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COAXIAL MICROWAVE SENSORS FOR LAYERED MEDIA BY SURESH KHANDIGE A D isse r t a t io n Su b m it te d i n P a r tia l F u lfillm en t o f t h e R e q u ir e m e n t s f o r t h e D egree o f D o c t o r o f P h il o so p h y E lec tr ica l E n g in e e r in g at U n iv e r sit y of W is c o n s in - M ilw aukee A u g u s t , 1996 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. UMI Number: 9637589 Copyright 1996 by Khandige, Suresh All rights reserved. UMI Microform 9637589 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. COAXIAL MICROWAVE SENSORS FOR LAYERED MEDIA by Suresh Khandige A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Electrical Engineering at University of Wisconsin - Milwaukee August, 1996 7 p i/? 6 Major Professor Graduate Scnooi Approval u>ate' n R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. COAXIAL MICROWAVE SENSORS FOR LAYERED MEDIA by Suresh Khandige The University of Wisconsin - Milwaukee, August 1996 Under the Supervision of Dr. Devendra Misra This dissertation presents a non-destructive technique for the characterization of two layered dielectric materials. An open-ended coaxial line is used as an electromagnetic sensor for this purpose. Two different methods of theoretical computation of aperture admittance are discussed. These theoretical results are compared with experimental data for many solid as well as liquid samples. This procedure is then extended to determine the electrical property of a dielectric layer. The possibility and limitations of determination of thickness of first medium and electrical property of second medium are also discussed. Date 111 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. © Copyright by Suresh Khandige, 1996 All Rights Reserved iv R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ACKNOWLEDGEMENTS Research and dissertation of doctoral level can only be completed by understanding the students’ strengths and weaknesses, by providing constant guidance and by encouraging positively at every difficult and disappointing moments of a doctoral student’s career. I am indebted to my major Prof. Devendra Misra for being such an important person in my life. Perhaps, often I was not a student who could stand up to his expectations, but he has always been a teacher of very high caliber. I am particularly grateful to the former Chairperson Prof. Joseph McPherson who has been a source of encouraging strokes through out my doctoral career. I learnt the basics of Electrical Engineering all over again by attending his undergraduate courses. By attending his undergraduate and graduate courses I learnt how to think and solve problems in both academic and real life. Also, I greatly benefited by learning his methods of teaching. H e placed major emphasis on solving the engineering problems based on concepts, rather than mechanically solving them. I felt deep sorrow at his demise. I wish to thank the members of my doctoral program committee, Professors Devendra Misra, Ali M. Reza, Chiu Tai Law, Len P. Levine and A1 Ghorbanpoor, for accepting to be in my committee. I acknowledge m y thanks to the present Chairperson Prof. David Yu and Prof. Ali M. Reza, who gave me emotional support during the difficult times of my degree. I would like to thank Prof. Chiu Tai Law for his thoughtful and thorough review of dissertation. I appreciate his help in correcting many of the mistakes in this dissertation. I would like to thank Mr. Paul Knauer, from whom I learnt the practical basics of electrical engineering. I worked under him as a teaching assistant for almost eight semester. I would like to thank Prof. George Pan and Prof. Devendra Misra for allowing me use their lab facilities to do my doctoral research. The v R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Milwaukee) using N etw ork Analyzer HP8752A and Prof. George Pan’s Lab (Room No. E280, EMS Building, University of Wisconsin, Milwaukee) using N etw ork Analyzer HP8510C. Further, I am sincerely grateful to Prof. Thomas Koruyu Ishii and Prof. James E. Richie from Marquette University for their interest and confidence in me and their indirect encouragement. I joined Marquette University to do my doctoral degree, I had to transfer due to financial considerations. I would like to thank my parents Mr. Khandige Krishna Bhat and Ms. Khandige Shakuntala Bhat. They have shown constant faith in my ability and encouraged me to get a doctoral degree. Also, I owe a great deal to my dear wife Sowmya Khandige for her help, patience and support. It has been very difficult on her to cope with emotional difficulties that we had to face during the final years of my doctoral program. My special thanks to my friends George Alexopoulos and his family, and Mathew P. Tharaniyil and his family for being helpful during this period. Finally, I would like to thank every one who directly or indirectly helped me during m y doctoral degree. vi R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. TABLE OF CONTENTS C h a p t e r 1 ................................................................................1 IN T R O D U C T IO N 1.1 Definition of Complex Permittivity or Complex Dielectric Constant........................................................................................................1 1.2 Brief History of Research on Nondestructive Measurement of Complex Permittivity of Materials.......................................... 1.2.1 Waveguides as Antennas............................................................... 3 1.2.2 Coaxial Lines as A ntennas............................................................5 1.2.3 Methods of Non-Destructive Measurement of Complex Permittivty.............................................................................................. 6 1.3 Coaxial Lines as Dielectric Sensors.......................................................6 1.3.1 Advantages and Disadvatages of Coaxial Line Dielectric Sensors.................................................................................................... 7 1.3.2 Applications of Coaxial Line Dielectric Sensors......................... 8 1.4 Motivation for Research on Coaxial Sensors...................................... 9 1.5 The organization of this Thesis.......................................................... 10 3 C h a p t e r 2 ..............................................................................12 REVIEW O F REASERCH PAPERS O N N O N DESTRUCTIVE MEASUREMENT O F COMPLEX PERMITTIVITY U SING O PEN -EN D ED COAQXIAL TRANSMISSION LINES 2.1 Introduction.................................................................................. vii R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 12 2.2 Electrical Characterization of Materials Using Coaxial Lines: Development in Early Years .................................................................14 2.2.1 Study of Reflection of An Open Ended Coaxial Line Sensor and their Application to Characterization of Materials.................... 16 2.2.2 Coaxial Probe Technique for Determining the Permittivity of Biological Tissues.................................................................................. 18 2.3 Electrical Characterization of Materials Using Coaxial Lines: Development in Recent Years................................................................. 20 2.3.1 Open Ended Coaxial Lines Terminated by Infinite Medium: Quasi Static Analysis............................................................................21 2.3.2 Open Ended Coaxial Lines Terminated by Infinite Medium: Improved Calibration Technique....................................................... 23 2.3.3 O pen Ended Coaxial Lines Terminating in a Conductor Backed Dielectric Layer: Startified Media, 2nd Medium Being a C onductor....................................................................................... 27 2.3.4 Open Ended Coaxial Lines Terminated by Two Layered Dielectric Media: Stratified Media, Quasi Static Analysis............28 2.3.5 Open Ended Coaxial Lines Terminated by Stratified Low Permittivity Dielectric Media: Spectral Domain Analysis...........30 C h a p t e r 3 ............................................................................. 34 COAXIAL LINE TERM INATED BY TW O LAYERED MEDIAFO R M U LA TIO N O F TH E PROBLEM 3.1 Introduction....................................................................................... 34 3.2 Waves Inside a Coaxial Line............................................................... 34 3.2.1 Characteristic Admittance of a Coaxial L ine.......................... 36 3.2.2 Static Electric Scalar Potential at the Aperture Cross Section of the Coaxial Line, <j>e(p )........................................................................ 38 3.2.3 Incident TEM Waves near the Aperture, inside the Coaxial Line........................................................................................................40 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 3.2.4 Scattered EM Waves near the Aperture, inside the Coaxial L ine...................................................................................................... 42 3.2.4.1 Electric Field at the Aperture, using Image M ethod........ 43 3.2.4.2 Fields inside the line, using Maxwell’s Equations.............45 3.2.4.3 Aperture Field Distribution - First Type of F orm ulation.....................................................................................56 3.2.4.4 Aperture Field Distribution - Second Type of Formulation......................................................................................60 3.2.5 Aperture Admittance and Reflection Coefficient of the Coaxial L ine.........................................................................................................62 3.3 Waves in Layered Media.................................................................... 64 3.3.1 Electromagnetic Fields in the Layered M edia........................... 65 3.3.1.1 Electric Field in the Layered Media....................................65 3.3.1.2 Electric and Magnetic Fields in the Layered Media, in Spectral D om ain..................................................... ....................... 67 3.3.1.3 Constants of Field Equations of equations (3.98) and (3.100)............................................................................................... 69 3.3.1.4 Total Electric and Magnetic Fields in layered m edia 70 3.3.2 A Variational Expression for Aperture Admittance.................72 3.3.3 An Integro-Differential Equation for Aperture Electric Field. 75 C h a p t e r 4...................................................................... 76 APERTURE ADM ITTA NCE BY VARIATIONAL PRINCIPLE 4.1 Introduction...................................................... 76 4.2 A Variational Expression for the Aperture Admittance of a Coaxial L ine..............................................................................................................80 4.2.1 The Poles of the Integrand of the Variational Expression.....83 4.2.1.1 Surface Wave Poles [s, > s2]................................................ 85 4.2.1.2 Guided Wave Poles [s, < s2] ............................................... 86 4.2.1.3 Three - dimensional Surface Plots O f the Integrand........ 86 ix R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.3 Numerical Solutions for the Aperture Admittance of a Coaxial L ine............................................................................................................. 94 4.3.1 Aperture Admittance of a Coaxial Line, Numerical Solution I ...................................................................................... 94 4.3.2 Aperture Admittance of a Coaxial Line, Numerical Solution II ................................................................................... 96 4.3.3 Aperture Admittance of a Coaxial Line, Numerical Solution El ......................................................... 99 4.3.3.1 Using Singularity Extraction Technique..........................101 4.3.3.2 Using Taylor’s Series Expansion......................................103 4.3.3.3 Using Approximate Substitutions For The Integrand, Near The Poles....................................................................... 104 4.3.4 Special Cases................................................................................104 4.3.4.1 Quasi Static Approximations........................................... 105 4.3.4.2 Conductor as Second M edium ......................................... 106 4.3.4.3 Coaxial Line Terminated By Infinite M edium ............... 107 4.4 Comparison of Theoretical and Experimental Aperture Admittance .................................................................................................................... 107 4.4.1 Discussion on Experimental Results ......... 109 4.4.1.1 Teflon/Air, 8.3 mm Coaxial Line, d =1.7 mm, / =0.5 to 3.5 G H z ....................................................................................... 109 4.4.1.2 Teflon/Air, 3.6 mm Coaxial Line, d = 1.7 to 13.7 mm, / = 1.5 G H z ..................................................................................109 4.4.1.3 Polyethylene/Air, 8.3 mm Coaxial Line, d =3.2 mm, / =0.5 to 3.5 G H z ....................................................................... 110 4.4.1.4 Free Space/Conductor, 8.3 mm Coaxial Line, d = 1 to 7 mm, / = 0.8 G H z ........................................................................ 110 4.4.1.5 Glycerol/Teflon, 8.3 mm Coaxial Line, d =0.5 to 12 mm, / =2 G H z ............................................................................ 110 4.4.1.6 Glycerol/Conductor, 8.3 mm Coaxial Line, d =0.5 to 10 mm, / =0.8 G H z ......................................................................... 110 4.5 Material Characteristics Using a Coaxial Sensor..............................I l l 4.5.1 Discussion on Experimental Results............... 112 x R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.5.1.1 Teflon/Air, 8.3 mm Coaxial Line, d =1.7 mm, / =0.5 to 3.5 G H z ......................................................... 112 4.5.1.2 Teflon/Air, 3.6 mm Coaxial Line, d = 1 to 13.7 mm, / = 1.5 G H z ...................................................................................... 112 4.5.1.3 Polyethylene/Air, 8.3 mm Coaxial Line, d =3.2 mm, / =0.5 to 3.5 G H z ................................................................... 112 4.5.1.4 Free Space/Conductor, 8.3 mm Coaxial Line, d =1 to 7 mm, / =0.4 G H z .........................................................................113 4.5.1.5 Glycerol/Teflon, 8.3 mm Coaxial Line, d =0.5 to 12 mm, / =2 G H z ............................................................................113 4.5.1.6 Glycerol/Conductor, 8.3 mm Coaxial Line, d =0.5 to 10 mm, / =0.8 G H z .........................................................................113 4.5.2 Three Percent [3%] Margin Test: [A Theoretical Study of Effect in Inversions Because of ±3% Difference Between Theoretical And Experimental Aperture Admittance].......................................114 4.6 Theoretical Study of Air-Dielectric and Dielectric-Air Termination of a Coaxial Sensor..................................................................................116 C h a p t e r 5........................................................................... 128 APERTURE ADM ITTANCE BY M ETH OD O F MOMENTS 5.1 Introduction....................................................................................... 128 5.2 Basics of Method of Moment Solutions...........................................130 5.3 Formulation of the Problem for MoM Solution .............................. 132 5.3.1 Description of A:cl(p,p') in Equation (5.10)............................ 133 5.3.2 Description of Z(p,p') in Equation (5.10)................................134 5.4 Method of Moments (MoM) Solution..............................................138 5.5 Comparison of Aperture Admittance Results Between MoM for Two Layered Media and O ther M ethods ........................................142 5.5.1.1 Air Termination, 3.6 mm Line, d —>oo, f = 1-40 G H z 142 xi R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 5.5.1.2 Methanol Termination, 3.6 mm Line, d —> oo, f = 1 - 4 0 G H z ..............................................................................................144 5.5.1.3 Water Termination, 3.6 mm Line, d —» o o , f = 1-40 G H z .............................................................. ...............................144 5.5.1.4 W ater/Teflon Termination, 3.6 mm Line, d —> oo, f = 1 40 G H z ........................................................................................ 145 5.6 Comparison Between Method O f Moment And Experimental Aperture Admittance.............................................................................. 154 5.6.1 Discussion on Experimental Results..................................... 157 5.6.1.1 Teflon/Air, 3.6 mm Line, d=6.5 mm, f = 5 - 40 G H z 157 5.6.1.2 Polyethylene/Air, 3.6 mm Coaxial Line, d =3.2 m m ,/ = 5 - 4 0 G H z .....................................................................157 5.6.1.3 Water/Teflon, 8.3 mm Coaxial Line, d = 1 -16 mm, / =0.5 G H z .........................................................................157 5.6.1.4 Air (Infinite Medium), 8.3 mm Coaxial Line,, / =0.5-3 G H z ......................................................................................... . 157 5.6.1.5 Teflon/Air, 8.3 mm Coaxial Line, d =3.2 mm, / =0.5-3 G H z ..............................................................................................158 5.6.1.6 Water/Teflon, 8.3 mm Coaxial Line, d = 1-13 mm, / =0.5 G H z ................................................................................158 C h a p t e r 6 ............................................................................165 C O N CLU SIO N B i b l i o g r a p h y ................................................................. 168 xii R ep ro d u ced with p erm ission o f th e copyright ow ner. 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A p p e n d i x A .....................................................................195 LISTING O F SYMBOLS USED IN THESIS Appendix B ................................................................ 208 LISTING O F FO R TR A N PROGRAM S IN VARIATIONAL FO R M U LA TIO N B.l Brief Explanation o Fortran Programs............................................ 208 B.2 Fortran Program Listing................................................................... 213 B.2.1 A D M T N C E .FO R .................................................................... 213 B.2.2 E PSLN -l.FO R .......................................................................... 213 B.2.3 EPSLN-2.FOR.......................................................................... 217 B.2.4 T H IC K -l.FO R .......................................................................... 221 B.2.5 C .F O R ........................................................................................226 B.2.6 H E A R T .FO R ........................................................................... 230 B.2.7 H E A R T Q .FO R ........................................................................ 233 B.2.8 IN M OD U LE.FO R...................................................................233 B.2.9 W RM ODULE.FOR.................................................................237 B.2.10 M AM ODULE.FOR...............................................................238 B.2.11 M IM ODULE.FOR.............................. 239 B.2.12 RTM O D U LE.FO R................................................................242 B.2.13 SEM ODULE.FOR.................................................................246 B.2.14 Q 1M O D U LE.FO R................................................................248 B.2.15 Q 2M O D U LE.FO R................................................................250 B.2.16 R1M ODULE.FOR ................................................................251 B.2.17 R2M ODULE.FOR.................................................................253 B.2.18 RDM ODULE.FOR .................................................... 255 R ep ro d u ced with p erm ission o f th e copyright ow ner. 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Appendix C ................................................................... 259 LISTING O F FO R TRA N PROGRAMS IN M E T H O D O F MOM ENTS FORM U LATION C .l Brief Explanation of Fortran Programs..........................................259 C.1.1 Computation of /c(p.y>P;c)............................... 260 C.1.2 Computation of Z^p^p^j........................................................ 260 C.1.3 Generation of Matrix A .......................................................... 261 C.2 Fortran Program Listing.................................................................. 263 C.2.1 MOM-Y.FOR...........................................................................263 C.2.2 M OM -Z.FOR...........................................................................269 C.2.3 M OM-KC.FOR........................................................................269 C.2.4 M OM -C.FOR...........................................................................270 C.2.5 M OM -M ICE.FOR...................................................................272 C.2.6 M O M -IN TG .FO R.................................................................. 277 C.2.7 M OM -INTS.FOR................................................................... 279 C.2.8 M OM -FUNC.FOR................................................................. 281 C.2.9 MOM -ZRT.FOR......................................................................282 C.2.10 M OM -KCRT.FOR............................................................... 286 C.2.11 M OM -HART.FOR............................................................... 287 A p p e n d i x D ..................................... 292 BASICS O F ELECTROM AGNETIC TH EO RY A N D TRANSM ISSION LINE THEORY D .l Introduction...................................................................................... 292 D.2 Basics of Electromagnetic Theory................................................... 292 D.2.1 Generalized Maxwell’s Equations in Differential F orm 295 D.2.2 Symmetry of Generalized Maxwell’s Equations and Duality Principle.............................................................................................. 296 xiv R ep ro d u ced with p erm ission o f th e copyright ow ner. 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D.2.3 Stoke’s Theorem & Divergence Theorem............................. 2.97 D.2.4 Generalized Maxwell’s Equations in Integral F orm ..............297 D.2.5 Fourier Transformer Equations..............................................298 D.2.6 Generalized Maxwell’s Equations in Time-Harmonic Case. 299 D.2.7 Equation of Continuity (Conservation of Charge)...............300 D.2.8 Constitutive Relations............................................................. 301 D.2.9 Complex Permittivity (Complex Dielectric Constant).........301 D.2.10 Boundary Conditions............................................................ 302 D.2.11 Lorentz Gauge..................................... 304 D.2.12 Coulomb G auge.................................................................... 305 D.2.13 Helmholtz’s Theorem ........................................................... 305 D.2 14 Magnetic Vector Potential, Electric Scalar Potential...........305 D.2.15 Electric Vector Potential, Magnetic Scalar Potential...........307 D.2.16 EM Fields in terms of Vector Potentials.............................. 309 D.2.17 Energy Relations between EM Fields (Differential-Integral Forms)..................................................................................................310 D.2.18 Energy Relations between EM Fields (Time Harmonic Form) and Poynting Rheorem..........................................................312 D.2.19 Uniqueness Theorem............................................................. 313 D.2. 20 Lorentz Reciprocity T heorem .............................................314 D.2. 21 Time Harmonic Wave Equations and Plane Wave Propagation......................................................................................... 315 D.2. 22 Time Harmonic Solution to the Wave Equations in a Waveguide........................................................................................... 318 D.2.22.1 TE or H Modes Electromagnetic Waves in a Waveguide.....................................................................................320 D.2.22.2 TM or E Modes Electromagnetic Waves in a Waveguide.................................................................. 324 D.3 Basics of Transmission Line T heory...............................................326 D.3.1 Differential Length of a Transmission Line........................... 329 D.3.2 Wave Equations for a Transmission Line.............................. 330 D.3.3 Wave Propagation in a Transmission Line............................ 332 D.3.4 Characteristic Impedance of a Transmision L ine..................336 D.3.5 Input Impedance of a Transmision Line................................ 338 xv R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. D.3.6 Reflection Coefficient of a Transmission Line, at Length z = d ......................................................................................................341 D.3.6.1 At the Input of Transmission Line (d=0)...................... 342 D.3.6.2 At the Load End of Transmission Line (d=l)................342 D.3.6.3 At any Point on the Transmission Line (z=d)..............343 D.3.6.4 Reflection Coefficient of a Lossless Transmission Line 344 D.3.7 VSWR of a Lossless Transmission Line.................................347 xvi R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. LIST O F FIGURES Figure 2 -1 Coaxial line sensor configuration involving a single semi infinite medium............................................................................14 Figure 2 - 2 Block Diagram of a semiautomated network Analyzer system......................................... 19 Figure 2 -3 Coaxial line sensor configuration involving a stratified semi infinite media, the second medium being a conductor............. 27 Figure 2 - 4 Coaxial line sensor configuration involving a two layered stratified semi-infinite media.......................................................29 Figure 2 - 5 Coaxial line sensor configuration involving a stratified semi infinite media, the third medium being a conductor............... 31 Figure 3 -1 Coaxial transmission lin e ............................................................37 Figure 3 - 2 Coaxial line, with aperture terminated by a perfect conductor.....................................................................................44 Figure 3 -3 Coaxial line of Figure 3-2, replaced by a single continuous line................................................................................................ 44 Figure 3 -4 Two layered media termination of a coaxial sensor................... 66 Figure 4 -1 Experimental set up for measurement of electrical property. ...77 Figure 4 - 2 The incident, reflected, guided (surface) and radiated electromagnetic waves in the coaxial sensor configuration......83 Figure 4 - 3 3-D picture of the intergand of equation (4.50), / = 1 GHz, d = 3 b , 8.3 mm coaxial line terminated by s, = 100,s 2 = 1...... 89 Figure 4 - 4 3-D picture of the intergand of equation (4.50), / = 1 GHz, d = 3 b , 8.3 mm coaxial line terminated by s, =100-yi,s2 = 1 ........................................................................ 89 Figure 4 - 5 3-D picture of the intergand of equation (4.50), / = 1 GHz, d = 3 b , 8.3 mm coaxial line terminated by s, = 100-y‘10,s2 = 1 ...................................................................... 90 Figure 4 - 6 3-D picture of the intergand of equation (4.50), / = 1 GHz, d - 3b, 8.3 mm coaxial line terminated by s, = l,e2 = 100...... 90 xvii R ep ro d u ced with p erm ission o f th e copyright ow ner. 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Figure 4 - 7 3-D picture of the intergand of equation (4.50), / = 1 GHz, d = 3 b , 8.3 mm coaxial line terminated by e, = 1 -/1 ,s 2 = 100........................................................................ 91 Figure 4 - 8 3-D picture of the intergand of equation (4.50), / = 1 GHz, d = 3b, 8.3 mm coaxial line terminated by s, = 1-/10, s 2 =100......................................................................91 Figure 4 - 9 3-D picture of the intergand of equation (4.50), / =40 GHz, d = 3 b , 8.3 mm coaxial line terminated by s, = 100, s 2 = 1.............................................................................. 92 Figure 4 -10 3-D picture of the intergand of equation (4.50), f =40 GHz, d = 36, 8.3 mm coaxial line terminated by s, = 100-/1,s 2 = 1 ....................................................................... 92 Figure 4- 1 1 3-D picture of the intergand of equation (4.50), / =40 GHz, d = 3 b , 8.3 mm coaxial line terminated by s, = l,s2 = 100............................................................................... 93 Figure 4 - 1 2 3-D picture of the intergand of equation (4.50),/ =40 GHz, d = 3 b , 8.3 mm coaxial line terminated by s, = l - / l , s 2 =100........................................................................ 93 Figure 4 - 13 Aperture Admittance of a Coaxial Line, Medium 1 = Teflon, Medium 2 = Air, 8.3mm Line, d = 1.7 m m ............ 118 Figure 4 - 1 4 Aperture Admittance of a Coaxial Line, Medium 1 = Teflon, Medium 2 = Air,8.3 mm Line, d = 1.5 G H z ........... 118 Figure 4 - 15 Aperture Admittance of a Coaxial Line, Medium 1 = Polyethylene, Medium 2 = Air, 8.3 mm Line, d = 3.2 mm 119 Figure 4 - 16 Aperture Admittance of a Coaxial Line, Medium 1 = Free Space (Air), Medium 2 = Conductor, 8.3 mm Line, / = 0.8 G H z......................................................................................119 Figure 4 -17 Aperture Admittance of a Coaxial Line, Medium 1 = Glycerol, Medium 2 = Teflon, 8.3 mm Line, / = 2 GHz.. 120 Figure 4 -18 Aperture Admittance of a Coaxial Line, Medium 1 = Glycerol, Medium 2 = Conductor, 8.3 mm Line, / = 0.8 G H z............................................................................................ 120 xviii R ep ro d u ced with p erm ission o f th e copyright ow ner. 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Figure 4 -19 Dielectric Constant of Medium 1, Medium 1 = Teflon, Medium 2 = Air, 8.3 mm Line, d = 1.7 m m ........................121 Figure 4 - 20 Dielectric Constant of Medium 2, Medium 1 = Teflon, Medium 2 = Air, 8.3 mm Line, d = 1.7 m m ........................121 Figure 4- 2 1 Thickness of Medium 2, Medium 1 = Teflon, Medium 2 = Air, 8.3 mm Line, d = 1.7 mm.......................................... 122 Figure 4 - 2 2 Dielectric Constant of Medium 1, Medium 1 = Teflon, Medium 2 = Air, 3.6 mm Line, / = 1.5 GFfz......................122 Figure 4 - 23 Dielectric Constant of Medium 1, Medium 1 = Polyethylene, Medium 2 = Air, 8.3 mm Line, d = 3.2 mm .......................................................................................123 Figure 4 - 24 Dielectric Constant of Medium 1, Medium 1 = Free Space (Air), Medium 2 = Conductor, 8.3 mm Line, / =0.8 G H z............................................................................................ 123 Figure 4 - 25 Thickness of Medium 1, Medium 1 = Free Space, Medium 2 = Conductor, 8.3 mm Line, / =0.8 G H z ........................124 Figure 4 - 2 6 Dielectric Constant of Medium 1, Medium 1 = Glycerol, Medium 2 = Teflon, 8.3 mm Line, / = 2 G H z ....... 124 Figure 4 - 27 Dielectric Constant of Medium 1, Medium 1 = Glycerol, Medium 2 = Conductor, 8.3 mm Line, / = 0.8 G H z 125 Figure 4 - 28 Three Percent Margin Test of Inversion of Dielectric Constant of Medium 1, Medium 1 = Teflon, Medium 2 = Air, 8.3 mm Line, / = 2 G H z ........................................... 125 Figure 4 - 29 Three Percent Margin Test of Inversion of Dielectric Constant of Medium 2, Medium 1= Teflon, Medium 2 = Air, 8.3mm Line, / = 2 G H z............................................. 126 Figure 4 - 3 0 Three Percent Margin Test of Inversion of Thickness of Medium 1, Medium 1= Teflon, Medium 2 =Air, 8.3mm Line, / = 2 G H z......................................... 126 Figure 4 - 31 Comparison of theoretical aperture admittance of a coaxial line sensor by variational expression for infinite medium Teflon termination, with 0 mm, 0.1 mm, 0.3 mm air gap, 3.6 mm line.............................................................................. 127 xix R ep ro d u ced with p erm ission o f th e copyright ow ner. 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Figure 4 32 Comparison of theoretical aperture admittance of a coaxial line sensor by variational expression for infinite medium Teflon termination, with 0 mm, 0.1 mm, 0.3 mm air gap, 8.3 mm line................................ 127 Figure 5 1 Comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory) for Air, 3.6 mm line........................ 147 Figure 5 ■2 at the aperture of a coaxial line sensor, by method of Figure 5 Figure 5 Figure 5 Figure 5 Figure 5 moments (two layered media) for Air, 3.6 mm line, / =40 G H z............................................................................................ 148 3 Angle of E p at the aperture of a coaxial line sensor, by method of moments (two layered media) for Air, 3.6 mm line, / =40 G H z ....................................................................... 148 4 Comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory) for methanol, 3.6 mm line............................................................. 149 at the aperture of a coaxial line sensor, by method of ■5 moments (two layered media) for methanol, 3.6 mm line, / =40 G H z ................................................................................150 6 Angle of E p at the aperture of a coaxial line sensor, by method of moments (two layered media) for methanol, 3.6 mm line, / = 4 0 G H z................................ 150 7 Comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory) for Water, 3.6 mm line.................................................................... 151 xx R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Figure 5 -8 at the aperture of a coaxial line sensor, by method of moments (two layered media) for Water, 3.6 mm line, / =40 G H z ................................................................................152 Figure 5 - 9 Angle of E p at the aperture of a coaxial line sensor, by method of moments (two layered media) for Water, 3.6 152 mm line, / = 4 0 G H z................................. Figure 5 -10 Comparison of theoretical aperture admittance (conductance) of a coaxial line sensor)by variational expression, method of moments (two layered media theory) for Water followed by Teflon, 8.3 mm line............. 153 Figure 5-11 Comparison of theoretical aperture admittance (susceptance) of a coaxial line sensor) by variational expression, method of moments (two layered media theory) for Water followed by Teflon, 8.3 mm line............. 153 Figure 5 -12 Aperture admittance of a 3.6 mm coaxial line sensor when terminated by Teflon (d = 6.5mm), followed by Air..............159 Figure 5 - 13 Aperture admittance of a 3.6 mm coaxial line sensor when terminated by Polyethylene (d =3.2 mm), followed by Air......................................... 160 Figure 5 -14 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by Water ( / =3.0 GHz),followed by Teflon 161 Figure 5 -15 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by Air (infinite medium), calibration of Network Analyzer was doneby time domain gating..............162 Figure 5 -16 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by Teflon (d =3.2 mm) followed by Air, calibration of Network Analyzer was done by time domain gating............................................................................ 163 Figure 5 -17 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by Water ( / =0.5 GHz) followed by Teflon, calibration of Network Analyzer was done by time domain gating.................. 164 Ep xxi R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. Figure B -1 Flowchart depicting the forward process of determining the aperture admittance of a coaxial line terminated by two layered media............................................... 210 Figure D -1 A uniform waveguide of arbitrary cross section, aligned toward z direction...................................................... .............. 319 Figure D - 2 A complete transmission line circuit....................................... 329 Figure D - 3 A unit differential length of a transmission line......................330 Figure D - 4 Characteristic impedance of a transmission line......................336 Figure D - 5 Input impedance of a transmission line .............................338 Figure D - 6 Voltage standing wave in a transmission line..........................347 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. LIST O F TABLES Table 2 - 1 Dispersion parameters used with equation (2-10) to calculate the dielectric properties of the fluids used in this stu d y ..........26 Table 5 -1 Summary of comparisons among four different techniques for measuring dielectric properties of materials, for a 3.6 mm coaxial line sensor, single medium termination...............146 Table A -1 Symbols used in Electromagnetic T heory................................ 195 Table A - 2 Symbols used in Transmission Line T h eo ry ............................197 Table A - 3 Symbols used in The Research of Coaxial Line Sensors.......... 200 Table A - 4 Definitions Commonly used in Electromagnetic Theory.......206 Table B -1 Main Fortran Programs for Chapter 4 ...................................... 212 Table B - 2 Subroutine Modules used in Programs of Table B -l................ 212 262 Table C -1 Main Fortran Programs for Chapter 5....... Table C - 2 Subroutine Modules used in Programs of Table C - l............... 262 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 1 Chapter 1 IN T R O D U C T IO N 1.1 D E FIN IT IO N O F COMPLEX PERMITTIVITY O R COMPLEX DIELECTRIC CONSTANT It is important to understand the meaning of complex dielectric constant in a unique manner in order to avoid confusion regarding the terminology. The most commonly used definition has been employed in this thesis, which is defined in this section. Using the following set of generalized Maxwell’s equations1for a time-harmonic case of ejal, V x E = -y'coB - J m (1. 1) V x H = ;<bD + J ( ( 1. 2) V •D = p t, ( 1. 3) V • B = Pm ( 1. 4) it is possible to define the complex permittivity of a material. The Maxwell’s equation that relate the field quantities E and H, can be expanded as follows: ! The sym bols used in this Chapter 1 and Chapter 2 are defined in Table A -l and A-2. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. V x H = (ycoe08r +ct)E - ./©£„ s r - j The term cos,,/ cos, (1.5) in equation (1.5) is known as complex relative perm ittivity or complex dielectric constant. It depends on frequency of the EM fields, and the medium in which EM fields exist. Therefore, the following definitions come into picture, as related to the complex permittivity of the material: s t. = s 0s z r - J- <7 = coe0y s 0( s ' - y ' s " ) s ' = s r ,s' CD£r tan(8) = —- = a ( 1. 6) (1.7) (1. 8) where, tan(5) is loss tangent. If tan(§) « 1, then the medium is a good dielectric and if tan(S)»l, then the medium is good conductor. Using equation (1.6), equation (1.5) can be written as (1.9) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 1.2 BRIEF HISTORY O F RESEARCH O N N O N D ESTRUCTIV E MEASUREMENT O F COMPLEX PERMITTIVITY O F MATERIALS The study of coaxial lines, or the research on coaxial lines, mainly involves the studying of the nature of electromagnetic fields inside the coaxial lines, under different circumstances such as when terminated by free space or multiple layers of dielectrics. Once the nature of electromagnetic fields inside the coaxial lines (or, in the media in which it radiates electromagnetic waves) is known, this knowledge of fields can be used for various applications such as coaxial line antennas, coaxial line dielectric sensors etc. The study of coaxial lines uses, the basics of electromagnetic theory and transmission line theory. 1.2.1 Waveguides as A ntennas Open-ended waveguides have been studied by many researchers, both for their use as antennas and for applications of non-destructive measurement of dielectric properties. One of the most relevant of open-ended waveguides can be found in reference [147]. This paper was published by Wu. in 1969. This paper analyzes the radiation from a waveguide through a die lectric slab by using the integral equation formalism. The author explains a very imaginative and intuitive understanding of the physical interpretations based on the procedure described in the paper, they are as follows [147]: • The spectral representation given in this paper indicates that the discontinuity at the antenna aperture excites in the semi-infinite dielectric slab region waves with transverse wave number a ranging from - oo to °o R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. • O ut of this spectral distribution, only constitutes the homogenous portion satisfying [or; > k1 (or propagating) part which contributes to the far-field radiation • The range k 2 < a 2 < k2e corresponds to the surface wave region • The remaining portion belongs to the spectrum of inhomogenous (or evanescent) waves which exponentially attenuated away from the plane of excitation. This interpretation holds good for all of the waveguide (or coaxial line) configurations that are terminated by a dielectric medium. The integral equations are solved by the generalized method of moments to obtain the aperture electric field, from which reflection coefficients, radiation patterns, and the degree of surface wave excitation are calculated. This paper describes that when thin dielectric slabs of high dielectric constant are placed in front of a waveguide, it causes the higher order modes to be strongly excited over the aperture. Therefore, it is important to include higher order modes in determination of aperture field distributions accurately. One of the useful information that emerges from this research is that only those waves which are multiply reflected between the antenna aperture and air-dielectric interface can significantly influence the aperture field distribution. Though, this paper analyzes waveguides for applications as antennas, the configuration of this study is very similar to the ones that are used for the measurement of complex permittivity. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. As far as the measurement of dielectric properties using waveguides are concerned the papers of interest can be found in references [9, 54]. 1.2.2 Coaxial Lines as Antennas Coaxial lines have been studied by many researchers for over fifty years. Initially the coaxial lines were studied as antennas (i.e., radiation of microwave signals into a semi infinite dielectric materials such as free space), later the applications of coaxial lines as antennas were extended to applications involving the dielectric characterization of materials. One of the earliest studies on coaxial lines was published in 1951 [82]. This paper by Levine and Papas formulates an integral equation for the aperture admittance by using the continuity of tangential magnetic field at the aperture plane of the coaxial line. The numerical computations of aperture admittance been done in this work after assuming that the dominant fields at the aperture of the coaxial line are those of principal modes. The reference [52] summarizes this work again. The study of coaxial lines described in this paper forms the basis for numerous researches that were published during the last 44 years, for applications such as coaxial lines as antennas and coaxial lines for determination material characterization. Another paper of interest, from the early years of coaxial line researches can be found in the reference [58]. This paper describes a straight forward method of finding aperture admittance of a coaxial line terminated by semi infinite free space that uses a modal expansion technique. The principle of duality simplifies the derivations of expressions of fields. This paper describes its application in calibrating near-zone field R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. strength instrumentation and in determining the response of an unsheathed coaxial cable connector that is mounted flush with the skin of missile and subjected to intense plane-wave electromagnetic field pulses. This paper was originally published in 1971, the formulations described in this paper has been used by great many researchers, specially in applications involving the characterization of coaxial line sensors. Another publication of relevance can be found in literature [16]. All of the above mentioned publications study coaxial lines for the purpose of antennas. 1.2.3 M ethods of Non-Destructive M easurement of Complex Perm ittivity There are numerous methods of measuring the dielectric properties of materials, non-destructively. These methods can be divided into two categories, based on the basic concepts they use - the free-space method and transmission line method. Open-ended coaxial lines for characterization of dielectric materials belong to the second of the above two categories, they are gaining importance in recent years. More detailed study of literature on open-ended coaxial line sensors can be found in Chapter 2. 1.3 COAXIAL LINES AS DIELECTRIC SENSORS Developments of low cost, miniature microwave solid state devices, in recent years, have attracted many microwave engineers to do research on coaxial line dielectric material sensors. In this section, the coaxial line sensors are compared with other sensor configurations. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 7 1.3.1 Advantages and Disadvantages of Coaxial Line Dielectric Sensors Coaxial line sensors have many advantages over other techniques of measurement of complex permittivity, some of these advantages are listed below [7,8,37, 50]: 1. Ability to make non-destructive measurements 2. Simple and convenient geometry 3. Small size ( potentially as small as 0.5 mm) 4. Broad frequency range of operation 5. Compatibility with time domain 6. Applicability to frequency domain and resonant measurement techniques 7. Cost effective and fast measurement 8. Ease of fabrication. However, there are some disadvantages for the coaxial sensors, which need to be mentioned here. They are: 1. Presence of higher order modes limit the operating frequency 2. The reflections at junctions (other than the aperture) limit the accuracy R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 3. The air-gap between the sensor and the sample reduces the accuracy. 1.3.2 Applications of Coaxial Line Dielectric Sensors Typical applications of nondestructive measurements at microwave frequencies are [37, 107] 1. Biomedical microwave diagnostic applications 2. Measurement of moisture within the walls of buildings (in particular historical ones) 3. Finding local inhomogeneities in microstrip substrates 4. Measurement of electrical properties of agricultural products for moisture measurement 5. Monitoring of industrial process where continuous dielectric data at microwave frequencies are required (for example, moisture contents of food products), etc. The applications mentioned above do not permit destruction of any part of the material to be tested. For these applications, coaxial line terminated by a dielectric material has been studied by many researchers. Further, in recent . days, the applications can be found in such diverse fields as Food Industries [60, 76, 109, 120, 125, 158], Agricultural Industries [4, 110, 120], Oil Industries [26, 43], Physics [27, 59, 73, 75, 88, 91], Chemical Engineering [5, 30, 114, 148], Oceanic Engineering [141], Civil Engineering [17, 136, 153] and variety of other fields [6, 20, 42,123, 124, 146, 157]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. As discussed above, one of the important application of coaxial lines with infinite dielectric terminations is as sensors in non-destructive measurement of complex permittivity of dielectrics at microwave frequencies. When used with a computer controlled network analyzer, this method offers convenience and provides good accuracy of measurements [7.8]. Knowledge of permittivity of tissues at radio and microwave frequencies is of importance for two main reasons. The accurate dielectric properties are needed for variety of biological/biomedical applications as the ones listed below [14, 77]: 1. Exploring interactions of electromagnetic fields with living matter (in order to evaluate potential hazard of RF radiation) 2. Electromagnetically induced hyperthermia for cancer treatment 3. Radiometry for cancer detection 4. Electromagnetic thawing of cryogenically preserved tissue and organs 5. Investigations of physiological processes such as organ activity or inactivity, induced physiological changes, diseases etc. 1.4 M O TIV A TIO N F O R RESEARCH O N COAXIAL SENSORS As described in previous sections, nondestructive measurement of electrical characteristics are preferred in wide ranging areas such as biomedical engineering, measurement of moisture, finding local inhomoginities in R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. various materials, and monitoring of industrial process where continuous dielectric data at the microwave frequencies are required. Coaxial lines have attracted great many researchers for this purpose because of their simplicity and ease with which one can obtain the complex permittivity. The arrangement of coaxial lines radiating into infinite medium has been thoroughly studied by many researchers, many papers and theses can be found in literature. These techniques have limitations in that they can not characterize the material characteristics of stratified media, which is an important factor in many applications. When the author decided to take up this study, the works that were done so far had one or other limitations. The studies previously done on material characterization of stratified media were based on certain conditions such as on quasi static basis [89, 94, 103], or a thin layer of dielectric followed by conductor [31, 46]. Therefore, the author found an opportunity to study coaxial lines terminated by two layered media in detail. The goal of this research was to come up with a technique that would characterize the 2 layered media, and further be able to find the material thickness of first medium. 1.5 TH E O R G A N IZA TIO N O F THIS THESIS As described in previous sections, the research on coaxial lines have been going on since past 50 years. In Chapter 2 of this thesis, the study of coaxial line sensors done by other researchers, their advantages and limitations are discussed in detail. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Chapter 3 explains the basic formulation of problem that was chosen for the dissertation. The equations required for this purpose are derived from the basics of electromagnetic theory and transmission lines. Specifically, a formulation based on variational technique is derived. This formulation has been studied by many researchers before, however, these solutions have had one or other constraint. Therefore, it was the objective of this doctoral research to solve this formulation without any restrictions imposed on it. Chapter 4 examines many possible solutions of this problem. The important difficulty in solving this problem is the singularities that this formulation poses because of the poles that are caused by either surface waves or guided waves that occur in the layered media. The solution has been successfully evaluated with many experimental verifications. Also, in Chapter 3 an alternate formulation can be found. This formulation allows one to find the aperture admittance via numerical techniques. Chapter 5 explains solution by Method of Moments, this method is slow in terms of computer time required. However, it has no assumptions on the nature of electric field distribution, therefore accurate in its evaluation. Appendix A lists all the important symbols that is used in this research. The FORTRAN program listing can be found in Appendix B. Appendix B also explains the development of this programming. Appendix C explains the FORTRAN programming used in Method of Moment solutions. Also, the listing of this programming is given in Appendix C. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 12 Chapter 2 REVIEW O F RESEA RCH PAPERS O N N O N -D ESTRU CTIV E M EA SUREM ENT O F CO M PLEX PERM ITTIVITY U SIN G OPENE N D E D CO A X IA L TRA N SM ISSIO N LINES 2.1 IN T R O D U C T IO N Open-ended coaxial line dielectric constant measurement method forms an excellent technique for applications in which speed, accuracy and the ability to measure non-destructively are important. Therefore, coaxial transmission line sensors have gained importance in research in the past 20 years. In this technique of measurement of complex permittivity of materials, an open ended transmission line such as a waveguide or a coaxial line is located next to the material to be tested, or in actual contact with it. The electromagnetic field distribution in the vicinity of the aperture is then computed by matching the boundary conditions for field components at the aperture, on the basis of electromagnetic and transmission line theories. Researchers, commonly use the following developmental procedure to come up with a practical technique for non-destructive measurement (using open-ended coaxial lines, at microwave frequencies): 1. Develop a coaxial sensor configuration based on past experience (imagination would be of immense help in coming up with a new sensor configuration). For example, in order to find the electrical property of any substance one should develop a coaxial line R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. configuration such that there is a precise mechanism for measuring the interaction between electromagnetic waves and the substance 2. Develop a theoretical model for the coaxial line sensor configuration 3. Use the theoretical relationship of step 2 and the practical measurements of the reflection coefficients of the test samples, to find the material characteristics. Normally, the computations mentioned in step 3 above involve lengthy and repeated calculations. Therefore, the complete open ended coaxial line measurement set up involve a transmission line sensor, automatic network analyzer and computer. Further, each practical technique developed will have accuracy, frequency limitations based on the model used and speed limitations based on the time required to compute material characteristics. Configurations involving rectangular waveguide probes were used by Decreton et al. [35-37] circular waveguides (azimuthally independent modes in) by Gex-Fabry et al. [55]. Waveguides present frequency limitations at the lower end of the frequency spectrum the operating band is limited by the dominant mode cutoff, and on the upper end the frequency range is limited by the onset of higher order modes [107]. Coaxial line sensors, on the other hand, provide broad operating frequency range for measurements. There are limitations to coaxial line sensors, these were discussed before in Chapter 1, section 1.3.1. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 14 A brief survey of the study of coaxial lines previously done by some selected researchers is presented in this Chapter. The review of literature in this Chapter is limited to forming background for this research, therefore this review may not include all the related papers. The material presented in this Chapter is divided into three sections, section 2.2 being on developments of coaxial sensors in early years such as a coaxial line sensor with semi-infinite medium termination. The section 2.3 explains in brief, the research of recent years such as the coaxial sensors followed by multiple stratified layers. 2.2 ELECTRICAL CHARACTERIZATION O F MATERIALS USING COAXIAL LINES: DEVELOPMENTS IN EARLY YEARS r. . . . i 2b 2a a 4..........I Coaxial Guide Figure 2 - 1 Coaxial line sensor configuration involving a single semi infinite medium In this section, few relevant studies of open-ended coaxial line sensors (refer to Figure 2-1) are discussed. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. As far as the principles behind the use of open ended coaxial sensors for the purpose of measurement of complex permittivity are concerned, there are two basic approaches [56]: 1. Methods that have used a lumped equivalent circuit description of sensor’s fringing fields, and 2. Methods that attempt a rigorous solution of the electromagnetic field equations appropriate for a coaxial line exposed to a dielectric material. Examples for the methods of step 1 can be found in a paper by Stuchly et al, in reference [132]. They provide a critical review and analysis of coaxial line structures using lumped equivalent circuit descriptions as given in step 1, in the frequency range of 100 MHz to few GHz. This method of using lumped equivalent circuit has the advantage of yielding simple closed-form equations based on a model of the discontinuity at the termination of the coaxial line [56]. Grant et al [56] examine, in another study, the theoretical basis of the coaxial line sensor technique. They developed improved method of obtaining complex permittivity with calculable uncertainties. They have considered both equivalent circuit model as well as point matching theory, experimental values of reflection coefficients and the inverse solution to realize RF and microwave complex permittivity measurements of various materials. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 16 2.2.1 Study of Reflection of An O pen Ended Coaxial Line Sensor and their Application to Characterization of Materials Mosig et al. describes a permittivity measurement technique using an open ended coaxial line in paper [107], taking into account both the effects of radiation and higher order modes. This paper was published in 1981. By using the boundary condition that the azimuthal component of magnetic field must be continuous across the boundary, the following relationship is derived: 2X t , = i (2 - 1) where. (2 - 2) T = Yo \ £cY In COS<t>C%/p' (2-3) /„(p) in equation (2-2) are the radial functions for the transverse electric field in the coaxial line. When n is greater than 1, the radial functions imply reflections because of higher order modes at the aperture. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. In equation (2-1), the term R„ means the generalized reflection coefficient of n ,h mode inside the coaxial line sensor. Other symbols of equations (2-1) through (2-3) can be obtained by referring to [107]. The integral expressions of equation (2-3) can only be solved using numerical integration methods. The pole at r = 0 must be extracted before numerical integrations can be carried out. The authors use a point matching approach to satisfy the continuity of the magnetic field on the aperture, having radii p, N circles within = 1 , 2 The values of complex reflection coefficients R, are then obtained for the modes used in the point matching calculation. For the reflection measurement method considered in paper, only the TEM reflection factor R 0 is of interest. Further, for the cases where the approximation that modulus of the quantity cobyje^, /c tends toward infinity can be used, an asymptotic approximation is derived. In this paper, the authors validate the above mentioned calculation technique by comparing the reflection coefficients obtained from equation (2-1) for N = 5 with the values found by Tanabe and Jones [138] and Marcuvitz [85]. The derivation of this asymptotic equations for the reflection coefficient, according to Mosig et al., allows calculations of reflection coefficients of material with very high permittivities, to be computed very fast. Mosig et al. conclude in this study that the presence of higher order evanescent modes has a significant influence on the reflection coefficient. Further, according to the authors, the measurements of reflection R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. coefficients can be carried out by standard reflection techniques such as reflectometer, slotted line, TDR [35-37]. 2.2.2 Coaxial Probe Technique for Determ ining the Perm ittivity of Biological Tissues Ability to measure the permittivity of biological substances - nondestructively is an extremely important factor in many biomedical applications. Therefore, many researchers have studied coaxial line sensors for applications in biomedical engineering. Among the advantages of this coaxial transmission line technique has over many other techniques, in biomedical applications, are [14]: 1. An ability to perform living (in vivo ) tissue dielectric measurement 2. Elimination of the need for tedious sample preparation 3. The ability to obtain continuous dielectric property data from below 100 M Hz to above several tens of GHz, and 4. The ability to process data on a real time basis. Burdett et al. [14] explain in one such study (published in 1980), a practical measurement technique that is suitable for biomedical applications. The authors fabricated and experimentally evaluated a number of infinitesimal monopole measurement probes, both with and without small circular ground planes. According to them, these probes will permit the accurate measurement of in vivo properties of samples of volume as small as 0.13 cm3 over a frequency range of 0.1 GHz to over 10 GHz. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 19 The experimental measurement were conducted by authors on many types of tissues, in order to validate this technique. These samples include canine muscle, canine kidney, canine fat, rat muscle, rat blood, and rat brain. The effects of temperature and drugs on the above mentioned samples were also tested. Similarly, in another paper (published in 1982), Athey et al. [7, 8] describe a measurement system consisting of a sensor and a computer controlled network analyzer, as shown in Figure 2-2. j H P 2100S I M inicom puter Directional C oupler Source H P 8620C Test Set H P 8745A or H P 8743A Frequency C ounter H P 5340A Test Device Digital V oltm eter j D ata Precision ^ 3400 j A /D Converter N etw o rk A nalyzer H P 8410A 8411A 8412A M agnitude Phase Figure 2 - 2 Block diagram of semiautomated network analyzer system The sensor translates changes in the permittivity of a test sample into changes in the reflection coefficient of the sensor. These impedances are measured by the network analyzer. An open ended coaxial line placed in contact with the test sample is used as a sensor. The automated network analyzer system is based on a Hewlett Packard HP8410A network analyzer and covers a frequency range of 0.1 G H z to 12.4 GHz. The network R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. analyzer system is controlled by HP2100 microcomputer, and phase information is taken from the HP8412A polar display unit. The system can average results over 100-200 repetitive measurements. The coaxial line sensor is represented by a simple equivalent circuit consisting of one capacitor. This convenient approximation is valid only if the relative dimensions of the line are such that changes of the capacitance with frequency can be neglected. The equivalent circuit used in computation of permittivity of biological substances are best suited for frequency ranging from 50 M Hz to 1 GHz. Athey et al. report that they conducted permittivity measurement of samples such as distilled water, saline solutions, and methanol over frequency range of 0.1 G Hz to 1 GHz. O n the biomedical front, they performed in vivo measurements on two types of muscle, on kidney, liver, and spleen of a cat. Athey et al. report that they obtained highly reproducible and accurate results, and that the method is very fast, typical time to obtain data at one frequency being 10 seconds. 2.3 ELECTRICAL CHA RA CTERIZA TIO N O F MATERIALS U SING COAXIAL LINES: DEVELOPMENTS IN RECEN T YEARS In recent days the research on characterization of materials using coaxial lines has gained prominence among researchers because of its simplicity, small size. Also, availability of fast personal computers have made researchers pursue more computationally demanding techniques, using better and more accurate models. Misra et al. have been prominent R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. researchers, among others, in recent years. Since their works form the basis for the research of this thesis, some of their works (that are of interest to this thesis) are discussed in detail, here. The works of Misra et al. can be found in references [2, 18, 19, 31, 46-48, 89. 92-103, 115-119, 155, 156 and 158]. Also, some of the recent developments in coaxial line measurement techniques considering stratified media and advanced computational techniques, by other researcher, are briefly described in this section. 2.3.1 O pen Ended Coaxial Lines Terminated by Infinite Medium: Quasi Static Analysis A paper published in 1987 by Misra [94] presents a simple method for calculating the capacitance of an open ended coaxial line. The technique is developed based on the quasi static modeling of the configuration shown in Figure 2-1. The development of the relationship between measured reflection coefficient to the permittivity of the material can be explained as follows [94]: 1. Using the theoretical methods explained by Galejas in [52], derive the expressions for the angular component ((J) component)of magnetic field both inside coaxial line (near the vicinity of the aperture) and in the media whose permittivity need to be measured 2. Equate them at the aperture in accordance with the boundary condition that the tangential components of magnetic fields must be continuous across the boundary R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 3. Assuming that the field distribution at the aperture is that of dominant TEM mode, form a stationary formulation for the aperture admittance. By following the above procedure, Misra derives the expression for normalized aperture admittance as follows: r‘ = 1 L(b / a) / | | C0S* 'dpdp' P - 4) The symbols defined in equation (2-4) are available in [94] and hence omitted for the sake of brevity. When the coaxial line opening is electrically small, equation (2-4) can be approximated to the first three terms of the series expansion of the exponential term in (2-4), and rearranged as follows: y _ A - 7'2cos [ln(6 / a)] (2-5) where, A= Jff^ r^ y p rfp ' a a 0 (2 - 6) b b n / 3 = j j §rcos$'d$'dpdp' (2 - 7) a a 0 Since /, and / 3 are independent of the material whose characteristic need to be found, and only dependent of the nature and size of the coaxial line, they can be pre-computed and stored. In evaluation of equation (2-1), the source R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. singularity needs to be taken care of. Once the capacitance in Figure 2-1 is measured, the permittivity can be found using equation (2-5). Misra compares the capacitance obtained from the equation (2-5) with that given in reference [77] in order to validate the results. The results show good agreement. However, this quasi static model will work with coaxial lines' with electrically small apertures. This, Misra observes by comparing the results with that given by MoM in Mosig et al. [107]. 2.3.2 O pen Ended Coaxial Lines Terminated by Infinite Medium: Improved Calibration Technique In another paper, Misra et al. [93] evaluate an approximate model due originally to Marcuvitz, on the basis of measured probe impedances from 1 to 18 G H z with samples consisting of water, methanol, and dioxane-water mixtures. In order to calibrate Automatic Network Analyzer and measure dielectric properties, a good modeling of the probe is important. In the limit of lower frequencies, the admittance YL of the coaxial sensor can be approximated by YL =C0 +j<t>C,(e'-Je") (2-8) where, C0 and Cf are constants that depend on the dimensions of the probe, and (s' - js" ) is the complex permittivity of the sample. This model fails at higher frequencies because of presence of higher order modes. The references [82, 85] expressed the admittance of a coaxial line configuration shown in Figure 2-1 as an integral over its aperture; this approximation was again rederived by Misra [94]. Marcuvitz approximation given in reference R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 24 [85] can be expanded in a series that is convenient for numerical solution, and in this form has been used in the experimental studies employing probe technique [107]. Alternatively, the probe admittance can be numerically calculated using mode-matching technique [107]. Mode matching methods are more accurate, but computationally very time consuming. Another equivalent form of equation (2-4) can be written as follows (refer to Misra [93] or Marcuvitz [85])l: Y, = G + jB /g G= B= (2-9) rJ 2 j l ~ ^ [ Jo (ko 4 ^b^ Q ) - J 0(koyl^asm Q )j dQ In(b/t Ypyf^m f 7t In ( b l a ) y f f Y j k0-Jem(a2 +b2 - labcosQ dQ - Si{lk0 sin(0 / 2)) - Si{lkQ-J%,b sin(0 / 2)) where, Si is the sine integral, other symbols are as defined in [93]. The measurements done using equations (2-4) or (2-9), are approximate in that the following assumptions are used: 1. The equations derived with the assumption that probes have infinite conductive flange, as shown in Figure 2-1 2. The equations derived with the assumption that the electric field distribution at the aperture of the probe is that of TEM mode 1 T he equation (2-9) is written in terms o f conductance and susceptance (i.e., G + jB) w ith the assumption that s m is real. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. In this study, they evaluated equations (2-4) and (2-9) numerically, using either series expansion or by numerical integration. Experimentally, Misra et al. measured the reflection coefficients of the probes using H P 8510 ANA. They calibrated ANA by using factory standard loads such as open circuit, short circuit, and 50 Q sliding and fixed loads, at the end of the precision test cable. They took care not to flex the cable after calibration, until the measurements are completed. The reference plane of the measurement was defined by shorting the end of the probe by an aluminum foil and adjusting the electrical delay of the ANA until constant 180° phase angle was observed. The measured reflection coefficients were transformed into the time domain, the connector reflection was electronically gated out, and the data was then transformed back into the frequency domain. They conducted experiments in the frequency range 0.1 to 18 G Hz on several liquids: water, methanol, and various water-dioxane mixtures. The authors summarized the results of measured permittivity using the following empirical formula + il + s;(ycox) . V with parameter values summarized in Table 2-1. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (2 ' 10> 26 Table 2 - 1 Dispersion parameters used with equation 2-10 to calculate the dielectric properties of the fluids used in this study [93] Substance e, t x 10" (Sec) a Water 5.2 78.5 8.3 0 80% Water, 20% Dioxane 3.75 61.9 1.1 0 60% Water, 40% Dioxane 3.35 44.5 1.5 0.1 40% Water, 60% Dioxane 3.1 27.2 1.7 0.1 20% Water, 80% Dioxane 3.75 11.9 2.3 0.1 Dioxane 2.2 2.2 - - Methanol 5.6 32.6 4.8 0 Further, they use the well known calibration procedure using S-Parameters for two port networks (refer to Page 10, equation 5 in [93]) to derive the following alternate formulation as given below: fc -ip fe -r ,) (iy -r .H n -r ,) ( V k ) ( i; - i;) ( r„ - r ;) ( r j- r ,) where, Y and r means aperture admittance and reflection coefficient, respectively; the subscripts 1,2 and 3 represent three standard materials, and the subscript M represents test substance. The authors experimentally verify R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. the calibration technique given in equation (2-11) and conclude that the errors appear to be small when the electrical properties of the standard materials are close to that of the test substance. 2.3.3 O pen Ended Coaxial Lines Term inating in a C onductor Backed Dielectric Layer: Stratified Media, 2nd M edium Being a Conductor Another paper by Fan and Misra [46] considers a study of coaxial probes in order to determine the dielectric properties of a thin layered substance, followed by a conductor. The coaxial line arrangement is as shown in Figure 2-3. Metal Flange r. . . . i 2b 2a x 4-.... I Coaxial Guide Conductoi Figure 2 - 3 Coaxial line sensor configuration involving a stratified semi infinite media, the second medium being a conductor The study of reference [46] can be considered as beginning of study of coaxial probes measuring the electrical properties of stratified media. After R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. matching the boundary conditions at the aperture, Misra et al. derive a variational stationary formula for the aperture admittance as follows2 ^ 2 8 CO b h - - -ikR J + j n[In(6/ I h f CZ t yJ f ^ (2 - 12) „-jk\R2+4n2d2 4g (q a^ a ^0 a £ ^3? -R +4/7 v7 / ? ' r c o s <^ w pw p where, the first term in equation (2-12) is same as (2-4). O ther symbols This means that the second term in equation (2-12) corresponds to the finite thickness of the material. When d > 2b , the second term reduces to cosmfot (b2 - a 2) e-j2nM 8[ln(Z? / <ar)]~ d 2 ^ n2 . (“ ) Misra et al. calculate capacitance and aperture admittance at various frequencies and thickensses for various dielectrics. Further, equations (2-12) and (2-13) show that when d>2b, the equation (2-12) gets close to infinitely thick dielectric medium. 2.3.4 O pen Ended Coaxial Lines Terminated by Two Layered Dielectric Media: Stratified Media, Quasi Static Analysis In another paper McKelvey and Misra [103] investigate a coaxial probe for stratified media. The probe arrangement is as shown in Figure 2-4. Misra et al. develop a quasi static formulation for the aperture capacitance of a coaxial probe arrangement shown in Figure 2-4. Following a spectral 2 T he aperture admittance o f the coaxial line is analyzed in spectral domain [104] in order to determine the fields in the stratified media, further references can be found in publications related to coaxial lines as antennas, such as [52, 82]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. domain analysis for this problem, an expression for the static capacitance at the annular aperture may be obtained as follows [103]: C= 27is, xre, + s. tanh(?ui) [ln(6 / a )]2 o £i + s2 tanh(/W) o \ j x{kp)dp dk (2 - 14) Metal Flange Coaxial Guide Material Second Material Figure 2 - 4 Coaxial line sensor configuration involving a two layered stratified semi-infinite media The assumptions in derivation of equation (2-14) is same as those in Misra et al. papers described before, that the coaxial line is excited in its principal mode, which exhibits no angular variations and has only radial component of the electric field and angular component of the magnetic field. Alternatively, the aperture capacitance can be derived in a form similar to that in equation (2-12), as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 30 C= 2s, b h 7t « 2 J J / — cos<t) 'd§ 'dp' dp ( 2 . 15) + r[\n(b/a)] / S| St;Ui 822+£,J ~ L | f f f /■■■, 1 = f =r H i ' J r 2 +4n d C0S<l) 'gftt>'*/ P ' ^ P The first part of equation (2-16) can be considered as the static capacitance, seen by the probe, while second term accounts for the finite thickness of the first of the layered media. Misra et al. claim in this paper that this formulation is useful up to 2 GHz, in measuring the electrical characteristics of the terminating media. However, there is limitations in measuring the thickness of the dielectric layer as well as the permittivity of a medium behind the layer. The dielectric layer should be only a small fraction of the probe diameter and the permittivity contrast between the two media should below. 2.3.5 O pen Ended Coaxial Lines Terminated by Stratified Low Perm ittivity Dielectric Media: Spectral-Domain Analysis In literature, one can find variety of techniques developed to measure electrical characteristics of different types of materials. Most of these probes are suited to measure the permittivities of substances with high permittivities, for example, the permittivity of an organic tissue. A paper by De Langhe et al. [34] describes a probing technique that is useful to measure the electrical characteristics of thin sheet materials of low dielectric constants. They use the formulation given by Levine and papas [82], for the basic probe configuration given in Figure 2-1, as reference for their work. The formulation given by Levine and Papas is as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. y - me'" - 0 16) “» p r -fr f- The authors demonstrate using the Levine and Papas formulation that probes with small dimensions are not suitable for measuring substances with low dielectric. Therefore, the authors developed a new type of probe with larger dimensions. Coaxial Guide First Material Second Material Figure 2 - 5 Coaxial line sensor configuration involving a stratified semi infinite media, the third medium being a conductor Further, using the spectral domain approach they develop a method to generalize an expression for aperture admittance of multilayered media, made up of n layers. They derive an expression for two layered dielectric media, followed by a conductor, as shown in Figure 2-5. The aperture admittance of the configuration shown in Figure 2-5 is given as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. y _ . COS, „ J Cy[zc ln(6 / a) j 08, y ■J A, r,A. • ■ — j h + jyl J “i2 \t- i V / [&/0(M)tf, (^&) - aJa{^a)R\ (Qa) ]^ where, 1+ coth r,G?, co th r 2<i2 A*U) = — ^ -----f - -----------1 cothr.J, + ---cothr,^, r 2s, (2-18) O ther symbols can be found in the paper. Since, there is no data available for multilayered media, authors use the data obtained from Mosig et al. [107] for infinitely thick material. The data obtained from Mosig validates De Langhe et al. approach. The authors performed experiments using HP8510C network analyzer. They used open circuit, short circuit and Teflon as standards for calibration, assuming a zero loss factor for Teflon. They compare in this paper, the results gotten by equation (2-17) for 1 cm thick Teflon layer to that from Levine and Papas formulation (equation (2-16)), in the frequency range 0.5 GHz to 1 GHz. The conclusion is that while the Levine and Papas formulation deviates by 15% from that of known value of permittivity of 2.1, the authors model deviates only by 5% to 8%. Similarly, they apply this approach of measurement on many different materials. The authors attribute whatever discrepancies found in measurement to calibration (lack R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. of standard permittivities available in literature) and because of unavoidable air gap that comes in between the probe and the test material. They study experimentally, using the model given in equation (2-17), the effect of air gap by purposefully introducing an air gap of 0.5 mm in between Teflon layer of 1 cm thickness. They found the resulting permittivity of Teflon in the frequency range of 300 MHz to 2 G Hz as 1.8, instead of 2.1. This means, an error of up to 20% can be introduced by neglecting air gap, in cases of low permittivity materials. In another paper by Ching-Leih et al. [24], the authors present an accurate full-wave theory for the flanged open-ended coaxial probe for nondestructive measurement of the EM properties of materials, in the range of 0.3 to 4 GHz. The technique described in this paper allows simultaneous evaluation of both permittivity and permeability of materials. The authors derive an integral equation for the unknown aperture electric field (EFIE), considering N stratified layers terminating the coaxial line. This equation is then solved using Method of Moments and the electric field at the aperture is then obtained. The coaxial eigenfunctions are used as the set oi basis functions for the unknown aperture electric field, and Galerkin’s technique is used to convert the EFIE into a set of simultaneous equations, which can then be solved by a typical numerical subroutine. The EM properties can be determined from the measured input impedance or reflection coefficient of the coaxial probe held against the material layer. However, two input impedances of sample materials are needed in order to determine both permittivity and permeability. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 34 C h a p ter 3 COAXIAL LINE TERMINATED BY TWO LAYERED MEDIAFORMULATION OF THE PROBLEM 3.1 INTRODUCTION The problem for research was that of a coaxial line terminated by two layered media, in terms of its applications in measurements of dielectric properties of the terminating material. This chapter explains the methodical development that was taken in order to arrive at the problem for dissertation, together with complete material background. A list of symbols used in this research can be found in Appendix A. The topics covered in this chapter form the background for the research, they include Waves inside a coaxial line - section 3.2 and Waves in layered media - section 3.3. 3.2 WAVES INSIDE A COAXIAL LINE In order to find the reflection coefficient and the aperture admittance of a coaxial line sensor simple voltage and current wave equations are not enough. The alternative is to study the propagation of electric and magnetic field waves using EM theory. In this section, EM theory is used to analyze coaxial lines, while some of the aspects of transmission line theory is utilized in some parts of the analysis. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. The first step in developing a background for the study of coaxial lines terminated by layered media would be to understand the nature of fields in the coaxial line, near the aperture. In terms of EM theory, the electromagnetic waves propagating in a coaxial line may be split into different modes. Any mode of EM wave transmission along a coaxial line, except that of TEM mode, requires a minimum separation of half wavelength between conductors. Any discontinuity in coaxial lines, in practice, will cause higher order modes to exist near the discontinuities. This section deals with following topics, in order to find the nature of EM waves inside and near the aperture of coaxial line sensor: • Characteristic Admittance of a Coaxial Line: An equation for characteristic admittance is derived for a uniform coaxial line in terms of its conductor radii and the dielectric EM constants. This can be found in section 3.2.1. Derivation of scalar electric potential can be found in section 3.2.2. • Incident TEM Waves in a Coaxial Line: The incident EM waves in a coaxial line are assumed to be made up of only TEM mode. These field equations are derived in section 3.2.3. • Image M ethod and Scattered EM Fields: Near the aperture of coaxial line sensors, there exists a discontinuity. This discontinuity causes higher order modes to exist near this region. These are evanescent waves and do not propagate far into the coaxial line. However, in order to find the EM fields at the aperture, these R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. modes will have to be taken into consideration. Because, the incident TEM mode wave has only angular component of magnetic field, higher order modes that exist near the aperture are made up of TM modes. These waves are called as scattered waves. Image method is used in order to analyze higher order modes, these derivations can be found in section 3.2.4. The fields near the aperture are presented in two different forms for further use in Chapters 4 and 5. • A perture Admittance and Reflection Coefficient of the Coaxial Line: A relation between aperture admittance and reflection coefficient is derived in section 3.2.5, which will be used later in Chapter 5. 3.2.1 Characteristic Admittance of a Coaxial Line: Consider a coaxial line with inner and outer radii a and b respectively, truncated in an infinite ground plane as shown in Figure 3-1. Assume that a sinusoidal voltage with a magnitude of V0 is applied as a source for the coaxial line, with a matched load termination. Then the characteristic admittance of a coaxial line is same as input admittance at any point with in the coaxial line looking forward toward load, given by where, F0 is the peak voltage difference between two conductors and /„ is the corresponding current that flows through the coaxial line. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 37 Infinite Ground Plane Coaxial Line Z=0 Figure 3 - 1 Coaxial transmission line The sinusoidal voltage causes a TEM wave to exist inside the coaxial line. The current /„ is given by integrating angular component of magnetic field (assuming no angular variation of magnetic field) as 2;t 2rr dif = H^p J<# = 2npHi = h = 2npE 0 (3.2) where, (3.3) is the impedance of TEM waves inside the coaxial dielectric medium. The voltage V0 is given by integrating radial component of electric field as = a a ~ I o') I («)]- u b- 2n R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3.4) Therefore, the characteristic admittance of the coaxial line is given by dividing I0 by equation (3.4) as follows: 2k J o3 a \a ) 2k r, ruj 2 n/ In k (3.5) Ho The result of equation (3.5) can be verified by referring to page 12 of [58]. 3.2.2 Static Electric Scalar Potential at the Aperture Cross section of the Coaxial Line, ^^(p): Static electric scalar potential as a function of radial distance can be gotten by solving equation for electric scalar potential, with co =0 and p, =0 (since there are no electric charges inside the coaxial line). The resulting equation for electric scalar potential is known as Laplace’s equation and is as follows: (3.6) v 2<Mp) = o Considering the cross section of coaxial line, in two dimension, equation (3.6) becomes 15 p 5p 5 , r i 1 d2 , , v . P — <}>e[ p + — TTT<t>c( p ) = 0 5p L J p 5<j) (3.7) Since, there is an angular symmetry in cross section of the coaxial line, the second term of equation (3.7) can be dropped as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 39 15 d , r - — p — <t)c[p] = 0 p dp dp (3.8) The solution of equation (3.8) can be written as *,(p) = Cl[b(p)] + C2 (3.9) where, Cl and C2 are constants. Imposing the following boundary conditions ♦,(“) = (3. 10) ♦,(») = OJ the equation (3.9) becomes V0 = Cl[ln(a)] + C2 0 = Cl[ln(6)] + C2 where, Vg (3. 11) is the applied D C voltage. Solving for constants Cl and C2 and putting in equation (3.9), the electric scalar potential can be written as follows: <Mp ) : K In Therefore, the corresponding static electric field is given by R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 12) 40 In E0 = In K d W4 In V0 In In a ) dp a [ln (p )-ln (6 )] (3. 13) d a ) dp [ln (p )] = In The result of equation (3.13) can be verified by referring to page 9 of [58]. 3.2.3 Incident TEM Waves near the Aperture, inside the Coaxial Line: The general equations for a TEM wave propagating in any arbitrary direction i is given by E = E 0e ' JKr H= ix E Tl/ (3. 14) K = ki r = x x + y y + zz In this particular case, the direction of propagation is z. Therefore, the electric field of equation (3.14) simplify as E(p,z) = E0e - ^ (3. 15) where, E0 is the electric field when either z = 0 or k, = 0 , with the corresponding direction of field orientation. From equation (3.13), it can be seen that the electric field when k, = 0 is R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 41 K (3. 16) E 0 = P- In Now since this is not the static case, V0 has to be replaced by the corresponding TEM peak voltage of equation (3.4) as follows: E0 = P- In K In =P :s ! r. a 2 tc ( t 'A l P In ■ =P 2np (3. 17) \a s Next, assuming an excitation of 1 ampere, i.e., I0 = 1 ampere, incident TEM electric and magnetic fields are given by (3. 18) H (p ,z) = - = z x p - d h — e ~A- _ ^ _ i _ g A - ri, 27tpr|/ 27tp (3. 19) In terms of field components, the incident TEM fields can be written as £ (0 A = Jk_g-A2np H J p ,z ) = — ’ 2np (3. 20) e -Jk'z The result of equation (3.20) can be verified by referring to page 11 of [58]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 3.2.4 Scattered EM Waves near the Aperture, inside the Coaxial Line: In this section, image method is applied in order to find the fields inside a coaxial line. Consider a coaxial line with inner and outer radii a and b, respectively, as shown in Figure 3-1. The coaxial line is aligned in z direction. Therefore, the TEM waves incident on the aperture of the coaxial line have electric and magnetic fields in transverse directions, perpendicular to the direction of propagation. The electric field £ p(p.z) is directed in outward radial direction, while the magnetic field /^(p,^) is directed in angular direction. Because of geometrical symmetry of the coaxial configuration (refer to Figure 3-1), there will be no angular variations of either electric field or magnetic field. However, near the aperture (the region of discontinuity), there exist TM0n modes due to scattering from the discontinuity of the aperture. These are evanescent waves, do not travel back into the coaxial line, nevertheless they exist near the aperture. The total EM fields, therefore, at any observation point P(p,z ) with in the coaxial line, near the aperture, can be written as sum of incident and scattered waves, as follows: E(p,z) = E'(p,z) + r (p ,z ) (3.21) H (P>2) = H '(p,z) + H '(p,z) (3.22) where, E ', H ', E ' and H v are incident electric field, incident magnetic field, scattered electric field and scattered magnetic field, respectively. Determination of the total fields near the region of aperture (within the R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. coaxial line), i.e., equations (3.21) and (3.22), and thereby the fields at the aperture of the coaxial line is the objective of this section. Scattered fields near the aperture can be found by using image method or image theory. In order to find fields at the aperture using image method, the following steps need to be followed: A. Find the electric field using image theory and a conceptual magnetic source B. Find the electric field inside the line using Maxwell’s equations C. Equate the electric field of Steps - A and B, and find aperture field distribution with out the magnetic source. Following the sections 3.2.4.1 through 3.2.4.3 will lead, step by step, toward the solutions for equations (3.21) and (3.22). The derivations mentioned in sections 3.2.4.1 - 3.2.4.3 can be found in reference [58], which have later been used in research of coaxial line sensors, by many other researchers such as [2, 18, 19, 31, 46-48, 93-95, 101, 103, 155]. The derivations oi section 3.2.4.4 can be found in chapter 3 of reference [52]. 3.2.4.1 Electric Field a t the Aperture , using Image Method: Assume that there exists a perfect conductor at the aperture as shown in Figure 3-2. Then, the tangential component of electric field at the aperture can be replaced by its equivalent magnetic current source as follows: J„(p.0) = k > , 0 ) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3.23) 44 Infinite Ground Plane Perfect Conductor Termination Coaxial Line Z=0 Figure 3 - 2 Coaxial line, with aperture terminated by a perfect conductor 2 r* Coaxial Line Z«0 Figure 3 - 3 Coaxial line of Figure 3-2, replaced by a single continuous line R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. where, ■ 7 ,„ > (p ,0 )= £ p( p ,0 ) ( 3 .2 4 ) Now, using image method, the coaxial line terminated by a perfect conductor can be replaced by a single continuos line, as shown in Figure 3-3. The problem will now reduce to a simpler one consisting of a forward incident wave and a backward image wave and a doubled magnetic source. Referring to Figure 3-2, from symmetry E'(p,z) = - E '( p - z ) (3.25) W {p,z) = -W {p ,-z) Also, at the aperture , z (3.26) = 0 and 61™0[£;(p's ) " £' (p’' 5)] =2'/-*(p’0) (3- 27) Equation (3.27) means that £ „ '(p .O )= ^ (p .O ) 3. 2.4.2 (3 .2 8 ) Fields inside the line, using M axwell 3s Equations: The scattered fields are the evanescent waves that exist near the discontinuities in the coaxial lines. In this particular case, scattered waves exist near the aperture. Because of the physical shapes involved, the scattered waves are made up of TM modes. These scattered fields, therefore, R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. can be expanded in terms of all TM0n modes. It should be noted that there are no sources with in the coaxial line. Therefore, the electrical current source J , , and the magnetic current source J„, do not exist inside the coaxial line. Consider Maxwell’s equation V x H'(p,z) = /(os0s,E*(p,z) (3. 29) The equation (3.29) can be expanded in cylindrical coordinates as follows: p r (p ,z ) = - r - L - V X JCOSqS, d H'(p,z) = — — yCOSoE, P dp h ; { p ,z ) p<i> d 5(j) ph ; { p ,z ) oz h i {p , z) Since, magnetic field has only angular component, the above equation can be rewritten as E'(p ,z) = 1 1 J(DS0S, p dp o 3<)> ph ; { p , z ) dz o Further simplification of the above equation leads to the components of scattered electric field as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 47 I , d -p e '( p -’) = — : e, — j cos0 pv 8z .8 Hz— pHl{p,z) dp. i -HZ(p,z) + ---- -p 8 j(£>s0£i 8z 15 (3. 30) ycoSoS, p dp = P^p(P.z) + 2^/(P»2) Equation (3.30) shows that the scattered electric field with in the coaxial line, near the aperture, is given by partial derivatives of angular component of magnetic field. In order to derive a single equation in terms of angular component of magnetic field, Maxwell’s equation will have to be considered as follows: (3. 31) V x EJ(p,z) = yco|i0i r ( p , z ) Since, magnetic field has only angular component, considering only <j) component of equation (3.31), the following equation can be written f f ( p , z ) = $tf;(p,z) 1 1 V x E '( p ,z ) 1 op e 0 pcj) 0 d_ dp 8_ 3<j) 8_ dz ; ( q , z )p£;(p,r) £ ;(p ,z ) * R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 32) Next, replacing the scattered electric field components E * (p ,z ) and E t(p ,z ) from equation (3.30) in equation (3.32), the angular component of the scattered magnetic field near the aperture can be written as follows //; ( p ,z ) = — dz ycoM-o 1 1 d dp jo )e0e, p dp (3. 33) th*'M o - 4dz — j(o s0s, dz 1 © -p0s0s, dp p dp ’ dz2 ’ Equation (3.33) can be written in the form of second order differential equation in cylindrical coordinates as follows: '5 1 5 d2 --------------P + — T + dp p dp dz~ h ;{ p ,z) = o (3. 34) In order to solve the above equation, the angular component of magnetic field can be assumed to be made up of product of two functions, i.e.. h (3. 35) ; { p ,z ) = R { p ) Z ( z) Putting equation (3.35) in (3.34) yields the following equation: d id dp p dp P+ d2 dz ,2 r- + k, R (p )Z {z) = 0 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 36) Next, the following steps, in order will reduce the equation (3.36)1 + |r [ * ( p)z W]+ *?[«P)ZW] * o z(z) ^ ^ pS(p) + *(p) ^ n z M + H K pW ) =0 # (p ) d p p dp Z (z) dz W ) T P l i pR{p)^ d 1 =>----- + (* /- dp p r/p ^ pl Jdp? p! k' - - ° i/p y;)* (p ) = o p +p £dp A( p>+ p ^ +[pi(*‘ ■i’•) - *]*(p) =° *<p >+ [<p - p >2 _ ' H p ) = 0 1 The m eaning o f the sym bol “ => ” is “im plies”. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 50 Let, 0 = p„p, then 0 , dQ p = — ,d p = — P. , , 2 dQ2 and dp = —— P. P. Therefore 9! *(p)+ 0 ^ - ' H 5) = 0 ( 3 - 37) The equation (3.37) is a standard Bessel equation o f first kind. The solution for this type of Bessel equation can be found in Mathematical Handbooks as 09 (0) +6,^(0)]. Noting that 0 = p„p, the solution for equation n-0 (3.37) will be as follows: n=Q Also, another equation that was used in deriving equation (3.37) was ^ Z ( z ) +Y:z(z) = 0 (3.39) The solution to the differential equation of (3.39) is given by (3-40) Z(z) - K e ±jy": where, K is a constant. Using equations (3.35), (3.38) and (3.40) the solution for the angular component of the scattered magnetic field can be written as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 51 (3. 41) n=0 where, (3. 42) V l= k ? -Y l and J,((5„p) ,^(p„p) are first order Bessel and Neumann functions, respectively. More details on these functions can be found in Mathematical Handbooks. Further, an and bn are constants that need to be resolved by the physical situations that exist at the boundaries. Note that absorbed together with these two constants an and K has been bn. The applicable boundary conditions are as follows: • A t p = a , El = 0, therefore from equation (3.33), dp pa ;( p ,z) or, ^ - dp = 0 P~a h ; ( p ,z) + - p h ; ( p ,z) = 0 (3. 43) p=a A t p = b , £/ = 0, therefore from equation (3.33), dp ph ;( p,z) = 0 or, |-f/;(p ,.-)+ -//;(p ,z ) dp p = 0 p =b R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 44) By putting equation (3.41) in equation (3.43), the following equation can be obtained: = <3pt P /i=0 + ^ .P - riP ." ) + n=0 a. M P,a) a + p ,4 P - a) 0 p=a o a a + b„ = (3. 45) 0 Similarly, by putting equation (3.41) in equation (3.44), the following equation can be obtained: a ., ■/.(M) + M i(P nb) + b, ^ >:(p.4) + p „j;(p .6) = 0 (3. 46) The following relations for the derivatives of Bessel and Neumann functions are obtained from Mathematics Handbooks2: ■6!P.p W o(P.p ) - ^ - /,(P..p ) (3. 47) i ; ( P . p ) = T 0( p „ p ) - - p - j ; ( p , p ) (3. 48) r'wr' 2 In Chapter 3, the sym b ol Y w ith various numerical subscripts are used to represent The N uem a n n F unction, where as in Chapters 3, 4 and 5, the same sym bol w ith various numerical and alphabetical subscripts are used to represent The Characteristic and Aperture A dm ittance o f the Coaxial Line. T he correct representation should be read in its proper context. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Using equations (3.47) and (3.48), equations (3.45) and (3.46) can be written as a„J0(p„a) + £„r0(p„a) = 0 (3.49) a„/0(p„6) + 6„r0(p„6) = 0 (3. 50) ■ From equation (3.49), one constant can be eliminated as follows: U „ ^o(Pna) K = - a' m 7 ) /I C1\ (3- 51) Further, solving the equations (3.49) and (3.50) to eliminate the constants, it can be seen that p„ is the solution of the following equation, for n > 1, y0(P„6)70(P „ a )- J0(p„a)r0(p„6) = 0 (3. 52) Incorporating equations (3.51) and (3.52) in equation (3.41) ensure that both boundary conditions of equations (3.43) and (3.44) are satisfied. Putting equation (3.51) in (3.43), (3 ' 53) where, ♦ .(p )-ro (M 4 (p .p W o (M i;(p .p ). R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3- 54) 54 Still, the constant an is left unresolved. The boundary conditions of axial component of electrical field are already used to resolve the constants bn , p„ and y„, as given in equations (3.51), (3.54) and (3.42) respectively. These boundary conditions were derived from equation (3.30). In equation (3.30), there is the radial part of electric field which can now be used to eliminate the constant an. Consider the radial part of electric field from equation (3.30). as follows: <3- 55) J COS0£ , dz Now, putting the magnetic field of equation (3.53) in equation (3.55) yields £ p (P ’z) = ■ 1 dz M P„a) ( 3- 56) Consider, the equation (3.56) at the aperture (i.e., z = 0) as follows: (3 - 57) Multiply both sides of equation (3.57) by p<t>„,(p) and integrate the resulting equation from a to & as follows: j£;(p ,o)t> m(p)prfp = — — j r a COEqE/ „=o j4>„(p>l>„,(p)p^P a R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3- 58) Now, using the orthogonality property of TM modes, it can be shown that where, 5m„ is Kronecker delta function given by 8 -"= f 1, m - n 0, m * n ,, (3- 60) and 4, Is a constant which is yet to be resolved. Therefore, when right hand side of equation (3.58) is zero and when m=n m* n, the equation (3.58) becomes (3 ' 61) where, s« = 7 F J<t>n( p ) ^ P ( p , 0 ) p d p (3 . 6 2 ) From equations (3.53), (3.61) and (3.62), the scattered fields can be summarized as follows: (3- 63) n =0 Yn R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. -1 cosfoS,, d K(p*z) = ytOEgS, dz ~ JY n JCOSqS/ ,7Y»= H=0 rI« -I COSoS^,, m=0 (3- 64) Yn = i X i ( p y r"'' £;(p.-’) = J(£)£ - 4q£-i “p op [p^(p.^)] (3. 65) where, y „= i *, > p. *, <p„ (3. 66) .4,, is a constant yet to be determined. The results of equations (3.63), (3.64) and (3.65) can be verified by referring to [58]. 3.2,43■ A perture Field D istribution - First Type o f Formulation: By equating the electric field obtained in section 3.2.4.1 and section 3.2.4.2, the total magnetic field distribution at the aperture can be found. The resulting equation for /^(p,0) can be put in two different but equivalent forms. The first of these two types of formulations is derived in this section. The result of this section is further considered in Chapter 5. The next section will cast the same result in a different format, incorporating reflection coefficient. Equating the electric field equations of (3.28) and (3.63), it can be seen that R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 57 I> A ( p) = 4 *( (3. 67) p , o) ;;=0 Multiplying equation (3.67) by p4»„,(p) and integrating from a to b, the following result is obtained (3. 68) I X Ji,(pX(p)p^p = Ji,(p)j »*(p)p^p n=0 TM0n modes are orthogonal, therefore using (3.59) and (3.60) when m=n equation (3.68) becomes b A2 = J<I>»(p)p^p a Using equation (3.54), after some manipulations A,, can be found as follows (also, see reference [58]): J llJ M A *15. 1 B il (3. 69) Therefore, the scattered magnetic field is given by (using equations (3.69), 8 CO (3.62) and (3.63)), 4 r jf,(p )£ P(p>0Wp ;E o <j)„(p>^”-' 7 n _ « a = -cos /s0£ <L(p) f<j>„(p')£p(p',0)p'c/p' ,vy*- 4,7 „ R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 70) Therefore, using equations (3.20), (3.22) and (3.70), the total magnetic field near the aperture can be written as follows: II e -jk'z + ejk': t 0' 11' 1 .m p o ^ p ' ^ p '^p ' 2^P (3. 71) S 4 ; y. It should be noted that for TEM mode, n = 0 and, Po = 0 Y o = k, (3. 72) <t>o(p) = - P Also, define S as follows: b S = l E p{p',0)dp' (3. 73) a Therefore, for TEM mode, equation (3.71) becomes „ I x H,tem{ P,z) = e~jk‘: + e jk': cos , s 0Se Jk': r—---------2k p k,p Inf \a Therefore, the total field near the aperture can be written as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 74) 59 H${p’z) ~ H$rEM{p,z) + H^TEM{p,z} + H^maii{p,z) _ e~jk,: + ejk,: cos0e,Sejk‘: 2"p (3. 75) *,p Inf ^ \a . —O £ 0S; YI^£(P) ’' h n o . M p 'K C p ' ^ p '^p ' In Chapter 5, the equation (3.71) will made use of in order to find the complex dielectric constant of the medium terminating the coaxial line, by a numerical technique called Method of Moments. In particular, the total magnetic field will be equated to that found in the layered medium (using the boundary conditions). The total magnetic field can be found at the aperture from equation (3.71), by putting z = 0 as follows: o /^(p,0) = — + ycoe0s, |£:p(p',o)^cl(p,p')p'rfp' (3. 76) 7 ip where, U p >p ') = j 1L n=0 ^ ( p M p ') A ll „ (3. 77) In equation (3.77), it should be remembered that n = 0 corresponds to TEM mode and therefore equations of (3.72) will have to be used when n = 0. The results of equation (3.76) can be verified by referring to [58]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 3.2.4.4 A perture Field D istribution - Second Type o f Formulation: In this section, the results obtained in section 3.2.4.3 are modified and put in terms of voltage reflection coefficient (or, TEM electric field reflection coefficient) at the aperture. Equation (3.74) gives the TEM part of the total magnetic field, which is th e , sum of incident and reflected TEM waves. The TM mode part of equation (3.75) is evanescent in nature and attenuates very fast, inside the coaxial line. Therefore, this part can be neglected in finding the voltage reflection coefficient. Next, considering only the incident TEM waves in equation (3.74), the following equation can be written: JW c( M = 4 ,(3- 78) Again, considering only the reflected TEM waves in equation (3.74), the following equation can be written: ik - Co 11 t) = — ________ r, ® e.‘Sl S ...e Jk ' : K N - _______ 1 - 2tico£ , s 0S _ __________________ e J k r- (3 7 9 ) Consider the following definition of voltage reflection coefficient: „ _ \ ^ T F M . r e / { P ’ Z )\ _ \E tem m A p A TEM ,re/ ( P ’ Z )[ \H t e m , U P ’ z )\ R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. gQ\ Using equations (3.78) through (3.80), the reflection coefficient at any length within the coaxial line (looking toward load) can be found as follows: ^ In[ —| -2711(06,8 0S 27t£,pln| r„=- Jk,= a 2 tkos ,s 0S - jk ,z k, In 27tp -1 2jk,s (3. 81) a Therefore, the reflection coefficient at the aperture, looking toward the load can be written by putting z = 0 in equation (3.81), as follows: _ 2 jT(OS,80S m , (3. 82) T Also, equation (3.82) can be rewritten as follows: 038,8 0S ( b'] k , In — \a 1 + r, 2n (3. 83) Therefore, the equation for TEM waves with in the coaxial- line in equation (3.74) can be modified by putting equation (3.83) in (3.74) as follows: p - j k r- , p Jk,z i / ♦. m .(p ,z)) = 2np i , r- 2np - e J '* R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Therefore, total magnetic field with in the coaxial line, but near the aperture can be written as follows: ^ (P’Z) ~ ^TEm {P’Z) + -^TE.w(P’Z) + ^7M0„(P’Z) = HfyTEMjrtc (p» 2ftp -r, + EW.re/(P’Z) + ( p ’Z) (3. 85) 27lp & -cos0s J4>» ( p O ^ p (p ' . o ) p r€ / p f 'vi /i n=I A w 3 TV,I Therefore, in terms of reflection coefficient, the aperture magnetic field can be written as follows: u H*^ = 2rcp ^ ~ r '^ ~ 70)S°S/ j£ p(p',o)ii:c2(p,p')p'rfp' (3. 86) where, (3.87) Ki(p, p') = «—1 n The results of equation (3.86) can be verified by referring chapter 3, page 40 of [52]. 3.2.5 Aperture Admittance and Reflection Coefficient of the Coaxial Line: A relation between aperture admittance Yt , i.e., the load seen by the coaxial line at z = 0, and the electrical property of the substance is useful in solving for aperture admittance by Method of Moments. This relation is derived in R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. this section. Using equation (3.3), equation (3.82) can be rewritten as follows: (3. 88) ln| Also, voltage reflection coefficient at the aperture can be written as follows: r = h z lo . = Z/ + Z0 (3 89) Y0 + Yt y } Equating the reflection coefficients of equations (3.88) and (3.89), 2n S Yo + Y, *1/ in - K dJ _ ^i, = > Y ,= Y0 - n S Y 0 \a J 71S (3. 90) Using the equations (3.3) and (3.5), the characteristic admittance, the aperture admittance of equation (3.90) can be written as follows: Y,i = -s~ Y o 0 (3. 91) The results of equation (3.86) can be verified by referring page 12 of [58]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 3.3 WAVES IN LAYERED MEDIA In section 3.2, the fields within and at the aperture of a coaxial line were found. This section deals with fields in the layered media. For analysis, here, two layered media terminating the coaxial sensor is considered. The first layer has a thickness of d while the second layer extends up to infinity. The fields excited in the layered media will be made up of TEM and symmetrical TM modes. The field quantities in layered media are found using spectral domain analysis. This section deals with following interconnected topics: • EM Fields in the Layered Media: Electromagnetic fields in the layered media are found in section 3.3.1, starting from Maxwell’s equations and then transforming them into spectral domain. • A Variational Expression for A perture Admittance: By assuming the nature of electric field at the aperture to be inversely proportional to the radial distance from the z axis, i.e., Ep(p,0) cc —, an integral expression can be derived for the aperture admittance of a coaxial sensor, when terminated by two layered media. The research on the numerical solutions for the aperture admittance and the resulting inversions for the terminating media EM characteristics are done in Chapter 4. This topic can be found in section 3.3.2. • A n Integro-Differential Equation for A perture Electric Field: Instead of assuming that the electric field at the aperture to be R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 65 inversely proportional to the radial distance from the z axis, these fields can be found numerically by using the Method of Moments. Research on the numerical solutions for the electric field at the aperture and its admittance are done in Chapter 5. This topic can be found in section 3.3.3. 3.3.1 Electromagnetic Fields in the Layered Media: Fields generated by a coaxial line in the layered media will be of symmetrical TM modes. These field equations will be derived in this section by using spectral domain techniques. The following sections present a step by step derivation of the total field in the layered media. 3.3.1.1 Electric Field in the Layered Media: The geometry of the problem in 3 dimensional space is shown in Figure 3-4. Consider the Maxwell’s equation in the layered media given by V x H (p,z) = y'coscE (p,z) (3. 92) where, sc = s lc or s c = s 2c, depending on which medium is considered. The equation (3.92) can be expanded in cylindrical coordinates as follows: E (p ,z ) = — V x H (p ,z ) = - J C0 £ c 1 1 ;c o e c p p p<j> z d_ d_ d_ dp 5<j) dz H : ( P, z) #p(p , z) p # * (p ,z) Since, magnetic field has only angular component, the above equation can be rewritten as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 66 E (p ,z) = 1 1 p p P<t> d d z a dp 3(j) dz 0 p H, ( p , z ) 0 C oaxial Line M edium 2 Figure 3 - 4 Two layered media termination of a coaxial line sensor Further simplification of the above equation leads to the components of electric field as follows: E(p,r) = - 1 1 .5 .5 - p ^ + z— dz dp. i:H* ( p ’z) = ycosc dz (p»' z) + jc osc ~p dp (p^z ) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Therefore, Ep(p'z) = —— t ~h AP’z) P j ( O S c OZ v (3- 93) E: ( P’Z) = — — ■ -~-pH^(p,z) J o e c p dp 3.3.1.2 Electric an d Magnetic Fields in the Layered Media, in Spectral D om ain: Consider the wave equation, with out any sources given by ( v 2 + £ 2) //* (p ,z ) = 0 \d _ f d_ ' pSpV dp) 1 d2 ■*—~ : p 2 5(j) 2 d2 trr + h H,{p,z) = 0 dz2 (3. 94) In order to transform the differential equation of (3.94) to spectral domain, consider the following Fourier Bessel transform pair: fH (p ,z)J ,(x p )p d p (3. 95) 00 H ( p , z ) = j H ( x , z M ( x p ) x ‘iX Using equation (3.96), equation (3.94) is transformed as follows: d z2 i k2- r ) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 96) 68 or, d2 % T dzT + ot' M =0 (3.97) where a 2 = k/ 2 ~x 2 This equation can be solved for the two different media. The two solutions will look different because of the associated boundary conditions in two regions. It can be written in terms of sine and cosine functions, for the first medium. In the second medium, the EM waves propagate up to infinity and therefore can be written in exponential terms. Therefore, the solution for the two regions are constructed as follows: x _ j 4 x ) c o s ( a 1z) + 5 ( x ) s i n ( a Iz), 0 <z<d ( z>d where, a (x ) , b (x ) and c(y) are constants, which will be resolved later by matching the boundary conditions. Next, consider £p(p,^) of equation (3.97) in spectral domain, as follows: ( 3 - 99) Equation (3.98) in (3.99) yields R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 69 EPU ’Z) = J^c [^(x)sin(a,z)- 5(x) cosfct.z)], 0 < z < d (3. 100) a, z>d coe. 3.3.1.3 Constants o f Field Equations o f Equations (3.98) an d (3.100): In order to find the constants of equation (3.98), three equations arerequired. The applicable boundary conditions yield two equations as follows: • At z=d, is continuous, therefore 4 x ) cos(aId)+5(x)sin(a1j) = C(x) • At z = d , E p( x , z ) (3.101) is continuous, therefore T ^ -[4 x )sin (a^ )-5 (x )co s(a,j)] = - ^ C ( x ) yC0S,c (3. 102) The third equation can be gotten by considering the magnetic field of equation (3.100) at z = 0, as follows: B(x ) = : f p ± E ' ( X,0) (3. 103) a l Solving the equations (3.101) and (3.102), the following equation can be obtained: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 70 ^ ( x ) c o s ( a , ^ ) + 5 ( 5 c ) s i n ( a , ^ ) = ^ ^ L [ ^ ( x ) s i n ( a , c / ) - JS ( x ) c o s ( a 1t/)] i £ka 2 - s in ( a ,^ ) - F’2‘a ' co s(a ,£ /) ^ A{x) = 5 ( x ) ------------7 £ v a l • ( j \ c o s l a . a ) - ---s in la .a ) / s lca 2 ,t \ n/ \ ~ J e \ca 2 s in ( a ,r f ) - E 2ca 1 c o s (a ,t/) => 4 x ) = % ) —--------- 7— x -------- . / _/slfa 2 c o s ( a ,a ) - s 2ca , sin^ ajfl) S 12ca 2 + 7 a i t a n l a i“ ) COSi, a, +ye,2ca2tan(a,c?) 2 +Jaitan (a,<5?) £p(x.o) (3. 104) 4x)= la, y 8uca where, Next, putting equations (3.103) and (3.104) in constant C can be written as follows: , v C(x)= a , + / s p , a 2 ta n (a .rf) V , } / ; cos(a4)-ysm(a4) s 12ca 2 + j a x ta n (a ,d ) C08, a, equation (3.101), the £p(x.o) (3. 106) 3.3.1.4 Total Electric and Magnetic Fields in layered media: Using the equations (3.103), (3.104) and (3.106), the magnetic field in the layered media given by equation (3.92) can be written as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 71 ^(xjcosfoz) - j sin(a,z)]ic (x), 0 <z <d <;(x)cos(aI^)-ysin(a1j)]K(x)e"ja:(-"'/), z > d (3. 107) where, r t \ a >+yS|2ca 2tan(a i^) W = --------■ .t a n((a ,a.)\ y s12ca 2 + ya, (3. 108) k ( x) = ^ (3. 109) £ p(x,o) Using the definition of Fourier Bessel transforms, u (3. 110) (x>°) = f^ p (p > 0 )/,(x p )p ^ p , k (x ) can be rewritten as, (3. I ll) K ( x ) = ^ u K ( p , o ) y 1( x p ) p r f P 1a i Similarly, the radial component of electric field can be written as follows: [“ ^(x)sin(a1z) + cos(a1r)]fp(x,0), 0 < z <d £p(x>z) = [^(x)cos(a,rf)-y'sin(ald)]^12^ 2- f p(x>0)e“-/“,(-"‘,), z > d Oti Now, the inverse Fourier Bessel transform of equations (3.107) and (3.112) can be obtained by using equation (3.96) as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 03 \\c, ( x) c o s ( a ,z) - j s i n ( a ,z ) ] k ( x ) J, ( xp) yd/_, 0 <z<d 0 (3. 113) 00 }[<;(x)cos(a^)-ysin(al^)]K(x)e'M(-'"t/)yi(xp)x^, z> d 50 {[-X fx J sin fa ^ J + co sfa .z^ p fx ^ J .fx p J x ^ X . £P(jC>*) = 0<r<t f 0 CO {[C( x )cos(a,d) - ysin(a,c/)j £p( x ,0)e_J“2 J ,( x p ) x ^ X > (3. 114) The results of equation (3.86) can be verified by referring to [147]. 3.3.2 A Variational Expression for Aperture Admittance: Assuming that the electric field at the aperture is inversely proportional to the radial distance, i.e., (3. 115) Ep{ p , 0 ) ^ ~ P an equation for the aperture admittance can be derived using the magnetic field at the aperture from equations (3.86) and (3.113). This expression will be solved numerically in Chapter 4. Equating the magnetic fields at z = 0 from equations (3.86), (3.87) and (3.117), the following equation can be obtained: ^ + a>e0e;£ foptphO)^ Z7lP n=\ a ^ ^ ^nl n p'dp' = ^ ( x M x W x p ) * ^ Q R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 116) 73 Using equation (3.82), i.e., COE (Z qS 2n K + r, ) ’ equation (3.116) can be modified as follows: i-r , . , ^ f , ( p > t ) n( p ' ) f j , ----------- +03£q£/2 , J ^ p (p ';Q) 2 9 d9 n=l a nYn *| h { ^ p l + r ' C08/eo5 (3. 117) Knowing £p(p',0), the reflection coefficient at the aperture can be found. The equation (3.117) can be brought into stationary form after multiplying both sides of equation (3.117) byp£p(p,0) and integrating the resulting equation from a to b (also, use the approximation of equation (3.115)), as follows: J a Ir In -Jp i T t -P £ p ( p » ° ) 4 > / +rog0s/ Z"=• faf P ^ P P '£ p (p ^ )£ p (p ,o M x /p ' a AnYn so b =0JJ[4(xM xWxp)£p(p>0)px]dp<ft a (0£,E0 i - r , i + r, a CO 1 + COS0E n=l I it a 4 ,n (3.118) = W x ) ^ M ( xpVp yAi 0 u l _a R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. From the orthogonality property, and equation (3.54), it can be shown that 0 = a (3. 119) 4n, Further, assuming that (3. 120) .Mxpjrfp equation (3.118) becomes 1-r, (oslek, i + r, (os0e; K (x) a Be{ l h dX 0j ^1 (3. 121) K \aJ -)l Therefore, from the definition of normalized aperture admittance, the following equation for normalized aperture admittance can be written: i-r, Y,= i + r , K (x) z Xek , In 0 ai (3. 122) a Finally, the aperture admittance of coaxial line is In r = I0 r'-=± Jl i+ r, = 2^ In In | M Po 80s I In Eoe/ B h)xdl 0 «! a R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (3. 123) The result of equation (3.123) has been used by many researchers such as [18, 19, 31, 46-48, 93-95, and 101], in order to apply a coaxial transmission line as a sensor. 3.3.3 An Integro-Differential Equation for Aperture Electric Field: Instead of assuming that the electric field at the aperture to be inversely proportional to the radial distance from the z axis (as was done in section 3.3.2), the electric field at the aperture can be found numerically by using the Method of Moments. This will take more computer time, but at higher frequencies it is hard to neglect the higher order modes. Equating the angular magnetic fields at the aperture as given by equations (3.76) and (3.113), it can be seen that o 1 — + /®s 0e, j,£p(p\0)Kcl(p,p')ptfp' = J^(xM xW xp)x^X 7Cp where, £ cl( p , p ' ) , C,{y) and k (x ) (3.124) are as given in equations (3.77), (3.108) and (3.111) respectively. This equation is solved for £p(p,0) using the Method of Moments in Chapter 5. Once the radial electric field at the aperture is known, equations (3.73) and (3.91) can be used to evaluate the aperture admittance. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 76 Chapter 4 A PER TU R E A D M ITTA N C E BY V A RIA TIO N A L PR IN C IPL E 4.1 IN T R O D U C T IO N Coaxial lines are widely used as sensors for measuring electric and magnetic characteristics of various materials. Some of the important practical applications of coaxial sensors are measurement of dielectric constants of materials such as biological tissues, paper and other industrial samples. Also, by measuring the electrical characteristics of any material, it is possible to measure the moisture content or temperature of that material. This is possible because the dielectric constant of that material depends on the moisture content or temperature of that material. In this research, an arrangement of a coaxial sensor terminated by two layered media is studied. By monitoring the relationship between incident and reflected signals in the coaxial sensor, it is possible to measure the dielectric constant of either of the two media or the thickness of the first medium. However, in order to measure the dielectric constant of medium 2 or the thickness of medium 1, the thickness of the medium 1 has to be much smaller in comparison with the dimensions of the coaxial sensor. The measurement of electrical characteristics of the terminating media of a coaxial line is done by applying a single tone microwave signal to the R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. terminating media, as shown in Figure 4-1. Then, the resulting reflection coefficient (or, the aperture admittance) of the coaxial line sensor is measured. Based on this measured reflection coefficient, it is possible to find the electrical characteristics of the terminating media. The theory required for this purpose has been explained briefly in Chapter 3. N etw ork Analyzer Coaxial Cable Second Material e, Coaxial Probe Figure 4 - 1 Experimental set up for measurement of electrical property R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Logically, the first step in this research would be to find aperture admittance of the coaxial line theoretically. The second step would be to verify these theoretical results with corresponding experimental results. Finally, after both theoretical and experimental methods of finding the aperture admittance are studied and developed for an unknown material termination, the equation I»W -4(<p)=0 (4.1) can be used to find the electrical characteristics of the termination, where Ylh(<.p) is the theoretical aperture admittance, ^(<p) is the measured aperture admittance and <p is one of the following three possible unknowns (assuming, the other two are known) • s 1- dielectric constant of medium 1, • si - dielectric constant of medium 2, • d - thickness of medium 1. This chapter presents the numerical solution of variational expression for aperture admittance and verification of these results by comparing the theoretical results with the corresponding experimental results. Also, the determination of electrical characteristics of medium 1, electrical characteristics of medium 2 and thickness of medium 1 using equation (4.1) are presented in detail. At the end of this chapter, a theoretical study of a coaxial line termination by air followed by dielectric and vice versa are discussed briefly. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 79 This Chapter deals with the following topics: • A Variational Expression for the A perture Admittance of a Coaxial Line: A variational expression for the aperture admittance has already been derived in Chapter 3. From the point of view of mathematical solutions, section 4.2 investigates the > surface wave and guided wave poles of the integrand of the variational expression, section 4.2 explains in detail the nature of integrand when terminated by different types of materials. • Num erical Solutions for the A perture A dm ittance of a Coaxial Line: Once the expression for aperture admittance is studied, all possible numerical solutions need to be investigated, with the point of view of ease of computation and least computational time. In section 4.3, four different numerical solutions of the variational expression for aperture admittance are also discussed. The relative advantages and disadvantages of these numerical techniques are discussed. Special cases of these solutions (i.e., Quasi Static Approximations, Conductor as Second Medium, Coaxial Line Terminated by Infinite Medium) are also presented in section 4.3. • Comparison between Theoretical and Experimental A perture Admittance: In section 4.4, the comparison between theoretical results of aperture admittance using variational expression and the experimental results of aperture admittance is given. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. • M aterial Characteristics using a Coaxial Sensor: In section 4.5, the experimental determination of material characteristics using a coaxial sensor is listed. This is done by solving equation (4.1) and the experimental results. • Theoretical Study of Air-Dielectric and Dielectric-Air T erm ination of a Coaxial Sensor: In section 4.6, a theoretical study of air followed by Dielectric and Dielectric followed by air, when terminated by a coaxial sensor are studied. 4.2 A VARIATION A L EXPRESSION F O R TH E APERTURE A D M ITTA N CE O F A COAXIAL LINE The problem defined in the previous chapter was that of a coaxial sensor terminated by a two layered non-magnetic media. The complete set up included a coaxial sensor of inner radius a and outer radius b, a medium (first of two layered media terminating the sensor) with a complex permittivity of s \ and a thickness of d ; a medium (second of two layered media terminating the sensor) with a complex permittivity of c‘ and extending up to infinity. The geometry of the complete set up and the nature of materials terminating the line are depicted in Figure 3-4. The aperture admittance of coaxial sensor with two layered media termination had been found in Chapter 3 (refer to equation 3.123) as V - 10 Y 1-r, i + r, 27tcoslc 'K ( x ) \ - Z ~ Be{l)%dx m(* n til \a R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (4.2) Equation (4.2) gives a theoretical means to solve for aperture admittance of a coaxial line. It should be remembered here that this equation uses one approximation that the electric field at the aperture is inversely proportional to the radial distance from the center of the coaxial line. Equation (4.2) can not be solved analytically because of the mathematical complexity involved with the integrand. Further, in case of lossless material termination, the integrand has surface wave or guided wave poles in the integrand. Equation (4.2) can be rewritten, for the sake of simplicity, as follows: oa i (4. 3) where, u B.. = _ P i( xpV p 2ncoe lc In (4.4) (4.5) a a , + y s I2ca 2 t a n ( a ,d ) (4.6) s , 2ca 2 + j a , t a n ( a xd ) S.2c = (4.7) (4.8) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 82 «2 = - / V x 2 - a2 (4.9) k{ = ( o 2n 0s lc (4.10) k I = co2p.0s , c (4. 11) 81c = s 0£[ (4.12) £ 2c = s o (4.13) e 2 Further, in this section and rest of this thesis, when the materials terminating the dielectric are purely lossless, the following symbols are used: s;=s, (4.14) s*2 = e 2 (4-15) Physical insights into the entire structure, that are provided by understanding the results of the mathematical symbolic manipulations, are presented through out this chapter, as the need for explanation arise. The above formulation for aperture admittance has been well studied in many papers such as [18, 19, 31, and 94]. However, they impose one or more conditions while solving equation (4.3). In this section, author has successfully evaluated the integrand with no limitation on the nature of material that terminate the coaxial line. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 83 4.2.1 The Poles of the Integrand of the Variational Expression The discussion in this section applies mainly to lossless dielectric termination of coaxial line. However, if the materials are slightly loss, then these cases might still be applicable, with little modification. Z=0 Z=d M edium 2 SURFACE W A VES IN CIDENT W A VES R A D IA T E D W A VES REFLECTED W A VES SURFACE W A V ES Figure 4 - 2 The incident, reflected, guided (surface) and radiated electromagnetic waves in the coaxial sensor configuration R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Consider the case of the coaxial line exciting EM waves into two layered, lossless, dielectric termination. As is well known, a part of the EM energy excited into the layered structure will be radiated in medium 2, while another part gets transmitted along the layered structure in radial directions, as depicted in Figure 4-2. The waves transmitted along the layered structure can either be guided waves or surface waves, depending on the dielectric properties of layered media. Also, it is often convenient to divide the solutions for guided or surface waves into two modes of propagation, Transverse Magnetic-TM mode (or E mode) and Transverse Electric-TE mode (or H mode). TM modes do not have magnetic field component in the direction of propagation while TE waves do not have electric field component in the direction of propagation. EM fields for TM mode can be derived from the electric-type Hertzian potential, while for TE modes can be derived from magnetic-type Hertzian potential. The poles of the integrand of Aperture Admittance of equation (4.3) indicate the transmission of these waves in the stratified media. For the purpose of studying the poles of equation (4.3), it can be re-written as follows: a, + j - La.2ta n (a ,i/) ^2 M xpV p (4. 16) a , —- + y'a, tan(a,< i) A careful observation of equation (4.16) shows that, along the path of integration (i.e. real axis of x ) there occur many poles for some particular X. These poles need to be examined and a suitable technique need to be adopted for the purpose of numerical integration. But beyond that, these R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. poles give valuable insights into the ways in which the electromagnetic waves emerging from the coaxial line distribute. There is one pole in the integrand of equation (4.16) because of a , , at % = kx. This pole indicates the guided wave along the first medium. Also, there exist poles because of the denominator of the second term in equation (4.16), these poles correspond to either guided waves or surface waves, depending on the dielectric properties of the stratified media. Analysis of this term (i.e., denominator of the second term in equation (4.16)) shows that zeros exist only in the region kx < x < k2 when s, >s, and in the region k2 < i < k x when s, < s , . The sections 4.2.1.1 and 4.2.1.2 discuss in detail about these poles, while section 4.2.1.3 presents a brief graphical view of the nature of these poles, when terminated by different hypothetical stratified media. 4.2.1.1 Surface W ave Poles [ s x > s , / : When the first medium is, in terms of dielectric properties, denser than second medium (i.e., e, > s 2), the transmitted energy is the combination of guided and surface waves. These waves travel along x -direction, as depicted in Figure 4-2. The poles are the roots of the following Eigen value equation. (4. 17) This equation can be re-written as follows: ^ - X u m X = Jv* - X 1 s R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (4. 18) 86 where, X = a xd (4.19) V 2 = ( k 2 - k 2) d 2 (4.20) For (N - 1)71 there exists <V < N Nn , there are only N number of solutions, therefore, trapped surface-wave modes. Also, for the structure to support only one surface wave mode, the thickness and frequency should be below certain value, given by, 2 y s, - e 2 In equation (4.21), A.0is free space wave length, and N is a positive integer, N = 1,2,3...etc. Also, it can be seen that zeros exist only in the region k2 < x < kx. The number of surface wave modes that exist in the structure is decided by N . 4.2.1.2 G uided W ave Poles [ e x < e 2J: When the second medium is, in terms of dielectric properties, denser than first medium (i.e., s, <e2), the transmitted energy is in the form of guided waves. Guided waves travel along x -direction, as depicted in Figure 4-1. Also, it can be seen that zeros exist only in the region kx < i < k 2. 4.2.1.3 Three - dimensional Surface Plots O f the Integrand: In order to solve the integral of equation (4.3), it is important to understand the behavior of the integrand, and in particular, the poles of the integrand R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. and their movement in the complex x -plane. Also, it gives an in depth insight into the electrical nature of the termination of coaxial line. This section describes the movement of poles, when the medium 1 / medium 2 are dielectric, lossy materials etc. The plots shown in Figures 4-3 to 4-12 depict the nature of the integrand of equation (4.50) *, that is, — - j . a \ The Figures 4-3, 4-4 and 4-5 show the nature of the integrand when £,>£,, when frequency is 1 GHz. Figure 4-3 shows the nature of the integrand in the complex %-plane, when e, = 100 and s 2 = 1. In this case, there exists surface waves at the interface between medium 1 and medium 2. The corresponding poles can be seen in Figure 4-3 on the x = 0 axis. Figure 4 4 and 4-5 show the integrand when s, = 100-jl and e, = 100-jl0 respectively. The surface plots here show that the surface wave poles move away from the real axis and integration of equation (4.50) can be handled directly. The Figures 4-6, 4-7 and 4-8 show the nature of the integrand when e , < e,, when frequency 1 GHz. Figure 4-6 shows the nature of the integrand in the complex x-plane, when e , = 1 and s 2 = 100. In this case, there exists guided waves in the first medium similar to the ones in the waveguides. The corresponding poles can be seen in Figure 4-6 on the x = 0 axis. Figure 4-7 and 4-8 show the integrand when s, = 1-jl and e , = 1-jlQ respectively. The surface plots here show that the guided wave poles move away from the real axis and integration of equation (4.50) can be handled directly. 1 Refer to section 4.3.3 for detailed discussion on equation (4.50). R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. The Figures 4-9 and 4-10 show the nature of the integrand when s, >s, , when frequency is 40 GHz. Figure 4-9 shows the nature of the integrand in the complex x-plane, when s, = 100 and s, = 1. In this case, when frequency is increased to 40 GHz, the number of modes or poles on the real axis increase significantly. The difficulty in handling these integrands in equation (4.50) also increases significantly. Figure 4-10 shows the integrand in complex %-plane, when s, == 100-jl and s 2 = 1. The surface plots here show that the surface wave poles move away from the real axis and integration of equation (4.50) can be handled directly. The Figures 4-11 and 4-12 show the nature of the integrand when s, < e2, when frequency is 40 GHz. Figure 4-11 shows the nature of the integrand in the complex %-plane, when s, = 1 and s 2 = 100. In this case, when frequency is increased to 40 GHz, the number of guided wave modes or poles on the real axis increase significantly. Figure 4-12 shows the integrand in complex %-plane, when s , = 1 and e 2 = 100-jl. The surface plots here show that the poles move away from the real axis and integration of equation (4.50) can be handled with out much difficulty. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 89 Imaginary Part of X Figure 4 - 3 3-D picture of the integrand of equation (4.50), / = 1 GHz, d = 3b, 8.3 mm coaxial line terminated by e, = 100,e2 = 1 100 - 200 150 ..." >. 100 ; 200 150 -100 -f -5 0 5 V0 Imaginary Part of X Figure 4 - 4 3-D picture of the integrand of equation (4.50), / * 1 GHz, d = 3b, S.3 mm coaxial line terminated by e, = 100-yi,e, = 1 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 90 Imaginary Part of % Figure 4 - 5 3-D picture of the integrand of equation (4.50), / = 1 GHz, d = 3b, 8.3 mm coaxial line terminated by e, = 100-y'10,s2 = 1 Imaginary Part of % Figure 4 - 6 3-D picture of the integrand of equation (4.50), / = 1 GHz, d = 3b, 8.3 mm coaxial line terminated by s, = l,s2 = 100 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 91 200 £ - 20- 7 40 7 -30 -20 7-------- 7 -10 r 200 SP - 2 0 - Imaginary Part of x Figure 4 - 7 3-D picture of the integrand of equation (4.50), / = 1 GHz, d = 3b, 8.3 mm coaxial line terminated by e, = 1- j l , s 2 = 100 -50 -40 -30 -20 -10 0 10 Imaginary Part of 20 30 40 50 % Figure 4 - 8 3-D picture of the integrand of equation (4.50), / = 1 GHz, d = 3b, 8.3 mm coaxial line terminated by s, = 1- j 10,8, = 100 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 92 t:rt d. <u 1 -500 -400 -300 7-------- 7-------- 7 -200 -100 -5 0 - j 7 -------- 7-------- 7-------- 7 -500 -400 -300 -200 -100 ~T 0 7 0 7 7 100 200 ~7 100 300 7-------- T 400 500 7 200 Imaginary Part of 7 300 T 400 / 7000 500 % Figure 4 - 9 3-D picture of the integrand of equation (4.5Q), / =40 GHz,rf = 36, 8.3 mm coaxial line terminated by s, = I00,e, = 1 ' ~?-------- 7-------- 7-------- 7 -500 -400 -300 -200 -100 1 0 7 1W 200 T 7 300 400 i 7000 500 r 50 u rt 9000 C ‘5b 9000 0 0000 " 50 -500 / ------- 7------ "7-------- 7-------- 7-------- 7-------- 7-------- 7-------- 7-------- T / -400 -300 -200 -100 0 100 200 300 400 500 Imaginary Part of 7000 % Figure 4 -10 3-D picture of the integrand of equation (4.50), f =40 GHz, d = 36, 8.3 mm coaxial line terminated by e >~ 100- A s , = 1 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. -6000 -4000 -2000 0 2000 Imaginary Part of 4000 6000 % Real Part Figure 4 -1 1 3-D picture of the integrand of equation (4.50), / =40 GHz, d = 3 b , 8.3 mm coaxial line terminated by e, = l,s2 = 100 -4000 -2000 0 2000 4000 6000 Imaginary Part -6000 -, 0 / ---------- -6000 -4000 Imaginary Part of x Figure 4 -12 3-D picture of the integrand of equation (4.50),/ =40 GHz, d = 3b, S.3 mm coaxial line terminated by e, = \ - j \ , z 2 = 100 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.3 NUMERICAL SOLUTIONS FOR THE APERTURE ADMITTANCE OF A COAXIAL LINE The problem defined in equation (4.3) can be solved analytically if some assumptions are made on the basis of nature or thickness of either or both media (or on Quasi Static basis). However, a general analytical solution for this problem is very hard or impossible to find. This problem gets even more complicated by the fact that poles that appear in the integrand. These poles represent the surface waves or guided waves as decided by the electrical characteristics of the media in front of the sensor. Therefore, the alternative is to use a numerical technique. Four different numerical solutions obtained as part of dissertation are listed here. O ut of all these possible numerical solutions, only one has been perused for further study. 4.3.1 Aperture Admittance of a Coaxial Line, Numerical Solution I: Here the aperture admittance of the coaxial line is split into two parts as follows • because of infinite medium in front of the sensor • because of finite thickness of medium 1. Therefore, the aperture admittance of equation (4.3) can be split into two parts as Y, = Yx + Y2 where, R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (4. 22) 95 Y' = — f j x o ^ ' +iUR. ^ ' Vj p^P' 7i aJ a| If v0 0 ' b b (<& \ s - s S J J u x „d% \dpdp' /J=l a a ^-0 (4. 24) ' f - l Y V i ? e~2Ja'ndy Xn=' > ^ _ S lc« 2 p a, (4- 23) ^ (4.26) £ 2ea 1 (4. 27) Slca 2 + S2ca . Bp = Jx{ip)Jx(rfi') (4.28) R = 7 p 2 + P '2 -2ppcos<j) (4. 29) Symbols that are not defined here exclusively are same as those in equations (4.5) through (4.15). Yx can be solved easily using a computer, though the singularity at the origin needs to be taken care of. The singularities in Y2 are hard to solve in this case, since the singularities become more serious as n is increased. Besides, the time required for this type of numerical solution was found very impractical. Therefore, this method is not studied beyond this point. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.3.2 A perture Admittance of a Coaxial Line, Numerical Solution II: Here again the aperture admittance of the coaxial line is split into two parts, but the second part is not expanded exponentially, as follows (4. 30) Y, = Yx +Y2 where •g b b i 7t 71 -a "r 1 10 7L , „ Vn n & J dpdp' (4. 31) (4. 32) K = 4 /^ - 5 ,^ oa i Here, C,2 is the part extracted from C, of equation (4.3) such that ^ becomes the part of Yt that is coming from infinite medium in front of the coaxial line. C,2 is given by _ (a i - 8i2ca 2)(Wtan(a,^)) £ 2 Also, L is given by L = Rel ,1-^5+ *,1 Here (4. 33) 8 !2 ca 2 + J a i tan(a ,<i) L (4. 34) is chosen in such a way that the range of x within 0 to I has all the poles of the integrand, in case of dielectric followed by second medium. Also, L has been chosen in such a way that tan(a {d ) « - j in equation (4.33). R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Yx can be solved easily using a computer. Next, Y2 is solved using singularity extraction technique. Assume that there exists one complex pole Xp = x ' + j%" j near the real axis of integration, between %= x, and %= x2• Then, the integration of equation (4.32), i.e., from 0 to L , can be split into two parts - one that contains singularities along the path of real axis (^ 2), while the other does not contain any singularities Xi j+ L° or, I \dX T2 (Y2l) Xi JV2r/x (4.35) x2 (4.36) Y2 = Y2l + Y22 where, (4.37) Y2X can be solved directly while Y22 has to be solve using the singularity extraction technique as follows y12 = 4 J K - n 2)dx +$ }n 2dX (4.38) Here, the term ( ^ - n 2)does not contain any singularity. The function n 2 is given by n 2= —— R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (4.39) 98 where j xp = %' + j%" is the complex pole of W2, the integrand of Y2, and R' is the residue T2 the pole. ^2 nuJ(lp)Be{Xp)xP (4. 40) dA^p) dm where, g(xP) is the denominator of i.e., the product of a, and the denominator of C,2 = (4. 41) 4 * + / “ i tan(a irf)] and dglxp) d% —)—- = u' + yu" Si«i X P u —— -------------------— (4. 42) e ia 2X p (4. 43) j --- — (4. 44) a xd s e c 2 (a ,* /) + 2 ta n (a , ^ )] Also, ^2„um[xP) is the numerator of C,2. With this, the first part of equation (4.38), i.e., Y22, is solved. Now, the second part of Y22 can be solved analytically as follows (4.45) Xi R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 99 ?' = f l n X"2+(%2-X')2 (4. 46) x"2+(xi + x')2 q ['= R ' tan'1 X2-X q ”= R ' tan (4. 47) X" X ~Xi X" (4. 48) Symbols that are not defined here exclusively are same as those in equations (4.5) through (4.15). Yx can be solved easily using a computer. The singularities in Y2 can be extracted. However, the time required for this type of numerical solution was not found very practical. Therefore, this method is not studied beyond this point. 4.3.3 Aperture Admittance of a Coaxial Line, Numerical Solution III: Here the aperture admittance of the coaxial line is solved numerically, by direct integration. Integration along the real %-axis requires consideration for the possible singularities on this axis. If s, >e, , then the poles along the real axis in the region kx to k2 correspond to the surface wave modes excited by the coaxial line, while for s, < s2, these poles correspond to the guided wave modes supported by the medium 1. Number of modes supported by the structure depends on the thickness of the first medium, incident signal frequency, and the permittivities of the two media. In order to facilitate the evaluation, the range of integration can be split into two parts as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 100 L Y,=\ ao J+ I.w (4. 49) .0 Here, L has been chosen to be sufficiently large such that that given in equation (4.34), and that that when x = L > a i ~ a i * ~JX L » £, and L is greater than L » k2. L should be such • With these approximations, equation (4.49) can be improved as L, (4. 50) JWx+7'C '¥ = (4. 51) B. *fl R (4. 52) y Next, assume that there exists one complex pole %p = x ' + j l " , near the real axis of integration, between x = X i a n d X = X 2 • Then the integration of equation (4.50) can be divided into two parts as follows: (4. 53) where, Xi L J+ f 'VdX + jC Xi R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (4. 54) 101 and (4. 55) K =tj'i-dZ Here, it should be noted that C is a constant for a given coaxial line and therefore, it does not require repeated evaluation. Once the constant C is computed for a given coaxial line, Yx can be solved by direct numerical integration. Next, in order to overcome the problem of integrating along real axis of y hi the presence of singularities, many techniques available in the literature are investigated here. However, it should be noted here that if the first medium is lossy, the integration of equation (4.50) can be done directly without having to resort to singularity extraction technique. 4.3.3.1 Using Singularity Extraction Technique: As discussed in section 4.3.2, again, singularity extraction is used here for integration between X.2 y = y x and y = y 2. Then Y2 can be solved as follows X2 (4. 56) n= Yx R' (4. 57) x-xP does not have any poles and can be integrated directly using numerical techniques. Here, R' is the residue of ¥ at the pole. It is given by R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 102 _ ^num(xp)ffi;(Xp)xf (4. 58) g'(xP) g(xp) is the denominator of 4* when x = %p, and g'(xp) is the derivative of g(xp) with respect to x . g'(xp) is given by (4. 59) + jv g S|Cta P eia iXP u ———;—- + —;—~ (4. 60) o" = - x p[ct,c? sec2(a,rf) + 2tan(a,£/)j (4. 61) s ,a , s2a. Also, C,,UOT(xp) is the numerator of £ . With this, the first part of Y3 is solved. Now, the second part of Y2 can be solved analytically as Xz $ jn A = 5 [ 9'+x?r+?;')] (4. 62) x ,,2+ ( * ' - x 'P R\ 4F'=— ln| (4. 63) X"2+(a' + X')2 q\'= R' tan' q%=R' tan'1 b '- r X" X" R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (4. 64) (4. 65) 103 This method was found very useful in terms of computer speed, the results were found to be accurate and time efficient. 4 .3 .3 .2 U sin g T a y lo r's Series E x p a n sio n : Here the integrand of Y2 of equation (4.50) can be expanded by Taylor’s series, around the pole as follows (4. 66) where, <^(xP) is C evaluated at the root x p, and C'(x„) is the derivative of ^ evaluated at the root yp. c ( x P) is given by (4. 67) where, s,um(x) Is the numerator of C, , and <^„(x) is the denominator o f^ . The prime ( ') indicates their derivatives, given by 1 . Qum{l) = - 1 — + J'zn (Xi f ta n (a ,i/) d a 2 s e c 2( a ,i/) > a (4. 68) (4. 69) This solution is not further investigated in this research. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.3.33 Using Approxim ate Substitutions For The Integrand, Near The Poles: Here the C, of integrand of Y2 of equation (4.50) can be approximated as follows (4. 70) where, C, = l-2 -S-12-- 2— -^ e -2ja,‘/ +2 £ I2®12 ^ 1 e12a , + a VSi2^2 ^ 1^ e -4ja (4. 71) , or, another approximation that can be used here is a , + y'8s ]2a [a 2d 1 , 1 | ti2 -4(a,c?)2j |9 ti2 -4(a,<3?)‘ j (4. 72) 1 1 (ti2 -4 (a ,rf)2) (9k 2 -4 (a ,rf)2) s I2a 2 + j% a ]d This solution is not further investigated in this thesis. 4.3.4 Special Cases The solution given in section 4.3.3 is general, special cases have been studied by many researchers before. When the appropriate conditions or approximations are introduced, this general solution given in equation (4.50) reduces to special cases like Quasi Static Approximations, Conductor as Second Medium and Coaxial Line Terminated by Infinite Medium. These cases are detailed here. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.3.4.1 Quasi Static Approximations: This case has been studied by several researchers before. The solution suggested for this case involves triple integrals which consume lot of computer time. However, when the solution of equation (4.50) is simplified for this case, the solution becomes one simple direct numerical integration which is very easy to solve. This solution is explained here. In quasi static approximations, the static capacitance of the coaxial line is found by dividing equation (4.50) by /co and enforcing co = 0 in the resulting equation, as follows (4.73) Y ,=j<oC Therefore, C = "“ MS i l GO - » 0 where, Cs /co (4.74) is the static capacitance of the coaxial line at the aperture. When equation (4.74) is applied on equation (4.50), the term C, of equation (4.50) reduces to i+ ^ ta n h fe ) s12c +tanh(x^) This is because, when co -» 0, a , -» -/co and a 2-> -/co . Therefore, equation (4.50) can be rewritten as R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. The term inside the bracket in equation (4.76) (except / ) is a cable dependent constant and can be kept pre-computed. The constant C is defined in equation (4.52). This approximation is valid coaxial probes with electrically small apertures and are very fast to compute. 4.3.4.2 Conductor as Second Medium: Here, the second medium is assumed to be a conductor. Therefore, e12c -» 0. With this, the term £ of equation (4.50) reduces to Therefore, Y, of equation (4.50) reduces to (4. 78) This is a direct integration from 0 to L , the time required to compute is much less that the methods suggested before. The constant C is defined in equation (4.52). This analysis is valid for all the frequencies where equation (4.50), i.e., variational formula, is valid. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.3.4.3 Coaxial Line Term inated By Infinite Medium: In this case, d - » oo, the aperture admittance of equation (4.50) reduces to the following equation r imfl V \ a, ; B ' d y + j C 4.4 C O M PA R ISO N O F THEORETICAL AND EXPERIMENTAL APERTURE ADM ITTANCE23 Several possible numerical procedures for the computation of aperture admittance of a coaxial line based on the variational expression of equation (4.2) have been discussed in section 4.3. Computer programs have to be written in order to compute the aperture admittance from these solutions. O ut of these possible techniques, the solution of equation (4.50), section 4.3.3 was chosen for numerical programming. The computer program was written in FORTRA N using MICROSOFT FORTRAN COMPILER VERSION 5.0. This program is given in Appendix B. In order to solve equation (4.50) numerically, the singularity needs to be extracted. Singularity extraction technique [106, 122] needs the determination of poles of the integrand of the aperture admittance. Mueller’s Method [108] was used to determine the poles of the integrand, in case of dielectric termination. The limitation of equation (4.50) need to be 2 The experimental results for the sections 4.4.1.3, 4.4.1.5, 4.5.1.3 and 4.5.1.5 were obtained from reference [ 89]. 3 The experimental results for the sections 4.4.1.4, 4.4.1.6, 4.6.1.4 and 4.6.1.6 were obtained from reference [31]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. determined accurately for dielectric termination, otherwise the error in aperture admittance increases rapidly. The results obtained by these solutions were compared with the experimental results. Aperture admittance of most of the experiments showed very good agreement with that of theory, while many others' compared very poorly with theory because of experimental limitations. Some of the limitations of the experiments are • air gap between the terminating material and coaxial line • inaccurate measurement of thickness of the terminating materials ® inaccurate calibration. 4 All the experimental results were calibrated by 3 - standard calibration method [93], with known standards such as air, Teflon, Methanol etc. It should be noted that the experiments that did not yield good results are not listed here. Among the solids tested for aperture admittance were Teflon followed by air, Polyethylene followed by air, air followed by Conductor. Among the liquids tested experimentally were Glycerol followed by Teflon and Glycerol followed by Conductor. Teflon material thickness was d =1.7 mm, d =3.2 mm, d =6.5 mm and d = 13.7 mm. Polyethylene thickness was d =3.2 mm, d =6.7 mm and d =13.3 mm. Glycerol layer thickness used was in the range of d =0.5 mm to d =10 mm. Coaxial line dimension was 8.3 mm, i.e., inner conductor radius a =1.124 mm and outer conductor R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. radius b =3.62 mm. Some of the materials were also tested with 3.6 mm coaxial lines with, a =0.455 mm and b =1.499 mm. Frequencies used were in the range of 0.5 G Hz to 3.5 GHz. The results of these experiments were then compared with the theoretical results. Some of these results are discussed here. 4.4.1 Discussion on Experimental Results: 4.4.1.1 Teflon/Air, 8.3 m m Coaxial Line, d = 1.7 mm, f =0.5 to 3.5 G H z : In one of the experiments Teflon of thickness d = 1.7 mm was used as first terminating medium of the coaxial line. Second medium was air. Frequency was varied between 0.5 GHz to 3.0 GHz. The dimensions of the probe used 'were that of 8.3 mm coaxial line. The comparison between the theoretical aperture admittance of equation (4.50) and that of experiments showed very good agreement between theory and experiment, as shown in Figure 4-13. Experimentally, the aperture conductance was found be zero since Teflon is a good dielectric. However, theoretically this was found to be of very small value. The aperture susceptance was very close to that of theory. 4.4.1.2 Teflon/Air, 3.6 m m Coaxial Line, d = 1.7 to 13.7 m m , f =1.3 G H z: In another experiment, 3.6 mm coaxial line was used, with Teflon as first medium and air as second medium. Frequency used was 1.5 GHz. Teflon .thickness was varied in steps as d = 1 .0 mm, 3.2 mm, 6.5 mm, and 13.7 mm. Here also the aperture admittance showed very good agreement with theory, as depicted in Figure 4-14. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 110 4.4.1.3 Polyethylene/Air, 8.3 m m Coaxial Line, d - 3 .2 m m , f =0.5 to 3.5 G H z : In this experiment, 8.3 mm coaxial line was used with 3.2 mm thick Polyethylene as terminating medium. Second medium was air. Frequency was varied between 0.5 G H z to 3.5 GHz. In this case, the results were in good agreement with theory up to 2 GHz, beyond this frequency, the experimental result deviated slightly from that of theory. The reason could be attributed to the effect of air gap and inconsistencies in experimental set up, at higher frequencies. These results are shown in Figure 4-15. 4.4.1.4 Free Space/Conductor, 8.3 m m Coaxial Line, d = 1 to 7 m m , f = 0.8 G H z : Similarly in another experiment, air followed by Conductor was used, with thickness varying from 1 mm to 7 mm. Frequency used was 0.8 GHz with a 8.3 mm coaxial line sensor. For theoretical purposes, sj = 0 - j 106 was used for conductor. The experimental results showed very close agreement with theory, as depicted in Figure 4-16. 4.4.1.5 Glycerol/Teflon, 8.3 m m Coaxial Line, d =0.5 to 12 m m , f = 2 G H z: In this case, 8.3 mm coaxial line was used with Glycerol as terminating medium with thickness d = 0.5 mm to 12 mm. A Teflon container was used to hold Glycerol. Frequency used was 2 GHz. In this case, as thickness was increased, the experimental values were found deviate slightly from that of theory, as shown in Figure 4-17. 4.4.1.6 G lycerol/C onductor, 8.3 m m Coaxial Line, d =0.5 to 10 m m , f = 0.8 G H z : In this case, 8.3 mm coaxial line was used with Glycerol as terminating medium with thickness d = 0.5 mm to 12 mm. Conductor was used as the R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. second medium. Frequency used was 0.8 GHz. Here, the results were found to be in very good agreement with that of theory, as shown in Figure 4-18. 4.5 MATERIAL CHARACTERISTICS U SIN G A COAXIAL SENSOR The aperture admittance found theoretically together with the experimental results can be used to invert and find the electrical characteristics of the terminating materials, as given by equation (4.1). The inversion can be done with respect to s ‘,S2 and d (i.e. the permittivity of medium 1, permittivity of medium 2, or the thickness of medium 1), provided the other two variables are known. The theoretical value of aperture admittance for equation (4.1) was found by solving the equation (4.50). Mueller’s method [108] was used to find the zeros of equation (4.1). This method does not need exclusive specification of initial guesses and the results are computed very fast. The inversion technique proved to be very time efficient and accurate. In all of the experiments discussed in the preceding section, the inversion for material characteristics and thickness was carried out. The results were found very satisfactory in case of s \ . However accuracy of s i and d were directly dependent on the thickness of the material and size of coaxial line used. This part is theoretically studied and discussed separately in section 4.6.2. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 112 4.5.1 Discussion on Experimental Results: 4.5A .1 Teflon/Air, 8.3 m m Coaxial Line, d = 1 .7 mm, f =0.5 to 3.5 G H z : Experimental results of the aperture admittance used for the purpose of inversion was that from section 4.4.1.1. The results of inversions for s' were well within the limits of 5% error in 3 out of 8 different frequencies at which measurements were made, as shown in Figure 4-19. However the inversions for si and d were found to be all over the range and was not satisfactory, as shown in Figures 4-20 and 4-21. These results show that the determination of sj, si and d does not depend on frequency (within the frequency range of these experiments). 4.5.1.2 Teflon/Air, 3.6 m m Coaxial Line, d =1 to 13.7 m m , f = 1.5 G H z: The experimental aperture admittance used for the purpose of inversion was that from section 4.4.1.2. The results of inversions for si was well within the limits of 5% error in all 4 different frequencies at which measurements were made, as shown in Figure 4-22. However the inversions for s i and d were found to be all over the range and was not satisfactory, in this case it was because of the small dimensions of the coaxial line used (3.6 mm coaxial line sensor). 4.5.1.3 Polyethylene/Air, 8.3 m m Coaxial Line, d =3.2 m m , f =0.5 to 3.5 G H z : The experimental aperture admittance used for the purpose of inversion was that from section 4.5.1.3. In case of Polyethylene, the results of inversion for e[ was within the limits of 5% error, up to 1.5 GHz, as shown in Figure 4-24. Again, the inversions for sj and d were found unsatisfactory. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.5.1.4 Free Space/Conductor, 8.3 m m Coaxial Line, d =1 to 7 mm, f = 0.4 G H z : The experimental aperture admittance used for the purpose of inversion was that from section 4.4.1.4. In the case of free space followed by conductor, the inversion for dielectric constant of medium 1 was found to be in agreement with the actual value. In this case, second medium is Conductor and therefore, results of inversion for s j was not converging. The thickness of air gap here (i.e. thickness of medium 1) was found satisfactory up to 5 mm. These results are depicted in Figures 4-24 and 4-25. 4.5.1.5 Glycerol/Teflon, 8.3 m m Coaxial Line, d =0.5 to 12 m m , f =2 G H z : The experimental aperture admittance used for the purpose of inversion was that from section 4.4.1.5. This was an example of measuring the electrical characteristics of liquids. The results of inversion for e\ were found to be in agreement with actual value, both in terms of real and imaginary parts. The second medium used here was Teflon. The results of inversion for electrical characteristics of medium 2 was unsatisfactory, so was it for thickness of medium 1. These results are shown in Figure 4-26. 4.5.1.6 Glycerol/Conductor, 8.3 m m Coaxial Line, d =0.5 to 10 m m , f = 0.8 G H z : The experimental aperture admittance used for the purpose of inversion was that from section 4.4.1.6. In this case, the dielectric constant of Glycerol was found very accurately, as shown in Figure 4-27. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.5.2 Three Percent Margin Test (A Theoretical Study O f Effect In Inversions Because O f ±3% Difference Between Theoretical And Experimental Aperture Admittance): Here it was intended to find the amount of error occurred during the process of inversion, when the experimentally measured aperture admittance was varied ±3% over that of theoretical aperture admittance. Since, the error is maximum with smaller dimensions of the coaxial line, here a 2.2 mm coaxial line was chosen. The first medium was Teflon and second medium was air. Frequency chosen was 2 GHz. The results show that when the thickness d is more than twice as much as the coaxial dimension, a variation of ±3% in experimental aperture admittance over that of theoretical one will cause an error of similar amount in e\. This can be observed in the results of Figure 4-28. This result holds good for medium 1 being any material. Therefore, careful measurement of aperture admittance, experimentally, should yield dielectric property of medium 1 very accurately. In case of dielectric property of medium 2, the results started to deviate by more than ±12%, when d was less than 0.2 mm and increased very rapidly. This shows that accurate measurement of e 2’ is possible only when d is very small in comparison with the coaxial line dimensions, for the case of thin material as first medium. The result is depicted in Figure 4-29. In case of inversion for the thickness of medium 1, the results were very similar to that of dielectric property of medium 2. Therefore, only the thickness of thin materials as the first medium is possible, as shown in R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Figure 4-30. Even with very high experimental accuracy and statistical average of number of results would yield a result of s and d to a limited amount of accuracy, when d is large. The following conclusion can be derived from above results. The error caused in the inversion of complex permittivity of medium 1 (e j) appears to be directly proportional to that of difference between experimental and theoretical aperture admittance. This means that, when d is greater than 2b, difference in real part of aperture admittance, between theoretical and experimental values, causes equal amount of error in imaginary part of the s * and vice versa. The reason behind this could be attributed to the fact that EM fields interact directly with first medium at the aperture and the aperture admittance is directly proportional to the complex permittivity of medium 1. However, the same can not be said about inversions of thickness of medium 1 (d) or complex permittivity of medium 2 (sj). When a small different is introduced between the theoretical and experimental aperture admittance of the coaxial line the error in s \ and d increase very rapidly (exponential in nature), as thickness of medium 1 is increased. The reason can be attributed to the fact that EM fields interact less and less with medium 2 as the thickness is increased. The EM fields travel along medium 1 with exponential decay in field strength, as d is increased. This causes an exponentially increasing error as d is increased, even though, the difference between experimental and theoretical aperture admittance kept constant at ±3%. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 4.6 THEORETICAL STUDY O F AIR-DIELECTRIC AND DIELECTRIC-AIR TERM IN ATIO N O F A COAXIAL SENSOR In this part, a theoretical study of coaxial line terminated by 2 different dielectric materials was done, when / = 10 GHz. 8.3 mm coaxial line was chosen. 40 different thickness values were chosen between 0 and 15 mm and aperture admittance were found theoretically. In first part of study, real part of e | was kept constant as 100-jX, where X was varied from -10 to -0.1. The second medium was air. The idea here was to see how well the singularity extraction part of the FORTRAN Program works, as the medium 1 is reduced from a lossy medium to close to a perfect dielectric. The results of this theoretical study showed that there are surface waves in medium 1. The singularity strength increases as X becomes smaller. In second part of study, si was chosen to be a perfect dielectric. Second medium was chosen to be air. The dielectric constant was varied in steps from 1 to 100 as 1, 2, 10, 20, 70 and 100. These results show that the effect of surface waves become clearer, to a significant extent, as e[ becomes larger. In third part, sj was varied between 1 to 100, while first medium was chosen to be air. This study would also help to find out the effect of air gaps in coaxial lines. The results showed that when d = b , the admittance becomes approximately same as that of infinite medium. When thickness of air gap is very small, the effect of air gap becomes significant, specially as s', becomes larger. These results are not shown here. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Finally, in another theoretical study, the effect of air gap on the measurement of aperture admittance was studied. The results are explained in Figures 4-31 and 4-32. The results of equation (4.49) were evaluated with d = 0.1 mm and 0.3 mm. The frequency was varied from 1 GHz to 40 GHz, in steps of 1 GHz. The terminating media were Teflon and water, respectively. The results clearly show that as operating frequency of the sensor is increased, the effect of air gap becomes significant and can not be neglected. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. -x—Y of eqn. (4.50) - Aperture Conductance —x—Y of eqn. (4.50) - Aperture Susceptance ■ Y Experimental - Aperture Conductance • Y experimental - Aperture Susceptance 0.0025 0.002 0.0015 0.001 3 a, 0.0005 < 0 0.5 1.5 2.5 •0.0005 Frequency [GHz] Figure 4 - 1 3 Aperture Admittance of a Coaxial Line, Medium 1 = Teflon Medium 2 = air, 8.3 mm Line, d = 1.7 mm Y of eqn. (4.S0) - Aperture Conductance —x—Y of eqn. (4.50) - Aperture Susceptance ■ Y Experimental - Aperture Conductance • Y experimental - Aperture Susceptance s ~ 0.0006 I■o < <u 3 0.0004 Z a, < 0.0002 Thickness of Medium 1 [mm] Figure 4 - 1 4 Aperture Admittance of a Coaxial Line, Medium 1 = Teflon Medium 2 = air, 8.3 mm Line, d = 1.5 G Hz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. - x - Y of eqn. (4.50) * Aperture Conductance ■ Y Experimental • A perture Conductance • Y of eqn. (4.50) - A perture Susceptance Y experimental - Aperture Susceptance 0.0035 Aperture Admittance [S] 0.003 0.0025 0.002 0.0016 0.001 0.0005 ■ -X - -a - 1.5 -» 2 Frequency [GHz] Figure 4 - 1 5 Aperture Admittance of a Coaxial Line, Medium 1 Polyethylene, Medium 2 = air, 8.3 mm Line, d = 3.2 mm Y of eqn. (4.50) - Aperture Conductance -x —Y of eqn. (4.50) - A perture Susceptance ■ Y Experimental - Aperture Conductance • Y experimental - Aperture Susceptance Aperture Admittance [S] 0.0006 3 4 Thickness o f Medium 1 [mm] Figure 4 - 1 6 Aperture Admittance of a Coaxial Line, Medium 1 = air, Medium 2 = conductor, 8.3 mm Line, / =0. 8 GHz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 120 -x —Y of eqn. (4.50) - Aperture Conductance a Y Experimental - Aperture Conductance • Y of eqn. (4.50) - Aperture Susceptance Y experimental - Aperture Susceptance 0.006 Aperture Admittance [S] 0.005 0.004 0.003 0.002 0.001 0 2 6 4 8 10 12 Thickness of Medium 1 [mm] Figure 4 - 17 Aperture Admittance of a Coaxial Line, Medium 1 = Glycerol, Medium 2 = Teflon, 8.3 mm Line, / = 2 GHz Y o f eqn. (4.50) - Aperture Conductance —x - Y of eqn. (4.50) - Aperture Susceptance a Y Experimental - A perture Conductance • Y experimental - A perture Susceptance 0.007 Aperture Admittance [S] 0.006 0.006 0.004 0.003 0.002 0.001 • 0 1 2 3 4 6 6 7 8 9 Thickness o f Medium 1 [mm] Figure 4 -1 8 Aperture Admittance of a Coaxial Line, Medium 1 = Glycerol, Medium 2 = conductor, 8.3 mm Line, / = 0.8 G Hz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 10 Actual Dielectric Constant - Real P art -x —Actual Dielectric Constant - Imag Part ■ Experimental Dielectric Constant - Real P art • Experimental Dielectric Constant - Imag Part KB a o U O £ °-5 & Frequency [GHz] Figure 4 - 19 Dielectric Constant of Medium 1, Medium 1 =» Teflon, Medium 2 = air, 8.3 mm Line, d = 1.7 mm —x—Actual Dielectric Constant - Real P art a Experimental Dielectric Constant - Real P art Actual Dielectric Constant - Imag Part Experimental Dielectric Constant - Imag Part N E "39 0.6 e U 0.2 JS 0 ------------- X-------------X-------------X-------------X------------- X-------------X *5 ? 2.5 3 a Frequency [GHz] Figure 4 - 20 Dielectric Constant of Medium 2, Medium 1 = Teflon, Medium 2 = air, 8.3 mm Line, d = 1.7 mm R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 122 Actual Thickness ■ Experimentally Determined Thickness m 1,4 S s 1 5 |5 0.8 cu 1 0,6 ou» I 0.4 1.S Frequency [GHz] Figure 4 -2 1 Thickness of Medium 2, Medium 1 = Teflon, Medium 2 = air, 8.3 mm Line, d = 1.7 mm Actual Dielectric Constant - Real Part ■ Experimental Dielectric Constant - Real Part -x- Actual Dielectric Constant - Imag P art • Experimental Dielectric Constant - Imag Part xm e 3 =3 o O 0 h .S- M *3 Q — *-H 14 Thickness of Medium 1 [mm] Figure 4 - 22 Dielectric Constant of Medium 1, Medium 1 = Teflon, Medium 2 = air, 3.6 mm Line, / = 1 .5 G H z R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. — Actual Dielectric Constant - Real P art ■ Experimental Dielectric Constant - Real Part -x- Actual Dielectric Constant - Imag Part • Experimental Dielectric Constant - Imag Part 3 ■3 V s S « 1.5 • Frequency [GHz] Figure 4 -2 3 Dielectric Constant of Medium 1, Medium 1 = Polyethylene, Medium 2 = air, 8.3 mm Line, d = 3.2 mm —x—Actual Dielectric Constant - Real P art —x—Actual Dielectric Constant - Imag Part ■ Experimental Dielectric Constant - Real P art • Experimental Dielectric Constant - Imag P art e *C 0.4 Thickness of Medium 1 [mm] Figure 4 - 24 Dielectric Constant of Medium 1, Medium 1 = air, Medium 2 = conductor, 8.3 mm Line, / = 0.8 G H z R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. — Actual Thickness ■ Experimentally Determined Thickness E E ■0o> .3 i M o 1 2 3 4 5 6 Actual Thickness of Medium 1 [mm] Figure 4 -2 5 Thickness of Medium 1, Medium 1 = air, Medium 2 = Conductor, 8.3 mm Line, / = 0 .8 GHz — Actual Dielectric Constant - Real P art —x—Actual Dielectric Constant - Imag Part ■ Experimental Dielectric Constant - Real P art • Experimental Dielectric Constant - Imag Part x ) Thickness of Medium 1 [mm] Figure 4 -2 6 Dielectric Constant of Medium 1, Medium 1 = Glycerol, Medium 2 = Teflon, 8.3 mm Line, / = 2 G Hz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 7 — Actual Dielectric Constant - Real P art —x—Actual Dielectric Constant - Imag Part ■ Experimental Dielectric Constant - Real Part • Experimental Dielectric Constant - Imag Part 10 8 6 4 2 0 •2 -4 -8 Thickness of Medium 1 [mm] Figure 4 -2 7 Dielectric Constant of Medium 1, Medium 1 = .Glycerol, Medium 2 = Conductor, 8.3 mm Line, / = 0 .8 GHz - Exact Value of Y - o — b 3% E rror in Y —-a - -3% E rro r in Y 2.2 2.18 2.14 2.12 2.1 2.08 ££ 2.06 T3 S 2.04 -a a 2.02 0 2 4 6 8 10 12 Thickness of Medium 1 [mm] Figure 4 - 28 3% Margin Test for Inversion of Dielectric Constant of Medium 1, Teflon followed by air, 8.3 mm Line, / = 2 GHz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Exact Value of Y - -o - + 3% E rro r in Y - -a - -3% E rro r in Y j 8 7 C4 6 S a 5 4 08 1o 3 U #o ir 2 at s 1 0 •1 Thickness of Medium 1 [mm] Figure 4 -2 9 3% Margin Test for Inversion of Dielectric Constant of Medium 2, Teflon followed by air, 8.3 mm Line, / = 2 GHz —o — Exact Value of Y - o — I- 3% E rro r in Y - -a - -3% E rro r in Y 1.4 •• 1.2 • 0.8 - 3 06 at 0.4 • £ 0.2 • 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Actual Thickness of Medium 1 [mm] Figure 4 -3 0 3% Margin Test for Inversion of Dielectric Constant of Medium 1, Teflon followed by air, 8.3 mm Line, / = 2 GHz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 2 Conductance [no gap] -------Susceptance [no g a p ] Conductance [0.1 mm] Susceptance [0.1 mm] ......... Conductance [0.3 m m ]............Susceptance [0.3 mm] 0.016 0.014 5T 0012 4M 1 a < 5 ra 6 < 0.01 0.008 o.ooe 0.004 0.002 0 5 10 15 20 25 30 35 40 Frequency [GHz] Figure 4 -3 1 Comparison of theoretical aperture admittance of a coaxial line sensor by variational expression for infinite medium-Teflon termination, with 0 mm, 0.1 mm, 0.3 mm air gap, 3.6 mm line Conductance [no gap] ------- Susceptance [no g a p ] Conductance [0.1 mm] Susceptance [0.1 mm] ------- Conductance [0.3 m m ]........... Susceptance [0.3 mm] 0.03 0.025 0.02 +3 ■* 1< 0.015 £3 IS 0.01 & < 0.005 0 5 10 15 20 25 30 35 40 Frequency [GHz] Figure 4 -3 2 Comparison of theoretical aperture admittance of a coaxial line sensor by variational expression for infinite medium Teflon termination, with 0 mm, 0.1 mm, 0.3 mm air gap, 8.3 mm line. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 128 Chapter 5 A PERTU RE A D M ITTA N C E BY M E T H O D O F M OM ENTS 5.1 IN T R O D U C T IO N Coaxial line sensors terminated by two layered media has been studied in Chapter 4. A complete solution for the variational formulation of the aperture admittance was given. However, it was assumed there that the reflection caused within the coaxial line is purely because of TEM mode at the aperture. This means that the electric field distribution at the aperture is inversely proportional to the radial distance from the origin. Therefore, it was important to be able to verify this assumption This can be done by evaluating the integral equation given in section 3.3.3, by using numerical techniques. In this Chapter, a solution procedure is presented to evaluate the aperture fields and admittance. Numerical techniques are one of the widely used solution techniques in Electromagnetics and Microwave Engineering. They are very popular because of the easy availability of computers. Among these numerical techniques, Method o f Moments (MoM) is predominant in Electromagnetics and Microwave Engineering. In Electromagnetics, this terminology, i.e., Method of Moments, was first used by R. F. Harrington in 1968 to specify a certain general method for reducing linear operator equations to finite matrix equations [57]. Some other common names for the general concept R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. of solution of equations by projection on to subspaces are Method o f Weighted Residues, Method o f Projections and Petrov-Galerkins Method. The reference for this topic can be found in [ 90, 92, 104]. This Chapter deals with the following topics: • Basics of M ethod of M oments Solution: Method of Moments is a numerical technique used for solving integro-differential equations and is based on the idea of converting linear functional equation (with one unknown quantity) into an equivalent linear matrix equation and then solving it using computers, section 5.2 explains briefly the use of this numerical technique. • M ethod of M oments Solution for A perture Admittance of a Coaxial Line: Instead of assuming the electric field to be inversely proportional to the radial distance from the origin, it is possible to find the actual electric field distribution at the aperture of the coaxial line using the Method of Moments. This field distribution will include all the higher order modes. Once the field distribution at the aperture is known, it is possible to find the aperture admittance of the coaxial line. The problem for Method of Moments solution is described completely in section 5.3, while section 5.4 describes in detail the solution. • Comparison between M ethod of Moments and Experimental A perture Admittance: In section 5.5, the comparisons between theoretical (i.e., the Method of Moments) results of aperture R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. admittance are compared with the corresponding experimental results. 5.2 BASICS O F M ETH O D O F M OM ENT SOLUTIONS The general concept of Method of Moments is solution of linear equations by projection on to subspaces, the basics are given in this section [90]. Consider the following deterministic equation (5.1) -Gf = g where, J3 is any linear operator, / is a known function and g is an unknown function to be determined. Let / be represented a set of functions { /,,/2, / 3......}in the domain of linear operator ^ as a linear combination, as follows: f = Y . a jfj (5- 2) j where, a j are scalar quantities, f } are called as expansion functions or basis functions. Substituting equation (5.2) in (5.1) and using the property of linearity of the operator J 3 , it can be seen that 2 X -£ /, = g (5.3) j Defining a set of testing functions or weighing functions {w,,w2,w3......} in the range of J 2 , and introducing them in equation (5.3), and taking inner products, equation (5.3) becomes R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 131 (5.4) where, / = 1,2,3,....... and (a,b) is the inner product between a and b. Using the linearity of the inner product, equation (5.4) can be written as > y(w<s^ / y) = (w„g; (5.5) Now equation (5.5) forms a set of linear equations, putting these into matrix form, the following equation can be obtained: \( ( W2’-G fl) \ \ a, \ wng? a . = ( wi , g ) : V: ^ : J (5.6) Symbolically, the equation (5.6) can be written as La = g (5.7) If L in equation (5.7) is non-singular, its inverse exists and a in equation (5.7) can be found as follows: a =L g (5. 8) Once the unknown a is found from equation (5.8), the unknown function / of equation (5.1) can be found from equation (5.2). Choice of expansion functions and testing functions plays an important role in the speed of convergence and the accuracy of the solution. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 5.3 FO R M U LA TIO N O F TH E PROBLEM F O R MoM SOLUTION Using the basic theory of electromagnetics, the magnetic fields can be computed inside the coaxial line, as well as in the dielectric medium close to the opening of the coaxial sensor. The boundary condition for magnetic fields require that its tangential components fields have to be continuous across the interface. Using this condition, the fields found inside the coaxial line can be equated to that found in first layer of the terminating media, at the interface, i.e., at z = 0. Thus, an integro-differential equation had been derived in section 3.3.3 as follows (refer to equation (3.124)): T ip + ycos0e; J£ p(p,.°)k.i(p>p')p',,p’ - f c h y ( x ) J i ( w h di (5.9) where, £ cl(p,p'),<;(x) and k ( x) are as given in equations (3.77), (3.108) and (3.111), respectively. The equation (5.9) can be rewritten for the purpose of Method of Moment solution, as follows: u If + yo)s0S/ j£ „(p ',0 )A :ri( p ,p ’) p ’</p' = <oel t j £ l>( p \0 ) z ( p ,p ’) p ,rfP' up (5. 10) where, the terms £ cl(p,p') and Z(p,p') are described briefly in the following sections 5.3.1 and 5.3.2. The left hand side of equation (5.10) pertains to the fields analyzed from inside the coaxial line, while the right hand side of equation (5.10) pertains to the fields analyzed in the first terminating medium of the coaxial line. £ p(p',0) is the electric field distribution along the radial direction (i.e., from a to b) of the coaxial line. £ p(p\0) is the R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 133 unknown in (5.1) that needs to be determined by numerical technique called Method of Moments. 5.3.1 Description of £ cl(p,p') in Equation (5.10): In equation (5.10), the term £ cl(p,p') can be described mathematically as follows: n=Q / n N is the number of modes to be considered at the aperture, k, is the propagation constant in the line, k, is given by k, = c o V p 0e, (5.13) and e, is the dielectric constant of the material in between the conductors of the coaxial line s , = E lre 0 (5.14) P„ is the solution of the following equation, for n > 1, J0(P„6)F0(p„«)-J0((3„fl)7o(p„6) = O R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (5. 15) Here n is an integer that varies between 1 to N as given in equation (5.11). In equation (5.1), the function 0„(p) is given by <t-„(p)s r o(P„«M(P„p)-^o(PH«K(P„p)> and the function A = f . m $ A„ (5. 16) is defined as follows: r hn>- ] It should be noted that for TEM mode ( 5 - 17) n =0 and Po = 0 y o = */ (5. 18) <Mp) = ^ A0 = Jin] a Therefore equation (5.11), for TEM modes can be written as follows: (5. 19) * C1(P,P') = PP'*, 5.3.2 Description of Z(p,p') in Equation (5.10): The Z(p,p') in the right side of equation (5.10), can be written as follows (using the same type of mathematical and logical development that yields the equation (4.50), section 4.3.3): R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (5. 20) z(p,p')= j v d x + J C where, the functions on the right hand side of equation (5.20) are defined as given below Cx_ . ^a, ^j (5. 21) B„ a , + j s l2ca 2 tan(a xd ) (5. 22) s 12ca , + j a , tan(a xd ) £„ = -A(xpM xp') (5. 23) C = ^ ( J x 0rf<|>'+ (5. 24) X0 = COS<}>'-1 (5. 25) R (5. 26) R = Vp2 + p '2 -2pp'cos(j)' The equation (5.20) will have to be solved using singularity extraction technique, as was done in section 4.3.3. Here, L has to be chosen such that L is greater than that given in equation (4.34), and that L should be such that when x = L, L» kx and L » k2 . a, « a 2 « - j % . The integration of equation (5.10) can be split into two portions such that one portion contains singularities while the other does not, as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 136 Z(p,p') = z , ( p , p ' ) + Z2(p,p') (5. 27) where, Xi I. (5. 28) z .(p , p ') = {+ [ v d t+ jC and X: z 2(p,p') X2 J(>p-n)dx+ jnrfx (5. 29) Here, n represents part of the integrand that extracts singularity from T , the integrand of the equation (5.20). Z, does not have poles and can be integrated directly, while Z, can be evaluated as described in the rest o f the portions of this section. In equation (5.29), n is given by n =- R’ (5. 30) X~XP where, %p = %' + j%" is the complex pole of T , that exists near the real axis of integration, between x = Xi and %= x2- Here, R' is the residue o f 'f at the pole. It is given by _ Cmm ( X p ( Xp ) %p R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (5. 31) g(xp) is the denominator of T(p,p') when x = 'LP-> and g'(xp) is the derivative of g(xP) • g'(x„) is given by (5. 32) +j u g Si «,Xp Si «2%f, u =— (5. 33) u" = -Xp [°l\d sec2(a 1af) + 2tan(aI£/)] (5. 34) e2oc2 s2a, and, ^ ( x p ) in equation (5.31) is the numerator of £ . Next, the second part of Z2 can be solved analytically as, X2 4|nrfx=[9'+y(?;'+?;')j R' q '= y i n x " M » ’- x f X''2+(a' + x f . g"= i?'tan 1 (5. 35) (5. 36) fr'-X' X" (5. 37) x '- g ' (5. 38) . x" . Symbols that are not defined here exclusively are same as those in equations (4.5) through (4.15). R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. The formulation of the problem of coaxial line terminated by two layered media for solution by Method of Moments is now complete. 5.4 M E T H O D O F MOM ENTS (MoM) SOLUTION Method of Moment solution for the problem of a coaxial line terminated by two layered media is described in this section, step by step. The problem has been defined before, the solution is developed in accordance with the basics given in section 5.2. The integral of equation (5.10) is a linear deterministic equation. Equation (5.10) can be rearranged and put in following form: u 71CO u ii,Jz(p,p■)Ep(p^O)pyp-_re0sJ/v ,(p ,p ’)£l>(p^O)p■^^p■ I (5.39) In equation (5.39), the unknown £p(p',0) can be represented by a set of linear functions as follows: M Ep(p'.0) = I > ./.( P ') (5.40) x=\ Substituting equation (5.40) in (5.39) and using the property of linearity of the integrating operator, it can be seen that s,c Jz(p,p')/x(p')p'c/p - j e 0s, \ Kc,(p,p’) f x(p')p'dp' a, (5. 41) P where, At is the xth segment along p'. When the expansion functions and weighing functions are the same, the special case of Method of Moment solution is called as Ga.lerk.ins Method. The simplest specialization of R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Method of Moments is called Point Matching. This basically involves satisfying the approximate representation of equation (5.41) at discrete points of region of interest. In terms of Method of Moments, this is the same as choosing Dirac Delta functions as weighing function. The integration represented by inner products in equation (5.41) now becomes a trivial case, which is a major advantage of this method. Next, the continuous variable p is discretized by point matching equation (5.41) at M discrete points, i.e., M x =\ Sic j z ( p v,p')fx(p')p'dp-jEoei j Kcl(py,p')fx(p')p'dp' A A, , (5. 42) Py where, y = 1,2 — M . Each integer value of v is a point at which the left hand side of equation (5.42) is matched with right hand side of equation (5.42). Now, choosing pulse as the expansion function, i.e., the expansion function f x( p') is regarded as constant (with magnitude one) along each partition, equation (5.42) can be reduced to M .r=l E|Cjz(p^P ')p'4> -/£0e/ JX,(p,,p')p'Jp' A, (5. 43) P, Equation (5.43) can be rewritten in the following manner, in order to put it into a matrix form j M 7twp,AZ a4 £icz (p.''’pO _ -7’s ')s /^ '(p ^ p -)j = P V JC=1 — R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (5. 44) where, A = b- a (5. 45) Now, equation (5.44) can be put in a matrix form as follows: Aa = R (5. 46) where, R= P> (5. 47) J_ Pm a. a= (5. 48) aM . A\\ A■\M A= (5. 49) XM\ The elements of matrix A are defined as follows: A , = I"“ P,A [sltZ ( p ,,p 1) - y e 0s,X el( p ,,p ,) ] (5. 50) Now, if A is non-singular then its inverse exists and solution for a in equation (5.46) can be written as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 141 a = A 'R (5. 51) Once a is found, the electric field distribution across the aperture of the coaxial line can be found as follows: £„(p',o) = a ' In equation (5.52), a is a column vector of M (5.52). elements, higher the integer the more closer is the approximation for the continuously varying M, electric field at the aperture, along the radial axis. Further, the aperture admittance of the coaxial line can be found as follows: n = f-IS ‘ (5.53) where, S = J.£p(p',0)c/p' (5.54) a In terms of a , S can be approximated as S^A^a, *=i In equation (5.55), (5.55) ax are the elements of column matrix a . Finally, Ya equation (5.53) is the characteristic admittance of the coaxial line, given by R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. in 142 2 tc (5. 56) In 5.5 CO M PA RISO N O F APERTURE ADM ITTANCE RESULTS BETWEEN MoM F O R TW O LAYERED MEDIA A N D O TH ER M ETHODS Method of Moment solution of equation (5.10) requires numerical programming. The computer program was written in FORTRAN using MICROSOFT FORTRAN COMPILER VERSION 5.0. This program is listed in Appendix C. This section gives a brief idea about how the results of aperture admittance compare with other methods. The methods chosen for comparison of aperture admittance of a coaxial sensor were as follows: • Quasi static approximation of equation (4.73) • Variational expression of equation (4.50) • Method of moments (infinite medium theory1) • Method of moments (two layered media theory) of equation (15.53). 5.5.1.1 A ir Termination, 3.6 m m Line, d —> oo, / = 1 - 40 G H z : Figure 5-1 shows one such comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory, with d-+ oo) for air. The coaxial line 1 These results were the results of MoM computer programs of reference [117] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. sensor used was 3.6 mm line. For the case two layered media, d = 40 mm was used in place of infinite medium. Figures 5-2 and 5-3 show the electric field distribution at the aperture of the sensor, at / = 40 GHz. The aperture admittance of the coaxial sensor by the method of moments (infinite medium theory) and the method of moments (two layered media' theory, with d - » oo) show very good agreement with each other, as shown in Figure 5-1. This is expected since both techniques use the same kind of approximations. However, the aperture admittance from quasi static analysis starts to deviate from that of Method of Moments early in terms of frequency (~10 GHz), specially in case of aperture conductance. This happens because of the assumption used in the formulation of the mathematical model, which is that the static aperture capacitance (i.e., the capacitance of the aperture evaluated when the frequency tends to zero) is constant across the range of frequency considered. O n the contrary, at low permittivities such as that of air, the variational formulation gives a fairly close results (both in terms of aperture conductance and susceptance), when compared to the results from the Method of Moments. The slight difference that can be observed between the Method of Moments technique and the variational technique is due to the assumption that the magnitude of aperture electric field is inversely proportional to the radial distance. However, the computation of electric field distribution at the aperture by the Method of Moments (refer to Figures 5-2, 5-3) show that this is not true. The Method of Moments computations use the existence of higher order modes at the aperture. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 5.5.1.2 Methanol Termination, 3.6 m m Line, d —> oo, / = / - 40 G H z : Figure 5-4 shows comparison of theoretical aperture admittances of a coaxial line sensor for the case of infinite medium Methanol term in at inn The coaxial line sensor used was 3.6 mm line. For the case two layered media, d = 40 mm was used in place of infinite medium. Figures 5-5 and 5-6 show the corresponding electric field distribution at the aperture, at / = 40 GHz.. Again, the aperture admittance of the coaxial sensor by the method of moments (infinite medium theory) and the method of moments (two layered media theory, with d -» oo) show very good agreement as shown in Figure 54. Quasi static approximation deviates from that of Method of Moments much earlier than that for air («5 GHz), in terms of frequency. At permittivities such as that of Methanol, the variational formulation and the Method of Moments give fairly close results (both in terms of aperture conductance and susceptance). As can be seen from Figures 5-5, 5-6 the aperture electric field distribution show s significant deviation from that of TEM mode approximation. 5.5.1.3 W ater Termination, 3.6 m m Line, d < x > , f = 1 - 40 G H z : Figure 5-7 shows comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory, with d -> oo) for water. The coaxial line sensor used was 3.6 mm line. For the case two layered media, d = 40 mm was used in place of infinite medium. Figures 5-8 and 5-9 show the electric field distribution at the aperture of the sensor, at / = 40 GHz. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . Water termination forms an example of the high complex permittivity material termination. Again, here the aperture admittance of the coaxial sensor by the method of moments (infinite medium theory) and the method of moments (two layered media theory, with d oo) show very good agreement with each other, as shown in Figure 5-7. However, the aperture admittance from quasi static analysis deviates from that of Method of Moments much lower frequency (*2 GHz) when compared to air termination. Figures 5-8, 5-9 show that the electric field distribution at the aperture is quite different from that of TEM mode approximation. This means that the effect of higher order modes at the aperture on the aperture admittance is quite significant. This shows up specially in the frequency range from 10 GHz onwards. 5.5.1.4 W ater/Teflon Termination, 3.6 m m Line, d -» oo, f = 1 - 40 G H z : Figures 5-10 and 5-11 show comparison of theoretical aperture admittance of a coaxial line sensor between variational expression and method of moments (two layered media theory) for water followed by Teflon. The coaxial line sensor used was 8.3 mm. The thickness of water layer used was d = 1 mm and d = 5 mm. Frequency was varied between 1 to 40 GHz. Here, when the thickness of the water layer is thin, variational formulation shows very good agreement with the aperture admittance of the Method of Moments. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Table 5-1 Summary of comparisons among four different techniques for measuring dielectric properties of materials, for a 3.6 mm coaxial line sensor, single medium termination Configuration Used Equation Used Q uasi Static A pproxim atio n fo r tw o L a yered M edia V ariational A pproxim atio n fo r tw o layered M edia M ethod o f M oments fo r Infinite M edia M ethod o f M oments fo r tw o layered M edia Figure 24 Figure 34 Figure 2-1 Figure 34 Equation (2.14) Equation (3.123) Important Aperture Aperture admittance is approximation electric field is directly involved in order inversely to simplify the proportional proportional model to the static to the radial capacitance of distance the coaxial line Method o f solution Numerical Numerical Integration Integration Operational Range Up to 10 GHz Up to 40 GHz o f Frequency fo r low dielectric materials Operational Range ofFrequency fo r high dielectric materials Computational Speed Computational Limitations None, higher order modes at the aperture is considered, Equation (3.124) None, higher order modes at the aperture is considered £ Method of Method of Moments Moments Above 40 GHz Above 40 GHz Up to 2 GHz Up to 10 GHz Above 40 GHz Above 40 GHz Very fast Fast Slow Very slow None Evaluation of guided wave and surface wave poles None Evaluation of guided wave and surface wave poles R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. —x— Y Vari. - Aperture Conductance-----------------— h • - Y Quasi - Aperture Conductance------------- ---■ Y MoM [Infinite] - Aperture Conductance • □ Y MoM [2 Layer] - Aperture Conductance o Y Vari. - Aperture Susceptance Y Quasi - Aperture Susceptance Y MoM [Infinite] - Aperture Susceptance Y MoM [2 Layer] - Aperture Susceptance 0.007 Frequency [GHz] Figure 5 -1 Comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory) for air, 3.6 mm line R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 148 Electric Field S tren g th Normalized 1/RHO 350000 300000 250000 200000 150000 100000 50000 0.2 .4 0.6 0.8 1 1.2 1.4 1.6 Thickness of Medium 1 [mm] Figure 5 - 2 |isp| at the aperture of a coaxial line sensor, by method of moments (two layered media) for air, 3.6 mm line, / =40 G Hz 0.2 •10 0.4 0.6 0.8 1.2 1.4 1.6 •• •14 •18 - •20 Thickness of Medium 1 [mm] Figure 5 - 3 Angle of Ep at the aperture of a coaxial line sensor, by method of moments (two layered media) for air, 3.6 mm line, / =40 G H z R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. —*— Y Vari. - Aperture Conductance Y Quasi - Aperture Conductance ■ Y MoM [Infinite] - Aperture Conductance □ Y MoM [2 Layer] - Aperture Conductance • o Y Vari. - Aperture Susceptance Y Quasi - Aperture Susceptance Y MoM [Infinite] - Aperture Susceptance Y MoM [2 Layer] - Aperture Susceptance 0.035 0.03 0.025 • Qi O 0.02 § •O < 4» U ra 0.015 & < 0.005 0 5 10 15 20 25 30 35 40 Frequency [GHz] Figure 5 - 4 Comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory) for Methanol, 3.6 mm line R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 150 Electric Field S trength Normalized 1/RHO 250000 200000 150000 100000 50000 0.2 0.4 0.6 0.8 1 1.2 1.4 Thickness of Medium 1 [mm] Figure 5 - 5 £ p at the aperture of a coaxial line sensor, by method of moments (two layered media) for methanol, 3.6 mm line, / =40 G Hz 0.2 0.4 0.8 1.2 •20 -60 Thickness of Medium 1 [mm] Figure 5 - 6 Angle of Ep at the aperture of a coaxial line sensor, by method of moments (two layered media) for methanol, 3.6 mm line, / =40GHz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. —x— Y Van. - Aperture Conductance y Quasi - Aperture Conductance ■ Y MoM [Infinite] - Aperture Conductance □ Y MoM [2 Layer] - Aperture Conductance • o Y Van. - Aperture Susceptance Y Quasi - Aperture Susceptance Y MoM [Infinite] - Aperture Susceptance Y MoM [2 Layer] - Aperture Susceptance 0.10 0.16 0.14 0.12 0.1 S •o < Z & < 0.06 0.04 0.02 0 10 15 25 35 - 0.02 -0.04 Frequency [GHz] Figure 5 - 7 Comparison of theoretical aperture admittance of a coaxial line sensor between quasi static approximation, variational expression, method of moments (infinite medium theory) and method of moments (two layered media theory) for water, 3.6 mm line R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 152 Electric Field S trength Normalized 1/RHO 90000 80000 70000 60000 SOOOO 40000 30000 20000 10000 0.2 0.4 0.6 0.8 1 1.2 1.4 Thicness of Medium 1 [mm] Figure 5 - 8 ji?p| at the aperture of a coaxial line sensor, by method of moments (two layered media) for water, 3.6 mm line, / =40 G Hz 30 S > © 0> 'eh a < I 0.2 0.4 0.6 0.8 1.2 1.4 -10 -15 Thickness of Medium 1 [mm] Figure 5 - 9 Angle of Ep at the aperture of a coaxial line sensor, by method of moments (two layered media) for water, 3.6 mm line, / =40GHz R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 153 0.3 0 .2 5 o « 0.2 S Y V a n ., d * 1 m m Y V a n '., d * 5 m m y £ t:a & L < 0 .1 5 9 # Y M oM , d » 1 m m O Y M oM , d = 5 m m 0.1 0 .0 5 0 5 15 10 20 25 30 35 40 Frequency [GHz] Figure 5 - 1 0 Comparison of theoretical aperture admittance (conductance) of a coaxial line sensor)by variational expression, method of moments (two layered media theory) for water followed by Teflon, 8.3 mm line 0.2 0 .1 5 0.1 Y V a n ., d = 1 m m oi | Cfl £ I < • - • -• Y V a n .. d = 5 m m 0 .0 5 0 □ 0 Y M oM , d = 1m m O Y M oM . G =5 m m 40 15 Frequency [GHz] Figure 5- 1 1 Comparison of theoretical aperture admittance (susceptance) of a coaxial line sensor) by variational expression, method of moments (two layered media theory) for water followed by Teflon, 8.3 mm line R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 5.6 C O M PA R ISO N BETWEEN M ETH O D O F M OM ENT AND EXPERIMENTAL APERTURE ADM ITTANCE Several theoretical as well as experimental studies were conducted to validate Method of Moment results. Most important of them are discussed here. The experiments were conducted by using Network Analyzer HP8752A. The reflection coefficients of well known standards were measured first, followed by the reflection coefficients of the samples. The following procedure was used in order to measure the reflection coefficient of a coaxial line: 1. Calibrate the N etwork Analyzer with factory supplied calibration standards, (i.e., open circuit, short circuit and 50 Ohm termination) at the port where the sensor would be connected 2. Connect the sensor at the calibrated port, taking care to see that the cable connecting the Network Analyzer and the sensor is not flexed from this position 3. Using time domain gating option of the Network Analyzer, remove the reflections from unwanted discontinuities such as coupling joints. Further, electrical delay should be added to ensure that with a short circuit, the reflection coefficient the sensor shows a uniform 180 degrees phase shift. Alternatively (if time gating option is not available) external calibration may be done by using three samples, which will remove the effects of reflections from unwanted discontinuities. However, in case of samples, it is very R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. important to have accurate aperture admittance values over the required frequency range to avoid errors 4. Do measurement with samples. Further, the following care will ensure the accurate measurement of reflection from the coaxial sensor: 1. All solid samples and the coaxial line sensor aperture should be washed in soap water every time before starting actual measurements 2. The sensor aperture and solid samples should be are dry while conducting the experiment ; 3. All connections should be tight and well coupled, loose connections can change the measurements over the period of measurement 4. In case of solids, air gap should be minimized by pressing the sensor against the sample as hard as possible, air gap can have serious effects on overall measured value as can be seen from Figures 4-31, 4-32 5. The liquid samples used should not be contaminated by repeated experiments with various liquids 6. Care should be taken that the reflection coefficient for open circuit case returns back to the original open circuit values after R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. each measurement. This ensures that there are no residues left at the aperture of the sensor from previous measurement 7. Take care not to exceed the upper TEM cutoff frequency for a given coaxial line sensor, since higher order modes traveling through the sensor is not considered in theoretical modeling 8. In case of external samples, well distributed samples in terms of complex permittivities ensure more accurate results. For example, in case of 3 standard technique explained in section 2.3.2, air, methanol, water can be considered as well distributed in terms of material electric characteristics. Several experiments were conducted on many samples, both solids and liquids. The range of frequency used were between 0.5 G Hz to 40 GHz. Among the solids tested for aperture admittance were Teflon followed by air, Polyethylene followed by air, free space (infinite medium). Among the liquids tested experimentally were Glycerol, Methanol, water and 0.1N Saline Solution. Two types of coaxial sensors were used, with 8.3 mm and 3.6 mm coaxial transmission line. The experiments were conducted with both calibration technique, using 3 standards, and time domain gating of ANA. In former case, the question of accuracy of knowledge of complex dielectric constants comes into picture, specially, when one has to compare the two methods, variational expression for aperture admittance and MoM for aperture admittance. In terms of time required for analyzing the characteristics of a sample material, variational formula is much faster. Some R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. of these results are discussed below, in comparison with that of MoM results. 5.6.1 Discussion on Experimental Results: 5.6.1.1 Teflon/A ir, 3.6 m m Line, d = 6 .5 m m , f = 5 - 40 G H z : In one experiment, a 6.5 mm thick Teflon was used as first terminating, medium of the coaxial line. The second terminating medium was air. Frequency was varied between 5 G Hz to 40 GHz. The coaxial line dimensions were that of 3.6 mm. The comparison of this result with that of MoM results showed very good agreement between theory and experiment. This result can be seen in Figure 5-12. 5.6.1.2 Polyethylene/Air, 3.6 m m Coaxial Line, d = 3.2 m m , f = 5 - 40 G H z : In another experiment, 3.6 mm coaxial line was used with Polyethylene as terminating medium with d = 3 . 2 mm, followed by air. Frequency was varied between 5 G H z to 40 GHz. The experimental results were in good agreement with theory as can be seen in Figure 5-13. 5.6.1.3 W ater/Teflon, 8.3 m m Coaxial Line, d = 1 • 16 m m , f =0.5 G H z: In this experiment, the first terminating medium was chosen as water and second terminating medium was chosen as Teflon. The frequency chosen was 0.5 GHz. The thickness was varied in steps from 0.5 mm to 16 mm. The results are shown in Figure 5-14, the admittance can be seen in agreement with MoM theoretical results. 5.6.1.4 A ir (Infinite Medium), 8.3 m m Coaxial Line,, f = 0 . 5 - 3 G H z: Figure 5-15 shows the experimental results in comparison with MoM results. The sensor used was a 8.3 mm coaxial transmission line. The sensor R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. was left open (i.e., air as infinite medium). The calibration of Network Analyzer was done by time domain gating as explained above. The results show an excellent agreement with MoM theoretical results. The fact that the experimental results were gotten independent of theory is an important factor in building confidence in the theoretical results. These results are shown in Figure 5-15. 5.6.1.5 Teflon/A ir, 8.3 m m Coaxial Line, d = 3.2 m m , f = 0.5 - 3 G H z : In this case, a 8.3 mm coaxial line sensor was used to measure the aperture admittance of Teflon sheet with a thickness of 3.2 mm, the second medium being air. The calibration of N etwork Analyzer was done by time domain gating. The results are shown in Figure 5-16. Also, in this graph, the aperture admittance measurement was taken with sensor kept tightly pressed to the Teflon layer (in order to reduce air gap) and with sensor just kept lightly pressed to the Teflon layer. Since, in these experimental results no theoretical results are involved these results clearly show the effect of air gap. The air gap indeed makes significant change in the measurement, as can be seen in Figure 5-16. 5.6.1.6 W ater/Teflon, 8.3 m m Coaxial Line, d = 1 - 1 3 m m , f =0.5 G H z : In this case, 8.3 mm coaxial line was used with water as terminating medium with thickness d = 0.5 mm to 12 mm. Teflon was used as container for water, therefore the second medium is Teflon. Frequency used was 0.5 GHz. Here, the results were found to be in very good agreement with that of theory, as shown in Figure 5-17. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. —x— y ■- ■+■■• Y ■ Y □ Y V an. - Aperture Conductance Quasi - Aperture Conductance MoM - Aperture Conductance • [expt.] - Aperture Conductance o Y Van. - Aperture Susceptance Y Quasi - Aperture Susceptance Y MoM - Aperture Susceptance Y [expt.] - Aperture Susceptance 0.016 0.014 0.012 Aperture Admittance [S] 0.01 0.008 0.006 0.004 0.002 •0.002 Frequency [GHz] Figure 5 - 1 2 Aperture admittance of a 3.6 mm coaxial line sensor when terminated by Teflon (d = 6.5mm), followed by air R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. —x— Y V an. - Aperture Conductance • Y Quasi - Aperture Conductance ■ Y MoM - Aperture Conductance □ Y [expt.] - Aperture Conductance • o Y Vari. - Aperture Susceptance Y Quasi - Aperture Susceptance Y MoM - Aperture Susceptance Y [expt.] - Aperture Susceptance 0.016 0.014 0.012 Aperture Admittance [S] 0.01 0.008 0.006 0.004 0.002 0 5 10 15 20 25 30 35 40 Frequency [GHz] Figure 5 -13 Aperture admittance of a 3.6 mm coaxial line sensor when terminated by Polyethylene (d =3.2 mm), followed by air R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. — x — Y V a n . - A p e r tu r e C o n d u c ta n c e ....................Y Q u a s i - A p e r t u r e S u s c e p t a n c e □ Y [ e x p t] • A p e r tu r e S u s c e p ta n c e ------------- Y V a n . * A p e r t u r e S u s c e p t a n c e ■ O • -+- • - Y Q u a s i - A p e r t u r e C o n d u c t a n c e Y M oM • A p e rtu re C o n d u c ta n c e • Y M oM • A p e r tu r e S u s c e p t a n c e Y [ e x p t] - A p e r tu r e S u s c e p ta n c e 0.12 0.1 Aperture Admittance [S] 0.08 0.06 :xx' .x-*x*x~x“X—x—xxx—3 •X X X ' 0.04 0.02 +.+++. 0 2 4 6 +++•+.*+•+•■+++•+.*+•+• 8 10 12 14 16 Thickness [mm] Figure 5 - 1 4 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by water ( / =3.0 GHz), followed by Teflon R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 162 • Y M oM ♦ A p e r tu r e C o n d u c ta n c e □ -| < Y [ e x p t] - A p e r tu r e C o d u c ta n c e Y M oM * A p e r tu r e S u s c e p ta n c e o Y [ e x p t] • A p e rtu re S u s c e p ta n c e 0.0006 9 1 < □ 1 □ □ □ 1.5 Frequency [GHz] Figure 5- 15 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by air (infinite medium), calibration of Netw ork Analyzer was done by time domain gating R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. — - * — -------------- Y v a r i . - A p e r t u r e S u s c e p t a n c e Y v a ri. - A p e r tu r e C o d u c ta n c e — — — Y M oM - A p e r tu r e S u s c e p ta n c e ....................Y M o M • A p e r t u r e C o d u c t a n c e X Y [ e x p t, w ith a ir g a p ] • A p e r tu r e C o d u c ta n c e O Y [ e x p t., w ith a ir g a p ] - A p e r tu r e S u s c e p ta n c e 4 Y [e x p t., w ith n o a i r g a p ] • A p e r tu r e C o d u c ta n c e ■ Y [ e x p t , w ith n o a ir g a p ] • A p e r tu r e S u s c e p t a n c e 0.0025 Aperture Admittance [S] 0.002 0.0015 0.001 0.0005 0 0.5 1 1.5 2 2.5 3 Frequency [GHz] Figure 5 - 1 6 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by Teflon (d =3.2 mm) followed by air, calibration of N etw ork Analyzer was done by time domain gating R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ■ • -+■ ♦ • Y v a r i . - A p e r t u r e C o d u c t a n c e O Y M oM - A p e r tu r e S u s c e p ta n c e ■■ ■■ Y v a ri. * A p e r tu r e S u s c e p t a n c e ■ Y [ e x p t.] • A p e r t u r e C o d u c t a n c e x Y M oM • A p e r tu r e C o d u c ta n c e A Y [ e x p t.] - A p e r t u r e S u s c e p t a n c e 0 .0 1 4 0.012 Aperture Admittance [S] 0.01 0 .0 0 8 0 .0 0 6 0 .0 0 4 0.002 0 2 4 6 +-■-+- -5- ■+-1-+-*•+- 10 12 8 • 14 16 Thickness of Medium 1 [mm] Figure 5 - 17 Aperture admittance of a 8.3 mm coaxial line sensor when terminated by water ( / =0.5 GHz) followed by Teflon, calibration of N etw ork Analyzer was done by time domain gating R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 165 C h a p t e r 6 CONCLUSION This dissertation presents a non-destructive technique for the characterization of stratified dielectric materials. An open-ended coaxial line is used as an electromagnetic sensor for this purpose. Its two electrical models are studied for this purpose. The first technique uses a spectral domain analysis and variational technique in order to arrive at an integral expression for the aperture admittance of the coaxial line sensor configuration followed by two layered media. The magnitude of the aperture electric field was assumed to be inversely proportional to the radial distance of the coaxial line (as in the case of TEM mode). Various possible solution for this expression were researched. One of these solutions was further studied and used for subsequent development. The integrand of this expression was studied in detail by plotting 3dimensionl pictures in order to understand the effect of guided wave and surface wave poles on the aperture admittance. The poles due to guided or surface waves were resolved using singularity extraction technique. O n the contrary, the second technique uses spectral domain analysis in order to arrive at an integral equation at the aperture of the coaxial line. This equation takes into account existence of higher order modes at the aperture. It was solved by the Method of Moments. Again, the poles due to R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. guided or surface waves were resolved using singularity extraction technique. Numerical procedures were developed in order to evaluate the aperture admittance by both of the above mentioned techniques. Mueller’s method was used to find the poles of the integrand. These results are compared with experimental data for both solid and liquid samples in the frequency range from 0.5 G H z to 40 GHz. The results validate the correctness of the aperture admittance by both of these techniques. This procedure is then extended to determine the complex property of a dielectric layer or its thickness. O n the other hand, if the layer parameters are specified, we can determine the complex permittivity of the large medium behind it. Numerical programs were developed in order to invert the aperture admittance and find the electrical characteristics of the materials. The characterization procedure is validated by experimental results. It was found that the time required to find the electrical property of a substance was typically less than 2 seconds in a Pentium 133 MHz PC, when variational technique was used. Determination of complex permittivity of medium-2 and thickness of medium-1 were sensitive to the thickness as well the complex permittivity of the medium-1. If the dielectric layer is highly lossy then the aperture fields may not reach up to the interface. A similar situation will exist if the thickness of medium-1 is larger than the aperture diameter, even though the medium may be lossless. These factors cause significant limitations to the dielectric characterization procedure presented in this thesis. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. This w ork has many research possibilities for the future. There are applications involving the determination of electrical property of materials, when these materials have electrical characteristics as a function of distance from the probe. This problem needs to be researched. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 168 BIBLOGRAPHY 1) Akyel, C., and R. G. Bosisio, “New Developments on Automated Active Circuits for Permittivity Measurements at Microwave Frequencies,” IEEE Transactions, Instrumentation and Measurements, Volume 38, Number 2, Page 496-504, April 1989 2) Alexopoulos, George., Analysis of an Open Ended Coaxial Line by the Method of Moments, Master's Dissertation, Electrical Engineering Department, University of Wisconsin, Milwaukee, 1992 3) Anderson, J.M., C.L. Sibbald, and S.S. Stuchly, ; “Dielectric Mersurement Using Rational Functional Model,” IEEE Transactions, Microwave Theory and Techniques, Volume 42, Number 2, Page 199204, February 1994 4) Anis, M.K., and A.K. Jonscher, “Frequency and Time Domain Measurements on Humid Sand and Soil,” Journal of Material Science, Volume 28, Page 3626-3634, July 1st 1993 5) Archer, D.G., and Peiming Wang, “The Dielectric Constant of Water and Debye-Huckel Limiting Law Slopes,” Journal of Physical and Chemical, Volume 19, Page 371-411, March, April 1990 6) Article - “Dielectric Probe Analyses Materials,” Design News, Volume 46, Page 32, September 17th 1990 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 169 7) Athey, T.W., M.A. Stuchly, and S.S. Stuchly, “Measurement of Radio Frequency Permittivity of Biological Tissues with an Open Ended Coaxial Line: Part I,” IEEE Transactions, Microwave Theory and Techniques, Volume 30, Number 1, Page 82-87, January 1982 8) Athey, T.W., M.A. Stuchly, and S.S. Stuchly, “Measurement of Radio Frequency Permittivity of Biological Tissues with an Open Ended Coaxial Line: Part H,” IEEE Transactions, Microwave Theory and Techniques, Volume 30, Number 1, Page 87-92, January 1982 9) Bakhtiari, Sasan., Stoyan I Ganchev, and Reza Zoughi, “Open-Ended Rectangular Waveguide for Nondestructive Thickness Measurement and Variation Detection of Lossy Dielectric Slabs Backed by a Conducting Plate,” IEEE Transactions, Instrumentation and Measurements, Volume 42, Number 1, Page 19-24, February 1993 10) Belhadj-Tahar, N., A. Fourier-Lamar, and Helie De Chanterac, “Broad-Band Simultaneous Measurement of Complex Permittivity and Permeability using a Coaxial Discontinuity,” IEEE Transactions, Microwave Theory and Techniques, Volume 38, Number 1, Page 1-7, January 1990 11) Bianco, B., A. Corana, L. Gogioso, S. Ridella, and M. Parodi, “OpenCircuited Coaxial Lines as Standards for Microwave Measurements,” Electronics Letters, Volume 16, Number 10, Page 373-374, May 8th 1980 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 170 12) Boifot, A.M., “Broadband Method for Measuring Dielectric Constant of Liquids using an Automatic Network Analyzer,” IEE Proceedings Part H, Microwaves, Antennas and Propagation, Volume 136, Page 492-498, December 1989 13) Booton Jr., Richard C., Computational Methods for Electromagnetics and Microwaves, John Wiley & Sons, Inc., 1992 14) Burdette, E.C., F.L. Cain, and J. 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Anderson, “A New Aperture Admittance Model for Open-Ended Waveguides,” IEEE Transactions, Microwave Theory and Techniques, Volume 42, Number 2, Page 192198, February 1994 135) Stuchly, S.S., Gregory Gajda, Lily Anderson, and Andrzej' Kraszewski, “A New Sensor for Dielectric Measurements,” IEEE Transactions, Instrumentation and Measurements, Volume 35, Number 2, Page 138-141, June 1986 136) Su, W., O.A. Hazim, and Imad L.Al-Qadi, “Permittivity of Portland Cement Concrete at Low RF Frequencies,” Materials Evaluation, Volume 52, Page 496-502, April 1994 ‘ 137) Su, Wansheng., and Sedki M Riad, “Calibration of Time Domain Network Analyzers,” IEEE Transactions, Instrumentation and Measurements, Volume 42, Number 2, Page 157-161, April 1993 138) Tanabe, E., and W.T. Joines, “A Nondestructive Method for Measuring the Complex Permittivity of Dielectric Materials at Microwave Frequencies Using an Open Transmission Line Resonator,” IEEE Transactions, Instrumentation and Measurements, Volume 25, Number 3, Page 222-226, September 1976 139) Tanguay, L., and R. Vaillancourt, “Numerical Solution of the Dielectric Equation for a Coaxial Line,” IEEE Transactions, Instrumentation and Measurements, Volume 33, Number 2, Page 88- R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 191 90, June 1984 140) Teodoridis, V., T. Sphicopoulos, and F.E. Gardiol, “The Reflection from an Open Ended Rectangular Waveguide Terminated by a Layered Dielectric Medium,” IEEE Transactions, Microwave Theory and Techniques, Volume 33, Number 5, Page 359-365, May 1985 141) Tinri, Martti E., Anitt Sihvola, and Ebbe G Nyfors, “The Complex Dielectric Constant of Snow at Microwave Frequencies,” IEEE Journal of Oceanic Engineering, Volume 9, Page 377-382, December 1984 142) Tsai Leonard, L.,“A Numerical Solution for the Near and Far Fields of an Annular Ring of Magnetic Current,” IEEE Transactions, Antennas and Propagation, Volume AP-20, Num ber 5, Page 569-576, September 1972 143) Von Hippel, A., Dielectric Materials and Applications, Cambridge, MA: MIT Press, 1954 144) Wei, Yan Zhen., and S. Sridhar, “Radiation Corrected Open-Ended Coax Line Technique for Dielectric Measurements of Liquids Up to 20GHZ,” IEEE Transactions, Microwave Theory and Techniques, Volume 39, Num ber 3, Page 526-531, March 1991 145) Weng, Cho Chew., Waves and Fields in Inhomogenous Media, Van Nostrand Reinhold, New York, 1990 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 146) White, Raymond., and Reza Taherian M, “Measuring the Dielectric Permittivity of Filter Cakes,” Measurement Science and Technology, Volume 4, Page 1021-1023, September 1993 147) Wu, C.P., “Integral Equation Solutions for the Radiation from a Waveguide Through a Dielectric Slab,” IEEE Transactions, Antennas ' and Propagation, Volume AP-17, Number 6, Page 733-739, November 1969 148) Xiping Hu, Harvey A Buckmaster, and Oscar Barajas, “The 9.355GHZ Complex Permittivity of Light and Heavy Water From 1 to 90°C Using High Precision Instrumentation System,” Journal of Chemical and Engineering Data, Volume 39, Page 625-638, October 1994 149) Xu, D., L. Liu, and Z. Jiang, “Measurement of the Dielectric Properties of Biological Substances Using an Improved Open Ended Coaxial Line Resonator Method,” IEEE Transactions, Microwave Theory and Techniques, Volume 35, Number 12, Page 1424-1428, December 1987 150) Xu, Y., R.G. Bosisio, and T.K Bose, “Some Calculation Methods and Universal Diagrams for Measurement of Dielectric Constants Using Open-Ended Coaxial Probes,” IEE Proceedings-H, Volume 138, Page 356-360, August 1991 151) Xu, Yansheng., and R.G. Bosisio, “Nondestructive Measurements of R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. the Resistivity of Thin Conductive Films and the Dielectric Constant of Thin Substrates Using an Open-Ended Coaxial Line,” IEE Proceedings, Part H , Microwaves, Antennas and Propagation, Volume 139, Page 500-506, December 1992 152) Xu, Yansheng., Fadhel M Ghannouchi, and Renato G Bosisio,' “Theoretical and Experimental Study of Measurement of Microwave Permittivity Using Open Ended Elliptical Coaxial Probes”, IEEE Transaction on Microwave Theory and Techniques, Volume 40, Page 143-150, January 1992 153) Yoon, S.S., S.Y. Kim, and H.C. Kim, “Dielectric Spectra of Fresh Cement Paste Below Freezing Point Using an Insulated* Electrode,” Journal of Material Science, Volume 29, Page 1910-1914, April 1994 154) York, Richard A., and R.C. Compton, “An Automated Method for Dielectric Constant Measurement of Microwave Substrates,” Microwave Journal, Volume 33, Page 115, March 1990 155) Zandron, S.D., Chris Pournaropoulos, and D.K. Misra, “Complex Permittivity Measurement of Material by the Open-Ended Coaxial Probe Technique,” Journal of wave-Material Interaction, Volumes 5 and 6, N um ber 4, October 1991 156) Zandron, Steven David., Determination of Complex Permittivity Using the Open-Ended Coaxial Line Method, Senior Thesis, Electrical Engineering Department, University of Wisconsin, Milwaukee, 1990 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 157) Zhou, Zheng-Ou., Ding Yi-Yhan, and Gong Yao-Huan, “Geological Measurements of Dielectric Constant,” Microwave Journal, Volume 27, Page 159-160, July 1984 158) Zuercher, J., L. Hoppie, R. Lade, S. Srinivasan, and D. Misra, “Measurement of the Complex Permittivity of Bread Dough by a n ' Open Ended Coaxial Line Method at Ultrahigh Frequencies,” International Microwave Power Institute, Volume 25, Number 3, Page 161-167,1990 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 195 A p p e n d i x A LISTING OF SYMBOLS USED IN THESIS Table A -1 Symbols used in electromagnetic theory # S ym bol D escription a n d U nit 1 E electric field [volts/meter] 2 H magnetic field [amperes/meter] 3 D electric displacement [coulombs/meter2] 4 B magnetic displacement [webers/meter2] 5 J. electric current density [amperes/meter2] 6 J. magnetic current density [volts/meter2] 7 Pe electric charge or electric monopole [coulombs/meter3] 8 Pm magnetic charge or magnetic monopole [webers/meter3] 9 E = e 0e r permittivity of lossless medium [farads/meter] 10 S c = E 0E* permittivity of lossy medium [farads/meter] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 11 s* =e' - ye" complex permittivity or complex dielectric constant in time harmonic form (defined in frequency domain alone) 12 e'=er real part of s* (defined in frequency domain alone) imaginary part of s* (defined in frequency domain 13 cos0 alone) loss tangent (defined in frequency domain alone) 14 ta n (S ) = 7 7 relative permittivity of lossless medium 15 16 s 0 = 8.854 x 10-1 2permittivity of free space [farads/meter] permeability of.the medium [henrys/meter] 17 18 Mr relative permeability of the medium 19 p.0 = 47t x 10“ 7 permeability of free space [henrys/meter] 20 a conductivity of the medium [siemens/meter] 21 c = 2 9 9 ,7 9 2 ,4 5 8 velocity of light in free space [meters/second] 22 / frequency of the fields [hertz/second] 23 CO angular frequency of the fields [radians/second] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Table A - 2 Symbols used in transmission line theory # S ym b o l Description a n d U n it 1 V voltage [volts] 2 i current [amperes] 3 R resistance [ohms-Q/meter] 4 G conductance [siemens/meter] 5 C capacitance [farads/meter] 6 L inductance [henrys/meter] 7 Z = R + jX impedance [Q/meter] 8 Y = G + jB admittance [siemens/meter] 9 X =oL reactance [Q/meter] 10 B 11 V j? ) = (oC susceptance [siemens/meter] incident voltage at length d from the beginning of transmission line [volts] 12 reflected voltage at length d from the beginning of transmission line [volts] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 13 V{d) = Vinc{d) + Vnf{d) total voltage at length d from the beginning of transmission line [volts] 14 incident current at length d from the 4 X d) beginning of transmission line [volts] 15 reflected current at length d from the L f {d) beginning of transmission line [volts] 16 total current at length d from the l(d) = Iinc( d ) ~ I Kf{d) beginning of transmission line [volts] z»=J! 21 22 ir characteristic admittance of II T3 K transmission line [siemens/meter] 1 a"" •3. 8 "ST ^ III II 20 transmission line[Q/meter] [^J_N 18 19 characteristic impedance of II 17 voltage reflection coefficient p = 20iog,0{ /[» (r)f+ [3 (r)]i } magnitude of reflection coefficient [dB] J 3 (r)l A= tan T S = l+P 180 . ; x— 5R(r) 7i angle of reflection coefficient [degrees] voltage standing wave ratio 1 -p R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 23 24 v 26 [Q/meter] i i +r 0 z0 i - r 0 z0 23 input impedance of transmission line z„ = 1 + r° i-r0 Y _ 1 _ !-ro in z in i1+ ^ 1r 0 y Y>n yQ 1i-r0 Y0 \ + r 0 normalized input impedance of transmission line input admittance of transmission line [siemens/meter] normalized input admittance of transmission line 27 y = yfZY = a + j p propagation constant [ /meter] 28 201ogIO(a) attenuation constant [dB/meter] 29 a x 180 P 71 propagation constant [degrees/meter] 30 v„ = — p = Jfk phase velocity in transmission line p II 31 >> [meters/second] wavelength of signals or waves in transmission line [meters] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 200 Table A - 3 Symbols used in the research of coaxial line sensors # S ym b o l D escription a n d U n it 1 E(p,z) electric field inside the coaxial line or in the layered media [volts/meter] 2 H ( P>~) magnetic field inside the coaxial line or in the layered media [amperes/meter] 3 £P(p>o) radial component of electric field at the aperture of coaxial line [volts/meter] 4 # * ( p .o) angular component of magnetic field at the aperture of coaxial line [amperes/meter] 5 £ p( p ,z ) ,£ ; ( p ,z ) ,£ _ ( p ,z ) field components of E (p,z) in cylindrical coordinates, in respective directions [volts/meter] 6 / / p( p ,z ) ,/ / ^ ( p ,z ) ,/ £ ( p ,z ) field components of H (p ,z) in cylindrical coordinates, in respective directions [amperes/meter] 7 E '(p ,z) incident electric field inside the coaxial line, near the aperture [volts/meter] 8 H '(p ,z) incident magnetic field inside the coaxial R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. line, near the aperture [amperes/meter] 9 £ ; ( p ?z ) , £ ; ( p , z ) , £ ' ( p , z ) field components of E'(p,z) in cylindrical coordinates, in respective directions [volts/meter] 10 field components of H'(p,z) in cylindrical coordinates, in respective directions [amperes/meter] 11 scattered electric field inside the coaxial line, near the aperture [volts/meter] 12 H '(p ,z ) scattered magnetic field inside the coaxial line, near the aperture [amperes/meter] 13 e ‘p{ p , z ) , e ; ( p , z ) , e : ( p , z ) field components of E f(p,z) in cylindrical coordinates, in respective directions [volts/meter] 14 h ;( p , z), h ;{ p , z), h s: {p , z) field components of H'(p,z) in cylindrical coordinates, in respective directions [amperes/meter] 15 J«(p ,o) magnetic current density at the aperture of coaxial line, at z = 0 [volts/meter2] 16 components of J„,(p,0) in cylindrical R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 202 coordinates [volts/m eter] 17 e ( x ,*) electric field in layered media, in spectral domain as a function of variable [volts/meter] 18 magnetic field in layered media, in spectral domain as a function of variable [amperes/meter] 19 field components of E(x,z)in cylindrical coordinates, in respective directions [volts/meter] 20 4 field components of fi(x,z) in cylindrical coordinates, in respective directions [amperes/meter] 21 P,<M unit vectors in cylindrical coordinates 22 p,<)),Z distances from origin, in cylindrical coordinates, in respective directions 23 z direction of propagation of incident EM waves in medium 1 and 2, z — 0 being the aperture of coaxial line 24 a inner radius of coaxial line [meters] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 25 b outer radius of coaxial line [meters] 26 d thickness of medium 1 [meters] 27 ja0 = 471 x 10" 7 permeability of free space [henrys/meter] 28 s 0 = 8 .854x 1O'12 permittivity of free space [farads/meter] 29 c =299,792,458 velocity of light in free space [meters/second] 30 / frequency of the fields [hertz/second] 31 co = 2 n f angular frequency of the fields [radians/second] 32 V 2% - --. . . . inM L ^ J v £os / 33 *i(co) = Y0 1 _ r ' /W 0 i + r, characteristic admittance of coaxial line [siemens] aperture admittance of coaxial line (i.e., the load seen by the coaxial line, at z=0) as a function of frequency [siemens] R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 34 r ' * U p ,o ) M p >o ) Voltage reflection coefficient at z = 0. 4 , c(p> o) ^ (p ,o ) ^pTEM.inc(p^) Epli:M.ref{P^) __ ^0/'£A/.mc(P’0 ) ^*7£A /.rc/(p’0 ) 35 capacitance of coaxial line at the c(») = i aperture [farads] lim 36 = co 37 K — —> 0 j o static capacitance of coaxial line at the aperture [farads] permeability of the medium inside the M' = M'oM'r = Po coaxial line or in the layered media [henrys/meter] 38 relative permittivity of the dielectric in E/ between tw o conductors of coaxial line 39 40 E lc - E 0E l — E rl 41 propagation constant inside the coaxial line kj = co tJ P-oS qE/ E1 •a i J permittivity of medium 1 [farads/meter] COSo complex permittivity or complex dielectric constant of medium 1 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. propagation constant in medium 1 o CO 42 3II S 205 43 • a 2 £ 2c 44 £ 0£ 2 £ r2 * £ 2 J permittivity of medium 2 [farads/meter] cos0 complex permittivity or complex dielectric constant of medium 2 45 k2 = 0 3 ^ oz 2c propagation constant in medium 2 46 £i ratio of complex permittivities of 8,2 47 2 a, = - A / x 2-£,2 medium 1 and medium 2 a variable obtained while doing spectral domain transformation [ / meter] 48 a 2 = - j h 2~ kl a variable obtained while doing spectral domain transformation [ / meter] R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Table A - 4 Definitions commonly used in electromagnetic theory Sym bol D e s c r ip tio n 1 *(A) real part of a time harmonic vector A 2 3(A) imaginary part of a time harmonic vector A 3 A = A(r,f) time domain variables are functions of space and time 4 A = A(r,co) frequency domain variables are functions of 5 r = xx + yy + zz the distance in 3 dimensional space, from origin of 6 A any arbitrary vector quantity in either space and frequency 3 dimensional space time or frequency domain 7 * (4 real part of a frequency domain (phasor) quantity 8 3(A) imaginary part of a frequency domain (phasor) quantity 9 A = A (d,t) A A time domain variables are functions of length and time 10 A = A ( d , co) frequency domain variables are functions of length and frequency R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 207 11 d the distance in length, from origin of the transmission line 12 A any arbitrary scalar quantity in either time or frequency domain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2C8 A p p e n d i x B LISTIN G O F FO R T R A N PROGRAM S IN V A RIA TIO N A L FO R M U LA TIO N B.l BRIEF EXPLANATION O F FO R TRA N PROGRAM S The theory behind the study of coaxial line sensors terminated by two layered media has been given in Chapter 4. Section 4.3.3.1 describes the solution of variational expression based on singularity extraction technique, section 4.3.4.1 describes the solution for aperture admittance based on quasi static approximation. Further section 4.6 describes the material characteristics of terminating stratified media, specifically, the complex permittivity of medium 1 (s' ), complex permittivity of medium 2 (s^) and the thickness of medium 1 (d ). In order to determine the complex permittivity of either media or the thickness of first medium from a measured value of aperture admittance, equation (4.50) needs to be inverted accordingly. This was done iteratively, using Mueller’s method. In this Appendix these programs listed. The programming was done using FORTRAN language and compiled using MICROSOFT COMPILER VERSION 5.0. Figure B-l depicts the flow of logic behind the programming of aperture admittance solution of equation (4.50). The symbols used in this flow chart need to be understood in conjunction with the theory explained in section 4.3.3.1. The process of determination of the complex permittivity of either media or the thickness Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of first medium from a measured value of aperture admittance can be better understood by following the example given below. Consider the inversion process of the complex permittivity of medium 1, given the measured aperture admittance, complex permittivity of the second medium and the thickness of medium 1. The following two steps, now, illustrate the process of determination of complex permittivity of medium 1, by using Mueller’s method: 1. Mueller’s method requires 3 initial values to start the iterative process, 1+jO, 1.5+j0 and 2+jO were used as initial guess. Aperture admittance was initially computed to an approximation value using the following relation: % (£ * -')* • X = 0 . — ,••• A 20 1 m where, the symbols are same as that in section 4.3.3, I is a discrete summation variable 2. After the inverse process is complete with respect to the approximate aperture admittance of equation (1) is complete, the initial values are updated with one obtained from the first iteration as s \ , £ | + 0.5 and s \ - 05. Following the algorithm given in Figure A-l, the aperture admittance is computed from equation (4.50) this time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 C L Start ~ ^ > Set initial variables Using Mueller's method find the poles near the real axis Do the integration of equation (4.50) and store result in Y Set j = number of poles found Arrange poles in increasing order Set a' = Min(kl, k2 ) (Refer to equations (4.53) through (4.56)) Set b' = Re(n-th root + del), where del = a small no. Do the integration of equation (4.56), add result to Y2 Do the integration of equation (4.54), store result to Y1 Y = Y1 + Y2 End Figure B - 1 Flowchart depicting the forward process of determining the aperture admittance of a coaxial line terminated by two layered media Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The three initial values are generated quickly by step 1, to start iterations. These are refined subsequently in step 2. A similar procedure may be developed for determination of complex permittivity of medium 2 or thickness of medium 1. It should be noted here that the number of poles increase as thickness of medium 1 is increased, in case of loss less media termination. This may increase the numerical error caused at each of these points. The FORTRAN programs were constructed on modular basis, that is, the entire program is broken into number of well defined modules, then required modules are later included in the corresponding main program. The main programs are described in Table B-l, the modules or group of subroutines used in these main programs are given in Table B-2. Cable dependent constant C given in equation (4.52) need not be computed repeatedly, therefore a separate program called C.FOR was written to compute this value. This value will be later used in other main programs. The program ADM TNCE.FOR also finds the quasi static approximation value of aperture admittance, since it was necessary for comparisons. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table B - 1 Main Fortran Programs for Chapter 4 s e c ti o n # M a in P ro g ra m D e s c r ip tio n B .2 .1 ADM TN CE .FOR Aperture admittance (Figure B-l) B .2 .2 EPSLN-l.FOR Complex permittivity of medium 1 B .2 .3 EPSLN-2.FOR Complex permittivity of medium 2 B .2 .4 THICK-l.FOR Thickness of medium 1 B .2 .5 C.FOR Constant C of equation (4.52) Table B - 2 Subroutine Modules used in Programs of Table B-l # M o d u le s s e c ti o n # M o d u le s s e c ti o n B .2 .6 HEART.FOR B .2 .1 3 SEMODULE.FOR B .2 .7 HEARTQ.FOR B .2 .1 4 Q 1MODULE.F OR B .2 .8 INM ODULE.FOR B .2 .1 5 Q2MODULE.FOR B .2 .9 WRMODULE.FOR B .2 .1 6 R1MODULE.FOR B .2 .1 0 MAMODULE.FOR B .2 .1 7 R2MODULE.FOR B .2 .1 1 MIMODULE.FOR B .2 .1 8 RDM ODULE.FOR B .2 .1 2 RTMODULE.FOR Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 B.2 FO R TR A N PRO G RA M LISTING B.2.1 A D M T N C E .F O R $DECLARE $INCLUDE:'INMODULE.F O R ' $INCLUDE:'W R M ODULE.F O R ' $INCLUDE:'M A M O D U L E .F O R ' $INCLUDE:'M I M O D U L E .F O R ' $INCLUDE: 'R T M O D U L E .FOR 1 $INCLUDE:1SEMODULE.F O R ' $INCLUDE:'Q 2 M O D U L E .F O R ' program a d m t n c e .for double complex e l ,e 2 ,e l 2 ,k l ,k 2 ,k ,r t s ,y,sum,lmin,ct,y q , j ,d double precision e o , f ,a,b,c,1,om,ko,11,12,r k l ,r k 2 ,a y , i l ,fr integer s z ,n r t s ,u , y e s ,dd,dbg,stage,min integer*2 hour,minute,sec dimension rts (20) open(unit = 1, file = 'c . d a t ',status = 'unknown') !INITIALIZE call init(f,j ,eo,el,e2,e l 2 ,ko,kl,k2,rkl,rk2,a,b,c,d,k,1 ,yes,sz,om, +hour,m i n u t e ,sec,ay,dbg,il) dd = 1 $INCLUDE:'H E A R T .F O R ' $INCLUDE:'H E A R T Q .F O R ' ! FINALIZE AND WRITE ALL THE RESULTS call wr(el , e 2 ,f ,kl,k2,rkl,rk2,a,b,d,1,y e s ,h o u r ,minute,sec, +nrts,r t s ,y ,c t ,yq,dd,k) stop end ! MAIN PROGRAM UNIT -- END B.2.2 EPSLN-1.FOR $DECLARE $INCLUDE:'M A M O D U L E .F O R ' $INCLUDE:'M I M O D U L E .F O R ’ $INCLUDE:'R T M O D U L E .F O R ' $INCLUDE:'R 1 M O D U L E .F O R ' $INCLUDE:'S E M ODULE.F O R ' program epsln-l.for double complex ysnsr,ytemp,j,e2,mainrt,epsll,ges,d double precision a ,b,f ,r,i,c,pi,tempi,temp2,om integer s z ,l o s ,error,last,lastl,material,n c ,m i ,me2 integer*2 thl, t m l ,t s l ,tssl, th.2,t m 2 ,t s 2 ,tss2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. real ttm,ttml,x external mainrt open(unit = 1, file = 1c i l .d a t 1,status = 'unknown') ’ WRITE ALL INITIAL STUFF write (*,*) 'COAXIAL LINE - MEDIUM-1 EPSILON COMPUTATION:' write (* *) *????????????????????????????????????????????????????? +?????????????????????' write write write write write (*,*) (*,*) (*,*) (*,*) (*,*) 'Y ( w ) ,a,b,d,f,epsilon2 ========> epsilonl' 'MEANT FOR SINGLE RUN [0 GHz < f < 10 G H z ] ' ' [d < 15 mm] ' ' [RESULTS ARE STORED IN CI1.DAT] ' 1????????????????????????????????????????????????????? +?????????????????????' write (*,*) ! READ ALL STUFF FROM KEY BOARD write (*,*) 'Select Medium 2: ' write (*,*) '___________________________________________________________ + ' write (*,*) ' [1] Methanol [2] Water [3] Acetone [4] Ethanol [5] +Glycerol [6] Air' write (*,*) ' [7] Teflon [8] Nylon - 66 [9] Polyurethane [10] Po +lyethylene [11] Derlin' write (*,*) ' [12] CarbonTetraChloride [13] An y other material [1 +4] Conductor' write (*,*) '___________________________________________________________ write write (*,*) ' Note: In Cases of Solids, Select [13] if f > 3 GHz' (*,*) '___________________________________________________________ read (*,*) me2 if (me2.gt.14) then write (*,*) '==========> PROGRAM IS TERMINATED, ERROR IN INPUT!' stop endif if (me2.eq.13) then write (*,*) 'Enter e2 [Real Part , Imaginary' P a r t ] : ' read (*,*) r ,i e2 = dcmplx(r,i) endif write (*,*) 'Enter Coaxial Admittance' read (*,*) r ,i ytemp = dcmplx(r,i) ysnsr = ytemp write (*,*) 'Enter ==> 0 - If Medium-1 is a Dielectric' write (*,*) ' 1 - If Medium-1 is Lossy or Not Known' read (*,*) material write (*,*) 'Enter Frequency [GHz]' read (*,*) r write (*,*) 'Enter d [mm] ' read (*,*) i write (*,*) 'Enter ==> 0 - +- x% Admittance Calculations,' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. write (*,*)' 1 - If Not' read (*,*) lastl if (lastl.eq.O) then write (*,*) 'Enter the Percentage Variation Allowed in Admittan +ce: ' read (*,*) x x = x/lOO.dO last = 3 else last = 1 endif write (*,*) 'Enter Cable Size [Enter 1-5]:' write (*,*) ' [1] 2.2mm; [2] 3.6mm; [3] 6.4mm; [4] 8.3mm' write (*,*) ' [5] Any Other Dimensions of Cable' read (*,*) sz if (sz.eq.5) then write (*,*) ' ============>' write (*,*) ' Enter a [mm] : ' read (*,*) a write (*,*) ' Enter b [mm] : ' read (*,*) b write (*,*) ' Enter C [Use C.EXE to find Constant C ] : ' read (*,*) c a = a* l.d-3 b = b* l.d-3 endif write (*,*) 'Enter Maximum Limit for Iterations: ' read (*,*) mi write (*,*) ’___________________________________________________________ ! INITIALIZE if (sz.eq.l) then a = 0.255d-3 b = 0.838d-3 = 3.212895842309772e-4 c elseif (sz.eq.2) then a = 0.455d-3 b = 1.499d-3 = 5.758110164705024e-4 c elseif (sz.eq.3) then a = 0.824d-3 b = 2.655d-3 = 1.002538968584023d-3 c elseif (sz.eq.4) then a = 1.124d-3 ■ b = 3.62d-3 = 1.366445980459857d-3 c endif f = r * l .0d9 pi = 4.do * datan(l.dO) om = 2 * pi * f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j d = d c m p l x (0.Od O ,1.OdO) = d c m p l x (i * l .Od-3,0.do) SET PERMITTIVITY if (me2.eq.l then e2 = 5.7d0 + 27.4d0 / ( l.dO + j * f / 3.d9) !methanol elseif (me2.eq.2) then e2 = 5.dO + 73.dO / (l.dO + j * f / 19.7d9) iwater elseif (me2.eq.3) then e2 = 1.9d0 + 19.3d0 / (l.dO + j * om * 3.3d-12) !acetone elseif (me2.eq.4) then e2 = 4.2d0 + 20.9d0 / (l.dO + j * om * 1.4d-10) lethanol elseif (me2.eq.5) then e2 = 4 . 18d0 + 38.3d0 / ((l.dO + j * om * 2.49d-9)**0.6)!glycerol elseif (me2.eq.6) then e2 = d c m p l x (1.d O ,0. d o ) elseif (me2.eq.7) then e2 = d c m p l x (2.ldO,0 .dO) elseif (me2.eq.8) then e2 = d c m p l x (3.14d0,-0.05d0) elseif (me2.eq.9) then e2 = d c m p l x (3.4 d 0 ,0.d O ) elseif (me2.eq.10) then e2 = d c m p l x (2.2 S d 0 , 0 .dO) elseif (me2.eq.ll) then e2 = d c m p l x (2.8d0,O.dO) elseif (me2.eq.12) then e2 = d c m p l x (2.17d0,0.dO) elseif (me2.eq.14) then e2 = dcmplx(O.dO,-10.d6) endif do error = 1,last if (error.eq.2) then tempi = dreal(ytemp)+x*dreal(ytemp) temp2 = dimag(ytemp)+x*dimag(ytemp) ysnsr = d c m p l x (tempi,temp2) elseif (error.eq.3) then tempi = dreal(ytemp)-x*dreal(ytemp) temp2 = dimag(ytemp)-x*dimag(ytemp) ysnsr =dcmplx(tempi,temp2) endif ! FIND M I APPROXIMATE GUESS FOR EPSILON 1 call gettim(thl,tml,tsl,tssl) los = 0 ges = 1.dO + j call mainrtsolver(e2,a,b,d,f,ysnsr,c,los,g e s ,epsll,nc,mi) write (*,10) dreal(epsll),dimag(epsll) 10 format (' Initial Guess = ',F14.7,', ',F14.7,'i') call gettim(th2,t m 2 ,t s 2 ,tss2) ttml = (th2*3600.+tm2*60+ts2+tss2/l00.) + - (thl*3600.+tml*60+tsn-tss2/l00.) write (*,15) nint(ttml) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 format (' ===> Initial Guess t ook',13,' Sec.===> Please Wait' write (*,*) los = 1 ! FIND EPSILON 1 ges = epsll call mainrtsolver(e2,a,b,d,f,ysnsr,c,los,g e s ,e p s l l ,nc,mi) call gettim(th2,t m 2 ,ts2 ,ts s 2 ) ttm = (th2*3600.+tm2*60+ts2+tss2/100.) + - (thl*3600.+tml*60+tsl+tss2/100 .) ! WRITE ALL STUFF IN THE END write (1,*) write (1,*) write (1,*) write (1,30) a,b,dreal(d),f 30 format (3P,' a = ',F14.3,/,' b = ',F14.3,/, + ' d = ',F14.3,/,1P,' f = ',E 1 4 .3 E 2 ) write (1,40)dreal(e2),dimag(e2) 40 format (' e2= ',F 1 9 .5, ', ',F 1 9 .5, 'i ',/) write (1,45) nint(ttm) 45 format (' Inversion t o o k ',13,' seconds') if (nc.eq.l) then write (1,*) ' *** RESULTS DO NOT CONVERGE ***' write (1,*) ' *** ITERATIONS ALLOWED =',mi endif write (1,*) write (1,*) ' THE EPSILON OF MEDIUM 1 IS FOUND TO BE -' if (material.eq.0) epsll = dcmplx(dreal(epsll),0.dO) write (1,50) dreal(epsll),dimag(epsll) 50 format (' epsilonl = ',F14.6,', ',F 1 4 .6,'i ',/) write (1,60) dreal(ysnsr),dimag(ysnsr) 60 format (' Ysensor = ',F14.9,', ',F 1 4 .9,'i ',/) write (1,*) los = 0 enddo stop end ! MAIN PROGRAM UNIT -- END B.2.3 EPSLN-2.FOR $DECLARE $INCLUDE:'M A M O D U L E .F O R ' $INCLUDE:'M I M O D U L E .F O R ' $INCLUDE:'R T M O D U L E .F O R ' $INCLUDE:'R 2 M O D U L E .F O R ' $INCLUDE:'SEMODULE.F O R ' program epsln-2.for double complex ysnsr,ytemp,j,el,mainrt,epsl2,ges,d double precision a,b,f,r,i,c,pi,tempi,temp2,om integer sz,los,error,last,lastl,material,mel,nc,mi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. integer*2 thl,tml,tsl,tssl,th2,tm2,t s 2 ,tss2 real ttm,ttml,x external mainrt open(unit =1,' file = 1c i 2 .d a t 1,status = ’u n k n o w n 1) ! WRITE ALL INITIAL STUFF write (*,*) ’COAXIAL LINE - MEDIUM-2 EPSILON COMPUTATION:’ write (* *) *?????????????????????????????????????????????????'?'::>'::)? +?????????????????????1 write (*,*) ’Y ( w ) ,a,b,d , f ,epsilonl ========> eps i l o n 2 ’ write (*,*) ’ [MEANT FOR SINGLE RUN [0 GHz < f < 10 GHz] ’ WRITE (*,*) ’ [d < CABLE SIZE] [RESULTS ARE STORED IN C I2.DAT]’ write (*,*) *T??????????????????????????????????????????????????*3? +?????????????????????1 write (*,*) ! READ ALL STUFF FROM KEY BOARD write (*,*) ’Select Medium 1: ’ write (*,*) ’ [1] Methanol [2] Water [3] Acetone [4] Ethanol [5] +Glycerol [6] A i r ’ write (*,*) ’ [7] Teflon [8] Nylon - 66 [S] Polyurethane [10] Po +lyethylene [11] D e r l i n ’ write (*,*) ’ [12] CarbonTetraChloride [13] A ny other m a t e r i a l ’ write (*,*) ’_______________________________________________ '___________ write write (*,*) 1 Note: In Cases of Solids, Select [13] if f > 3 G H z ’ (*,*) ’___________________________________________________________ read (*,*) mel if (mel.gt.13) then write (*,*) ’==========> PROGRAM IS TERMINATED, ERROR IN INPU T ! ’ stop endif if (mel.eq.13) then write (*,*) ’Enter el [Real Part , Imaginary P a r t ] : ’ read (*,*) r,i if (dabs(i).le.O.OldO) then write (*,*) ’Enter a Small Negative Number if Medium l 1 write (*,*) ’ is a Dielectric (Lossless) Ex: - 0 . 0 1 ’ read (*,*) i endif el = dcmplx(r,i) endif write (*,*) ’Enter Coaxial Admittance’ read (*,*) r ,i ytemp = dcmplx(r,i) ysnsr = ytemp write (*,*) ’Enter ==> 0 - If Medium-2 is a D ielectric’ write ( * , * ) ’ 1 - If Medium-2 is Lossy or Not K n o w n ’ read (*,*) material write (*,*) ’Enter Frequency [GHz]’ read (*,*) f f = f * 1.0d9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. write (*,*) 'Enter d [mm]' read (*,*) i d = dcmplx(i * l.d-3,O.dO) write (*,*) 'Enter ==> 0 +- x% Admittance Calculations,' write (*,*)' 1 - If Not' read (*,*) lastl if (lastl.eq.O) then write (*,*) 'Enter the Percentage Variation Allowed in Admittan +ce: ' read (*,*) x x = x/lOO.dO last = 3 else last = 1 endif write (*,*) 'Enter Cable Size [Enter 1-5]:' write (*,*) ' [1] 2.2mm; [2] 3.6mm; [3] 6.4mm; [4] 8.3mm' write (*,*) ' [5] Any Other Dimensions of Cable' read (*,*) sz if (sz.eq.5) then write (*,*) ' ==============>' write (*,*) ' Enter a [mm] : ' read (*•, *) a write (*,*) ' Enter b [mm] : ' read (*,*) b write (*,*) ' Enter C [Use C.EXE to find Constant C ] : read (*,*) c a = a* l.d-3 b = b* l.d-3 endif write (*,*) 'Enter Maximum Limit for Iterations: read (*,*) mi write (*,*) '_______________________________________ ! INITIALIZE if (sz.eq.l) then a = 0.255d-3 b = 0.838d-3 c = 3.212895842309772e-4 elseif (sz.eq.2) then a = 0.455d-3 b = 1.499d-3 C = 5.758110164705024e-4 elseif (sz.eq.3) then a = 0.824d-3 b = 2.655d-3 C = 1.002538968584023d-3 elseif (sz.eq.4) then a = 1.124d-3 b = 3.62d-3 c = 1.366445980459857d-3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 endif pi = 4.do * datan(l.dO) j = d c m p l x (0.O dO,1.OdO) om = 2 * pi * f ! SET PERMITTIVITY if (mel.eq.l) then el = 5.7d0 + 27.4d0 / ( l.dO + j * f / 3.d9) 'methanol elseif (mel.eq.2) then el = 5.dO + 73.dO / (l.dO + j * f / 19.7d9) Iwater elseif (mel.eq.3) then el = 1.9d0 + 19.3dO / (l.dO + j * om * 3.3d-12) !acetone elseif (mel.eq.4) then el = 4.2d0 + 20.9d0 / (l.dO + j * om * 1.4d-10) !ethanol elseif (mel.eq.5) then el = 4.18d0 + 3 8.3d0 / ((l.dO + j * om * 2.49d-9)* * 0 . 6 ) 'glycerol elseif (mel.eq.6) then el = d c m p l x (1.dO,-O.OOOOldO) elseif (mel.eq.7) then el = d c m p l x (2.ldO,-0.0000ldO) elseif (mel.eq.8) then el = d c m p l x (3.14d0,-0.05d0) elseif (mel.eq.9) then el = dcmplx(3.4d0,-O.OOOOldO) elseif (mel.eq.10) then el = dcmplx(2.26d0,-O.OOOOldO) elseif (mel.eq.ll) then el = dcmplx(2.8d0,-O.OOOOldO) elseif (mel.eq.12) then el = d c m p l x (2.17d0,-O.OOOOldO) endif do error = l,last if (error.eq.2) then tempi = dreal(ytemp)+x*dreal(ytemp) temp2 = dimag(ytemp)+x*dimag(ytemp) ysnsr =dcmplx(tempi,temp2) elseif (error.eq.3) then tempi = dreal(ytemp)-x*dreal(ytemp) temp2 = dimag(ytemp)-x*dimag(ytemp) ysnsr = d c m p l x (tempi,temp2) endif ! FIND A N APPROXIMATE GUESS FOR EPSILON 2 los = 0 call gettim(thl,tml,tsl,tssl) ges = d c m p l x (1.dO,0.dO) call mainrtsolver(el,a,b,d,f,ysnsr,c,los,ges,epsl2,nc,mi) write (*,10) dreal(epsl2),dimag(epsl2) 10 format (' Initial Guess = ',F14.7,', ',F14.7,'i') call gettim(th2,t m 2 ,ts2, tss2) ttml = (th2*3600.+tm2*60+ts2+tss2/100.) + - (thl*3600.+tml*60+tsl+tss2/100.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 write (*,15) nint(ttml) format (' ===> Initial Guess took',13,' write (*,*) Sec.===> Please Wait') los = 1 ! FIND EPSILON 2 ges = epsl2 call mainrtsolver(el,a,b,d,f,ysnsr,c,los,g e s ,e p s l 2 ,nc,mi) call gettim(th2,tm2,ts2,tss2) ttm = (th2*3600.+tm2*60+ts2+tss2/100.) + - (thl*3600.+tml*60+tsl+tss2/100.) ! WRITE ALL STUFF IN THE END write (1,*) write (1,*) write (1,*) write (1,30) a,b,dreal(d),f 30 format (3P,1 a = ',F14.3,/,' b = ',F14.3,/, + d = ',F14.3,/,1P,1 f = 1,E 1 4 .3E2) write (1,40) dreal(el),dimag(el) 40 format (' el= ',F 1 9 .5,',',F 1 9 .5,'i ',/) write (1,45) nint(ttm) 45 format (' Inversion took',13,' seconds') if (material.eq.0) epsl2 = dcmplx(dreal(epsl2),0.d o ) if (nc.eq.l) then write (1,*) ' *** RESULTS DO NOT CONVERGE ***' write (1,*) ' *** ITERATIONS ALLOWED =',mi endif write (1,*) write (1,*) write (1,*) ' THE EPSILON OF MEDIUM 2 IS FOUND TO BE -' write (1,50) dreal(epsl2),dimag(eps12) 50 format (' epsilonl = ',F14.6,', ',F 1 4 .6,'i ',/) write (1,60) dreal(ysnsr),dimag(ysnsr) 60 format (' Ysensor = ',F14.9,', ',F 1 4 .9,'i ',/) write (1,*) los = 0 enddo stop end ! MAIN PROGRAM UNIT -- END B.2.4 TH ICK-1.FOR DECLARE $INCLUDE:'M A M O D U L E .F O R ' $INCLUDE:'M I M ODULE.F O R ' $INCLUDE:'R T M ODULE.F O R ' $INCLUDE:'R D M ODULE.F O R ' $INCLUDE:'SEMODULE.F O R ' program thick-1.for double complex y s n s r ,ytemp,j,e l , e 2 ,mainrt,d,ges Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. double precision a,b,f ,r,i,c,pi,tempi,temp2,om integer sz,error,last,lastl,mel,nc,mi,los,me2 integer*2 t h l ,t m l ,t s l ,tssl,t h 2 ,t m 2 ,ts 2 ,tss2 real ttm,x external mainrt open(unit = 1, file = 'c d . d a t 1,status = 'unknown') ! WRITE ALL INITIAL STUFF write (*,*) 'COAXIAL LINE - MEDIUM-1 THICKNESS COMPUTATION:' writs (*, *) 1T?????????????????????????????????????????????????'5'?'5 +?????????????????????1 write write write write write (*,*) (*,*) (*,*) (*,*) (*, *) ’Y(w),a,b,f,epsilonl,epsilon2 ========> d [mm]' ' [MEANT FOR SINGLE RUN [0 GHz < f <10 GHz] ' ' [d < CABLE S I Z E ] ' ' [RESULTS ARE STORED IN CD.DAT] ' '??????????????????????????????????????????????????■5,p-3 +?????????????????????1 write (*,*) ! READ ALL STUFF FROM KEY BOARD write (*,+) 'Select Medium 1 & Medium 2: ' write (*,*) '___________________________________________________________ write (*,*) ' [1] Methanol [2] Water [3] Acetone [4] Ethanol [5] +Glycerol [6] Air' write (*,*) ' [7] Teflon [8] Nylon - 66 [9] Polyurethane [10] Po +lyethylene [11] Derlin' write (*,*) 1 [12] CarbonTetraChloride [13] Any other material [1 +4] Conductor' write (*,*) '___________________________________________________________ write write write (*,*) ' Note: In Cases of Solids, Select [13] if f > 3 GHz' (*,*) ' Do not Select [14] for Medium 1' (*,*) '___________________________________________________________ read (*,*) mel if (mel.eq.13) then write (*,*) 'Enter el [Real Part , Imaginary P a r t ] ' read (*,*) r,i if (dabs(i).l e .0.OldO) then write (*,*) 'Enter a small negative number if Medium 1' write (*,*) ' is a Dielectric (Lossless) Ex: - 0.01' read (*,*) i endif el = dcmplx(r,i) endif read (*,*) me2 if (mel.gt.13) then write (*,*) '==========> PROGRAM IS TERMINATED, ERROR IN INPUT!' stop endif if (me2.gt.14) then write (*,*) '==========> PROGRAM IS TERMINATED, ERROR IN INPUT!' Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. stop endif if (me2.eq.13) then write (*,*) 'Enter e2 [Real Part , Imaginary P a r t ] : ' read (*,*) r,i e2 = dcmplx(r,i) endif write (*,*) 'Enter Coaxial Admittance' read (*,*) r,i ytemp = dcmplx(r,i) ysnsr = ytemp write (*,*) 'Enter Frequency [GHz]' read (*,*) r write (*,*) 'Enter ==> 0 - +- x% Admittance Calculations,' write ( *,*) ' 1 - If Not' read (*,*) lastl if (lastl.eq.O) then write (*,*) 'Enter the Percentage Variation Allowed in Admittan +ce: ' read (*,*) x x = x/lOO.dO last = 3 else last = 1 endif 1 write (*,*) 'Enter Cable Size [enter 1-5] :' write (*,*) ' [1] 2.2mm; [2] 3.Smm; [3] 6.4mm; [4] 8.3mm' write (*,*) ' [5] A n y Other Dimensions of Cable' read (*,*) sz if (sz.eq.5) then write (*,*) ' ===============>' write (*,*) ' Enter a [mm] : ' read (*,*) a write (*,*) ' Enter b [mm] : ' read (*,*) b write (*,*) ' Enter C [Use C.EXE to find Constant C ] : ' read (*,*) c a = a* l.d-3 b = b* l.d-3 endif write (*,*) 'Enter Maximum Limit for Iterations: read (*,*) mi write (*,*) '_______________________________________ ! INITIALIZE if (sz.eq.l) then a = 0.255d-3 b = 0.838d-3 c = 3 .212895842309772e-4 elseif (sz.eq.2) then a = 0.455d-3 b = 1.499d-3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. c elseif a b c elseif a b c endif f pi j om = = = = = 5.758110164705024e-4 (sz.eq.3) then = 0.824d-3 = 2.655d-3 = 1.002538968584023d-3 (sz.eq.4) then = 1.124d-3 = 3.62d-3 = 1.366445980459857d-3 r*1.0d9 4.do * datan(l.dO) d c m p l x (0.O d O ,1.OdO) 2 * pi * f SET PERMITTIVITY FOR BOTH MEDIA if (mel.eq.l) then el = 5.7d0 + 27.4d0 / ( l.dO + j * f / 3.d9) !methanol elseif (mel.eq.2) then el = 5.dO + 73.dO / (l.dO + j * f / 19. 7d9) !water elseif (mel.eq.3) then el = 1..9d0 + 19.3d0 / (l.dO + j * om * 3.3d-12) !acetone elseif (mel.eq.4) then el = 4.2d0 + 20.9d0 / (l.dO + j * om * 1.4d-10) !etha'nol elseif (mel.eq.5) then el = 4.18d0 + 38.3d0 / ((l.dO + j * 2.49d-9)**0.6)!glycerol elseif (mel.eq.6) then el = d c m p l x (1.d O , -0.OldO) elseif (mel.eq.7) then el = d c m p l x (2.ldO,-0.OldO) elseif (mel.eq.8) then el = d c m p l x (3.14d0,-0.05d0) elseif (mel.eq.9) then el = d c m p l x (3.4d0,-0.OldO) elseif (mel.eq.10) then el = d c m p l x (2.26 d 0 ,-0.OldO) elseif (mel.eq.ll) then el = d c m p l x (2.8d0,-0.OldO) elseif (mel.eq.12) then el = d c m p l x (2.17d0,-0.OldO) endif if (me2.eq.l) then e2 = 5.7d0 + 27.4d0 / ( l.dO + j * f / 3.d9) Imethanol elseif (me2.eq.2) then !water e2 = 5.dO + 73.dO / (l.dO + j * f / 19. 7d9) elseif (me2.eq.3) then 3.3d-12) !acetone om e2 = 1.9d0 + 19.3d0 / (l.dO + j elseif (me2.eq.4) then 1.4d-10) !ethanol om e2 = 4.2d0 + 20.9d0 / (l.dO elseif (me2.eq.5) then * 2.49d-9)**0.6)Iglycerol e2 = 4.18d0 + 3 8.3d0 / ((l.dO elseif (me2.eq.6) then e2 = d c m p l x (1.d O ,0.d o ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elseif e2 = elseif e2 = elseif e2 = elseif e2 = elseif e2 = elseif e2 = elseif e2 = endif (me2.eq.7) then d c m p l x (2.l dO,0.dO) (me2.eq.8) then d c m p l x (3.14d0,-0.05d0) (me2.eq.9) then d c m p l x (3.4 d 0 ,0.dO) (me2.eq.10) then d c m p l x (2.26d0,0.dO) (me2.eq.ll) then d c m p l x (2.8 d0,0.dO) (me2.eq.12) then d c m p l x (2.17d0,0.dO) (me2.eq.14) then dcmplx(0.d0,-10.d5) do error = l,last if (error.eq.2) then tempi = dreal(ytemp)+x*dreal(ytemp) temp2 = dimag(ytemp)+x*dimag(ytemp) ysnsr = d c mplx(tempi,temp2) elseif (error.eq.3) then tempi = dreal(ytemp)-x*dreal(ytemp) temp2 = dimag(ytemp)-x*dimag(ytemp) ysnsr =dcmplx(tempi,temp2) endif ! FIND A N APPROXIMATE GUESS FOR THICKNESS call gettim(thl,tml,tsl,tssl) los = 0 ges = d c m p l x (1.d - 3 ,0.dO) call mainrtsolver(el,e2,a,b,f,ysnsr,c,ges,d,n c , m i ,los) write (*,10) dreal(d) 10 format (3P,' Initial Guess = ',F 7 .3,' [mm]') call gettim(th2,t m 2 ,t s 2 ,tss2) ttml = (th2*3600.+tm2*60+ts2+tss2/100.) + - (thl*3600.+tml*60+tsl+tss2/100.) write (*,15) nint(ttml) 15 format (' ===> Initial Guess t o o k ',13,' Sec.===> Please W a i f write (*,*) ges = d los = 1 call gettim(thl,tml,tsl,tssl) call mainrtsolver(el,e2,a,b,f,ysnsr,c,ges,d,nc , m i ,los) d = d*1. d3 call g e t t i m (th2,tm2,t s 2 ,tss2) ■ttm = (th2*3600.d0+tm2*60.d0+ts2+tss2/100.dO) + - (thl*3 6 0 0 .d0+tml*60.d0+tsl+tss2/l00.dO) ! WRITE ALL STUFF IN THE END write (1,*) write (1,*) write (1,*) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. write (1,30) a,b,f format (3P,' a = ',F14.3,/,' b = ',F14.3,/, + IP,' f = ',E 1 4 .3E2) write (1,40) dreal(el),dimag(el) 40 format (' el= 1,F 1 9 .5, ', ',F 1 9 .5, 'i ',/) write (1,50) dreal(e2),dimag(e2) 50 format (' e2= ',F 1 9 .5, ', ',F 1 9 .5, 'i ',/) write (1,45) nint(ttm) 45 format (' Inversion t o o k ',13,' seconds') if (nc.eq.l) then write (1,*) ' *** RESULTS DO NOT CONVERGE ***' write (1,*) ' *** ITERATIONS ALLOWED =*,mi endif write (1,*) write (1,*) ' THE d OF MEDIUM 1 IS FOUND TO BE write (1,60) dreal(d) 60 format (' d = ',F14.9,' [mm]',/) write (1,70) dreal(ysnsr),dimag(ysnsr) 70 format (' Ysensor = ',F14.9,', ',F 1 4 .9, ’i ',/) write (1,*) los = 0 enddo stop end ! MAIN PROGRAM UNIT -- END 1 30 B .2.5 C .F O R ! THIS PROGRAM COMPUTES CONSTANT C FOR Y(W) program C .for double precision mpi,maa,mbb,mdelt,mrho,y,msum double precision mso,msl,ms2,mterm,integl integer mn,rhosteps,sz external integl write (*,*) 'Enter integration steps:' read (*,*) rhosteps write (*,*) 'Enter cable size [enter 1-4] :' write (*,*) ' [1] 2.2mm; [2] 3.6 mm; [3] 6.4mm; [4] 8.3mm' write (*,*) ' [5] any other dimensions of Cable' read (*,*) sz if (sz.eq.l) then maa = 0.255d-3 mbb = 0.838d-3 elseif (s z .e q .2) then = 0.455d-3 maa = 1.499d-3 mbb elseif (sz.eq.3) then = 0.824d-3 maa = 2.655d-3 mbb elseif (sz.eq.4) then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 maa = l.l24d-3 mbb = 3.62d-3 elseif (sz.eq.5) then write (*,*) 'Enter a [mm] : read (*,*) maa write (*,*) 'Enter b [mm] : read (*,*) mbb maa = maa* l.d-3 mbb = mbb* l.d-3 endif mpi = 4.do * d a t a n (1.d o ) = rhosteps mn msl = 0 .do = O.dO ms 2 = O.dO mso = (mbb - maa) / (2.d0 * mn) mdelt i = 0 = i + 1 i = (maa + i * mdelt) mrho mterm = integl(mrho,m a a ,m b b ,m p i ,rhosteps) msl = msl + mterm = i + 1 i = (maa + i * mdelt) mrho mterm = integl(mrho,maa,mbb,mpi,rhosteps) ms 2 = ms2 + mterm write (*, *) 'yl(w) ==> iteration no. ',i,' out of'. if ( i.lt .(2 * mn - 2)) then goto 60 else = (maa + (2.d0 * mn - l.dO) * mdelt) mrho msl = msl + integl(mrho,maa,mbb,mpi,rhosteps) = maa mrho = integl(mrho,maa,mbb,mpi,rhosteps) mso = mbb mrho = mso + integl(mrho,maa,mbb,mpi,rhosteps) mso = (mdelt/3.d0) * (mso + 4.d0 * msl + 2.dO msum endif y = (msum / mpi) write write write stop end (*,*) 'C is' (*,*) y (*,*) double precision function integl(trho,taa,tbb,tpi,trhosteps) double precision integ2,tso,tsl,ts2,tterm,tsum double precision trho,trhod,tpi,tst,ten,tdelt, taa, tbb integer tn,trhosteps external integ2 tn tsl = trhosteps = O.dO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 ts2 = O.dO tso = O.dO tst = taa ten = tbb tdelt = (ten - tst) / (2.dO * tn) i = 0 i = i + 1 trhod = (tst + i * tdelt) tterm = integ2(trho,trhod,tpi) tsl = tsl +■ tterm i = i + 1 = (tst + i * tdelt) trhod tterm = integ2(trho,trhod,tpi) = ts2 + tterm ts2 if ( i .It .(2 * tn - 2 ) ) then goto 300 else trhod = (tst + (2.dO * tn - 1. do) * tdelt) tsl = tsl + integ2(trho,trhod,tpi) trhod = tst tso = integ2(trho,trhod,tpi) trhod = ten tso = tso + integ2(trho,trhod,tpi) tsum = (tdelt/3.d0) * (tso + 4.dO * tsl + 2.d0 endif integl = tsum return end double precision function integ2(srho,srhod,spi) double precision fnl,sso,ssl,ss2,sterm,ssum,integ2p double precision s r h o ,srhod,spi,s s t ,e llip,tiny double precision sen,sphid,sdelt,sp2,srl,ising integer sn,stl external f n l ,ellip stl tiny integ2p do while sn ssl ss2 sso sst sen sdelt i i sphid sterm ssl i sphid = 2 = l.dO = d c m p l x (0.d o ,0.do ) (tiny.gt.l.d-4) = stl = O.dO = O.dO = O.dO = O.dO = spi = (sen - sst) / (2.do * sn) = 0 = i + 1 = (sst + i * sdelt) = fnl(srho,srhod,sphid) = ssl + sterm = i + 1 = (sst + i * sdelt) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sterm = fnl(srho,srhod,sphid) ss2 = ss2 + sterm if ( i .I t .(2 * sn - 2)) then goto 400 else sphid = (sst + (2.dO * sn - l.dO) * sdelt) ssl = ssl + fnl(srho,srhod,sphid) sphid = sst sso = fnl(srho,srhod,sphid) sphid = sen sso = sso + fnl(srho,srhod,sphid) ssum = (sdelt/3.d0) * (sso + 4.d0 * ssl + 2.do ’ ss2) endif sp2 = (4.d0*srho*srhod)/((srho+srhod)**2) srl = srho + srhod if (sp2.eq.1.do) then sp2 = 0.99999 endif ising = (2.dO/srl) * ellip(sp2) integ2 = ssum + ising stl = stl + 8 tiny = cdabs(integ2 - integ2p) if (stl.gt.40) tiny = 0.0 integ2p = integ2 end do return end double precision function fnl(zrho,zrhod,zphid) double precision zrho,zrhod,zphid,zr,zrl, zr2 zrl = (zrho**2) + (zrhod**2) zr2 = 2.dO * zrho * zrhod * dcos(zphid) zr3 = zrl - zr2 if (zr3.I t .1.d-18) then zr3 = O.dO endif zr = dsqrt(zr3) if (zr.eq.0.0) then fnl = O.OdO else fnl = (dcos(zphid)-1.d O ) / zr endif return end double precision function ellip(el) double precision el,ec,ed,eaO,eal,ea2,ea3,e a 4 ,ebO double precision ebl,eb2,e b 3 ,e b 4 ,e t l ,et2 ed = 1.dO - el ec = 1.do - el if (ed.It.1.d-20) ed = l.d-20 then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e n d if eaO eal ea2 ea3 ea4 ebO ebl eb2 eb3 eb4 = = = = = = = = = = etl et2 = (((((((ea4*ec)+ea3)*ec)+ea2)*ec)+eal)*ec)+ea0 = (((((((eb4*ec)+eb3)*ec)+eb2)*ec)+ebl)*ec)+eb0 1.3862944 0.096663443 0.035900924 0.037425637 0.014511962 0.5 0.12498594 0.068802486 0.033283553 0.0044178701 ellip = etl + d l o g (1.do/ed) return end ! MAIN PROGRAM UNIT -- END * et2 B.2.6 H E A R T .F O R ! HEART -- BEGIN if (d b g .e q .0) then write (*,1100) a,b,dreal(d),1/1000,f format (3P,' a = ',F8.3,’ b = ',F8.3,' d = ',F8 .3 ,/ , 1100 + L = 1,F 1 4 .3,I P , ' f = 1,E l l .4 E 2 ) write (*,1200) dreal(el),dimag(el),dreal(e2),dimag(e2) format (' el = ',F l l .5 , 1, ',F l l .5, 'i ',/, 1200 e2 = ',F l l .5,',',F l l .5,'i ' ) + write (*,1300) r k l ,dimag(kl),r k 2 ,dimag(k2) format (' kl = ',F l l .5,',',F l l .5,'i ',/, 1300 k2 = ',F l l .5, ', ',F l l .5, 'i ’) pause 'intialization is done, waiting for prompt' d if ! FIND ALL ROOTS ALONG THE REAL AXIS fr = 0.0001 call rootmain(f,d,kl,k 2 ,r k l ,r k 2 ,e l ,e l 2 ,fr,min,lmin,nrts,rts) if 1400 (dbg.eq.0) then if (nrts.gt.0) then write (*,*) ' the following roots were found' do u = l,nrts write (*,1400) u,dreal(rts(u)),dimag(rts(u)) formate ',17,'] ',F14.9,', ',F14.9,'i') enddo else write ( *,*)' no roots found close to real axis, ' write (*,*) ' direct integration is done' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 endif pause endif 'rootmain is done, waiting for prompt' ! DO INTEGRATION OF MAIN FUNCTION FROM 0 TO 11 y = d c m p l x (0.d o ,0.do) 11 = O.dO 12 = d m i n l (r k l ,r k 2 ) if (nrts.ne.0) then if (dreal(rts(1)).It.12) 12 = d r e al(rts(1))-1.do endif stage = 1 call d i r e c t (11,12,a,b,d,kl,k2,e l 2 ,sum,ay,stage) y = sum if (dbg.eq.0) then write (*,1600) dreal(sum),dimag(sum) format (' STAGE 1 : y = ',F 1 9 .15, ',' ,F19 .15, 'i ') 1600 write (*,1700) 11,12 format (' 11,12 = ',F l l .5,’,',F l l .5) 1700 pause 'stage 1 is done, waiting for prompt' endif ! IF ROOTS EXIST, THEN DO INTEGRATION OF FUNCTION IN THE REGION OF ROOTS if (nrts.ne.0) then do u = l,nrts 1 11 = 12 12 = dmaxl (12,dreal (rts (u) ) - 0.3d0) stage = 2 call d i r e c t (11,12,a, b,d,kl,k 2 ,e l 2 ,sum,ay,stage) y = y + sum if (dbg.eq.0) then write (*,1800) dreal(sum),dimag(sum) format (' STAGE 21 : y = ',F 1 9 .15,',',F 1 9 .15,'i ') 1800 write (*,1900) 11,12 format (' 11,12 = ',F l l .5,',',F l l .5) 1900 pause 'stage 21 is done, waiting for prompt' endif 11 = 12 12 = dreal(rts(u)) + 0.3d0 call s e (11,12,a,b,d,kl,k2,el2,rts(u),sum,ay) y = y + sum if (dbg.eq.0) then write (*,2000) d r e a l (su m ) ,d i m a g (sum) format C STAGE 22 : y = ',F19.15,',',F19.15,'i ') 2000 write (*,2100) 11,12 format (' 11,12 = ',F l l .5,',',F l l .5) 2100 pause 'stage 22 [SE] is done, waiting for prompt' endif enddo if (rk2.1t.l) then stage = 2 11 = 12 12 = dmaxl(rkl,rk2) , call d i r e c t (1 1 ,12,a,b,d,kl,k 2 ,e l 2 ,sum,ay,stage) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2200 2300 ! IF 2400 2500 2600 2700 2800 2900 y = y + sum if (dbg.eq.0) then write (*,2200) dreal(sum),dimag(sum) format (' STAGE 23 : y = 1,F 1 9 .15,',',F 1 9 .15,'i ') write (*,2300) 11,12 format (' 11,12 = ',F l l .5,',',F l l .5) pause 'stage 23 are done, waiting for prompt' endif endif NO ROOTS, THEN DO THE CORRESPONDING INTEGRATION else if (min.eq.l) then 11 = 12 12 = lmin stage = 2 call d i r e c t (11,12,a,b,d,kl,k2,el2,sum,ay,stage) y = y + sum if (dbg.eq.0) then write (*,2400) dreal(sum),dimag(sum) format (' STAGE 21 :y = ',F 1 9 .15,',',F 1 9 .15,'i ') write (*,2500) 11,12 format (' 11,12 =',F l l .5,',',F l l .5) pause 'stage 21 is done, waiting for prompt' endif endif if (rk2.1t.l) then 11 = 12 12 = d m a x l (r k l ,rk2) stage = 2 call d i r e c t (11,12,a,b,d,kl,k2,el2,sum,ay,stage) y = y + sum if (dbg.eq.0) then write (*,2600) d r e a l (su m ) ,d i m a g (sum) format (' STAGE 22 : y = ',F19.15,',',F19 . 1 5 , 'i ') write (*,2700) 11,12 format (' 11,12 = ',F l l .5,',',F l l .5) pause 'stage 22 is done, waiting for prompt' endif endif endif stage = 3 call d i r e c t (12,1,a,b,d,kl,k 2 ,e l 2 ,sum,ay,stage) y = y + sum if (dbg.eq.0) then write (*,2800) dreal(sum),dimag(sum) format (' STAGE 3 : y = ',F 1 9 .15,' , ',F19 .15 , 'i ') write (*,2900) 12,1 format (' 12,1 = ',F l l .5,',',F l l .5) pause 'stage 3 are done, waiting for prompt' endif ! ADD THE PART CORRESPONDING "TO THE INT FROM L TO INF y = y + j * c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. if 3000 (dbg.eq.0) then write (*,3000) dreal(j*c),dimag(j*c) format (' STAGE 4 : y = ',F 1 9 .15,',1,F 1 9 .15,'i ’) pause 'stage 4 is done, waiting for prompt' endif y = k * y ! HEART -- END B.2.7 H E A R T Q .F O R ! HEARTQ -- BEGIN ! DO THE INTEGRALS OF QUASI STATIC APPROXIMATIONS if (yes.eq.0) then 11 = O.dO call q d i r e c t (11,1,a,b,d,el2,yq,ay) yq = k * (yq+j *C) ct = yq / (j*om) if (dbg.eq.0) then write (*,200) dreal(ct),dimag(ct) 200 formate Cquasi= ',E l l .4 E 2 , ', ',Ell .4E2 , 'i ') write (*,210) dreal(yq),dimag(yq) 210 formatC yquasi= ',F19 .15, ', ',F19 .15 , 'i ') pause 'quasi is done, waiting for prompt' endif endif ! HEARTQ -- END B .2 . 8 I N M O D U L E . F O R ! INITIALIZE MODULE -- BEGIN subroutine init (f,j ,e o, e l ,e 2 ,e l 2 ,ko,kl,k2,rkl,rk2,a,b,c,d,k,l, +q,sz,om,hour,minute,sec,accuracy,dbg,il) double complex el,e2,el2,kl,k2,k , j ,d double precision pi,eo,mo,f,a,b,c,1,om,ko,rkl,rk2,p i e , r ,i , +accuracy,il integer*2 hour,minute,sec,secs integer q , s z ,dbg,mel,me2 ! WRITE THE INITIAL STUFF write (*,*) 'COAXIAL LINE ADMITTANCE COMPUTATION:' write (*,*) '????????????????????????????????????????????????????? +?????????????????????' write (*,*) 'INTEGRATION OF EQUATION (1)' write (*,*) 'MEANT FOR SINGLE RUN [0 GHz < f < 10 G H z ] ' write (*,*) ' [d < 15mm] ' write (*,*) ' [RESULTS ARE STORED IN C.DA T ] ' write (*, *) '????????????????????????????????????????????????????? +?????????????????????' write {*,*) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ! READ ALL VARABLES REQUIRED FROM KEYBOARD write (*,*) 'Select Medium 1 & Medium 2: ' write (*,*) '________________________________ write (*,*) 1 [1] Methanol [2] Water [3] Acetone [4] Ethanol [5] +Glycerol [6] A i r 1 write (*,*) ' [7] Teflon [8] Nylon - 66 [9] Polyurethane [10] Po +lyethylene [11] Derlin' write (*,*) ' [12] CarbonTetraChloride [13] Any other material [1 +4] Conductor' write (*,*) '___________________________________________________________ write write write (*,*) ' Note: In Cases of Solids, Select [13] if f > 3 GHz' (*,*)' Do not Select [14] for Medium 1' (*,*) '___________________________________________________________ read (*,*) mel if (mel.eg.13) then write (*,*) 'Enter el [Real Part , Imaginary P a r t ] : ' read (*,*) r,i if (dabs(i).l e .0.OldO) then write (*,*) 'Enter a small negative number if Medium 1' write (■*,*) ' is a Dielectric (Lossless) Ex: - 0.01' read (*,*) i endif el = dcmplx(r,i) endif read (*,*) me2 if (me2.eq.13) then write (*,*) 'Enter e2 [Real Part , Imaginary P a r t ] : ' read (*,*) r,i e2 = dcmplx(r,i) endif write (*,*) '__________________________________________________ write (*,*) 'Enter read (*,*) f f = f * l.d9 write (*,*) 'Enter write (*,*) ' [1] write (*,*) ' [5] read (*,*) sz if (sz.eq.5) then write (*,*) ’ write (*,*) ' read (*,*) a write (*,*) ' read (*,*) b write (*,*) 1 read (*,*) c a = a* l.d-3 b = b* l.d-3 endif write (*,*) 'Enter Frequency [GHz]: ' Cable Size [Enter 1-5]:' 2.2mm [2] 3.6mm [3] 6.4mm [4] 8.3mm' Any other Dimensions of Cable' ==================>’ Enter a [mm] : ' Enter b Enter C d [mm] [mm] : ' [Use C.EXE to find Constant C ] : : Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. read (*,*) r d = dcmplx(r+1.d - 3 ,0.do) write (*,*) 'Enter Accuracy Required: 1 read (*,*) accuracy write (*,*) 'Enter 0 if Debug is Required: ' read (*,*) dbg il = O.dO write (*,*) 'Enter 0 if Quasi Static Approximation is Required: read (*,*) q write (*,*) '_______________________________________________________ ! INITIALIZE ALL VARIABLES call gettim(hour,minute,sec,secs) if (sz.eq.l) then a = 0.255d-3 b = 0.838d-3 c = 3.212895842309772e-4 elseif (sz.eq.2) then a = 0.455d-3 b = 1.499d-3 c = 5.758110164705024e-4 elseif (sz.eq.3) then a = 0.824d-3 b = 2.655d-3 c = 1.002538968584023d-3 elseif (sz.eq.4) then a = 1.124d-3 b = 3.62d-3 C = 1.366445980459857d-3 endif j = d c m p l x (0.dO,1 .dO) pi = 4.dO * datan(l.dO) eo = l.dO / (36.dO * pi * l.d9) mo = 4.dO * pi / 1.d7 om = 2 * pi * f ! SET PERMITTIVITY FOR BOTH MEDIA if (mel.eq.l) then el = 5.7d0 + 27.4d0 / ( l.dO + j * f / 3.d9) Imethanol elseif (mel.eq.2) then el = 5.dO + 73.dO / (l.dO + j * f / 19.7d9) !water elseif (mel.eq.3) then el = 1.9d0 + 19.3d0 / (l.dO + j * om * 3.3d-12) !acetone elseif (mel.eq.4) then el = 4.2d0 + 20.9d0 / (l.dO + j * om * 1.4d-10) !ethanol elseif (mel.eq.5) then el = 4.18d0 + 38.3d0 / ((l.dO + j * om * 2.49d-9)**0.6)!glycerol elseif (mel.eq.6) then el = d c m p l x (1.d o ,-0.OldO) elseif (mel.eq.7) then el = d c m p l x (2.l dO,-0.OldO) elseif (mel.eq.8) then Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. el = elseif el = elseif el = elseif el = elseif el = endif d c m p l x (3.14d0,-0.05d0) (mel.eq.9) then d c m p l x (3.4 d 0 ,-0.OldO) (mel.eq.10) then d c m p l x (2.2 6d0,-0.OldO) (mel.eq.ll) then dcmplx(2.8d0,-0.OldO) (mel.eq.12) then d c m p l x (2.17d0,-0.OldO) if (me2.eq.l) then !methanol e2 = 5.7d0 + 27.4d0 / ( l.dO + j * f / 3 .d 9 ) elseif (me2.eq.2) then e2 = 5.dO + 73.dO / (1.dO + j * f / 19-.7d9) !water elseif (me2.e q .3) then !acetone e2 = 1.9d0 + 19.3d0 / (l.dO + j * om * 3.3d-12) elseif (me2.eq.4) then e2 = 4.2d0 + 20.9d0 / (l.dO + j * om * 1.4d-10) !ethanol elseif (me2.eq.5) then e2 = 4.18d0 + 38.3d0 / ((l.dO + j * om * 2.49d-9)**0.6)[glycerol elseif (me2.eq.6) then e2 = d c m p l x (1. d o ,0. d O ) elseif (me2.eq.7) then * e2 = d c m p l x (2.l d O ,0.d O ) elseif (me2.eq.8) then e2 = d c m p l x (3.14d0,-0.05d0) elseif (me2.eq.9) then e2 = d c m p l x (3.4 d 0 ,0.d O ) elseif (me2.eq.10) then e2 = d c m p l x (2.26d0,O.dO) elseif (me2.eq.ll) then e2 = d c m p l x (2.8d0,0. d O ) * elseif (me2.eq.12) then e2 = d c m p l x (2.17d0,O.dO) elseif (me2.eq.14) then e2 = d c m p l x (0.d O ,-10.d 6 ) endif el2 = el/e2 ko = om * dsqrt(mo*eo) kl = ko * cdsqrt(el) k2 = ko * cdsqrt(e2) rkl = dreal(kl) rk2 = dreal(k2) k = (2.dO*pi*om*eo*el)/ (dlog(b/a))**2 1 = dnint(dsqrt((10.0/d)**2 + rkl**2)) pie = 1 / dsqrt(1**2 - rkl**2) do while ((ple-1).gt.le-3) 1 = 1 + 100 pie = 1 / d s q r t (1**2 - rkl**2) end do return end R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ! INITIALIZE MODULE -- END B.2.9 W R M O D U LE .F O R ! WRITE & FINALIZE MODULE -- BEGIN subroutine wr(el,e2,f ,k l , k 2 ,r k l ,r k 2 ,a,b,d,1,y e s ,h o u r ,minute,sec, + nrts,rts,y,ct,yq,fno,k) double complex el,e2,k l ,k 2 ,y , c t ,yq,rts,d,k double precision a,b,1,f ,rkl,rk2 integer*2 hour,minute,sec,hour2,minute2,sec2,secs2 real ttm integer n r t s ,y e s ,g,fno dimension rts (20) 10 20 25 30 40 60 70 80 call gettim(hour2,minut e 2 ,se c 2 ,secs2) ttm = (hour2*3600+minute2*60+sec2)- (hour*3600+minute*60+sec) write (fno,*) write (fno,*) write (fno,*) write (fno,*) 'RESULTS:' write (fno,10) a,b,dreal(d),1/1000,f format (3P,' a = ',F8.3,' b = ',F8.3,' d = ',F 8 .3,/, +' L = ',F 1 4 .3,I P , ' f = ',E l l .4 E 2 ) write (fno,20) dreal(el),dimag(el),dreal(e2),dimag(e2) ' format C el = ',F 1 9 .5,’,',F 1 9 .5,'i ', /, + ' e2 = ',F19 .5, ', ',F19 .5, 'i ’) write (fno,25) dreal(k),dimag(k) format (' zeta = ',F 1 9 .5,',',F 1 9 .5,'i ') write (fno,30) r k l ,dimag(kl),r k 2 ,dimag(k2) format (' k l = ',F 1 9 .5,',',F 1 9 .5,'i ',/, + ' k2 = ',F19.5,',',F19.5,'i ') write (fno,*) if (nrts.gt.0) then write (fno,*) ' the following roots were found' do g = 1,nrts write (fno,40) g,dreal(rts(g)),dimag(rts(g)) formatC ',17,'] ',F14.9,\ ',F14.9,'i') enddo else write (fno,*) ' no roots found close to real axis, ' write (fno,*) ' direct integration is done' endif write (fno,*) write (fno,*) ' Integration took ',nint(ttm),' seconds' write (fno,*) write (fno,60) dreal(y),dimag(y) format (' y= ',E 1 5 .8 E2,',',E 1 5 .8 E2,'i ') if (yes.eq.0) then write (fno,70) dreal(ct),dimag(ct) formatC Cquasi = ',E15 .8E2 , ', ',E15 .8E2 , 'i ') write (fno,80) dreal(yq),dimag(yq) f o r m a t (' yquasi = ',E 1 5 .8 E 2 ,',',E 1 5 .8 E 2 ,'i ') R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. write (fno,*) write (fno,*) endif return end ! WRITE & FINALIZE MODULE -- END B.2.10 M A M O D U LE .F O R ! MAIN INTEGRATION MODULE -- BEGIN subroutine direct(x,y,a,b,d,kl,k2,e l 2 ,sum,limit,stage) double complex e l 2 ,k l ,k 2 ,y t ,sum,yy,d double precision a,b,x,y,xl,yl,d e l ,limit integer stage del = 100.dO !GENERATE STEP SIZE if (stage.eq.4) del = 1000.dO yt yy xl yl do = dcmplx(O.dO,O.dO) !DO THE INTEGRATION = d c m p l x (0.d o ,0.d o ) = x = x while ( y l .It.y) yl = yl + del if (yl.gt.y) yl = y call intmain(xl,yl,a,b,d,kl,k2,el2,yy,limit,stage) yt = yt + yy xl = yl enddo sum = yt return •end ! GAUSSIAN QUADTURE METHOD OF INTEGRATION ON MAIN PART subroutine intmain(s,e,a,b,d,kl,k2,e l 2 ,sum,limit,stage) double complex sum,sump,kl,k2,e l 2 ,fnmain,d double precision s ,e,x,w,a,b,error,limit integer st,i,j,incr,stage dimension x(1000),w(1000) external fnmain st = 2 incr = 2 stage = stage !DUMMY if (limit.It.l.d-9) limit = l.d-9 sump = d c m p l x (0.d O ,0.d o ) do i = 1,200 !REPEAT UNTIL RESULTS ARE ACCURATE ENOUGH sum = d c m p l x (0.dO,0.do) call evalwght(s,e,x,w,st) !THIS PART IS do j =1,st !GAUSSIAN sum = sum +w(j) *fnmain(x(j),a,b,d,kl,k2,el2) !QUADRATURE enddo !INTEGRATION R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. error = cdabs(sum-sump) !CHECK FOR ERROR sump = sum !STORE PREVIOUS INTEGRAL RESULT if (error.It.limit) exit !IF INTEGRATION IS CONVERGED THEN EXIT st = st + incr !IF RESULT DIDN'T CONVERGE INCREASE STEP SIZE if (st.gt.950) exit if (st.gt.100) st = 950 enddo return end ! DO THE INTEGRATION ON FOLLOWING FUNCTION double complex function f n m a i n (1,a,b,d,kl,k 2 ,e l 2 ) double complex al,a2,dm, d e , j ,k l ,k 2 ,e l 2 ,tn, d double precision l,a,b,intjlr,be external intjlr j = d c m p l x (0.d O ,1.d o ) al = -j*cdsqrt(l**2-kl**2) a2 = -j*cdsqrt(l**2-k2**2) be = (intjlr(1,a , b ) )**2 tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn de = el2*a2+j *al*tn fnmain = ((l*be*dm)/ (de*al)) - (j*be) return end ! MAIN INTEGRATION MODULE -- END B.2.11 M IM O D U L E .F O R ! MICELLANEOUS MODULE -- BEGIN ! EVALUATE WEIGHTS & ABSCISSAS FOR GAUSSIAN QUAD. INTEGRATIONS subroutine evalwght(xl,x2,x,w,n) double precision xl,x2,x(n),w ( n ) ,pi,xm,xl,pl,p 2 ,p 3 ,z ,z l ,pp,eps integer i ,j ,m parameter (eps=3.d-14) 10 m = (n+1)/2 xm = 0 . 5 * (x2+xl) xl = 0.5 * (x2-xl) pi = 4. * datan(l.) do i=l,m z = dcos(pi * (i-0.25) / (n+ 0.5)) pi = l.dO p2 = O.dO do j = l,n P3 = p2 P2 = pl pi = ((2.dO*j - 1.dO)*z*p2 - (j - 1.dO)*p3) enddo pp = n * (z*pl - p 2 ) / (z*z - l.dO) zl = z / j R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 240 z = zl-pl/pp if(dabs(z-zl).gt.eps) goto 10 x(i) = xm-xl*z x(n+l-i) = xm+xl*z w(i) = 2.dO * x l / ((1.d0-z*z)*pp*pp) w(n+l-i) = w(i) enddo return end ! DO THE INTEGRATION ON BESSEL FUNCTION IF FUNCTION IS REAL double precision function i n t j l r (1,a,b) double precision 1,a,b,s,r,j lr,t,tl,t2,t3,t4 integer lp external jlr s = (b-a.) /30 .do intjlr = O.dO r = a do lp = 1,10 tl = jlr(r*l) t2 = 3.do * jlr((r+s)*l) t3 = 3.dO * j l r ((r+2.d0*s)*1) t4 = j l r ((r+3.d0*s)*1) t = tl + t2 + t3 + t4 intjlr = intjlr + (3.d 0 / 8 .dO)*s*t r = r + s * 3 .do end do return end 1 ! BESSEL FUNCTION J 1 ( x ) , x = a REAL NUMBER double precision function jlr(x) double precision y,pl,p2,p3,p4,p5,ql,q2,q 3 ,q 4 ,q5, r l , r 2 ,r 3 ,r4,r5, + r 6 ,s i ,s 2 ,s 3 ,s 4 ,s 5 ,s 6 ,x, ax,z,xx data rl,r2,r3,r4,r5,r6/72362614232.do,-7895059235.do,2423 96 853.1 + d 0 ,-2972611.439d0,15704.48260d0,-30.16036606d0/, +S1, S2,S3,S4,s5,S6/144725228442.dO,2300535178.dO,18583304.74d0, +99447.43394d0,376.9991397d0,1.dO/ data p i ,p 2 ,p 3 ,p 4 , p 5 / l .d O , .183105d-2,-.3516396496d-4, .2457520174 + d - 5 ,- .240337019d-6/,ql,q2,q3,q4,q5/.04687499995d0,-.2002690873d-3, +.8449199096d-5,-.88228987d-6,.105787412d-6/ if(dabs(x).I t .8.) then y=x**2 jlr=x*(rl+y*(r2+y*(r3+y*(r4+y*(r5+y*r6) ) ) ) ) + / (sl+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))) else ax=dabs(x) z = 8 ./ax y=z**2 X X = a x - 2 .356194491 jlr=dsqrt(.636619772/ax)* (dcos(xx)* (pl+y*(p2+y*(p3+y*(p4+y R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. + *p5))))-z*dsin(xx)* (ql+y*(q2+y*(q3+y*(q4+y*q5)))))* d s i g n (1. endif return end ! DO THE INTEGRATION ON BESSEL FUNCTION IF THE FUNCTION IS COMPLEX double complex function intjlc(l,a,b) double complex 1,jlc,t,tl,t2,t 3 ,t4 double precision a,b,s,r integer lp external jlc s = (b-a)/30.do intjlc = O.dO r = a do lp = 1,10 tl = jlc(r*l) t2 = 3.do * jlc((r+s)*l) t3 = 3.dO * j l c ((r+2.d0*s)*1) t4 = j l c ((r+3.d0*s)*1) t = tl + t2 + t3 + t4 intjlc = intjlc + (3.d 0 / 8 .do)*s*t r = r+s*3.d0 end do return end ! BESSEL FUNCTION J1(x), x = a COMPLEX NUMBER double complex function jlc(xc) double complex xc,sum,crl,c r 2 ,c r 3 ,c r 4 ,cr5,del double precision pi integer n 10 pi = 4. * datan(l.) sum = xc/2.0 crl = xc cr2 = 2.0 cr4 = 1.0 n =1 crl = crl*(xc**2) cr2 = cr2+2.0 cr3 = (cr2-2.0)**2 cr4 = cr4*cr3 cr5 = cr4*cr2 del= {(-1)**n)*crl/cr5 sum= sum+del n = n+1 if (dabs(del).gt.le-18) goto 10 jlc = sum return end ! COMPLETE ELLIPTIC INTEGRAL OF FIRST KIND double precision function elp(x) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. double precision x, c, d , a O ,al, a 2 ,a 3 ,a 4 ,b o ,b l ,b 2 ,b 3 ,b 4 ,t l ,t2 d = 1. 0 - x c = 1. 0 - x if (d.It.1.0d-20) d = 1.0d-20 aO = 1.3862944 al = 0.096663443 a2 = 0.035900924 a3 = 0.037425637 a4 = 0.014511962 bO = 0.5 bl = 0.12498594 b2 = 0.068802486 b3 = 0.033283553 b4 = 0.0044178701 tl = ({(((((a4*c)+a3)*c)+a2)*c)+al)*c)+a0 t2 = (((((((b4*c)+b3)*c)+b2)*c)+bl)*c)+bO elp = tl + d l o g (1.OdO/d) * t2 return end ! MICELLANEOUS MODULE -- BEGIN B.2.12 R T M O D U L E .F O R ! ROOT MODULE -- BEGIN 4 subroutine r o o tmain(f ,d , k l ,k 2 ,r k l ,r k 2 ,e l ,e l 2 ,f r ,min,lmin,m,rts) double complex el,el2,kl,k2,r,rp,rt,rts,fn,rmin,lmin,l,ges,rtfn,d double precision f ,rkl,rk2,fr,x,z,nl,n2,dl integer s,m,sort,min,yes,cnl,cn2 dimension r t s (20) external rtfn •if (f.I t .10.d9) cnl = 0 !DO NOT SEARCH AGAIN IN THIS CASE if (r k l .g t .r k 2 ) cn2 = 1 !DO NOT SEARCH FOR MINIMA IN THIS CASE do S = 1,20 !SET ALL rts VARIABLES TO 0 rts(s) = d c m p l x (0.dO,0 .dO) end do m = 0 !INITIALIZE yes = 0 min = 0 x = dmaxl(dminl(rkl,rk2)- 3 .dO,2 .dO) z = d m a x l (r k l ,rk2)+3.dO if (dreal(el).gt.1000.do) z = rkl+lOO.dO 1 = dcmplx(x,0 .do) rp = rtfn(l,d,kl,k2,el2) rmin = rp lmin = 1 dl = (z-x)/(1.9**dlogl0(f)) ! SEARCH FOR A GOOD INITIAL GUESS, ALSO SEARCH FOR ! MINIMUM VALUE OF RTFN IN THE INTERVAL {x,z} ! SEARCH - I R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. do w hile(dreal(1).I t .z) 1 = 1 + dcmplx(dl, 0 .dO) r = rtfn(l,d,kl,k2,el2) if (cdabs(r).lt.cdabs(rmin)) then rmin = r lmin = 1 endif nl = dsign(l.OdO,dreal(rp)) 'FIRST SIGNAL FOR POLE CLOSE BY n2 = d s i g n (1.OdO,dreal(r)) if (nl.ne.n2) yes = 1 nl = d s i g n (1.O d O ,dimag(rp)) !SECOND SIGNAL FOR POLE CLOSE ] n2 = dsign (1. OdO, dimag (r) ) if (nl.ne.n2) yes = 1 if (yes.eq.l) then 'VERIFY THE GUESSES FOR A POLE ges = dcmplx(1-dl,0.do) call rtsolver(d,kl,k 2 ,e l 2 ,g e s ,r t ,fn) do s = l,m if (dabs(rts(s)-rt).I t .le-3) then fn = d c m p l x (10.do,0.do) endif end do if (cdabs(fn).I t .l.d-9) then if (dabs(dimag(rt)).I t .fr) then m = m + 1 rts (m) = rt endif endif yes = 0 endif rp = r .end do * 'SEARCH - II if (cnl.eq.l) then yes = 1 1 = dcmplx(x,0 .dO) dl = (z-x)/20.dO rp = rtfn(l,d,kl,k2,el2) do while( d r e a l (1).lt.z) ges = d c m p l x (1,0.d O ) call rtsolver(d,kl,k 2 ,e l 2 ,g e s ,r t ,fn) if (cdabs(fn).It.1 .d-9) then do s = l,m if (dabs(rts(s)-rt).I t .le-3) then yes = 0 endif end do if (yes.eq.l) then if (dabs(dimag(rt)).I t .fr) then m = m + 1 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. rts(m) = rt endif endif endif 1 = 1 + dcmplx(dl,0.d O ) yes = 1 end do endif if (cn2.eq.1) then !FINE TUNE THE MINIMA FOUND BEFORE if (m.eq.O) then 1 = lmin - dcmplx(dl,0.dO) dl = 2.dO * dl/500 rmin = r t f n (1,d,kl,k2,el2) do s = 1,500 1 = 1 + dcmplx(dl,0 .dO) r = rtfn(l,d,kl,k2,el2) if (cdabs(r).I t .cdabs(rmin)) then rmin = r lmin = 1 endif end do min = 1 endif endif do sort = l,m ISORT ALL POLES IN INCREASING ORDER do s = 1, (m-1) if (dreal(rts(s)) .gt.dreal (rts (s+1))) then r = rts(s) rts(s) = rts(s+1) rts(s+1) = r endif end do end do return end ! USE MUELLER'S METHOD TO SOLVE FOR POLES subroutine rtsolver(d,kl,k2,el2,ges,rt, fn) double complex rtfn,ges,kl,k2,el2,x0,xl,x2,hi,h2,f O ,f1, f 2 ,fdl, +fd2,ft,c,sqr,den,rt,fn,d double precision epl,ep2 integer i external rtfn epl ; = l.d-18 ep2 ■ = l.d-20 xO = ges + 0.5 xl = ges - 0.5 x2 = ges fO = rtfn(xO,d , k l ,k 2 ,e l 2 ) fl = rtfn(xl,d,kl,k 2 ,el2) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 245 10 f 2 = rtfn(x2,d,kl,k2,el2) hi = xl-xO h2 = x2-xl fdl = (f2-fl)/h2 fd2 = (fl-fO)/hi do i = 2,100 if (hl.eq.h2) exit ft = (fd2-fdl)/ (h2-hl) c = fd2 + h2*ft sqr = cdsqrt(c*c - 4*f2*ft) if (dreal(c)*dreal(sqr)+dimag(c)*dimag(sqr).It.O.dO) den = c - sqr else den = c + sqr end if if (cdabs(den).l e .0.do) den = l.dO hi = h2 h2 = -2 * f2/den xO = xl xl = x2 x2 = x2 + h2 fO = fl fl = f2 f2 = rtfn(x2,d,kl,k2,el2) fdl = fd2 fd2 = (f2-fl)/h2 if (cdabs(x2).gt.10000.dO) exit if (cdabs(h2).It.epl*cdabs(x2)) exit if (cdabs(f2).It.ep2) exit if (cdabs(f2).ge.10.dO*cdabs(fl)) then h2 = h2/2 x2 = x2 - h2 goto 10 endif enddo rt = x2 fn = rtfn(x2,d,kl,k2,el2) return end then ! FUNCTION FOR POLE double complex function rtfn(r,d,kl,k2,el2) double complex al,a2,j,kl,k2,el2,tn,r,dm,de,d j = d c m p l x (0.d O ,l .d o ) al = -j * cdsqrt(r**2-kl**2) a2 = -j * cdsqrt(r**2-k2**2) tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn de = el2*a2+j*al*tn rtfn = a l / ((dm/de)*r - j * al) return end ! ROOT MODULE -- END R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. B.2.13 SEM O D U LE.FO R ! SINGULARITY EXTRACTION MODULE -- BEGIN subroutine se(x,y,a,b,d,kl,k2,e l 2 ,r t ,sum, limit) double complex e l 2 ,k l,k 2 ,r t ,y s ,y y , y t ,rd,is,r e s ,sum,d double precision a,b,x,y,del,xl,yl,limit external rd,is res = rd(rt,a,b,d,kl,k2,el2) !FIND THE RESIDUE ys = is(rt,x,y,res) !FIND THE PART TO BE ADDED TO SE PART del = 10.dO yt yy xl yl !GENERATE STEP SIZE = d c m p l x (0.d o ,0.d o ) = d c m p l x (0.d O ,0.d o ) =x =x !DO THE INTEGRATION do w h i l e ( y l .It.y) yl = yl + del if (yl.gt.y) yl = y call intse(xl,yl,a,b,d,kl,k 2 ,e l 2 ,res,rt,yy,limit) yt = yt + yy xl = yl enddo sum = (yt+ys) return end 4 ! GAUSSIAN QUADTURE METHOD OF INTEGRATION ON SE PART subroutine intse(s,e,a,b,d,kl,k2,el2,res, rt, sum, limit) •double complex sum,sump,kl,k2,e l 2 ,r e s ,rt,fnse,d double precision s,e,x,w,a,b,error, limit integer st,i,j dimension x(900),w(900) external fnse st = 2 if (limit.It.l.d-9) limit = l.d-9 sump = d c m p l x (0.d o ,0.d o ) do i = 1,200 !REPEAT UNTIL RESULTS ARE ACCURATE ENOUGH sum = d c m p l x (0.d o ,0.d o ) call evalwght(s,e,x,w,st) 'THIS PART IS do j=l,St !GAUSSIAN QUADRATURE INTEGRATION sum = sum + w(j) * fnse(x(j),a,b,d,kl,k2,el2,res,rt) enddo St = st + 2 !IF RESULT DIDN'T CONVERGE INCREASE STEP SIZE error = cdabs(sum-sump) !CHECK FOR ERROR sump = sum !STORE PREVIOUS INTEGRAL RESULT if (error.It.limit) exit !IF INTEGRATION IS CONVERGED THEN EXIT enddo return R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 247 en d ! DO THE INTEGRATION ON FOLLOWING FUNCTION double complex function f n s e (1,a,b,d,kl,k2,e l 2 ,res,rt) double complex al,a2,dm,de,j,kl,k2,e l 2 ,tn,res,rt,d double precision 1,a,b,intjlr,be external intjlr j = d c m p l x (0.d o ,1.d o ) al = -j*cdsgrt(l**2-kl**2) a2 = -j*cdsqrt(l**2-k2**2) be = (intjlr(l,a,b))**2 tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn de = el2*a2+j*al*tn fnse = ((l*be*dm)/ (de*al)-res/(1-rt)) return end - (j*be) ! THE PART TO BE ADDED SINGULARITY EXTRACTED FUNCTION double complex function i s ( r t ,ad,bd,res) double complex j , p l ,p 2 , p 3 ,r t ,res double precision tl,t2,ad,bd j = d c m p l x (0.d O ,1.d O ) tl = (dimag(rt))**2 + (bd-dreal(rt))**2 t2 = (dimag(rt))**2 + (ad+dreal(rt))**2 pi = (res * dlog(tl/t2))/2.do p2 = j * res * d a t a n ((bd-dreal(rt))/dimag(rt)) p3 = j * res * d a t a n ( (dreal(rt)-ad)/dimag(rt)) is = pi + p2 + p3 return ■end 5 ! USE THE FOLLOWING FUNCTION TO FIND RESIDUE double complex function r d ( r t ,a,b,d,kl,k2,e l 2 ) double complex al,a2,dm,j,k l , k 2 ,e l 2 ,g l , g 2 ,g3,gd,rt,intjlc,tn,d,be double precision a,b external intjlc j = dcmplx(O.dO, 1 .dO) al = -j * cdsqrt(rt**2 - kl**2) a2 = -j * cdsqrt(rt**2 - k2**2) be = (intjlc(rt,a,b))**2 tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn gl = - (al*el2*rt)/a2-(a2*el2*rt)/al g2 = -j*al*d*rt*(1/cacos(al*d))**2 g3 = -j * 2 ,dO*rt*tn gd = gl+g2+g3 rd = (rt*be*dm)/gd return end ! SINGULARITY EXTRACTION MODULE -- END R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. B.2.14 Q 1M O D U LE .F O R ' QUASI MODULE -- BEGIN subroutine quasimain(eo,el,el2,f ,a,b,dd,sz,ct,yq,bch,x,il) double complex e l 2 ,j ,i 2 ,ct,yq,yqp,el,dd double precision p i , f ,a,b,d,ql,i l ,eo,ac,il integer sz,n,bch,x external ql pi = 4.dO * datan(l.dO) 'INITIALIZE ALL CONSTANTS j = d c m p l x (0.d o ,1. d o ) d = dreal(dd) if (sz.eq.1) il = 1.008353231642491d-3 if (sz.eq.2) il = 1.807179364733595d-3 if (sz.eq.3) il = 3.146126322998121d-3 if (sz.eq.4) il = 4.288115038096473d-3 if (bch.eq.O) then i2 = dcmplx(O.dO,O.dO) yqp = d c m p l x (0.d o ,0.d o ) ac = 3 0.dO 10 'DO SUMMATION OF TRIPPLE INTEGRALS do n = 1,100 if (n.eq.2) ac = 18.dO 4 if (n.eq.3) ac = 18.dO if (n.gt.3) ac = 27.do i2 = i2 + ql(a,b,d,n,ac) * ((I.a0-el2)/ (I.d0+el2))**n ct = 2.d 0 * e o*el/((dlog(b/a))**2)*(il+2.d0*i2) y q = j * 2.dO * pi * f * ct write (*,10) n,dreal(yq),dimag(yq) format (1 n = 1,13,1 y q so far is 1,F 1 9 .15, 1, ',F 1 9 .15, 'i ' if (cdabs(yqp-yq).It.l.d-6) exit yqp = yq end do ct = 2 .d0*eo*el/((dlog(b/a))**2)*(il+2.d0*i2) bch = 1 endif yq = j * 2.dO * pi * f * ct return end i D O FIRST OF TRIPPLE INTEGRALS double precision function ql(a,b,d,n,ac) double precision q 2 ,s,a,b,d,r,tl,t2,ac integer n,j external q2 s = (b-a)/ac ql = O.dO r = O.dO do j = 1,idnint(ac/3.do) tl = q2(a,b,d,n,r)+ 3 ,d0*q2(a,b,d,n,r+s) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. t2 = 3.d0*q2(a,b,d,n,r+2.dO*s)+q2(a,b,d,n,r+3.dO*s) ql = ql + .375d0 * s * (tl+t2) r = r + s*3.d0 end do return end ! DO SECOND OF TRIPPLE INTEGRALS double precision function q2(a,b,d,n,r) double precision q3,q2p,s/a,b,d,r,rd,tl,t2 integer n,j,p external q3 q2p = 0. do do p = 30,150,9 s = (b-a)/p q2 = 0. dO rd = O.dO do j = 1,idnint(p/3.dO) tl = q3 (d, n, r, rd) +3 .d0*q3 (d, n, r, t2 = 3 .d0*q3 (d, n, r ,rd+2 .d0*s) +q3 q2 = q2 + .375d0 * s *. (tl+t2) rd = rd + s*3.d0 enddo if (dabs(q2-q2p).lt.l.d-6) exit q2p = q2 enddo return end rd+s) (d, n, r ,rd+3 .d0*s) i ! DO THIRD OF TRIPPLE INTEGRALS double precision function q3(d,n,r,rd) ■double precision qfn,q3p,s,d,r,rd,t,pi,tl, t2 integer n,j,p external qfn q3p = 0.dO pi = 4.dO * d a t a n (1.dO) do p = 3,150,3 s = pi/p q3 = 0. dO t = O.dO do j = 1,idnint(p/3.do) tl = qfn(d,n,r,rd,t)+ 3 ,dO*qfn(d,n,r,rd,t+s) t2 = 3.d0*qfn(d,n,r,rd,t+2.d0*s)+qfn(d,n,r,rd,t + 3 .d0*s) q3 = q3 + .375d0 * s * (tl+t2) t = t + s*3.d0 enddo if (dabs(q3-q3p).lt.l.d-6) exit q3p = q3 enddo return end R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ! DO THE FUNCTION OF TRIPPLE INTEGRAL double precision function qfn(d,n,r,rd,t) double precision d,r,rd,t integer n qfn = dcos(t)/dsqrt(r*r+rd*rd-2*r*rd*dcos(t)+4*n*n*d*d) return end ! QUASI MODULE -- END B.2.15 Q 2M O D ULE.FOR ! QUASI MODULE -- BEGIN subroutine qdirect(x(y,a,b,d,el2,sum,limit) double complex e l 2 ,yt,sum,yy,d double precision a , b ,x, y,xl,y l ,d e l ,limit del = (y-x)/100.dO !GENERATE STEP SIZE yt yy xl yl do = d c m p l x (0.d O ,0.dO) !DO THE INTEGRATION = d c m p l x (0.d O ,0.dO) = x = x while(yl.I t .y) yl = yl + del if (yl.gt.y) yl = y call qintmain(xl,yl,a,b,d,el2,yy,limit) yt = yt + yy xl = yl enddo sum = yt return end ! GAUSSIAN QUADTURE METHOD OF INTEGRATION ON MAIN PART subroutine qintmain(s,e,a,b,d,el2,sum,limit) double complex sum,sump,el2,qfnmain, d double precision s ,e ,x,w,a,b,error,limit integer st,i,j,incr dimension x (1000),w(1000) external qfnmain st = 2 incr = 2 if (limit.I t .1.d-9) limit = l.d-9 sump = d c m p l x (0.do,0 .dO) do i = 1,200 'REPEAT UNTIL RESULTS ARE ACCURATE ENOUGH sum = d c m p l x (0.d O ,0.d O ) call evalwght(s,e,X,w,st) in!!!!!!!!!!!!!!!!!!!! !THIS PART IS do j =1,St !GAUSSIAN sum = sum + w(j) * qfnmain(x(j),a,b, d, e l 2 ) !QUADRATURE enddo !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!INTEGRATION R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 251 error = cdabs(sum-sump) !CHECK FOR ERROR sump = sum !STORE PREVIOUS INTEGRAL RESULT !IF INTEGRATION IS CONVERGED THEN EXIT if (error.I t .limit) exit st = st + incr !IF RESULT DIDN'T CONVERGE INCREASE STEP SIZE if (st.gt.950) exit if (st.gt.100) st = 950 enddo return end ! DO THE INTEGRATION ON FOLLOWING FUNCTION double complex function q f n m a i n (1,a,b,d,eI2) double complex al,a2,dm,de,j,e l 2 ,tn,d double precision 1,a,b,intjlr,be external intjlr j = d c m p l x (0.d o ,1.do) al = -j*l a2 = -j*l be = (intjlr(1,a,b))**2 tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn de = el2*a2+j*al*tn qfnmain = ( (l*be*dm)/ (de*al)) - (j*be) return end ! QUASI MODULE -- END B.2.16 R 1M O D U L E .F O R ! EPSILON 1 ROOT MODULE -- BEGIN subroutine mainrtsolver(e2,a,b,d,f,adm,c,los,ges,rt,nc,mi) double complex mainrt,g e s ,x O ,x l ,x 2 ,h i ,h 2 ,f0,f1, f2,fdl, +fd2,f t ,cc,sqr,den,rt,a d m , e 2 ,d double precision e p l ,e p 2 ,a,b,c,f integer i,nc,mi,los external mainrt epl = l.d-7 ep2 = l.d-7 xO = ges + 0.5 xl = ges - 0.5 x2 = ges fO = mainrt(xO,e 2 ,a,b,d,f,adm,c,los) fl = mainrt(xl,e 2 ,a,b,d,f,adm,c,los) f2 = mainrt(x2,e 2 ,a,b,d,f,adm,c,los) hi = xl-xO h2 = x2-xl fdl = (f2-fl)/h2 fd2 = (f1-f0)/hi nc = 0 do i = 2,mi R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 252 99 10 20 if (hl.eq.h2) exit ft = (fd2-fdl)/ (h2-hl) cc = fd2 + h2*ft sqr = cdsqrt(cc*cc - 4*f2*ft) if (dreal(cc)*dreal(sqr)+dimag(cc)*dimag(sqr).I t .0.dO) then den = cc - sqr else den = cc + sqr end if if (cdabs(den).le.O.dO) den = l.dO hi = h2 h2 = -2 * f2/den xO = xl xl = x2 x2 = x2 + h2 fO = fl fl = f 2 f2 = mainrt(x2,e 2 ,a,b,d,f,adm,c,los) if (los.ne.O) then write (*,10) dreal(x2),dimag(x2) format (' zero so far is ',F19.15,', ',F 1 9 .15,1i ') write (*,20) dreal(f2),dimag(f2) format (' fn. value is ',F19.15,', 1,F 1 9 .15, 1i 1,/) endif fdl = fd2 fd2 = (f2 -f1)/h2 if (cdabs(h2).I t .epl*cdabs(x2)) exit if (cdabs(f2).I t .ep2) exit if (cdabs(f2).ge.10.d0*cdabs(fl)) then h2 = h2/2 x2 = x2 - h.2 goto 99 endif if (i.eq.mi) nc = 1 enddo rt = x2 return end ! USE THE FOLLOWING AS MAIN ROOT FUNCTION double complex function mainrt(el,e 2 ,a,b,d,f,adm,c,los) double complex e l ,e 2 ,e l 2 ,k l ,k 2 ,k,rts,y,sum,lmin,j,fnmain,adm,d double precision e o , f ,a,b,c,1,om,ko,11,12,r k l ,r k 2 ,ay,xx,ww +,pi,mo,pie,fr integer n r t s ,u,stage,dbg, los dimension r t s (20),xx(30),ww(3 0) external fnmain j = d c m p l x (0.dO,1.d O ) UNITIALIZE pi = 4.dO * datan(l.dO) eo = l.dO / (36.do * pi * i.d9) mo = 4.do * pi / l.d7 ay = l.d-8 om = 2.dO * pi * f R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. el2 = el/e2 ko = om * dsqrt(mo*eo) kl = ko * cdsqrt(el) k2 = ko * cdsqrt(e2) rkl = dreal(kl) rk2 = dreal(k2) k = (2.dO*pi*om*eo*el)/ (dlog(b/a))**2 I = dnint(dsqrt((10.dO/dreal(d))**2 + rkl**2)) pie = 1 / d s q r t (1**2 - rkl**2) do while ( (ple-1) .g t .1.d - 3 ) 1 = 1 + 100 pie = 1 / d s q r t (1**2 - rkl**2) end do if (los.eq.O) goto 599 dbg = 1 $ INCLUDE:'H E A R T .F O R ' ! GENERATE A GOOD GUESS 599 if (los.eq.O) then sum = d c m p l x (0.d o ,0.dO) y = sum II = 0 .dO. 12 = 1 u = 20 call evalwght(ll,12,xx,ww,u) do u=l,u sum = sum + ww(u) * fnmain(xx(u),a,b,d,kl, k 2 ,el2) enddo y = k * (sum + j * c) end if ! COMPUTE THE DIFFERENCE BETWEEN GIVEN & COMPUTED ADMITANCE mainrt = adm - y return end ! EPSILON 1 ROOT MODULE -- END B.2.17 R 2M O D U L E .F O R ! EPSILON 2 ROOT MODULE -- BEGIN subroutine mainrtsolver(el,a,b,d,f,adm,c,los,g e s ,r t ,nc, mi) double complex mainrt,ges,x O ,x l , x 2 ,h i ,h 2 ,f0, f1,f2,f d l , +fd2,f t ,c c ,sqr,den,rt,adm,el,d double precision epl,ep2,a,b,c,f integer i,nc,mi,los external mainrt epl = l.d-6 ep2 = l.d-7 xO = ges + 0.5 xl = ges - 0.5 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 254 99 10 20 x2 = ges fO = mainrt(xO,el,a,b,d,f,adm,c,los) fl = mainrt(xl,e l ,a ,b , d , f ,a d m , c ,los) f2 = mainrt(x2,e l ,a,b, d , f ,adm,c,los) hi = xl-xO h2 = x2-xl fdl = (f2 -f 1) /h.2 fd2 = (fl-fO)/hi nc = 0 do i = 2,mi if (hl.eq.h2) exit ft = (fd2-fdl)/ (h2-hl) cc = fd2 + h2*ft sqr = cdsqrt(cc*cc - 4*f2*ft) if (dreal(cc)*dreal(sqr)+dimag(cc)*dimag(sqr).I t .0 .dO) then den = cc - sqr else den = cc + sqr end if if (cdabs(den).l e .0.do) den = 1.dO hi = h2 h2 = -2 * f2/den xO = xl xl = x2 x2 = x2 + h.2 fO = fl fl = f 2 f2 = mainrt(x2,e l ,a,b,d,f,adm,c,los) if (los.ne.O) then write (*,10) dreal(x2),dimag(x2) format (' zero so far is ',F19.15,', ',F 1 9 .15,1i ') write (*,20) d r e a l (f2),d i m a g (f2) format (' fn. value is ',F19.15,', ',F 1 9 .15,'i ',/) endif fdl = fd2 fd2 = (f2-fl)/h2 if (cdabs(h2).I t .epl*cdabs(x2)) exit if (cdabs(f2) .It.ep2) exit if (cdabs(f2 ) .ge.10.d0*cdabs(fl)) then h2 = h2/2 x2 = x2 - h.2 goto 99 endif if (i.eq.mi) nc = 1 enddo rt = x2 return end ! USE THE FOLLOWING AS MAIN ROOT FUNCTION double complex function mainrt(e2,e l ,a,b,d,f,a d m , c ,los) double complex el,e2,e l 2 ,k l ,k 2 ,k,rts,y,sum,Imin,j,fnmain,adm,d double precision e o ,f ,a,b,c,1,om,ko,11,12,r k l ,r k 2 ,ay,xx,ww +,pi,mo,pie,fr R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. integer n r t s ,u , s p s ,stage,dbg,los dimension r t s (20), x x (30),ww(30) external fnmain j = d c m p l x (0.dO,1 .dO) UNITIALIZE pi = 4.dO * datan(l.dO) eo = l.dO / (36.dO * pi * l.d9) mo = 4.dO * pi / 1.d7 ay = l.d-8 om = 2.dO * pi * f el2 = el/'e2 ko = om * dsqrt(mo*eo) kl = ko * cdsqrt(el) k2 = ko * cdsqrt(e2) rkl = dreal(kl) rk2 = dreal(k2) k = (2,d0*pi*om*eo*el)/ (dlog(b/a))**2 I = dnint(dsqrt((10.dO/d)*+2 + rkl**2)) pie = 1 / dsqrt(1**2 - rkl**2) do while ( (ple-1) .gt.1.d - 3 ) 1 = 1 + 100 pie = 1 / dsqrt(1**2 - rkl**2) end do if (los.eq.O) goto 599 dbg = 1 $INCLUDE: 1H E A R T .F O R ' ! GENERATE A GOOD GUESS 599 if (los.eq.O) then sum = d c m p l x (0.d o ,0.do) y = sum II = 0.do 12 = 1 sps = 2 0 call e v a l w g h t (11,12,xx,ww,sps) do u=l,sps sum = sum + ww(u) * fnmain(xx(u),a,b,d,kl,k 2 ,el2) enddo y = k * (sum + j * c) end if ! COMPUTE THE DIFFERENCE BETWEEN GIVEN & COMPUTED ADMITANCE mainrt = adm - y return end ! EPSILON 2 ROOT MODULE -- END B .2 . 1 8 R D M O D U L E . F O R ! THICKNESS ROOT MODULE -- BEGIN R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. subroutine mainrtsolver(el,e2 ,a, b, f ,a d m , c ,g e s ,r t ,n c ,m i ,lcs) double complex m a i n r t ,g e s ,x O ,x l ,x 2 ,h i ,h 2 ,f 0,f1, f2 , f dl,fd2 ,f t , +cc,sqr,den,rt,adm,el,e2 double precision e p l ,e p 2 ,a,b,c,f integer i,nc,mi,los external mainrt 99 10 20 epl = l.d-7 ep2 = l.d-7 xO = ges + 0.id-3 xl = ges - 0.1d-3 x2 = ges fO = m a i n rt(el,e 2 ,a ,b,x O ,f ,adm,c,los) fl = mainrt(el,e 2 ,a ,b,x l ,f ,adm,c,los) f2 = mainrt(el,e 2 ,a ,b,x 2 ,f ,adm,c,los) hi = xl-xO h2 = x2-xl fdl = (f2 -f1)/h2 fd2 = (fl-fO)/hi nc = 0 do i = 2 ,mi if (hl.eq.h2) exit ft = (fd2-fdl)/ (h2-hl) cc = fd2 + h2*ft sqr = cdsqrt(cc*cc - 4.a0*f2*ft) if (dreal(cc)*dreal(sqr)+dimag(cc)*dimag(sqr) .I t .0.dO) then den = cc - sqr else den = cc + sqr end if if (cdabs(den).l e .0.d o ) den = l.dO hi = h.2 h2 = -2.dO * f2/den xO = xl xl = x2 x2 = x2 + h2 fO = fl fl = f 2 f2 = mainrt(el,e2,a ,b, x 2 ,f ,a d m , c ,los) if (los.ne.O) then write (*,10) dreal(x2),dimag(x2) format (' zero so far is ',F19.15,', ',F 1 9 .15,1i ') write (*,20) dreal(f2),d i m a g {f2) format C fn. value is ',F19.15,', ',F 1 9 .15,'i ',/) endif fdl = fd2 fd2 = (f2-fl)/h2 if (cdabs(h2).I t .epl*caabs(x2)) exit if (cdabs(f2).I t .e p 2 ) exit if (cdabs(f2 ) .ge.10.d0*cdabs(f1)) then h2 = h.2/2 .dO x2 = x2 - h2 goto 99 endif R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. if (i.eq.mi) enddo rt = x2 return end nc = 1 ! USE THE FOLLOWING AS MAIN ROOT FUNCTION double complex function mainrt(el,e 2 ,a,b,d,f,a d m , c ,los) double complex e l ,e 2 ,e ! 2 ,k l ,k 2 ,k , r t s ,y,sum,lmin,j,fnmain,adm,d double precision e o , f ,a , b , c ,1,om, ko,11,12,rk l ,r k 2 ,ay, +pi,mo,ple,xx,ww,fr integer nrts,u,stage,los,dbg dimension r t s (20),x x (30),w w (30) external fnmain j = d c m p l x (0.dO,l.dO) 'INITIALIZE pi = 4.do * datan(l.dO) eo = l.dO / (36.dO * pi * l.d9) mo = 4.dO * pi / 1.d7 ay = l.d-8 om = 2 * pi * f el2 = el/e2 ko = om * dsqrt(mo*eo) kl = ko * cdsqrt(el) k2 = ko * cdsqrt(e2) rkl = dreal(kl) rk2 = dreal(k2) k = (2.d0*pi*om*eo*el)/ (dlog(b/a))**2 I = dnint(dsqrt((10.dO/dreal(d))**2 + rkl**2)) pie = 1 / d s q r t (1**2 - rkl**2) do while ((ple-l).gt.1.d - 3 ) 1 = 1 + 100 pie = 1 / d s q r t (1**2 - rkl**2) end do if (los.eq.O) goto 599 dbg = 1 $INCLUDE:'HEART.FOR1 ! GENERATE A GOOD GUESS 599 if (los.eq.O) then sum = d c m p l x (0.d O ,0.d O ) y = sum II = 0.dO 12 = 1 u = 20 call e v a l w g h t (11,12,xx,ww,u) do u=l,u sum = sum + ww(u) * fnmain(xx(u),a,b,d,kl,k2,e l 2 ) enddo y = k * (sum + j * c) end if ! COMPUTE THE DIFFERENCE BETWEEN GIVEN & COMPUTED ADMITANCE R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. mainrt = adm - y return end ! THICKNESS ROOT MODULE -- END R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 259 A p p e n d ix C LISTING OF FORTRAN PROGRAMS IN METHOD OF MOMENTS FORMULATION C .l BRIEF EXPLANATION OF FORTRAN PROGRAMS T h e M e t h o d o f M o m e n t s f o r m u l a t i o n o f t h e p r o b l e m o f c o a x ia l lin e s e n s o r t e r m i n a t e d b y t w o la y e r e d m e d ia a n d t h e c o r r e s p o n d in g s o l u t i o n h a s b e e n d e s c r ib e d i n s e c tio n s 5.3 a n d 5 .4 , C h a p t e r 5. I n o r d e r t o s o lv e t h e e q u a ti o n (5.10) f o r e le c tr ic f ie ld d i s t r i b u t i o n a t t h e a p e r t u r e f o r a g iv e n f r e q u e n c y , f o r a g iv e n d i m e n s io n o f c o a x ia l s e n s o r a n d f o r a g iv e n s e t o f t e r m i n a t i n g m e d ia , t h e m a t r i x e le m e n t s A xy o f e q u a ti o n (5.49) h a s t o b e c o m p u t e d . E a c h m a t r ix e le m e n t g e n e ra tio n o f m a trix A needs £d(p, ,pt) and z ( pv, p t) to b e c o m p u t e d . T h e s e t w o f u n c t i o n s a re d e s c r ib e d i n s e c tio n s 5 .3 .1 a n d 5 .3 .2 . T h e f u n c t i o n A.'d ( p v, p t ) c a n b e c o m p u t e d d i r e c tl y w i t h o u t a n y d if f ic u lty , how ever z( pt , p t) n e e d s a n i n t e g r a t i o n t o b e d o n e . I n cases o f lo ss less m a te r ia ls t e r m i n a t i n g t h e c o a x ia l s e n s o r , it is e s s e n tia l t h a t s in g u la r itie s be e x tr a c t e d f r o m i n te g r a n d . F o r t h is p u r p o s e , in t h e f o ll o w i n g p r o g r a m s , t h e lo g ic g iv e n i n F i g u r e B - l w a s u s e d . A ls o t h e s y m m e t r y o f m a t r ix A c a n be u s e d t o g e n e r a te l o w e r o r u p p e r t r i a n g u l a r p a r t o f m a t r i x a n d fill t h e m u p a p p r o p r i a t e l y i n t h e o t h e r h a lf o f t h e m a tr ix . T h i s A p p e n d i x lis ts t h e F O R T R A N p r o g r a m s , f o r t h e f o r w a r d p r o c e s s o f d e t e r m i n a t i o n o f e le c tr ic fie ld d i s t r i b u t i o n a t t h e a p e r t u r e a n d t h e a p e r tu r e a d m itta n c e . R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. T h e M e t h o d o f M o m e n t s o l u t i o n n e e d s g e n e r a ti o n o f m a t r i x A as g iv e n e q u a ti o n (5 .4 9 ). T h i s m a t r i x c a n b e g e n e r a te d as d e ta ile d b e lo w . C.1.1 C om putation o f K l , ( p ,. p r ) : 1. F i n d p „ , i.e . m u l t i p l e r o o t s 1 o f e q u a ti o n (5.15) 2. F in d y„ and A„ f r o m e q u a ti o n s (5.12) a n d (5.17) r e s p e c tiv e ly 3. F i n d 4>„ ( p r ) a n d 4>„(p v) f r o m e q u a ti o n (5.16) 4. C o m p u te /q .,(p i> P .t) f r o m e q u a ti o n (5 .1 1 ), b y d o i n g s u m m a tio n fro m n = 1 to N 5. C o m p u t e £ cl( p v.,p_r ) f r o m e q u a ti o n (5 .1 9 ), f o r n =0 (i.e., f o r T E M M o d e ) , a n d a d d t h e r e s u l t t o t h a t o f s te p 4. C.1.2 Com putation o f z ( p „ , p r ): 1. C o m p u t e c ( p , , p t ) f r o m e q u a ti o n (5.24) 2. I n te g r a t e t h e f u n c t i o n o f e q u a t i o n (5.21) f r o m 0 t o L , u s e s i n g u l a r i ty e x t r a c t i o n if n e c e s s a ry 3. C o m p u t e Z ( p v, p t ) f r o m e q u a t i o n (5.20). 'The num ber of roots to be found, depends on the number o f modes to be considered at the aperture o f the coaxial line. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. C.1.3 Generation o f M atrix V a ry of A ; x a n d y f r o m 0 t o A/, a t e a c h (x ,y) f in d t h e v a lu e Ary f r o m e q u a t i o n (5.50). I n o r d e r t o sa v e t im e , t h e m a trix A e q u a tio n c a n b e g e n e r a te d as f o llo w s . D i v i d e (5.50) b y in te rc h a n g in g p v . N o w , m a trix A An o f is s y m m e t r i c a n d p t a n d p v d o n o t m a k e a n y d if f e r e n c e , i.e ., A xy = Ayx. U s i n g t h i s s y m m e t r y , g e n e r a te o n l y l o w e r tria n g u la r m a trix of A and fill u p u p p e r tria n g u la r m a t r i x o f A f r o m s y m m e t r y . F i n a ll y , e a c h r o w o f A c a n b e m u ltip lie d b y c o rre s p o n d in g p , . O nce, A is g e n e r a te d , it c a n b e i n v e r t e d a n d f o l l o w i n g e q u a ti o n s (5.52), (5.53) a n d (5.55), t h e a p e r t u r e a d m i tt a n c e w a s f o u n d . T h e F O R T R A N p r o g r a m s w e r e w r i t t e n b a s e d o n m o d u le s , as w a s d o n e in A p p e n d i x B . T h e m a i n p r o g r a m s a re g iv e n i n T a b le C - l , t h e m o d u le s u s e d i n th e s e m a i n p r o g r a m s a re d e s c r ib e d i n T a b le C -2 . R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 262 T a b l e C - 1 M a in F o r t r a n P r o g r a m s f o r C h a p t e r 5 section # M ain Program Description C.2.1 M O M - Y .F O R F i n d t h e a p e r t u r e a d m i tt a n c e b y M O M T a b l e C - 2 S u b r o u t i n e M o d u le s u s e d i n ° r o g r a m s o f T a b l e C - l section # Modules section # Modules C.2.2 M O M - Z .F O R C.2.9 M O M - Z R T .F O R C.2.3 M O M - K C .F O R C.2.10 M O M - K C R T .F O R C.2.4 M O M - C .F O R C.2.11 M O M - H A R T .F O R C.2.5 M O M - M I C E .F O R C.2.6 M O M -IN T G .F O R C.2.7 M O M - IN T S .F O R C.2.8 M O M - F U N C .F O R R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. C .2 F O R T R A N P R O G R A M L I S T I N G C.2.1 M O M -Y.F O R $DECLARE $INCLUDE:'M O M - Z .F O R ' $INCLUDE:1M O M - C .F O R ’ $INCLUDE:'M O M -MICE.FOR' $INCLUDE:'M O M -INTG.F O R ' $INCLUDE:'MOM-INTS.F O R ' $INCLUDE:'M O M -FUNC.F O R ' $INCLUDE: 'M O M-ZRT.F O R 1 $INCLUDE:’M O M - K C .F O R ' $INCLUDE:'M O M -K C R T .F O R ' ! MOM -- Y BEGIN program mom-y.for double complex e l ,e 2 ,z ,d,kc,axy,det,j ,y, s ,yo double precision f ,rx,ry,accuracy,i,r,el,a,b,pi,delta,om,eo,m o , + k l ,k o ,zero,r h o ,T a u ,gaman integer dbg,n,m,ii,j j ,ijob,mel,m e 2 ,s z ,i nt,rowlen character*64 filename dimension a x y (100,101),r h o (100),g a m a n (200) open(unit open(unit open(unit open(unit open(unit = 2, = 3, = 4, = 5, = 6, file file file file file = 'r e a l .d a t ',status = ’unknown') = 'imag.dat',status = 'unknown') = 'a b s .d a t ',status = 'unknown') = 'a n g .d a t ',status = 'unknown') = 'n r h o .d a t ',status = 'unknown') 1 READ ALL VARABLES REQUIRED FROM KEYBOARD write (*, ' (A\)') ' Enter the Result Data Filename: ' read (*, ' (A)') filename open(unit = 1, file = filename,status = 'unknown') write (*,*) 'Select Medium 1 & Medium 2: ' write (*,*) '____________________________________________ write (*,*) ' [1] Methanol [2] Water [3] Acetone [4] Ethane- )5] +Glycerol [6] Air' write (*,*) ' [7] Teflon [8] Nylon - 66 [9] Polyurethane [101 Po +lyethylene [11] Derlin' write (*,*) ' [12] CarbonTetraChloride [13] Any other material [1 +4] Conductor' write (*,*) '___________________________________________________________ write write write (*,*) ' Note: In Cases of Solids, Select [13] if f > 3 GHz' (*,*) ' Do not Select [14] for Medium 1' (*,*) '___________________________________________________________ read (*,*) mel if (mel.eq.13) then write (*,*) 'Enter el [Real Part , Imaginary P a r t ] : ' R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 264 read (*,*) r,i if (dabs(i).I t .0.OldO) then write (*,*) 'Enter a small negative number if Medium 1' write (*,*) ' is a Dielectric (Lossless) Ex: - 0.01' read (*,*) i endif el = dcmplx(r,i) endif read (*, *) me2 if (me2.eq.13) then write (*,*) 'Enter e2 [Real Part , Imaginary P a r t ] : ' read (*,*) r,i e2 = dcmplx(r,i) endif write (*,*) '__________________________________________________ write (*,*) 'Enter Frequency [GHz]: ' read (*,*) f f = f * l.d9 write (*,*) 'Enter Cable Size [Enter 1-5] write (*,*) ' [1] 2.2mm [2] 3.6mm [3] 6.4mm [4] 8.3mm' write (*,*) ' [5] Any other Dimensions of Cable' read (*,*) sz if (sz.eq.5) then write (*,*) ' ==================>' write (*,*) ' Enter a [mm] : ' read (*,*) a write (*,*) ' Enter b [mm] : ' read (*,*) b endif write (*,*) 'Enter d [mm] : ' read (*,*) r d = dcmplx(r*l.d - 3 ,0.dO) write (*,*) 'Enter 1: For Simpson"s 1/3 Adaptive Integration' write (*,*) 1 2: For Gaussian Quadrature Integration1 read (*,*) int if (int.eq.l) then write (*,*) 'Enter Accuracy Required [%] , Ex: 2 Percent:' read (*,*) accuracy accuracy = accuracy/100 else write (*,*) 'Enter Accuracy Required : ' read (*,*) accuracy endif write (*,*) 'Enter number of modes [Max = 199] : ' read (+,*) n write (*,*) 'Enter number of partitions required [Max = 99]: ' read (*,*) m write (*,*) 'Enter dielectric Constant of the Material in between + the Conductors: ' read (*,*) el write (*,*) 'Enter1: if Debug is Required ' write (*,*) ' 0: if Detailed Debug is Required ' write (*,*) 1 Any Other Integer if Debug is Not Required' R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 265 read (*,*) dbg write (*,*) ' ! INITIALIZE CABLE SIZES if (sz.eq.l) then a = 0.255d-3 b = 0.838d-3 elseif (sz.eq.2) then a = 0.455d-3 b = 1.499d-3 elseif (sz.eq.3) then a = 0.824d-3 b = 2.655d-3 elseif (sz.eq.4) then a = 1.124d-3 b = 3.62d-3 endif ! INITIALIZE ALL CONSTANTS/VARIABLES pi = 4.d O * d a t a n (1.dO) j = d c m p l x (0.d O ,1.d O ) delta = (b-a)/m eo = 1.d O / (36.d O *pi*l.d9) mo = 4.d0*pi/l.d7 om = 2*pi*f ko = om * dsqrt(mo*eo) kl = ko * dsqrt(el) Tau = dsqrt(mo/(el*eo)) yo = d c m p l x ((2.dO*pi)/ (dlog(b/a)*Tau),0.do) 'FIND n GAMMA AND STORE THEM IN gaman call momkcrt(a,b,n,gaman) ' INITIALIZE PERMITTIVITY FOR BOTH MEDIA if (mel.eq.l) then el = 5.7d0 + 2 7.4d0 / ( I . d 0 + j * elseif (mel.eq.2) then el = 5.dO + 73.dO / (l.dO + j * f elseif (mel.eq.3) then el = 1.9d0 + 19.3d0 / (l.dO + j * elseif (mel.eq.4) then el = 4.2d0 + 20.9d0 / (l.dO + j * elseif (mel.eq.5) then el = 4.18d0 + 38.3d0 / ((l.dO + j elseif (mel.eq.6) then el = d c m p l x (1.dO,-O.OOldO) elseif (mel.eq.7) then el = d c m p l x (2.ld O ,-0.OOldO) elseif (mel.eq.8) then el = d c m p l x (3.14d0,-0.05d0) elseif (mel.eq.9) then el = d c m p l x (3.4 d 0 ,-0.OOldO) elseif (mel.eq.10) then f/3.d9) / 19.7d9) 'methanol 'water om * 3.3d-12) 'acetone om * 1.4d-10) !ethanol * om * 2.49d-9)**0.6)'glycerol R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 266 el = d c m p l x (2.26d0,-0.OOldO) elseif (mel.eq.ll) then el = d c m p l x (2.8d0,-0.OOldO) elseif (mel.eq.12) then el = d c m p l x (2.17d0,-0.OOldO) endif if (me2.eq.l) then e2 = 5.7d0 + 27.4d0 / ( I . d 0 + j * elseif (me2.eq.2) then e2 = 5.dO + 73.dO / (l.dO + j * f elseif (me2.eq.3) then e2 = 1.9d0 + 19.3d0 / (l.dO + j * elseif (me2.eq.4) then e2 = 4.2d0 + 2 0.9d0 / (l.dO + j * elseif (me2.eq.5) then e2 = 4.18d0 + 38.3d0 / ((l.dO + j elseif (me2.eq.6) then e2 = d c m p l x (1.d O ,0.dO) elseif (me2.eq.7) then e2 = d c m p l x (2.ldO,0.dO) elseif (me2.eq.8) then e2 = d c m p l x (3.14d0,-0.05d0) elseif (me2.eq.9) then e2 = d c m p l x (3.4a0,0 .do) elseif (me2.eq.10) then e2 = d c m p l x (2.26d0,0 .dO) elseif (me2.eq.ll) then e2 = d c m p l x (2.8d 0 ,0.dO) elseif (me2.eq.12) then e2 = d c m p l x (2.17d0,0.d O ) elseif (me2.eq.14) then e2 = d c m p l x (0.dO,-10.d6) endif f/3.d9! / 19.7d9) !methanol Iwater om * 3.3d-12) !acetone om * 1.4d-10) !ethanol * om * 2.49d-9)**0.6) iglycerol ! GENERATE LOWER TRIANGULAR PART OF MATRIX A if ( (dbg.ne.1 ) .and.(dbg.ne.0)) then write (*,*) 1 MATRIX ELEMENT BEING GENERTARED:' endif ry = a rowlen = 1 do i i = 1,m if (ii.eq.l) then ry = ry + delta/2.dO else ry = ry + delta endif rx = a do j j = 1,rowlen if ((dbg.n e .1).and.(dbg.ne.0)) then write (*,50) ii,jj 50 format (' .... [1,12, ', ',12, '] ') endif if (j j .eq.1) then rx = rx + delta/2.do R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 500 600 700 800 900 else rx = rx + delta endif call m o mz(el,e 2 ,f ,rx,ry,d , z ,accuracy,dbg,int) call momkc(rx,ry,a,b,kl,n , k c ,gaman) axy(ii,jj) = j*delta*eo*om*(el*z-pi*el*kc) if ( (dbg.eq.1 ) .or.(dbg.eg.0)) then write (* 500) ii- j j format ( ITERATION N O : [' ,12, ', 1,12, '] ') write (* 600) dreal(kc),dimag(kc) format ( kc = ',F 2 5 .15,',',F 2 5 .15, 1i ' write (* 700) d r e a l (z ),d i m a g (z ) format ( z = ',F 2 5 .15,',',F 2 5 .15, 'i ' write (* 800) dreal(axy(ii,j j )),dimag(axy(ii,jj)) format ( axy = ',F 2 5 .15, ', 1,F 2 5 .15, 1i ' write (* 900) rx, ry format ( r x ,ry = 1,F 2 5 .15,',',F 2 5 .15) pause ' ... matrix element generation is completed, +ng for p r o m p t ' endif enddo rowlen = rowlen+1 enddo ! USE SYMMETRY TO GENERATE UPPER TRIANGULAR PART OF MATRIX A rowlen = 1 do ii = l,m-l rowlen = rowlen+1 do jj = rowlen,m a x y (ii,jj) = axy(jj,ii) enddo enddo waiti ' ! MULTIPLY MATRIX A BY rho rx = a do jj = l,m if (j j .eq.1) then rx = rx + delta/2.dO else rx = rx + delta endif do ii = l,m axy(ii,jj) = rx * axy(ii,jj) enddo enddo ! GENERATE l/RHO jj = m+1 ry = a do ii = l,m if (ii.eq.l) then ry = ry + delta/2.do else ry = ry + delta R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. endif axv(ii,jj) = d c m p l x (1.dO/ry,0.do) r h o (i i ) = ry enddo zero = l.d-200 ! DETERMINANT SINGULARITY PARAMETER ijob = -X ! SET FOR SOLUTION OF SET OF LINEAR EQUATIONS call cmatpac(ijob,axy,m,1,d e t ,zero) jj =m+l ! WRITE RESULTS IN A FILE, COMPUTE Y s = d c m p l x (0.dO,0 .dO) write (1,*) 1 METHOD OF MOMENTS RESULTS:' write (1,*) 1 write (1,*) 'Rho[mm],Re[E] ,Im [E],Abs[E],Ang[E] deg,Norm[1/rho] : write (2,*) m , 1 2' write (3, *) tn, 1 2 ' write (4,*) m , ' 2' write (5,*) m , 1 2 1 write (6,*) m, ' 2 ' do i i = 1,m write (1,1500) rho(ii)*1.d 3 ,dreal(axy(ii,j j )) , + dimag(axy(ii,jj)),cdabs.(axy(ii,jj) ) , + 18 0.d0*datan(dimag(axy(ii,jj))/dreal(axy(ii,jj)))/pi, + cdabs(axy(idnint(m/2.d o ),j j ))+rho(idnint(m/2.do))/rho(ii) 1500 format (F8.3,E 1 2 .5 E 2 ,E 1 2 .5 E2,E 1 2 .5 E2,F 8 .3,E 1 2 .5E 2 ) write (2,*) rho(ii),d r e al(axy(i i ,jj)) write (3,*) rho(ii),dimag(axy(ii,jj)) write (4,*) r h o (ii),cdabs(axy(i i ,jj ) ) write (5,*) rho(ii),18 0.d0*datan(dimag(axy(ii,j j ))/ + dreal(axy(ii,jj)))/pi write (6,*) rho(ii), + cdabs(axy(idnint(m/2.do),jj))*rho(idnint(m/2.d o ) )/rho(ii) s = s + axy(ii,j j ) ! INTEGRATE E-FIELD FROM a TO b enddo y = 2.d0/s/delta - yo write + 2000 '2100 2200 2300 2400 (1,*) 1______________________________________ _ ' write (1,2000) a * l .d 3 ,b * l .d 3 ,dreal(d)*1.d 3 ,n , m , f / I .d9 format ('a [mm] = ',F8.3,' b = ',F8.3,' d = ',F8.3,/, +' Modes [N] = ',13,' Partitions [M] = ',13, +' f [GHz] = ',F 8 .2) write (1,2100) dreal(el),dimag(el),dreal(e2),dimag(e2) format (' el =',F 1 9 .5,',',F 1 9 .5,'i ',/, + ' e2 = ' ,F19.5, ', ',F 1 9 .5, 'i ') write (1,2200) dreal(y),dimag(y) format (' Admittance =',E 1 5 .6 E 2 ,E 1 5 .6 E 2 ,'i ') write (1,23 00) dreal(s),dimag(s) format (' Sigma E rho =',E 1 5 .6 E 2 ,E 1 5 .6 E 2 ,'i ') write (1,2400) d r e a l (1.dO/yo),dreal(yo) format (' Zo, Yo = ',F 1 2 .6,F 1 2 .6) write (1,*) '__________________________________________________________ R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 269 stop end ! MOM - - Y END C .2 .2 M O M -Z.F O R ! MOM-Z MODULE -- BEGIN subroutine m o m z(el,e 2 ,f ,rx,ry,d,z,ay,dbg,int) double complex e l ,e 2 ,e l 2 ,k l ,k 2 ,r t s ,z ,sum,lmin,j ,d double precision e o , f ,rx,ry,1,om,ko,11,12,r k l ,r k 2 ,a y , f r ,p i , +mo,ple,momc,c integer n r t s ,u,min,dbg,int dimension r t s (20) external momc ! INITIALIZE j = d c m p l x (0.d O ,1.dO) pi = 4.d 0 * d a t a n (1.do) el2 = el/e2 eo = 1.d O / (36.d 0 * p i * l .d9) mo = 4.d0*pi/l.d7 om = 2*pi*f ko = om * dsqrt(mo*eo) kl = ko * cdsqrt(el) k2 = ko * cdsqrt(e2) rkl = dreal(kl) rk2 = dreal(k2) 1 = dnint ( d s q r t ((10.0/d)**2 + rkl**2)) pie = 1 / d s q r t (1**2 - rkl**2) do while ((ple-1).gt.le-3) 1 = 1 + 100 pie = 1 / d s q r t (1**2 - rkl**2) end do $INCLUDE: 'M O M - H A R T .FOR 1 return end ! MOM-Z MODULE -- END C .2 .3 M O M - K C . F O R ! MOM-KC MODULE -- BEGIN subroutine momkc(rx,ry,a,b,kl,n , k c ,gaman) double complex j,bn,betan,sum,kc double precision a,b,gaman,pi,k l ,an,phiofro,rx,ry,an2,phirx,phiry integer i ,n dimension g a m a n (200) external gn,fgn,an,bn,phiofro j = d c m p l x (0.d O ,1.dO) pi = 4.d 0 * d a t a n (1.do) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. sum = d c m p l x (0.dO,0 .dO) do i = 1,n an2 = an(gaman(i),a,b) betan = b n ( k l ,g a m a n (i )) phirx = phiofro(rx,gaman(i ) ,a) phiry = p h i o f r o (ry,gaman(i ) ,a) sum = sum + (phirx*phiry)/ (an2*betan) enddo kc = j * (sum + 1.do/(rx*ry*dlog(b/a)*kl)) return end ! THE FUNCTION FOR PHI OF RHO double precision function phiofro(ro,gn,a) double precision ro,gn, a ,j0,y O ,j1,yl external j 0,y O ,j1,yl phiofro = j1 (gn*ro)*y0(gn*a)-j 0 (gn*a)*yl(gn*ro) return end ! THE FUNCTION FOR BETA-r. double complex function bn(kl,gn) double precision kl,gn if (kl.gt.gn) then bn = dcmplx(dsqrt(kl**2 - gn**2),0.d0) else bn = d c m p l x (0.d o ,-dsqrt(gn**2 - kl**2) ) endif return end ! THE FUNCTION FOR A-n TO THE POWER 2 double precision function an(gn,a,b) double precision gn,a,b,k,m,o,p,pi,j0 external j 0 pi = 4.d 0 * d a t a n (1.dO) k = 2.d o / ((pi*gn)**2) m = (jO(gn*a))**2 o = (j 0 (gn*b))**2 p = m/o an = k*(p-1) return end ! MOM-KC MODULE -- END C.2.4 M O M -C .F O R ! MOM-C MODULE -- BEGIN double precision function momc(rx,ry) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 271 double precision p i , d l ,rx,ry,sum,so,si,s2,tm,p,elp,fun, + s t ,en,p2,r l ,ising,cp,tiny integer n,i,steps external elp,fun 100 pi = 4.d O * datan(1.d O ) steps = 2 tiny = l.dO cp = 0 .do do while (tiny.g t .1.d-4) n = steps si = 0.dO s2 = 0.dO so = O.dO st = O.dO en = pi dl = (en-st)/ ( 2 .dO*n) i = 0 i = i+1 p = (st+i*dl) tm = fun(rx,ry,p) si = sl+tm i = i+1 p = (st+i*dl) tm = fun(rx,ry,p) s2 = s2+tm if (i.I t . (2*n-2)) then goto 100 else p = (st+(2.d 0 * n - l .dO)*dl) si = si + fun(rx,ry,p) p = st so = fun(rx,ry,p) p = en so = so + fun(rx,ry,p) sum = (dl/3.dO)* (so+4,d0*sl+2.d0*s2) endif p2 = (4,dO*rx*ry)/ ( (rx+ry)**2) rl = rx + ry if (p2.eq.l.d0) then p2 = 0.99999 endif ising = (2.dO/rl)*elp(p2) momc = sum + ising steps = steps + 8 tiny = dabs(momc - cp) if (steps.gt.40) tiny = 0.0 cp = momc end do momc = momc/pi return end ! THE FUNCTION FOR C R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. double precision function fun(rx,ry,p) double precision rx,ry,p,r,r l ,r 2 ,r3 rl = (rx**2)+ (ry**2) r2 = 2.d0*rx*ry*dcos(p) r3 = rl - r2 if (r3.I t .1.d -18) then r3 = O.dO endif r = dsqrt(r3) if (r.eq.O.O) then fun = 0.OdO else fun = (dcos(p)-1.dO) endif return end ! MOM-C MODULE -- END / r C.2.5 M O M -M ICE.FOR ! MISCELLANEOUS MODULE -- BEGIN ! BESSEL FUNCTION J O ( x ) , x = a REAL NUMBER double precision function jO(x) double precision y ,p i ,p 2 ,p 3 ,p 4 ,p 5 ,q l ,q 2 ,q 3 ,q 4 ,q 5 ,r l ,r 2 ,r 3 ,r 4 ,r 5 , +r6,sl,s2,s3,s4,s5,s6,x,ax,z,xx data pl,p2,p3,p4,p5/l.d0,-.1098628627d-2, .2734510407d-4, + - ,2073370639d-5,.2093887211d-6/, q l ,q 2 ,q 3 ,q 4 ,q5/ + - .1562499995d-l,.1430488765d-3,-.691114 7651d-5, +.7621095161d-6, - .934945152d-7/ data r l ,r 2 ,r 3 ,r 4 ,r 5 ,r6/57568490574.d O ,-13362 5903 54.d O , + 6 51619640.7 d 0 ,-11214424.18d0,773 92.33 017d0 ,-184 .90524 56d0/ , +sl,s2,s 3 ,s 4 ,S5,s6/57568490411.dO,1029532985.d0, +94 94680.718d0,592 72.64853d0,267.8532712d0, 1 ,d0/ if(dabs(x).I t .8.) then y=x**2 j0=(rl+y*(r2+y*(r3+y*(r4+y*(r5+y*r6) ) ) ) ) + / (sl+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))) else ax=dabs(x) z = 8 ./ax y=z**2 x x = a x - 0 .785398164d0 j 0=dsqrt(.636619772/ax)* (dcos(xx)* (pl+y*(p2+y*(p3+y*(p4+y + *p5) ) ) ) -z*dsin(xx)* (ql+y*(q2+y*(q3+y*(q4+y*q5))))) endif return end R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ! BESSEL FUNCTION J 1 ( x ) , x = a REAL NUMBER double precision function jl(x) double precision y ,p i ,p 2 ,p 3 ,p 4 ,p 5 ,q l ,q 2 ,g 3 ,q 4 ,q 5 ,r l ,r 2 ,r 3 ,r 4 ,r 5 , +r6,sl,s2,s3,s4,s5,s6,x,ax,z,xx data rl,r2,r3,r4,r5,r6/72362614232.d O ,-789505 9235.d O ,242 3 968 5 3.1 + d 0 ,-2 972611.43 9d0,15704.4 826O d O ,-30.16036606d0/, +Sl, S 2 ,S3,s4,s5,s6/144725228442.d O ,2300535178.d O ,18583304.74d0, +9944 7.43 3 94d0,3 76.999139 7 d 0 ,1.dO/ data pl,p2,p3,p4,p5/l.d0,.183105d-2,-.3516396496d-4,.2457520174 +d-5,-.240337019d-6/,ql,q2,g3,q4,q5/.04687499995d O , -.20026 908 73d-3 + ,844 9199096d-5,-.88228987d-6,.105787412d-6/ if(dabs(x).I t .8.) then y=x**2 jl=x*(rl+y*(r2+y*(r3+y*(r4+y*(r5+y*r6) ) ) ) ) + / (sl+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))) else ax=dabs(x) z = 8 ./ax y=z* *2 x x = a x - 2 .356194491 j l = d s q r t (.636619772/ax)* (dcos(xx)* (pl+y*(p2+y*(p3+y*(p4+y + *p5) ) ))-z*dsin(xx)* (ql+y*(q2+y*(q3+y*(q4+y*q5)))))* d s i g n (1.,x) endif return end ! BESSEL FUNCTION Y 0 ( x ) , x = a REAL NUMBER double precision function yO(x) double precision y ,p i ,p 2 ,p 3 ,p 4 ,p 5 ,q l ,q 2 ,q 3 ,q 4 ,q 5 ,r l ,r 2 ,r 3 ,r4 ,r5 , + r 6 ,s i ,s 2 ,s 3 ,s 4 ,s 5 ,s 6 ,x , z ,xx,j 0 external jo data p i , p 2 , p 3 ,p 4 ,p5/l.d O ,-.1098628627d-2 , .2734510407d-4, + - ,2073370639d-5, ,2093887211d-6/, q l ,q2 ,q3 ,q4 ,q5,/-. 15624 99 c ; : .. +.1430488765d-3,-.6911147651d-5,.76210 95161d-6,-.934 94 5152c; data rl,r2,r3,r4,r5,r6/-2957821389.dO,7062834065,d0, + -512 359803.6 d 0 ,108 79881.2 9 d 0 ,-86327.92757d0,228.4622733d0, , +sl,s2,S3,S4,s5,S6/40076544269.dO,74524 9964.8 d 0 ,7189466.438dC +4744 7.26470d0,226.103 0244d0, l.dO/ if(x.It.8.) then y=x**2 y0=(rl+y*(r2+y*(r3+y*(r4+y*(r5+y*r6) ) ) ) ) / (sl+y*(s2+y + * (s3+y*(s4+y*(s5+y*s6)))))+.636619772d0*j0(x)*dlog(x) else z = 8 ./x y=z**2 x x = x - 0 .7853 98164d0 y 0 = d s q r t (.636619772/x)* (dsin(xx)* (pl+y*(p2+y*(p3+y*(p4+y + *p5))))+z*dcos(xx)* (ql+y*(q2+y*(q3+y*(q4+y*q5))))) endif return R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 274 end ! BESSEL FUNCTION y l ( x ) , x = a REAL NUMBER double precision function yl(x) double precision y , p i ,p 2 ,p 3 ,p 4 ,p 5 ,q l ,q 2 ,g 3 ,q 4 ,q 5 ,r l ,r 2 ,r 3 ,r 4 ,r 5 , +r6,sl,s2,s3,s4,s5,s6,s7,x,z,xx,jl external jl data pl,p2,p3,p4,p5/l.d0, .lS3105d-2,-.3516396496d-4 , +.2457520174d-5,-.2403 37019d-6/, q l ,q 2 ,q 3 ,g 4 ,q 5 / .04687499995 d 0 , +-.2002690873d-3,.844 919 90 96d-5,-.8822 8987d-6,.105787412d-6/ data rl,r2,r 3 ,r 4 ,r 5 ,r 6 / -.4 900604 94 3dl3,.127 5274 3 90dl3, + -.51534 3 813 9dll, .7349264 551d9,-.42 37922726d7, .8511937 935d4/, +Sl,S2,S3,S 4 ,S5,S6,S 7 / .24 99580570dl4,.4244419664dl2, +.3733650367dl0,.2245904002d8,.1020426050d6,.3549632885d3,1.dO/ if (x.It.8.) then y = x**2 yl=x*(rl+y*(r2+y*(r3+y*(r4+y*(r5+v*r6)) ) ) ) / (sl+y*(s2+y*(s3+y* + (s4+y*(s5+y*(s6+y*s7))))))+.636619772d0*(jl(x)*dlog(x)-l.dO/x) else z = 8 .dO/x y=z**2 x x = x - 2 .356194491d0 y l = d s q r t (.636619772d0/x)* (dsin(xx)* (pl+y*(p2+y*(p3+y*(p4+y + *p5) ) ) )+z*dcos(xx)* (ql+y*(q2+y*(q3+y*(q4+y*q5))))) endif return end ! BESSEL FUNCTION J l ( x ) , x = a COMPLEX NUMBER double complex function jlc(xc) double complex xc,sum,crl,c r 2 ,c r 3 ,c r 4 ,c r 5 ,del double precision pi integer n 10 pi = 4. * d a t a n (1.) sum = x c / 2 .0 crl = xc cr2 = 2 . 0 cr4 = 1.0 n = 1 crl = crl*(xc**2) cr2 = cr2+2.0 cr3 = (cr2-2.0)**2 cr4 = cr4*cr3 cr5 = cr4*cr2 del= ((-1)**n)*crl/cr5 sum= sum+del n = n+1 if (dabs(del).gt.le-18) jlc = sum return end goto 10 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ! COMPLETE ELLIPTIC INTEGRAL OF FIRST KIND double precision function elp(x) double precision x,c,d,aO,a l , a 2 ,a 3 ,a 4 ,b O ,b l ,b 2 ,b 3 ,b 4 ,t l ,t2 d = 1. 0 - x c = 1. 0 - x if (d.lt.1.0d-20) d = 1.0d-20 aO = 1.3862944 al = 0.096663443 a2 = 0.035900924 a3 = 0.037425637 a4 = 0.014511962 bO = 0.5 bl = 0.12498594 b2 = 0.068802486 b3 = 0.033283553 b4 = 0.0044178701 tl = (((((((a4*c)+a3)*c)+a2)*c)+al)*c)+a0 t2 = (((((((b4*c)+b3)*c)+b2)*c)+bl)*c)+b0 elp = tl + d l o g (1.OdO/d) * t2 return end ! MATRIX OPERATIONS PACKAGE subroutine cmatpac(ijob,a,n,m,det,ep) double complex a,b,det,const,s double precision ep,c,d integer ijob,n,m,npl,npm,nml,npi,i p l ,i ,j ,k,1,n p j ,j p l ,kpl dimension a (1 0 0 ,101) 1 3 2 6 det = dcmplx(l.do,0 .do) npl = n + 1 npm = n + m nml = n - 1 if (ijob) 2,1,2 do 3 i=l,n npi = n + i a(i,npi) = l.dO ipl = i + 1 do 3 j = ipl,n npj = n + j a (i ,npj ) = 0 .dO a (j ,npi) = 0.dO do 4 j=l,nml c = c d abs(a(j ,j )) jpl = j + 1 do 5 i=jpl,n d = c d a b s (a (i ,j )) if (c-d) 6,5,5 det = - det do 7 k = j ,npm b = a (i,k) a (i, k) = a (j ,k) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 276 5 15 4 14 16 17 18 11 10 19 13 12 c = d continue if (cdabs(a(j ,j ))-ep) 14,15,15 do 4 i=jpl,n const = a (i ,j ) / a ( j ,j ) do 4 k=jpl,npm a(i,k) = a(i,k) - const*a(j,k) if (cdabs(a(n,n))-ep) 14,18,18 det = O.dO if (ijob) 16,16,17 w r i t e (*,*)'THE DETERMINANT IS ZERO - CAN NOT BE HANDLED' return do 11 i=l,n det = det*a(i,i) continue if (ijob) 10,10,17 do 12 i=l,n k = n-i+1 kpl = k+1 do 12 l=npl,npm s = O.dO if (n-kpl) 12,19,19 do 13 j=kpl,n s = s + a(k,j)* a (j ,1) a (k, 1) = (a (k,1) -s) /a (k, k) return end ! EVALUATE WEIGHTS & ABSCISSAS FOR GAUSSIAN QUAD. INTEGRATION subroutine evalwght(xl,x 2 ,x,w,n) double precision x l ,x 2 ,x ( n ) ,w ( n ) ,p i ,xm,xl,p i ,p 2 , p 3 ,z ,z l ,p p ,eps integer i,j,m parameter (eps=3.d-14) 10 m = (n+1)/2 xm = 0.5 * (x2+xl) xl = 0.5 * (x2-xl) pi = 4. * datan(l.) do i=l,m z = dcos(pi * (i-0.25) / (n+ 0.5)) pi = 1.dO p2 = O.dO do j = 1,n p3 = p2 p2 = pl pi = ((2.d0*j - I.d0)*z*p2 - (j - I.d0)*p3) enddo pp = n * (z*pl - p2) / (z*z - l.dO) zl = z z = zl-pl/pp if(dabs(z-zl).gt.eps) goto 10 x(i) = xm-xl*z x(n+l-i) = xm+xl*z w(i) = 2.dO * x l / ((l.d0-z*z)*pp*pp) / j R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. w(n+l-i) = w(i) enddo return end ! MISCELLANEOUS MODULE -- END C.2.6 M O M -IN T G .F O R I MOM - GAUSIAN INTEGRATION MODULE -- BEGIN ! INTEGRATION BY GAUSSIAN QUADRATURE subroutine directg(x,y,rx,ry,d,kl,k2,e l 2 ,sum,limit) double complex e l 2 ,k l ,k 2 ,y t ,sum,yy,d double precision rx,ry,x,y,xl,y l ,d e l ,limit del = 100.dO !GENERATE STEP SIZE if ((y-x).gt.100.dO) del = (y-x)/10.d0 yt yy xl yl do = dcmplx{O.dO,O.dO) !DO THE INTEGRATION = d c m p l x (0.dO,0 .dO) = x = x w h i l e ( y l .It.y) yl = yl + del if (yl.gt.y) yl = y call intmain(xl,y l ,r x , r y ,d,kl,k2,e l 2 ,y y ,limit) yt = yt + yy xl = yl enddo sum = yt return end ! GAUSSIAN QUADTURE METHOD OF INTEGRATION ON MAIN PART subroutine intmain(s,e,rx,ry,d,kl,k 2 ,e l 2 ,sum,limit) double complex sum,sump,kl,k 2 ,e l 2 ,fnmain,d double precision s ,e,x,w,rx,ry,error,limit integer s t ,i ,j ,incr dimension x ( 1000),w(1000) external fnmain st = 20 incr = 20 if (limit.I t .1.d-9) limit = l.d-9 sump = d c m p l x (0.d O ,0.dO) do i = 1,200 'REPEAT UNTIL RESULTS ARE ACCURATE ENOUGH sum = dcmplx (0 .d O ,0 .d O ) call evalwght(s,e,x,w,st) 'THIS PART IS GAUSSIAN QUADRATURE INTEGRATION do j=l,st sum = sum + w(j) * fnmain(x(j),rx,ry,d,kl,k 2 ,el2) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. enddo error = cdabs(sum-sump) sump = sum !STORE PREVIOUS INTEGRAL RESULT if (error.I t .limit) exit !CHECK FOR ERROR st = st + incr !IF RESULT DIDN'T CONVERGE INCREASE STEP SIZE if (st.gt.300) exit !IF INTEGRATION IS CONVERGED THEN EXIT if (st.gt.100) st = 300 enddo return end ! SINGULARITY EXTRACTION PART subroutine seg(x,y,rx,ry,d,kl,k2,e l 2 ,rt, sum, limit) double complex e l 2 ,k l ,k 2 ,r t ,y s ,y y ,y t ,rd,is,r e s ,sum,d double precision rx,ry,x,y,del,x l ,y l ,limit external rd,is res = rd(rt,rx,ry,d,kl,k 2 ,el2) !FIND THE RESIDUE ys = is(rt,x,y,res) !FIND THE PART TO 3E ADDED TO SE PART del = 10.dO yt yy xl yl = = = = 'GENERATE STEP SIZE dcmplx (0 .d O ,0 .d O ) d c m p l x (0.dO,0 .dO) x x !DO THE INTEGRATION do w h i l e ( y l .I t .y) yl = yl + del if (yl.gt.y) yl = y call intse(xl, yl, rx, ry, d, kl, k2 ,e l 2 ,r e s ,r t ,yy,limit) yt = yt + yy xl = yl enddo sum = (yt+ys) return end ! GAUSSIAN QUADTURE METHOD OF INTEGRATION ON SE PART subroutine intse(s,e,rx,ry,d,kl,k2,e l 2 ,r e s ,r t ,sum,limit) double complex sum,sump,kl,k 2 ,e l 2 ,r e s ,r t ,fnse, d double precision s ,e ,x,w,rx,ry,error,limit integer s t ,i ,j dimension x (900),w (900) external fnse st = 10 if (limit.I t .l.d-9) limit = l.d-9 sump = d c m p l x (0.d O ,0.d O ) do i = 1,200 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. !REPEAT UNTIL RESULTS ARE ACCURATE ENOUGH sum = d c m p l x (0.d O ,O.dO) call evalwght(s,e,x,w,st) !THIS PART IS GAUSSIAN QUADRATURE INTEGRATION do j =1,s t sum = sum + w(j) * f n s e ( x (j ),rx,ry,d,kl,k 2 ,el 2 ,r e s ,rt) enddo st = st + 20 !IF RESULT DIDN'T CONVERGE INCREASE STEP SIZE error = cdabs(sum-sump) !CHECK FOR ERROR sump = sum !STORE PREVIOUS INTEGRAL RESULT if (error.It.limit) exit !IF INTEGRATION IS CONVERGED THEN EXIT if (st.gt.300) exit if (st.gt.100) st = 300 enddo return end ! MOM - GAUSIAN INTEGRATION MODULE -- END C.2.7 M O M -IN TS.FO R ! MOM - SIMPSON'S 1/3 RULE INTEGRATION MODULE -- BEGIN ! INTEGRATION BY ADAPTIVE SIMPSON'S 1/3 RULE subroutine directs(st,en,rx,ry,d,kl,k2,e l 2 ,sum,accuracy) double complex suml,e n d s ,even,odd,sum,fnmain,kl,k 2 ,e l 2 ,d double precision r,accuracy,st,en,h,v,x,rx, ry integer i n t ,n o i ,max,i external fnmain 2 max = 2 0 !LIMIT TO NUMBER OF ITERATIONS if ( (en-st).l e .0.dO) then sum = d c m p l x (0.d O ,0.dO) goto 32 endif noi = 0 !COUNTS ACTUAL NUMBER OF ITERATIONS TAKEN TO INTEGRA odd = d c m p l x (0.d o ,0.d o ) int = 1 v = 1. dO even = d c m p l x (0.d O ,0.dO) suml = d c m p l x (0.d o ,0.d O ) ends = fnmain(st,rx,ry,d,kl,k2,el2) + + fnmain(en,rx,ry,d,kl,k2,el2) h = (en-st) / v odd = even + odd x = st + (h/2.do) even = d c m p l x (0.d o ,0.d o ) do i = l ,int even = even + fnmain(x,rx,ry,d,kl,k2,el2) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 31 32 4 x = x + h enddo sum = (ends + (4.d0*even) + (2.d0*odd)) * h / 6.do noi = noi + l r = c d a b s ((suml-sum)/sum) if (noi-max) 31,32,32 if (r-accuracy) 32,32,4 'ACCURACY IS USED TO TEST IF GIVEN ACCURACY HAS BEEN ACHIEVED !r FINDS THE ACTUAL ACCURACY ACHIEVED !sum YIELDS IN INTEGRAL RESULT return suml = sum int = 2*int v = 2.dO * v go to 2 end ! SINGULARITY EXTRACTION PART subroutine ses(st,en,rx,ry,d,kl,k 2 ,e l 2 ,rt,sum,accuracy) double complex s u m l ,e n d s ,even,odd,sum,fnse,kl,k 2 ,e l 2 ,d,rd, +is,res,rt,ys double precision r,accuracy,st,en,h,v,x,rx,ry integer i n t ,n o i ,max,i external fnse,rd,is 2 31 32 res = rd(rt,rx,ry,d,kl,k 2 ,el2) !FIND THE RESIDUE ys = is(rt,st,en,res) !FIND THE PART TO BE ADDED TO SE PART max = 100 if ((en-st).l e .0.do) then sum = d c m p l x (0.d o ,0.dO) goto 32 endif noi = 0 odd = d c m p l x (0.d o ,0.do) int = 1 v = l.dO even = d c m p l x (0.d o ,0.dO) suml = d c m p l x (0.dO,0 .dO) ends = fnse(st,rx,ry,d,kl,k 2 ,e l 2 ,r e s ,rt) +• + fnse(en,rx,ry,d,kl,k 2 ,e l 2 ,res,rt) h = (en-st) / v odd = even + odd x = st + (h/2.d0) even = d c m p l x (0.d o ,0.do) do i=l,int even = even + fnse(x,rx, ry,d,kl,k 2 ,e l 2 ,r e s ,rt) x = x + h enddo sum = (ends + (4.d0*even) + (2.d0*odd)) * h / 6.d0 noi = noi + 1 r = c d a b s ((suml-sum)/sum) if (noi-max) 31,32,32 if (r-accuracy) 32,32,4 sum = sum + ys R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. return suml = sum int = 2*int v = 2.dO * v go to 2 end ! MOM - SIMPSON'S 1/3 RULE INTEGRATION MODULE -- END 4 C.2.8 M O M -FU N C.FO R ! MOM - INTEGRATION FUNCTIONS -- BEGIN ! DO THE INTEGRATION ON FOLLOWING FUNCTION \MAIN FUNCTION\ double complex function f n m a i n (1,rx,ry,d,kl,k 2 ,el2) double complex al,a2,dm,de,j,k l ,k 2 ,e l 2 ,tn,d double precision 1,rx,ry,j1,Bp external j1 j = d c m p l x (0.d O ,1.dO) al = -j*cdsqrt(l**2-kl**2) a2 = -j*cdsqrt(l**2-k2**2) Bp = jl(rx+1)*jl(ry+1) tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn de = el2*a2+j*al*tn fnmain = ((l*Bp*dm)/ (de*al)) - (j*Bp) return end ! DO THE INTEGRATION ON FOLLOWING FUNCTION ! /SINGULARITY EXTRACTION FUNCTIONS/ double complex function f n s e (1,rx,ry,d,kl,k2 ,e l 2 ,r e s ,rt) double complex a l , a 2 ,dm,de,j,k l ,k 2 ,e l 2 ,tn,res,r t ,d double precision 1,rx,ry,j1,Bp external j1 j = d c m p l x (0.d O ,1.d O ) al = -j*cdsqrt(l**2-kl**2) a2 = -j*cdsqrt(l**2-k2**2) Bp = jl(rx*l)*jl(ry*l) tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn de = el2*a2+j*al*tn fnse = ((l*Bp*dm)/ (de*al)-res/(1-rt)) return end - (j*Bp) ! THE PART TO BE ADDED SINGULARITY EXTRACTED FUNCTION double complex function is(rt,ad,bd,res) double complex j,pl,p2,p 3 ,r t ,res double precision tl,t2,ad,bd R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. j = d c m p l x (0.d O ,1.dO) tl = (dimag(rt))**2 + (bd-dreal(rt))**2 t2 = (dimag(rt))**2 + (ad+dreal(rt))**2 pi = (res * d l o g (tl/t2))/ 2 .do p2 = j * res * d a t a n ((bd-dreal(rt))/dimag(rt)) p3 = j * res * d a t a n ((dreal(rt)-ad)/dimag(rt)) is = pi + p2 + p3 return end ! USE THE FOLLOWING FUNCTION TO FIND RESIDUE double complex function r d ( r t ,rx,ry,d,kl,k2,el2) double complex a l ,a 2 ,d m , j ,kl,k2,e l 2 ,g l ,g 2 ,g 3 ,g d , r t ,j l c ,tn,d,Bp double precision rx,ry external jlc j = d c m p l x (0.d O ,1.dO) al = -j * cdsqrt(rt**2 - kl**2) a2 = -j * cdsqrt(rt**2 - k2**2) Bp = jlc(rx*rt)*jlc(ry*rt) tn = cdsin(al*d)/cdcos(al*d) dm = al+j*el2*a2*tn gl = - (al*el2*rt)/a2-(a2*el2*rt)/al g2 = -j*al*d*rt*(1/cdcos(al*d))**2 g3 = -j *2.d0*rt*tn gd = gl+g2+g3 rd = (rt*Bp*dm)/gd return end ! MOM - INTEGRATION FUNCTIONS - END C.2.9 M O M -Z R T .F O R ! Z-R00T MODULE -- BEGIN subroutine rootmain(f,d , k l ,k 2 ,rkl,rk2,e l ,e l 2 ,f r ,min,lmin,m,rtsi double complex el,el2,k l , k 2 ,r,rp,rt,rts,fn,rmin,lmin,1,g e s ,rtfn, double precision f ,r k l ,r k 2 ,f r ,x , z ,n l ,n 2 ,dl integer s ,m , s o r t ,min,yes,c n l ,cn2 dimension r t s (20) external rtfn if if (f.I t .10.d9) (rkl.gt.rk2) cnl = 0 !DO NOT SEARCH AGAIN IN THIS CASE cn2 = 1 !DO NOT SEARCH FOR MINIMA IN THIS CASE do s = 1,2 0 !SET ALL rts VARIABLES TO 0 rts(s) = d c m p l x (0.d O ,0.dO) end do m = yes min x = 0 !INITIALIZE = 0 = 0 dmaxl(dminl(rkl,rk2)-3.dO,2 .dO) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. z = dmaxl(rkl,r k 2 )+3.do if (dreal(el).g t .1000.dO) z = rkl+lOO.dO 1 = dcmplx(x,0.do) rp = r t f n (1,d,kl,k2,el2) rmin = rp lmin = 1 dl = (z-x)/(1.9**dlogl0(f)) ! SEARCH FOR A GOOD INITIAL GUESS, ALSO SEARCH FOR ! MINIMUM VALUE OF RTFN IN THE INTERVAL {x,z} ! SEARCH - I do w hile(dreal(1).I t .z) 1 = 1 + dcmplx(dl,0.do) r = r t f n (1,d , k l ,k 2 ,e l 2 ) if (cdabs(r).I t .cdabs(rmin)) then rmin = r lmin = 1 endif nl = d s i g n (1.OdO,dreal(rp)) 'FIRST SIGNAL FOR POLE CLOSE BY n2 = d s i g n (1.O d O ,dreal(r)) if (nl.ne.n2) yes = 1 nl = d s i g n (1.OdO,dimag(rp)) 'SECOND SIGNAL FOR POLE CLOSE : n2 = d s i g n (1.O d O ,dimag(r)) if (nl.ne.n2) yes = 1 if (yes.eg.1) then 'VERIFY THE GUESSES FOR A POLE ■ ges = dcmplx(l-dl,O.dO) call rtsolver(d,kl,k 2 ,e l 2 ,g e s ,r t ,fn) do s = 1,m if (dabs(rts(s)-rt).I t .le-3) fn = d c m p l x (10.d O ,0.do) endif end do then if (cdabs(fn).I t .1.d-9) then if (dabs(dimag(rt)).I t .fr) then m = m + 1 rts (m) = rt endif endif yes = 0 endif rp = r end do !SEARCH - II if (cnl.eq.l) then yes = 1 1 = dcmplx(x,0.do) dl = (z-x)/20.dO rp = r t f n (1,d,kl,k2,el2) do w hile(dreal(1).I t .z) ges = d c m p l x (1,0.dO) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. call rtsolver(d,kl,k 2 ,e l 2 ,g e s ,r t ,fn) if (cdabs(fn).I t .1.d- 9) then do s = l,m if (dabs(rts(s)-rt).I t .le-3) then yes = 0 endif end do if (yes.eq.l) then if (dabs(dimag(rt)).I t .fr) then m = m + 1 rts(m) = rt endif endif endif 1 = 1 + dcmplx(dl,0.do) yes = 1 end do endif if (cn2.eq.1) then !FINE TUNE THE MINIMA FOUND BEFORE if (m.eq.O) then 1 = lmin - dcmplx(dl,0.do) dl = 2.dO * dl/500 rmin = r t f n (1,d , k l ,k 2 ,e l2) do s = 1,500 1 = 1 + dcmplx(dl,0.dO) r = r t f n (1,d,kl,k2,e l 2 ) if (cdabs(r).I t .cdabs(rmin)) then rmin = r lmin = 1 endif end do min = 1 endif endif do sort = 1,m !SORT ALL POLES IN INCREASING ORDER do s = 1 , (m-1) if (dreal(rts(s)) .gt.dreal(rts(s+l))) then r = rts(s) rts(s) = rts(s+l) rts(s+l) = r endif end do end do return end ! USE MUELLER'S METHOD TO SOLVE FOR POLES subroutine rtsolver(d,kl,k 2 ,e l 2 ,g e s ,r t ,fn) double complex rtfn,ges,kl,k2,e l 2 ,x O ,xl,x2,h i ,h 2 ,f 0,f1,f2,f d l , +fd2,f t ,c,sqr,den,rt,fn,d R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. double precision epl,ep2 integer i external rtfn 10 epl = l.d- 18 ep2 = l.d- 20 xO = ges + 0 .5 xl = ges - 0 .5 x2 = ges fO = rtfn (x O ,d, k l ,k2 ,el2 fl = rtfn(xl, d , k l ,k2 ,el2 f 2 = rtfn (x 2 ,d, k l ,k2 ,el2 hi = xl-xO h2 = x2-xl fdl = (f 2 -fl) /h2 fd2 = (fl- fO) /hi do i = 2,100 if (hi.eq .h2 ) exit ft = (fd2 -fdl)/ (h2- hi) c = fd2 + h2 *f t sqr = cdsqrt (c*c - 4*f 2 if (dreal(c)*dreal(sqr)+dimag(c)*dimag(sqr).I t .0.dO) den = c - sqr else den = c + sqr end if if (cdabs(den).l e .0.do) den = l.dO hi = h2 h2 = -2 * f2/den xO = xl xl = x2 x2 = x2 + h2 fO = fl fl = f 2 f2 = rtfn(x2,d , k l ,k 2 ,e l 2 ) fdl = fd2 fd2 = (f2-f1) /h2 if (cdabs(x2).gt.10000.dO) exit if (cdabs(h2).I t .epl*cdabs(x2)) exit if (cdabs(f2).I t .ep2) exit if (cdabs(f2).g e .10.d0 * c d a b s (f1)) then h2 = h2/2 x2 = x2 - h2 goto 10 endif enddo rt = x2 fn = rtfn(x2,d , k l ,k 2 ,e l 2 ) return end then ! USE THE FUNCTION FOR POLE double complex function rtfn(r,d , k l ,k 2 ,e l2) double complex a l ,a 2 ,j ,k l ,k 2 ,e l 2 ,t n , r ,zeta,d R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 286 j = d c m p l x (0.d o ,1.do) al = -j * cdsqrt(r**2-kl**2) a2 = -j * cdsqrt(r**2-k2**2) tn = cdsin(al*d)/cdcos(al*d) zeta = (al+j*el2*a2*tn)/ (el2*a2+j*al*tn) rtfn = al/(zeta*r-j*al) return end ! Z-ROOT MODULE -- END C.2.10 M O M -K C R T .F O R ! KC-ROOT MODULE -- BEGIN subroutine momkcrt(a,b,n,gaman) double precision a,b,gn,fgn,fx,xz,f x l ,d e l ,r O ,r l ,x r ,gaman,guess integer i,n dimens ion g a m a n (200) external gn,fgn 10 del = 100.dO rO = del do i = 1, n fx = fgn(r0,a,b) rl = rO + del fxl = fgn(rl,a,b) xz = dabs(fx+fxl) - dabs(fx-fxl) if (xz.gt.O.dO) rO = rl !IF NO CHANGE IN SIGN THEN GO BACK TO 10 if (xz.gt.O.dO) go to 10 guess = (r0+rl)/2.d0 xr = gn(guess,a,b) g a m a n (i ) = xr rO = xr + del enddo return end ! SIMPSON'S METHOD OF FINDING ROOT FOR GAMMA-n double precision function gn(x0,a,b) double precision x O ,x l ,a,b,fgn,fdgn,f,fd integer i external fgn,fdgn xl = xO do i = l ,100 f = fgn(xl,a,b) if (dabs(f).l e .1.d-12) fd = fdgn(xl,a,b) gn = xl - (f/fd) xl = gn enddo exit R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 287 gn = xl return end ! THE FUNCTION FOR FINDING ROOT TO GAMMA-n double precision function fgn(gn,a,b) double precision gn,a,b,jO,yO external jO,yO fgn = j 0 (gn*b) *y0 (gn*a) -j 0 (gn*a) *y0 (gn*b) return end ! THE FIRST DERIVATIVE OF FUNCTION FOR FINDING ROOT TO GAMMA-n double precision function fdgn(gn,a,b) double precision a,b,gn,gnl,gn2,jO,yO,jl,yl external jO,yO,jl,yl gnl = a*j1 (gn*a)*y0(gn*b) gn2 = b * j 1 (gn*b)*y0(gn*a) fdgn = gnl - gn2 return end ! KC-ROOT MODULE -- END + b* j 0 (gn*a)*yl(gn*b) + a*j0 (gn*b)*yl(gn*a) C.2.11 M O M -H A R T .F O R ! HEART MODULE -- BEGIN if (dbg.eq.O) then write (*,1100) rx,ry,dreal(d),1/1000,f format (3P, ' rx = ',F 8 .3, ' ry = \F8.3,' d = ',F8 .3, /, 1100 L = 1,F 1 4 .3,I P , 1 f = 1,E l l .4 E 2 ) •«* write (*,1200) dreal(el),dimag(el),dreal(e2),dimag(e2) format (' el = ',F l l .5,',',F l l .5,'i ',/, 1200 ' e2 = ',F l l .5,',1,F l l .5,1i ') write (*,1300) r k l ,dimag(kl),r k 2 ,dimag(k2) format (' kl = ',F l l .5,',1,F l l .5,1i ',/, 1300 + k2 = ',F l l .5,',',F l l .5,'i ') pause 1intialization is done, waiting for prompt' en d if ! FIND ALL ROOTS ALONG THE REAL AXIS if (dabs(dimag(el)).I t .2.d o ) then fr = 0.1 call rootmain(f,d,kl,k2,r k l ,r k 2 ,e l ,e l 2 ,fr,min,lmin,nrts, rts) if (nrts.eq.0) then 11 = O.dO if (int.eq.l) then call d i r e c t s (11,1,rx,ry,d,kl,k2,el2,sum,ay) else call d i r e c t g (11,1,rx,ry,d,kl,k2,el2,sum,ay) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 1350 1375 1400 endif z = sum if (dbg.eq.O) then write (*,*) no roots found1 write (*,1350) dreal(sum),dimag(sum) format (' STAGE 21 : z = ',F 2 5 .15,1, 1,F 2 5 .15,'i ') write (*,1375) 11,1 format (' 11,1 = ',F 2 5 .15,',',F 2 5 .15) pause ' stage 1,2,3 are done, waiting for prompt' endif else if (dbg.eq.O) then if (nrts.gt.0) then write (*,*) ' the following roots were found' do u = l,nrts write (*,1400) u,dreal(rts(u)),dimag(rts(u)) fo r m a t C ',17,'] ',F14.9,', ',F14.9,'i') enddo else write (*,*)' no roots found close to real axis, ' write (*,*) ' direct integration is done' endif pause ■' rootmain is done, waiting for prompt' endif ! DO INTEGRATION OF MAIN FUNCTION FROM 0 TO 11 z = d c m p l x (0.d O ,0.d o ) 11 = O.dO 12 = d m i n l (r k l ,rk2) if (nrts.ne.0) then if (dreal(rts(1)).I t .12) 12 = dreal(rts(1))-1.do endif if (int.eq.l) then call d i r e c t s (11,12,rx,ry,d,kl,k2,e l 2 ,sum,ay) else call d i r e c t g (11,12,rx,ry,d,kl,k 2 ,e l 2 ,sum,ay) endif z = sum if (dbg.eq.O) then write (*,1600) dreal(sum),dimag(sum) format (' STAGE 1 : z = ',F 2 5 .15,',',F2 5 .15,'i ') 1600 write (*,1700) 11,12 format (' 11,12 = ',F 2 5 .15,',',F 2 5 .15) 1700 pause 'stage 1 is done, waiting for prompt' endif ! IF ROOTS EXIST, THEN DO INTEGRATION OF FUNCTION IN THE REGION OF ROOTS if (nrts.ne.0) then do u = l,nrts 11 = 12 12 = d m a x l (12,dreal(rts(u)) - 0.3d0) if (int.eq.l) then call d i rects(11,12,rx,ry,d,kl,k 2 ,e l 2 ,sum,ay) else R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 289 1800 1900 2000 2100 2200 2300 ! IF call d i r e c t g (11,12,rx,ry,d,kl,k2,e l 2 ,sum,ay) endif z = z + sum if (dbg.eq.O) then write (*,1800) dreal(sum),dimag(sum) format (' STAGE 21 : z = ',F 2 5 .15,1,',F 2 5 .15,'i ') write (*,1900) 11,12 format (' 11,12 = 1,F 2 5 .15,',’,F 2 5 .15) pause ' stage 21 is done, waiting for prompt' endif 11 = 12 12 = dreal(rts(u)) + 0.3d0 if (int.eq.l) then call s e s (11,12,rx,ry,d,kl,k2,e l 2 ,rts(u),sum,ay) else call s e g (11,12,r x , r y ,d,kl,k2,e l 2 ,rts(u),sum,ay) endif z = z + sum if (dbg.eq.O) then write (*,2000) dreal(sum),dimag(sum) format (' STAGE 22 : z = 1,F 2 5 .15,',',F 2 5 .15,'i 1) write (*,2100) 11,12 format (' 11,12 = ',F 2 5 .15,',',F 2 5 .15) pause ' stage 22 [SE] is done, waiting for prompt' endif enddo if (rk2.1t.l) then 11 = 12 12 = d m a x l (r k l ,r k 2 ) if (int.eq.l) then call d i r e c t s (11,12,rx,ry,d,kl,k2,e l 2 ,sum,ay) else call d i r e c t g (11,12,rx,ry,d,kl,k 2 ,e l 2 ,sum,ay) endif z = z + sum if (dbg.eq.O) then write (*,2200) dreal(sum),dimag(sum) format (' STAGE 23 : z = ',F 2 5 .15,',',F 2 5 .15,'i ') write (*,2300) 11,12 format (' 11,12 = ',F 2 5 .15,',',F 2 5 .15) pause 1 stage 23 are done, waiting for p r o m p t 1 endif endif NO ROOTS, THEN DO THE CORRESPONDING INTEGRATION else if (min.eq.l) then 11 = 12 12 = lmin if (int.eq.l) then call d i r e c t s (11,12,rx,ry,d,kl,k2,el2,sum,ay) else call d i r e c t g (11,12,rx,ry,d,kl,k 2 ,e l 2 ,sum,ay) endif z = z + sum R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. if 2400 2500 2600 2700 (dbg.eq.O) then write (*,2400) dreal(sura),dimag(sum) format (' STAGE 21 : z = ',F 2 5 .15, ', 1 ,F 2 5 .15, 'i 1) write (*,2500) 11,12 format (' 11,12 = ',F 2 5 .15,',',F 2 5 .15) pause 1 stage 21 is done, waiting for prompt' endif endif if (rk2.1t.l) then 11 = 12 12 = d m a x l (r k l ,rk2) if (int.eq.l) then call d i r e c t s (11,12,r x , r y ,d,kl,k2,e l 2 ,sum,ay) else call d i r e c t g (11,12,r x , r y ,d , k l ,k 2 ,e l 2 ,sum,ay) endif z = z + sum if (dbg.eq.O) then write (*,2600) dreal(sum),dimag(sum) format (' STAGE 22 : z = 1,F 2 5 .15, 1, ',F 2 5 .15, 'i ') write (*,2700) 11,12 format (' 11,12 = ',F 2 5 .15,',',F 2 5 .15) pause ' stage 22 is done, waiting for prompt' endif endif endif if 2800 2900 3000 3100 (int.eq.l) then call d i r e c t s (11,1,rx,ry,d,kl,k 2 ,e l 2 ,sum,ay) else call directg (11,1, rx, ry, d.,k l ,k 2 ,el2, sum, ay) endif z = z + sum if (dbg.eq.O) then write (*,2800) dreal(sum),dimag(sum) format (' STAGE 3 : z = ',F 2 5 .15,1,',F 2 5 .15,'i ') write (*,2900) 12,1 format (' 12,1 = ',F 2 5 .15,',',F 2 5 .15) pause ' stage 3 are done, waiting for prompt' endif endif else 11 = O.dO if (int.eq.l) then call d i r e c t s (11,1,r x , r y ,d, k l ,k 2 ,e l 2 ,sum,ay) else call d i r e c t g (11,1,r x , r y ,d , k l ,k 2 ,e l 2 ,sum,ay) endif z = sum if (dbg.eq.O) then write (*,3000) dreal(sum),dimag(sum) format (' STAGE 21 : z = ',F 2 5 .15,',',F 2 5 .15,'i ') write (*,3100) 11,1 format (' 11,1 = ',F 2 5 .15,',',F 2 5 .15) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. pause endif endif ' stage 1,2,3 are done, waiting for prompt' ! ADD THE PART CORRESPONDING TO THE INT FROM L TO INF c = momc(rx,ry) z = z + j * c z = -j * pi * z if (dbg.eq.O) then write (*,3200) dreal(j*c),dimag(j*c) 3200 format (' STAGE 4 : z = ',F 2 5 .15,1,',F 2 5 .15,'i ') pause 1 stage 4 is done, waiting for prompt' endif ! HEART MODULE -- END R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 292 A p p e n d i x D BASICS O F ELECTRO M A G NETIC TH EO R Y A N D TRA N SM ISSIO N LINE TH EO RY D .l IN T R O D U C T IO N The topics covered in this chapter form the background for the research, they are as follows: ® Basics of electromagnetic theory - section D.2 • Basics of transmission line theory - section D.3 Basics are explained as briefly and logically as possible, the symbols that are obvious for electrical engineering are explained briefly if necessary or not explained at all, if found not necessary. A list of symbols used in this research can be found in Appendix A. D.2 BASICS O F ELECTROMAGNETIC THEORY1 Electromagnetic fields are made up of two fundamental fields called electric and magnetic fields. When these fields are static they do not interact with each other, therefore EM waves do not exist. However, when these fields are time varying, electric and magnetic fields interact with each other in a complicated manner. This physical phenomenon causes electromagnetic 1 The references for the subject o f section D .2 and D .3 can be found in [28, 29, 62, 63, 64, 69, 83]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. fields to exist. The subject that deals with these interactions between electric and magnetic fields in terms of space, time and the medium is called electromagnetic theory or EM theory. The study can be done in differential and integral forms in time domain and in time-harmonic form in frequency domain. The electromagnetic theory required for the research of coaxial line sensors are covered in this section. They are as follows: • Maxwell’s Equations: Mathematically, the interaction between electric and magnetic fields in terms of space and time are known as Maxwell’s equations. Maxwell’s equations are the product of experiments from Ampere, Faraday and other prominent scientists, and they often form a starting point for most of the analysis involving EM waves. This topic is found in section D.2.1 through section D.2.7 ® Constitutive Relations: Constitutive relations explain the relationship between EM waves and the medium in which they interact. This topic is found in section D.2.8 and section D.2.9 • Boundary Conditions: When there are more than one medium involved, the boundary conditions explain the relationship between the EM fields in different media, at the boundary. Boundary conditions are derived from Maxwell’s equations. This topic is found in section D.2.10 • Scalar and Vector Potentials: Scalar and vector simplify the mathematical analysis, though they may not represent anything R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. that are physically definable. This topic is found in section D.2.11 through section D.2.16 • Energy Relations: Energy relations of EM waves explain the relationship between EM energy that enter into a volume, EM energy that leaves that volume and the EM energy that gets converted into other forms of energy within that volume. These relations are derived from Maxwell’s equations. This topic is found in section D.2.22 and section D.2.23 • Uniqueness Theorem and Reciprocity Theorem: These are useful conceptual tools in EM theory and provide simpler ways of understanding and solving EM problems. This topic, is found in section D.2.24 and section D.2.25 • Wave Equations: These equations explain the propagation of EM waves in lossy or lossless homogenous space. The solutions of these equations yield electric and magnetic fields in open space, dielectric or lossy medium. This topic is found in section D.2.26 ® Waveguides: Waveguides are metal tubes that carry EM waves, in practice they take cross sectional shapes of either a rectangle or a circle. The solutions for fields inside the waveguide are different from that of free space. This topic is found in section D.2.27. There are some assumptions made in this section regarding the media in which fields interact. If these assumptions are not true then the relations developed from Maxwell’s equations should be modified accordingly. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nature of the types of media, in which EM theory developed, in this section are as follows: • Homogenous (i.e., uniform in consistency of medium EM characteristic) • Either a pure dielectric or a lossy material (i.e.,s’ can be a complex). D.2.1 Generalized Maxwell’s Equations in Differential Form: Generalized Maxwell’s equations in differential form relate electric and magnetic fields, in microscopic space (i.e., at every point in space), in terms of time and 3-dimensional space as follows: dt (D. 1) VxH = 7 D + J (D.2) V •D = p (D.3) dt (D.4) where, the first two are most general and independent equations, other two can be derived from first two. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D .2.2 Symmetry of Generalized Maxwell’s Equations and Duality Principle: Note that in Maxwell’s equations - (D.l) through (D.4), magnetic charge and magnetic current J,„ p„, are purely conceptual, mathematical and theoretical. They do not exist in physical situations. These quantities have been added in Maxwell’s equations in order to develop a mathematical symmetry. This principle is called duality principle. When, analyzing or interpreting a physical situation, these quantities may be assumed to be zero. Because of symmetry, the following four set of transformations do not affect the Maxwell’s equations: E —>H , J,, —» Jm, pc — » p,„ and p s (D. 5) H - » - E , Jra—> - J e, p„,-> - p e a n d s -> p (D. 6) E (D. 7) J H, J e -> y—J„, and pe -> p„, H- (D- 8> O f course, in one transformation, all the relevant quantities (i.e., one complete set) must be transformed together. Among the above four transformations, the transformations of (D.7) and (D.8) do not require interchanging p. and s , and therefore is useful in dealing with quantities in the same medium. The duality principle, for example, helps to eliminate the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. derivation of equation for magnetic field, once the equation corresponding to electric field is found. D.2.3 Stoke’s Theorem & Divergence Theorem: Stake’s theorem relates a surface integral to a closed line integral and divergence theorem (or, Gauss’s law) relates a volume integral to a surface integral. These are important mathematical tools, often used to convert from integral form to differential form: j j v x A •ds = <|a ■dl (D . 9) JJJ(V •A )dv = cjijA •ds v (D . 10) .v where, A is any arbitrary vector. D.2.4 Generalized Maxwell’s Equations in Integral Form: Generalized Maxwell’s equations in integral form relate electric and magnetic fields, in large volume of space, in terms of 3 dimensional space and time. They are as follows: c^E •dl = - — B + J „, j •ds (D. 11) ^ H -dl= J J ^ D + j J - d s (D. 12) <jijl)-ds = jjjp t.dv (D. 13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 298 £[B -ds= \ \\ ? mdv S (D. 14) v These equations are derived from Generalized Maxwell’s equations in differential form, using Stoke’s and Divergence theorem. Word statements of Maxwell’s equations are readily obtained in integral form from equations (D .ll) through (D.14). Neglecting p„, and J m, which are purely conceptual, they can be stated as follows: • The electromotive force (electric voltage) around a closed path c is equal to the time derivative of the magnetic displacement (magnetic current) through any surface s bounded by path c • The magnetomotive force (magnetic voltage) around a closed path c is equal to the conduction current plus the time derivative of the electric displacement (electric current) through any surface s bounded by path c • The total electric displacement through the surface s enclosing a volume v is equal to the total electric charge within the volume v • The total magnetic flu x emerging through the surface s enclosing a volume v is equal to the total magnetic charge within the volume v. D.2.5 Fourier Transform Equations: These are mathematical tools used to convert time domain equations to frequency domain equations and vice versa, they are as follows: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 299 E(r) - -J- jE(co)e;“Vco (D. 15) E(co) = (D. 16) 2n -CJC l ’E(/)e-7“'^ 2n J -00 D.2.6 Generalized Maxwell’s Equations in Time-Harmonic Case: In time-harmonic case, each of the field quantity is assumed to be a wave of sinusoidal variations with time at single frequency. A signal or wave of single frequency sine wave is also known as monotonic or monochromatic or a single tone signal. In general terms, a monotonic vector field can be written as A (iv ) = ^ A ph(r)ej(a,+d) = A p/l(r)cos( a t +0) (D. 17) where, A is a monotonic vector field as a function of time and 3 dimensional space, and A ph is peak value of a complex phasor field at a single frequency© , as a function of space. Therefore, Maxwell’s equations in time-harmonic case can be written as follows: V x E = -ycoB - Jm (D. 18) V x H = jcoD + Je (D. 19) v - v = p e (D. 20) V-B = p„, (D .21) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 300 Here, it should be noted that, when time-harmonic case is considered all field quantities refer to peak values o f complex phasor field at a single frequency co, and should not be confused with field quantities of time domain. D.2.7 Equation of Continuity (Conservation of Charge): Equation of continuity relate electric current and electric volume charge, magnetic current and magnetic volume charge and is derived from Maxwell’s equations. Equation of continuity in differential form, integral form and for time harmonic case are given below: V-J, = - f Pe (D. 22) JIM 7 (D - 23) V-Je = -/coPe (D. 24) = (D- 25) P„ • * = - ! : JIfp-> s (D - 26) v V - J m=-7'®Pm (D - 27) Here, it should be noted that magnetic sources are purely theoretical, conceptual and do not exist physically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D.2.8 Constitutive Relations: Constitutive relations describe the relation between various fields to the medium in which they exist. These are given below: D = sE (D. 28) B = |j.H (D. 29) Je = cjE (D. 30) s , p. and a are the characteristic of the medium in terms of EM fields, they can be as simple as a scalar quantity to as complex as a dyadic quantity depending on the nature of the medium. However, in this work s , |i and ct are considered to be scalar quantities. D.2.9 Complex Permittivity (Complex Dielectric Constant): The Maxwell’s equation that relate the field quantities E and H , for a monotonic wave (in time-harmonic form), can be re-written as /■ \ V x H = (/cos0e r +ct)E = /ct)s0 s r - j —— E The term * r ~ J ' (D. 31) in equation (D.31) is known as complex perm ittivity or complex dielectric constant and depends on frequency of the EM fields, and the medium in which EM fields exist. Therefore, the following definitions come into picture, as related to the complex permittivity of the material, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 302 e ,-y - cos0; (D. 32) = e0( s '- /s " ) (D. 33) s' = e r5e" = COSr tan(8 ) = ^—= (D. 34) where, tan(8) is loss tangent. If tan(5) « 1 , then the medium is a good dielectric and if tan(5) » 1 , then the medium is good conductor. Using equation (D.32), equation (D.31) can be written as V x H = ycoscE 4 (D. 35) It should be noted here that the purpose o f this research is to measure the complex dielectric constant o f the medium that terminates a coaxial line sensor. D.2.1Q Boundary Conditions: At the interface of two media with different s or \x , field quantities follow certain conditions. These conditions are derived from Maxwell’s equations, they can be stated as follows: • The discontinuity in tangential component o f E across the boundary is given by magnetic current Jm/ (V /m ) at that boundary. If there exits no magnetic current at the boundary, then the tangential component o f E is continuous across the boundary • The discontinuity in tangential component o f H across the boundary is given by electric current Jw (A /m ) at that boundary. If Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 303 there exits no electric current at the boundary, then the tangential component o f H is continuous across the boundary • The discontinuity in normal component o f D across the boundary is given by electric surface charge density p„ (coulombs/m2) at that boundary. If there exists no electric surface charge at the boundary, then the normal component o f D is continuous across the boundary • The discontinuity in normal component o f B across the boundary is given by magnetic surface charge density pms (webers/m2) at that boundary. If there exists no magnetic surface charge at the boundary, then the normal component o f B is continuous across the boundary. Mathematically, these boundary conditions may be specified as follows: (E2 - E j) x n = J m/ (D. 36) (H2 - H , ) x n = - J (D. 37) (D2 - D , ) - n = p „ (D. 38) (B2- B , ) . n = pu p . 39) In these boundary conditions, subscripts “ , ” and “ 2” denote fields in medium-1 and medium-2, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D.2.11 Lorentz Gauge: There are certain situations in mathematics of EM theory in deriving various potentials, in which the choice of divergence of these potentials are undefined. In these situations, as will be seen later, using Lorentz or Coulomb’s gauge will help simplify equations involving potentials. Lorentz gauge or conditions are as follows. In differential form V -A = -|ae J~<t>e (D. 40) V . f = - hsJ hL (D. 41) In time-harmonic case i V •A = — _ /c o e (D. 42) V •F = -y'co(i£(|)m (D. 43) Also in case of hertzian potentials, v -n , (D. 44) (D -45) A , F , <|>e, <(»m, and in equation (D.40) through (D.45) are magnetic vector potential, electric vector potential, electric scalar potential, magnetic scalar potential, electric type hertzian potential and magnetic type hertzian Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. potential respectively. These symbols are described in detail in sections D.2.14, D.2.15, D.2.19 and D.2.20. D.2.12 Coulomb Gauge: This is another condition used in order to derive mathematically simpler equations involving various potentials. These are as follows: V- A = 0 (D. 46) V-F = 0 (D- 47) D.2.13 Helmholtz’s Theorem: Helmholtz’s theorem states that a vector may be completely specified by its curl and divergence. D.2.14 Magnetic Vector Potential, Electric Scalar Potential: Assume, in Maxwell’s equations (D.l) through (D.4), that there are no magnetic sources. Then, one of the Maxwell’s equations states that divergence of magnetic displacement is equal to zero. Therefore, mathematically, it implies that B=VxA (D. 48) where, A is any arbitrary vector called magnetic vector potential. Equation (D.48) in equation (D.l) yields (assuming no magnetic sources) (D. 49) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 306 This again implies that E + J^A = -V<|>e (D. 50) where, <)>e is any arbitrary scalar called electric scalar potential. Using Lorentz gauge of equation (D.40) it can be shown (by taking curl on equation (D.48)) that, A and <)>, are, respectively, the solution of the following equations: V2A - Ms |^ A = - MJ e (D. 51) (D. 52) For time-harmonic case, the above two equations become V 2A + k2A = —jj.Je V 2<j>e + k 2fye = s (D. 53) (D. 54) Once, the solutions for magnetic vector potential and electric scalar potential are found from equations (D.51), (D.52) or equations (D.53), (D.54), the fields can be found as follows, in differential form E=- ot H = —V x A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D. 55) (D. 56) In time-harmonic form, taking gradient of Lorentz gauge of equation (D.42), gradient of electric scalar potential can be written as v<j)c = — — V(V-A) ycofis (D. 57) Therefore from equations (D.55) and (D.56), in time-harmonic case, the fields in terms of these potentials, can be written as E = - ycoA + ^ —V(V-A) (D. 58) H = —V x A (D. 59) COJTS D.2.15 Electric Vector Potential, Magnetic Scalar Potential: Again, as in previous section, assume, in Maxwell’s equations (D.l) through (D.4), that there are no electric sources. Then, one of the Maxwell’s equations states that divergence of electric displacement is equal to zero. Therefore D = -V x F (D. 60) where, F is any arbitrary vector called electric vector potential. Equation (D.60) in (D.2) yields (assuming no electric sources) V x (h + | f )=0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D. 61) 308 This again implies that H + f F = - v *,„ (D. 62) where, <)>„, is any arbitrary scalar called magnetic scalar potential. Using Lorentz gauge of equation (D.41) it can be shown (by taking curl on equation (D.60)) that, F and <j)„, are, respectively, the solution of the following equations: V!F - mE|J -F = - sJ . Ot = p . 63) \l “ ( ° - 64) For time-harmonic case, the above two equations become V2F + k 2F - - s J m (D. 65) V2<f>+ £2<j)m= (D. 66) Once, the solutions for electric vector potential and magnetic scalar potential are found from equations (D.63), (D.64) or (D.65), (D.66), the fields can be found as follows, in differential form E = - —V x F s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D. 67) In time-harmonic form, taking gradient of Lorentz gauge of equation (D.43), gradient of magnetic scalar potential can be written as V f , = — — V(V-F) ycojae (D. 69) Therefore from equation (D.67) and p . 68), in time-harmonic case, the fields in terms of these potentials, can be written as E = - —V x F (D. 70) H = - ycoF + ^ - V ( V - F ) cojas (D. 71) s D.2.16 EM Fields in terms of Vector Potentials: Now, using Maxwell’s time-harmonic equations p . 18) through p.21), and the fact that the system consisting of fields is linear, the fields can be rewritten in terms of vector potentials. In this case, the sources are both electric and magnetic current densities. Using equations p.58), p . 59) and p .70), p .7 1 ), time-harmonic fields can be written as E = - V x F - y'coA + — V(V •A) (D- 72) H = —V x A - 7<oF + ^ - V ( V - F ) (D. 73) g fi ©fig ©|ag Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 310 Again, the magnetic vector potential A is the solution of equation (D.53) and the electric vector potential F is the solution of equation (D.65). Instead of Lorentz gauge, coulomb gauge can be used if the region is source free. D.2.17 Energy Relations between EM Fields (Differential-Integral Forms): Consider the following mathematical relation between magnetic field and electric field (D. 74) V- ( ExH) = H - V x E - E ' V x H Now, using equation (D.74) and Maxwell’s equations (D.l) and (D.2), the following equation can be written: . or, V' ( E x H) = -I sE- — E + (J.H-—H I - E •J e dt dt (D. 75) where, all fields are function of time t and space r . Considering the physical situation, magnetic current in equation (D.l) is omitted here. Also, note the following: E - — E = ——|E|2 8t 2 Btl ' (D. 76) v ' H — H = ——|H|2 dt 2 dr 1 (D. 77) v ' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Next, consider the following energy relations: + iv, p . 80) where W , We and Wm are the total energy density, electric energy density and magnetic energy density, respectively. Also, consider the definition S= ExH (D. 81) where, S is poynting vector and represents the power flu x density (W /m2). Now, in terms of equations (D.76) through (D.79), equation (D.75) can be written as = (D. 82) Integrating equation (D.82) within a volume of space v, and using equations (D.9) and (D.10) - § s ‘, s = | f J f ,r d v + JJKE ' J > <D - 83) Now, left hand side of equation (D.83) means the total power flowing into the volume v through the surface s. Right side of equation (D.83) means Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. increase in the EM energy inside the volume v (first term in the right side) plus total energy dissipated in volume v (the second term in the right side). Therefore equation (D.83) means that the total power flow ing into the volume per u n it tim e is equal to the sum o f the increase in total EM energy and the energy dissipation per u n it tim e in the volume. D.2.18 Energy Relations between EM Fields (Time Harmonic Form) and Foynting Theorem: The relations of energy in time harmonic form, is called Poynting Theorem. It is possible to derive the energy relations directly from that of equation (D.82). However, in order to avoid confusion, it is derived again here in terms of time harmonic Maxwell’s equations. Consider the mathematical relation of equation (D.74). Now, using equation (D.74) and Maxwell’s equations in time harmonic form equations (D.18) and (D.19), the following equation can be written: V •(E * H) = H •( - ycoB) - E •(ycoD + J e) o r, V •(E x H) = y'cosH•IT - ycosE •E* + c E ■E* ID. 84) where, all fields are complex phasors and are functions of frequency c» and space r. Again, considering the physical situation, magnetic current in equation (D.18) is omitted here. Next, consider the following energy relations: (D. 85) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 313 _ H'pH'r |H|2 (D. 86) a 'E|2 2o>s0 1 = 3 S = -j-ExH’ where We , IWm (D. 87) (D. 88) are the time averaged electric and magnetic stored energy densities. L is real and positive and represents the power dissipation per unit volume in a lossy medium. S is complex poynting vector and represents the complex power flu x density (W /m2). Now, in terms of equations (D.85) through (D.88), equation (D.84) can be written as V •S = -2 ja>(Wm - We) - L - | e •E* (D. 89) D.2.19 Uniqueness Theorem: It is known in circuit theory that for a passive circuit with N terminals, if N voltage sources are applied at these N terminals, all the voltages and currents inside the network can be uniquely determined. Similarly, for a passive circuit with N terminals, if N current sources are applied at these N terminals, all the voltages and currents inside the network can be uniquely determined. Uniqueness theorem answers a parallel question in terms of EM theory. The question is - considering a volum e v w ith a surface area s, what field quantities should be specified on the surface s in order to uniquely determine all the fields Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. inside (the field quantities on the surface s could be tangential or normal, electric or magnetic fields, electric or magnetic flu x densities)} Uniqueness Theorem states that one of the following three conditions is necessary and sufficient to uniquely determine all the fields inside • The tangential electric field (n x E) is specified on s • The tangential magnetic field (n x H) is specified on s • The tangential electric field (nx E) is specified on a part of s and the tangential magnetic field (n x H) is specified on rest of the s. D.2.20 Lorentz Reciprocity Theorem: It is known in terms of a linear passive electric circuit th a t’if an input voltage VA causes a short circuit current I A and if another voltage VB causes a short circuit current IB, then 1± = L l VA vB (D. 90) There is an analogous situation in EM theory. This is called Lorentz Reciprocity theorem and states as follows: I f a set o f source ( J tvf, JmA) causes the fields ( E a , H a) and another set o f source ( J eB, J mB) causes the fields ( E b ,H b), then P . 91) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D.2.21 Time H arm onic Wave Equations and Plane Wave Propagation: Consider the Maxwell’sequations (D.18) through (D.21)in time harmonic form. Assume that there exists no magnetic sources. The following mathematical steps based on Maxwell’s equations lead to wave equation: • take curl on equation (D.18) • substitute the resulting equation in equation (D.19) • use the vector identity V2 s VV •- V x V x on the resulting equation • and use equation (D.20) to further simplify, this yields (v2+ fc2)E = y©pje v ' s , (D. 92) A similar procedure on magnetic field yields (v2 + &2)h = - V x J e (D. 93) Equations (D.92) and (D.93) are called wave equations. In homogenous medium where there are no sources, therefore case of e q u a tio n s (D.92), (D.93) simplify as (V2 +A:2)E = 0 (D. 94) (V2+ £ 2)H = 0 (D. 95) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where, V2 = Laplacian operator. Equations (D.94) and (D.95) are wave equations fo r homogenous m edium. Knowing either electric or magnetic field is enough, the other one can be determined using Maxwell’s equation (D.18) and (D.19). The following definitions are related to wave equations in free space: k0 = coyj\i0s 0 = — = — (free space wave number) (D. 96) X0 = y (D. 97) c C= , X0 (free space wave length (m)) 1 =■= — CO= f k. 0 V^o£o k 0 (velocity of light (m/s)) • (D. 98) « 3 x 108 tl0 = — « 377 (intrinsic impedance of free space (Q)). (D. 99) Similarly, the following definitions come into picture, as related to wave equations in any other homogenous medium: k=(o^j[iec ~1 _ = ®1|M-Eol Er ~ j ~<£>~E0) (wave number in the medium) (D. 100) = p-ya V X 9 tt =-£■ = — (wave length in the medium (m)) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D. 101) v„ = 1 CO - j = = —= fk (phase velocity in the medium (m/s)) (D. 102) r| = J — (intrinsic impedance of the medium (Q)) (D. 103) n = yl\irs r (refractive index of the medium) (D. 104) 201og10[a] (attenuation constant (dB/m)) (D. 105) 180 P x — (phase constant (°/m)) (D. 106) 6 = — (skin depth (m)) (D. 107) Solving the relations of equations (D.100), (D.101), (D.102), (D.105) and (D.106) the following equations can be derived for attenuation constant, phase constant and phase velocity as applied to TEM mode electromagnetic waves, respectively, a = 201og1 P= — c V2 * vp = !----- (dB/m) + s " 2 +s'j x — * it = --^(Vs'2 + s " 2 +s') (°/m) (m/s) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D. 108) (D. 109) (D. 110) where, the medium dependent constants e ' and e " are as given in equations (D.32) and (D.33), \ir is the relative permeability of the medium, c is as given in equation (D.98), and © is the angular frequency of the fields. A transverse electromagnetic -TEM wave is the solution for the wave equations in homogenous medium. They have the property of E field, H field and the direction of propagation at any point are perpendicular to each other. A plane wave moving in any arbitrary direction i is given by E = E 0^ K r p . Ill) H=— (D. 112) K = fci (D. 113) r = xx + yy + m (D. 114) D.2.22 Time Harmonic Solution to the Wave Equations in a Waveguide: Waveguide is a conductor pipe that carries EM waves. EM waves in a waveguide do not propagate freely as in open space, instead they travel guided by conductor walls. Often in practice, conductor wall cross sections are either rectangular or circular in shape. The inside of a waveguide can be air or any other dielectric. The EM waves that propagate in free space are in TEM mode. However, it canbe shown that TEM waves cease to exist in a waveguide, when bound by conductors. Therefore the solutions for EM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fields in free space should be either transverse electric -TE modes (H modes), or transverse magnetic -TM modes (E modes). The modes that exist in a waveguide depend on the excitation and structure of the waveguide. For the purpose of study, cross section of the waveguide is assumed to be of any arbitrary shape. However, the shape has to be uniform along the axial direction of the waveguide. This type of waveguide is referred to as a uniform waveguide. In order to adopt the solutions of uniform waveguide to any other specific cross sectional shape, the corresponding assumptions need to be incorporated in the general solutions. The solutions for fields inside a waveguide can be gotten simply by using the most general field equations, in terms of Hertzian Potentials. Z Figure D -l A uniform waveguide of arbitrary cross section, aligned toward z direction. Considering electric type hertzian potentials for the source free regions of the wave guide, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 320 (v2 + yfc2)rit, =o (D. 115) H = jc o scV x Tle = jc oeV x n , (D . 116) E= v x V x n <= it2n e + w - n f (D . 117) Similarly, consider magnetic type hertzian potentials for the source free regions of the wave guide, (V2+A:2)nm= 0 (D. 118) E = -ycojaV x IT,, (D . 119) H = v x V x n m= jt2ix ,+ v v .n n (D. 120) where, k is the wave number of the medium. The following two sections deal with the solutions for EM fields of TM and TE modes in the waveguide. D .2.22.1 TE or H M odes Electromagnetic W aves in a Waveguide: This section deals with TE modes in a waveguide. TE modes do not have electric field in the direction along the axis of the waveguide, i.e., E. = 0 ,H .* 0 . Since, the uniform waveguide is aligned toward z direction, in order to generate electromagnetic waves with no axial or z component of electric field (TE modes) the potential IX, has to have only z component. Therefore n,,, has to be as follows: P . 121) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then using equations (D.l 18) through (D.120), the following field equations can be written: (V2 +£2)nm r =0 (D. 122) E = -yco^iV X(znm __) (D. 123) h = k2{znm:) + v[v •( a i j ] (D. 124) Next, the V operator can be split into transverse and axial operators as V= where, V, is transverse operator. Further, considering the propagation constant p along the direction of propagation, the hertzian potential becomes (assuming no losses inside the medium of the waveguide) ^ m:{x,y,^) = ^(x,y)e~J'i: (D. 126) where, p 2 = k 2 - k 2c (D. 127) In equation (D.127), the term kc is cutoff wave number. Therefore, (D.122) becomes (V 2 + £ 2)vj/(x,.y) = 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (D. 128) The solution of equation (D.128) yields^ (x,y). The magnetic field inside the i1 1 < .< + + N> N> i i8 *1^ i I..... waveguide becomes (using equations (D.124) through (D.127)) H = z k 2\y (x, jy)e 7p; + |-v(/(x,y)e'/P; _oz = 7 k \ { x , y ) e ~ j9c + i [-y'P¥ {x,y)e~J*: = z k \ r (x,y)e~3: + |v ,[— yPq/ (x,^)e_7p;] + z |- [ - y P y (x,y)<T7Pr] = z AV (x,y)e~3: + {-y p v t[\|/ (x, y)e~J?:] + zpfy ( x ,y ) e '3 : ^ (D. 129) = i k 1 - P2)v (x ,v )e ''p-' - ypvt[v ( x ,y ) e - jti:] = i k j y (x,y)e~J?: - ypV,[\|/ (x,^)e“jP-'] = zH. + H, where, H. and H, are axial and transverse components of magnetic field respectively. Since, the axial component of electric field E .= 0, the transverse components of electric field becomes (using equations (D.123) and p . 129)) E, = -y<Bjx{v x [zvy (x,.y)e'7P'*]} = -yco|ij[Vn/(x,,y)e'7P‘'] x z} = yco|J.|z x [Vy (x,>’)e '7P'']j = yap. jz x = ycofijz x [ v t(v (x,y)fT7p-')]} = ycop v , + z^)v(x,v)e'7p; H. (D. 130) ~j P. —CO(J. ( z x H, ) = Z*(H, xz ) where, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 323 (D. 131) The term Zh is TE mode wave impedance. The summary of TE mode field equations are as follows: H, = -y'Pv t[M'(*,3'K7|5r] H. E. = zk p y { x , y ) e ' J^ = Zh( H ,x z ) = (D. 132) 0 It should be noted that in equation (D. 132), if the propagation constant p is imaginary, the waves inside the waveguide attenuate very fast, and do not propagate, i.e., yjk2 - k ] =k (D. 133) 0= j \l k] - k 1 = jk which means, when frequency is less than f c, the cutoff frequency, the waves inside the waveguide do not propagate, instead they attenuate, these waves are called evanescent waves. Also, note the definitions of the following two wavelengths: k - ?7r R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 134) 324 K=y P 135) where, Xc is the cutoff wavelength (m) of the waveguide, and Xg is the guided wavelength (m) in the waveguide. D. 2.22.2 TM or E Modes Electromagnetic Waves in a Waveguide: This section deals with TM modes in a waveguide. TM modes do not have magnetic field in the direction along the axis of the waveguide, i.e., H. =0 ,E . * 0 . Since, the uniform waveguide is aligned toward z direction, in order to generate electromagnetic waves with no axial or z component of magnetic field (TM modes) the potential ri, has to have only z component. Therefore n e has to be as follows: ne = m e: (D. 136) Then using equations (D.115) through (D.l 17), the following field equations can be written: (v2+*2)n e. . = 0 (D. 137) H = ycoeV x (fn„) (D. 138) e = k2(m e:) + v[v •(zne:)] (D. 139) Again, the V operator is split into axial and transverse operatorsas given in equation (D.125). Further,considering the propagation constant p along R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 325 the direction of propagation, the hertzian potential becomes (assuming no losses inside the medium of the waveguide) -jP: where, p2 = k 2 - k 2c . (D. 140) Again, the term kc is cutoff wave number. Therefore, (D.137) becomes (D. 141) The solution of equation (D.141) yields <j>(x,_y) • The electric field inside the waveguide becomes (using equation (D.169)) E = z k 2§ { x ,y ) e J?: + V. + z — ->3- dz = i k 2$ (x,y)e~ J^ + V. + zdz = z k 2§ (x,y)e~ J^: + | v t[-yP<j)(x,_y)e~7f5:] + z ^ - \ - j $ $ ( x , y)e~J^ (D. 142) = z k 2^(x,y)e~3 '- + { - y p V .^ x ^ J e '^ J + zp2^ ^ , ^ ’713-'} = ^k2- f32)<J)(jc,j^)^_jP^—y'PVt[(j)(^,>;)e“7^] - ik^(x,y)e~ J^ -ypVt[(j>(x,.y)e~/[i-'] = zE. + E, where, E. and E, are axial and transverse components of electric field respectively. Since, the axial component of magnetic field H. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. = 0, the transverse components of magnetic field becomes (using equations (D.138) and (D.142)) H, = /cos j v x £z<t>(;c,_y)e ■ /P‘']j = /cos jjV()>(x,.y)e yP''j x zj = -/cos jz x = - /cos<z x Vt + z £ j ( ( i x , y ) e - * : (D. 143) = - /c o s jz x [ v t(<f>(x,.y)e *•')]) = - / c o s ( z x - ^ -iP. = y - ( z xE,) = Ze(zxE,) where, Z„ = cos (D. 144) The term Z, is TM mode wave impedance. The summary of TM mode field equations are as follows: E , = - /p V t[<t>(x,^)e_>p-'] H, = z k ^ ( x ,y ) e ~ J<i: = Ze( z x E ,) H. = E. (D . 145) 0 N ote that the definitions and physical interpretations of equations (D.133) through (D.135) remain same for TM mode. D.3 BASICS O F TRANSM ISSION LINE THEORY The circuit theory of electrical engineering deals with relations between electric voltages and currents. These electrical quantities are derived from R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. electric fields and magnetic fields of EM theory, but as applicable to circuits made up of electrical conducting wires. When electric voltages and currents are static, they are known as direct current (DC) quantities. The theory develops for DC circuits from O hm ’s law, and continues with a set of mathematical and conceptual tools called theorems. When electric voltages and currents are time varying, they are called as alternating current (AC) quantities. The development of this theory starts again with O hm ’s law and continues with a set of theorems as tools for solving AC problems. It is easier to deal with AC circuits using frequency domain techniques rather than time domain techniques, and frequency domain analysis is common in practice. However, when the frequency becomes higher, the circuit theory fails to yield proper results. The main reason for this is that the component sizes become comparable with that of wavelength of AC signals. The alternative for this is to use the knowledge and experience gained in circuit theory and EM theory and develop another circuit theory as applicable to voltages and currents at higher frequencies. This branch of electrical engineering is called Transmission Line Theory. Transmission line theory has applications in Microwave Solid State Circuits. Fundamental equations of transmission line theory is briefly explained in this section, with out going into the details of derivations. The transmission line theory covered in this section are as follows: • Differential Length of a Transmission Line: In transmission line theory, the circuit is assumed to be made up of millions of unit R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. differential lengths of a transmission line, which is represented by basic circuit elements like a unit resistance conductance ( G A z , siemens), a unit capacitance unit inductance (LAz, (RAz, (CAz, Q), a unit farads) and a henrys). Mathematically, the propagation of voltage and current waves in a transmission line are represented by wave equations. This topic can be found in sections D .D .l and D.D.2 • Wave Propagation in a Transmission Line: The solutions of the wave equations for a transmission line gives the voltage and current at any point along that transmission line. These solutions are later specialized for lossless transmission lines, which are very common in practice. This topic can be found in section D.D.3 • Parameters Related to a Transmission Line: There are parameters that help explain the nature of transmission lines, and help analyze and design solid state microwave circuits for various applications. These parameters are characteristic impedance and admittance (Z0,70), input impedance and admittance { Z :n.Ym), reflection coefficient ( r) and voltage standing wave ratio or VSWR ( S ). Characteristic impedance and admittance are constant for a given uniform transmission line irrespective of the source and load attached to the line. Input impedance and admittance are constant for a given transmission line and given load, irrespective of the source attached to it. Reflection coefficient has magnitude and phase, and depends on characteristic impedance of the line and R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. load attached to the line. For lossy lines, both magnitude and phase of the line vary all along the length of the transmission line. For lossless lines, magnitude of reflection coefficient remains constant, however, the phase changes all along the length of the transmission line. VSWR is defined only for lossless lines and has only magnitude, no phase. VSWR is constant for a given transmission line, depends on the magnitude of reflection coefficient. This topic can be found in sections D.D.4 through D.D.7. D.3.1 D ifferential Length of a Transmission Line: A transmission line carries the voltage signal of high frequency. In practice it takes the form of coaxial lines, striplines, microstrip lines, slot, lines, inverted microstrip lines, suspended microstrip lines, strip dielectric waveguides etc. _______ transmission line__________________ incident signals '1 source reflected signals z = 0 z = d load z=1 Figure D-2 A complete transmission line circuit. Even though uniform conductor waveguides can be considered as transmission lines, this theory deals mostly with solid state transmission R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. lines, uniform waveguides have EM theory for analysis and design. In this research a coaxial line is used as a transmission line. A complete transmission line circuit has a high frequency source, a transmission line and a load as shown in Figure D-2. In transmission line theory, a unit differential length o f a transmission line is represented by basic circuit elements like a unit resistance ( R A z , Q), a unit conductance (G A z, siemens), a unit capacitance (CAz, farads) and a unit inductance (lA z , henrys) , as shown in Figure D-3. A complete transmission line circuit is assumed to be made up of millions of such differential units connected back to back in cascade. One can observe the fact that this assumption holds good only if the transmission line is uniform till the other end. D.3.2 Wave Equations for a Transmission Line: Consider a unit differential length of a transmission line with voltage at the input F(z) and current at the input /(z ) , as shown in Figure D-3. > I(z) I(z +A z) unit differential length 4 \ 7 f\ V(z) V(z +A z) z -IA z Figure D-3 A unit differential length of a transmission line. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 331 Here, it should be noted that the voltages and currents are functions of length and frequency, i.e., the differential length is excited by a sinusoidal signal of single frequency. Then, the following two equations can be written for the differences in voltages and currents at the two ends of this unit differential length of transmission line, using Kirchoff’s voltage and current laws in frequency domain, V (z)-V (z + Az) = yco(ZAz)/(z) + (i?Az)/(z) (D. 146) /(z)-/(z +Az) = j(o(CA z)V(z) + (GAz)V(z) (D. 147) Simplifying the above two equations, V(z± ^ ) - V(z) = _(* + j(0L)i(z) (D. 148) = (D. 149) + ja C )V (z) Taking limits on equations (D.148) and (D.149), as J i m V(z + A z ) - V ( z ) = d Az -> U Az az Az -> 0, = _ {R + /(z), - (G + yMc r W ^ 150) P . 15!) Taking differentiation on equations (D.150) and (D.151) with respect to z, the following two equations can be gotten: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 332 £ rV (z) = -(R + j« ,L )j-l(z) P . 152) Using equations (D.140) and (D.141) in (D.142) and (D.143), the final wave equations for the transmission line can be gotten as follows: ^ V ( z ) = ZYV(z) (D. 154) j-jl(z ) =zn (z) (D. 155) where, Z = R+ jasL (D. 156) Y = G + y'coC (D. 157) D.3.3 Wave Propagation in a Transmission Line: The solutions of the wave equations (D.154) and (D.155) gives the voltage and current at any point along the transmission line, assuming that there is a sinusoidal voltage excitation at the beginning of the line, i.e., at z = 0. They are as follows: V(z) = V+e~<: + V_e+r- R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 158) 333 I(z) = i y <: + I_e+<: = — (v+e-y: + V_e+y:) = Y0(v+e~y: + F e +?:) (D. 159) where, l\ ,V_,I+ and /. are constants that need to be resolved depending on the conditions that exists at either ends of the transmission line. Z0 and Y0 are characteristic impedance and characteristic admittance, respectively. Also, y is the propagation constant (/m) of the transmission line. The propagation constant of the line is given by Y = a + y'P = yfZY (D. 160) where, a and p are the attenuation and phase constants of the line, respectively. Solving (D.160) together with (D.156) and (D.15?)yields the following results: CO- H e a = 20Iog,n — IL V2 Ri CO2J} 2/^2 G1 R*G + - — -■ + co 2C 2 - CO4I 2C 2 1 RG - co l LC. (D. 161) (dB/m) P= co 4 lc V2 1+ R2 2 rl co L R G RG ">s-t“> 4 t~> + 1 co 2L C co C' co LrC~ 180 n (°/m) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 162) 334 1 V, = ----- m p . 163) = 1 fi ^ ^ R 2G 2 ( _ RG \ 4 l VV + co2 1: + co2C 2 + co4 Z r C 2 + l co2I c J (m/s) where, vp is the p/wse velocity (m/s) of voltage waves along the transmission line. It should be noted that for an ideal transmission line, c = -^JLr, the speed of light. Constants in equations (D.158) and (D.159) have to be chosen such that they satisfy equations (D.154) and (D.155) together with the assumption that transmission line voltages are made up of two waves called incident and reflected waves. Therefore, these solutions should indicate that there are two waves, one is forward going (called incident wave , other is backward going (called reflected wave these two is what appears = vmc(z ) + Vre/( z ) ,I ( z ) = Iinc(z) - IKf( z ) ) as Vjnc( z ) ,I mc( z ) ) Vre/( z ) ,I re/( z ) ) . and the The sum of instantaneous value at any point along the line. Note the following special cases with regard to propagation constant, attenuation constant, phase constant and phase velocity of a voltage or current wave in a transmission line: ® W hen oo = 0, i.e., D C y = dZY = dRG R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 164) 335 a = 201og10^V^Gj (dB/m) (D. 165) (3 = 0 ( ° /m ) (D. 166) vp = 0 (m/s) (D. 167) • W h e n tf« co Z = V z l« W l +Gi l a « 20 log 10 W I - # & G «coC + yea V ic (dB/m) p =(co V ic ) x — (°/m) ' ' v- a i c (D. 168) (D. 169) (D. 170) 7t (D. 171) (m/s) • W hen R —>0 ,G = V z l = yco V IC -> 0, i.e., line is lossless (D. 172) a = 0 (dB/m) (D. 173) P=(co V I c ) x ^ p (°/m) (D. 174) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 336 =71c (D. 175) (m /s ) D.3.4 Characteristic Impedance of a Transmission Line: The input impedance of a uniform transmission line of infinite length is called characteristic impedance (Z0, Q). Referring to Figure D-4, assuming, that the line proceeds further till infinity, the ratio instantaneous voltage and current at the input of this unit differential length of transmission line is the characteristic impedance of that line. Since the transmission line proceeds till infinity, the rest of the line can be represented by a load of Z0. in /\ ZAz Zm= Zp, V in Figure D-4 Characteristic impedance of a transmission line. The is same as the input impedance of this unit differential length of a transmission line, given by Z .= 7 l = ( 1^ ) | ( Z A z+ Z0) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. p . 176) where, and/,,, are the input voltage and current of the unit differential Vjn length of a transmission line. Solving equation (D.176) for Z Q, the following equation can be obtained: z° = S = J H H < p - i77>. Note the following special cases with regard to characteristic impedance of a line: When co = 0 , i.e., DC z 0 \Z- I* \ r • When When ^0 = (D. 178) J (o L » R R 0,G & coC » G ->• 0 , i.e., line is lossless (D. 180) Note that K = J — is the characteristic admittance of the transmission line. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. D.3.5 In p u t Impedance of a Transmission Line: Input impedance of a transmission line is the ratio of input voltage and current of the transmission line, as shown in Figure D-5. Impedance at any point, looking forward, along a transmission line is given by Z. = V (z) I(z) V ^ + V ^ z) 7/nc( z ) - / re/(r) VinM ^ P + Vr A ° y r- (D. 181) /„lc( 0 > - - / „ , ( 0 ) e - where, Vmc, Vref, Iinc and Iref are the incident voltage, reflected voltage, incident current and reflected current, respectively, at z = 0. The subscript “ n” means input while the subscript “inc” means incident at any point d, the confusion between these two should be avoided. The sign in the denominator of equation (D.181) indicates the direction of the reflected current. i l l me /\ ref Ziin -L 'in load — — O z = 0 Z = 1 Figure D-5 Input impedance of a transmission line Therefore, impedance at the input of the line can be gotten by putting z = 0 in equation (D.181), as follows: R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 339 I+ M ° ) , r«(o) + r„(o) ^ ( o ) + ^ ( o ) , " ' U » ) - U o ) > . ( o ) ^ ( o ) _ ^° Z„ r_(o) M o) M °) ^ - 18^ Equation (D.182) is not found very useful in practice, it is- convenient to derive input impedance, in terms of load impedance and length of transmission line as follows: 7 mo+mo y^y'‘*v^y" , ' u o - u o z„ z, ,n km yM o) F„(0) ' where, I is the length of the transmission line.Therefore, ratios of reflected voltage to incident voltage of equating the equations (D.182) and (D.183), Z"~~ Z° = Zj„ + Z Q Z, + Z0 (D. 184) Next, solving equation (D.184) for input impedance, R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 340 z„ = z n 1+ ZL ZA e -ty Z/ + Z 0 = zr i _ Z[ ~ Z° e-2y‘ (Z; + Z 0) + (Z/ - Z 0)e (Z, + Z 0) - ( Z , - Z 0)e - 2 y / Z ,+ Z 0 „ Z,(l + e-J'') + Z0( l - e -« ) - Z;( l - e - 2T/)4-Z0(l + e - ^ ) f e+-'' _e-A z, + z 0 V e ^ '+ e ^ J = zn Z0 +Z, = zn z, + z0 = z„ 1-e-2" 1+ e-2^ (D. 185) 1- e z 0 +Z/ Y+~ -w Z, + Z0 tanh(y/) Z0 + Z, tanh(y/) \ e +yl + e ' y,J For a lossless line y = / P , therefore, equation (D.185) reduces to z; +yZ0 tan(p/) 7in= 7 0 (D. 186) Z0 + j Z , tan(p/) Note the following special cases with regard to input impedance, for lossless lines: • • When end of line is short circuited, i.e., Z, = 0 (D. 187) z,„ = j Z 0 tan(p/) - the input impedance is purely reactive, either inductive or capacitive • When end of line is open circuited, i.e., Z, = 00 z,„ = - JZp tan(p/) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 188) - the input impedance is purely reactive, either inductive or capacitive • When length of the line is equal to quarter wave, i.e., / = — =^ Z/ (D. 189) Note that Yin is the input admittance of a transmission line, given by D.3.6 Reflection Coefficient of a Transmission Line, at Length z = d: Reflection coefficient along a transmission line is the ratio of incident voltage to reflected voltage of the transmission line, i.e., where, p and <j> are magnitude and angle of the reflection coefficient, respectively. Equation (D.191) is general and is applicable to any length from the input of the line. Vmc, Vre/, Imc and ln{ are the incident voltage, reflected voltage, incident current and reflected current at length d from the input of the line. It is very useful in practice to get relations between reflection coefficient at the beginning of the line, at the load-end of the line and in general, at any point along the line, in terms of input and load R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. impedances. These relations, together with some special cases, are listed in the sections D.3.6.1 through D.3.6.4. D .3.6.1 A t the Input o f the Transmission Line (d = 0): When d = 0, the input impedance is given by , +M 2 m M 0 )+ M 0 ) ■. z" “ /(o) - /„ (o )- , J o) - z» °) M o) 7 i+ r» KJP) - z« 1 - rp ,n 192> K Jo) where, F0 is the reflection coefficient at the beginning of the transmission line. Therefore, solving equation (D.192) for r0 r0= P0^ o = y L—^r- (D. 193) An + A D.3.6.2 A t the Load E n d o f the Transmission Line (d = I): When d = /, the load impedance is given by , _n>) MO+MO Z' ~ W ) ~ 1+ m o , MO , 1+r, U O -U O ° 7 ~ M 0 -Zo! ^ ,n ( , 194) y jf) where, r, is the reflection coefficient at the beginning of the transmission line. Therefore, solving equation (D.194) for T, r, = p, ^ , = A^A R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. p . 195) 343 D.3.6.3 A t any Point on the Transmission Line (z = d): When z = d, the impedance looking forward into the transmission line is V(d) r,„(d)+v„,(d) v jd ) 4 - / ( r f ) - / _ ( r f ) - U r f ) - z» i + Tj 1- r , , . v* ,V ) where, is the reflection coefficient at length d from the beginning of the transmission line. Therefore, solving equation (D.196) for Td p . 197) A* o Again, when z = d, the impedance can also be written as i+ , r „ (o + ^ y^ (o j 0)) z' - U o k - - / * ( o K * - z \ _ Kvs M M e.v M °) , i+ r y * ' ’i- w * ( U - Therefore, solving equation (D.198) for r 0 = Z d Z 0 e - 2y d = T e -l* zd+zQ or> = r 0e+v Again, when z = d, the impedance can be written as R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 199) 344 V{d) Vinc(O y yd + K A ° y ili K ,M e+l(l~d) + Kf(l)e +y(l-d) = z, uo> -y(l-d) =7(^)" (D. 200) -ly(l-d) = zr l + r,e-2t (/-</) 1 -I> vjf) " MO Therefore, solving equation (D.200) for r, ZrfZp +2y (/-< /) 1I — ^ r, V Zrf + Z0 T- _ _ r ~1 +2y ( l - d ) 5 or, rrf = r> (D. 201) D.3.6.4 Reflection Coefficient o f a Lossless Transmission Line: ‘ This section is very useful in practice, since most of the lines designed in microwave solid state circuits can be approximated as lossless transmission lines. When the transmission line is lossless, the magnitude of reflection coefficient (p) do not change with length (p need not be 1), only the phase of reflection coefficient (<}>) changes, i.e., for a lossless transmission line P ~ Po —P/ - Pd (D. 202) Also, note that 0 < p < 1. If the transmission line is lossless, then equation (D.201) becomes Td = i 0e +2jM = PZ((j>0 + 2pJ) (D. 203) o r,r, = rie-2jw -d) = pZty, -2 $ { l-d )] (D. 204) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Summary of reflection coefficients, as related to a lossless transmission line are as follows r 0 = f * — ^ = r /e-2^' = r ,z - 2p/ = pzfo, - 2p/) An + 0 r, = = r 0e+2Jp/ = r 0z2 p / = Pz (^0 + 2p/) A o —rr 0„+ - r;e'2JP(,"‘,) = r,Z - 2 p(/ - d) = Pz[<j), - 2P(/ - d)] A# — e 2>P‘/ = (D. 205) The following are special cases with regard to reflection coefficient of a lossless transmission line: ® When end of line is short circuited, i.e., Z, =0 Z —Z r, = —------ =-1, therefore p = 1, and A + A> r0 = Z/n- - z-0 = -e~1M = -1Z - 2p/ = iz(n - 2p/) • An + A) r = A z z k = r e +2Jv = - 1 = I Z n A + A> Td = r 0e+27iW= - e-W-d) = -1Z - 2P(/ - d) = 1Z[tt - 2P(/- d)] ' - the reflection coefficient has a magnitude of 1, only phase changes depending on where in the line, one is looking forward toward the load • When end of line is open circuited, i.e., Z, = oo _ A—A _ ! therefore p = 1, and A + A> R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 346 = lZ -2 p / Z° = r ne+2jp/ = 1Z0 r,= Z/ Z, + Z0 (D. 207) r rf = r oe+2;pd = e~2Ml' d) = \ Z - 2P (/ - rf) - the reflection coefficient has a magnitude of 1, only phase changes depending on where in the line, one is looking forward toward the load • When line is matched, i.e., Z, = Z0 z —z r, = — = 0, therefore p = 0, and Z,+Z0 r0 = o r, = o • r„ = o (D. 208) the reflection coefficient is zero • P1= r. = When length of the line is equal to quarter wave, i.e., I = 2n i X XJ 4 A n 2 ’ = r,e-J’ = -r, = pzfr , +*) r <= TZy T+ ZTq = = p 4 * . + <■) rrf = r0e+2jpd = r>-^+27(W= ^ (tc + 2prf) = pZ(<j>, + it + 2prf) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (D. 209) D.3.7 VSWR of a Lossless Transmission Line: The solutions of equation (D.158) indicates that the total voltage wave that appears on the transmission line can be assumed as made up of two waves, one is incident voltage wave, Vmc(2 ) and the other is reflected voltage wave, r e transmission line in load max 'nun standing wave pattern Figure D-6 Voltage standing wave in a transmission line. The sum of these two voltage waves is what appears as the total voltage as a function of length V(z) = Vmc(z) + Vre/ (z), along the transmission line. The voltage waveform magnitude, if the line is lossless, appears like a wave that is standing still along the transmission line, as shown in Figure D-6. This waveform pattern, in a lossless transmission line, is called standing wave. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. Voltage Standing Wave Ratio or VSW R (S) of a lossless transmission line is defined as the ratio of maximum voltage to minimum voltage on that transmission line, i.e., „ K * 'L > I Mv /|lm. in (D. 210) where, z, and z, are two points along the length of the transmission line where magnitude of standing voltage waveform pattern happens to be maximum and minimum, respectively. Equation (D.210) is defined only for a lossless transmission line and is same for any length along a given lossless transmission line. and Fmin are the maximum and minimum values of £ the magnitude of total voltage wave that appears on a lossless transmission VmBX line. A relation between VSWR and magnitude of reflection coefficient can be derived as follows S = ^ ( 0 ) e*-' +F„/ (0)e-*'| r WI, V i Zl)\ ,+ m ■ ' 2yP-i rjo ) m ax i + r0<r2j'p*''|Im ax [1+ r 0e~2j^ J_______ __ 1+ pe7V |l + pe,j$e~2jP:2 M o) l +peX*-2^‘) |l + pZ((|)-2pz1)|max_ l + p 1+ pe |l + pZ(4>-2|Jz2)| (D. 211) 1 -p The reason for the last step of above derivation is that, irrespective of length z, or z2, the maximum and minimum of or 1Z(<|> -2(5z) is + 1 and -1, R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. respectively. The following are special cases with regard to VSWR of a lossless transmission line: • When end of line is short circuited, i.e., Z, =0 z —z T, = —---- - = -1, therefore p = 1, and Z, + Z 0 5 = oo (D . 212) - the VSWR is infinite • When end of line is open circuited, i.e., Z, = oo z —z r = _j 9. = i therefore p = 1, and Z/ + Z0 5 = oo <r (D . 213) - the VSWR is infinite • When line is matched, i.e., Z, = Z0 z —z r = -i 9. = o, therefore p = 0, and z, + z0 5=1 p . 214) Therefore, VSWR varies from 1 to infinity. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. VITA Title of Dissertation: COAXIAL MICROWAVE SENSORS FOR LAYERED MEDIA Name of the Author: Suresh Khandige Place of Birth: Sakaleshpur (near Bangalore), Karnataka, India Colleges and Universities: BMS College of Engineering (Affiliated to Bangalore University, Bangalore, India): M.E (Master of Engineering) in Electronics, passed in First Class Distinction (1987-1989) Bangalore Institute of Technology (Affiliated to Bangalore University, Bangalore, India): B.E (Bachelor of Engineering) in Electronics (1980-1985) Professional Positions Held: College of Engineering and Applied Sciences, Electrical Engineering Department, University of Wisconsin, Milwaukee, WI: Teaching Assistant - Electronic Circuits and Senior Lab (Fall 1991 Spring 1996) Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India: AD-HOC Scientific Officer in the project Adaptive Antenna Arrays for RADAR and Communications (March 1989 - May 1989) Center for Development of Telematics, Bangalore, India: Trainee Engineer in Design and Development of IF sub-systems for the VSAT project 0une 1989 - October 1989) Membership in Learned or Honorary Societies: The Institute of Electrical and Electronic Engineers, Inc. (IEEE) New York Academy of Sciences, N Y Publications: Khandige, Suresh., and D.K. Misra, “Characterization of the Layered Dielectrics Using an Open Ended Coaxial Line Sensor,” Conference of Precision Electromagnetic Measurements Digest, Boulder, Colorado, Page 65-66, June 27-July 1, 1994 Major Department Electrical Engineering Minors Computer Science Major Professo: R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. Date

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