water which-is sufficiently pronounced at 1.0 GHz to over- shadow the ionic conductivity of the salt solution. The overall results for dissipative media indicate that meaningful results can be obtained both experimentally and theoretically. It is evident from the excellent agreement with Scott's results that a sufficiently accurate measure of the effective conductivity at the desired Operating frequency would yield theoretical results which would compare extremely well with the experiment. Because of the high dissipation exhibited at the L— hand frequencies, it is seen that no use can be made of the cavity resonance coupling effects seen in the previous sections for the empty cavity. It appears that these effects would become useful 255 at frequencies where the lower order water filled cavity resonances occur (i.e. scaled by the square root of the effective dielectric constant) and where the dissipation is reduced. 256 1.0 V— A experiment (Scott) q .. H theory ~ (3 -40 -80 ~120 1 L 1 L 1 i l J J 0 .5 y/h 1.0 (a) 01/6 = 0.3 ab = 3.427 FIGURE 5.33 Current Distribution in Highly Dissipative Media Comparison of Theory with Experimental Literature 257 1.0 ‘ A experiment (Scott) P -o- theory ‘ '1' P e '1 . - a (b) _ k: __ 00 a/B = 0.592 . S b _ eh = 1.324 (A ‘ b ‘ '1 '20 . A #- ¢ u ‘ . . . -40 A o A A A A l A A A A . 5 y/h l. 0 FIGURE 5.33 Cont'd A experiment (Scott) ’ —+—- theory ‘ (C) I a/B = 0.592 I I ah = 4.487 -100 -200 1.0 ~. P4 —- 1.0 III 258 '» A expe riment (S cott) -+- theory A (d) . 0/6 = 0.97 3h = 2.564 5 -100 « -140 -l8O 1.0 FIGURE 5 .33 (C oncluded) A experiment (Scot t) -o- theory ’/ 'I .5 y/h a/B = 0.97 ah = 4.986 - -160 ~ -240 Ax -320 259 -200 -400 0 experiment, distilled water ._._. delta-gap o = 0.2 mho/m theory, -+- coax gap er = 83.5 cavity configuration: = 2b = 16.51 cm, L = 60.96 cm a d1 = 0.5;, h = 6 cm, f = parameter FIGURE 5.34 Current Distribution in Distilled Water at Different Frequencies 260 FIGURE 5.34 (Concluded) 0 . I“ ...... ..n G m ./ «J c v. 1 6 = = f h 5 \A\ \ [Gill z .nnu m 8 C 1 6 = = .t .n \ \Q 261 1.0 o_._d1 = 0.1334. P A--A--d1 = 0.2904, 1 1 ‘ L 0 O “‘0 a = 2b = 16.51 cm _ L = 60.96 cm (11 = parameter '1 I'200 h = 6 Cm f = 1.0 Ghz < -400 FIGURE 5.35 1.0 Current Distribution in Distilled water at Different Loca- tions 1.0 ‘ O-°--d1 = 0.448L _ z 4 A+ d1 = 0.5L F '1 ¢ Theory used: .0 a = 0.2 mho/m 0 ._ er - 83.5 -200 -400 262 b P- d14flh 4 L . 0 ‘ a = 2b = 16.51 cm _b00 L = 60.96 Cm d1 = 0.5L 5 h = parameter f = 1.0 Ghz {- ‘9‘0 4 '200 «z. p ngw o «4 \K. o . \ o . 4.00 ‘~‘Jlm~‘_ o A A A A l A A A A FIGURE 5.36 .5 y/h 1.0 Current Distribution in Distilled Water for Different Heights 0 experiment, distilled water '00 4» theory, a = 0.2 mho/m cr = 83.5 -200 -400 263 FIGURE 5.36 (Concluded) 05 ' o -l— 0 . -200 b \ o \w‘ " '400 Q _ K O \ q \ o \ A J A l J A A‘L’ 264 -lOO 0 experiment, 1 gm/liter NaCL solution 0 experiment, distilled water -o— theory, a = 0.2 mho/m, er = 83.5 cavity configuration: a = 2b = 16.51 cm, L = 60.96 cm (11 = 0.54,, h = 6 cm, f = 350 MHz FIGURE 5.37 Current Distribution for Distilled and Salt Water at 350 MHz 265 FIGURE 5.37 (Continued) 266 T I I 1 I I t T I 1.0L . ~ .. o - I \I\ F' _. in. --0- 00 .5_. _ (b . 1 -2000 . - 4.000 0 1.0 0 experiment, 1 gm/liter salt solution (3 experiment, distilled water -o-— theory, a = 0.2 mho/m, er = 83.5 cavity configuration: a = 2b = 16.51 cm, L = 60.96 cm (11 = 0.51,, h = 6 cm, f = 1.0 GHz FIGURE 5.38 Current Distribution for Distilled and Salt Water at 1.0 GHz 267 5.2.6 Cavity Filled with Dissipative Media; Input Admittance The input admittance for each case considered in the pre- vious section is given in Tables 5.21 through 5.26. As in other comparisons of experimental and theoretical admittance the lack of agreement is basically accounted for by the inadequacy of the excitation models. It should also be remembered that the theoretical results are determined only for a fixed approximate conductivity and thus should not be compared exactly to the experimental data. The comparison of the theory with Scott's results [35] in Table 5.21 shows the same degree of difference that the results for the air filled cavity do. This tends to support the theoretical results for dissipative media even though no attempt is made at determining a terminal zone correction network. The comparison between theory and experiment in Tables 5.22 through 5.26 is poor in all cases. However, the differences between coaxial-gap and delta-gap theories also is significant and indicates the sensitivity of the models to the properties of the medium in the dissipative case. The determination of proper terminal zone correction networks for the case of abrupt transitions from air filled coaxial line to dissipative media is not yet well understood and thus the seriousness of the differences seen in these results cannot be effectively evaluated. For the high frequency data in Tables 5.22, 5.23 and 5.24 the sign on the input susceptance is seen to change sign between the coaxial-gap and delta-gap theories. This seems tolqualitatively indicate that a correction network would need to compensate to a large degree in the susceptive component and 268 results comparable to the experimental data are not unreasonable. The low frequency data in Table 5.25 again shows results which differ on the order of the air cavity results and thus are felt to be acceptable. The differences between experimental data for the sodium chloride solution and distilled water indicates the sensitivity of the input admittance to changes in the conductivity of the medium. At 1.0 GHz (Table 5.26), on the other hand, where the effective conductivities of both media are comparable, the e experimental results agree well. The difference can easily be re- lated to the probable higher conductivity of the ionic NaCL solu- tion. 269 TABLE 5.21 Input Admittance in Highly Dissipative Media Comparison of Theory with Experimental Literature (see Figure 5.33 for current) input admittance, millimhos Theory Scott, exp. a/B = 0.3 21.55 + j3.25 27 + j2.48 ah = 3.427 a/B = 0.592 13.0 - j2.46 22.1 - j5.88 5h = 1.324 a/B = 0.592 16.4 -j4.6 17.6-j6.0 ah = 4.487 a/B = 0.97 18.7 - j11.6 35.8 - j21.9 ah = 2.564 ale = 0.97 19.0 - j11.5 35.2 - j22.l ah = 4.986 The ratio h/ra was not equal in this case. 270 TABLE 5.22 Input Admittance in Distilled Water at Different Frequencies theory based on o = 0.2 mho/m, er = 83.5 current shown in Figure 5.34 configuration: a = 2b = 16.51 cm, L = 60.96 cm (11 == 0.5L, h = 6 cm, f = parameter input admittance, millimhos frequency theory experiment 1.0 Ghz 93.8 + j52.6 (delta) 23.8 - jl.9 85.5 - j29.8 (coax) 1.2 Ghz 103 +-j85 (delta) 162 + j7.8 118 - j35 (coax) 1.8 Ghz 131 + j87 (delta) 183 + j16 82 - j74 (coax) 271 TABLE 5.23 Input Admittance in Distilled Water at Different Locations current shown in Figure 5.35 theory based on a = 0.2 mho/m, er = 83.5 2b = 16.51 cm, L = 60.96 cm configuration: a = d1 = parameter, h = 6 cm, f = 1.0 Ghz input admittance, millimhos d1 theory experiment 0.133L 96.1 + j58.l (delta) 23.2 - j4.1 90.7 - j26.2 (coax) 0.290; 92 +-j53.5 (delta) 25.6 - j6.3 84.4 - j28.2 (coax) 0.448L 93.6 + j53.0 (delta) 24.3 - j3.3 85.3 - j29.7 (coax) 0.500L 93 + j51.3 (delta) 23.8 - j1.9 85.5 - j29.8 (coax) 272 TABLE 5.24 Input Admittance in Distilled water for Different Heights current shown in Figure 5.36 theory based on o = 0.2 mho/m, er = 83.5 configuration: a = 2b = 16.51 cm, L = 60.96 cm (11 = 0.51,, h = parameter, f = 1.0 Ghz input admittance, millimhos h, cm theory experiment 5.0 77.8 + j90.5 (delta) 21.9 - j0.3 81.9 + jll.8 (coax) 5.5 96.4 + j87.8 (delta) 21.7 - j0.7 123.6 - j7.02 (coax) 6.0 93.0 + j51.3 (delta) 23.8 - jl.9 85.5 - j29.8 (coax) 6.25 81.0 + j48.6 (delta) 24.4 - j0.8 75.5 - j17.9 (coax) 273 TABLE 5.25 Input Admittance for Distilled and Salt Water at 350 MHz current shown in Figure 5.37 theory based on o = 0.2 mho/m, er = 83.5 configuration: a = 2b = 16.51 cm, L = 60.96 cm (11 = parameter, h = 6 cm, f = 350 MHz input admittance, millimhos experiment (11 theory NaCL Distilled (delta) 1 gm/L 0.133L 49.5 + j49.3 108 +-j75.7 118 +-j76.5 0.500; 50 + 356.5 93.9 + j85.7 55.7 +-189.2 274 TABLE 5.26 Input Admittance for Distilled and Salt Water at 1.0 GHz current shown in Figure 5.38 theory based on o = 0.2 mho/m, er = 83.5 configuration: a = 2b = 16.51 cm, L = 60.96 cm (11 = 0.51,, h = 6 cm, f = 1.0 Ghz input admittance, millimhos experiment theory NaCL Distilled 1 gm/L 93.6 + j52.7 (delta) 21.7 - j0.8 23.6 - j3.2 85.5 - j29.8 (coax) 275 5.3 Coupled Antennas In order to take full advantage of the antenna-waveguide- media system for potential application, an understanding must be had of the functioning of two coupled antennas in the waveguide or cavity. This section presents these results. Because the results of the single antenna presented in Section 5.2 show the coupling that exists between the antenna and the cavity, only a few representative results of varying pertinent parameters will be presented for the two~antenna case. Each physical and electrical configuration con- sidered in this section has its parallel in the single antenna case of the previous section and it is instructive to make the comparison to qualitatively evaluate the effect of the second parasitic antenna. In all the cases presented in this section, only the results of the delta gap theory are included, since the coaxial gap theory, as developed to date, is capable of only a single antenna solution. The current distributions shown are only those on the driven antenna (always designated #1) with the parasitic antenna (#2) loaded with a standard calibrated 50 ohm termination. As well as calculating the current (and admittance) for the loaded case, Y the theory also 50’ determines the admittance parameters, [Y], for this two port system. The theoretical admittance matrix is given in the tables of input admittance . The experimental measurements of admittance are three fold, xvith all measurements taken on the driven antenna. First, the input infinittance is measured with the 50 ohm load on the parasitic antenna (Y Second, the admittance is measured with the parasitic antenna 50)' electrically short-circuited (YSC). The admittance is finally 276 measured for an Open-circuited parasite (Yoc). These three measure- ments are than sufficient to determine the admittance parameters for the system. Although, as noted in King [17], these measurements are not the most desirable (since the calculation of the admittance para- meters involves differences of large nearly equal numbers), they are the simplest tO make with the eXperimental system at hand. The short- circuit and open-circuit conditions for the parasitic antenna were obtained with a sliding short attached to the second antenna's General Radio connector. The existence of the short or Open condi- tion was first determined from auxiliary measurements taken on a slotted line driving the connector-sliding short combination from the antenna terminal plane. Because of the problems already noted in the measurement of admittance (Section 5.1), the experimental data has not been converted to admittance parameters to avoid compounding errors. For comparison purposes, it is seen that Y =‘Y , 11 SC 211 = Z0C and, of course, the 50 Ohm loaded cases are equivalent. For calculation purposes it should be noted that, for the cavity case, the admittance parameters are all purely imaginary. Thus,measurement of YSC (Y11) and Y50 (G50 + jBSO) is sufficient to determine the three unknown admittance parameters: Y11 = jBll’ Y12 = Y21 = .1312, and Y22 = jBZZ. The necessary relationships eare readily determined from the admittance parameter two port network equations. Since only purely imaginary results are desired, the snalution of the resulting cubic equation for 812 or B22 in terms <3f GSO’ B50, and BSC is readily solved by algebraic methods. In general, the purpose of the coupled antenna meaSurements iJI‘this report is to test the theory and demonstrate the Operation of 277 the experimental system for potential application. The results for two antennas in a semi-infinite waveguide are presented first. In Sections 5.3.2 and 5.3.3 the results for an air filled cavity are shown for independent variations in antenna separation and the two antenna heights. For the case Of dissipative media, the theory is compared to measurements in distilled water and in sodium chloride solution with the results presented in Section 5.3.4. 278 5.3.1 waveguide with Matched Terminations For the case Of the semi-infinite waveguide studied and dis- cussed for the single antenna in Section 5.2.1, two representative configurations were tested with coupled antennas. The experimental system again used microwave absorber in the Open end of the short- circuited section of waveguide. The theory assumes a perfect short at one arm and a perfect match at the other arm. The driven antenna is located (as in the single antenna case) at approximately one- quarter of a guided wavelength from the short circuit. The second parasitic antenna is then placed approximately one-half of a guided wavelength from antenna number 1. The theoretical and experimental current distributions on the driven antenna at 1.05 and 1.3 Ghz are shown in Figure 5.39 (a) and (b). These results can be compared to the same single antenna results in Figures 5.5(a) and 5.6(a), respectively. In the coupled case the current is more nearly triangular and the phase of the current is more nearly quadrature than in the single case. This indicates that the parasitic antenna is acting to increase the stored energy in the vicinity Of the antennas while decreasing the energy radiated out Of the Open end of the waveguide. (This would also account for the better agreement between theory and eXperiment, since the crude experimental matched termination is not as important as in the single antenna case.) The input admittance for these two cases are given in Table 5.27. Comparing to Tables 5.1 and 5.2 shows a decrease in input conductance and hence a decrease in radiated power for the case of the 50 Ohm loaded parasite. The agreement between theory and 279 experiment (YSC to Yll’ Y50 to YSO) again is roughly acceptable within the usual factor of two for this system. The mutual re~ sistance is seen to be negative which is expected for antennas at one-half wavelength Spacing in free-Space. These few results for the semi-infinite waveguide serve as a first-order indication of the Operation of the experimental apparatus and theoretical model for the coupled antenna configuration. Since this type system was not envisioned as part of the applications a3pect, no further data is presented on the semi-infinite waveguide at this time. 280 1-0 ~—& (3) f = 1.05 Ghz 1 2 P x = 57.23 cm 7 ‘ 0 (1g " O 253 b d1 d3’- 11’! b 1 0 Kg «1 0 d = 0.503 7. L 2 8 ‘ a = 2b = 16.51 cm 0 - = L ‘ hl - 5 cm .175 x0 .5 _ o d h2 = 5.08 cm = .178 x0 . Ant. #1: driven ’ ‘ Ant. #2: 50 ohm load 70 r ‘ ' . 4 60 b ‘ ‘ - . d 50 - . A A A A l A A A L O .5 y/h 1.0 FIGURE 5.39 1.0 ' ’ Driven Antenna Current _ for Coupled Antennas in . , (b) f - 1'3 Ghz J Semi-Infinite Waveguide xg = 32.26 cm (f = 1.05; 1.3 Ghz) l 1 b ‘ 61 = 0-25 xg ‘ I I = . _ 0 d2 0 496 xg T O 05 h . d b o d 70 * °o ‘ 60 f ; ; _ . O 50 I- . .0 «1 L 1 AL A L l J A A 281 TABLE 5.27 Admittance for Coupled Antennas in Semi-Infinite Waveguide For configuration of Figure 5.39(a): f = 1.05 Ghz Theory: 0.84 + j4.59 -0.84 + j2.35 [Y] = millimhos -0.84 + j2.35 0.84 + j4.53 YSO = 1.35 + j4.75 m-mhos Experiment: = . + ' . 5 - Y50 2 84 310 2 m mhos YSC = 0.944 + j10.37 m-mhos Y = 3.4 + j6.12 m-mhos oc For configuration of Figure 5.39(b): f = 1.3 Ghz Theory: 1.21 + j7.80 -l.32 + j5.22 [Y]: millimhos -1.32 + j5.22 1.55 + j7.75 Y50 = 3.31 + j7.64 m-mhos Experiment: YSO = 13.3 + j26.2 m-mhos Y = 2.43 + j18.7 m-mhos sc Y = 8.03 + j9.58 m-mhos 0C 282 5.3.2 Air Filled Cavity; Current Distribution Some representative results for coupled antennas in the air filled cavity are presented in this section and the next. The cavity, as always, is Specified by dimensions a, b, and L for width, height and length, reSpectively. The current distributions for the case Of the parasitic antenna loaded with 50 ohms are pre- sented in Figures 5.40 through 5.43. Figure 5.40 shows the excellent agreement between theory (solid line) and experiment (circles) for various separations (d2) of the driven antenna (#1) and the parasitic antenna (#2). Since the antennas are contained in a theoretically lossless cavity, the only mechanism for dissipation is the coupling of excitation energy into the 50 Ohm load on the parasitic antenna. Therefore, a qualitative measure of the coupling between the antennas is the amount of phase variation of the current compared to the pure quadrature current that would exist in the unloaded case. (For comparison, the single active antenna in the same configuration is given in Figure 5.20.) The change in shape shown for the current in Figure 5.40 as the separation is varied is primarily due to the perturbation of the cavity resonance caused by the location of the parasitic antenna. As the resonant frequency is shifted relative to the Operating fre- quency (and effective Q lowered) the current distribution is eXpected to change characteristics in a manner similar to that dis- cussed for the single antenna (Section 5.2.2). Thus the tendency toward a triangular distribution for the larger separations results from the system resonant frequency being moved from above to below the Operating frequency (see, for example, Figure 5.23). 283 The variation in current distribution on the active antenna for changing active antenna heights is shown in Figures 5.41 and 5.42. In each case the antennas are located symmetrically in the cavity at locations as close as experimentally possible to the shorting end plates. The comparable single antenna results for the Operating frequency of 1.2 Ghz are given in Figures 5.19. Comparing Figure 5.19 to Figure 5.41 indicates a greater sensitivity Of the current to height changes for the coupled antenna case. This is again readily accounted for by the shift in cavity resonance such that small changes in antenna length result in larger variation in the resonant nature of the system. The shift of resonance is evident, since the quadrature component of current in the coupled case is inductive (-900) compared to the capacitive nature for the single antenna thus indicating a shift of the resonant frequency past the operating frequency. For an Operating frequency of 1.3 Ghz the results are given in Figure 5.42 (the comparable single case is in Figure 5.20). The differences between single and coupled antenna results is again directly relateable to the shift in resonant frequency. At either 1.2 Ghz or 1.3 Ghz the in-phase component of current is some measure of the coupling to the parasitic antenna and 50 ohm load as discussed in relation to Figure 5.40. For variations in the height of the parasitic antenna, the active antenna current is given in Figure 5.43u Comparing again to the single antenna results of Figure 5.20, it is seen that for the l" parasite the current is very similar to the single case and has only a small in-phase component (small coupling) as expected. As 284 the height of the parasite is increased the coupling becomes greater, the cavity resonance is increasingly shifted, and the characteristic shape and phase Of the current on the driven antenna changes. The overall agreement between theory and experiment for various changes in parameters of two coupled antennas in the cavity is excellent. Differences that are seen are primarily accountable by the approximation made in the theory for antenna shape which ~1' ultimately affects the effective shift in cavity resonances. As t.— has been shown, the coupling between antennas is most sensitive to variations in separation and heights in a manner directly relate- able to the coupling between each antenna and the cavity resonances. The variations in coupling will be more evident in the admittance results presented in the next section. 285 . -6O . -70 0 4 -80 0 A A A.— L A _L A 5 y/h l O l 2 a = 2b = 16.51 cm; L = 60.96 cm b d d l 2 h1 = 5 cm; h2 = 5.08 cm L A d1 = 8.08 cm = .133L; d2 = parameter f = 1.3 Ghz Ant. #1 driven Ant . #2 5012.10“ FIGURE 5 .40 Driven Antenna Current for Coupled Antennas as a Function Of Separation (f = 1.3 Ghz) 1.0 . l 1,! 1.0 l I 1.! 286 16 cm .262L 11. FIGURE 5.40 (continued) 287 1.0. ; , ‘ d2 = 32 cm . = .525L II.1 ' ‘ O .5 .. ' . O , ‘ 60 , O r ‘ . 40 . O O . O _ O .. . 20 .b _(_00 ¢ 0 A A 4 A l A A A A .5 y/h 1.0 FIGURE 5.40 1.0 ‘ . (continued) . d2 = 44.8 cm . = .735L 117 _ .5 >- ‘ . I- : Q II 60 ~ ' . ‘ 5 40 O . ° . 20 O 0O 9 IF droo A 1 l 1 l l J .l ’L 288 1.0 . 1 2 b.31fl*__' d2 ”fl IL! ' ” L a = 2b = 16.51 cm . L = 60.96 cm .5 .. d1 = 8.08 cm = .133L d2 = 44.8 cm = .734L . h1 = parameter h2 = 5.08 cm 80 . f = 1.2 Ghz 70 . Ant. #2: 50 ohm load 60 b FIGURE 5.41 Driven Antenna Current for Coupled Antennas as Function of h (1.2 Ghz) 1 289 FIGURE 5.41 (continued) II 1.0 y 60} 40. 20 - 290 CID 2b = 16.51 60.96 cm 8.08 cm 44.8 cm cm .133; .734L parameter 5.08 cm = 1.3 Ghz Ant. #2: 50 Ohm load 1.0 FIGURE 5.42 Driven Antenna Current for Coupled Antennas as a Function of h1 - (1.3 Ghz) .4 ‘ 60 . 40 o - 20 boo 291 1.0 ‘ FIGURE 5.42 (continued) 292 1.0 b 112 = 2.54 cm 1 2 = 1.0 " ‘ U 11.! ‘ L ' i a = 2b = 16.51 cm . ‘ L = 60.96 cm d1 = 8.08 cm = .133L .5 _ ‘ d2 = 44.8 cm = .734L = 5 _ q h1 cm h2 = parameter -70 r - f = 1.3 Ghz Ant. #2: 50 ohm load -80 ' . 0 ° 0 0 O0 -96- T. o A A A A l A A A .5 y/h 1.0 FIGURE 5.43 Driven Antenna Current for Coupled Antennas as Function of h2 (1 .3 Ghz) 293 1.0 - h = . , 2 6 35 cm = 2.5" 1 11,! .5 G 0 FIGURE 5.43 1.0 i 3. (continued) . h2 = 7.62 cm ‘ o = 3.0" W ‘ . d . 1 Q o 5 1" -l O := . O '80 b 2 . . - 7O - C . _ m . 3 460 - 4 A l l l l l J I ‘ 0 .5 y/h 1.0 294 5.3.3 Air Filled Cavity: Admittance The values of measured and calculated admittance for each case considered in Section 5.3.2 are presented in Tables 5.28 through 5.31 of this section. As stated earlier, the experimental data are given in unaltered form because of the uncertainty of a terminal zone network and the problems of calculating the admittance para- meters as differences Of large numbers. The raw data is given for ._ general comparison and future reference. Since the antenna-cavity-media system is reciprocal for all n- cases considered here (isotrOpic media) the theoretical admittance matrix shown in the tables is symmetric (le = Y21)° Although the two port admittance parameters are sufficient tO characterize the system, the driven antenna input admittance for the 50 Ohm load case is also included to provide a check with the experiment. It belabors the point to discuss again differences between the eXperi- mental and theoretical values shown in the tables, but it is mean- ingful to note that the overall tracking as a function of the various parameters is maintained. From the small sampling of cases given for the coupled antennas little can be concluded in a definitive manner. However, it is important to Observe that the mutual admittance is highly dependent upon both location and height of the antennas. (The relative insensitivity shown in Table 5.30 at 1.3,Ghz is readily related to the insensitivity of the cavity resonances to changes in height.) Both this and the variation shown in self admittances in- dicate at least qualitatively that the coupled antenna system could also be used to advantage in detecting changes in a medium contained in the cavity. The results for dissipative media follow in the next section. 295 TABLE 5 . 28 Input Admittance for Coupled Antennas as a Function Of Separation (f = 1.3 GHz) configuration of Figure 5.40 Theory: units: millimhos d2 Y11 Y12 = Y21 Y22 Y50 .105L = 6.4 cm -jl.82 +j4.15 +j7.52 1.10 - j2.65 .158L = 9.6 cm +j2.32 +j4.17 +j4.72 1.42 + j1.65 .262L = 16.0 cm +j7.52 +j7.12 +j6.98 3.90 + j4.79 .525L = 32.0 cm +j7.44 -j7.44 +j6.79 3.78 + j4.87 .735L = 44.8 cm '+j4.79 +j4.69 +j4.61 1.81 + j3.95 Experiment: units: millimhos (12 cm Ysc Y0C YSO 6.4 -j2.3 -- -- 9.6 +j5.5 -j3.5 2.3 + j2.4 16.0 +jl8.2 +321.7 8.8 +-j10.9 32.0 +j14.2 -j2.7 7.3 + j8.4 44.8 +j10.8 -j0.4 3.8 + j8.9 296 TABLE 5.29 Input Admittance for Coupled Antennas as a Function Of h1 (1.2 GHz) for configuration of Figure 5.41 Theory: millimhos h1 cm Y11 Y12 = Y21 Y22 Y50 4 + 39.97 - j6.25 + 39.97 1.96 + 38.01 5 + 340.9 - 33.73 1 + 340.8 7.89 + 38.72 6 - j11.6 + 315.7 - 312.8 9.34 + 30.3 7 - 33.58 + 37.78 - 35.14 4.78 - 31.12 Experiment: millimhos h1 cm Ysc Yoc Y50 4 - 390 + 312.9 22.1 + 310.3 5 + 324.4 + 311.3 6.1 + 316.8 6 - 310.8 + 314.9 12.5 - 32.7 7 - 33.3 + 315.4 6.2 - 31.2 297 TABLE 5.30 Input Admittance for Coupled Antennas as a Function of h (1.3 Ghz) 1 for configuration Of Figure 5.42 Theory: millimhos h1 C‘“ Y11 Y12 = Y21 Y22 Y50 4 + 34.10 + 34.25 + 31.28 1.77 + 33.87 5 + 34.63 + 34.67 + 34.45 1.82 + 33.82 6 + 33.88 + 34.64 + 35.58 1.64 + 32.97 7 + 33.78 + 34.45 + 36.69 1.36 + 32.87 Experiment: millimhos hl cm YSC Y0C Y50 4 + 38.0 - 38.8 2.7 + 37.2 5 + 38.5 -31.5 3.3 + 37.0 6 + 38.6 +-3l.l 3.2 + 36.5 7 + 38.6 + 32.5 2.5 + 36.6 298 TABLE 5.31 Input Admittance for Coupled Antennas as a Function of h 2 (1 .3 Ghz) for configuration Of Figure 5.43 The ory: mi llimhos h2 1“ Y11 Y12 = Y21 Y22 Y50 1 - 32.84 + 33.04 + 34.29 0.78 - 33.17 1.5 - 30.69 + 34.22 + 33.97 1.53 + 30.07 2 + 34.63 + 34.67 + 34.45 1.82 + 33.82 2.5 + 36.71 + 34.61 + 34.44 1.78 + 35.92 3 +-37.08 +—j4.l6 4-33.62 ln53 + 36.53 Experiment: millimhos h2 in YSC YoC Y50 1 ..33.0 3-37.0 1.1 ~33.4 1.5 -+33.1 -3446 2.6-F31.9 2 + 38.5 -jl.5 3.3 + 37.0 2.5 + 311.5 + 31.7 3.2 + 39.9 3 4—313.3 +-35.5 2.9-?312.0 299 5.3.4 CavityiFilled with Dissipative Media; Current and Admittance For coupled antennas in a dissipative medium, experimental measurements were taken on distilled water and sodium chloride solu- tion at two frequencies. The results in Figure 5.44 show the 350 MHz case with the driven antenna (#1) centered in the cavity and the parasite (#2) as near as experimentally possible (3.2 cm). The experimental results are compared in both curves to the theoretical current based on a dielectric constant Of 83.5 and a conductivity of 0.2 mhos/meter. These values were used even though the conductivity value does not represent that for distilled water at the low fre- quencies where dipolar relaxation effects are dominant. The experi- mental results for distilled water show the lower conductivity, since the dynamic range of the current is larger and the phase variation along the antenna is more rapid than predicted from the theory. The sodium chloride solution result in the lower curve shows a higher conductivity and loss than the 0.2 th/meter theoretical results, probably because of the combination of ionic effects (measured at low frequency to be 0.2 mhos/meter) and the beginning of the dipolar effects. Also shown in Figure 5.44 as a dashed line is the theoretical result for the same configuration except with the parasite removed (single antenna results, Figure 5.37). Apparently the conductivity of 0.2 mhos/meter used in the theory is somewhat unique in the particular configuration such that little difference occurs between the single and coupled cases. A greater difference is at least expected and comparison with the experimental results in Figure 5.37 shows that, indeed, an effect is seen. 300 Similar results for distilled water and salt solution at 1.0 Ghz are given in Figure 5.45. At this high frequency the dif- ferences between distilled water and salt solution become less distinct because Of the dominance Of the dipolar relaxation. The difference between the single antenna (dashed lines) and coupled antennas is quite pronounced. Although the experiment appears to be comparable with the single results, it is important to remember that the theoretical values used for dielectric constant and con- ductivity are not expected to be exact at this frequency and are used only as a reference. The input admittance and theoretical admittance parameters for the 350 MHz and 1.0 GHz cases are given in Tables 5.32 and 5.33, reSpectively. As shown, the eXperimental results for both fre- quencies are the same regardless of the loading of the parasitic antenna (where sc, oc, and 50 indicate short circuit, Open circuit and the 50 Ohm loaded parasite measurements, respectively). BaSed on the theoretical results (differing as usual by the approximate factor of two) for the same measurements (e.g. Y = Ysc)’ this 11 roughly indicates the sensitivity of the admittance measuring methods. The fact that the coupling term (Y12 = ) is small shows that Y21 the medium is reaponsible for the majority of the loss and approxi- mately shields the parasite from the driven antenna at the chosen separations of 3.2 cm (approximately 0.35 gm at 350 MHz; 1 Am at 1.0 GHz). The experimental results at 1.0 GHz in Table 5.33 are markedly different from the theory in the susceptance and it is suSpected that an error was made in recording the reference phase angle during calibration for these measurements. 301 These few examples investigated for coupled antennas in dissipative media are obviously not complete. They do, however, serve to show the ability Of the theory to predict results compar- able to experiment as well as qualitatively representing the experi- mental accuracy and results expected. This therefore concludes the study Of coupled antennas in known media. The next and final section considers the results from measurements on a sample of sewage treatment plant effluent. 1.0 1.0 1 -100 experiment distilled water V experiment 0.1 gm/‘ NaC{,solution 2b = 16.51 Cm 60.96 cm 6.0 cm 6.35 cm 0.5L 0.052L = 350 MHz 1 2 l 2 FIGURE 5.44 Driven Antenna Current for Coupled Antennas in Dissipative Media (350 MHz) Theory, both curves: er = 83.5 a = 0.2 th/m coupled antennas --- Ant. #1 alone 303 1.0 "Q 0 experiment; distilled ‘ water 1 2 r -—-d 1 7M- d b b 2 11,! _ L a = 2b = 16.51 cm _ L = 60.96 cm h = 5.0 cm O S b- 1 h2 = 5.08 cm -100 . d1 = 0542 d2 = 0.052L -200 ‘ f = 1.0 Ghz ® D -400 t 0 FIGURE 5.45 Driven Antenna Current 1 0 for Coupled Antennas in - % 0 experiment; 0.1 gm/L Dissipative Media (1.0 Ghz) -- NaCL solution a P II.1 " Theory, both curves: L gr = 83.5 a = 0.2 th/m .5 " coupled antennas --- Ant. #1 alone —-LOO b -200 ~ -300 - ¢ -400 ’ 304 TABLE 5.32 Input Admittance for Coupled Antennas in Dissipative Media at 350 MHz configuration of Figure 5.44 Theory 0 = 0.2 mhos/m , er = 83.5 F [Y] = 50 +351 -6.0 - 37.7 millimhos -6.0 - 37.7 58 + 344 YSO = 50 + 350 millimhos Experiment Distilled water Y = Y =‘Y = 74.6 + 3104 millimhos SC 0c 50 NaCL solution, 1 gm/liter Ysc = Yoc = YSO = 103 + 392 millimhos 305 Table 5.33 Input Admittance for Coupled Antennas in Dissipative Media at 1.0 GHz configuration of Figure 5.45 Theory 0 = 0.2 mhos/m , er = 83.5 [Y] = 70 +357 -2.5 - 316 millimhos -2.5 - 316 69 + 360 Y50 = 71 + 355 millimhos Experiment (error eXpected in recording data) Distilled water YSc = Yoc = YSO = 23.1 - 32.4 millimhos NaCL solution, 1 gm/liter YSC ='Y0C = YSO = 21.5 - 30.4 millimhos 306 5.4 Special Applications to Sewage Plant Effluent In the preceding section, the operation of the antenna- waveguide-media system.was studied for dissipative media by using solutions Of controlled amounts of pure sodium chloride. The question may legitimately be raised about the usefulness of the chosen con- centration in predicting performance in a natural environment medium. To approach the question, this section presents the results of studying one extreme medium: the effluent from a sewage treatment plant. This case study represents one example of the type of application oriented study which is necessary for this system, al- though it is of course far short of the comprehensive experimental and theoretical correlation that is required. The chosen sample for this one case was obtained from the first of a series of large ponds used to store the raw effluent from the final stage of the East Lansing, Michigan, sewage treatment plant.* Since very little natural breakdown Of the sewage occurs in the first pond, this sample closely represents the undiluted output Of a typical municipal waste water treatment plant. The chemical analysis of the sample taken is given in Table 5.34 along with the measured low frequency conductivity. When compared to the results given in Figure 4.8 it is seen that the sewage sample corresponds to a salinity (of NaCL) of 0.5 grams per liter in low frequency conductance. The 1.0 gram per liter concentration used in the previous section yielded twice the conductivity of the sewage. These storage ponds are part of the ongoing research under the Institute of Water Research, Michigan State university, whose assistance I gratefully acknowledge. 307 The sewage sample, therefore, shows that the dissipative media were realistically representative. (Of course, much more highly dissipative media are possible, for example, if sea water is con- sidered with a typical salinity of 35 grams per liter and con- ductivity of 4 mhos per meter.) Single antenna in cavity; current The results for the antenna current distribution for sewage filling the experimental rectangular cavity are given for four fre- quencies (0.35, 1.0, 1.2, 1.6 Ghz) in Figures 5.46 through 5.50. In all but the 1.6 Ghz case the theoretical results are included for two combinations of permittivity and conductivity: 3 = 83.0 a = 0.1 mho/m and e = 83.5 0.2 mho/m . Q N The former combination represents the measured low frequency con- ductivity Of the sewage sample (the permittivity was chosen arbitrarily), while the latter combination is that used fOr the comparison with the sodium chloride solution in Sections 5.2.5-6 and 5.3.4. The results at 350 MHz in Figure 5.46 best indicate the type of results expected. The experimental data appears to be bounded by the high and low conductivity theories. It is difficult to readily Obtain quantitative explanations for the results, since the effect Of the media on the interaction between antenna and 308 cavity is complex. This interaction effect may also eXplain the differences seen at 1.0 GHz in Figure 5.47 and 5.48. In that data as the height of the antenna is increased from 5.0 cm, the agreement between theory and experiment worsens. The interesting point of this example is that the theory exhibits greater dynamic range variation ("peak- to-peak” level) as the height of the antenna is increased. This is not immediately obvious, but one could also imagine that proper changes in dielectric constant could result in better agreement, since the cavity resonance structure would then be changed very rapidly (whereas a two-fold change in conductivity shows little effect). In Figure 5.48 it is seen that as the antenna location is changed, very little change occurs in either theory or eXperiment (realize that the theoretical curves are rough hand drawn representa- tions Of the twenty point numerical solution and are therefore irregular in smoothness). This is indicative of the small coupling between cavity and antenna at the high frequency and high effective dissipation. The unexpected variation of the current with antenna height found at 1.0 GHz is again seen at 1.2 GHz in Figure 5.49. Again it is reasonable to expect that changes in dielectric constant could bring the theoretical result more in line with the experiment. Un- fortunately, time and computing cost precluded further investigation into this. The final case presented for the single antenna is at 1.6 GHz in Figure 5.50. At this higher frequency the attenuation is so large that the phase shift along the antenna has become nearly 309 linear and the current has been smoothed to the point that very little information other than rate of decay can be Obtained from the data. This effect is seen even for the low conductivity theory, although it is more evident for the experimental results Of the effective (high frequency) higher conductivity sewage plant effluent. Single antenna in cavity; admittance The input admittance for the single antenna in sewage plant effluent for the configurations presented in the preceding paragraphs is given in Tables 5.35 through 5.38. In all cases only the theoretical results comparing the solutions for the two different conductivities are compared. NO experimental data is presented because of error in the phase angle calibration Of the admittance measuring system. A length of line of unknown length (and phase shift) was inserted inadvertantly in one cable between admittance measuring block and vector voltmeter. The phase calibration was not changed, however, and thus the conversion Of the data from the measuring plane to the antenna terminal plane was impossible. From the tables one can see that at 350 MHz the difference in conductivity has an effect on the level Of input admittance and the coupling to the cavity is evident by the sensitivity to changes in location. At the ikband frequencies, however, the differences are considerably reduced, although a sensitivity is still noticeable. The theory Of course does not account for dipolar relaxation effects, but the higher frequency increases the electrical size Of the cavity thus increasing the electrical distance between antenna and walls. The admittances at 1.6 GHz are given in Figure 5.50 instead Of in a table. 310 Coupled Antennas Very little data was taken for coupled antennas in sewage. The high loss and low coupling that was Observed for salt solution at the higher frequencies (above 1.0 GHz) indicated that little is gained from experimental measurements of current distribution for the coupled case. Therefore, only a few cases at 350 MHz are pre- sented in Figures 5.51 and 5.52 and Table 5.39. The theoretical and experimental cases in the two figures differ by the height of the driven antenna, but the effect Of varying separation is seen tO be the same for both sets of data. The input admittance parameters in Table 5.39 show that there is very little coupling but that the coupling definitely varies as the separation (d2) varies. The selfqadmittance terms remain relatively fixed, however, which indicates that the cavity-antenna coupling is not a strong function of the location Of the antennas. The value Of input admittance when the parasite is loaded with 50 Ohms (YSO) is seen to differ very slightly from the driven antenna selfqadmittance. The losses Of the medium are thus nearly masking the effect Of second antenna loading from the active antenna. Received Voltagg One possible use of coupled antennas other than for two-port terminal measurements is as an absorption measurer. The radiation, waveguide transmission, and reception of energy for absorption studies is used in electrochemistry analysis Of absorption spectra. In a similar manner, the data for received voltage was taken for different separations between two antennas in the cavity filled with sewage plant effluent. The case chosen was for a frequency of 1.0 311 GHz with the cavity and antenna having dimensions: a = 2b = 16.51 cm L = 60.96 cm d1 = 8.08 cm; d2 = parameter h1 = 6.0 cm; h2 = 5.08 cm The theoretical data points were Obtained by making individual program runs for each separation and noting the input current at the second antenna when loaded with a 50 Ohm load. The results are presented in Figure 5.53 where in 5.53(a) data is included for distilled water and in 5.53(b) the sodium chloride solution results are included. The data was taken for the active antenna located as close as experimentally possible to one shorting plate (d1 = 8.08 cm = 0.133L). The receiving antenna was then moved in increments of 3.2 cm (approximately 0.052L) and the received voltage noted. This voltage was measured on the vector voltmeter with reference taken to one arbitrary side Of the admittance sensing block. The data presented in Figure 5.53 is only the magnitude Of the complex voltage received. The phase information involves multiples of Zn ambiguities, because for water (er = 83) the 3.2 cm increment is nearly equal to a full wavelength. Furthermore, the magnitude information is normalized to the voltage received at the closest separation to simplify com- parison. Although this set Of single frequency data is only part of the measurements that are necessary for swept frequency absorption studies, they show the range of dissipation predicted (and measur- able) for these sample media over an increasing path length. 312 It is interesting to note that of the three water samples studied only the distilled water results tend to approximate the theory. It is also interesting that the sewage results are smoother with less fluctuation than the sodium chloride results. Of course this data is insufficient to suggest explanations, but the dif- ferences seen are significant enough to warrant further investiga- tion. This concludes Chapter 5 and the presentation Of experimental and numerical results of this report. The highlights Of these re- sults are summarized in the final chapter which follows. TABLE 5.34 Analysis Of Sewage Sample (sample taken 20 November 1974) Ag pH Dissolved oxygen Total phOSphorus Soluble phOSphorus Nitrite Nitrate .Ammonia Total Carbon Organic Carbon Alkalinity 7.9 12.6 mg/L - 0.0. 0.80 mg/L - P 0.65 mg/L - P 0.078 tug/L - N 5.1 lug/4. - N 9.3 mg/L - N 48 Ins/l. - C 6 mg/L - C 179 mg/L caco3 Conductivity measurements using method Of Figure 4.8 frequency, MHz 0 , mhos/m 1.5 2.0 5.0 10.0 15.0 0.104 0.100 O .100 0.106 0.096 314 4 "'d b 1’“ h ' L a = 2b = 16.51 cm L = 60.96 cm . h = 6.0 cm (11 = parameter 7 f = 350 MHz b-SO 4 (b d “-100 a & b A A A A l A A __2 AA 0 .5 y/h 1.0 FIGURE 5.46 Current for Single Antenna in Sewage (350 MHz) Theory: (a) at = 83.0 a = 0.1 mho/m (b) er = 83.5 a = 0.2 th/m 315 6], "flh b :5 L a = 2b = 16.51 cm L = 60.96 cm d1 = 0.5L h = parameter f = 1.0 GHz FIGURE 5.47 Current as Function 1 0 - of Height for Single . A I Antenna in Sewage h = 5-5 cm (1.0 0112) Theory: er = 83 o = 0.1 th/m "- er = 83.5 o = 0.2 th/m 316 1.0 - h ' 6.0 cm ‘0 - -100 ‘ ~200 ‘ 6 - -400 1.0 FIGURE 5.47 (continued) Theory: __ er 3 83.0 . a = 0.1 th/, 0 --- er a 83.5 -200 1.0 1.0 II! ‘D 317 d - 0.133L 8 8.08 cm d1 ' 0.185L = 11.28 cm r3 ,, L a = 2b = 16.51 cm L = 60.96 cm h a 6.0 cm d ' parameter 1 f I 1.0 Ghz FIGURE 5.48 Current as Function Of Location for Single Antenna in Sewage (1.0 Ghz) Theory: 3 = 83.0 —_ r O - 0.1 th/m ---- gr 8 83.5 a = 0.2 mho/m 1.0 III III 318 .9 r h ' 5.0 cm ‘ 1 b I- . -.‘ .3 \ ‘\ I . \ I /‘\ \ \\ I, O 1’ ‘\ /\‘ '7 o 0‘" O O ’ 1 S '3 ‘ O O a / O O ‘\ \ ’ -200 ‘ “Q o ‘ Q . o‘\ d 6 ° O b o 3 \O . ° . -400 ‘° . \\. ‘ (D o d l A j l U l A A l . 5 y/h 1. o h = 6.0 cm a = 2b = 16.51 cm L = 60.96 cm (11 = 0.5L h = parameter f = 1.2 GHz 1. FIGURE 5.49 Current as Function Of Height for Single Antenna in Sewage (1 .2 GHz) Theory: at = 83.0 a = 0.1 mho/m ---- er = 83.5 a = 0.2 th/m 319 . -200 - -4oo - -600 1.0 a = 2b = 16.51 cm b (11 AUh L = 60.96 L d1 = 0.5L h = 6.0 cm Theory: er = 83.0, o = 0.1 mho/m Input Admittance: Y“: 119 + 375 millimhos Y FIGURE 5.50 Current for Single Antenna in Sewage at 1.6 GHz 320 TABLE 5.35 Admittance for Single Antenna in Sewage (350 MHz) see Figure 5.46 for configuration and current theoretical admittance, millimhos e=83 r o = 0.1 th/m er - 83.5 a = 0.2 th/m d1 = 0.1331, 43.3 + 339.6 d1 = 0.54, 44.4 +359 49.5 + 349.3 50 + 356.5 F. 321 TABLE 5.36 Admittance as Function of Height of Single Antenna in Sewage (1 .0 Ghz) see Figure 5.47 for configuration and current theoretical admittance , millimhos at = 83 at = 83.5 h, cm a = 0.1 th/m a = 0.2 th/m 5.0 78 + 389 77.8 + 390.5 5.5 104 + 397 96.4-+ 387.8 6.0 104 + 350 93.0 + 351.3 6.25 86 + 346 81.0 + 348.6 322 TABLE 5.37 Admittance as Function of Location for Single Antenna in Sewage (1.0 Ghz) see Figure 5.48 for configuration and current theoretical admittance, millimhos St, = 83.0 at = 83.5 o = 0.1 th/m q = 0.2 mholm d1 = 0.133L 105 + 365 96 + 358 61 = 0.185L 85 + 353 - 323 TABLE 5.38 Admittance as Function of Height of Single Antenna in Sewage (1.2 GHz) see Figure 5.49 for configuration and current theoretical admittance, millimhos s = 83.0 e = 83.5 r r h, cm a = 0.1 th/m o = 0.2 mho/m 5.0 97 + 370 96.7 + 382 6.0 114 + 390 103 + 385 324 « 50 th 1 ( eory on y) .LO 3 -50 4 -100 A A A l A 1 A L .5 y/h 1.0 IL d ..[Hl— a=2b=16.51 cm 1 d 2 L = 60.96 cm L d1 = 0.5L, d2 = parameter h1 = 5.0 cm, h2 = 5.08 cm f = 350 MHz Theory: er = 83.0, c = 0.1 mho/m 50 ohm load on parasite (#2) FIGURE 5.51 Theoretical Driven Antenna Current for Coupled Antennas in Sewage (350 MHz, hl ' 5 Cm) I ‘ do“. 1.0 "* ‘ ' ' I ' £5 *' 3 A A r 8 A A 4 £5 (3 0 . £5 0 0 £5 , 1111 o A. O o [3 ‘5 O ,3 _ o A’4A £3 0 o O o 3 .5r 0 0 AG 3 (experiment only) . £1 3' 8 8 8 o ‘ 9 6 9 8 96869 A l l L l l A A A 0 .5 y/h 1.0 1 2 b d1 ..HL d2 a = 2b = 16.51 cm L = 60.96 cm L d1 = 0.54, (3 d2 = 0.052L A d2 = 0.1054, h1 = 6.0 cm, h2 = 5.08 cm f = 350 MHz 50 Ohm load on parasite (#2) FIGURE 5.52 Experimental Driven Antenna Current for Coupled Antennas in Sewage (350 MHz, h 1 = 6 cm) 326 TABLE 5.39 Theoretical Admittance for Coupled Antennas in Sewage at 350 MHz (h = 5 cm) 1 see Figure 5.51 for configuration and current (theory only) for d - 0.052L 2 [Y] = 24-+ 342.5 8.1 - 32.4 millimhos 8.1 - 32.4 28 + 341.5 YSO = 24 + 343.8 millimhos for d2 = OolOSL [Y] = 23 + 338.7 -1.9 - 39.0 millimhos _-1.9 + 39.0 28.5 + 335.6 Y = 23.6 + 337.2 millimhos 50 327 mane: opaumaHmmao 5:3 oozam hairdo 5 moccouo< @3960 you :Owumumaom moose? mwmuHo> cm>wmoom Amvmm.m mmaoom m.se 96 .aoauunmaom e.- 6.m 6.6 ~.m o . a . d . J 4 a . . a _ . _ moo o _u U n/ A .l I \.\D’a,0 IA H0.0 /, .\iu // U\ \0’ \\ I .l \\o A g \\\\ A I, O . I \\ . l I .... \xx \ /O\. 1 mo 0 IO\ . U W/ \ A 1 .I O 1H.O {b ~.o u o A 4 8 . A [ Inoo 1 l u use o.H u a . p p L O.H paznauuou ‘838310A paAIBDBH 328 «new: n>aomaammao 5:3 coaafim amt/mo ow mmocouo< eoaoooo you cofiumumaom mousse, mwmuao> woe/woos“ Anvmm.m mmaon w.qs 86 .aoaumnmaom 3.N~ 6.8 3.6 ~.m o E q — q a u q — u q q 4 q 1. A MP \ \ \4/ A II \\ I U I z/ \ .\@ I - (D X l I/ \D\\\ / II /mt\l\ll old I I \ \ ‘01! /< .. n: :4- . 01/. A t /D / mesom ./. 4 D/ . O I / \ .l W. \\\\\ o Oil/II. A O A H///// \\\\\ 1 r A“ . / JR? N.o ".0 \\\\\o . mapOOSu 0 mm H o u use o.a u u U moo.o Ho.o no.0 H.o m.o 0.4 pazrtemaou ‘BBBJIOA paAIBOBH CHAPTER 6 CONCLUSION; RECOMMENDATIONS This report has described the theoretical and experimental investigation of antennas in waveguide boundaries with general homo- geneous isotropic media. With an emphasis towards practical utilization of the relation between antenna characteristics and electrical prOperties of the medium, results have been presented for the antenna current distribution and input admittance as a function of both the waveguide configuration and the conductivity, permittivity and permeability of the medium. The theoretical development formally presents the mathe- matical model of N linear antennas in a rectangular waveguide with arbitrary reflection coefficients and medium. The set of coupled integral equations for the unknown current distributions are found by appropriate application of the Lorentz reciprocity relation to a modal analysis of the induced fields in the waveguide region. Parallel developments are presented which consider the antennas under the thin-wire and the strip approximations for an assumed delta-gap excitation. The single component of the dyadic Green's function which is derived by this method for the interior of the waveguide illustrates the modification necessary for consideration of arbitrary waveguide terminations compared to the infinite (matched) waveguide case. The integral equations developed are reduced to 329 330 Hallen-type equations in order to facilitate solution by numerical methods. The antenna problem is then reconsidered analytically for the case of a coaxial gap TEM mode excitation. For a single antenna under the thindwire approximation the integral equation is developed by evaluation of definite integrals over the coaxial region in the rectangular coordinate system of the waveguide. The theoretical development is concluded by showing the agreement between suitable special cases of the general results found here and results which have been published in the literature. The numerical solution of the general coupled set of integral equations is accomplished by the method of subsections and point matching according to the typical moment method approach. Results of this simplest approach are then shown to agree with solutions obtained by the more accurate Galerkin's method based on pulse func- tions. Attention is paid to the convergence properties of the numerical solution and it is shown that the strip approximation of the cylindrical antenna exhibits improved convergence properties over the conventional thin-wire approximation. Criteria are derived for Optimally truncating the involved modal series and a correSpondence is established between these criteria and sampling theory. Finally, representative cases are presented which compare the results for the various theoretical approximations. Relative current distributions are shown to be the same for all approximations with the input impedance varying between approximations. The experimental investigation of the antenna-waveguide- medium system is described in Chapter 4. Details are given of the construction of the probes for measuring antenna current distributions 331 and antenna input impedance. The design of the impedance sensing block for measuring input impedance with a vector voltmeter is dis- cussed and shown to be an effective, accurate, and convenient method compared with the conventional slotted line techniques. Descriptions are also given of the methods used to measure the electrical properties of the dissipative media studied; conductivity is measured via a low-frequency coaxial capacitor and permittivity by perturbation measurements in a cylindrical cavity. Included in the discussion of experimental equipment are con- siderations which proved significant in the measurement techniques. In particular, the orientation of the current sensing probe in the waveguide is shown to affect the voltage induced in the probe. The orientation in a transverse waveguide plane which most truly represents the current on the antenna is determined from a discussion of the dominant waveguide fields and supportive eXperimental data. Techniques are also described for calibration of the impedance sensing block which are unique to the set-up used. Discussions of the predominant sources of error in measurement of phase angle are included with an accuracy of 1;2° being typical. The numerical solutions of the theory and experimentally measured values of current distributions and antenna input admittance are compared in Chapter 5. The correlation of experimental and theoretical input admittance is shown to depend on the infinite susceptance introduced by the delta-gap model, whereas this effect is not seen in the coaxial gap model. Attempts to obtain satisfactory agreement between theoretical and experimental admittance typically involve terminal zone correction networks. The analysis 332 of a simple two element lumped equivalent network is presented as it applies to this problem. Of particular interest in this analysis is the unique relationship for determination of the Optimum matching network based on simple curve fitting techniques. The correction network is not used to alter the experimental data, however, for two reasons. First, the interaction that exists between the antenna feed and the cavity and the antenna itself is extremely complex, which requires a different correction network for virtually every configura- tion. Second, the presentation of results which have been made to agree by some correction network offers very little insight into either the true operation of the system or the importance of the antenna feed configuration in the cavity excitation process. The antennadwaveguide-media system is studied in Chapter 5 by systematically varying the waveguide/cavity, antenna height, antenna location, and the operating frequency for both the single antenna and two coupled antennas. For the open ended (i.e., semi- infinite) waveguide the results for the single antenna current are shown to differ significantly with the typically assumed zeroth order shifted sine distribution. The possibility of applying this theoretical development to characterizing coaxial line to rectangular waveguide couplers is well demonstrated. For the cavity configuration the air medium is stressed to show the interaction of the antenna with the resonant cavity. The most significant results are best summarized in Figures 5.23 and 5.24 for the dependence of the current distribution on the cavity resonance. This relationship forms the basis for understanding the effect of other parameter variations from a cavity perturbation vieWpoint. That is, as the antenna height, location, or 333 cavity dimensions are varied the current distribution changes accordingly as the cavity resonance is shifted with reSpect to the Operating frequency. For the study of the effect of the dissipative media on the antenna-cavity coupling, excellent agreement was obtained between the theory and both published and independently measured values. The ability to obtain measurements easily for the water medium supports the practicality of this system for potential application in this area. The study of coupled antennas for a representative subset of cases covered for the single antenna was primarily seen as a perturbation of the cavity by the second antenna. The coupling be- tween antennas, however, was more dependent on element Spacing and the distribution of fields in the cavity as was expected. The study of dissipative media for coupled antennas showed a much reduced coupling effect due to the high losses at the L-band frequencies. Reduction of the operating frequency by a factor of about 3 to 350 MHz resulted in increased coupling and stronger interaction with the cavity. The final study was done for sewage treatment plant effluent as the medium filling the cavity. As a feasibility study the results were highly successful showing that this extreme case exhibited losses sufficiently low to render meaningful measurements. The effects seen were primarily related to the macroscOpic properties (conductivity andtiielectric constant) rather than the microscopic electro-chemistry as would be eXpected. However, a more detailed study of other cases might easily depend on the microscopic pro- perties. In particular the absorption-type results presented show 334 a feasibility for this type of correlation. The differences noted between sewage, sodium chloride solution, and distilled water seemed to indicate the possibility of a relationship between differences in type or concentration of ions present and the energy transmitted through the solution As a brief summary of the conclusions of this report, the following main points stand out. First and foremost, the theory which was developed gives excellent agreement with all experimentally determined current distributions including those special cases pub- lished in the literature. The agreement obtained with the input admittance is only qualitative, however, with the absolute differences being relateable through terminal zone correction network analysis. In the numerical analysis, knowledge was gained of optimum truncation of series in modal analysis problems solved by the method of moments. The usefulness of the findings to the initial motivating problem.of developing a measurement tool is based primarily on the current distribution as a function of cavity resonance. The newly determined fact is that the purely quadrature current on an antenna in a cavity exhibits a zero crossing which is a strong function of the operating frequency compared to the cavity resonance frequency. Since the zero crossing is easy to detect, and since it is extremely sensitive to the cavity resonance (which is easily perturbed), the possibility exists for exploiting this feature as a detection or monitoring tool. The develOpment of this relationship is certainly worth further study. 335 Recommendations for Further Study Besides developing the relationship between current and cavity resonances just mentioned, some further study is also recommended in other areas that have been discussed in this report. In particular, the connection noted between sampling theory and the optimum numerical truncation offers the possibility of giving further understanding of these types of modal analysis problems. Also, the Observations and relationships noted between the experimental coaxial gap and the theoretical delta-gap model could be systematically in- vestigated using the rectangular cavity as a control on the type of coupling between transmission line and antenna. Such work would be invaluable in the development of well understood terminal zone correc- tion networks. Areas which this author feels deserve continuation or new effort are both in the purely theoretical as well as the applica- tions study areas. It is possible that a reworking of the coaxial theory develOped here (with the integral solutions in Appendix II) to include more than the TEM coaxial mode could result in better agreement in absolute input admittance values. The coaxial theory could Obviously also be expanded to include coupling between multiple antennas. Further, the theory could be simplified in an effort to Obtain an approximate analytical, closed form, two or three term expression for the current distribution to aid in separating the interactions between antenna and cavity. The experimental or applications work begun here bears con- tinuation in at least two areas. First, the short study of semi- infinite waveguides could be made complete to advance the knowledge 336 beyond the approximate analysis typically used today. Second, the application to pollution or dissipative media studies should be expanded based on the following areas. Electrochemis ry should be included in the theory at least in the form of the frequency de- pendent relation between complex permittivity and frequency for dipolar liquids. Studies should be expanded using controlled salt concentration and various salt types to look for possible correla- tion. Microscopic investigations should be continued of the two antenna absorption type done at the end of Chapter 5. And finally, and perhaps most interestingly, there remains a great deal of potential in the area of air pollution studies where low dissipation and low dielectric constant maintain the ability for reliable measurements in the 1.0 to 2.0 GHz range. B IBLIOGRAPHY 1' I s 10. 11. BIBLIOGRAPHY Chang, D.C., "End correction Of a monOpole antenna with arbitrary radius," IEEE Trans. on Antennas and Propagation, Vol. 16, pp. 606-607, September, 1968. Collin, R.E., Field Theory of Guided Waves, MCGraw Hill, New York, 1960. Collin, R.E., Foundations of Microwave Engineering, McGraw Hill, New York, 1966. COOper, R., "The electrical properties of salt-water solutions over the frequency range 1-4000 Mc/s.," IEE Journal, Vol. 93, Part III, No. 22, pp. 69-75, March 1946. Dorsey, N.E., Properties of Ordinary water Substance, American Chemical Society Monograph, Hefner Publishing Co., Inc., New York,1968. Duncan, R.E., "Theory of the infinite cylindrical antenna including the feedpoint singularity in antenna current," Journal of Research, N.B.S., Vol. 66D, NO. 2, pp. 181-188, MarcheApril 1962. Eisenhart, R.L. and P.J. Khan, "Theoretical and experimental analysis Of a waveguide mounting structure," IEEE Trans.'MTT, V01. 19, NO. 8, pp. 706-719, 19710 Galejs, J., “Admittance of a rectangular slot which is backed by a rectangular cavity," IEEE Trans. A.P., pp. 119-126, March 1963. Gradshteyn, 1.3. and I.M. Ryzhik, Table of Integral Series and Products, Academic Press, New York, 1965. Grant, E.H., T.J. Buchanan and H.F. Cook, "Dielectric behavior Of water at microwave frequencies," Journal Of Chemical Physics, Vol. 26, NO. 1, January, 1957. Harrington, R.F., Field Computation by Moment Methods, MacMillan Company, New York, 1968. 337 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 338 Iizuka, K., "An experimental study of the insulated dipole antenna immersed in a conducting medium," IEEE Trans. AnP., pp. 518-532, September, 1963. Iizuka, K» and R.W.P. King, "The dipole antenna immersed in a homogeneous conducting medium," IRE Transactions on Antennas and Propagation, pp. 384-392, July, 1962. Iizuka, Km and R.W}P. King, “An experimental study of the half-wave dipole antenna immersed in a stratified medium," IRE Trans. A.P., pp. 393-399, July, 1962. Jasik, H., Antenna Engineering Handbook, McGraw Hill, New York, 1961. Johnson, C.C., Field and Wave Electrodynamics, McGraw Hill, New York, 1965. King, R.W.P., Theory_of Linear Antennas, Harvard University Press, Cambridge, Massachusetts, 1965. King, R.W. P. and C.W. Harrison, Antennas and Waves, MIT Press, Cambridge, Massachusetts, 1969. King, R.W}P. and K. Iizuka, "The complete electromagnetic field of a half-wave dipole in a dissipative medium," IEEE Trans. AmP., pp. 275-285, May, 1973. King, R.W.P., R.B. Mack and 8.8. Sandler, Arrays of Cyligdrical Dipoles, Cambridge University Press, Cambridge, Massachusetts, 1968. King, R.W.P. and L.D. Scott, "The cylindrical antenna as a probe for studying the electrical properties of media," IEEE Trans. AJP., Vol. 19, NO. 3, pp. 406-416, May, 1971. Lee, 8.91., W.R. Jones and J.J. Campbell, 'Convergence of numerical solutions of iris-type discontinuity problems," IEEE Trans. MI‘T, Vol. 19, NO. 6, pp. 528-536, June 1971. Lepley, L.K. and W.M. Adams, "Electromagnetic diSpersion curves for natural waters," Water Resources Research, Vol. 7, NO. 6, pp. 1538-1547, December, 1971. Lewin, L., Advanced Theory_pf Waveguides, Iliffe and Sons, Ltd., London, 1951. Lewin, L., "A contribution to the theory of probes in wave- guides," Proc. of I.E.E. (London) Part C, pp. 105-116, 1958. MeLachlan, N.W., Bessel Functions for Engineers, Oxford Press, 1934. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 339 Mishra, S.R., L.D. Scott and R.W.P. King, 'Currents, charges, and admittance Of linear antennas in dissipative media," Radio Science, Vol. 9, NO. 4, pp. 487-495, April, 1974. Papoulis, A., The Fourier Integral and Its Applications, McGraw Hill, New York, 1962. Rao, B. Rama, “A two-element Yagi-type array in a parallel- plate waveguide - theoretical and experimental studies," IEEE Transactions on Antennas and Porpagation,‘Vol.13 , pp. 675-682, September, 1965. Rogers, J.C. and I.C. Peden, "The electrically short sheathed dipole: Experimental relationship between its measured admittance and the permittivity of the external medium," IEEE Trans. A;P., Vol. 21, No. 6, pp. 857-862, November, 1973. Sauter, G.D. and E.G. Chilton, "Air improvement recommendations for the San Francisco Bay Area," NASA contract NGR-05-020-409, School Of Engineering, Stanford University, 1970. Saxton, J.A., "Electrical properties of water," Wireless engineer, pp. 288-292, September, 1949. Saxton, JnA. and J.A. Lane, 'Electrical properties Of sea- water," Wireless Engineer, pp. 269-275, October, 1952. Scaife, B.K§P., Complex Permittivity, The English University Press Ltd., London, 1971. Scott, L.D., "Apparatus for studying the properties of antennas in an effectively infinite dissipative medium," NASA Scientific Report NO. 6, Division Of Engineering and Applied Physics, Harvard university Press, Cambridge, Massachusetts, December, 1969. Scott, L.D. and Mishra, S.R., "Theoretical and experimental investigations on coupled antennas in homogeneous isotropic media," Technical Report NO. 641, Division of Engineering and Applied Physics, Harvard university, Cambridge, Massachusetts, April, 1973. Sucher, M. and J. Fox (eds.), Handbook Of Microwave Measurements, Volume II, Polytechnic Press, New York, 1963. Tai, C.T., Dyadic Green's Functions in Electromagnetic Theory, Scranton, Pa., International Text Book Co.,1971 . VanBladel, J., "Some remarks on Green's dyadic for infinite space," IRE Trans. on Antennas and Propagation, pp. 563-566, November, 1961. Wu, T.T., "Input admittance of linear antennas driven from a coaxial line," J. Res. NBS, Vol. 67D, pp. 83, 1963. GENERAL REFERENCES Ament, W.S., J.C. Katzin, M. Katzin, and B.Y. Koo, "Impedance of a cylindrical dipole having a sinusoidal current distribu- tion in a homogeneous anisotropic ionosphere," Radio Science J. Res. N.B.S. 68D, 4, 379-405 (1964). Biggs, A.W., "Dipole antenna radiation field in stratified Antarctic media," Trans. IEEE AP-16, 445 (1968). Blaedel, W.J. and H.V. Malmstadt, "High frequency titrations," Analytical Chemistry, 22, 734 (1950). Blair, W.E., "Experimental verification Of dipole radiation in a conducting half-Space," Trans. IEEE AP-ll, 269 (1963). Brekhovskikh, L.M., waves in Layered Media, Academic Press (1960). Ginzburg, V.L., The Propagation of Electromagnetic waves in Plasmas, 2nd Edition, (Translated from Russian), Pergammon Press Ltd. (1970). Golterman, H.L., Methods of Chemical Analysis of Fresh Waters, International Biological Programme Handbook NO. 8, Blackwell Scientific Publications, Oxford (1969). Grant, F.S. and G.F. West, Interpretation Theory in Applied Geo- physics, McGraw-Hill Book.Company (1965). IiZuka, K., “An experimental investigation on the behavior Of the dipole antenna near the interface between the conducting medium and free Space," Trans. IEEE AP-12, 27 (1964). Iizuka, K., "Research on the properties of antennas in a conducting media," Scientific Report NO. 8, Gordon McKay Laboratory, Harvard University, Cambridge, Massachusetts (1964). IiZuka, K., "The circular loop antenna immersed in a dissipative medium," Trans. IEEE AP-13, 1, (1965). Iizuka, K., "Experimental study on the circular lOOp antenna immersed shallowly in a conducting medium," Radio Science J. Of Res. N.B.S., 69D, 9, (1965). 340 341 Johnson, P.L. (Ed.), Remote Sensing in Ecology, Univ. of Georgia Press, Athens (1969). Johnson, D.E. and C.G. Enke, Analytical Chemistry, Vol. 42, 329, (1970). King, R.w.P., C. Harrison, Jr., and D.H. Denton, Jr., "The electrically short antenna as a probe for measuring free electron densities and collision frequencies in an ionized region," J. Res. of N.B.S. Radio Propag., 65, 4, (1961). Kraichman, M.B., "Basic experimental studies of the magnetic field from electromagnetic sources immersed in a semi-infinite conducting medium," J. Res. NBS 64D, 21 (1960). Muller, O.H., The Polarographic Method of Analysis, Chem. Ed. Pub. Co., Easton, Pa. (1956). Olson, TnA. and F.J. Burgess (editors), Pollution and Marine Ecology, Interscience Publishers, New York, (1967). Saran, G.S. and Held, "Field strength measurements in fresh water," J. Research NBS 64D, 435 (1960). Siegel, M., "Linear antennas and dipole arrays in a dissipative half-Space," Technical Report NO. 608, Division Of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts (July 1970). Tannenbaum, B.S., Plasma Physics, McGraw Hill Book Company, Inc., New York (1967). Wait, J.R., Electromagnetic waves in Stratified Media, Pergamon Press, Oxford (1962). APPENDICES n. APPENDIX I DESCRIPTION AND DEFINITION OF NOTATION A list of the frequently used notation for this report is given in this appendix. Symbols which are not used repeatedly and those whose use is readily implied from the context are not included. If a symbol has been used for two different quantities, only the primary usage is given here. Where possible, the units are included as well as reference to defining equations or figures in the text. a, b transverse waveguide dimensions in the & and y directions, reSpectively. (meters) Figure 2.1. B antenna input susceptance. (mhos) A , B , C , D waveguide mode eXpansion coefficients, Equation (2.7). d Spacing between antennas. (meters) d positive distance from origin to waveguide termination in negative 2 direction. (meters) Figure 2.2. d2 positive distance from origin to waveguide termination in positive E direction. (meters) Figure 2.2. 342 DBES(p) DSi ex,v(x.y) 3t,v(x.y) e 4+ Fibre/.2) f (z‘z') V 343 gp - gpoag) difference of p-th order Bessel functions with Specified arguments. Sa(rb§) - Si(ra§) difference of sine integrals with Specified arguments. (Appendix II) normalized v-th mode x-component of electric field involving only x and y dependence. Equation (2.5). rF-!FF-r ai-eaulll transverse component Of normalized V-th mode electric field. Equation (2.6). superscript for TM.mOde. total electric field for v-th eigenmode, propagating in i_§ direction. (volts/ meter) Equation (2.6). reflection coefficient factor. Equation (2.13). antenna input conductance. (mhos). kernel Of integral equation, "Green's function". magnetic field with usage analogous to electric field, e. Equation (2.5). superscript for TE mode. height of antenna (monOpole). (meters). total magnetic field with usage analogous to electric field, E. Equation (2.6). I(y) J(x,y,z) 344 subscript for number of antenna used as source location. antenna line current density. (amps). zeroth order modified Bessel function. Equation (2.33). complex number, (-1)%. subscript for number of antenna used as g. field location. antenna volume current density. (amps/m2). p-th order Bessel function "(pe)%, wavenumber for general medium. (meters-1). nn/a, waveguide mode eigenvalue for x coordinate. (meters-1). mn/b, waveguide mode eigenvalue for y coordinate. (meters-1). (k: + k:)%, waveguide cutoff wavenumber. (meters-1). number of subsections on i-th antenna. Equation (3.2). length of cavity or cavity resonance mode index. zeroth order modified Struve function. Equation (2.33) and following Equation (3.21). waveguide mode index for y coordinate. NMAX Q(y) 60) in 3L St 345 maximum value of m for truncation Of mode summations. Equation (3.36). waveguide mode index for x coordinate. maximum‘value of n for truncation of mode summations. Equation (3.42). subscript for number of subsection Of antenna current. Equation (3.2). subscript for number of subsection of antenna for point matching. Equation (3.9). complex antenna line charge density. (coulombs/m) Equation (3.24). -jQ(y), complex antenna line charge density referenced in phase to the antenna driving voltage. (coulombs/m) Equation (3.25). radius of antenna, radius Of inner conductor of coaxial line. (meters) Figure (2.1). radius of outer conductor of coaxial line. (meters) Figure (2.1). approximate antenna input resistance for semi-infinite waveguide configuration. (ohms) Equation (5.5). superscript for longitudinal strip approxima- tion. Section 2.5.2. superscript for transverse strip approxima- tion. Section 2.5.2. surface area. Pq Si(x) in 50 346 matrix representing discretized form of kernel of integral equation. Section 3.1.4. I: sin u/u du, sine integral. (Appendix II). superscript for thin wire approximation. Section 2.5.1. pulse function for method of subsections. Equation (3.3). voltage across coaxial line at antenna terminals. Equation 2.35. f driving voltage for j-th antenna. volume around i-th antenna for Lorentz reciprocity relation. Figure 2.3. transverse waveguide coordinate. Figure 2.1. theoretical antenna input reactance. (Ohms) Section 5.1.2. reactance of terminal zone inductor. (Ohms) Equation (5.3). transverse waveguide coordinate. Figure 2.1. apparent (experimental) antenna input admittance. (mhos) general antenna input admittance. (mhos) input admittance of driven antenna with second parasitic antenna loaded with 50 Ohms. (mhos) SC 0C 6(y - 1/2) 347 input admittance of driven antenna with second parasitic antenna short circuited. (mhos) input admittance of driven antenna with second parasitic antenna Open circuited. (mhos) longitudinal waveguide coordinate. Figure 2.1. antenna input impedance analogous to admittance. (ohms). -jvv/we, wave impedance for v-th waveguide TM mode. (Ohms) +jwu/Yv’ wave impedance for u-th waveguide TE mode. (ohms) amplitude of current at p-th subsection on i-th antenna. (amps) Equation (3.3). 2 2 a . (kc - k0) , (n,m)-th mode propagation constant in waveguide. (meters-1). complex reflection coefficient of waveguide termination at location Of load, 2 = —d . l ~2y d F e , complex reflection coefficient 1L of waveguide termination referred in phase to the origin of coordinates. Equation (2.7). Dirac's delta function. Equation (2.2). 348 e - jo/w, complex permittivity Of e o r dissipative medium. (farads/m). permittivity of free space (8.85 X 10—9 farads/m). relative dielectric constant. 1 for n = 0; 2 for n # O. Neumann's number. 2 kx - 7:, dummy variable. following Equation (11.28). permeability Of general medium. (henry/m). permeability Of free space (4w X 10-7 henry/m). general mode index used as shorthand nota- tion to imply bo£h_ n and m indices for TE and/or TM modes. conductivity Of medium. (mhos/m). infinitesimal gap width for delta gap approximation. Equation (2.2). radian frequency. (sec.-l). APPENDIX II EVALUATION OF DEFINITE INTEGRALS From Section 2.5.2 In Section 2.5.2, the development Of the longitudinal strip approximation requires evaluation of the following integral: 2322. f (z.\z')dz' F (Z_\Z.) = I V l V J 1 z,-2r 2 , 2 i a n (Zra) - (Z ‘ 21) where z is the center of the antenna on which the boundary con- J dition is satisfied, and 21 is the center Of the antenna carrying (II.1) 'I the source current. With the expression for fv(zj\z') given in Equation (2.13) the integral becomes _ 1 -‘Y‘Zj'Z" FV(zj'zi) " "(Flvrzv_1) {Ire / (zra)2 - (z' - ZA)2 dz' + -y(z +z') E + Flufe j /J(2ra)2 - (z' - 21) dz' + +y(z,+z') ‘ + I‘zvje J /‘/(2ra)2 - (z' - zi)2 dz' + +y\Z.-z" J 2‘ 2 . A + rlvFZVJe / (Zra) — (z - 2,) dz } (11.2) with the limits of integration the same as in Equation (11.1). To evaluate Equation (11.2), consider first the middle two integrals. They may be written as 349 l' .5. 350 +2r iyz z a ' e j WY “ll/Q2 ) -zi)2 dz' (11.3) 2. -2ra i which with the change of variables 2 - z = 2r COS 9 i a results in _z, n’ +yz ifiyr cos 9 Jje 1 e a de (11.4) Now, from a table of integrals (e.g., formula 3.915.4 in Gradshteyn [9 '1) one obtains: + e_gcos e 0“): d8 = n JO(B) since the zeroth order modified Bessel function (4%) is an even function. Therefore, the integral (11.3) is evaluated as 2 +2 ra iyz, i _y ‘ e J I e +2/1/(2ra?') -zi)2 dz' = z. '2ra i iy(z.+z_) = e J 1 n.Jo(2Yra) (11.5) Now, consider the first and fourth integrals in Equation -(II.2). They may be written as +2 . 1 Zi ra iN‘z.’Z'\ 2: I e J / (Zra)2 - (z' - 2.) dz' (11.6) -2 1 2i ra In order to handle the absolute value, three cases must be considered: (1) the source point antenna is to the left Of the field point antenna; i.e., i < j or zi < zj which implies z' < z . J' 351 (2) the source point antenna is to the right of the field point antenna; i.e., i > j or z, > zj which implies 1 I z > z . j (3) the source point and field point are on the same antenna; i.e., i = j or 21 = zj which implies two cases: I _2 I (a) z < zj for 21 ra S 2 < zi (b) 2' > z. for z.l< z' s z +'2r J 1 i a Case (1): i j ; z' > zj The integral in (11.6) becomes a ;N(Z.'Z') ' e J /\/(2ra)2 - (z' - z )2 dz' z.-2r 1 a z,+2r 1 With Equation (11.5), this evaluates as ;&z zi+2ra e 3 f eiyz lqfi2ra)2 - (2' - z.)2 62' = a z.-2r 1 :y(zj-zi) = e n.JO(2yra) for i > j . (11.8) 352 The results Of Case (1) and Case (2) may now be recombined by using the absolute value. Thus, Equation (11.6) becomes 2 +2r i a iy\Z.-z" j iy\z,-z,\ I e J /:‘/(2ra)2 - (z' - zi)2 dz' = e J 1 1'1 Jo(2yra) -2 z1 ra for 1 ¥ j (II.9) Case (3): i = j The integral given in (11.6) must be divided into two inter- vals as follows: z.+2r 1 a :y\2.-Z'\ 2 2* I e J /‘/(2r) ' (2' - 2.) dz' = a 1 z,-%r 1 a 2i 1 - ' 2 2 I eiy(ziz)/‘/(2r) -(z' -z.) dZ'+ _2 a 1 z, r 1 a Zi+2ra :‘Y(zi-Z') 2 2. + JP 9 / (2r ) - (z' - 2.) dz' (11.10) 2 a 1 1 Making the change of variables 21 - z' = u in the first integral, but 2' - 21 = u in the second gives O 21.a 2 2‘ 2' 4‘ eiYu/ (2r) - u du +‘f eiYu/ ,‘/(2r )2 - u du = a a r O a 2ra 2 2‘ = 2] eiW/‘fi2r3) - u du (11.11) 0 This is evaluated with the aid of formula 3.387.S in Gradshteyn [9 j which gives 2 2 - (23%,, - u ) id. = (n/2)(JO(O'B) + some» (11.12) OL—JQ 353 Since the zeroth order modified Bessel function (Jo) is even, and the zeroth order modified Struve function (1%) is odd, the integral \ in (11.10) becomes z.+2r 1 a z.-2r 1 a = nKJg(2yra) i_1%(2vra)) for i = j (11.13) Combining (11.9) and (11.13), the integral in (11.6) is completely evaluated as zi+2ra iyjz.-z'\ 2 2‘ j. e J [WVQZra) - (z' - 2.) dz' = 1 zi-Zra iY‘zj-zi\ n e JO(2~{ra) for i 7‘ j = (11.14) "(Jo(2Yra) -_l_-_ {0(2yra)) for i = j All the integrals in Equation (11.2) for Fv(zj‘zi) have now been evaluated. Substituting the results (11.5) and (11.14) back into the desired integral (11.2) gives: For i ¥ j, -1 \Z~-z.\ _._ ____1_.__. v J 1 FV(zj\zi) I‘lvr‘zv-l {e J0(2'Y\)ra) + w (Z.+2.) v j 1 + Flve JO(2yvra) + +y (z.+z.) v j 1 + rzve Jo(2yvra) + +v ‘z -z \ v j i + 1l'l\)r2\) e . 'flo(2Yvra)} or, upon factoring and identifying the function fV(zj\Zi) from Equation (2.13), 2r iv z.-z'l I a fi I e ‘ 1 //‘/(2ra)2 - (z' - zi)2 dz' = 2r eiYu/¢(2ra)2 - u2 du = O 354 Fv(zj\zi) = Jo(2yvra)fv(zj\zi) for 1 1‘ j (II.15a) For i = j, _ ____l - Fv(zi‘zi> _ F P -l {Jo(2Yvra) £b(2Yvra) + 1V 2v -2szi + Flue Jo(2yvra) + +2szi + FZve Jo(2yvra) + + FIVFZvCflo(2Yvra) + ib<2Yvra))} or, upon factoring as above Fv(zi‘zi) =‘Jo(2Yvra)fv(zi‘zi) - ib(2Yvra) for i = j (11.15b) This completes the evaluation of the integrals in Section 2.5.2. From Section 2.6 For the development of the coaxial gap solution in Section 2.6, there are eight integrals which must be evaluated. They are 3:1) = ggmw x Fig) fiii dS (II.16a) 3:23;) =£ (1)6 x Egyfifii dS (II.16b) 8 fig) =£ (2)6: x Eg)-Hv- dS (II.16c) g 332%) = £g(2)(9 x Fig) fiti dS (11.16d) 355 In order to evaluate these, the integrals must first be reduced to functional form. Since E; and 88 are referenced to an origin of coordinates at the center of the coaxial gap (y = 0), the magnetic field fiv(x,o,z) (referenced to a coordinate system centered at the waveguide corner) must be translated in the x coordinate by the amount a/2. Therefore, the magnetic field to be used is fiv(x,o,z) where, i = x + a/Z (11.17) so that x is centered at the coaxial gap and x is the usual coordinate for waveguide fields. The coaxial gap field, Eg’ is radially directed and may be broken into a and % components as Eg? = Eg((x/r)x + (z/r)%) where r = (x2 + 22)% . (11.18) Then, the cross-product with the unit vector y yields 9 x fig = Eg((z/r)3: - (x/r)"z) . Since Eg varies inversely with r, from Equation (2.35), this becomes V A r _ o z . x Considering first the integrals in Equation (II.16a), 356 5'11 e+ ~e+ -— = . - —-dS = 3(1) £g(1)(y X g) Hv V _ 2 e + = o + " 0 yz mi _ (z/r )hxv(x, )e dS b a (l) 8 since h: E 0, and, using Equations (2.5b) this is v V k e+ o 2 X p 2 . ... :Yz y‘- = :_(z/r )s1n(k x)e dx dz (11.20) (1) ‘Wfirbira) 22 ab kc %g(1) X ~ Using (11.17) for x and (11.18) for r in the above yields n/Z —-2 2 2 (-1) G? g :_sin(k x)e z/(x + 2 )dx dz (1) X (11.213) g for n even Fifi; = 2:1 _ (‘1) 2 a? g i_COS(kxX)e+Yz z/(x2 + zz)dx dz 8(1) (11.21b) for n odd Now, since the integrand for even numbered modes n, above, is an odd function about x = O, and Sg(l) is symmetric about x = 0, Equation (11.21a) evaluates to zero. Applying the Euler Identity to Equation (11.21b) gives n-1 — - —-— -jk x + yz +jk x + yz fi = :(-1) 2 ago/2) [e X + e x 1 X g X z/(x2 + zz)dx dz for n odd Because of the shape of the region Sg(l) (see Figure 11.1) it is best to transform to polar coordinates as shown. >4 1 - -r cos 9 dx dz = -r dr d9 N ll r sin 9 Thus, 357 x = - r cosG z = r sine j dxdz : - rdr d9 x = rcose z = -rsin9 dxdz : -rdrd8 FIGURE II. 1 Transformation to Polar Coordinates for Gap Regions 358 n-l r . '—- b n jr(k cos 9 + jy Sin 9) e+> 2 - ag- = 1(4) gem/2m y sin 91. x + 1) r o a jr(-k cos 9 i jy sin e) +'e x 3dr de for n odd (11.22) From the above, now consider the single integral: rb n jr(k cos 9 + jy sin e) 1' = I I sin e e x dr d9 (II-23) r o I * This is evaluated by considering two other integrals ; namely, rb n jr(k cos 9 + jy sin e) 1' = I I (kxcos e +'jy sin 9)e x dr d9 r o a = j j g(e)eJr g(e)dr de (11.24) rb n jr(k cos 9 + jy sin 9) 15 = I I (-kxsin e + jy cos e)e x dr d9 r o a = I I d 3392 ejr g(e)drde (11.25) These two integrals combine to yield the desired integral 1' as: U _ 2 2 . I I 1 — (1/(kx - v ))[JY 11 - kxlz] (11.26) The integrals Ii and 15 are now evaluated as follows: The author is indebted to Dr. J.S. Frame, Department of Electrical Engineering, Michigan State University, for his many suggestions incorporated in the evaluation of this integral. 359 b n . Iisifil. eJr g(e) dr d9 H II '19—: "1 de a rb . . rb sin r k =j(1/jr)(ejr g(n‘) - eJr g(0))dr' = ~2J‘ ——-;——)£ dr r r a a rk rk ax = -2[jb xsin u/u du + I sin u/u du] o o = -2[Si(rbkx) - Si(rakx)] 15 = -2 D81 (11.27) where DSi is used to denote the difference of sine integrals. g(e)ejr g(e)dr d9 Fig—7 H 09—311 1' = 1 a n jrbgcoslp + E p odd + 2kX DSi] for n odd (11.31) 362 V0 2 dc =[nub/ta) V Ze ab v where 07k” DBES(p) = gph-bg.) - (91,0135) 2 2 2 E = kx - 'Y DSi = Si(rbkx) - Si(rakx) Y = arctan(kx/jv) There are now six other integrals to be evaluated in (11.16) of a similar nature. The integrals in (11.16b) are 11+ _ . h-_+-_ .. :16) -g (1) (9 x fig) HV (x,0,z)dS (II.16b) :1 Using (11.19) for (§ x E?) and Hzi' from Equation (2.6), this becomes v — — [i'(Z/r2)h:v(§’0)e+vz + (X/r2)h:v(§,0)e+Yz It =___£__ 3I(l) 9m(rb/ra) g (1) de g and, with Equations (2.5a) e e k ; 3hi = '—“—(/l—— 4%4111 E3; g [1(z/r2)sin(k x)e yz + (1) fm(rb ta) 22v ab C 8(1) X k2 ‘— - -£-(x/r2)cos(k x)e+Yz]dS k v x x With a = x + a/2 and a’1 for the coefficient, this becomes, for n even arbi- = (-1)‘“’2 c)“ [: s1n(k x>e+Yzcos