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Coaxial microwave sensors for layered media

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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
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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
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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
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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
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© Copyright by Suresh Khandige, 1996
All Rights Reserved
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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
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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.
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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..................................................................................
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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)
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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
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• 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.
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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
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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.
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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
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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].
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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
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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
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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.
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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
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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
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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
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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
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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
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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.
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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.
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(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
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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
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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].
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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:
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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:
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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:
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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
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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.
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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.
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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
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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.
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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
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(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:
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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
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(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
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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].
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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
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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 )
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(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
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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,
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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:
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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 )
*
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(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
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(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”.
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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:
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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
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(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.
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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 ).
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(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
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(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
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-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
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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 „
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(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:
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(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].
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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 )\
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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 '*
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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
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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].
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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
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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:
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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 )
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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 )
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(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
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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:
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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:
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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:
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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
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(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
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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
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(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.
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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
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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
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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.
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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.
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• 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
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(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)
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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.
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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
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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
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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
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(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
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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).
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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.
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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
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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
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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
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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
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-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
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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,
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(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.
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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).
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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= ——
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(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
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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:
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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
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(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
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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"
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(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.
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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.
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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
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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.
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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].
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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
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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.
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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
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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.
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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.
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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.
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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
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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%.
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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.
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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.
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-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
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- 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
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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
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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
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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
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— 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
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—
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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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(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):
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(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:
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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
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(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).
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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
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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
—
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(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:
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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
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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]
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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.
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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.
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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.
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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
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—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
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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
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—*— 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
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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
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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
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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
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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
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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.
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—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
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—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
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—
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
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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
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— - * —
-------------- 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
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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
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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.
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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.
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168
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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-
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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
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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
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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
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154) York, Richard A., and R.C. Compton, “An Automated Method for
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Constant
Measurement
of
Microwave
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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
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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]
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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]
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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]
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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
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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]
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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
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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
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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]
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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]
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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
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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]
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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
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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
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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)
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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
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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
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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
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mainrt = adm - y
return
end
! THICKNESS ROOT MODULE -- END
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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 .
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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.
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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 .
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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
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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 ] : '
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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'
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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
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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
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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
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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].
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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
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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
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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.
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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
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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)
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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:
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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)
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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,
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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
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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)
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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
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(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
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(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
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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
'
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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
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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)
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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
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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)
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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)
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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))
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(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)
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(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
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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,
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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)
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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
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(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,
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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
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(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
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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.
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= 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
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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
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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
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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
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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.
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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:
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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-
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(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)
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(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
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(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)
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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)
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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.
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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:
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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,
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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/)
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(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
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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
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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
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(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)
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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>
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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)
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(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.
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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,
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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.
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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:
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