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Computer-aided modeling and analysis of passive microwave and millimeter-wave high-temperature superconductor circuits and components

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Order Number 9418631
Computer-aided modeling and analysis of passive microwave and
millimeter-wave high-temperature superconductor circuits and
components
Antsos, Dimitrios, Ph.D.
California Institute of Technology, 1994
Copyright ©1994 by Antsos, Dimitrios. All rights reserved.
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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COMPUTER-AIDED MODELING AND ANALYSIS OF PASSIVE
MICROWAVE AND MILLIMETER-WAVE HIGH-TEMPERATURE
SUPERCONDUCTOR CIRCUITS AND COMPONENTS
Thesis by
Dimitrios Antsos
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1994
(Defended November 5, 1993)
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ii
© 1994
Dimitrios Antsos
All Rights Reserved
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ACKNOW LEDGM ENTS
Early in the morning o f June 15,1993, my thesis advisor Professor Edward C. Posner of
Caltech was killed in a traffic accident, while riding his bicycle to work. Part o f me died
with him that day and things are never going to be quite the same again. As I write this
emotional eulogy and acknowledgment, which admittedly has no place in any thesis, I
cannot but think how really essential it is to this one.
Without Ed (I have only
posthumously and lovingly started to call him Ed; when he was living the aura o f wisdom,
love and respect his person projected "forced" me to use the more formal Dr. Posner), or
another person o f his caliber, it simply wouldn't have been possible. Unfortunately - and
this is my own opinion - persons o f Ed's caliber represent a species that is virtually extinct
in today's society. Hence the huge vacuum that he left behind him will not be easy to fill.
This thesis is hiM em oriam of a worthy person with a great mind and heart.
With similar gratitude, I would like to thank and acknowledge Robert C. Clauss (Bob),
former manager o f the TDA Systems Development Program at JPL and current JPL
Member o f Technical Staff, without whose help and support (moral, judicial and
mechanical amongst others) I might not have even had a BS. today (he doesn't either; yet
his thoughts and suggestions induce academic nightmares to many "educated" Ph.D.'smyself not excluded).
From Caltech: I wish to thank professor R. J. McEliece who, after Dr. Posner's death,
became my advisor and helped me along in the critical final months of my graduate
studies. Additionally, I would like to acknowledge professors Vaidyanathan and Rutledge
o f the Electrical Engineering department and professor Brattkus and Sebius Doedler o f the
Applied Mathematics department for the time that they devoted to me, selflessly and
willingly, for discussions at various times during the development o f my thesis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
From JPL: I would like to thank my boss Dan Rascoe, Supervisor o f group 3363 (who
probably lost quite a few tenths o f a degree of his eyesight trying to untangle my Gordian
knot o f syntax and spelling) for helpful suggestions and discussions and for taking the
considerable time entailed in being part o f my thesis committee.
Also Wilbert Chew
provided valuable suggestions and references. Gratitude is also due to all my colleagues,
the group who tolerated me for 2.5 years, sharing their facilities and taking up office
space, with minimal productivity in return. Many thanks are due Section 336 in general
and its Manager, Tom Komarek, for the nurturing nest they provided me in these research
years.
Gratitude is due M arc M atzner who volunteered his professional proof-reading and editing
services. Without his comments, suggestions and edits, implemented in their entirety in
this thesis, the latter would have been considerably harder to read.
Support for the computing research presented in chapter 8 was provided by the JPL
Supercomputing Project on the CRAY computers at the Jet Propulsion Laboratory and at
Goddard Space Flight Center. The JPL Supercomputing Project is sponsored by the Jet
Propulsion Laboratory and the NASA Office o f Space Science and Applications. Finally, I
have been asked to mention Brian Hunt and his group at JPL who fabricated the HTS
circuits used for the experimental verifications in chapters 5, 6 and 7.
Conductus Inc., provided me with a "student-priced" HTS YBCO resonator, which is
used for the experimental verification in chapter 4, for which I am grateful.
Last and foremost I would like to thank and acknowledge my fiancee, Judith Din, who,
with her love and patience, provided the haven of peace and concentration necessary for
the successful completion o f my work.
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ABSTRACT
As their critical temperatures continue to rise, high-temperature superconductors (HTS)
promise applications in microwave, and to some extent in millimeter-wave circuits,
because they should exhibit lower loss, in these frequencies, than their normal metal
counterparts. However, in the case o f passive circuits, fundamental performance limits
(finite insertion loss) still exist and apply, as explored in this thesis.
Commercial computer-aided design (CAD) and analysis software tools exist, that permit
design and analysis o f normal metal microwave and millimeter-wave circuits. These tools
minimize design and manufacturing errors and the need for costly re-work and design
iterations. In the case o f HTS circuits these tools are insufficient because of three effects
present in HTS circuits that do not exist in normal metal circuits. First, because of
manufacturing practices, the HTS layers on substrates are usually very thin; o f the order of
the magnetic field penetration depth. Second, there is an additional internal inductance,
the kinetic inductance, which is due to the inertia o f the superelectrons. Third, high input
power induces high magnetic fields and current densities which drive the superconductor
into its normal state, in which it is an insulator.
This thesis is a study o f these phenomena and their effects on quasi-TEM transmission line
circuit performance. Methods for accounting for these effects and introducing them into
currently available CAD tools are presented. These methods are applied to three example
circuits for which modeled and measured performance is compared.
The viability and advantages o f HTS waveguides are also studied and analyzed. A finite
difference analysis program is presented.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
vi
CONTENTS
Acknowledgments
iii
Abstract
v
Contents
vi
List of Figures
xi
List of Tables
xvii
1 Introduction
1
......................................
1
..................................................................................
4
1.1 The Current State o f Development o f CAD Tools
1.2 An Outline o f the Thesis
2 Superconductivity and Passivity
2.1 Introduction
6
.......................................................................................................
2.2 An Analytical Statement o f Passivity
..........................................................
6
.......................................................
10
.........................................................................................
10
2.3 Implications o f Passivity. An Example
2.3.1 The Circuit
2.3.2 The Calculations
2.4 Conclusions
6
...............................................................................
12
.......................................................................................................
22
2.A Appendix: Sample Matlab File
.....................................................................
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24
3 Low Field M odeling of Quasi-Transverse Electric and M agnetic HTS
Transmission Lines
3.1 Introduction
26
........................................................................................................
3.2 The Two-Fluid Model o f a Superconductor
26
................................................
27
...........................................
32
3.3 Surface Impedance o f a Bulk Superconductor
3.4 A Phenomenological Loss Equivalence Model for Quasi-TEM HTS Microwave
Transmission lines
..........................................................................................
34
3.5 Algebraic Verification o f Equations (35) and (36)
39
3.6 References
40
........................................................................................................
3.A Appendix: MathCAD File that Algebraically Verifies Equations (35) and (36)
........................................................................................................
41
4 Validation and Application of the PEM Loss Model: An HTS M icrostrip Ring
Resonator
4.1 Introduction
44
........................................................................................................
4.2 The YBCO Microstrip Ring Resonator
4.3 The Model
.......................................................
44
........................................................................................................
45
4.3.1 The Modeling Methodology
..............................................................
4.3.2 Using Touchstone and Academy (TM) by E E sof Inc
4.3.3 The Modeling Strategy
45
.....................
47
.....................................................................
50
4.4 Comparison of Model versus Measurement
4.5 References
44
.............................................
52
........................................................................................................
60
4. A Appendix: MathCAD File Used to Calculate the Parameters o f the Two Types
o f Transmission Line Used in the Resonator C i r c u i t
4.B Appendix: Touchstone Circuit Filethat Models the HTS Resonator . . .
5 An Application of the PEM Loss Model: An HTS CPW Low Pass Filter
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62
68
73
5.1 The YBCO CPW LPF
5.2 The Model
...................................................................................
73
........................................................................................................
74
5.2.1 The Modeling M ethodology
...........................................................
74
5.2.2 The Touchstone Circuit File
..............................................................
76
.....................................................................
77
5.2.3 The Modeling Strategy
5.3 Comparison o f Model versus Measurement
.............................................
79
....................................................
79
...........................
85
........................................................................................................
87
5.3.1 S-parameters versus Frequency
5.3.2 Temperature Dependence o f the Insertion Loss
5.4 References
5. A Appendix: Sample MathCAD File Used to Calculate the Parameters o f the
CPW Lines o f the LPF
...................................................................................
90
5.B Appendix: Sample Touchstone Circuit F i l e ....................................................
98
6 An Application of the PEM Loss Model: An HTS Microstrip Band Pass Filter 104
6.1 The YBCO Microstrip BPF
6.2 The M odel
6.2.1
........................................................................ 104
............................................................................................................105
The Modeling M ethodology
............................................................. 105
6.2.2 The Model o f the Input/Output Stub Resonator
6.2.3
The Coupled Microstrip Resonators Section
6.2.4 The Touchstone Circuit File
6.3 The Modeling Strategy
................................... 108
............................................................. 110
....................................................................................... 112
6.4 Comparison o f Measurement versus Model
6.4.1
.................................105
The Case o f N o Dispersion
6.4.2 The Case o f Dispersion
.................................................112
............................................................. 112
.....................................................................117
6.4.3 Dispersion or No Dispersion? This is the Question
6.5 References
.........................122
........................................................................................................ 123
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6. A Appendix: Sample MathCAD File Used to Calculate the Parameters o f the
Microstrip Lines o f the BPF
.................................................... 125
6.B Appendix: Sample Touchstone Circuit File:The Case o f No Dispersion
144
6.C Appendix: Sample Touchstone Circuit File:The Case ofD ispersion . . .
153
7 A M odification of the PEM Loss M odel for High Loss Modeling. An Application
to High Power Modeling
163
7.1 A Modification o f the PEM Loss Model. The High-Loss Case
7.2 From a Complex to a Real Characteristic Impedance
.............. 163
.................................. 165
7.3 Application o f the High-Loss M odel to a LPF o f Chapter 5
169
7.4 Application o f the High-Loss M odel to High Power M o d e li n g ......................175
7.4.1 Introduction
.............................................................................................175
7.4.2 High Power Measurements
...............................................................176
7.4.3 The Power-Dependent Model
........................................................... 178
7.4.4 Discussion o f the Fit o f the Model to the Measured Data
7.5
First-Order Effects due to Collision Relaxation
7.5.1 The Analysis
7.5.3 Discussion o f the Results
............................................ 182
............................................... 183
......................................................................185
7.5.4 Testing the Variable Effective Line-Width Hypothesis
7.6
References
181
...........................................................................................182
7.5.2 The Fit o f the Model to Measurement
7.5.5 Conclusions
. . . .
......................186
...........................................................................................189
.......................................................................................................... 189
7. A Appendix: First-Order HTS CPW LPF Touchstone Model (Low Pow er
Response)
................................................................................... 190
7.B Appendix: High-Loss HTS CPW LPF Touchstone Model (Low Pow er
Response)
....................................................................................... 196
7.C Appendix: High-Loss Touchstone Model (5dBm Input Power)
. . . .
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202
7.D Appendix: New Model with Improved Conductivity Equation
................. 209
7.E Appendix: Touchstone Circuit File that Verifies the Conjecture o f Section
7.5.4
216
8 Closed Rectangular HTS Waveguides
8.1 Introduction
223
....................................................................................................... 223
8.2 The Cross-Over Frequency
............................................................................224
8.3 A Contrast o f the Exponential Attenuation of Normal and HTS Waveguides 228
8.4 A Finite-Difference Numerical Solution for the Modes o f HTS Waveguides 232
8.4.1 The Problem
........................................................................................ 232
8.4.2 The Solution
........................................................................................ 235
8.4.3
........................................................................................ 249
The Program
8.4.4 Running the Program
8.4.4.1
.......................................................................... 259
CPU Time and Memory Usage
............................................ 259
8.4.5 The Results
........................................................................................ 260
8.4.6 Conclusions
............................................................................................279
8.5 Power Handling Capability
8.6 References
............................................................................. 280
.......................................................................................................281
8.A Appendix: MathCAD File Used to Calculate and Plot the Cross-Over
Frequency
......................................................................................284
8.B Appendix: MathCAD File Used to Calculate and Plot the Exponential
Attenuation versus Frequency
....................................................287
8.C Appendix: Mathematica Results on Characteristic Equation of32-by-32
Lossless A-Matrix
........................................................................ 290
8.D Appendix: C-code Listing o f the wg_plot.c Program
...................................298
8.E Appendix: Sample Output (for a WR90 HTS Waveguide) o f wg sweep
. 327
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xi
LIST OF FIGURES
2.1
A 3-port network
.......................................................................................................
2.2
A branch-line coupler
2.3
Trade-off between insertion loss & isolation
2.4
The power, in each o f the ports, for zero dissipation
2.5
Trade-off between insertion loss & isolation
7
............................................................................................
11
..........................................................
14
............................................
15
..........................................................
17
2.6
The power, in each of the ports, for zero d is s ip a tio n ............................................
18
2.7
The power, in each o f the ports, for zero d is s ip a tio n ............................................
19
2.8
Trade-off between insertion loss & isolation
..........................................................
21
2.9
The power, in each o f the ports, for zero d is s ip a tio n ............................................
22
3.1
Identical results o f equations (35), (36) and ( 3 2 ) ...................................................
40
4.1
The layout o f the YBCO microstrip ring r e s o n a t o r ...............................................
44
4.2
Atwater's dispersion model compared to the measured S21 o f the gold resonator
4.3
A schematic representation o f the model
.............................................................
46
49
4.4 Measured (WIDE for ITTS circuit, WIDE_AU for gold circuit) versus modeled
(RINGP) magnitude o f S21
53
4.5 Measured (WIDE for HTS circuit, WIDE_AU for gold circuit) versus modeled
(RINGP) angle o f S21
54
4.6 Measured (WIDE for HTS circuit, WIDE_AU for gold circuit) versus modeled
(RINGP) magnitude o f SI 1
55
4.7 Measured (WIDE for HTS circuit, WIDE_AU for gold circuit) versus modeled
(RINGP) angle o f S 11
4.8 Measured (WIDE) versus modeled (RINGP) magnitude o f S21
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56
57
xii
4.9 M easured (WIDE) versus modeled (RINGP) angle o f S21
58
4.10 M easured
(W IDE) vs. modeled (RINGP) magnitude o f SI 1
59
4.11
(WIDE) versus modeled (RINGP) angle o f S 11
60
M easured
5.1
The layout o f the HTS CPW LPF
...........................................................................
5.2
Definitions o f CPW dimension variables
5.3
Comparison o f insertion loss o f silver and YBCO filters
73
................................................................
77
.....................................
80
5.4 M easured (YBCO) versus modeled (FIL) magnitude o f S21
81
5.5 M easured (YBCO) versus modeled (FIL) angle o f S21
82
5.6 M easured (YBCO) versus modeled (FIL) magnitude o f SI 1
83
5.7 M easured (YBCO) versus modeled (FIL) angle o f SI 1
84
5.8 Measured (YBCO) versus modeled (FIL) S21 plotted on a Smith chart
o f unit radius
...............................................................................................................
85
5.9 M easured magnitude o f the insertion loss at 50, 60, 70 and 80 K
.....................
86
5.10 Predicted magnitude o f the insertion loss at 50, 60 70 and 80 K
.....................
87
6.1
The layout o f the HTS microstrip BPF
.................................................................... 104
6.2
The input section o f the HTS BPF with the stub r e s o n a t o r ..................................... 106
6.3
An equivalent input stub resonator section employed for m o d e lin g ........................106
6.4
A schematic representation o f the input/output stub resonator section
6.5
The coupled microstrip resonator section
6.6
Definitions o f coupled line dimension variables
6.7
A schematic representation o f the coupled line resonator section o f the filter . . I l l
6.8
Measured (YBCO) versus modeled (FLTRBSC) magnitude of S21
113
6.9
M easured (YBCO) versus modeled (FLTRBSC) angle o f S21
114
.................107
................................................................. 109
...................................................109
6.10 M easured
(YBCO) versus modeled (FLTRBSC) magnitude o f S 11
6.11
(YBCO) versus modeled (FLTRBSC) angle o f S 11
Measured
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115
116
6.12 Measured (YBCO) versus modeled (FLTRBSC) S21 plotted on a Smith chart of
unit radius
...................................................................................................................... 117
6.13 Measured (YBCO) versus modeled (FLTRBSC) magnitude o f S21
118
6.14 Measured (YBCO) versus modeled (FLTRBSC) angle o f S21
119
6.15 Measured (YBCO) versus modeled (FLTRBSC) magnitude o f SI 1
120
6.16 Measured (YBCO) versus modeled (FLTRBSC) angle o f SI 1
121
6.17 Measured (YBCO) versus modeled (FLTRBSC) S21 plotted on a Smith chart of
unit radius
...................................................................................................................... 122
7.1 Unit cell of ladder-network model
.........................................................................165
7.2 Magnitude o f S 11 o f ladder and transmission lines
................................................ 166
7.3 Magnitude o f S21 o f ladder and transmission lines
.....................................................167
7.4 S-parameter differences between LAD and LINE 1
168
7.5 S-parameter differences between LAD and LINE2
................................................. 168
7.6 S-parameter differences between LAD and LINE3
................................................. 169
7.7 The three candidate impedances plotted versus frequency
.................................. 170
7.8 First-order model, magnitude o f S21, measured (YBCO) versus modeled (FIL) . 171
7.9 High-loss model, magnitude o f S21, measured (YBCO) versus modeled (FIL)
. 171
7.10 First-order model, phase o f S 21, measured (YBCO) versus modeled (FIL)
. 172
7.11 High-loss model, phase o f S 2 1, measured (YBCO) versus modeled (FIL)
. . 172
7.12 First-order model, magnitude o f SI 1, measured (YBCO) versus modeled (FIL). 173
7.13 High-loss model, magnitude o f S 11, measured (YBCO) versus modeled (FIL) . 173
7.14 First-order model, phase o f SI 1, measured (YBCO) versus modeled (FIL)
7.15 High-loss model, phase o f S21, measured (YBCO) versus modeled (FIL)
7.16 High-power measurement setup
. 174
. . 174
................................................................................176
7.17 Measured magnitude o f S21at input p o w ers-10,-5, 0 and 5 dBm
.................. 177
7.18 Measured magnitude o f S21 at input powers 5, 10, 15 and 20 dBm
..................177
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xiv
7.19 Magnitude o f S21 of model versus measurement at 5 dBm input power
7.20 Phase o f S21 o f model versus measurement at 5 dbm input power
7.23 Magnitude ofS 21, model versus measured
7.24 Phase o f S21, model versus measured
......................180
........................................................... 183
....................................................... 184
........................................................... 187
.................................................................. 187
7.28 Phase o f S21, model versus measured
7.29 Magnitude o f S 11, model versus measured
8.1 Cross-section o f the HTS waveguide
........................................................... 188
......................................................................188
.........................................................................223
8.2 Cross-over frequency plotted versus zero-temperature penetration depth . . . .
8.3
180
......................................................................185
7.27 Magnitude o f S21, model versus measured
7.30 Phase o f SI 1, model versus measured
. . . .
......................................................................184
7.25 Magnitude o f S 11, model versus measured
7.26 Phase o f SI 1, model versus measured
179
......................179
7.21 Magnitude o f SI lo f model versus measurement at 5 dBm input power
7.22 Phase o f S 11 o f model versus measurement at 5 dBm input power
. . . .
226
Cross-over frequency plotted versus critical te m p e ra tu re .........................................227
8.4 Cross-over frequency plotted versus normal conductivity
..................................... 227
8.5 Normalized attenuation o f different types of HTS and gold w a v e g u id e
230
8.6
Minimum attenuation of different types o f HTS waveguides......................................232
8.7
The argument (angle) of the surface impedance o f an HTS
8.8
The magnitude of the surface impedance o f an HTS
. ..
............................233
............................................... 234
8.9 The cross-section o f the HTS waveguide sub-sectioned
by a uniform rectangular grid
.......................................................................................236
8.10 Definitions o f the cut-planes o f the field p l o t s ........................................................... 260
8.11 Cross-sectional view of the electric field o f the TE10 mode in a WR90 HTS
waveguide o f average HTS parameters at 12 GHz
.................................................261
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XV
8.12 Cross-sectional view o f the magnetic field o f the TE10 mode in a WR90 HTS
waveguide o f average HTS parameters at 12 GHz
................................................. 261
8.13 Surface view o f the magnetic field o f the TE10 mode in a WR90 HTS waveguide o f
average HTS parameters at 12 GHz
......................................................................... 262
8.14 Longitudinal view of the magnetic field o f the TE10 mode in a WR90 HTS
waveguide o f average HTS parameters at 12 GHz
................................................. 263
8.15 Longitudinal view o f the electric field o f the TE10 mode in a WR90 HTS waveguide
o f average HTS parameters at 12 GHz
.................................................................. 263
8.16 Cross-sectional view o f the electric field o f the TM32 mode in a WR90 HTS
waveguide o f average HTS parameters at 40 GHz
................................................. 264
8.17 Cross-sectional view o f the magnetic field o f the TM32 mode in a WR90 HTS
waveguide o f average HTS parameters at 40 GHz
................................................. 264
8.18 Surface view o f the magnetic field o f the TM32 mode in a WR90 HTS waveguide of
average HTS parameters at 40 GHz
......................................................................... 265
8.19 Longitudinal view o f the magnetic field o f the TM32 mode in a WR90 HTS
waveguide o f average HTS parameters at 40 GHz
................................................. 266
8.20 Longitudinal view o f the electric field o f the TM32 mode in a WR90 HTS
waveguide o f average HTS parameters at 40 GHz
................................................. 266
8.21 Cross-sectional view o f the electric field o f the TE10 mode in a WR3 HTS
waveguide o f worst-case HTS parameters at 380 GHz
.......................................... 267
8.22 Blow-up o f the region o f figure 20 below and to the right o f the middle o f the top
wall
......................................................................................................................... 268
8.23 Deviation angle o f the electric field vectors from the vertical, along a line parallel to
the y-axis
......................................................................................................................269
8.24 Deviation angle o f the electric field vectors from the vertical, along a line parallel to
the x-axis
......................................................................................................................269
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xvi
8.25 Cross-sectional view o f the magnetic field o f the TE10 mode in a WR3 HTS
waveguide o f worst-case HTS parameters at 380 GHz
..........................................270
8.26 Surface view o f the magnetic field o f the TE10 mode in a WR3 HTS waveguide of
worst-case HTS parameters at 380 GHz
.............................................................. 271
8.27 Longitudinal view o f the magnetic field o f the TE10 mode in a WR3 HTS
waveguide o f worst-case HTS parameters at 380 GHz
..........................................272
8.28 Longitudinal view o f the electric field o f the TE10 mode in a WR3 HTS waveguide
o f worst-case HTS parameters at 380 GHz
...........................................................272
8.29 Blow-up o f the region o f figure 25 below and to the right o f the middle o f the top
wall
273
8.30
Attenuation versus frequency in a WR90 HTS waveguide
8.31
Propagation constant versus frequency in a WR90 HTS waveguide
8.32
Attenuation versus frequency in a WR28 HTS waveguide
8.33
Propagation constant versus frequency in a WR28 HTS waveguide
8.34
Attenuation versus frequency in a WR10 HTS waveguide
8.35
Propagation constant versus frequency in a WR10 HTS waveguide
8.36
Attenuation versus frequency in a WR5 HTS waveguide
8.37
Propagation constant versus frequency in a WR5 HTS waveguide
8.38
Attenuation versus frequency in a WR3 HTS waveguide
8.39
Propagation constant versus frequency in a WR3 HTS waveguide
................................. 274
................ 274
................................. 275
................275
................................. 276
................276
................................. 277
................277
................................. 278
................278
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LIST OF TABLES
1.1 Software used in this thesis
......................................................................................
4
.......................................................
32
..............................................................
45
3.1 Dependencies o f losses on physical variables
4.1 The physical parameters of the microstrip
4.2 The electrical parameters o f the two types o f line
................................................
5.1 The four types o f line o f the filter and their properties
.........................................
45
75
6.1 The three line types of the filter and their properties
.................................................108
6.2 The two types o f coupled lines and their properties
.................................................109
6.3 The optimum extracted values for the penetration depth
and the normal conductivity
.......................................................................................112
8.1 Three cases for the cross-over frequency....... ................................................................. 225
8.2 Waveguide cutoff frequencies
...................................................................................... 230
8.3 Fractional Perturbation o f the Propagation Constant
.............................................. 248
8.4 Maximum Powers o f HTS Waveguides
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281 .
1
CHAPTER 1
INTRODUCTION
1.1 The Current State of Development of CAD Tools
This thesis is devoted to Computer Aided Design (CAD) and Modeling o f HighTemperature Superconductor (HTS) Microwave Circuits.
Microwave circuits, which are important and widely used in communications, are, unlike
their lower frequency counterparts, difficult to model by equations that involve simple
circuit parameters like voltage and current. M uch research, during the latter half o f this
century, has been devoted to understanding these circuits and overcoming this difficulty.
Today, with the advent o f digital computers, CAD and modeling o f "conventional"
microwave circuits is a very developed science.
Kirchhoffs voltage and current laws
together with Ohm's law go a long way in modeling low frequency circuits, circuits for
which the wavelength o f the electric and magnetic fields is large compared to the linear
dimensions o f the elements of the circuit.
Conversely, in microwave circuits the wavelength o f the excitation is comparable to one
or more of the linear dimensions of the elements o f the circuit.
If additional linear
dimensions o f the circuit are comparable to the wavelength, it becomes harder to model
the circuit because of the greater number o f field-modes that have to be considered. Thus,
a microstrip patch antenna ,which has its width and length comparable to a wavelength, is
harder to model than a microstrip transmission line, which has only its length comparable
to a wavelength.
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Modeling is important in the design and fabrication o f microwave circuits because it helps
avoid expensive rew ork and manufacturing iterations. Yet today, microwave engineering
is still considered, by many within the engineering community, "black magic," as it remains
one o f the fields o f engineering in which there is no substitute for intuition and experience.
The advent o f powerful computers has had an impact here in recent years.
Hewlett
Packard (HP) has, in the last three years, released a CAD software package, the High
Frequency Structure Simulator (HFSS), which analyzes the steady-state response of
circuits o f arbitrary shape (o f which all dimensions may be comparable to a wavelength) to
electromagnetic sinusoidal excitation. It performs a finite difference solution of Maxwell's
equations in arbitrary volumes by sub-sectioning these volumes into elementary sub­
volumes (bounded by tetrahedra) in which the fields are assumed constant.
Nonetheless, while most modern microwave CAD programs are successful in predicting
the direction o f change o f the response of circuits with respect to their design variables, I
have yet to see one that accurately predicts the response o f the circuit given the design
parameters. Although the partial derivative o f the response parameters with respect to the
design variables can, usually, be predicted fairly accurately, the absolute values o f the
response parameters are more elusive.
This is because, among other reasons, material
uniformity over small spatial dimensions (that are comparable to the short wavelength of
high frequencies) is usually poor, and boundary condition assumptions, used in various
analyses, are usually over-idealized and do not correspond to actual laboratory
measurement conditions.
By comparison, CAD and modeling o f HTS microwave circuits postdate the discovery, by
Bednorz and Muller, o f superconductors with transition temperatures above the 77 K
temperature o f liquid
nitrogen.
These promise many practical applications
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of
3
superconductor microwave circuits in today's communications, given the abundance o f
nitrogen and the relatively cheap methods o f its liquefaction. Superconductors have the
property o f zero DC resistance to currents. Hence at DC they are advantageous to use
(less lossy) over regular conductors. This proves true also, as it turns out, at microwave
frequencies (and even for some materials into the millimeter-wave frequencies). Since the
use o f HTS microwave circuits is not yet widespread in communications systems, there is
comparatively less work in the area o f modeling o f these types o f circuits.
HTS and conventional microwave circuits are similar in many ways, but certain critical
differences cannot be neglected since they differentiate responses o f identical looking
circuits. With the aid of the plethora o f CAD and modeling tools which exist today for
conventional microwave circuits one can, however, model many HTS microwave circuits,
taking care to properly modify relevant parameters.
This thesis first sets some upper bounds on expectations for the lossless behavior o f HTSs
and then presents a loss model, which exhibits good success in modeling the highfrequency electromagnetic behavior o f HTSs, and further explains how to use it to model
the behavior o f different microwave circuits. Three applications are presented by way of
examples o f real HTS circuits for which measurements are compared with theory. The
idea o f HTS waveguides is also explored, showing some o f the possibilities in this area.
A wide variety o f CAD software packages have been used in writing this thesis.
Whenever I could, I avoided writing my own programs and preferred to modify existing
programs and CAD tools in order to use them in my application. Wherever I use a CAD
software package I try to explain why I used that particular one and point out possible
caveats for the use o f alternatives (which in many cases I have tried to use). M ost o f these
software packages are popular and well known either in the microwave modeling
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community or in the mathematics community.
I attempt a partial list, with a short
description, o f these in table 1:
Name
Company
Use
Touchstone
E E sof
Microwave CAD and modeling using "recipes."
EM
Sonnet
Planar Microwave Circuit Analysis using the
method o f moments.
MathCAD
MathSoft
Mathematical CAD.
Matlab
Mathworks
Matrix mathematical CAD.
Mathematica
Wolfram Research
Symbolic Mathematical CAD.
cc under
CRAY Research
The standard C compiler 3.0 on a CRAY Y-
UNICOS 7.0
cc under
MP2E main-frame computer system.
CRAY Research
UNICOS 7.C.3
The standard C compiler 3.0 on a CRAY C98
main-frame computer system.
Table 1 Software used in this thesis.
1.2 An Outline of the Thesis
The areas discussed in this thesis are as follows.
A mathematical statement o f the
implications o f passivity is presented in chapter 2. Chapter 3 is devoted to the modeling of
quasi-TEM transmission lines under different ratios of conductor thickness to field
penetration depth.
Chapter 4 is devoted to an experimental verification of the model
presented in chapter 3. Chapter 5 presents the model o f chapter 3 fit to the measured data
from a real microwave low-pass filter (LPF). Chapter 6 presents the model o f chapter 3 fit
to the measured data from a real microwave band-pass filter (BPF). Chapter 7 presents
some measurements o f the non-linear behavior, with respect to input power, o f the device
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5
of chapter 5, and a possible extension o f the model of chapter 3 into the non-linear
regions. Chapter 8 presents a finite difference solution o f Maxwell's equations in a HTS
closed rectangular waveguide. The modeling is performed using a CRAY supercomputer
and employs a finite difference approximation o f the Helmholtz electromagnetic wave
equations.
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CHAPTER 2
SUPERCONDUCTIVITY AND PASSIVITY
2.1 Introduction
This chapter will attempt to clear up some misconceptions that the prefix "super" in the
name o f superconductors may have created. Frequently, the fallacious assumption is made
that, by substituting normal conductors for superconductors in passive microwave circuits,
all unwanted losses will magically disappear and the ideal minimum "noise figure"
contribution for a given circuit can be achieved. While it is true that in the majority of
cases the losses (o f ohmic nature) in a given circuit decrease when superconductors are
substituted for normal conductors, there are fundamental physical limitations imposed by
passivity (i.e., the lack o f active, energy-producing devices in the circuit) on the
performance o f a given circuit. These limitations will be quantified in the chapter below
and a numerical example, o f what these limitations mean for the case o f a 3-port network
will be furnished. All superconducting circuits considered in this thesis are passive. The
following analysis will show that they are superconducting; yet not supernatural.
2.2 An Analytical Statement of Passivity
A network (superconducting or not) is considered passive when the power incident onto it
is greater than or equal to the power reflected from it, for all possible excitations.
A
network described by the 'a' and 'b' wave parameters is shown in figure 1 below (using a
3-port network for depiction purposes).
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The a-waves are the incident waves
a3
and the b-waves are the reflected
waves, at each port.
b3
These are
normalized so that, for example,
1 , 2,
( 1)
corresponds to the power carried by
bl <
b2
Figure 1 A 3-port network.
the wave incident at port 1. It is easy
to see, then, that the total power incident to this network from all ports is
patter
(2)
1=1
where N is the total number o f ports o f the network.
Similarly, the total power scattered by the network is
(3)
An alternate way o f writing the above equations in matrix notation is
= - a +a
P»
(4)
(5)
where a - ( a 1,a2,...,aN)1 and b=(bi,b2,...,b ^ )r and the "dagger" notation is used to denote
conjugate-transpose. By the definition o f the S-matrix w e have
b = Sa
(6 )
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where S is an N by N matrix characteristic of the network.
Combining the last 3 equations gives the total dissipated power as
(V)
where I have defined the dissipation matrix, Q, o f the network. For any passive network
(superconducting or not) it must be
(8)
Equations (8) and (7) imply that the matrix Q must be non-negative real (i.e., the
quadratic form a +Q a must be a non-negative real number V a , or Q must be a positive
semi-definite matrix).
Let us examine the implications o f this on Q.
To draw our
conclusions we will use the following theorems from matrix theory:
T heorem 1
Every hermitian matrix has real eigenvalues.
Theorem 2
A hermitian matrix has non-negative eigenvalues if and only if it is positive semi-definite.
Theorem 3
For every hermitian matrix there exists a complete set o f orthonormal eigenvectors.
Clearly,
S 'S
is
a
hermitian
(S+S)+ = S " (S +) ' = S +S and
and
positive
a+(s+s)a = ( a +S H)(S a)
semi-definite
matrix
= (S a )+(Sa) = b b > 0
(proof:
Va, since
the last expression is the square of the norm of the vector b and therefore non-negative).
Hence, by Theorems 1 and 2, the eigenvalues o f S +S are all real and non-negative, i.e.,
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9
A f s >0,
Vi .
(9)
The matrix Q is also hermitian (proof: Q + = (l-S +s)+= I +- ( s +s)+ = I - S +S = Q ) and
its eigenvalues are given by
J l? = l-tfs
Vi .
(10)
Proof:
Assume
is an eigenvalue o f Q.
Hence it must be that
det(Q - 2 ? l) = det( I —S +S - t i } l ) = - d e t[s+S - (l - X f ) l] = 0
i.e., 1-/1? is an eigenvalue o f S+S, q.e.d.
Using (9) and (10) we conclude that
2?<1,
V/ .
(11)
Hence, using Theorems 1 and 2, the hermiticity o f Q and equation (11) we can state the
necessary and sufficient condition o f passivity (equation (7)) in the following theorem.
Theorem
Passivity <=> 0 <
< 1 Vi .
(12)
(Actually the right part o f the right-hand-side o f equation (12) is guaranteed, as has been
proven in equation (11), but it is a good check on any calculations).
Equation (12) states, in words, that the eigenvalues o f Q (which are real since it is
hermitian) must be between 0 and 1 (greater than or equal to zero because of passivity and
less than or equal to one because o f the positive-semi-definiteness o f S+S).
An alternate proof o f the above theorem, which gives more insight into the physical
significance o f the eigenvalues and eigenvectors o f Q is as follows:
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10
Let us assume that we excite the network o f figure 1 with an incident wave a,, an
eigenvector o f matrix Q, o f power Pjnc, that corresponds to an eigenvalue 2?.
By
definition, a , must obey the following two equations:'
—a»a, = Pinc
(13)
and
Q a .= / ! ? a . .
(14)
Substituting (13) and (14) into (7) we obtain
Pdis = ~ a»Q a* ^ a U ? a , = ^ Q - a L a . j = ?%Pin
(15)
By the definition o f dissipated power we have
OZPoZP*
( 16)
•
Substituting (15) into (16) we obtain the desired result, as expressed in equation (12).
Equation (15) provides a good physical interpretation o f the significance o f the
eigenvalues of matrix Q. An eigenvalue of Q is the fraction o f the incident power that is
dissipated in the network, when the latter is excited by the eigenvector corresponding to
that eigenvalue. Equation (15) also tells us how to minimize the power dissipated in the
network:
Excite the network with an incident wave that is an eigenvector of Q that
corresponds to its minimum eigenvalue.
In fact, if Q has a zero eigenvalue, i.e., it is
singular, it is possible to excite the network in a way that no power is dissipated (with the
eigenvector that corresponds to the zero eigenvalue).
2.3
2.3.1
Im plications of Passivity. An Example.
T he C ircuit
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11
This example is relevant to a current effort in industry to fabricate multi-way power
dividers and combiners for array-antenna applications.
The unit-cell o f most o f these
dividers is a 3-port or a 4-port with one port terminated (branch-line, rat-race, Wilkinson
couplers, etc.). As the size o f the array-antenna increases, so do the required "levels" of
power division (as the base-two logarithm o f the size). Therefore, the insertion loss of
these unit-cell devices, which are cascaded in "levels," becomes an important concern and
design parameter. The examples o f unit-cell devices mentioned above all provide isolated
output ports which are matched to 50 Ohms. In other examples o f corporate multi-port
power dividers without isolation, the output ports are not matched to 50 Ohms. These
work well with passive arrays, where the antennas have input impedances close to 50
Ohms, but are not suitable, say, for driving the amplifiers o f active arrays, since the
amplifiers want to "see" a 50 Ohm impedance at their input. Therefore, these devices,
which are not 3-ports, will not be considered.
Following the trend, an idea which JPL is considering for reducing the size o f the conical
receiving horns in its Goldstone huge (34 m diameter) antennas, is using microstrip patch
array antennas with superconducting beam forming networks at cryogenic temperatures.
As calculations will show there is a minimum insertion loss, largely dictated by the
geometry o f the circuit used, below
which
a
passive
circuit
cannot
Port 1
Zo
operate.
Let us consider, then, a 3-port
network as an example.
A branch-
Port 2
line coupler with its isolated port
terminated with a 50 Ohm load
(figure
2).
One
o f the
three
Figure 2 A branch-line coupler.
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Port 3
i
12
remaining ports (say port 1) is the input and the other tw o (say ports 2 and 3) the outputs.
In an idealized model, where all the lines are exactly one quarter o f a wavelength long at
the design frequency and lossless (no ohmic losses), the isolation between ports 2 and 3
can be analytically shown to be infinite (i.e., S23=0). However, the realities o f building
the circuit on a substrate are different. On the actual mask which is used to fabricate the
circuit one can see that, at the point where the (mutually) perpendicular quarter-wave lines
join, there is a T-junction. This T-junction has dimensions itself and cannot be considered
a lumped element.
W hat this means is that the microwave current density is spread
throughout the T-junction and does not go through it only at one point. There are an
infinite number o f linear paths along which the phase length o f the current is 90 degrees
and there are also an infinite number of linear paths along which the phase length is slightly
different from 90 degrees. In the actual response o f the circuit this effect shows up as a
"broadening" and a "shallowing" o f the infinite well that the graph o f S23 ideally exhibits
about the design frequency. This effect is independent o f ohmic losses (i.e., occurs on
both normal and superconducting circuits) and means that S23 is very small, not 0, at the
design frequency. This rationale, which is geometry dependent but not material properties
dependent, allows us to set a lower bound on S23.
This bound, together with the
passivity constraints on the Q matrix, yield constraints on the insertion loss of this circuit
which are not material dependent. It does not matter how lossless the material is, the
constraints will be there.
2,3.2 The Calculations
The software package Matlab, by Mathworks Inc., is used for the analysis. Matlab is
preferred because it is optimized for matrix computations and quite accurate in eigenvalue
problems. An initial attempt to use MathCAD, by MathSoft Inc., failed because the rootfinding accuracy o f the package, to solve the characteristic equation and obtain the
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13
eigenvalues, was not good enough. Three different cases are analyzed. A sample Matlab
file for the first case is included as appendix A o f this chapter. The methodology is the
same in all three cases:
1.
Assume a form for the S-matrix o f the network.
2.
Compute the eigenvalues of the Q-matrix as a function of insertion loss and
isolation.
3.
Plot, in the two-dimensional space defined by the insertion loss and the isolation,
the curve demarcating the region where the 3-port is passive (i.e., realizable with
passive components) from the region where it is not. (i.e., plot the locus o f points
that lie where the minimum eigenvalue o f Q crosses from positive to negative
values).
An alternative method to determine the physically achievable region, where Q is passive, is
to find the locus o f points, in the isolation-insertion loss space, that make the matrix Q
singular. This approach should, however, be taken with caution to avoid trivial roots of
the characteristic polynomial of Q.
Case 1. Perfectly matched 3-dB coupler w/ finite isolation and insertion loss.
The (symmetric) S-matrix is assumed to have the form
a
0
n
s =
a
12
a
a
VT
0
X
X
0
a : Insertion loss, x: isolation
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14
The diagonal elements o f the matrix (SI 1, S22 and S33) are assumed zero, (i.e., the
device is perfectly matched).
S21 and S31 would ideally have the value 1/V 2 (3 dB
coupler) and a is the additional loss in excess o f the ideal division loss (insertion loss).
(dB)
-2
-3
-4
-5
-5 0
-4 5
-4 0
-3 5
-3 0
-2 5
-2 0
-1 5
-1 0
-5
isolation (dB)
Figure 3 Trade-off between insertion loss & isolation.
The result o f the analysis is shown in figure 3. The horizontal axis is the negative o f the
isolation (i.e., S32, since isolation is defined positive) in dB.
The vertical axis is the
negative o f the insertion loss. On the plotted curve, the minimum eigenvalue o f matrix Q
is exactly zero. Therefore, the minimum possible dissipated power, for this network, may
be achieved by exciting the network with an eigenvector o f Q that corresponds to this
zero eigenvalue.
The equation o f the zero-eigenvalue locus o f points that make zero
dissipation possible, for this form o f S-matrix, is
a= Jh^x
.
(17)
The corresponding normalized, unit-power eigenvector is
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The "achievable" region for a passive circuit is below the curve.
A typical value of
isolation to be expected by the circuits mentioned in above is about 20 dB (seldom more
than 30 dB). Here I treat the isolation as the "known" and read what the achievable
insertion loss is for this 3-port. The point of comparison will be 18 dB isolation. In this
case, for 18 dB isolation, the minimum achievable insertion loss is 0.59 dB.
As the
isolation tends to infinity (i.e., x=0 in the S-matrix) the minimum insertion loss tends to 0
dB, as is to be expected for an ideal circuit.
Figure 4 is a plot of the power in each of the three components of the unit-power,
zero-dissipation eigenvector (equation (16)) versus the isolation. Typically, in microwave
0.9
Pori 1
Pori 2
0.7
Port 3
S 0.5
0.3
0.2
-50
-45
-40
-35
-30
-25
-20
15
-10
•5
Isolation (dB)
Figure 4 The power, in each o f the ports. for zero dissipation.
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16
circuits, the excitation to the network is a wave incident to port one, the input port.
Hence, the closer the minimum-loss eigenvector (equation (16)) is to the vector
( a/2
0 o), the closer we can come to realizing the zero-dissipation condition.
Case 2. Imperfectly matched 3-dB coupler.
The S-matrix was assumed to have the form
0.1
s=
a
41
a
a
a
a
0.1
X
X
0.1
a
a : Insertion loss, x: isolation
for sub-case i. and
0.1
S=
a
a
41
Z -9 0
Z -9 0
- j= Z .- 9 0
a/2
0.1
x Z -1 8 0
-^ L z-9 0
a/2
x Z -1 8 0
a: Insertion loss, x: isolation
0.1
for sub-case ii.
In this case, a 20 dB return loss on all ports is assumed. Figure 5 shows the results of the
analysis for these matrices. The lower and upper curves show the analysis results for sub­
cases i. and ii. respectively.
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17
case n
case
1
(dB)
-2
—3
-4
-5
-5 0
-4 5
-4 0
-3 5
-3 0
-2 5
-2 0
-1 5
-1 0
-5
isolation (dB)
Figure 5 Trade-off between insertion loss & isolation.
_________________________
Sub-case i.
The minimum achievable insertion loss at 18 dB isolation increases to 1.56 dB. However,
this case is too restrictive as all the components o f the S-matrix are "forced" to be in
phase.
It is instructive however to note that the trade-off between insertion loss and
isolation also depends on the required phase through the circuit. In this case the minimum
insertion loss tends to 0.92 dB.
The equation o f the zero-eigenvalue locus o f points plotted for this sub-case is
a = — V S1-90X
(19)
10
and the unit-power, zero-dissipation eigenvector is
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18
' 11 8 - 2 0 * "
V 1 8 -1 0 *
J
■Vl 8 - 1 Ox
J
vV l8 - 1 0 x y
Figure 6 is a plot o f the power in each o f the three components o f the unit-power,
zero-dissipation eigenvector (equation (20)) versus the isolation.
0.9
Port 1
0.8
Port 2
0.7
—
Port 3
Q_
0.5
§ 0.4
0.3
0.2
-50
-45
-40
-35
-30
-25
-20
-15
-10
Isolation (dB)
Fi»ure 6 The power, in each o f the ports, for zero dissipation.
Sub-case ii.
The phases o f the S-matrix components are set to the values o f a 071 SO1' rat-race coupler.
The minimum achievable insertion loss at 18 dB isolation is 0.56 dB.
As the isolation
tends to infinity (i.e., x=0 in the S-matrix) the minimum insertion loss tends to 0.04 dB
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19
(which corresponds to the expected loss, in the ideal case, due to the reflected power
)S 11 p). Hence, an imperfectly matched 3-port will always have some insertion loss higher
than the ideal which corresponds to the reflection losses at the input.
The equation o f the zero-eigenvalue locus o f points plotted for this sub-case is
1
a = — ~j99-9Q x
(21)
10
and the unit-power, zero-dissipation eigenvector is
.
-J
1 1 - 10 *
\
1 0 -5 *
(22)
^ 2 0 - 10 *
a/ 2 0 - 1 0 x
0.9
Port 1
0.8
Port 2
0.7
Port 3
0.6
Q_
0.5
0.4
0.3
0.2
0.1
-50
-45
-40
-35
-30
-25
-20
-15
-10
Isolation (dB)
Figure 7 The power, in each o f the ports. For zero dissipation.
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Figure 7 is a plot o f the pow er in each o f the three components of the unit-power,
zero-dissipation eigenvector (equation (22)) versus the isolation.
Case 3. 2:3 coupler w/ phases from a measured Wilkinson type coupler.
To relax the constraint that all the elements o f the S-matrix are in phase, the measured
phases o f all the S-matrix components o f an actual 2:3 Wilkinson power divider, centered
at 30 GHz, are used. The assumed S-matrix is
with the usual definitions o f x and a .
The results o f the analysis are shown in figure 8,
below
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21
-1
(dB)
-2
-3
-4
-5
-5 0
-4 5
-4 0
-3 5
-3 0
-2 0
-2 5
-1 5
-1 0
-5
isolation (dB)
Figure 8 Trade-off between insertion loss & isolation.
The minimum insertion loss at 18 dB isolation is 1.1 dB. The minimum as isolation tends
to infinity is 0.88 dB. The above results are not sensitive to "adding line lengths at the
input and output ports."
The equation o f the zero-eigenvalue locus o f points plotted for this sub-case is
, ,
,
19. 9 9 -10 - 8 1 3 . V - 1. 00-10 .v - 5 1 3 . r a
- j .
3 . 7 3 - 1 0 - 1 . 4 7 • 10 .V+ 8 . 5 6 - 1 0 . v + 2 . 9 5 - 1 0 .v
/
V-1.84 • 1O' V +1.20 • 10sXs + 9.5 0 ■10V
,= ...=
=
-------------------------------------------------------------------------------------------------
V99080-164.1x-96000:t2
(23)
The algebraic expression for the zero-dissipation eigenvector is too complicated and is
therefore not included. Figure 9 is a plot of the power in each o f the three components o f
the unit-power, zero-dissipation eigenvector versus the isolation.
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22
0.9
Port 1
Port 2
0.7
“ Port 3
Q.
£ 0.4
0.3
0.2
0.1
-50
-45
-40
-35
-30
-25
-20
-15
-10
Isolation (dB)
Figure 9 The power, in each o f the ports, for zero dissipation.
2.4 Conclusions
This chapter has attempted to show that there exist more fundamental considerations than
just dielectric loss tangent and conductivity o f metallization limiting the performance of
passive multi-port networks. The example o f 3-port couplers was used. In cases where
bounds or restrictions can be set on certain parameters as a result o f considerations
independent o f ohmic losses, there are frequently additional restrictions on the
performance o f the network, imposed by passivity, that need to be considered.
In the
example above, insertion losses are imposed on matched 3-ports by passivity requirements
and there is nothing that can be done about them. If they are unbearable to the design
engineer, then other alternatives have to be considered. In particular the above results
also show that a matched divider without isolation is very lossy (see figs. 2-4 @ 5 dB
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23
isolation). The contra-positive o f the above statement is that if a divider without isolation
has low insertion loss, it cannot have low return loss on all ports.
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A ppendix A
Sample Matlab File
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25
clear;
dg;
i=l;
S(1,1)=0;
S(2,2)=0;
S(3,3)=0;
for x=-50:l:-5
for alpha=-5:0.02:0
alpha_m ag= 10 A(alpha/20);
S(2, l)=alpha_mag/sqrt(2);
S(3,l)=alpha_m ag/sqrt(2);
S(1,2)=S(2,1);
S(1,3)=S(3,1);
x_m ag=10A(x/20);
S(2,3)=x_niag;
S(3,2)=S(2,3);
Q=eye(3)-S'*S;
i=eig(Q);
if 1(1) <=0 11(2) <=0 11(3) <=0
g (i,i)= x ;
g(i,2)=alpha;
break;
end;
end;
i= i+ l;
end;
g
axis([-50 -5 -5 0]);
pIot(g(:,l).g(:.2));
xlabel('isolation’);
ylabeI('insertion loss');
grid;
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26
CHAPTER 3
LOW FIELD M ODELING OF QUASI-TRANSVERSE ELECTRIC
AND M AGNETIC (TEM ) HTS M ICROW AVE TRANSM ISSION
LINES
3.1 Introduction
M ost commonly used types o f microwave transmission line are o f the TEM or quasi-TEM
type.
Examples are microstrip, stripline, coplanar waveguide (CPW), slotline and
microshield. For all the above examples o f transmission lines, the fundamental (lowest
order) propagating electromagnetic mode has small field components along the direction
o f propagation o f the wave.
Two advantages of TEM modes are that they have good
dispersion characteristics (i.e., they are suitable for distortionless broadband transmission)
and they have no low frequency cut-off.
This chapter presents a phenomenological
mathematical model which permits effective modeling o f HTS transmission lines using
CAD tools designed for non-HTS circuits.
In usual microwave circuits the transmission line conductor thickness is large compared to
the skin depth (or depth o f penetration o f the fields into the conductor) As an example, a
common metallization thickness is 17.8 pm (0.0007 inches or 0.7 mils) and the skin depth
o f copper at 10 GHz is 0.66 pm. In this limit the surface resistance of the transmission
line is proportional to the square root o f frequency, a simple explicit function o f frequency
[1]. In a similar limit in HTS transmission lines, surface impedance is proportional to the
square o f frequency, again a simple explicit function o f frequency, as will be shown in
section 3.3. Unfortunately, typical film thicknesses o f HTS circuits (e.g., 500 nm) are,
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27
because o f manufacturing process limitations, o f the order o f the penetration depth o f the
fields into the superconductor at cryogenic temperatures (e.g., for Yttrium Barium Copper
Oxide (YBCO) with zero temperature penetration depth, Z0, o f 140 nm and critical
temperature, Tc, o f 85 K the penetration depth at 77 K, A(77), is 429 nm). In this case the
range o f integration o f the integral in section 3.3 cannot cover the whole semi-plane and
the dependence o f surface resistance on frequency deviates from the square law and
becomes more involved.
The response of this typical type o f HTS transmission line is
addressed in section 3.4.
3.2
The Two-Fluid Model of a Superconductor
A simple, yet powerful and commonly used model o f a superconductor is presented in this
section. It is fundamental to an understanding o f the rest o f this thesis and will therefore
be presented fully. In this model the superconductor is visualized as tw o fluids made up of
two kinds o f charge carriers. One fluid consists o f the "normal" electrons, which will be
denoted by the subscript n, and are the electrons found in a normal conductor. They are
responsible for scattering and therefore for Ohmic losses. The second fluid consists o f the
"superconducting" electrons, which will be denoted by the subscript s, and are the lossless
carriers responsible for superconductivity. They do not scatter but are accelerated by the
electric field in the same way as normal electrons. The purpose o f this section is to arrive
at an expression for the conductivity o f a two-fluid modeled superconductor.
Newton's law applied to the accelerating superelectrons due to the electric field E gives
d\
m — - = -eE
dt
(1)
where m is the mass o f the super-electrons, e is the magnitude o f their electric charge, E is
the applied electric field, vs is the velocity o f the super-electrons and / is time.
The same law applied to the normal electrons can only be applied in an average sense,
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28
(denoted by the <>), because o f the randomly varying velocity o f the electrons due to
collisions, and has to include a "damping" term due to scattering:
m
d
<v >
< v>
_
2— + m
— = -eE .
dt
(2)
t
Here z is the characteristic time between collisions.
Assuming a steady-state sinusoidal excitation, o f the form e J<0‘, where (o is the angular
frequency o f the excitation, the above equations become
jm c o \s = -e E
(3)
jm c o < \n > +m
<v >
— = -eE .
T
(4)
By the definition o f current density we have
J s = -"A V,
(5)
J n =- / / „e< v n >
(6)
where J is current density and ii is carrier density (i.e., number per unit volume).
Although superconducting electrons are paired (in Cooper pairs [2]), here we count two
electrons per pair, i.e., we still consider the electron, and not the pair, to be the
superconducting carrier.
The total current is, therefore, given by the sum o f the super-current and the normal
current, as calculated using equations (3), (5) and (4), (6) respectively.
v
*
,
( >Ke
J = J n + J , = e -r*-------
\jm < o
^
E .
(7)
m {j(0 + t ) J
Hence, the conductivity, which is by definition the ratio o f current density to electric field,
is given by
n a2 t
J
g = ~ = —E ffl(l+(BT)
0 /
. (>
t'~“ rf
----111(0
2-> 2o
(o r “
/l +, o)2 r 2)\
(1
\1
(8)
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29
where (7) w as factored into its real and imaginary parts.
The collision characteristic time constant, x, is o f the order of 1 0 '14 seconds so for
frequencies below 100 GHz we have (cur)2 « 1 . Also, within the scope o f the two-fluid
model it will never be /?s « nn (although we may approach this condition in chapter 7).
Hence, (8) may be simplified as
o
- =
e 2m„ j r e 2
r , - ; ( T ■
m
nm
c
The normal bulk conductivity of a material is given by
ne2r
crn =
.
m
, ,
(9)
(10)
Hence, with the help of (10), we can rewrite the real part o f (9) as
n
■
(1 1 )
Here n is the total carrier density, so n = ns + nn. Experimentally, we know that [2]
( rn\4
Wn -
( 12)
where "1" is the temperature and "7^" is the critical temperature o f the superconductor,
i.e., that temperature above which superconducting phenomena disappear. Hence (11)
can be expressed as
(13)
The real part o f the conductivity, cr,, has now been expressed in "readily measurable"
explicit physical parameters. In order to do the same for the imaginary part, cr,, we need
to do a little more work.
The starting point is London's first equation for superconductors [3],
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30
( 14)
where
A=
and
(15)
jli0
is the magnetic permeability o f vacuum. Equation (15) may be used to substitute
some o f the parameters o f the imaginary part o f (9) that do not correspond to explicit
physical parameters, provided A itself has an intuitive physical interpretation.
To show
this we need Maxwell's curl B equation, assuming displacement and normal currents
negligible.
V x B = //0J s
(16)
Taking the curl o f (14) and using the vector identity V x V x B = V ( V - B ) - V 2B and
Maxwell's equation V • B = 0 we obtain
-V 2B = //0V x J
.
(17)
Substituting (14) into (17) gives
Assuming a one-dimensional coordinate system where, say, V2 —>— - , the classic
dz~
solutions to this second-degree equation are o f the form e^-.
Assuming z > 0 , the
solution corresponding to the plus sign does not make physical sense, and the solution
corresponding to the minus sign describes an exponentially attenuating wave that
attenuates to 1/e o f its original value at depth A.. A. is called the penetration depth o f a
superconductor. It is a function of temperature, but not o f field strength or frequency. It
is important to distinguish between this penetration depth, which is a DC phenomenon and
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persists even when co = 0, and the skin depth o f a normal conductor. The skin depth of a
superconductor, associated with AC fields, will be derived in the next section.
Substituting now (15) into (9) we get
1
1
cr,
co/i0A2 2 nff.iQX-
(19)
Experimentally, it has been shown [3] that X has the following temperature dependence
An
A(T) = 1where
(20)
Ty
is the zero tem perature penetration depth. Finally substituting (20), (19) and
(13) into (9) we have the sought after expression for the conductivity o f a superconductor
< 7= C7„
/
\4
i - i
T
.
( 21 )
■J:
In fH o K
which is only valid when T <TC. (21) has to be qualified before being used. The model
and theory used to derive it are linear, local and low field theories [3]. The validity o f the
expression should, therefore, not be pushed beyond these limits when accuracy is required,
although, as will be shown in chapter 7, (21) can be stretched into these limits with some
partial success.
To better understand what the significance o f the complex conductivity postulated by (21)
is, we take a look at Poynting's equation [1],
V - ( e x h ) = - — f - |e |2V —
aW ) a
ej
(22)
Here I have used lower case letters to represent real quantities (as opposed to phasors),
with the usual field notation.
The last term represents the power density converted to
heat. When we switch to phasor quantities we can time average the quantity c -j by taking
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32
^-Re(E- J"). Hence, the average power density lost to heat becomes
- R e E - ( a ,- ./c r 2)‘ E ‘ = —cri|E|2 .
(23)
2
Hence, the smaller cr,, the less power is dissipated into heat. For a given magnitude o f
conductivity, |aj =
is.
the closer the phase to —j , the closer to lossless the circuit
Examination o f equation (21) reveals that a lossy transmission line has the
dependencies listed in table 1. Equation (21) will be used in the next section to derive a
result for the surface resistance o f a bulk superconductor.
When this variable increases...
The circuit losses...
<*«
Increase
T
Increase
Tr.
Decrease
f
Increase
Increase
T able 1 Dependencies o f losses on physical variables.
3.3 Surface Impedance of a Bulk Superconductor
Let us consider a HTS material filling the half-space z > 0.
Let a uniform x-directed
current flow in the material, with the current density at the surface being J0. Then the
equation governing the distribution of J into the superconductor is [4]
•
— ^ = JojjuaJ^
oz~
(24)
The (bounded) solution to this equation is
-K, =
(25)
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33
where
( 26 )
G= y l M ‘0v = 0 + i ) V xfMo& ■
Substituting (9) into (26), after some algebraic manipulations we get
. (o)r)nn
1+ / - —
2n.
X
( f in X ] _ 1
C = \ / * o — f a + j ( w K ) = J / ‘o - ”, 1+7
m
\|
m
ns
7
where the rest o f the terms o f the Taylor expansion can be neglected since
(27)
(cot)2 «
1.
After algebraic manipulations and substitutions from (10), (11), (12) and (15) we get
r j,\4
(28)
T j
Inspection o f (28) reveals that the skin depth of the superconductor is equal to the
penetration depth, and is independent o f frequency.
To find the surface impedance we need to integrate the current density in (25) from zero
to infinity to find the total current per unit width, J„„ flowing in the region z > 0.
(29)
£
But, by the definition o f conductivity we have J 0 = o E ^ , therefore
(30)
C
IW o
and by the definition o f surface impedance
jm >
— { n „ a z - jn t )
mco
J”s
mco
1 + j — cor
ti.
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34
f
\
,
( r^4
Z, = jco/.i0k 1 - j —^—cot = —//g2®2/l.3c — +jo)ju0X s R s +j(oLs
2»s J 2
"U ,
(31)
where a Taylor expansion has again been used and quadratic and higher terms discarded.
As promised, the surface resistance is proportional to the square o f the frequency. The
surface inductance is proportional to the penetration depth.
Equation (31) holds true when two limits are satisfied. The first is obvious from the above
analysis (equation (29)), and is that the thickness o f the HTS film must be infinite.
Effectively what this means is that, if d is the HTS film thickness, the ratio d /k must be
large (for example if d/k=2, R s is underestimated by 15.9% and L s is underestimated by
3.7%). The second assumption that must hold is that the HTS film must lie on a semiinfinite dielectric material. If this is not the case, the presence o f a ground plane (in the
case o f a two conductor quasi-TEM transmission line) on the bottom o f the dielectric slab
changes the effective surface impedance further [5], and leads to complicated expressions
that are impractical for design use.
The next section presents a phenomenological loss equivalence model that solves this
problem and can be used with any microwave CAD program.
3,4 A Phenomenological Loss Equivalence Model for Quasi-TEM HTS Microwave
Transmission Lines
Due to the similarity between a normal conductor and a superconductor the lossless Sparameters o f two identical circuits, one made with a "perfect" conductor, i.e., one that
exhibits zero surface impedance, and the other with a superconductor, are analogous.
This means that one may start to model an HTS transmission line and calculate its
electrical length and impedance from its geometrical dimensions using the standard
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35
formulas for perfect conductors [6], This analogy breaks down because there are losses,
resulting from the non-zero surface resistance o f equation (31), which must be included in
the calculation o f the microwave S-parameters o f the transmission line.
Because the
importance o f HTS transmission lines lies in reducing losses relative to normal conductors,
this is a very important difference.
The surface resistance o f a regular conductor is
proportional to the square root o f the frequency [1], However, the surface resistance o f a
superconductor, according to equation (31), is proportional to the square o f the
frequency. The superconductor may be modeled as a "normal" conductor with a complex
conductivity. The real and imaginary parts of the conductivity are the result o f the normal
electrons and superconducting electron pairs (Cooper pairs) respectively, as posed in the
two-fluid theory.
For convenience I reproduce here equation (21) for the complex
conductivity.
rTV
4
<r= cr
1-
'T '
2
(21>
where o n is the normal part o f the conductivity, T is the absolute temperature, Tc is the
critical temperature o f the superconductor, f is the frequency, p 0 is the magnetic
permeability o f vacuum and XQ is the zero-temperature penetration depth o f the magnetic
and electric fields into the superconductor. Using this conductance one may use Lee &
Itoh's phenomenological loss equivalence (PEM) method to calculate the additional
distributed internal impedance Zj, in Ohms/meter, due to the penetration of the fields into
the superconductor and the related surface impedance [7] as follows.
Z, = ZsGcoth(CGA)
(32)
where
£ = V - W ) 0- = 0 + J ) ^ . U 1oa
(26)
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36
is the complex propagation constant o f an electromagnetic wave propagating in a bulk
superconductor (see above discussion in section 3.3),
(33)
is the complex surface impedance o f an electromagnetic wave propagating in a bulk
superconductor (see equation (31) above), G is the incremental inductance geometric
factor, i.e., the partial derivative o f the inductance o f the line with respect to the receding
walls o f the line and A is the cross-sectional area o f the line under characterization. The
exponential attenuation coefficient follows directly from the above, as [1]
in Nepers/meter, where Z0' is the impedance corrected using Zj.
This series of calculations is easy to perform numerically, with any mathematical CAD
program for every different set o f values o f the parameters, but gives no insight into how
each individual parameter affects the attenuation. M oreover they are impossible to enter
into most popular microwave CAD software packages, which cannot handle complex
algebra, for circuit design and optimization.
The equations were, therefore, reduced
algebraically to obtain the following explicit formulae for the additional distributed internal
resistance R j and reactance X j .
Ri ~ Re(Z,.) =
,22JcosO
(35)
and
(36)
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37
where
(
|o| = V o f + o f is the magnitude o f the conductivity, </>= arctan
cr,
^
is the phase of the
\ °\ J
conjugate o f the conductivity,
4
2
B = G A ^ 2 n f/j0\c\
y /= ^j[e2Bcos0cos{2Bsm e ) - \ f + [e2Bcos0sin(25sin d ) f
and
X = arctan
e”
sin (2 £ sin 0 )
e 2Bcos0 cos(2B sin 6>)-l
To calculate Z0', the corrected characteristic impedance of the transmission line, we
proceed as follows. Let L ' be the corrected distributed inductance o f the transmission line.
Then
•
<37)
where C is the distributed capacitance o f the transmission line, which remains unaffected
by the field penetration into the superconductor. But,
L' = L +L = L +- ^ ~
2n f
,
(38)
where L is the distributed inductance o f the transmission line, as calculated assuming a
perfect conductor, before applying the PEM. Now, to calculate L we need to express
them as
i =j-V Ic =^- =^/fsL
(39)
and
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where Z 0 is the characteristic impedance o f the perfect conductor transmission line, s cff is
the effective relative dielectric constant o f the perfect conductor transmission line and c is
the velocity o f light.
Substituting (38), (39) and (40) into (37) gives
X,
Z ' = Z J 1+
2S
Sc f f f
(41)
Z 0
Likewise, using (38), (39) and (40), the corrected phase velocity may be calculated as
1
' ,,h
c
417c
£eff
1
(42)
X,
1+.
f
2 ^ / f cjy f
and the corrected effective relative dielectric constant as
(
\
(
\2
c
,
c
X,
---1 + ----- = = ----- £ eir
e€
{
2 ^ Se fff Z0 )
<’> )
(43)
From (42) the corrected propagation constant may be calculated as
p i,
~
y
X: = 0 1 +
Z,
°
X
(44)
Z0
It is important to note that equations (34), (41), (42), (43) and (44) are only approximate
equations that work well only when R j is small relative to coL. This is true o f most HTS
transmission lines in their low-power linear region. However, examples o f some cases in
which this condition may be violated are if T « Tc or if, say due to high transfer currents,
n
d
— ~ 1, or if the HTS film isso thin that 0 < — « 1. In such cases,it is worth usin" the
n
X
°
more exact relations for TEM lines whichwill be presented in chapter 7.
The next chapter presents an experimental verification o f the model presented in this
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
section.
3.5 Algebraic Verification of Equations (35) and (36)
Equations (35) and (36) are the key equations from which the loss and phase parameters
o f the transmission lines are calculated. They are derived from (26), (32) and (33) after
algebraic manipulation. O f pivotal interest here is the proper choice o f the phase plane o f
the square root o f a in (26) and (33), since equations (26) and (32) have a branch-cut that
extends from zero to infinity on the cy-complex plane. To assure that this is properly done
MathCAD is used. A test additional internal impedance calculation for three different line
widths is performed using the MathCAD worksheet included in appendix A of this
chapter. The chosen line widths, taken from the filter o f chapter 5, are 6, 50 and 200
microns, with corresponding incremental inductance factors o f 125500, 23500, 17340 and
parameter values
T= 11K ,
TC= 8 5 K ,
f = 5G H z, i o = 4 0 0 nm,
a n = 1.6-106 S / m ,
(45)
where T is the absolute temperature, Tc is the critical temperature o f the HTS line, / i s the
frequency, X0 is the zero-temperature penetration depth and on is the normal conductivity
of the HTS line. The additional distributed internal resistance, Rj and reactance, X t, are
computed twice, for the same above parameters, once using (35) and (36) and again using
the real and imaginary parts o f (32). The results are identical, as shown in figure 1 and in
appendix A o f this chapter. This is a
numerical
verification
of
the
134.539
algebraic manipulations leading to
equations (35) and (36).
134.539
7 .0 55 - 10
947.013
947.013
4 85 .8 6 5
4 85.065
Figure 1 Identical results o f equations (35), (36) and (32).
3.6 References
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[1] S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication
Electronics, Wiley, New York, 1965.
[2] T. Van Duzer and C. W. Turner, Principles o f Superconductive Devices and Circuits,
Elsevier, New York, 1981.
[3] A. C. Rose-Innes and E. H. Roderick, Introduction to Superconductivity, Pergamon,
Oxford, 1969.
[4] S. E. Schwarz, Electromagnetics fo r Engineers, Saunders, Philadelphia, 1990.
[5]
P.
Hartemann,
"Effective and Intrinsic
Surface
Impedances
o f High-Tc
Superconducting Thin Films," IEEE Transactions on Applied Superconductivity, Vol. 2,
pp. 228-235, December 1992.
[6] K. Gupta, R. Garg and I. Bahl, Microstrip Lines and Slotlines, Artech, Dedham, iMA,
1979.
[7] H. Lee and T. Itoh, "Phenomenological Loss Equivalence Method for Planar QuasiTEM Transmission Lines with a Thin Normal Conductor or Superconductor," IEEE
Transactions on Microwave Theory and Techniques, Vol. 37, pp. 1904-1909, December
1989.
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41
A ppendix A
MathCAD File that Algebraically Verifies Equations (35) and (36)
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42
i := 1 , 2 . . 3
t := 0 .5 - 1 0
6-10
5 0 -1 0
-6
200-10
:= 4 *ic-10
-7
£ 0 - 8 .8 5 4 - 1 0
Ho :=
12
r i0 = 3 7 6 .7 3 4
c = 2 .9 9 8 - 1 0
XQ ■■= 4 0 0 - 1 0
8
-9
T := 77
T c := 85
f := 5 - 1 0
a v := 1 . 6 - 1 0
T
Tc
o
:=
o
V
T,
2 -* -f-H o V
o = 1 .0 7 7 -106 - 5 .1 7 -1 0 ? i
« i :=
1255001
23500
17340
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43
Aj := wj-t
5 := (1+j
Z i i : = Z s -Gi -coth[?-Gi - Ai ]
<j> := - a r g [ o ]
0 :=
5 *jc <j)
4
V;
2
A|[ c x p [ 2 - B i -c o S[ e ] ] * c o S[ 2 ' B i - s i n [ 0 ] ] - l ]
+ c x p [^ 4 - B j -c o s fe ] J*sinj^ 2 *B ; * s in [ e ] j2
cxp|^2 'B j ‘c o s f o j j ’s i n ^ ‘B j • s i n [ 0 ] J
%i := atan
exp[^2 • B i , c o s [ 0 ] j , cos|^2 ’B ^ s i n f © ] ! - :
B:
R: :=
cos
v k l ’Vi
-
A i * k l *Vi
R:
1 3 4 .5 3 9
1 3 4 .5 3 9
1 6 .2 7 3
1 6 .2 7 3
5 .4 1 6
5 .4 1 6
2
1
% (!)
i
■I
+
[
sin
Rc Z j . l
+ cxpj^2 * B j - c o s [ 0 ] ] "cos 2 *B j -s i n [ 0 ] + - + 4
r
B;
% <b
4
2
‘
+ ex p 2 *B ; *cos[ 0 j \sin 2 *B- -s i n [ 0 ] + - + 1
L J 4 2
Im
X;
7 .0 5 5 *10'
7 .0 5 5 *10'
9 4 7 .0 1 3
9 4 7 .0 1 3
4 8 5 .8 6 5
4 8 5 .8 6 5
1
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44
CHAPTER 4
VALIDATION AND APPLICATION OF THE PEM LOSS MODEL:
AN HTS M ICROSTRIP RING RESONATOR
4.1 Introduction
In this chapter the PEM loss model presented in the previous chapter is applied to an HTS
microstrip ring resonator. A resonator is chosen because it is a simple structure, whose
electrical behavior depends on a minimum number o f physical parameters, yet yields a
large amount o f information.
4.2 The YBCO Microstrip Ring Resonator
The layout of the YBCO microstrip ring resonator is shown in figure 1 below.
dimensions are in millimeters (mm).
The
It is
purchased from Conductus Inc. at a very
0 .5 0 0
discounted "student" price, for which the
author is grateful.
10.000
o.eoo T
1 .7 9 2
208 -
It consists of a YBCO microstrip ring, o f 5
0 5 .0 0 0 0
mm inner and 6 mm outer diameter (crosshatched in figure 1) and an input and
output gold microstrip straight-line section
■ 10 . 0
00-
Figure 1 The layout o f Uic YBCO microstrip ring
resonator.
o f length 1.792 mm (single hatched in
figure 1). The substrate is lanthanum aluminate and measures 10x10x0.508 mm.
The
input and output lines are electromagnetically coupled to the ring via a 0.208 mm gap.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
The characteristic impedance o f the ring is 32 Q. and that o f the input and output line
sections is 47 iQ. The S-parameters o f the circuit are measured using a Hewlett-Packard
HP 8510 C Network Analyzer with the circuit dunked in liquid nitrogen (77 K).
4.3 The Model
4.3.1 The M odeling Methodology
The physical parameters o f the resonator circuit, listed in table 1, are supplied by
Conductus.
relative
These are the thickness and
dielectric
constant
of
the
substrate, and the thickness and critical
P aram eter
V alue
Substrate Thickness
0.508 mm
Substrate Dielectric
Constant
YBCO Film Thickness
24
0.4-0.6 pm
temperature o f the YBCO film. These are
YBCO Film Critical
Temperature
85 K
used in conjunction with the various
T able 1 The physical parameters o f the microstrip.
transmission line dimensions (i.e., width,
w) to arrive at the electromagnetic parameters o f each type o f line used in the circuit.
There are two types o f lines used in the resonator circuit, the input and output straight
line sections and the ring line.
The
electromagnetic parameters of these lines,
i.e., effective relative dielectric constant,
~
sf
and characteristic impedance, Z0,
L ine...
w (mm)
sf
z 0 (O)
47.0
32.1
In/Out
0.2
14.7
Ring
0.5
15.7
T able 2 The electrical parameters o f the two types
o f line.
listed in table 1, are first calculated, using standard microstrip formulas [1], These are
entered into a MathCAD worksheet, included as appendix A of this chapter, to arrive at
the parameters o f table 2. Although the value given for the relative dielectric constant of
the lanthanum aluminate by Conductus is 24.0, there is a range of values used in the
literature, from 23.5 to 24.5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
Accordingly, to gain some insight into how the line parameters are affected by this
uncertainty, appendix A includes the calculation o f the line parameters listed in table 2, for
a range o f values o f relative dielectric constant from 23.0 to 24.9 in increments o f 0.1. In
the final analysis (as well as table 2) the value 24 is used.
Microstrip is a quasi-TEM transmission line and is therefore dispersive.
Hence an
appropriate dispersion equation has to be used to correct the low frequency values o f the
effective relative dielectric constant listed in the first column o f table 2, and add a
frequency dependence. The characteristic impedance is also a weak function o f frequency,
but in the case o f this resonator the impedance dispersion effect is negligible and is
therefore not included. There are several dispersion models available in the literature that
are widely used today ([2]-[7]) but Atwater's model [2] is determined to be the most
accurate and is used in this analysis. To determine the best dispersion model the measured
S-parameters in the frequency range from 4 to 13 GHz of a ring resonator o f the same
EEsof - Libra - Wed Aug
□
-
DB[S2i]
WIDE_AU
+
4 0 8 :2 0 :3 1 1993 - r in g 3
DB[ S21]
RINGP
22 . 00
dB
-5 6 . 00
4 .0 0 0
B. 500
FHEQ-GHZ
1 3 .0 0
Figure 2 Atwater's dispersion model compared to the measured S21 o f the gold resonator.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
layout as the one shown in figure 2, but made out o f gold, are used.
Figure 2 is a plot o f the dB-magnitude o f S21 versus frequency o f the gold ring resonator
and also a simple model using Atwater's dispersion equation. Notice here how the peaks
o f the modeled and measured responses (resonance frequencies) match almost perfectly.
With the rest o f the dispersion models ([3]-[7]) this is not the case. Hence Atwater's [2]
microstrip dispersion model is adopted in the rest o f this thesis.
It is applied to the
effective relative dielectric constants, listed in column 1 o f table 2, to add the appropriate
frequency dependence and yield correct effective relative dielectric constants at each
frequency.
The electromagnetic parameters listed in table 2, after compensation for dispersion as
mentioned above, are then plugged into equations (3.34)-(3.43) to obtain the final
"corrected" values, which include complex conductivity and field penetration effect. The
necessary parameters to model the two types of transmission lines used in the ring
resonator circuit are the exponential loss coefficient ac (in dB/mm), the effective relative
dielectric constant s f
and the characteristic impedance Z0 (in Ohms).
Given these
parameters and the physical length o f each transmission line the circuit S-parameter
response may be modeled by "connecting," via signal flow graphs, the S-parameters of
each individual type o f line and calculating the overall response using Mason's rule, or
simply solving a system o f linear equations.
4.3.2 Using T ouchstone and A cadem y (TM ) by E E sof Inc.
The process described above, o f combining the modeled S-parameters o f each type of
transmission line at each frequency to arrive at an aggregate modeled response o f a circuit,
may be simply executed by a computer given the appropriate program. Using a program
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
that is customized to the particular circuit it models forces modification (if not re-write)
for every different circuit.
M ost programming languages also have a "tenuous
relationship" with the computer's graphics routines, and hence plotting the analysis results
can add undesired hassle.
In addition, changing the plotted parameters would require
modifying and re-compiling the program every time such a change would be desired, a
time consuming process.
The alternative is to use an existing microwave CAD software package to perform the
algebraic combination o f the S-parameters and the plotting o f the results. In this thesis I
have chosen to use two popular microwave CAD software packages by EEsof Inc. that
are widely used in the industry: Touchstone (TM) and Academy (TM). Touchstone is a
microwave computer-aided analysis and design software package analyzes linear
microwave circuits.
Its input is a netlist or circuit file which contains a number o f
microwave elements connected in a number o f nodes.
Touchstone includes its own
element library which contains models o f many commonly used linear elements.
The input netlist file is subdivided into several blocks each o f which contains a different
kind of information. The three most important blocks are the CAT, the VAR and the E O N
block. The CKT block contains the nodal description o f the circuit to be analyzed. The
VAR block contains the definitions of constants used in the CKT block. The EQN block
contains definitions o f new variable related to the constants defined in the VAR block.
The program sweeps the frequency variable over a specified (in the FREQ block) range of
frequencies at specified increments and plots the calculated measurements versus
frequency (as specified in the O U T block) in an output graphics window. Academy is a
schematic capture and layout tool that does away with the need for an input netlist file. It
uses Touchstone as its simulator and enables versatile plotting o f many output parameters
at a time, by the effective use o f windowing environments in many platforms.
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49
Within the EQN block o f Touchstone there is a pre-defmed variable called FREQ which
always contains the instantaneous value o f the sweep frequency. Using FREQ in the EQN
block, frequency dependent calculations, like dispersion or the PEM model, may be
implemented. One o f the elements available in the libraries is the TLINP element. This
element models a physical transmission line and takes the effective relative dielectric
constant, the characteristic impedance, the loss per unit length and the physical length o f
the line as its inputs. Hence, it is ideally suited to our calculated variables (see previous
section) and the circuit may be modeled by combining a number o f such elements. Using
Touchstone fast analysis and efficient plotting is possible.
Equations (3.34)-(3.43) are frequency dependent and therefore need to be calculated in
the EQN block o f the Touchstone circuit file. A sample circuit file is included as appendix
B of this chapter. The EQN block o f the YBCO microstrip ring resonator circuit file is
separated into two sub-sections, each containing equations (3.34)-(3.43) for one of the
narrow and wide lines respectively.
Equations (3.35) and (3.36) are too long for
Touchstone, so they are broken down into many smaller equations, in the circuit file. The
VAR block contains the variables listed in table 2, with values calculated in the MathCAD
sheet o f appendix A.
The convention en tloyed for naming variables in the VAR and
EQN blocks is last letter n for the narrow
lines and
for the wide lines. The results
MSUB
P
3'eos
EH
H
.500065
T‘0
-0.0
W0 -0
RGH-0
FILE/how/crafty/diiitri/eescf/hlsjipf/«del/t
of the calculations are the loss due to the
surface resistance o f the line, in dB/mm,
Pa P H .
TLINP
the corrected characteristic impedance and
the corrected effective dielectric constant,
for each type o f line. These are fed into a
number
of
inter-connected
elements in the circuit block..
TLINP
Each
h —t i
q
HGAP
TI4
T
il
2L*“Z
K
U
**W
L
in
p
u
t
S
0Q
.5ip
00
K‘k
*KQ
Ni
A
F-0
~~=-
TLINP
tio
Z‘L
*2X
L
half
K
‘K
K
A‘A
C
M
F-0
TLINP
rh T9
2**Z
LW
half
Lir-*IK
W
A**K
A
W
F-0
PLP H HH~I D~^=”
T
L‘
T1L3INP H
T1G
2AP
2L‘Z
N
v
*
w
g
a
p
lnput S-0.500
<
K"‘L
KN
<
A
FILE/hone/crafty/dltltri/eesof/htaJipt/nodftl/t
F**A
0OJ
Figure 3 A schematic representation o f the model.
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50
TLINP element emulates a propagating mode o f the form e{~a2~iP2) o f a given
characteristic impedance. Four TLINP elements, each given the true length o f the line it
models, model the four lines o f the ring resonator. The ring is modeled via two lines, each
o f half the true length o f the ring.
The tw o lines are interconnected via a 3-port S-
parameter matrix which emulates a lumped (dimensionless) lossless T-junction (see section
6.2.2 for a more detailed discussion).
This is done so that the M G AP Touchstone
element, which models a gap in a microstrip line, may be connected to the third port o f the
T-matrix to model the coupling gap between the input and output lines and the ring.
Figure 3 shows a schematic representation o f the model circuit, as described above. It is
plotted by the Academy schematic capture utility.
It is important to stress again that Touchstone is provided with all the pre-calculated
parameters and models and hence its own built-in transmission line models are not used,
with the exception of the model o f the microstrip gap (MGAP) which is identical in the
HTS and the normal circuits.
The function of Touchstone is to perform the algebraic
combination of the S-parameters o f the different types o f transmission lines o f the filter
and conveniently plot the results in formats familiar to microwave engineers.
4.3.3 The M odeling S trategy
The two most important unknowns used in modeling the YBCO HTS, are the normal
conductivity, <r„ and the zero-temperature penetration depth, A0.
Applied physics
researchers who grow extra pure single crystal YBCO report a A0.of 140 nm and a
<j „
of
1.14-10° S/m [8], This is, however, the penetration depth in the very pure, single crystal
limit.
The YBCO crystal fabricated at Conductus to make the HTS microstrip ring
resonator is not a single crystal and the controlled laboratory conditions under which it is
deposited are not state-of-the-art. As a result the crystal grows in many separate grains
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
and there is a surface energy associated with the boundaries between different grains. The
existence o f grains and grain boundaries causes the penetration depth to be non-uniform
over the surface o f the circuit and larger than in the pure single crystal case [9],
Imperfections and contaminants in the crystal also increase the penetration depth.
However, if the variations occur in an area that is spatially small compared to the
wavelength, they can be averaged out and an overall effective penetration depth may be
used. Polakos et al. from AT&T Bell Labs report an effective penetration depth o f 450
nm for a similarly deposited HTS microstrip circuit [10],
In the initial modeling attempts the value of Polakos was used for the penetration depth
and the value o f reference [8] for the normal conductivity. The fit between modeled and
measured response data was already very close, within 2.1 dB in magnitude o f S21 and
0.2 radians in the angle o f S21.
used.
Subsequently the optimizer feature o f Touchstone was
The optimizer performs a gradient search in the N-dimensional Euclidean space
defined by the optimized variables for the optimum vector that minimizes the integrated
squared error, over frequency, between the measured and the modeled S-parameters. The
zero-temperature penetration depth and real part o f the conductivity are permitted to
optimize. The optimum extracted values for these parameters are
/l0=438 nm
cr„=5.67xl06 S/m
(see appendix B, variables LD and Sn in the VAR block).
Both values are close to Polakos' values.
Another parameter that is optimized for
minimum integrated square error between the modeled and measured S-parameters is the
line length o f the 50 Q input and output lines, for best phase o f S21 match.
This is
because the Thru-Reflect-Line (TRL) calibration standards that are used to calibrate the
HP 8510C netw ork analyzer that is used to measure the resonator circuit have an
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
unknown phase reference plane. Also, because the electromagnetic coupling between the
end o f the input/output lines and the ring is greater than if the lines on both sides o f the
gap were the same width (due to the larger fringing capacitance at the end o f the line) the
width o f the MGAP element is permitted to optimize. The resulting optimized value for
the effective width is 0.34 mm, 1.7 times its real width o f 0.2 mm.
4.4 Comparison of Model versus M easurement
The following figures contain a comparison o f the results of the modeling and the
measured data, using the optimum parameter values reported above.
Figures 4-7 are
plotted versus a wide frequency range o f 4-13 GHz, whereas figures 8-11 are plotted
against a 30 MHz frequency range centered about the lowest order resonance frequency
(4.36088 GHz). Figure 4 shows a comparison o f the insertion loss (magnitude o f S21
plotted in a log vertical scale) o f the two ring resonators o f identical layout, one made o f
YBCO and the other o f gold, as discussed above. In the same figure the modeled value o f
S21 is plotted. Obviously, in this case, the HTS circuit has much higher Q (i.e., lower
loss) than the corresponding gold one.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
EEsof - L ibra - Thu J u l 29 1 5 :4 2 :4 9 1993 - r in g 2 a
DB[ S21]
WIDE
DB[S21]
WIDE.AU
DB[ S21]
RINGP
-15. 00
dB
-45. 00
-75. 00
4. 000
8. 500
FREQ-GHZ
13. 00
Figure 4 Measured (WIDE for HTS circuit, W IDE_AU for gold circuit) versus modeled (RINGP)
magnitude o f S2I.
Figure 5 is a plot o f the measured versus modeled angle o f S21 o f the HTS resonator and
also, for comparison, the angle o f S21 o f the gold resonator.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
EEsof - Libra - Thu Jul 29 15:42:49 1993 - ring2a
o ANGIS21]
WIDE
+ ANG[ S21] <> ANG[S2i]
WIDE_AU
RINGP
3.500
rad
0 . 000
-3.500
4. 000
B. 500
FREQ-GHZ
13. 00
Figure 5 Measured (WIDE for HTS circuit, WIDE_AU for gold circuit) versus modeled (RINGP) angle
o f S2I.
Figure 6 is a plot o f the measured versus modeled magnitude o f S l l o f the HTS
resonator, and also includes the measured magnitude o f S l l o f the gold resonator for
comparison..
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
EEsof - Libra - Thu Jul 29 15:42:50 1993 - ring2a
□ DBIS11]
WIDE
+ DB[ S ll]
WIDE_AU
o DB[ S ll]
RINGP
0.200
dB
-1.600
-3.400
4.000
B. 500
FREQ-GHZ
13. 00
Figure 6 Measured (WIDE for HTS circuit, W IDE_AU for gold circuit) versus modeled (RINGP)
magnitude o f S l l .
Figure 7 is a plot o f the measured versus modeled angle o f SI 1 o f the HTS resonator and
also, for comparison, the angle o f S21 o f the gold resonator.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
EEsof - Libra - Thu Jul 29 15:42:50 1993 - ring2a
□ ANG[S11] + ANG[ S ll] 0 ANG[ S ll]
WIDEWIDE_AU
RINGP
3.500
rad
0 . 000
-3. 500
4. 000
FREQ-GHZ
13. 00
Figure 7 Measured (WIDE for HTS circuit. WIDE_AU for gold circuit) versus modeled (RINGP) angle
of Sll.
Figures 4-7 are plotted over a wide frequency range, from 4 to 13 GHz. The figures that
follow, numbered 8 through 11, focus in a narrow interval (30 MHz) about the lowest
order resonance frequency o f the HTS resonator (4.361 GHz). Figure 8 is a plot of the
measured versus modeled dB magnitude o f S21.
The agreement between model and
measurement is good. The calculated loaded Q ( 0 L) o f the HTS resonator is 1697, while
the model yields a value o f 1707, an excellent agreement which verifies that the PEM
model, presented in chapter 3, successfully accounts for surface "ohmic" losses in the
HTS.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
EEsof - Libra - Thu Jul 29 15:49:21 1993 - ring2a
DB[ S21]
WIDE
DB[ S21]
RINGP
-IB. 00
dB
-30.00
-42.00 _____
4.350
L J ___ __ __ __ __ __
4. 365
FREQ-GHZ
4.3B0
Figure 8 Measured (WIDE) versus modeled (RINGP) magnitude o f S21.
Figure 9 is a plot o f the measured versus modeled angle o f S21 o f the HTS resonator.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
EEsof - Libra - Thu Jul 29 16:02:29 1993 - ring2a
ANGIS21]
WIDE
ANG[ S21]
RINGP
0. BOO
rad
-
1. 200
-3.200
I I
4.350
1 1 I
I
I
I
I
4.3651
L
FREQ-GHZ
4. 3B0
Figure 9 Measured (WIDE) versus modeled (RINGP) angle o f S21.
Figure 10 is a plot o f the measured versus modeled magnitude o f S l l of the HTS
resonator.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
EEsof - Libra - Thu Jul 29 IB: 02:30 1993 - ring2a
DBIS11]
WIDE
DB[ S ll]
RINGP
0.000
dB
-0. BOO
- 1.
200
4.350
4. 365
FREQ-GHZ
4.380
Figure 10 Measured (WIDE) vs. modeled (RINGP) magnitude o f S l l .
Figure 11 is a plot o f the measured versus modeled angle o f SI 1.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
EEsof - Libra - Thu Jul 29 16:02:30 1993 - ring2a
ANG[ S ll]
WIDE
ANG[ S ll]
RINGP
-1. 050
rad
-
1. 120
-1.190___
4.350
1
L_
4.3651
FREQ-GHZ
4. 380
Figure 11 Measured (WIDE) versus modeled (RINGP) angle o f S l l .
4.5
References
[1] K. Gupta, R. Garg and I. Bahl, Microstrip Lines and S/ot/ines, Artech, Dedham, MA,
1979.
[2] H. A. Atwater, Introduction toMicrowave Theory, McGraw-Hill, New York, 1962.
[3] T. G. Bryant and J. A. Weiss, "Parameters o f Microstrip Transmission Lines and
Coupled Pairs ofM icrostrip Lines," IEEE Transactions on Microwave Theory and
Techniques, Vol. MTT-16, pp.1021-1027, December 1968.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
[4] M. V. Schneider, "Microstrip Lines for Microwave Integrated Circuits," The Bell
System Technical Journal, Vol. 48, No. 5, pp. 1421-1444, May/June 1969.
[5] E. 0 . Hammerstad, "Equations for Microstrip Circuit Design," Proceedings o f the
European Microwave Conference, Hamburg, W. Germany, pp. 268-272, September 1975.
[6] H. A. Wheeler, "Transmission Line Properties o f a Strip on a Dielectric Sheet on a
Plane," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-25, Mo. 8,
pp. 631-647, August 1977.
[7] E. O. Hammerstad and 0 . Jensen, "Accurate Models for Microstrip Computer-Aided
Design," IEEE M TT-S Symposium Digest, pp. 407-409, June 1980.
[8] D. R. Harshman et a l, "Magnetic Penetration Depth in Single Crystal YBa7Cu 30 7,"
Physical Review, Vol. B39, p. 2596.
[9]
T. L. Hylton and M. R. Beasley "Effect o f Grain Boundaries on Magnetic Field
Penetration in Polycrystalline Superconductors," Physical Review, Vol B39, pp. 90429048, May 1989.
[10]
P. A. Polakos, C. E. Rice, M. V. Schneider and R. Trambarulo, "Electrical
Characteristics o f Thin-Film Ba2YCu307 Superconducting Ring Resonators," Microwave
& Guided Wave Letters, Vol. 1, 1991.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix A
MathCAD File Used to Calculate the Parameters of the Two Types of
Transmission Line Used in the Resonator Circuit
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
P hysical C o n sta n ts
E0 := 8 .8 5 4 * 1 0
H 0 := 4*it*10
-12
-7
'JEO’^0
•\jEo
M i c r o s t r ip P a r a m e t e r s
h := 2 0 ' 1 0 ~ 3 ■2.54*10_2
h = 5.08*10
-4
t:= 5 -10’ 7
i := 1 , 2 . . 20
crj := 2 2 . 9 + i*0.1
M i c r o s t r ip D e s i g n E q u a t i o n s
R ing
W := 0.5 *10 3
W
1
1.25
>r
.,W +
*t i 1 +ln
h [2-ic]
K
L
t
l]
1.25
W
, W + --------1* 1 + ln 4
4 *jc*—
t j.
Jt
fr
[2
W ,
2j r
h
F := if — > 1 , 1 + 1 2 * — ;
1+12 —
h
W;
L
w
I
w I2
+ 0 .04 •: 1 — i
;
hj
[efi
+ 1] [Cri_1] [CV l ]-V
------- - + --------------------4.6
W cm = 5 . 0 1 7 * 1 0
-4
W
'j I'
w cm
Zomj =if
W cm
+ 1.393+ 0.667'ln
h
+ 1.444
!
f rci
2 *jc* J c r c ,
i.
W
8*—
+ 0 .2 5 -—
W „„
h
C a l c u l a t e n e w i m p e d a n c e s u n d e r r e c e s s i o n o f w a l l s by d n o r m a l :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
d no rinal . = -------
1000
t := t —2 •dnorinal
t = 4 . 9 9 '1 0
W - W - 2 'd n o r i n a l
F
if
W
>1
W = 5-1 O'4
1+ 12-
,
-7
W
i1+12- hj
L
+ 0.04
1-
W
-2
WJ
t
eri " i ] . h
4.6
w
^h
cred.
i
W>- 1 r ,
if —
h
2 ’it
Wom
>'25 1
W + -------it
1 +ln
1.25
, W + -------- 1
W
I +In 4x ’i t ’—
I
W cm = 5 . 0 1 7 ’10
W,
Z om d:
W,
- + 1 . 3 9 3 + 0 . 6 6 7 -In
if
A/crcdj
W,
- + 1 .4 44
2 'it • /cred.
In 8 ---------- + 0.25 •
W
J CTClli ’z „ m d i ->jCTCi
m ;
r
1 iH0-c
-]
d n o r m al
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
crc
23
1 5 .024 3 2 .8 2 4
2 4 0 6 .1 6 9 5
23.1
1 5 .0 8 8 3 2 .7 5 5
2 4 0 6 .1 6 9 5
23.2
15.151 3 2 .6 8 6
1406.1695
23.3
15.215 3 2 .6 1 7
2 4 0 6 .1 6 9 5
23.4
1 5 .2 7 9 3 2 .5 4 9
2 4 0 6 .1 6 9 5
2 4 0 6 .1 6 9 5
23.5
15.3 4 3 32 .481
23.6
15.4 0 6 3 2 .4 1 4
2 4 0 6 .1 6 9 5
23.7
3 2 .3 4 7
23.8
15.47
15 .534 32.281
2 4 0 6 .1 6 9 5
23.9
15.5 9 8 32 .215
2 4 0 6 .1 6 9 5
24
15.661 3 2 .1 4 9
2 4 0 6 .1 6 9 5
24.1
15.725 3 2 .0 8 4
2 4 0 6 .1 6 9 5
24.2
1 5 .7 8 9 3 2 .0 1 9
2 4 0 6 .1 6 9 5
24.3
15.853 3 1 .9 5 5
2 4 0 6 .1 6 9 5
24.4
1 5 .9 1 6 31.891
2 4 0 6 .1 6 9 5
24.5
15.98
3 1 .8 2 7
2 4 0 6 .1 6 9 5
2 4 .6
1 6 .0 4 4 3 1 .7 6 4
2 4 0 6 .1 6 9 5
24.7
1 6 .108 31 .701
2 4 0 6 .1 6 9 5
24.8
16.171 3 1 .6 3 8
2 4 0 6 .1 6 9 5
24.9
16.235 3 1 .5 7 6
2 4 0 6 .1 6 9 5
2 4 0 6 .1 6 9 5
In pu t/ Ou tp ut L i n e s
W := 0 . 2 - 10“3
1
, , 125
+------ •t •: U1 + lni
>r
v
[ 2 - jc]
It
[
W cm := i f
’l]
r
F := if
W
h
> 1 , 1+12—
i
2
:
: ,
h
1 + 12 —
i
W
t
[ cr i + I ] [ cri _ 1 ] c [ cri _ 1 ]
^+-------- * r -------- *
2
2
W
T)
->1
h
r
L
4 .6
1.25
,\V + ------ - f
it
j r
i2
,i
wi
crcj ::
i -
„ h
2 '-
i
w ’
L
11
W
1 +]n 4, *jc' —
t
+ 0 .0 4 •: 1 - -
W cm = 2 .0 1 7 - 1 0
-4
w
a] h
W
W.
- + 1 .3 9 3 + 0 .6 6 7 -In
-1
cm
+ 1.444
2 - it- J c r c ;
i,
W
•In K— - — + 0 .2 5 — —
h
C a l c u l a t e n e w i m p e d a n c e s u n d e r r e c e s s i o n of wa l ls by d n o rm al :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
d n o rm al = -------
1000
t := t ~ 2 ‘d n o rm al
t = 4 .9 8 -1 0
W - W - 2 ‘d n o rm a l
-7
W = 2 -1 0
;11
r
i
i2 i
-■
jw ,
h U 2'
F:= if1— > 1 , 1 + 1 2 — j .! 1 + 1 2 —h :I' J + 0 .0 4 -i 1 W;I
I h
wj L W i
I
h J
jr
ercdj :=
4 .6
jW
y h
W 1
125
Wcm :=* — >?------ ; , W + -------'t
ll
[2 ’)[]
1 +In
.w 3 ,.
It
W
1+I n 4 ‘it ‘
it
W cm = 2 .0 1 7 - 1 0
W
+ 1.393 + 0 .6 6 7 -In
^omd J c red\
■
cm
-+ 1 .4 4 4
2 ‘ic‘J c rc d ;
8 ---------- + 0 .2 5 W
em
Ic rc d -’Z
j ~ x r c 'Z
y
i o m d j a/
i om j
1V C
d n o rm a l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
23
2 3 .1
2 3 .2
2 3 .3
2 3 .4
2 3 .5
2 3 .6
2 3 .7
2 3 .8
2 3 .9
1 4 .1 1 5 4 8 .0 0 6
1 4 .1 7 4 4 7 .9 0 5
1 4 .2 3 4 4 7 .8 0 5
1 4 .2 9 4 4 7 .7 0 5
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
1 4 .3 5 3 4 7 .6 0 6
1 4 .4 1 3 4 7 .5 0 7
6 3 0 0 .0 6 9 6
1 4 .4 7 2 4 7 .4 0 9
1 4 .5 3 2 4 7 .3 1 2
6 3 0 0 .0 6 9 6
1 4 .5 9 2 4 7 .2 1 5
1 4 .6 5 1 4 7 .1 1 9
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
1 4 .7 1 1 4 7 .0 2 3
1 4 .7 7 1 4 6 .9 2 8
6 3 0 0 .0 6 9 6
2 4 .1
2 4 .2
1 4 .8 3
4 6 .8 3 4
6 3 0 0 .0 6 9 6
2 4 .3
1 4 .8 9
4 6 .7 4
24
2 4 .4
2 4 .5
2 4 .6
2 4 .7
2 4 .8
2 4 .9
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
1 4 .9 4 9 4 6 .6 4 7
1 5 .0 0 9 4 6 .5 5 4
6 3 0 0 .0 6 9 6
1 5 .0 6 9 4 6 .4 6 2
1 5 .1 2 8 4 6 .3 7
6 3 0 0 .0 6 9 6
1 5 .1 8 8 4 6 .2 7 9
1 5 .2 4 7 4 6 .1 8 9
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
6 3 0 0 .0 6 9 6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix B
Touchstone Circuit File that Models the HTS Resonator
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
8-4-93
FINAL VERSION
WITH ER=24.0
DIM
FREQ
RES
COND
IND
CAP
LNG
TIME
ANG
VOL
CUR
PWR
VAR
GHZ
OH
/OH
NH
PF
MM
PS
RAD
V
MA
DBM
LDO =437.524700000
T = 77
Tc = 85
Sn =5668582.00000
! CONSTANTS
cO = 8.854E-12
!TEMPERATURE OF MEASUREMENT
ICritical Temperture o f Sample
IPermittivity o f free space
eps =24.0000000000
KN00 =14.7100000000
ZNO =47.0200000000
KWOO =15.6600000000
ZW0 =32.1500000000
AGN = lc-10
AGW = 2.5E-10
GN = 6300
GW = 2406.2
wgap =0.34256800000
Llialf =8.40539300000
Linput =1.33346500000
SigAu =200000000.000
EQN
FDN =4*0.508*FR E Q /300*sqrt(eps-l)*(0.5+sqr(l+2*L O G (l+0.2/0.508)))
FDW =4*0.508*FR E Q /300*sqrt(eps-l)*(0.5+sqr(l+2*L O G (l+0.5/0.508)))
KNO =K N 00*SQ R (l+(SQ R T (eps/K N 00)-l)/(l+4*FD N **(-1.5)))
KWO=KWOO*SQR(l+(SQRT(cps/KWOO)-l)/(l+4*FDW**(-1.5)))
LD =LD 0*lc-9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
! Computation o f losses for first, narrow spaced, coupled lines
!
Constants
U0 = 4*PI* 1c-7
c = l/sqrt(e0*U 0)
h0=sqrl(U 0/c0)
(Magnetic Permeability o f vacuum
!Velocity o f light
(Impedance o f free space
f = FREQ* le9
(Frequency in Hz
Sr = Sn*(T /T c)**4
(Real Part o f conductivity o f YBCO (Sigma 1)
Si = (l-(T /T c)**4)/(2*PI*f*U 0*L D **2) (Imaginary Part o f conductivity (Sigma2)
P = A TA N (Si/Sr)
(Angle o f conductivity (Phi)
Th= 5*PI/4-P/2
(Auxiliary angle definition (Theta)
Sigm ag= SQRT(SQR(Sr)+SQR(Si)) (Norm o f conductivity
! MICROSTRIP LINE PARAMETERS
! WIDE RING LINE
BW = GW *AGW *SQRT(2*PI*f*U0*sigmag)
!B
CW= EXP(2*BW *COS(Th))
DW = COS(2*BW *SIN(Th))
EW = SIN(2*BW *SIN(Th))
UW= SQRT(SQR(CW *DW -1)+SQR(CW *EW ))
(Psi
WW= ATAN(CW *EW /(CW *DW -1))
(Chi
FW= BW /(AGW *sigmag*UW )
(Prefactor o f Ri and Xi
MW = 2*BW *SIN(Th)
NW = COS(PI/4+P/2-W W )
RPW= COS(M W +PI/4+P/2-W W )
RiW= FW *(NW +CW *RPW )
(Internal Resistance /M eter
NIW = SIN(PI/4+P/2-W W )
RDW= SIN(M W +PI/4+P/2-W W )
LiW = 1/(2*PI*I)*FW*(NIW+CW*RDW)
CORRW = 1+(c/sqrt(KW 0))*(Li W/ZWO)
ZW = ZWO*sqrt(CORRW)
KW = KWO*CORRW
(Internal Inductance / Meter
(Correction Factor (3.41)
(Corrected Char Impedance
(Corrected Dielectric Const (3.43)
ACW =(8.686e-3) * RiW/(2*ZW)
! NARROW INPUT/OUTPUT LINE
Th2= 5*PI/4
(Auxiliary angle definition (Tlicta) for Gold
BN =GN*AGN*SQRT(2*PI*f*U0*SigAu)
CN= EXP(2 *BN*COS(Th2))
D N= C OS(2*BN*SIN(Th2))
EN= SIN(2*BN*SIN(Th2))
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71
U N = SQRT (SQ R (C N *D N -1)+SQ R(CN*EN))
W N= A T A N (C N *EN /(C N *D N -1))
FN =BN /(A G N *SigA u*U N )
M N = 2*BN*SIN (T h2)
N N = C 0S(P I/4-W N )
RPN= C 0S(M N +PI/4-W N )
RiN = FN*(NN +CN *RPN )
!Psi
!Chi
!Internal Resistance / Meter
N IN = SIN(PI/4-W N)
RDN= SIN(M N+PI/4-W N)
LiN = 1/(2*PI*I)*FN*(NIN+CN*RDN)
!Internal Inductance / Meter
CORRN = l+(c/sqrt(KNO))*(LiN/ZNO)
[Correction Factor (3.41)
ZN = ZNO*sqrt(CORRN)
K N = KNO*CORRN
ACN = (8.686e-3) * R iN /(2*ZN )
[Corrected Char Impedance
[Corrected Dielectric Const (3.43)
[Loss Coefficient, in [dB/mm]
CKT
S2P_S1 1 2 0 /home/crafty7diinitri/eesof/hts_ring/ringl2
DEF2P 1 2 WIDE
MSUB_P3 ERAeps H =0.50800000000 T=0.0()050000000 RHO=0.0000()000000 &
R G H =0.00000000000
S3P_S2 2 13 3 /home/crafty/dimitri/cesof/hts_bpf/niodel/t MTEE W l= 1.00000000000 &
W 2= 1.00000000000 W 3=1.00000000000
S3P_S3 12 1 1 6 /homc/crafty/dimitri/eesof/hts_bpf/model/t MTEE &
W l= 1.00000000000 W 2= 1.00000000000 W 3= 1.00000000000
M GAP_T11 3 7 WAwgap S=0.50800000000
MGAP_T12 6 8 W A\vgap S=0.50800000000
TLINP_T9 11 2 ZAZW LAL half KAKW AAACW F=0.00000000000 MLIN W = l.00000000000 &
L =5.00000000000
TLINP_T10 13 12 ZAZW LALhaIf K AKW A AACW F=0.00000000000 MLIN W =1.00000000000 &
L=5.00000000000
TLINP_T13 8 10 ZAZN LALinput KAKN AAACN F=0.00000000000 MLIN W =1.00000000000 &
L=5.00000000000
TLINP_T14 7 9 ZAZN LALinput KAKN A AACN F=0.0()000000000 MLIN W = l.00000000000 &
L =5.00000000000
DEF2P 10 9 RINGP
RES_R1 1 0 RAKW
DEF1P 1 TEST
S2P_S1 1 2 0 /home/crafty/diinitri/eesof/hts_ring/rngau_\v4
DEF2P 1 2 W IDE_AU
TERM
PROC
MODEL
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
SOURCE
DCTR
FREQ
SWEEP 4.35 4.38 7.50E-5
ISWEEP 4 13 0.0225
POWER
FILEOUT
OUTVAR
OUTEQN
OUT
w ideD B[s21] grl
!\vide_au D B[s21] grl
ringp DB[s21] grl
wide ANG[s21] gr2
!widc_au ANG[s21] gr2
ringp ANG[s21] gr2
wide D B [s ll] gr3
!wide_au D B [ s ll] gr3
ringp DB[s 11] gr3
wide A N G [sll] gr4
!wide_au A N G [sl 1] gr4
ringp A NG fsl 1] gr4
TEST R E fZ ll] SCN
GRID
HBCNTL
OPT
RANGE 4.3575 4.365
RINGP MODEL WIDE
YIELD
TOL
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
CHAPTER 5
AN APPLICATION OF THE PEM LOSS MODEL:
AN HTS CPW LOW PASS FILTER (LPF)
5.1 The YBCO CPW LPF
In this chapter the model described in
Chapter 3 is applied to an HTS CPW LPF.
The layout of the filter is shown in figure 1
below.
The dimensions are in microns
(pm).
The YBCO HTS is laid on a
lanthanum
aluminate
substrate
measures 10x10x0.508 mm.
10000.0
which
The large
enclosed areas around the narrow winding
10000.0
-1
Figure 1 The layout o f the HTS CPW LPF.
line represent the coplanar ground-plane.
The narrow winding line consists o f alternating high and low impedance sections of HTS
CPW transmission line (narrow and wide line sections respectively), the sum o f the
electrical lengths o f which is an odd multiple o f one quarter o f a wavelength, at the stop
band of the filter.
The CPW has a lower ground plane (i.e., the bottom face o f the
substrate is metalized and serves as an additional RF ground).
The input and output
width-tapered lines are designed to maintain a 50 Q impedance and act as a transition
from a IC-connector coax-to-microstrip launch to coplanar waveguide.
The CPW
transmission line is meandered to minimize the area required for the circuit (since most
commercially available lanthanum aluminate substrates are o f this standard size and
growing large uniform YBCO crystals presents a manufacturing problem). This filter was
designed by W. Chew and A. L. Riley o f the Spacecraft RF Development Group and the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
HTS was deposited by B. D. Hunt, L. J. Bajuk and M. C. Foote o f the Thin Film Physics
Group at JPL for Phase I o f the Naval Research Laboratory (NRL) High Temperature
Superconductor Space Experiment (HTSSE).
Several filters, o f the same design, were
fabricated. The filter 3 dB cutoff frequency varies from 7 to 9.5 GHz and the maximum
stop band rejection from 40 to 50 dB, between different devices, depending on the quality
o f the YBCO film.
5.2 The Model
5.2.1 The M odeling Methodology
The model o f the HTS CPW LPF includes several effects, for each of which material is
drawn from CPW-related papers, cited in the references o f this chapter.
These are
combined with the complex-conductivity PEM model to arrive at physical parameters that
fully describe each type o f CPW line used in the filter (four different types according to
cross-sectional dimensions). The relevant physical parameters are the effective dielectric
constant, the characteristic impedance and the exponential loss coefficient per unit length.
Because o f the TEM nature o f the fundamental propagating mode o f CPW, the first two
are constant with respect to frequency, whereas the last is a function o f frequency. From
these physical parameters the S-parameter matrix o f each type o f line is derived. All of
these are combined to produce an S-parameter matrix that models the overall response of
the filter.
The latter is done using Touchstone (TM), by E E sof Inc., a widely-used
microwave CAD software package. It is important to stress that Touchstone is provided
with all the pre-calculated parameters and models and its own built-in CPW models are
not used. The function o f Touchstone is to perform the algebraic combination of the Sparameters o f the different types o f transmission lines of the filter and conveniently plot
the results in formats familiar to microwave designers. There are four types of CPW lines
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
used in the filter (the fourth type is an
average o f the dimensions o f the 50 Q
Ground
|
h
input/output
taper).
The
| Signal
| t
|
Ground
gap-sjc— w — % - g a p -) |
dimension
variables are defined in figure 2
and the
four line types and their dimensions and
physical properties are listed in table 1,
Figure 2 Definitions o f CPW dimension variables.
where Z0 is the characteristic impedance of
the
line,
G
is
the
L in e T ype
w (pm)
g a p (p m )
Narrow
6
122
Zn (0)
83.4
125500
50 Q
50
100
49.6
23500
12.52
inductance geometric
Wide
Avg. o f taper
200
25
17340
12.49
96.3
219.6
22.6
49.7
12860
12.85
factor, as defined in
T abic 1 The four types o f line o f the filter and their properties.
incremental
G
E/>rr
12.46
chapter 3 and sef f is the effective dielectric constant o f the line. The results listed in table
1 can also be found, in more detail, in appendix A o f this chapter.
Appendix A is a
MathCAD file which calculates the characteristics o f each type o f line, given the physical
parameters o f the line (i.e., width, gap, substrate thickness, substrate dielectric constant
and frequency).
To arrive at the parameters listed in table 1, four effects are included. First the parameters
o f a lossless CPW line without a lower ground plane are calculated using equations from
K. Gupta's book [1], Then the existence of a lower ground plane is accounted for, using
equations from G. Ghione's paper [2], Subsequently the loss, corrected impedance and
dielectric constant are calculated using equations (3.34)-(3.43). The final effect which is
included is dielectric loss. This is calculated assuming a dielectric loss tangent of 0.0001
(an approximate value reported in literature). The effect of dielectric loss is, however, in
this case, negligible and is only included for completeness of the model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
5.2.2 T he T ouchstone C ircuit File
Equations (3.34)-(3.43) are frequency dependent and therefore need to be calculated
internal to Touchstone. They are entered in the E O N block o f the Touchstone circuit file.
A sample circuit file (also called a netlist) is included as appendix B of this chapter. The
EQN block is the part of the circuit file where the user can define variables using
equations relating variables from the VAR block (the part of the circuit file where constant
variables are defined) and constants. The EQN block has access to FREO, the sweep
frequency variable. Hence the frequency dependent variables are re-calculated for each
frequency o f the sweep, as Touchstone calculates and plots the frequency response o f the
circuit under analysis. The EQN block of the HTS CPW LPF circuit file is separated into
three sections, each containing equations (3.34)-(3.43) for one o f the narrow, wide and 50
£2 lines respectively. Equations (3.35) and (3.36) are too long for Touchstone, so they are
broken down into many smaller equations, in the circuit file. The VAR block contains the
variables listed in table 1, with values calculated in the MathCAD sheet o f appendix A.
The convention employed for naming variables in the VAR and EQN blocks is last letter n
for the narrow lines, w for the wide lines and 5 or 50 for the 50 Q. lines. The calculations
provide results of the loss due to the surface resistance o f the line, in dB/pm, the corrected
characteristic impedance and the corrected effective dielectric constant, for each type of
line.
These are fed into the TLINP Touchstone element, which models a physical
transmission line of known impedance, effective dielectric constant, length and attenuation
coefficient. The TLINP element emulates a propagating mode o f the form e{' az"iP:) o f a
given characteristic impedance. Six TLINP elements, each given the true length of the line
it models, model the six lines o f half o f the filter. They are connected via two-port Sparameter files that model the discontinuity that is presented to the propagating wave by
the changes o f the line widths. These two-port S-parameter files where calculated using
EM by Sonnet Software Inc [3],
EM is an electromagnetic analysis software package
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
which uses an analytical full-wave numerical approach to analyze stratified planar enclosed
microwave circuits.
This analytical full-wave numerical approach invokes a Galerkin
method o f moments technique which develops the dyadic Green's function operator as a
bi-dimensional infinite vectorial summation o f homogeneous rectangular waveguide
eigenfunctions, resulting in a subsectional technique making use o f roof-top expansion and
testing functions that are closely related to spectral domain techniques. A similar analysis,
using EM, is performed for the input and output 50 Q. tapers, and they are included as Sparameter files as well. However, since the EM analysis assumes perfect conductors (or
loss proportional to the square root o f frequency, which would be inappropriate in this
case), the input and output tapers contribute some loss which would be ignored.
To
model this unaccounted-for loss two extra TLINP elements are included at the input; one
has the negative length of the other.
One o f the TLINP elements contributes the
calculated loss while the other is made lossless, but one cancels the phase o f the other so
that the total phase o f the two is zero. The total filter is modeled by a cascade o f the half
filter model back-to-back with itself. This is possible because the filter is symmetric about
an axis through its mid-point.
5.2.3 T he M odeling Strategy
The two most important unknowns used in modeling the YBCO HTS, are the normal
conductivity,
g„
and the zero temperature penetration depth, X0.
Applied physics
researchers who grow extra pure single crystal YBCO report a X0.o f 140 nm and a on o f
1.14-106 S/m [4], This is, however, the penetration depth in the very pure, single crystal
limit. The YBCO ciystal deposited at JPL to make the CPW LPF is not a single crystal
and the controlled laboratory conditions under which it was deposited are not state-of-theart. As a result the crystal grows in many separate grains and there is a surface energy
associated with the boundaries between different grains. The existence o f grains and grain
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
78
boundaries causes the penetration depth to be non-uniform over the surface o f the circuit
and larger than in the pure single crystal case [5], Imperfections and contaminants in the
crystal also increase the penetration depth. However, if the variations occur in an area
that is spatially small compared to the wavelength, they can be averaged out and an overall
effective penetration depth may be used. Polakos et al. from AT&T Bell Labs report an
effective penetration depth o f 450 nm for a similarly deposited HTS microstrip circuit [6],
In the initial modeling attempts the value o f Polakos was used for the penetration depth
and the value o f reference [4] for the normal conductivity. The fit between modeled and
measured response data was already very close, within 0.6 dB in magnitude o f S21 and
0.5 radians in the angle of S21. However, the uncertainty o f a precise value for Tc, the
critical temperature o f the YBCO, imposes a corresponding uncertainty window on the
zero temperature penetration depth, X0 and the normal conductivity, <r„ over which the
response is optimized for a minimum integrated squared error best fit. More specifically,
the critical temperature of the YBCO films was found to be in the range from 83 to 88 K,
using DC measurements that were made before packaging. However the precise value for
each film was not recorded.
All measurements o f the S-parameters of the HTS CPW
LPFs, which are used for modeling and fitting, were made at liquid nitrogen (LN?)
temperature (77 K). For modeling, a value o f 77 K is assumed for 7 'and 85 I< for Tc. The
inaccuracy in the latter assumption is about an 4.9% difference in X0 and 4.7% difference
in a„ per degree of difference of Tc from its true value.
Hence the 5 K window o f
uncertainty in Tc.corresponds to a 110 nm window o f uncertainty in X0 and a 270000 S/m
window o f uncertainty in an. The optimum extracted values o f these parameters, for one
of the devices are
7Vj=483 nm
cr„=l ,8xl06 S/m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Both values are within their uncertainty window from Polakos' values. Another parameter
that is optimized for minimum integrated square error between the modeled and measured
S-parameters is the line length o f the 50 Q. input and output tapers, for best phase match.
It is important to point out that the qualities o f the various YBCO films, as extracted by
the above optimization method performed on the various devices, vary considerably. The
zero-temperature penetration depth varies in the range from 400 to 700 nm and the normal
conductivity in the range 0.5-106 to 4-106. These ranges are relatively large and point to
the need for more repeatability in the YBCO deposition process, but they may be smaller
than they seem, if variations in the critical temperature of the YBCO films are taken into
account.
5.3 Comparison of Model versus Measurement
5.3.1 S-parameters versus Frequency
The following figures contain a comparison o f the results o f the modeling and the
measured data, using the optimum values reported above. Figure 3 shows a comparison
of the insertion loss (magnitude o f S21) o f two CPW LPFs o f identical layout, one made
of YBCO and the other o f silver. To make the comparison fair, both the silver and the
YBCO are at the LN2 temperature o f 77 K. Obviously, in this case, the best efforts o f the
normal metal do not match the performance o f the HTS.
The difference in cutoff
frequency is due to the kinetic inductance o f the HTS, which is successfully modeled by
equation (3.36).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
EEsof - L ibra - Man Sep 24 1 4 :2 5 :4 5 1990 - h4jun26
□
DB[ S21]
YBCO
+
DB[ S21]
SILVER
0 . 000
dB
-5 . 000
-
1 0 . 00
1.000
6 . 000
FREQ-GHZ
11.00
Figure 3 Comparison o f insertion loss o f silver and YBCO filters.____________________________________
Figure 4 shows a comparison o f the measured versus modeled magnitude o f the insertion
loss (S21) plotted on a vertical dB scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
EEsof - Libra - Tue Sep 25 10:35:30 1990 - h6jun26
□ DB[ S21]
FIL
+ DB[ S21]
YBCO
0.000
dB
-5. 000
-10. 00
1.000
6 . 000
FREQ-GHZ
11.00
Figure 4 Measured (YBCO) versus modeled (FIL) magnitude o f S21._________________________________
Figure 5 shows a comparison o f the measured versus modeled phase (angle) of S21. The
vertical scale is in radians.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
EEsof - Libra - Tue Sep 25 10:35:40 1990 - hBjun2B
□ ANGIS21]
FIL
+ ANG[ S21]
YBCO
3. 500
rad
0 . 000
-3. 500
1.000
6 . 000
FREQ-GHZ
Figure 5 Measured (YBCO) versus modeled (FIL) angle o f S 2 1._____________________________________
Figure 6 shows a comparison o f the measured versus modeled magnitude o f the return
loss (SI 1) plotted on a vertical dB scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
EEsof - Libra - Tue Sep 25 10:35:44 1990 - h6jun26
□ DB[ SI 1]
FIL
+ DB[ Sll]
YBCO
0.000
dB
-25. 00
-50. 00
1.000
6 . 000
FREQ-GHZ
11.00
Figure 6 Measured (YBCO) versus modeled (FIL) magnitude o f S 1 1.________________________________
Figure 7 shows a comparison of the measured versus modeled angle o f S 11. The vertical
scale is in radians.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
EEsof - Libna - Tue Sep 25 10:35:50 1990 - hBj un2B
□ ANG[ Sll]
FIL
+ ANG[ SllJ
YBCO
3. 500
rad
0 . 000
-3. 500
1.000
6.000
FREQ-GHZ
11.00
Figure 7 Measured (YBCO) versus modeled (FIL) angle o f S l l . ______________________________________
Figure 8 shows a comparison o f the measured versus modeled S21, plotted on a Smith
chart o f radius 1.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
EEsof - Libra - Tue Sep 25 10:36:02 1990 - h6jun26
S21
FIL
S21
YBCO
fl:
f2:
1 . 00000
11. 0000
Figure 8 Measured (YBCO) versus modeled (FIL) S21 plotted on a Smith chart o f unit radius.
5.3.2 Temperature Dependence of the Insertion Loss
In addition to measurements taken at a constant 77 I< temperature, with the hermetically
packaged HTS CPW LPFs immersed in LN 2 , another set o f measurements was taken at
different temperatures in the range from 15 to 95 K, in the vacuum jacket o f a closed cycle
refrigerator. However, in these measurements the connecting cables inside the refrigerator
and the air-tight connectors could not be calibrated out, so their insertion loss and
insertion phase is included in the measured S-parameters.
This insertion loss, which is
about 4 dB and increases slowly with frequency, may be neglected at frequencies where
the insertion loss of the HTS CPW LPF is itself sufficiently greater than 4 dB. Therefore,
this measured data should only be used beyond the pass band of the filter (say 9.5-11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
GHz). The data is used for a comparison o f the measured versus predicted tem perature
dependence o f the insertion loss o f the filter, with the understanding that because o f the
lack o f an accurate calibration, the comparison is qualitative only. For this reason the
measured and predicted insertion losses are not superimposed, as is done hitherto, but are
plotted in two separate figures. It is important to stress that no optimization is performed
in this case. The extracted optimized parameters from section 5.2.3 are used and the only
variable that is varied is T. Figures 9 and 10 show the measured and predicted insertion
losses o f the same filter respectively, at temperature 50, 60, 70 and 80 K.
EEsof - Libra - Thu Sep 27 13:34:07 1990 - temp
DB[ S21]
T50
DBIS21]
T60
DB[ S21]
T70
DB[ S21]
TBO
0 . 000
-10. 00
-
20 . 00
9. 000
FREQ-GHZ
11.00
Figure 9 Measured magnitude o f the insertion loss at 50, 60, 70 and 80 K.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
EEsof - Libra - Thu Sep 27 13: A 3 : 5B 1990 - temp
□ DB[S21]
T50
+ DB[ S21]
T60
<> DB[S21]
T70
x DB[S21]
TBO
0 . 000
dB
-
10 . 00
-
20 . 00
9. 000
FREQ-GHZ
11.00
Figure 10 Predicted magnitude o f the insertion loss at 5 0 ,6 0 70 and 80 K.____________________________
The essential features o f figure 9 are captured in figure 10. The non-uniform shift o f the
cut-off frequencies with respect to temperature and the almost parallel slopes o f the
response in the stop band are similar in the two figures. The slopes in figure 9 are a little
steeper than those in figure 10, but this is to be expected since the cable and connector
losses, which are not calibrated out o f the data of figure 9, are an increasing function of
frequency.
5.4 References
[1] K. Gupta, R. Garg and I. Bahl, Microsfrip Lines and Slot lines, Artech, Dedham, MA,
1979.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[2] G. Ghione and C. Naldi, "Parameters o f Coplanar Waveguides with Lower Ground
Plane," Electronics Letters, Vol. 19, pp. 179-181, September 1983.
[3] J. C. Rautio and R. F. Harrington, "An Electromagnetic Time-Harmonic Analysis of
Shielded Microstrip Circuits," IEEE Transactions on Microwave Theory and Techniques,
Vol. MTT-35, pp. 726-730, Aug. 1987.
[4] D. R. Harshman et al., "Magnetic Penetration Depth in Single Crystal YBayQ^Oy,"
Physical Review, Vol. B39, p. 2596.
[5]
T. L. Hylton and M. R. Beasley "Effect o f Grain Boundaries on Magnetic Field
Penetration in Polycrystalline Superconductors," Physical Review, Vol B39, pp. 90429048, May 1989.
[6]
P. A. Polakos, C. E. Rice, M. V. Schneider and R. Trambarulo, "Electrical
Characteristics of Thin-Film Ba 2YCu 3C>7 Superconducting Ring Resonators," Microwave
& Guided Wave Letters, Vol. 1, 1991.
[7] D. Antsos, "Modeling o f Planar Quasi-TEM Superconducting Transmission Lines,"
JPL New Technology Report, NASA Case No. NPO-D-18418, P L Case No. 7950,
January 1991.
[8] D. Antsos, "Equations for Designing Superconducting Transmission Lines," NASA
Tech Briefs Journal, Vol. 16, No. 8, p. 30, August 1992.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[9]
D. Antsos, W. Chew et al., "Modeling o f Planar Quasi-TEM Superconducting
Transmission Lines," IEEE Transactions on Microwave Theory and Techniques, Vol. 40,
No. 6, pp. 1128-1132, June 1992.
[10] W. Chew, A. L. Riley et al., "Design and Performance o f a High-Tc Superconductor
Coplanar Waveguide Filter," IEEE Transactions on Microwave Theory and Techniques,
Vol. 39, No. 9, pp. 1455-1461, September 1991.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
A p p en dix A
Sample MathCAD File Used to Calculate the Parameters of the CPW Lines
of the LPF
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
91
Coplanar waveguide parameters with losses for thin conductor
20 Sept 1990
The following effects are included:
EFFECT:
REFERENCE:
CPW (No loss, no ground plane)
Lower Ground Plane
K. C Gupta
G. Ghione
Loss for cond thickness
of order of penetration depth
H. Lee
Define the elliptic integral:
1
K (k ) :=
;d(j)
K ’(k ) := K V l - k
K K '(k ) :=
K (k )
K '( k )
Parameters to use:
Substrate permittivity
ep •- 2 4
i := 1 . . 4
CPW Line Widths of Center Conductor
Narrow,
6-10
50*10
200-10
9 6 .2 9 * 1 0
-6
high imp line
50 Ohm line
Wide,
lo imp line
Average of 50 Ohm taper
Substrate thickness
h := 0 .5 0 0 * 1 0 ~ 3
Conductor thickness
t := 0 .5 -1 0 ~ 6
Size of differential for calculating
the incremental .inductance geometric factor G
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8 voppa?t,
: = --------
1000
t|5 := h -i-8 vopp.a?i.
x8 := t ~
*8 vop(iaX,
2
t8
= 4 .9 9 * 1 0
Thickness effect
4 *tt *w •
1 .25 -t"
D elta - :=
1
-t-ln
it
4 • i t ^ w i - 2 , 8 v o p )ia ^ j
1 .2 5 -t8
A£?t/raS.
-t-ln
1
t8
w - -f-Delta-
2 5 0 -1 0
a cffj ;=
b e ff; ;
-W ;
g a P;
wi
D elta ;
+ g a p . - -----
2
g a p 4 := 2 1 9 .5 5 - 1 0
2
a c ff;
k i :=
k 'i :%
beff,-
H
ki
tfa e lT ;
tanh
2
klj :=
-h
Jt’b e f f ;
tanh
2
-h
w - + A eta a 5 .
aetjxjtSj
8
vop p e d
w;
Ae^-taS;
Pcp<j)8. := —-+ g a p ;----------2
cce(J)08;
k8 .
:=
2
8
voppcd
yattS; := g a p ; + 2 - 8 vop p a?i
Pe^S;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
tc- otetfxjjSj
tan h
2
k 18.
-h
tc• pe<jxf>5i
tan h
2
-h
Constants
c A y := 6 . 1 7 - 1 0
Conductivity of Silver
o X v := 5 . 8 - 1 0
Conductivity of Copper
p.0 := 4 - j r lO
-7
CO := 8 . 8 5 4 - 1 0
-12
1
c :=
c = 2 .9 9 8 - 1 0
|j.0-£0
8
10 := r
'VtO
■nO = 3 7 6 . 7 3 4
■10
A .0 \p B x o ■“ 5 6 6 0 ‘ 10
T := 7 7
T c := 85
f :=
1 -1 0
c v := 1 .1 4 - 1 0
JL
T„
t
^'I'B X O := o v ‘
■j
■-
2 -ji *1 -|i0 "X.0 >j/gXQ
cr := a
X'BXO
<7 = 7 . 6 7 7 *10 5
Zs :=
j
- 1 .2 9 1 -1 0 8 i
-2 -Jt-f •—
o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
R s := R e ( Z s )
Z s = 2 . 3 2 5 *10
+ 0 .0 0 8 i
Find effective permittivity and impedance
+—
KK'
r
1
• K K |k l .J
— t!
KK
k l/
1 ~i----------
KK’ k;
M
EEftj^XJCCOj
1 2 .5 5 8
1 2 .5 9 4
1 2 .6 2 7
1 2 .8 8 4
0 .7 ^EE^XlCCOj - 1 j ’
gap;
K K ' k . + 0 .7
L J
gap;
t5
0 .7 | EE^XrtCOj - 1 j ’
yaitSj
e e <p 6 t
S-
:=
e e tp o x itW j- -
tS
K K ’ x 8 . + 0 .7
ycatS.
Zcpwj :=
6 0 ■%
KK’fk jj+ K K 'fk li]
ZxrccoSj :=
e£<?<j)'r5j
G;
1_________
6 0 *jc
K K ’[ k 5 .J + K K ' [ k 18.]
l
ce^dTj •[ Zx^coSj - Zcpwj
1)0
SvopuaX
Ai := w j-i
C :=
(1
+ j ) *-^/re*f *ja.0 *c
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95
Zi; := Zs *G; -coth [C 'G i'A i]
w•
Zcpwj
6- 10
5-10
2-10
e e ^ ij
Zij
_______
-6
-5
-4
83.382
12. 455
49.555
12. 519
22.642
12. 488
49.711
12 . 84 7
9.629-10
15.358+2.705i-10'
1. 847 -4-345.3151
0.519+1441
1.255 GO5
2 . 3 5 -104
1.734-104
0 . 9 6 + 1 8 1 . 307i
1 .2 8 6 -104
ee^xjjXj
Im[2ii]
Lj := Zcpwj
2 *jc*f
Ci =
Zcpwj
Ri := Rc[Zij
R:
2. 705 - 1 0 3
1 5 . 3 58
345.315
1. 847
144
0.519
181.307
0.96
100.01
51.831
23.594
50.904
Ohmic,
penetration depth-induced losses
R cfzi,]
a X,- ■•= - -
aXj
0. 0 7 7
0. 0 1 8
0.011
0. 0 0 9
a^SB. = 8.686 -ax.
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96
X '■=-
f
xav8 := 0 .0 0 0 1
Dielectric Losses
2 7 .3 'ep T ee<j><j>x. - 1 ] *xav8
aSSBj := --------L -------- =!------Je«M>x.*[ep-l]*fc
a 8 8 B j> jij := 10~ 6 -a8S B ;
a%8 B_i)p.j := 10
6
'a%SB;
0
'
3 . 0 8 4 -1 0 - 8
a 8 8 B _uji =
6 . 6 6 9 -1 0 -7
f
3 .0 9 3 -1 0 - 8
3 .0 8 9 -10
-
a%8 B_up. =
1 . 5 4 7 -1 0 _ 7
io 9
-8
9 .5 4 7 -1 0
3 .1 4 1 -1 0 ~ 8
2 ’i f f
0
-R
.
f
io 9
8 . 1 8 7 -1 0 _8
________
4 .3 0 5 -1Q ~ 7
5 .4 9 6 -1 0 ~ 8
2 .2 9 2 -1 0 "
2 .8 8 6
-
8
1 0 '8
L,
E FF j :=
E FF -
L i-L ii
1 .1 9 9
1 .0 4 6
1 .0 4 2
1 .0 2 4
ee^xveWj := e e ^ x ^ E F F jJ 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
e e ^ t O K i V j
• - c
ee^^TvecOj
' L j ' C j
ee ^
t o k iv .
1 7 .9 1 7
1 7 .9 1 7
1 3 .6 9 5
1 3 .6 9 5
1 3 .5 6 1
1 3 .5 6 1
1 3 .4 7
1 3 .4 7
Z cp w sk irij :=
Z cp w sk irij
j ‘2 * i r f *C1 0 0 .0 1 - 0 . 0 8 7 i
5 1 .8 3 1
- 0 . 0 12i
2 3 .5 9 4 - 0 .0 0 3 i
5 0 . 9 0 4 -O.OO 61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix B
Sample Touchstone Circuit file
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
! MODEL OF THE YBCO FILTER FIT TO D ATA USING LOSS A ND LOSS TANGENT
! ALSO USING S2P FILE FOR 50 OHM TAPER K-CONNECTOR TO CPWG INTERFACE
! USING EXACT EQUATIONS FOR ALL ATTENUATIONS AS A FUNCTION OF LD AND Sn
! BY DIMITRIOS ANTSOS ( SEPTEMBER 2 0 ,1 9 9 0 )
DIM
LNGUM
ANGRAD
VAR
LD# 1000E-10 4.83c-07 10000E-10
!PENETRATION DEPTH FOR YBCO
T = 77
1TEMPERATURE OF M EASUREMENT
Tc = 85
'Critical Temperlure o f Sample
S n # lE 4 1791776. 1E8
INormal Conductivity o f Sample
LI # 2000 3730.566 3800
! Docs not affect phase, only taper loss
L50 # 1800 2241.989 2400
! = 2 006.4 on circuit
ACI0 = 8.30e-08
! COPPER LOSS OF INPUT TAPER (8.3e-8)
ADI00 = 0.000500
1 = 3 .1 4e-4
! CONSTANTS
eO = 8.854E-12
AGn = 5 c -12
Gn = 8.43479e4
A G \v = le-1 0
G\v = 1,7336e4
AG5 = 2 .5 e - ll
G5 = 2.350 le4
IPermitlivity o f free space
'.Dimension Variable ( Narrow L in e )
llncremental Inductance Rule Var (Narr
Z500 = 49.56
ZN0 = 8 3 .3 8
ZW0 = 2 2 .6 4
KI = 12.847
K500 = 12.52
KN0 = 12.455
KW0 = 12.49
LI = 7 2 0 .8
L2 = 997.0
L3 = 1369.7
L4 = 7 6 1 .3
L5H = 924.0
T AN D = 0.000100
EQN
ADN0 = 3.086E-4*TAND
A D500 = 3.093E-4*TAND
AD WO = 3 ,089E-4*T AND
ADI0 = ADI0O*TAND
! DIELECTRIC LOSS OF INPUT CPW 50 OHM TAPER
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
LI1 = -LI
A l = ACIO * FREQ**2 + ADIO * FREQ
! Compulation o f exact loss for narrow line
!
Constants
U0 = 4*P I*le-7
[Magnetic Permeability o f vacuum
f = FR E Q *le9
'.Frequency in Hz
Sr = Sn*(T/Tc)**4
.'Real Part o f conductivity o f YBCO
Si = (1-(T/Tc)**4)/(2*P1*P|!U0*LD**2) llmaginary Part o f conductivity
P = ATAN(Si/Sr)-2*PI
!Angle o f conductivity
T h= PI/4-P/2
(Auxiliary angle definition
r = SQRT(SQR(Sr)+SQR(Si))
[Norm o f conductivity
c = l/sqrl(e0*U 0)
'.Velocity of light
!
Narrow Line Parameters
B n= Gn*AGn*SQRT(2*PI*f*U0*r) .'Fudge Factors
Cn= EXP(2*Bn*COS(Th))
D n= COS(2*Bn*SIN(Th))
En= SIN(2*Bn*SIN(Th))
Un= SQRT(SQR(Cn*Dn-1)+SQR(Cn*En))
Wn= A TA N(Cn*En/(Cn*D n-l))
Fn= Bn/(AGn*r*Un)
M n= 2*Bn*SIN(Th)
N n= COS(PI/4+P/2-W n)
Rn= COS(Mn+PI/4+P/2-W n)
ReZn = Fn*(Nn+Cn*Rn)
'.Real Part o f Internal Impedance / Meter
NIn= SIN(PI/4+P/2-W n)
RIn= SIN(Mn+PI/4+P/2-W n)
InrZn = Fn*(NIn+Cn*RIn)
Zn = SQRT(SQR(ZN0)-(c*ZN0)/(2*PI*SQRT(KN0)*f)*ImZn)
ACN = -( 8 .6 8 6 c- 6 ) * RcZn/(2*Zn)
EFFN = ZN / ZNO
!
Wide Line Parameters
B w = Gw*AGw*SQRT(2*PI*f*U0*r) [Fudge Factors
Cw= EXP(2*Bw*COS(Th))
Dw= COS(2*Bw*SIN(Th))
Ew= SIN(2*Bw*SIN(Th))
Uw= SQRT(SQR(Cw*Dw-l)+SQR(Cw*Ew))
W w= A T A N (C w *Ew /(C w *D w -l))
Fw= Bw/(AGw*r*Uw)
M w= 2*Bw*SIN(Th)
N w = COS(PI/4+P/2-Ww)
Rw= COS(Mw+PI/4+P/2-W w)
RcZw = Fw*(Nw+Cw*Rw)
N Iw= SIN(PI/4+P/2-W w)
[Real Part o f Internal Impedance / Meter
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
RIvv= SIN(M w+PI/4+P/2-W w)
IrnZvv = F\v*(NI\v+C\v*RIw)
Zw = SQRT(SQR(ZwO)-(c*ZwO)/(2*PI*SQRT(KwO)*f)*ImZw)
ACw = -( 8 .6 8 6 e- 6 ) * ReZw/(2*Zw)
EFFw = Zw / ZwO
!
50 Olmi Line Parameters
B 5= G5*AG5*SQRT(2*PI*f*U0*r) IFudge Factors
C5= EXP(2*B5*COS(Th))
D 5= COS(2*B5*SIN(Th))
E5= SIN(2*B5*SIN(Th))
U 5= SQRT(SQR(C5*D5-1)+SQR(C5*E5))
W5= A TA N(C5*E5/(C5*D 5-1))
F5= B5/(AG5*r*U5)
M 5= 2*B5*SIN(Th)
N 5= COS(PI/4+P/2-W 5)
R5= COS(M 5+PI/4+P/2-W 5)
ReZ5 = F5*(N5+C5*R5)
N I5= SIN(PI/4+P/2-W 5)
RI5= SIN(M 5+PI/4+P/2-W 5)
ImZ5 = F5*(NI5+C5*RI5)
!Real Part o f Internal Impedance / Meter
Z50= SQRT(SQR(Z500)-(c*Z500)/(2*PI*SQRT(K500)*f)*ImZ5)
AC50 = -( 8 .6 8 6 e- 6 ) * RcZ5/(2*Z50)
EFF50 = Z50 / Z500
K50 = K500 * EFF50 * EFF50
KN = KNO * EFFN * EFFN
KW = KWO * EFFW * EFFW
A 50 = AC50 + A D 500 * FREQ
AN = ACN + ADNO * FREQ
AW = ACW + ADWO * FREQ
CKT
S2PA 1 2 0 /iiscr/dimitri/em/hts/costepI.s2p
DEF2P 1 2 BIG_STEP
S2PB 1 2 0 /uscr/dimitri/em/hts/htsl/costep2.s2p
DEF2P 1 2 SML_STEP
S2PC 1 2 0 ./Ii4jun26.s2p
DEF2P 1 2 YBCO
IS2PD 1 2 0 /uscr/dimitri/ckl/hts/silver/wc0523a.s2p
IDEF2P 1 2 SILVER
S2PE 1 2 0 /uscr/dimitri/em/hts/hls_50.s2p
DEF2P 1 2 FIFTY
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
TLINP 1 2 Z=50 LAL IK AK I A AAI F=0
TLINP 2 3 Z=50 LALI1 KAKI A =0 F=0
FIFTY 3 4
T L IN P T 1 4 5 ZAZ50 LAL50 KAK50 A AA50 F=0.0000000
SML_STEP 5 6
TLINP_T2 6 7 ZAZW LAL1 KAKW A AAW F =0.0000000
BIG_STEP 7 8
TLINP_T3 8 9 ZAZN LAL2 KAK N A AA N F=0
BIG_STEP 9 10
TLINP_T4 10 11 ZAZW LAL3 KAKW A AAW F=0.0000000
BIG_STEP 11 12
TLINP_T5 12 13 ZAZN LAL4 K AKN A AA N F=0.0000000
BIG_STEP 13 14
TL1NP_T6 14 15 ZAZW LAL 5 H K AKW A AAW F=0.0000000
DEF2P 1 15 HALF
HALF 1 2
HALF 3 2
DEF2P 1 3 FIL
IRES 1 0 RArez5
IDEF1P 1 TEST
FREQ
SWEEP 1 11 .125
OUT
FIL D B [S 211 GR1
YBCO D B[S21] GR1
FIL D B [S 11] GR3
YBCO DB[S11J GR3
FIL A N G |S21] GR2
YBCO ANG[S21] GR2
FIL A NG |S11] GR4
YBCO A N G [S 11] GR4
FILS21 SC2
YBCO S21 SC2
ITEST RE[Z11] GR5
! FIL DB[S21] GR6
! YBCO DB1S21] GR6
GRID
RANGE 1 1 1 1
GR1 -10 0 1
GR5 .001 .003 .0001
IRANGE 7 9 .2
! GR6 -2 0 .5
OPT
RANGE 2 11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
YBCO MODEL FIL
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
104
CHAPTER 6
AN APPLICATION OF THE PEM LOSS MODEL:
AN HTS M ICROSTRIP BAND PASS FILTER (BPF)
6.1 The YBCO Microstrip BPF
In this chapter the model described in
Chapter 3 is applied to an HTS
microstrip BPF. The layout o f the filter
is shown in figure 1 below. The
dimensions o f the mask shown in figure 1
are 10x10 mm. The YBCO HTS is laid
on a lanthanum aluminate substrate
which measures 10x10x0.508 mm. The
design is a combination o f a parallel
coupled resonator filter and a stub filter.
Figure 1 The layout o f the HTS microstrip BPF.
There are three parallel coupled line
resonators, two o f them symmetric about a center axis o f symmetry that is the
perpendicular bisector o f the middle coupled line resonator. There is also an open stub
line connected to the input and output of the filter. The line is actually connected to the
filter via a smoothly width-tapered line section (which improves the sharpness o f the skirts
o f the filter). At the frequency that the electrical length o f the open stub line is 90 degrees
the line acts as a transformer and transforms the open circuit at its end to a short circuit at
the input o f the filter, thus providing a zero, or null to the thru response. This zero is at a
frequency higher than the pass band o f the filter and improves the sharpness o f the roll-off
o f the latter. There are, scattered beyond the end o f the stub line, small squares o f metal
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
which are meant to provide a degree o f tunability o f this response null (if the frequency o f
the null needs to be reduced, the little pieces o f metal may be shorted to the main stub).
This filter was designed by W. Chew o f the Spacecraft RF Development Group and the
HTS was deposited by B. D. Hunt and M. C. Foote o f the Thin Film Physics Group of
JPL for Phase II o f the M IL HTSSE.
6.2 The Model
6.2.1 The M odeling Methodology
There are three main challenges in modeling this HTS microstrip BPF: The model o f the
tapered line which connects the stub to the input o f the filter, the model o f the coupled
lines and the inclusion o f dispersion (in contrast to the type o f transmission line employed
in the design o f the filter presented in chapter 5, microstrip supports a quasi-TEM
propagation mode which exhibits measurable distortion). In the particular case o f this
design the modeling frequency range o f interest (6-8.5 GHz) has a fractional bandwidth o f
34%, and it is arguable that dispersion may be neglected. However, in my opinion, it is a
borderline case and therefore both analyses with and without dispersion will be included.
6.2.2 The Model o f the Input/Output Stub Resonator
The stub resonator, which is connected to the input o f the filter via a tapered-width line is
shown in figure 2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
The reason for the tapered width line is that
Touchstone, which was used to design the
filter, does not model large T-elem ents
accurately. By tapering the width o f the line
from 0.5 mm down to 0.05 mm the size o f
the T is reduced and the accuracy of the
modeling is increased. However, the tapered
line presents a modeling problem because o f
its non-constant cross-section. Its
parameters vary continuously along its
length. Such a line cannot be modeled using
F igure 2 The input section o f the HTS BPF with
the stub resonator.
the PEM model o f chapter 3, which assumes a constant cross-section. The solution for
modeling the filter is shown in figure 3.
The stub line is made o f constant width (and
can now be modeled) and a normal large T is
employed to connect it to the input o f the
filter. Figure 3 is only a representation of the
concept o f the solution. In the actual model,
an ideal, lumped T element is used, in the
form o f a 3 by 3 S-parameter matrix, and the
lengths o f the stub and the 50 Q connecting
lines are increased to compensate. The 3 by
3 S-parameter matrix o f an ideal, lumped T
can be easily derived using four properties:
The equipotential character o f the T
F igure 3 An equivalent input stub resonator
section employed for modeling.
(1 + Sn = S2] because it is assumed
dimensionless), the energy conservation principle (S 'S = I, see chapter 2), the symmetry
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
o f the S-matrix (,S21 = S n , S 3l = S ]3 and S 32 = S 23) and the 3 way symmetry o f a T
( S n = S r = S 33 and S 2] = S 3] = S 32). It is included here because o f its general usefulness.
1
S=
2
0)
3
1
3
2
Figure 4 shows a schematic representation o f the actual input/output stub resonator
section that is used in the HTS BPF model. This schematic representation is produced by
Academy (TM), another software package by EEsof, which integrates Touchstone's
analysis capabilities with a schematic and layout capture and entry utility.
TLINP
P2
T19
'Z50
TLINP
T16
Z *ZP
L =0.1
K ‘ KP
A ‘ ACP
F =0
CH
TLINP
'L ex tra 2
K ‘ K50
AC50
MSTEP
MSTEP
=0
T2
W1 = 0 .3
W2 *w50
T15
W1 "W50
W2 *Wef
T17
Z ‘ ZS
L ‘ L stub2
K ‘ KS
A ‘ ACS
F =0
MLEF
TIB
W “ Wef
L =0
- O
-c = >
TLINP
T14
S3P
Z
L
K
A
F
S2
‘ Z50
‘ Linp2
‘ K50
*AC50
=0
FILE /h a m e /c ra f t y / d i m i t r i / e e s o f / h t s j p f / m o d e l / t
Fipjurc 4 A schematic representation o f the input/output stub resonator section
Each o f the elements in figure 4 can be seen in the layout o f figure 3, although the
rectangular T seen in figure 3 is represented as a lumped T, by an S-parameter matrix
(S3P), in figure 4. To compensate for the dimensions o f the T, the lengths o f TLINP
(physical transmission lines, see chapter 5 for explanation) elements T19, T14 and T15
(see figure 4) are optimized for best match (minimum integrated square error) between the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S-parameters o f the input/output resonator sections o f figures 2 and 4. The optimization
is performed assuming the ideal case o f perfect conductors. The compensated line lengths
are Lexlrci2, L in p l and Lslub2 (see figure 4). Once the appropriate line lengths are
chosen, the new input/output stub resonator section (figure 4) comprises only straight,
constant-width microstrip lines o f known length and can therefore be modeled using the
theory developed in chapter 3. The low frequency physical parameters o f the microstrip
lines are calculated from their dimensions and the characteristics o f the substrate, using
formulas from Gupta's book [1] and dispersion is accounted for with formulas from
Atwater's book [2], There are several microstrip dispersion models proposed in the
literature ([3]-[7]) but Atwater's model is found to be simple and yet o f adequate
accuracy.
There are three types o f microstrip lines used in the input/output stub
resonator section
and in the filter as a whole. These, together with their dimensions and physical properties,
are listed in table 1, as calculated in a MathCAD file, included as appendix A of this
chapter.
Line Type
The wide line is the open stub line, the
50 Q line is the thin line that connects
the stub to the coupled line part of the
w (pm)
50 Q
172
Wide
500
Input Pad
300
Z h (« )
49.0
31.8
39.9
G
A ir
14.89
2406
15.98
4160
15.37
Tabic 1 The Ihree line types o f the filter and their
properties.
filter and the input p a d is the little
rectangular input line that is used for contact purposes. W here is the width of the line, Z0
is the characteristic impedance o f the line, G is the incremental inductance geometric
factor and eef f \ s the effective dielectric constant o f the line. In this analysis dielectric loss
is negligible and is therefore neglected.
6,2.3 The Coupled M icro strip Resonators Section
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109
The layout o f the coupled microstrip resonators section o f the YBCO HTS BPF is shown
in figure 5.
It consists o f three coupled line resonators
in cascade. The first and last are identical
while the middle one is longer in length
than the other two. Hence there are two
types o f coupled lines in the filter. The
dimension variables o f the coupled line
pairs are shown in figure 6. Table 2
F ig u r e 5 T h e c o u p le d m icrostrip reson ator se c tio n .
contains their dimensions and physical
properties.
|e- w-He— s —
w -3|
W and S are dimensions shown in figure 6,
G is the incremental inductance geometric
factor o f the mode, Z is the zerofrequency characteristic impedance o f the
Ground
mode, £eJI is the zero-frequency effective
F ig u r e 6 D e fin itio n s o f c o u p le d lin e d im e n sio n
relative dielectric constant o f the mode, Zt
varia b les.
W
s
(m n )
(p m )
N a rro w
500
559
2554
W id e
500
1267
2522
C o u p le d
L in e
Go
G0
Ac
Z.„
(Cl)
(« )
(G P L )
1 3 .3 5
4 3 .3
3 8 .5
14.1
4 4 .7
1 4 .1 4
4 1 .2
4 0 .7
13.2
4 9 .3
Ze
(Q )
(fl)
1827
3 6 .0
2 8 .5
1 7 .3 4
1743
3 3 .8
3 1 .4
1 7 .1 5
(G & z)
T a b le 2 T h e tw o ty p e s o f c o u p le d lin e s an d th eir p rop erties.
is the infinite-frequency characteristic impedance o f the mode and f p is the frequency at
which the characteristic impedance o f the mode is approximately equal to the average o f
its zero and infinite-frequency values and also the frequency at which the effective relative
dielectric constant o f the mode is approximately equal to the average o f its zero and
infinite-frequency values (the infinite frequency value being equal to the relative dielectric
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
110
constant o f the substrate). In the previous notation the subscript 'e' denotes a parameter o f
the even mode and the subscript 'o' denotes a parameter o f the odd mode. Examination o f
the last two columns o f table 2 shows that odd mode dispersion can be neglected at the
frequency o f modeling (6-8.5 GHz). Appendix A includes a hard copy of the MathCAD
file that is used to calculate the parameters listed in table 2. Equations from Gupta's book
[1] and Garg's paper [8] are used for the calculations.
A word o f caution to the reader is in order here. Some equations o f references [1] and
[8], for microstrip coupled line calculations, are found to be wrong and are corrected in
appendix A. Specifically, equations (8.85), p. 338 o f [1] and (7a), p. 701 o f [8] for Cga
are wrong by a factor o f 2. They are shown corrected in appendix A. Equations (8.86)
and (8.87), p. 338 o f [1] for Cga, which use an approximation to the elliptic integral, are
correct. However, the corresponding approximation in [8] (equation (7b), p. 701) is
incorrect. Also, equation (18), p. 702 o f [8] for Z; is incorrect; the numerator and
denominator o f its fraction should be interchanged. This is also corrected in appendix A.
6.2.4 T he T ouchstone C ircu it File
The Touchstone circuit files used for the analysis o f the HTS BPF are included as
appendices B and C o f this chapter. In the circuit file o f appendix B dispersion is
neglected whereas in appendix C it is included. As in chapter 5, equations (3.34)-(3.43)
are included in the EQN block o f the Touchstone circuit file. (3.35) and (3.36) are broken
down in smaller sub-equations because they are too long for Touchstone. The EQN block
is sub-divided into five logical sections, each corresponding to one o f the two types o f
coupled lines and three types o f microstrip lines, respectively (see tables 1 and 2). The
parameters o f tables 1 and 2 are also included in the Touchstone file as constants in the
VAR block. The resulting parameters, which are re-calculated for each individual
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
frequency o f the sweep, are three: the loss due to the surface resistance o f the line, in
dB/pm, the corrected characteristic impedance and the corrected effective dielectric
constant, for each type o f line. Each of these is calculated for both the even and the odd
mode. These are fed into a combination o f TLINP elements, which are described in
section 5.2.2, and CL1NP elements. The CLINP element models a pair of coupled
physical transmission lines of known odd and even mode impedance, effective relative
dielectric constant, length and attenuation coefficient. The CLINP and TLINP elements
are interconnected via M STEP elements. These model the step discontinuity in the width
o f the microstrip transmission line. Since these elements are lumped, dimensionless
elements, they are perfectly conducting and do not contribute to the loss of the circuit.
Hence they can be used without modifications in modeling HTS circuits.
Four TLINP and two M STEP elements are used in modeling the input/output stub
resonator section, as shown in figure 4. A zero-length M LE F element, which models the
open end capacitance o f stub lines, is used for terminating the TLINP element which
models the stub line o f the input/output stub resonator section. Because it is zero-length,
it may be used in HTS modeling, for the same reason as the MSTEP elements. Figure 7
shows the schematic representation
CLINP
T5
ZE ‘ ZNe
ZO ‘ ZNa
L ‘ LFIL1
KE ‘ KNe
KO *KNo
AE ‘ ACNa
AO ‘ ACNo
generated by Academy for the
MSTEP
T3
HI 'Wextra
W2 'HFIL1
coupled line resonator section of
the filter.
MLEF
78
W‘ HFILl
L -0
O H " I
Pi
"HH
I—
MLEF
C
3 —
h—
h — 1
MLEF
HHi I W‘WFIL1
,-------- ,
,_____,
1 a°
;%FIL1
Three CLINP, two MSTEP and six
L =0
MLEF
Ti3
,_____,
i|L)H
H -WFIL2
11
L -0
CLINP
MLEF elements are used in this
main section o f the HTS BPF. The
MSTEP elements model the
microstrip line width change
T7
ZE ‘ ZWc
ZO ‘ Z«o
L ‘ LFIL2
KE *KWe
KQ ‘ KHo
»E 'JCWc
AO *ACWa
.
.
\- ~ 4
1
MLEF
CLINP
HI
T6
H ' “ FIL> ZE 'ZNe
L
ZO ‘ ZNa
L ‘ LFIL1
KE ‘ KNe
K0 ‘ KMD
AE ‘ ACNe
AD ‘ ACNo
„
1— 1
MSTEP
^
H
M 'NFIL1
*2
Figure 7 A schematic representation o f the coupled line
resonator section o f the filter.
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P2
112
discontinuity from the 50 Q line o f the input/output stub resonator section to the coupled
line resonator section. The MLEF elements are used to terminate the open end of the
CLINP elements. The length o f the first and last CLINP elements is 1815 pm and is
stored in the variable LFIL1 (in the VAR block) and the length o f the middle CLINP
element is 3004 pm, stored in LFIL2.
6.3 The M odeling Strategy
The strategy is to start the analysis with typical initial values for the zero-temperature
penetration depth, X0, and the normal conductivity, an. The same group that deposited
the YBCO film o f the LPF o f chapter 5 deposited the YBCO film o f the BPF o f this
chapter. Hence, it is reasonable to assume initial values in the range o f values seen in the
LPF o f chapter 5. As in the latter case, the critical temperature, Tc o f the YBCO film is
again not known accurately (the uncertainty is 83 to 88 K), so a Tc of 85 K is assumed for
the analysis. The assumed temperature of the measurements, rI\ is 77 K, the temperature
o f LN 2. The variables X0 and an.are then permitted to assume their optimum values,
within a reasonable domain of values (see section 5.2.3), that minimizes the integrated
square error between the measured and the modeled BPF S-parameters. Two cases are
analyzed. In one case dispersion is neglected
whereas in the other case it is included. The
optimum values extracted for X0 and c n.for
Case
K (nnt)
No dispersion
756.9
Dispersion
642.8
<7 „.(S/m)
1.35-106
-1 0 °
6 .0 0
Tabic 3 The optimum extracted values for the
penetration depth and the normal conductivity.
the two cases are shown in table 3.
6.4 Comparison of Measurement versus Model
6.4.1 The Case of No Dispersion
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113
EEsof - Libra - Wed Jun 23 13:37:19 1993 - bpfm4
□ DB[ S21]
+ DB[ S21]
YBCO
0. OOD
-35. 00
-70. 00
6.000
Figure
8
7.250
FREQ-GHZ
8.500
Measured (YBCO) versus modeled (FLTRBSC) magnitude o f S21.__________________________
Figure 8 shows a comparison of the measured versus modeled magnitude of the insertion
loss (S21) plotted on a vertical dB-scale.
The unexpected notch in the graph o f S21 at 7.63 GHz is thought to be real and not a
measurement error, but it could not be modeled. It is thought to be due to coupling
between the input/output stub resonator and the first coupled line pair o f the main section
of the BPF (see figure 1). A similar notch appears in the measured phase of S21 at that
same frequency, which is plotted in figure 9 together with the modeled phase of S21.
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114
EEsof - Libra - Wed Jun 23 13:37:19 1993 - bpfm4
o ANG[S21]
FLTFBSC
+ ANG[ S21]
YBCO
3.500
rad
0 . 000
-3.500
6 . 000
7. 250
FREQ-GHZ
Figu re 9 Measured (YBCO) versus modeled (FLTRBSC) angle o f S 2 1 .
Figure 10 shows a comparison of the measured versus modeled magnitude of the return
loss (SI 1) plotted on a vertical dB-scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
EEsof - Libra - Wed Jun 23 1 3 :3 7 :1 9 1993 - bpfm4
o DB[ S ll]
FLTRBSC
+ DB[ Sll]
YBCO
5.000
- 20 .
00
-45. 00
6 . 000
7 .2 5 0
FREB-GHZ
B. 500
Figure 10 Measured (YBCO) versus modeled (FLTRBSC) magnitude o f S l l .
Figure 11 shows a comparison o f the measured versus modeled angle o f SI 1. The vertical
scale is in radians.
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116
EEsof - Libra - Wed Jun 23 13:37:20 1993 - bpfm4
D ANG[ S ll]
FLTRBSC
+ ANG[Sll]
YBCO
3. 500
rad
0. 000
-3. 500
6 . 000
7.250
Figure 11 Measured (YBCO) versus modeled (FLTRBSC) angle o f S 11.
Figure 12 shows a comparison o f the measured versus modeled S21, plotted on a Smith
chart of unit radius.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
117
EEsof - Libra - Wed Jun 23 13:41:31 1993 - bpfm4
S21
FLTRBSC
S21
YBCO
B. 00000
B. 50000
Figure
6.4.2
12
Measured (YBCO) versus modeled (FLTRBSC) S21 plotted on a Smith chart o f unit radius.
The C ase of D ispersion
Figure 13 shows a comparison o f the measured versus modeled magnitude of the insertion
loss (S21) plotted on a vertical dB-scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
118
EEsof - Libra - Thu Jun 24 13:51:33 1993 - bpfmdl
o DB[S21]
FLTRBSC
+ DB[S2i]
YBCO
0.000
-35. 00
-70.00
6 . 000
7.250
FREQ-GHZ
B. 500
Figure 13 Measured (YBCO) versus modeled (FLTRBSC) magnitude o f S21.
Figure 14 shows a comparison of the measured versus modeled phase (angle) of S21. The
vertical scale is in radians.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
119
EEsof - Libra - Thu Jun 24 13:51:33 1993 - bpfmdl
□ ANG[ S21]
FLTRBSC
+ ANG[ S21]
VBCD
3. 500
rad
0. 000
-3.500
6 . 000
7. 250
FREQ-GHZ
B. 500
Figure 14 Measured (YBCO) versus modeled (FLTRBSC) angle o f S 2 1.
Figure 15 shows a comparison of the measured versus modeled magnitude of the return
loss ( S l l ) plotted on a vertical dB-scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
EEsof - Libra - Thu Jun 24 13:51:34 1993 - bpfmdl
DB[S11]
FLTRBSC
DB[ Sll]
YBCO
2.000
- 14 .
00
-30. 00
6.000
7.250
FREQ-GHZ
B. 500
Figure 15 Measured (YBCO) versus modeled (FLTRBSC) magnitude o f S 11.
Figure 16 shows a comparison o f the measured versus modeled angle of SI 1. The vertical
scale is in radians.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
121
EEsDf - Libra - Thu Jun 24 13:51:34 1993 - bpfmdi
□ ANG[ Sll]
FLTRBSC
+ ANG[ Sll]
YBCO
3.500
rad
0.000
-3.500
6 . 000
7.250
FREQ-GHZ
a 500
Figure 16 Measured (YBCO) versus modeled (FLTRBSC) angle o f SI I.
Figure 17 shows a comparison o f the measured versus modeled S21, plotted on a Smith
chart of unit radius.
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122
EEsof - Libra - Thu Jun 24 13:51:34 1993 - bpfmdl
S21
FLTRBSC
S21
YBCO
6 . 00000
B. 50000
Figure 17 Measured (YBCO) versus modeled (FLTRBSC) S21 plotted on a Smith chart o f unit radius.
6.4.3 Dispersion or No Dispersion? This is the Question
In the low fractional band width o f this model, dispersion could be neglected. However, it
does add some verisimilitude to some o f the S-parameter curves, compared to the
no-dispersion case. It is important to point out that most dispersion models (including the
one used in this analysis, which is not perfect) incorporate an error on the order o f a few
per cent. Hence, if the correction that the PEM model itself applies to the line parameters
is o f the same order (i.e., a few per cent), then it does not make sense to incorporate any
dispersion model in calculating these parameters. The correction that the PEM model
applies is the factor in parenthesis multiplying t\,lf in equation (3.43)
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123
\
(i.e.,
1
+
-
-
c
A
). In the case o f the BPF o f this chapter, the correction factors
2 ^ V % / Z0
are slightly, but not much, greater than this "noise floor" o f the dispersion model. There
may, therefore, be incosistencies, although some improvements are expected, by the
application o f the distortion model.
One such inconsistency is observed in table 3. The analysis with the dispersion model
yields a more believable value for the zero-temperature penetration depth (in the range
observed for the various LPFs described in chapter 5 and closer to values reported in the
literature), yet an excessively large value for the normal conductivity. Figures 8 and 13, 9
and 14, and 10 and 15 show the no-dispersion model is slightly closer to the measured
data, yet in figures 11 and 16 the dispersion model is significantly closer to the measured
phase o f SI 1 than the no-dispersion model. Both analyses are useful, each for different
purposes, but for larger bandwidth modeling a dispersion model should be included.
6.5 References
[1] K. Gupta, R. Garg and I. Bahl, M icrostrip Lines a n d SIolfines, Artech, Dedham, MA,
1979.
[2] H. A. Atwater, Introduction to Microwave Theory, McGraw-Hill, New York, 1962.
[3] T. G. Bryant and J. A. Weiss, "Parameters o f Microstrip Transmission Lines and
Coupled Pairs o f Microstrip Lines," IEEE Transactions on Microwave Theory and
Techniques, Vol. MTT-16, pp. 1021-1027, December 1968.
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124
[4] M. V. Schneider, "Microstrip Lines for Microwave Integrated Circuits," The Bell
System Technical Journal, Vol. 48, No. 5, pp. 1421-1444, May/June 1969.
[5] E. 0 . Hammerstad, "Equations for Microstrip Circuit Design," Proceedings o f the
European Microwave Conference, Hamburg, W. Germany, pp. 268-272, September 1975.
[6] H. A. Wheeler, "Transmission Line Properties o f a Strip on a Dielectric Sheet on a
Plane," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-25, Mo. 8,
pp. 631-647, August 1977.
[7] E. 0 . Hammerstad and 0 . Jensen, "Accurate Models for Microstrip Computer-Aided
Design," IEEE MTT-S Symposium Digest, pp. 407-409, June 1980.
[8] R. Garg and I. J. Bahl, "Characteristics o f Coupled Microstriplines," IEEE
Transactions on Microwave Theory and Techniques, Vol. MTT-27, No. 7, pp. 700-705,
July 1979.
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125
Appendix A
Sample MathCAD File Used to Calculate the Parameters of the Microstrip
Lines of the BPF
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
Microstrip C o u p le d Lines P a ra m e te rs
D e f i n e t h e el li pti c i n t eg r al :
T O L := 10
-5
'it
2
K ( k ) :=
*
d(j)
1 ~ k 2 * sin [ <f>]2
0
1 -k
K ’ (k) := K
Physical C onstants
e 0 := 8 . 8 5 4 - 1 0 1 2
:= 4 - i t - 10 7
1
•n :=
eo
9
f := 8 - 1 0
Microstrip P a ram eters
h := 2 0 • 1 0 ~ 3 - 2 .5 4 - 1 0 _2
h = 5 .0 8 -1 0
W := 0 . 5 - 1 0
-4
-3
S := 0 . 5 5 9 - 1 0
L
-3
1 .8 1 5 -1 0
t .= 5 -1 0
-3
-7
cr := 2 4 .5
C a p a c i t a n c e s of c o u p l e d Li nes
W
—
h
W cm
1 = ,W + ---------1
L25
------[ 2 - reJ
jc
1 .2 5
vV+------ -i-
1 +In
1 + ln
4„
W
—
t J.
1
F := if
2
W
>1
,
1 + 12-
1+ 12W
- 0 .0 4 '
W
W
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
W em = 5.0171425-10
-4
(cr+1) (cr-1) n (cr-1)
er e .--------------+ ------------ * F _ er e = 1 5 . 9 8 0 1 0 9 8
W
Z o m := i f
W
cm
+ 1. 3 93 + 0 . 6 6 7 *ln
h
L
cm
-+1.444
^ere
2 ’ it '\ l e r e
w cm
Z om = 31.8268821
VV
C P : = £ o ,cr
w
Ajere
C f := 0. 5
jc-Z,
-e0 - c r h
W1
A := e x p - 0 . 1 -exp 2. 3 3 - 2 . 5 3 — i
h j.
r<
Cf
■-
, h
1 +A •-•tanh
S
si"
10-
^crc
C c := C p + C f + C ' f
C c = 3.8556597-10
-10
k :=
(S+2-W)
k = 0.3585632
K ‘( k)
- 1. 5 1 3 8 7 1 8
K(k)
K’(k)
C ga
= co'
K(k)
C „ , = 1.3403821*10
-11
6d
e o ‘cr
-gd
In cot h
+ 0.65 * Cf
( 4* h)
o.o2 r
'A|cr +
S
W
■hi J . j ! _ * 0 . 2 5 - 5 2 -
i--L
^
gt '
- 2 •£ —
£o s
2
cr
h
C () . Cj - + ( . p + ( - g d + ^ ' g a + ^ g t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
C0 =4.2710151*10
-10
er := 1
t
c« s !f£ ll> t !£ L 2 ).F . ! 2 ^ ) .l l U
2
Z om
2
:= if: V
46
W cm
J */ere
r
h
ere = 1
iw
W
+ 1 . 3 9 3 + 0 . 6 6 7 -In
cm
•In
-+1.444
2 *tc•''Jere
+ 0.25 •
W
cm
Z om = 1 2 7 . 2 2 8 3 7 3 5
r
CP
- c
.
W
~ V cr t
'Jere
C f := 0.5
W
E^'cr —
“ ' Z om I
!>
er
P'
V* f
‘■'jCre
1 +A •-•tanh 1 0 *
S
C ’c
C p + c f + c 'f
C ’c = 2 . 2 2 3 6 3 8 3 '10
e 0 -cr
In
'gd =
colh
+ 0 . 6 5 -C
(4-h)
0.02 r
is'
:
—
;
-11
i
•■ycr+: 1 —
cr
2
[h j
C ' o := C f + C p + C g ( 1 + C g a + C g l
C 0 = 3 . 2 0 0 0 5 7 9 MO- 1 1
" Zero f r e q u e n c y " r e s u l t s :
c er e
Z o c :=
c\
c 'Cc
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129
Z oo :=
core
c*./C'o -C 0
C\
Z o e = 3 6 .0 2 4 1 1 1 1
e e re = 1 7 .3 3 9 4 1 9 8
e o r e = 1 3 .3 4 6 6 8 0 9
Z o o = 2 8 .5 3 1 8 6 9
Z oe
f pc
f p c = 1 4 .1 0 7 8 1 3 4 *10"
4 - p 0 -h
Z oo
po
f p o = 4 4 .6 9 4 7 6 3 6 - 1 0 '
H0 ,h ]
G d c := 0 .6 + 0 . 0 0 4 5 -Z o e
G dc = 0 .7 6 2 1 0 8 5
G d o := 0 . 6 + 0 . 0 1 8 -Z o o
G do = 1.1 1 3 5 7 3 6
"jtfW
k c
tanh
4
tanh
'iu .(W + S )’
4
h
h
it_ W
4
h
n (W + S )
tanh
4
Z lc := 6 0
•tanh
h
er := 2 4 .5
. ^ . Kt k e ]
Z tc = 4 3 .3 3 0 4 7 0 5
K| k
A/er
[ Z tc - Z o e
Z e := Z
tc
1.6
1 + G d e ’
pc J
Z e = 3 7 .7 4 2 3 7 2 6
K k0
k
Z to
:= 6 0 - —
Z o e = 3 6 .0 2 4 1 1 11
•
J cr
K k
Z t0 = 3 8 .4 4 6 1 5 3 9
Z U) - Z o o
Z o := Z
to
l l .6
1 +( ’ d o
'
P°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
Zo = 2 9.1 89 109 9
Zoo = 28.531869
(cr-ecrc)
e r e .= c r _
1
f
1+G dc' f
L Pe
j
eere = 1 7 . 3 3 9 4 1 9 8
er e = 1 8 . 7 4 8 8 2
(er-corc)
e r o .= er
•2
f
1 + G do
P°
core = 13.3466809
ero = 13.7308881
C a l c u l a t e n e w i m p e d a n c e s u n d e r r e c e s s i o n of w a l l s by d n o r m a l
d n o r m a l := — -—
1000
t := t ~2 ’d n o r m a l
t = 4 . 9 9 -10~7
W := W - 2 ’d n o r i n a l
W = 4.99999-10 ^
S := S + 2 ’d n o r m a l
S = 5 . 5 9 0 0 1 *10
C a p a c i t a n c e s of c o u p l e d Lines
W
—
W em
=if
h
>
1
,u+ -----125.t
__yj
2 ’j t :
l]
2 -; r
h ;
.
1 + 12’—
h
. . . 1.25
W +
'i
t
ju
-
F := if
I+In
1+ 12*
W
1
2
w:"
1 +ln 4 ’i t ’— :
t ij
W
+0.04
W
t
(cr+1)
(cr-1)
( e r - 1)
2
2
4.6
W cm = 5.0171022-10
ere . ------------ + ----------- ’p - ----------
w
Zom := if
h
r
er e = 1 5 . 9 8 0 1 1 7
W
W
cm
- + 1 . 3 9 3 + 0 . 6 6 7 -In
cm
i.
+ 1. 444
— ’ In
W
2 ’i t ’-’i e r c
W
S .J L - ,0 .2 5 - 5 ! ! .
cm
Z 0| n = 3 1 . 8 2 6 9 8 8 4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-10
Ce = 3.8556486-10
S
k :=
(S+2-W)
C
ga
- Lo
z '
k = 0.3585641
K’(k)
= 1.5138702
K'(k)
K(k)
K( k)
C _ , = 1.3403807-10
-11
ga
Eo ' cr
•In cot h
-gd -
+ 0 . 6 5 -C
(4-h)
0.02
(
*\!cr+
S
cr
h
C gt := 2 ’£ o s
C() .- Cf +Cp +C„j + c ga + c gt
C Q = 4.271002-10
-io
cr := 1
ere = 15. 9801 17
crc;=t££±l) +{££li).F-(£Ill2.
4.6
jw
•\jh
W
W
cm
+ 1.393+0.667-ln
' h
I
vc r c
-l
u
W
; ' ln 8 — -— + 0 . 2 5 ■— —
cm
+ 1. 444
2 -Tt’Vcr c
VV cm
Z o m = 1 2 7 . 22 8 8 2 6 7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C'c : = C p + C f + C ’f
+ 0.65 ’ C j
In coth
-g d
C 'o
C’ e = 2.2236311-10
0.02
'■ V CT +
(4 -h )
-11
1- cr
C f + C p + C g d + C g a + C gt
C’ 0 = 3.2000473-10
-11
" Zero f r e q u e n c y " r esul ts:
C,
1
Z o ed
c c re d :=
'f c 'C e
1
Z o o d :=
c o r e d :=
c ' CV
C ’,
C o
Z o ed = 3 6 .0 2 4 2 2 0 6
cc r e d = 1 7 .3 3 9 4 2 5 6
Z ood = 2 8 .5 3 1 9 5 9 9
c o r c d = 1 3 '3 4 6 6 8 4
C a l c ul a t e the i n c r e m e n t a l i n d u c t a n c e fact or :
G
1
A/ccrcd -Z o e d -A /eerc ‘Z o e
G c = 2 .5 5 4 0 3 9 - 1 0 “
dnorm al
G.
c o r c d -Z o o d -A /eorc "Zoo
dnorm al
G 0 = 1 .8 2 6 6 3 3 7 -1 0 “
2nd T Y P E O F C O U P L E D LINE
Mi cr ostr ip P a r a m e t e r s
h = 5 . 0 8 -1 0 ~4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
W := 0.5*10
-3
S := 1.267*10"
L : = 3 . 0 0 4 *10"3
t := 5 *10 7
c r := 2 4 . 5
C a p a c i t a n c e s o f co u p led Lines
W
Wcm -=if
1
1.25
-------W + ----------- 1'
[2*it:
n
— >P
h
h
1 +ln 2 * t
Y
2
h i
->1, 1 + 1 2 *
1 + 12 —
W
+0.04
W
h
W em = 5.0171425*10
( c rtl) ^ (er-l)
~
Y
2j
W
F = if
i
1.25
W
, W + -------- 1* 1 +l n 4 * i r — ;
it
t :
-4
( c r -1)
er e = 1 5 . 9 8 0 1 0 9 8
2 * 2 " " 46 Y
h
W
ZOITl ^
11
W
W
cm
- + 1.393 + 0 . 6 6 7 *ln
-1
cm
•In
-+1.444
I
h
■\ierc
2*it*Jcrc
w cm
Z om = 31. 8268821
Cp =Co *cr*
W
A^crc
C f := 0.5
W
'er —
c ' Z om I
h
wT
e x p -0.1 *cxp 2. 3 3 - 2 . 5 3 —
h j.
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
C’f :=
s
10h
1+A ---tanh
S
Cc
i ‘' h ™
C p + C f + C ’f
10
C e = 4.0919793-10
k (S+2-W)
k = 0.5588884
K ’( k)
= 1.1954202
K'(k)
^ga
Eo
K(k)
£ o ‘cr
'gd
^
K(k)
C p. = 1.0584251-10
ga
S 11
In co t h It --------+ 0. 65 -C
( 4 *h) J
-11
0.02
■>,/cr +
1
- —
cr
2
= 2 ''» 's
Co
C f + c p + c g d + c g a + c gt
10
C 0 = 3.9888096-10
cr := 1
. ( c r + 1) ( c r - 1)
( c r - 1)
cr c . - -----------+ ----------- - p ------------2
2
4.6
h
ere = 1
jw
Jh
W
Zom := if
- + 1. 393 + 0 . 6 6 7 -In
r
h
W
cm
\ i e rc
rl
t,
cm
-+1.444
w
■In! S — - — + 0.25 — “
2
' K ’\lcTC
w cm
Z om = 127.2283735
r
■
c p
•= E o ' c r T
w
Aicrc
W
c O m
: " £ o ' Cr' 7
!
h
C f := 0.5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
C\
C\
■
■
1 + A *—-ta n h 1 0 - S
h
C ’e
C p + C f + c ’f
C 'e = 2 . 3 8 6 2 3 3 6 ’10
V cr]
'gO
In c o th
+ 0 .6 5 ’C f •
0.02
( 4-h)
-11
er +
er
C o := C f + C p + C g d + C g a + C g t
C 0 = 2 . 8 2 1 4 9 7 8 -10
-11
"Zero f r e q u e n c y " r e su l t s :
1
Z o c :=
e c r c .=
C\
c Wc v
c c
c o r e .=
C,
< F C\> - c 0
Z o e = 3 3 .7 5 6 0 4 0 5
e e r e = 1 7 .1 4 8 2 7 6 4
Z o o = 3 1 .4 4 2 2 1 8 2
° ° re = 1 4 -1 3 7 2 0 6 1
f pe
Z oe
f
[ 4 ^ o -h ]
= 1 3 .2 1 9 5 8 8 3 - 1 0 '
Z oo
po
f po = 4 9 .2 5 3 7 8 3 9 -1 0 '
JJ-o *h I
G dc := 0 .6 + 0 .0 0 4 5 -Z o e
G d e = 0 .7 5 1 9 0 2 2
G d o := 0 . 6 + 0 . 0 1 8 - Z o o
G d o = 1 .1 6 5 9 5 9 9
It
tanh - . W
_4 h
■tanh
it (W + S)
4
h
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
136
J tW
ta n h
4
tanh
h
e r := 2 4 . 5
t t . (W + S)
4
h
Z k , 60. J L . ^ ]
£
KM
Z le = 4 1 .2 3 2 3 7 9 4
[ z te
Z c := Z"te
,
Z oe]
1.6
■f
1+G d e’
, f Pe .
Z e = 3 5 .6 3 8 9 6 7 3
«
Z to
Z o e = 3 3 .7 5 6 0 4 0 5
K '[k 0 ]
=60
£
Z to = 4 0 .6 8 7 3 5 5 6
K[ k o]
[ Z to - Z o o
Z o := Z to
1.6
1 +G
do
po
Z o = 3 1 .9 9 5 3 6 8 7
Z o o = 3 1 .4 4 2 2 1 8 2
(e r - c c r c )
ere := er ■
2
r f
’
1 + G de'
f pc .
c e r e = 1 7 .1 4 8 2 7 6 4
ere = 1 8 .7 3 5 5 8 3 1
( c r - co re)
ero := cr -
1 + G do
f
po
c o r e = 1 4 .1 3 7 2 0 6 1
J
ero = 1 4 .4 4 6 4 5 1 9
■ C a lc u la te n e w i m p e d a n c e s u n d e r r e c e s s i o n o f w a l l s b y d n o r m a l
d n o rm a l :=
t
1000
t : - t - 2 ’d n o r m a l
t = 4 .9 9 - 1 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
137
W :r W - 2 •dnormal
W = 4.9 9 9 9 9 110"4
S := S+2 'dnormal
S = 0.001267001
er := 24.5
C a p a c i t a n c e s of c o u p l e d Lines
W
1
1.25
h'
1.25
W em = ^ — >P------=, W + -------1* 1 + In 2 ' - ,W + ------- f
h [2-jcj
it
t
it
W
F := if
~>1, 1 1 2
+
h
W
W em = 5.0171022-10
W
1+ 12*
' —
W il
1 +ln 4 ' i t ' — :
t ij
+0.04
W
t
(cr+1) (e r - 1 ) ^ ( e r - 1)
ere .------------+---------- * F ----------2
2
4.6
h
ere = 15.980117
W
"j h
W
Zom
h
.
"
Wcm
1
1
W
-+ 1 .3 9 3 + 0 .6 6 7 'in
u
W
In 8 ' —- — +0.25 — ——
cm
-+1.444
r
\'crc
2
ere
Wcm
Z om = 31.8269884
W
C p := c0 'cr
C f := 0.5
•'•ere
W
"e o
c r ’
A - exp -0.1 -exp 2 .3 3 -2 .5 3
W
cr
C f :=
:
h
S :
i 1 + A ’- ' t a n h : 1 0 ' - I
h ;
i
S
L
Cc
dCrc
Cp +C j-+C (■
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
''
138
-10
C c = 4.0919668-10
S
k :=■
(S + 2'W )
k = 0.5588891
K’(k)
K’(k)
C ga ~ e o
K(k)
C„„ = 1.0584242-10
£0 ' er
+ 0 .6 5 -C f *
In coth
-gd
= 1.1954193
K(k)
-11
0.02 r
•Ajer + 1 -
S
(4*h)
er
h
C g l " 2 ’£ o ’ <
C o :~ C f + C p + C g d + C g a + C gt
C 0 = 3.9887977-10
10
c r 1
(cr+1)
;C •=------- +
( c r - 1)
( c r - 1)
*t ----------*
4,6
!w
crc = 1
jh
Zom := if
W
W cm
cm
W
11
— > 1, — • --------+1.393 +0.667-1.1 --------+1.444
h
h
!
h
•\/crc
:-l
,
11
.
j
2 -K'\lcrc
\
L
, ;„
n
-In: 8+ 0.23"
i w cm
.m
W
Z om = 127.2288267
r
W
C p - £o ' c r ' ~
C f := 0.5
'.icrc
W
-c r —
‘ •Zom !
h
1+A ---lanh
S
lo­s i
ll
h
•\icrc
C c = 2.3862336-10
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cm
h
139
In c o th
'g d
C ’D
rc*-
+ 0 .6 5 - C f •
0 .0 2
W cr +
(4 * h )
1- ~
er-*
- c f + c p + c g d + c g a + c gt
C ’0 = 2 . 8 2 1 4 9 * 1 0
11
"Z ero fr e q u e n c y " r e s u lt s :
C,
e e r c d ::
Z oed
C \
c \l< V C c
Z o o d :=
e o r c d :=
C \
Z o ed = 3 3 .7 5 6 1 4 9 5
e c r e d = 1 7 .1 4 8 2 8 2 1
Z o o d = 3 1 .4 4 2 3 0 8 9
c o r e d = 1 4 .1 3 7 2 0 3
C a lc u la t e th e in c r e m e n ta l in d u c t a n c e fa c to r :
1
A /ccred *Z ocd -'J c c r e *Z oc
G,
2 .5 2 1 7 0 8 -1 0 3
d n o rm a l
e o r e d * Z o o d ~ A/corc* Z o o
G
G.
d n o rm a l
F o ‘c
= 1 .7 4 3 3 8 9 1 - 1 0 3
M ic r o s t r i p L in e C a l c u l a t i o n s
50 O hm
M ic r o s t r i p P a r a m e t e r s
W := 0 .1 7 2 * 1 0 ” 3
t := 5 *10
er
-7
2 4 .5
F := if
W
>1
1+ 12
*—
W
1
+
12*
+ 0 .0 4
W
1 --
W
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
140
. . . - (er + 1 )
( C r _ 1 ) T- ( C r _ 1 )
ere
.------------- + ----------»F“ ----------2
2
4.6
P
■\i h
Wem " if
W> r
—
h
1
w+
W
L25
t
It
-4
W
cm
- > l , i
h
nr
W
W + U 5 11 1 +l n 4a *ju*—
1 +l n
2 -ji '
W c m = 1. 7371425*10
Z om ^
1
ere = 1 4 . 893395 5
+ 1. 393 +0. 667 i n
Wcm
+ 1.444
in
f ~
ycre
2 • ir-' j crc
Z om = 4 9 . 0 3 6 2 5 4 6
C a l c u l a t e n e w i m p e d a n c e s u n d e r r e c e s s i o n of w a l l s b y d n o r m a l :
d n o r ma l := — —
1000
t := t - 2 ’d n o r ma l
-7
t = 4. 99*10
W = W - 2 ‘d n o r ma l
F := if
W = 1. 71999 i O
W
h
h
W
—>1. 1+ 12 —
1 + 12 —
-4
W
1-
+ 0. 04
W
crc<l
4-‘
s
.jh
-v > i , 4 h
it
:
W
4*ir—
t
-4
W
Z o m d := i f
!
+
W cm = 1. 7371022*10
1.25
' l*
W
1
1.25
!
:
h
W cm - =i f — >;------ , , W + ------- ' t i 1 + 111; 2 *h 2 -it
L
!
t
crcd = 14. 8 934085
,'crcd
W
cm
+ 1. 393+ 0 . 6 6 7 -in
i
W
K-—t —+0.25 — - -
cm
- +1. 444
2 *it*ycrcd j
w „.„
Z omd = 4 9 . 0 3 6 5 9 1 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
h
141
m
1
-"v/cred 'Z 0 m<i ~A/crc ‘z ora
F0 ' c
dnormal
G m = 7.3326405 *10
Mi cros trip L i n e C a l c u l a t i o n s
W i d e Microstrip P a r a m e t e r s
W := 0 . 5 - 1 O’ 3
t := 5 *10
-7
cr :r 24.5
W
F := if - > 1 ,
1+12 •—
h
+0.04
1+ 12*
W
W
]2
h
W
(er + 1) (c r - 1 )
( c r - 1) |h
ere .= --------- +
*F--------------=—
4.6
crc = 15.9801098
Wcra =»
1 w +------‘'25 v
-W
> r------,,W
h
2 -it
it
w
h'
1*25
1 + 111 2 * - ,W +------*t * 1 + lm 4*Jt*—
t
it
[
1.
_
W cm = 5.0171425*10
W ,
^'om '
—
tl
>1 , ------------
i>
W
-4
W,
cm
-+ 1 .3 9 3 +0.667 *ln
r
vcrc
W,
•1.444
i
2 •jf'Jcrc
-In 8 ----------- + 0 . 2 5 '
W _„
Z om = 31.8268821
C a l c u l a t e n e w i m p e d a n c e s u n d e r r e c e s s i o n o f w a l l s by d n o r m a l :
dnormal := -----1000
t
t-2-dnorm al
t = 4.99 *10 7
W : - W - 2 'dnormal
W = 4.99999*10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4-6
s
Jh
W
1
>;
h [ 2 •%
W cm
1.25
W + -------- 1
crcd = 1 5 . 980 117
1+In
w 1.25
,W +
f
W
1 +l n A
4 - ji —
t
W em = 5 . 0 1 71 0 22 - 10
jw
;
'om d
,
-
>
11
l
! w em ,
— : ---------+ 1 . 3 9 3 + 0 . 6 6 7 - l n
------ +1.444
h
h
i h
w cm
-1
,
n
-In
2 , it*\?crcd
+0. 25 W,
z omd = 31.8269884
1
V c r c d - Z o m d - Vcro ’Z
otn
dnor mal
M
G m = 2.4061695-10
Mi cros t r i p L i n e C a l c u l a t i o n s
I nput P a d Mi c r o s t r i p P a r a m e t e r s
W := 0.3 - 10_3
t
5 -10 7
cr .= 24.5
W
F := if
->
r1
•)
1,
+ 12-
1 + 12 —
W
+ 0.04
W
t
( c r + 1 ) ( c r - 1)
( c r -1)
ere . = ---------- + ------------ F -------------2
2
4.6
crc = 1 5 . 3 669965
_h
jw
■'i h
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
W cn] = 3 . 0 1 7 1 4 2 5 - l ( f 4
->1,-3-
Z om ^
Wcm
r
h
W
L
W
‘ -*0.25
cm
+ 1 . 3 9 3 + 0 . 6 6 7 -In
+ 1.444
In
yerc
W
2 , JC,y e r e
cm
Z om = 39. 9424912
C a l c u l a t e n e w i m p e d a n c e s u n d e r r e c e s s i o n o f w a l l s by d no rma l :
d n o r ma l : = -------
1000
t t - 2 ' d n o r ma l
t = 4. 99*10
W .= W - 2 ‘d n o r ma l
iw
F = ill — > 1,
cr cd
W = 2.99999'10
h ]
1+ 1 2 ' - :
Wj
[2_
,
h :
1+12-— 1
Wj
(cr+1) (cr-1)
( c r - 1)
---------- + -----------' F ------------2
2
4.6
-4
2j
w
+ 0.04- 1 - h
!w
Jh
c r cd = 15 . 3670063
W
W cm : = i f
1
w
1.25
i'
1 h
>*------ : , W + -------- 1 : l + l n . 2 h [l
[
t
1.25
:
, W + --------f j 1+l n
W
4-ir—
t
W cm = 3. 0171022-10
Z omd
* > 1 ,4
W
cm
- + 1.393 + 0. 667 *ln
-1
cm
- + 1.444
in
h
crcd
2 , n ,\ c r e d
W
L
W
h
- + 0. 25 •—
cm
Z omd = 39.9426784
•\ ! crcd' Z( )ln(j - v c r c - Z
■Vc
om
G m = 2.4061695- 10
dnor mal
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
144
Appendix B
Sample Touchstone Circuit File:
The Case of No Dispersion
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
145
[MODEL OF YBCO BPF FILTER
[DIMITRIOS ANTSOS 5-4-93
DIM
FREQ
GHZ
RES
OH
COND
/OH
IND
NH
CAP
PF
LNG
MM
TIME
PS
ANG
RAD
VOL
V
CUR
MA
PWR
DBM
VAR
LDO #140.000000000 756.8542 8000.00000000
T = 77
[TEMPERATURE OF MEASUREMENT
Tc = 85
!Critical Temperture o f Sample
Sn #100000.000000 1351382. 8000000.00000
! CONSTANTS
cO = 8.854E-12
[Permittivity o f free space
[COUPLED LINE PARAMETERS
AGN =0.00000000025
GNc = 2.55404e3
GNo = 1,826634e3
ZN0c= 36.02
ZN()o= 28.53
KN0c= 17.33
KN0o= 13.35
[Incremental Inductance (Narrow gap, even mode)
[Incremental Inductance (Narrow gap, odd mode)
[Even M ode Characteristic Impedance
[Odd Mode Characteristic Impedance
[Even M ode Effective Dielectric Constant
[Odd M ode Effective Dielectric Constant
AGW =0.00000000025
GWc = 2.52171e3
GWo = 1.743389c3
ZW0c= 33.76
ZW0o= 31.44
KW0c= 17.15
K\V0o= 14.13
[Incremental Inductance (Wide gap, even mode)
[Incremental Inductance (Wide gap, odd mode)
[Even M ode Characteristic Impedance
[Odd M ode Characteristic Impedance
[Even Mode Effective Dielectric Constant
[Odd M ode Effective Dielectric Constant
[MICROSTRIP LINE PARAMETERS
[50 Ohm Line
AG5 =0.00000000009
G5 = 7.33264c3
Z5 = 49.03
K5 = 14.89
[Incremental Inductance
[Characteristic Impedance
[Effective Dielectric Constant
[Stub Line
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
AGS =0.00000000025
GS = 2.40617e3
ZSO = 31.83
KSO = 15.98
!Incremental Inductance
!Characteristic Impedance
!Effective Dielectric Constant
llput Pad Line
AGP =1.5e-10
GP = 4160.03
ZPO = 39.94
KPO = 15.37
ilncremental Inductance
!Characteristic Impedance
lEffective Dielectric Constant
Lextra2 #0.30000000000 0.788364 1.50000000000
LFIL1 = 1.815
! 1.815
LFIL2 = 3.004
! 3.004
Wextra = 0.172
WFIL1 = 0 .5
WFIL2 = 0.5
Lstub2 #1.00000000000 1.699751 3.50000000000
Linp2 #0.10000000000 0.294983 0.35000000000
W 50 =0.17200000000
W ef =0.50000000000
eps =24.5000000000
EQN
LD=LDO*le-9
! Computation o f losses for first, narrow spaced, coupled lines
!
Constants
U0 = 4*P I* le-7
c = l/sqrt(e0*U0)
h0=sqrt(U0/e0)
IMagnetic Permeability o f vacuum
!Velocity o f light
!Impedance of free space
f = F R E Q *lc9
IFrequency in Hz
Sr = Sn*(T/Tc)**4
IRcal Part o f conductivity o f YBCO (Sigma 1)
Si = (1-(T/Tc)**4)/(2*PI*P,:U0*LD**2) Hmaginary Part o f conductivity (Sigma2)
P = A TA N(Si/Sr)
lAnglc o f conductivity (Phi)
Th= 5*Pl/4-P/2
lA uxiliaiy angle definition (Theta)
Sigm ag= SQRT(SQR(Sr)+SQR(Si)) INorm o f conductivity
! NARROW -GAP COUPLED-LINE EQUATIONS
! Even Mode
BNe= G N c*A G N *SQ R T(2*Pl*PU 0*Sigm ag)
!B
CNe= EXP(2*BNc*COS(Th))
D N e= COS(2*BNe*SIN(Th))
ENe= SIN(2*BNe*SIN(Th))
U Nc= SQ R T(SQ R (C Nc*D Nc-1)+SQR(CNc*ENc))
!Psi
W Nc= AT A N (C N c*E N e/(C N c*D N c-1))
!Chi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
FNe= B N e/(AGN*sigm ag*UNe)
M N e= 2*BNe*SIN(Th)
N N e= COS(PI/4+P/2-W Ne)
RPNe= COS(M Ne+PI/4+P/2-W Ne)
RiNe = FNe*(NNe+CNe*RPNe)
!Prefactor o f Ri and Xi
llnternal Resistance / Meter
N IN e= SIN(PI/4+P/2-W Ne)
RDN e= SIN(MNe+PI/4+P/2-W Ne)
L iN e = l/(2*PI*f)*FNe*(NINe+CNe*RDNe) llnternal Inductance / Meter
C O R R N e= l+(c/sqrl(KNOe))*(LiNe/ZNOe)!Correction Factor (3.41)
Z Ne = ZNOe*sqrt(CORRNe)
KNe = KNOe*CORRNe
ACNe = (8.686e-3) * RiNe/(2*ZNe)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/mm]
! Odd Mode
BN o=
CNo=
D N o=
E No=
GNo*AGN*SQRT(2*PI*f*U0*sigmag)
EXP(2*BNo*COS(Th))
COS(2*BNo*SIN(Th))
SIN(2*BNo*SIN(Th))
IB
U No= SQRT(SQR(CNo*DNo-l)+SQR(CNo*ENo))
IPsi
W No= AT A N (C No*EN o/(C N o*DN o-1))
IClii
FN o= BNo/(AGN*sigm ag*UNo)
IPrcfactor o f Ri and Xi
M N o= 2*BNo*SIN(Th)
N N o= COS(PI/4+P/2-W No)
RPNo= COS(M No+PI/4+P/2-W No)
R iNo = FNo*(NNo+CNo*RPNo)
llnternal Resistance / Meter
N IN o= SIN(PI/4+P/2-W No)
R DNo= SIN(MNo+PI/4+P/2-W No)
LiNo = l/(2*PI*f)*FNo*(NINo+CNo*RDNo) llnternal Inductance / Meter
CORRNo = l+(c/sqrt(KN0o))*(LiNo/ZN0o) ICorrection Factor (3.41)
ZNo = ZNOo*sqrt(CORRNo)
KNo = KNOo*CORRNo
ACNo = (S.686e-3) * RiNo/(2*ZNo)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in |dB/mm)
I W1DE-GAP COUPLED-LINE EQUATIONS
I Even Mode
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
148
BW e= GW e*AGW *SQRT(2*PI*f*U0*sigmag)
!B
CWe= E XP(2*BW e*C 0S(Th))
DW e= C 0S(2*B W e* SIN (T h»
EWe= SIN(2*BW e*SIN(Th))
UW e= SQRT(SQR(C We*D W e-1)+SQR(CW e*E We))
!Psi
WWe= ATAN(CW e*EW e/(CW e*D W e-l))
!Chi
FWe= BW e/(AGW *sigmag*UW e)
IPrefactor o f Ri and Xi
MWe= 2*BW e*SIN(Th)
NW c= COS(PI/4+P/2-WWe)
RPWe= COS(M W e+PI/4+P/2-W W e)
RiWe = FW e*(NW e+CWe*RPW e)
llnternal Resistance / Meter
NIWe= SIN(PI/4+P/2-WWc)
RDWe= SIN(MW e+PI/4+P/2-W W e)
LiWe = l/(2*PI*f)*FW e*(NIW e+CW e*RDW e) llnternal Inductance / Meter
CORRWe = l+(c/sqrt(KWOe))*(LiWe/ZWOe) ICorreclion Factor (3.41)
ZWe = ZWOe*sqrt(CORRWe)
KWe = KWOe*CORRWe
ACWe = (8.686e-3) * RiW e/(2*ZW e)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/mm]
I Odd Mode
BW o= GW o*AGW *SQRT(2*PI*f*U0*sigmag)
IB
CWo= EXP(2*BW o*COS(Th))
DWo= COS(2*BW o*SIN(Th))
EWo= SIN(2*BW o*SIN(Th))
UWo= SQRT(SQR(CW o*DW o-l)+SQR(CW o*EW o))
IPsi
WWo= ATAN(CW o*EW o/(CW o*DW o-l))
IChi
FWo= BW o/(AGW *sigmag>|;UW o)
IPrefactor o f Ri and Xi
MWo= 2*BW o*SIN(Th)
NWo= COS(PI/4+P/2-WWo)
RPWo= COS(M W o+Pl/4+P/2-W W o)
RiWo = FWo*(NWo+CWo*RPWo)
llnternal Resistance / Meter
NIWo= SIN(PI/4+P/2-WWo)
RDWo= SlN(M W o+Pl/4+P/2-W W o)
LiWo = l/(2*Pl*l)*FW o*(NIW o+CW o*RDW o) llnternal Inductance / Meter
CORRWo = l+(c/sqrl(KAV()o))*(LiWo/ZWOo) ICorreclion Factor (3.41)
ZWo = ZWOo*sqrl(CORRWo)
KWo = KW()o*CORRWo
ACWo = (8.686c-3) * RiW o/(2*ZW o)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in |dB/mm|
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
! MICROSTRIP LINE PARAMETERS
! 50 OHM LINE
B 5= G 5*AG5*SQ RT(2*PI*PU0*sigm ag)
!B
C 5= EXP(2*B5*COS(Th))
D 5= COS(2*B5*SIN(Th))
E 5= SIN(2*B5*SIN(Th))
U 5= SQRT(SQR(C5*D5-1)+SQR(C5*E5))
!Psi
W 5= A TA N(C5*E5/(C5*D 5-1))
!Chi
F5= B 5/(A G 5*sigm ag*U 5)
IPrefactor o f Ri and X i
M 5= 2*B5*SIN(Th)
N 5= COS(PI/4+P/2-W 5)
RP5= COS(M5+PI/4+P/2-W 5)
Ri5 = F5*(N5+C5*RP5)
llnternal Resistance / M eter
N I5= SIN(PI/4+P/2-W 5)
R D5= SIN(M5+PI/4+P/2-W 5)
Li5 = l/(2*PI*f)*F5*(NI5+C5*R D5)
CORR5 = l+(c/sqrt(K5))*(Li5/Z5)
Z50 = Z5*sqrt(CORR5)
K50 = K5*CORR5
llnternal Inductance / Meter
ICorrection Factor (3.41)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
A C50 = (8.686c-3) * Ri5/(2*Z50)
ILoss Coefficient, in fdB/mm]
I STUB LINE
BS= GS*AGS*SQ RT(2*PI*FU0*sigm ag)
IB
CS= EXP(2*BS*COS(Th))
D S= COS(2*BS*SIN(Th))
ES= SIN(2*BS*SIN(Th))
U S= SQRT(SQR(CS*DS-1)+SQR(CS*ES))
IPsi
W S= A TAN(CS*ES/(CS*DS-1))
IChi
FS= BS/(AG S*sigm ag*US)
IPrefactor o f Ri and Xi
M S= 2*BS*SIN(Th)
N S= COS(Pl/4+P/2-W S)
RPS= COS(M S+PI/4+P/2-W S)
RiS= FS*(NS+CS*RPS)
llnternal Resistance / Meter
N IS= SlN(PI/4+P/2-W S)
RDS= SIN(MS+PI/4+P/2-VVS)
LiS = l/(2*PI*i)*FS*(NIS+CS*RDS)
C O R R S= I+(c/sqrt(KSO))*(LiS/ZSO)
ZS = ZSO*sqrt(CORRS)
llnternal Inductance / Meter
ICorreclion Factor (3.41)
ICorrected Char Impedance
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
KS = KSO*CORRS
ACS = (8.686e-3) * RiS/(2*ZS)
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/mm]
I INPUT PAD LINE
B P= GP*AGP*SQRT(2*PI*f*U0*sigmag)
IB
C P=EXP(2*BP*CO S(Th))
D P= COS(2*BP*SIN(Th))
EP= SIN(2*BP*SIN(Th))
UP= SQRT (SQR(CP*DP-1)+SQR(CP*EP))
IPsi
WP= AT AN(CP*EP/(CP*DP-1))
IChi
FP= BP/(AGP*sigmag*UP)
IPrefactor o f Ri and Xi
M P= 2*BP*SIN(Th)
NP= COS(PI/4+P/2-WP)
RPP= COS(M P+PI/4+P/2-W P)
RiP= FP*(NP+CP*RPP)
llnternal Resistance / Meter
NIP= SIN(PI/4+P/2-W P)
RDP= SIN(M P+PI/4+P/2-W P)
LiP = l/(2*PI*f)*FP*(NIP+CP*RDP)
llnternal Inductance / Meter
CORRP = l+(c/sqrt(KP0))*(LiP/ZP0)
ICorreclion Factor (3.41)
ZP = ZPO*sqrl(CORRP)
ICorrected Char Impedance
KP = KPO*CORRP
ICorrected Dielectric Const (3.43)
ACP = (8.686c-3) * RiP/(2*ZP)
ILoss Coefficient, in [dB/mniJ
Rbend =W 50/2
CKT
IRES 1 0 RArcz5
IDEF1P 1 TEST
M SUB_P 1 ERAeps H =0.50800000000 T=0.0005()0000()0 RHO=O.OOt)()OOOOQOO &
RGH=().()0000000000
MSTEP_T3 2 5 WK'Wcxtra W2A\VFIL1
MSTEP_T4 6 3 W1AWFIL1 W 2AWcxtra MLIN W=2.00000000000 L =2.00000000000
CLINP_T5 5 7 8 9 ZEAZNe ZOAZNo LALFIL1 KEAKNc KOAKNo AEAACNc AOAACNo MCFIL &
W =2.00000000000 S=3.00000000000 L=4.00000000000 W l=5.000000()0000 &
W 2=6.00000000000
CLINP_T6 14 1 1 6 12 ZEAZNc ZOAZNo LALFIL1 KEAKNe KOAKNo AEAACNc A O AACNo MCFIL &
W =2.00000000000 S=2.0000()000000 L=2.0000000000() \V1 =2.()0()()()0(K)(K)0 &
\V2=2.00000000000
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
CLINP_T7 8 13 14 15 ZEAZW e ZOAZWo LALFIL2 KEAKWe KOAKW o AEAACW c AOAACWo MCFIL
&
W =2.0()000000000 S=2.00000000000 L =2.00000000000 W l= 2.00000000000 &
W 2=2.00000000000
MLEF_T8 9 WAWFIL1 L =0.00000000000 CPWEGAP W =2.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T9 7 WAWFIL1 L =0.00000000000 CPWEGAP W = l.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T10 12 WAWFIL1 L =0.00000000000 CPWEGAP W =2.00000000000 G=2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T11 11 WAWFIL1 L=0.00000000000 CPWEGAP W =2.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T12 15 WAWFIL2 L=0.00000000000 CPWEGAP W =2.00000000000 G=2.00000000000 &
S=3.00000000000 L =4.00000000000
MLEF_T13 13 WAWF1L2 L =0.00000000000 CPWEGAP W =2.00000000000 G=2.00000000000 &
S=3.00000000000 L=4.00000000000
DEF2P 2 3 MAIN
MSTEP_T2 2 3 W l= 0.30000000000 W2Aw50
TLINP_T17 7 8 ZAZS LALstub2 KAKS AAACS F=0.00000000000 CONN_T15
MLEF_T18 8 WAW ef L =0.00000000000
TLINP_T19 10 9 ZAZ50 LALcxtra2 KAK50 AAAC50 F =0.00000000000 CONN_S2
TLINP_T14 3 4 ZAZ50 LALinp2 KAK50 AAAC50 F=0.00000000000 CONN
S3P_S2 5 4 10 /home/crafty/diniitri/ccsof/IUs_bpf/inodcl/l MTEE W l= 0.50000000000 &
W 2=0.50000000000 W 3=0.50000000000
MSTEP_T15 5 7 W1AW50 W 2AW cf
TLINP_T16 1 2 ZAZP L = 0 .10000000000 KAKP AAACP F =0.00000000000 M.LIN WAw50 &
LALextra2
DEF2P 1 9 LEFTB
MAIN_X 1 1 2
L EFT B _X 23 1
L EFT B _X 34 2
DEF2P 3 4 FLTRBSC
S2P_S1 1 2 0 /honie/crafty/dimitri/ccsof/hls_bpf/hls5cldni MTEE W 1=0.50000000000 &
W 2=0.50000000000 W 3=0.50000000000
DEF2P 1 2 YBCO
RES R4 4 0 RACORRWo ML1N WAW50 LALcxtra2
RES_R6 6 0 RACORRS MLIN W V 5 0 LALcxtra2
RES_R5 5 0 RACORR5 MLIN WA\v50 LALextra2
DEF3P 4 5 6 TEST
TERM
PROC
MODEL
SOURCE
DCTR
FREQ
SWEEP 6 8.5 0.01
POWER
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
FILEOUT
OUTVAR
OUTEQN
OUT
fllrbsc DB[S21] GR1
Y BC O D B [S21] GR1
fltrbsc DB[S11] GR3
YBCO DB[S11] GR3
fllrbsc ANG[S21] GR2
YBCO ANG[S21] GR2
(fltrbsc ANG[S11] GR4
! YBCO ANG[S11] GR4
(MAIN S21 SC2
(YBCO S21 SC2
!TESTRE[Z11] GR5
(FIL DB[S21] GR6
(YBCO DB[S21] GR6
fltrbsc A N G fsll]g r 4
YBCO A N G [sll]g r 4
TESTRE[Z11] SCN
TEST RE[Z22] SCN
TEST REfZ33J SCN
(TEST RE[Z44] SCN
(TEST RE[Z55] SCN
(TEST RE[Z66] SCN
fltrbsc S21 sc2
YBCO S21 sc2
GRID
! RANGE 1 1 1 1
! GR1 -10 0 1
! GR5 .001 .003 .0001
(RANGE 7 9 . 2
! GR6 -2 0 .5
(RANGE 6 8.5 .5
!gr4 -10 0 1
HBCNTL
OPT
YBCO MODEL FLTRBSC
YIELD
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A p p en d ix C
Sample Touchstone Circuit File:
The Case of Dispersion
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
154
'.MODEL OF YBCO BPF FILTER
(DIMITRIOS ANTSOS 5-4-93
DIM
FREQ
GHZ
OH
RES
COND
/OH
IND
NH
CAP
PF
MM
LNG
TIME
PS
RAD
ANG
VOL
V
MA
CUR
PWR
DBM
VAR
LDO #140.000000000 642.7823 1000.0000000
Sn #400000.000000 5999998. 6000000.00000
Lcxtra2 #0.30000000000 0.761918 1.50000000000
Lslub2 #1.00000000000 1.681639 3.50000000000
Linp2 #0.10000000000 0.259163 0.35000000000
T = 77
!TEMPERATURE OF M EASUREM ENT
Tc = 85
(Critical Temperture o f Sample
! CONSTANTS
cO = 8 .8 5 4 E -12
(Permittivity o f free space
!COUPLED LINE PARAMETERS
AGN =0.00000000025
GN c = 2.55404e3
GNo = 1,826634c3
ZNO0c= 36.39
Z N 00o= 28.81
KN0()c= 16.99
KN()0o= 13.08
Z TN c= 43.78
Z TN o= 38.84
!Incremental Inductance (Narrow gap. c\'en mode)
!Incremental Inductance (Narrow gap, odd mode)
'.Even Mode Characteristic Impedance
!Odd Mode Characteristic Impedance
!Even M ode Effective Dielectric Constant
!Odd Mode Effective Dielectric Constant
[Dispersive Impedance Correction
AGW =0.00000000025
GW c = 2.5217 le3
GWo = 1.743389C.3
ZWOOe= 34.10
Z W 00o= 31.75
K W 00c= 16.81
KVV()0o= 13.86
ZTW e= 41.66
ZTW o= 41.11
!Incremental Inductance (Wide gap. even mode)
!Incremental Inductance (Wide gap, odd mode)
lEvcn M ode Characteristic Impedance
!Odd M ode Characteristic Impedance
lEven Mode Effective Dielectric Constant
!Odd Mode Effective Dielectric Constant
(Dispersive Impedance Correction
(MICROSTRIP LINE PARAMETERS
(50 Ohm Line
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
AG5 =0.00000000009
G5 = 7.33264e3
Z5 = 49.53
K500 = 14.60
llncremental Inductance
!Characteristic Impedance
1Effective Dielectric Constant
IStub Line
AGS =0.00000000025
GS = 2 .4 0 6 17e3
ZS0 = 32.15
KS00 = 15.66
llncremental Inductance
!Characteristic Impedance
(Effective Dielectric Constant
!Iput Pad Line
AGP =1.5e-10
GP = 4160.03
ZP0 = 40.35
KP00 = 15.06
LFIL1 = 1.815
LFIL2 = 3.004
llncremental Inductance
!Characteristic Impedance
lEffective Dielectric Constant
! 1.815
! 3.004
Wextra = 0.172
WFIL1 = 0 .5
WFIL2 = 0.5
H=0.508
W50 =0.17200000000
W ef =0.50000000000
Wpad = 0.3
eps =24.0000000000
EQN
L D =L D 0*le-9
! Computation o flo sse s for first, narrow spaced, coupled lines
!
Constants
U0 = 4 * P l* le-7
c = l/sqrt(e0*U0)
hO=sqrl(UO/eO)
(Magnetic Permeability of vacuum
!Velocity o f light
(Impedance o f free space
f= F R E Q * le 9
(Frequency in Hz
Sr = Sn*(T/Tc)**4
IRcal Part o f conductivity of YBCO (Sigma 1)
Si = (1-(T /T c)**4)/(2*PI*PU 0*L D **2) !Imaginary' Part o f conductivity (Sigma2)
P = ATAN(Si/Sr)
!Angle o f conductivity (Plii)
Th= 5*PI/4-P/2
lAuxiliary angle definition (Tlieta)
Sigmag= SQRT(SQR(Sr)+SQR(Si)) INorm o f conductivity
! NARROW-GAP COUPLED-LINE EQUATIONS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
156
GDNe = 0.6+0.0045*Z N 00e
GDNo = 0.6+0.018*Z N 00o
FPNe = 0.3915*ZN00e
FPNo = 1.566*ZN00o
!Even M ode Dispersion Correction Factor
!Odd Mode Dispersion Correction Factor
IDispersion Scaling Frequency, in GHz
IDispersion Scaling Frequency, in GHz
KN0e=eps-(eps-KN00e)/(l+GDNe*sqr(FREQ/FPNe)) IDispersive eff. diel. const.
KN0o=eps-(eps-KN00o)/(l+GDNo*sqr(FREQ/FPNo)) IDispersive eff. diel. const.
ZN0e=ZTNe-(ZTNe-ZN00e)/(l+G DNe*((FREQ /FPNe)** 1.6))
ZNOo=ZTNo-(ZTNo-ZNOOo)/(l+GDNo*((FREQ/FPNo)**1.6))
I Even Mode
BNe= GNe*AGN*SQRT(2*PI*f*U0*Sigmag)
IB
CNe= EXP(2*BNe*COS(Th))
D N e= COS(2*BNe*SIN(Th))
ENe= SIN(2*BNe*SIN(Th))
UNe= SQRT(SQR(CNe*DNe-1)+SQR(CNc*ENe))
IPsi
\VNe= A TA N (C N e*EN e/(C N e*D N e-1))
IChi
FNe= B N e/(A G N *sigm ag*U N e)
IPrefactor o f Ri and Xi
MNe= 2*BNe*SIN(Th)
NNe= COS(PI/4+P/2-W Ne)
RPNc= COS(M Ne+PI/4+P/2-W Ne)
RiNe = FNe*(NNe+CNe*RPNc)
IInternal Resistance / Meter
N lN e= SIN (PI/4+P/2 -WN c)
RDNc= SIN(M Ne+PI/4+P/2-W Ne)
L iN e= l/(2*PI*f)*FN e*(N IN e+CN e*R DN e) IInternal Inductance / Meter
CORRNe= l+(c/sqrt(K N0e))*(LiNe/ZN0e) ICorrcction Factor (3.41)
ZNe = ZNOc*sqrt(CORRNc)
KNe = KNOe*CORRNe
ACNe = (8.6S6c-3) * RiN e/(2*ZN e)
ICorrccted Char Impedance
(Corrected Dielectric Const (3.43)
ILoss Coefficient, in |dB/mm]
I Odd Mode
BNo= G N o*A G N *SQRT(2*PI*FU0*sigm ag)
IB
CNo= EXP(2*BNo*COS(Th))
DNo= COS(2*BNo*SIN(Th))
ENo= SIN(2*BNo*SIN(Th))
UNo= SQRT(SQR(CNo*DNo-1)+SQR(CNo*ENo))
IPsi
WNo= A TA N (C N o*EN o/(C N o*D N o-l))
IChi
FNo= BNo/(AGN*sigm ag*UNo)
IPrefactor o f Ri and Xi
MNo= 2*BNo*SIN(Th)
NNo= COS(Pl/4+P/2-W No)
RPNo= COS(M No+PI/4+P/2-W No)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
RiNo = FNo*(NNo+CNo*RPNo)
llnternal Resistance / Meter
NINo= SIN(PI/4+P/2-W No)
R DN o= SIN(M No+PI/4+P/2-W No)
LiNo = l/(2*PI*f)*FNo*(NINo+CNo*RDNo) '.Internal Inductance / Meter
CORRNo = l+(c/sqrt(K N0o))*(LiNo/ZN0o) 'CorrectionFactor (3.41)
ZNo = ZNOo*sqrt(CORRNo)
KNo = KNOo* CORRNo
ACNo = (8.686e-3) * RiNo/(2*ZNo)
!Corrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/mm]
! WIDE-GAP COUPLED-LINE EQUATIONS
GDW e = 0.6+0.0()45*ZW 00e
lEven Mode Dispersion Correction Factor
GDWo = 0.6+0.018*Z W 00o
!Odd Mode Dispersion Correction Factor
FPWe = O.3915*ZWO0e
IDispersion Scaling Frequency, in GHz
FPWo = 1.566*ZW 00o
IDispersion Scaling Frequency, in GHz
KWOe=eps-(eps-KWOOc)/(l+GDWe*sqr(FREQ/FPWe)) IDispersive eff. diel. const.
KWOo=eps-(eps-KWOOo)/(l+GDWo*sqr(FREQ/FPWo)) 'Dispersive eff. diel. const.
ZW 0e=ZTW e-(ZTW e-ZW 00e)/(l+GDW e*((FREQ/FPW e)** 1.6))
ZW 0o=ZTW o-(ZTW o-ZW 00o)/( l+GDW o*((FREQ/FPW o)** 1.6))
! Even Mode
BW e= GW e*AGW *SQRT(2*Pl*f*U0*sigmag)
IB
CWe= EXP(2*BW c*COS(Th))
D W e= COS(2*BW c*SIN(Th))
EW e= SIN(2*BW c*SIN(Th))
UW c= SQRT(SQR(CW c*DW e-1)+SQR(CW c*EW c))
!Psi
W We= ATAN(CW e*EW c/(CW e*DW e-l))
!Chi
FW e= BW e/(AGW *sigmag*UW c)
IPrefactor o f Ri and Xi
M W e= 2*BW c*SlN(Th)
NW e= COS(PI/4+P/2-W W e)
RPWc= COS(M \Vc+Pl/4+P/2-W W c)
RiW c = FW c*(N\Ve+CW cI,<RPWe)
llnternal Resistance / Meter
NIW e= SIN(Pl/4+P/2-W W e)
RDW e= SIN(MW e+PI/4+P/2-W W c)
LiWe = ]/(2*Pl*f)*FW c*(NIW c+CW c*RDW c) llnternal Inductance / Meter
CORRW e= l+(c/sqrt(KWOc))’|:(LiVVc/ZWOe) ICorreclion Factor (3.41)
ZWe = ZW()c*sqrl(CORRWe)
ICorrected Char Impedance
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158
K W e = KWOe*CORRWe
ACW e = (8.686e-3) * RiW e/(2*ZW e)
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/mrn]
! Odd Mode
B W o= GW o*AGW *SQRT(2*PI*f*U0*signtag)
!B
CW o= EXP(2*BW o*COS(Th))
D W o= C O S(2*BW o*SIN (Th»
EW o= SIN(2*BW o*SIN(Th))
UW o= SQRT(SQR(C Wo*D W o-1)+SQR(C Wo*E Wo))
!Psi
WWo= A TA N(CW o*EW o/(CW o*D W o-l))
ICIti
FW o= BW o/(AGW *signtag*UW o)
IPrefactor o f Ri and Xi
M W o= 2*BW o*SIN(Th)
N W o= COS(PI/4+P/2-W W o)
RPW o= COS(MWo+PI74+P/2-WWo)
RiWo = FW o*(NW o+CW o*RPW o)
llnternal Resistance / Meter
NIW o= SIN(PI/4+P/2-W W o)
RDW o= SIN(M W o+PI/4+P/2-W W o)
LiWo = l/(2*PI*f)*FW o*(NIW o+CW o*RDW o) llnternal Inductance / Meter
CORRWo = l+(c/sqrt(K W 0o))*(LiW o/ZW 0o) ICorrection Factor (3.41)
ZWo = ZWOo*sqrt(CORRWo)
KWo = KWOo*CORRWo
ACW o = (8.686C-3) * RiW o/(2*ZW o)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/ntml
I MICROSTRIP LINE PARAMETERS
I 50 OHM LINE
I Dispersion
FD50=4*H *FR E Q /300*sqrt(eps-l)*(0.5+sqr(l+2*LO G (l+W 50/H )))
K 5=K 500*SQ R (l+(SQ R T (cps/K 500)-l)/(l+4*F D 50**(-1.5)))
IPEM Equations
B5= G5*AG5*SQRT(2*PPP=U0*sigmag)
IB
C5= EXP(2*B5*COS(Th))
D 5= COS(2*B5*SIN(Th))
E5= SIN(2*B5*SIN(Th))
U 5= SQRT(SQR(C5*D5-1)+SQR(C5*E5))
IPsi
W5= A TA N (C 5*E 5/(C 5*D 5-1))
IChi
F5= B5/(A G 5*sigm ag*U 5)
IPrefactor of Ri and Xi
M 5= 2*B5*SJN(Th)
N 5= COS(PI/4+P/2-W 5)
RP5= COS(M 5+PI/4+P/2-W 5)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
159
Ri5 = F5*(N5+C5*RP5)
llnternal Resistance / Meter
NI5= SIN(PI/4+P/2-W 5)
RD5= SIN(M 5+PI/4+P/2-W 5)
Li5 = l/(2*PI*f)*F5*(NI5+C5*RD5)
C 0RR 5 = l+(c/sqrt(K 5))*(Li5/Z5)
Z50 = Z5*sqrl(CORR5)
K50 = K5*CORR5
llnternal Inductance / Meter
(Correction Factor (3.41)
ICorrected Char Impedance
ICorrected Dielectric Const (3.43)
AC50 = (8.686e-3) * Ri5/(2*Z50)
ILoss Coefficient, in [dB/mm]
I STUB LINE
IDispersion
FDS=4*H *FREQ/300*sqrl(eps-l)*(0.5+sqr(I+2*LO G(l+W ef/H)))
KSO=KSOO*SQR(l+(SQRT(eps/KSOO)-l)/(l+4*FDS**(-1.5)))
I PEM Equations
.
BS=GS*AGS*SQ RT(2*PI*f*U0*sigm ag)
IB
CS= EXP(2*BS*COS(Th))
D S= COS(2*BS*SIN(Th))
ES= SIN(2*BS*SIN(Th))
US= SQRT(SQR(CS*DS-1)+SQR(CS*ES))
IPsi
WS= A TAN(CS*ES/(CS*DS-1))
IChi
FS= BS/(AGS*sigm ag*US)
IPrefactor o f Ri and Xi
MS= 2*BS*SIN(Th)
N S= COS(PI/4+P/2-WS)
RPS= COS(M S+Pl/4+P/2-W S)
RiS=FS*(N S+CS*R PS)
llnternal Resistance / Meter
NIS= SlN(PI/4+P/2-W S)
RDS= SlN(M S+PI/4+P/2-W S)
LiS = l/(2*PI*f)!,:FS*(NlS+CS*RDS)
llnternal Inductance / Meter
CORRS = l+(c/sqrl(KSO))*(LiS/ZSO)
ICorrection Factor (3.41)
ZS = ZS0*sqrt(CORRS)
KS = KS0*CORRS
ACS = (8.686c-3) * RiS/(2*ZS)
(Corrected Char Impedance
ICorrected Dielectric Const (3.43)
ILoss Coefficient, in [dB/inni]
I INPUT PAD LINE
IDispersion
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160
FDP=4*H *FREQ/300*sqrt(eps-l)*(0.5+sqr(l+2*LOG (l+W pad/H )))
KP0=K P00*SQ R (l+(SQ R T(eps/K P00)-l)/(l+4*FD P**(-1.5)))
! PEM Equations
BP= GP*AGP*SQRT(2*PI*Pt=U0*sigmag)
CP= EXP(2*BP*COS(Th))
DP= COS(2*BP*SIN(Th))
EP= SIN(2*BP*SlN(Th))
UP= SQRT (SQ R(CP*DP-1)+SQR(CP*EP))
WP= ATAN(CP*EP/(CP*DP-1))
FP= BP/(AGP*sigmag*UP)
MP= 2*BP*SIN(Th)
NP= COS(PI/4+P/2-W P)
RPP= COS(MP+PI/4+P/2-W P)
RiP= FP*(NP+CP*RPP)
!B
!Psi
!Chi
IPrefactor o f Ri and Xi
llnternal Resistance / Meter
NIP= SIN(PI/4+P/2-W P)
RDP= SIN(M P+PI/4+P/2-W P)
LiP = l/(2*PI*f)*FP*(NIP+CP*RDP)
llnternal Inductance / Meter
CORRP = l+(c/sqrt(KP0))*(LiP/ZP0)
ICorrection Factor (3.41)
ZP = ZPO*sqrt(CORRP)
ICorrected Char Impedance
KP = KPO*CORRP
ICorrected Dielectric Const (3.43)
ACP = (S.6S6c-3) * RiP/(2*ZP)
ILoss Coefficient, in [dB/mm]
Rbcnd =W 50/2
CRT
IRES 1 0 RArcz5
IDEF1P 1 TEST
MSUB_P1 ERAeps H=().50800000000 T=().00050000000 RHO=0.0000000()00() &
RGH=0.00000000000
MSTEP_T3 2 5 W lAWcxtra W2AWFIL1
MSTEP_T4 6 3 W1AWFIL1 W2AWe.\tra
CLINP_T5 5 7 8 9 ZEAZNe ZOAZNo LALFIL1 KEAKNc KOAKNo AEAACNe AOAACNo MCFIL &
W =2.00000000000 S=3.00000000000 L=4.00000000000 W 1=5.00000000000 &
W 2=6.00000000000
CLINP_T6 14 1 1 6 12 ZEAZNe ZOAZNo LALF1L1 lvEAKNc KOAKNo AEAACNc AOAACNo MCFIL &
W=2.0()0000()0000 S=3.000000()0000 L=4.00000000000 W 1=5.00000000000 &
W2=6.0000()()000()0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
161
C L IN P T 7 8 13 14 15 ZEAZWe ZOAZWo LALFIL2 KEAKWe KOAKWo AEAACWe AOAACWo MCFIL
&
W =2.00000000000 S=3.00000000000 L=4.00000000000 W l= 5.00000000000 &
W 2=6.00000000000
MLEF_T8 9 WAWFIL1 L=0.00000000000 CPWEGAP W= 1.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T9 7 WAWFIL1 L=0.00000000000 CPWEGAP W= 1.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T10 12 WAWFIL1 L =0.00000000000 CPWEGAP W = l.00000000000 G =2.00000000000 &
S=3.00000000000 L =4.00000000000
M LEF_T11 11 WAWFIL1 L=0.00000000000 CPWEGAP W = l.00000000000 G =2.00000000000 &
S=3.00000000000 L =4.00000000000
MLEF_T12 15 WAWFIL2 L =0.00000000000 CPWEGAP W= 1.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
MLEF_T13 13 WAWF1L2 L =0.00000000000 CPWEGAP W = l.00000000000 G =2.00000000000 &
S=3.00000000000 L=4.00000000000
DEF2P 2 3 M AIN
MSTEP_T2 2 3 W l= 0.30000000000 W2A\v50
TLINP_T17 7 8 ZAZS LALstub2 KAKS AAACS F=0.00000000000 CONN_T15
MLEF_T18 8 WAW ef L =0.00000000000
TLINP_T19 10 9 ZAZ50 LALexlra2 KAK50 AAAC50 F=0.00000000000 CONN_S2
TLINP_T14 3 4 ZAZ50 LALinp2 KAK50 AAAC50 F=0.00000000000 CONN
S3P_S2 5 4 1 0 /home/crafty/dimitri/cesof/hts_bpf/inodeI/t MTEE W 1=0.50000000000 &
W 2=0.50000000000 W 3=0.50000000000
MSTEP_T15 5 7 W1AW50 W2AW ef
TLINP_T16 1 2 ZAZP L = 0.10000000000 KAKP A AACP F=0.00000000()00 ML1N WA\v50 &
LALcxtra2
DEF2P 1 9 LEFTB
MAIN_X1 1 2
L E F T B _X 23 1
L E F T B _X 34 2
DEF2P 3 4 FLTRBSC
S2P_S1 1 2 0 /hom e/ciafly/diinitri/cesof/hls_bpl7hls5cldin MTEE W 1=0.50000000000 &
W 2=0.50000000000 W 3=0.50000000000
DEF2P 1 2 YBCO
RES_R4 4 0 RAKWOO MLIN WAW50 LALcxtra2
RES_R6 6 0 RAKW0E MLIN WAw50 LALcxlra2
RES_R5 5 0 RAKNOO MLIN WA\v50 LALcxlra2
DEF3P 4 5 6 TEST
TERM
PROC
MODEL
SOURCE
DCTR
FREQ
SWEEP 6 8.5 0.01
(STEP 8
POWER
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162
FILEOUT
OUTVAR
OUTEQN
OUT
fltrbsc DBIS21] GR1
YBCO DB[S21] GR1
fltrbsc DB[S11] GR3
YBCO D B [S 111 GR3
fltrbsc ANG[S21] GR2
YBCO ANG[S21] GR2
Ifltrbsc A N G [S 11] GR4
! YBCO A N G [S 11] GR4
IMAIN S21 SC2
!YBCO S21 SC2
ITEST R E[Z11] GR5
!FIL DB[S21] GR6
IYBCO DB[S21] GR6
fltrbsc A N G [sl 1] gr4
YBCO A N G [sl 1] gr4
TEST REJZ11] SCN
TEST RE[Z22J SCN
TEST RE[Z33] SCN
ITEST RE|Z44] SCN
ITEST RE|Z55] SCN
ITEST RE[Z66| SCN
GRID
I RANGE 1 1 1 1
I GR1 -10 0 1
I GR5 .001 .003 .0001
IRANGE 7 9 .2
! GR6 -2 0 .5
IRANGE 6 8.5 .5
Igr4-10 0 1
HBCNTL
OPT
YBCO MODEL FLTRBSC
YIELD
TOL
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163
CHAPTER 7
A M O DIFICATIO N OF THE PEM LOSS M ODEL FOR HIGH-LOSS
M ODELING.
AN APPLICATION TO HIGH-POW ER MODELING
7.1 A M odification of the PEM Loss Model. The High-Loss Case
Beyond the transmission line’s inductance and capacitance per unit length the PEM model,
presented in chapter 3, gives an additional impedance per unit length, Z,-, that is due to the
field penetration into the conductors.
It can be used with TEM and quasi-TEM (e.g.,
microstrip) lines. Equations (3.34) and (3.41)-(3.44) are, as was pointed out in chapter 3,
only first-order approximations and work well when R h the additional distributed internal
resistance due to the field penetration, is small relative to (oL. This is true of most HTS
transmission lines in their low-power linear region. However, examples of some cases in
which this condition may be violated are if T « Tc , or if, say due to high transfer currents,
— « 1 , or if the HTS film is so thin that 0 < — «
n
X
1, or if the width o f the transmission line
is very small. The last case is encountered in the HTS LPF presented in chapter 5, where
the high-impedance lines are 33 times narrower than the low-impedance lines. One device
of this design will be used in this chapter as a case-point for application of the improved
PEM loss model. The chosen device has an HTS film with a particularly high-penetration
depth to make the effect o f the loss more pronounced.
The more accurate equations, to substitute (3.34) and (3.41 )-(3.44), are derived using the
following general equations that describe a wave propagating in a TEM medium [1]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
a 'c + jP " = -JjcoCiRj +ja>L')
(1)
and
A +JcoL'
Zo =
.
'
(2)
jd C
assuming G, the "leakage" conductance per unit length o f the transmission line, to be zero.
H e re !,'is the total distributed inductance o f the transmission line (see equation 3.38), C is
the distributed capacitance o f the transmission line and co is the angular frequency of the
excitation. When (1) and (2) are used in conjunction with the model presented in chapter
3, the following more accurate equations are obtained for a'c, the exponential attenuation
per unit length, ft", the corrected propagation constant o f the wave and Z"0, the corrected
characteristic impedance o f the transmission line
< = P 1+ T
2
6
sin
v2
P ’= P 1 +
cos
f e
v2
kQ .
(3)
(4 )
and
'-0
7/
'-Q 1 +
f0
IsJ
2”
e Ji
(5)
where
6 = arctan
(6 )
and
0=
coL'
(7)
~R~
From (4) and (5) corresponding expressions for v"p/l, the corrected phase velocity o f the
wave and s"eff, the corrected effective relative dielectric constant of the transmission line
may also be derived as follows
.1
e
Vph, = Vph. l +
sec
(3)
\ Q.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The variables /?', 7J0, L \ Rh v'p/l and e'ej j in equations (3)-(9) are the first-order
approximations for these physical quantities, as defined in equations (3.44), (3.41), (3.38),
(3.35), (3.42) and (3.43) respectively. Clearly, as 0 becomes large compared to unity, the
double-primed variables o f equations (4), (5), (8) and (9) tend to their single-primed
counterparts o f equations (3.41), (3.35), (3.42) and (3.43) respectively. This relationship
is not as obvious in the case o f a 'c and a c (equations (3) and (3.34)) but it can be easily
shown using first-order Taylor expansions to evaluate the square root and the sine o f
equation (3).
7.2 From a Complex to a Real Characteristic Impedance
The main difference between the primed and double-primed equations is that Z"0 in
equation (5) is a complex quantity. This has the physical interpretation that the current
and voltage are out of phase throughout the transmission line. The TLINP transmission
line modeling element of Touchstone and the corresponding elements o f most other
commonly used microwave CAD software packages (DragonWave, SuperCompact, Puff),
however, can only take a real impedance as an input. In the example o f the HTS LPF of
chapter 5, the narrow, high-impedance line has a corrected impedance, the real part and
magnitude o f which may be different by
as much as 5%.
Hence the question
what is the best impedance to use for
RES
I NO
INO
RES
RI
R 'Rzn
LI
L 'Lzn
L?
l ‘Lzn
P2
P Ten
t > W A r ' ' VYY v v ' * A / W < i
~ CAP
modeling the lossy transmission line
arises.
■ C!
C 'Cz
Candidates are the impedance
as given by equation (3.41) and the
Figure 1 U n i t cell o f l a d d e r n e t w o r k m o d e l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
166
magnitude or the real part o f the impedance as given by equation (5). To determine the
best alternative the narrow transmission line used in the HTS CPW LPF of chapter 5 is
modeled using a ladder RLC network o f 128 elementary cells o f length Dz. The CPW
impedance o f the line is 83 Q and its length is 997 pm. The unit cell o f the ladder network
is shown in figure 1. Equation (3.35) is used to obtain R, the series resistance o f a length
Dz o f the transmission line. Equations (3.36) and (3.38)-(3.40) together with the
knowledge o f the effective relative dielectric constant and characteristic impedance of the
line are used to obtain L and C, the inductance and capacitance, respectively, o f a length
Dz o f the transmission line. The calculated S-parameters o f this model are compared to
the S-parameters o f the transmission line model with each o f the three candidate real
impedances mentioned above. Figures 2 and 3 show'the plots o f SI 1 and S21 for each of
the 4 calculated responses respectively (LAD is the ladder network and LINE1, LfNE2,
EEsof - Touchstone - Mon Sep 14 1 5 :3 0 :5 5 1992 - narrow
DB[ S il]
LAD
DB[ S ll]
LINE1
DB[ S ll]
LINE2
DB[ S ll]
LINE3
-5 . 000
dI3
•18 .
00
-2 8 . 00
0 . 000
7. 000
FREQ-GHZ
15. 00
F i« u r e 2 Magnitude o f S l l o f ladder and transmission lines.
LTNE3 are the transmission line models with each o f the candidate impedances mentioned
above, respectively), plotted versus frequency.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
167
EEsof - Touchstone - Mon Sep 1 A 1 5 :3 0 :5 5 1992 - narrow
□
DB[ S21 ]
LAD
+
DB[ S21]
LINE1
<> DB[S21]
LINE2
x
DB[S21]
LINES
0. OOO
-1 . 500
-3 . 000
0. 000
7. 000
FREQ-GHZ
15. 00
Figure 3 Magnitude o f S 2 1 o f ladder and transmission lines.
Figures 4,5 and 6 show the difference in magnitude (left scale, dB) and phase (right scale,
rad) o f S l l and S21 o f the ladder network from that of each of the three candidate
transmission lines. As may be seen by comparison of figures 4-6, LINE1 best models the
behavior o f the ladder network (which faithfully emulates the complex impedance given by
equation (6)). This corresponds to the impedance given by equation (3.41), which is used
in all subsequent modeling.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
168
EEsof - Touchstone - Mon Sep 14 15: 30: 55 1992 - narrow
+ P211
□ PHI
OUTEQN
DB
R211
OUTEQN
ANG
OUTEQN
DB
R ill
OUTEQN
ANG
1.000
'0.100
rad
dB
0 . 000
= s
at----- ■Ait--- ■■** --- BS- ----- i
—
—
—
S==
----- < - 0 . 000
- 1. 000
-
0 .0 0 0
7 .0 0 0
0 .1 0 0
15. 00
FREQ-GHZ
Figure 4 S-paramctcr differences between LAD and LINE1.
EEsof - Touchstone - Mon Sep 14 1 5 :3 0 :5 6 1992 - narrow
□
P112
OUTEQN
DB
P212
x
OUTEQN
DB
R212
OUTEQN
ANG
o
R112
OUTEQN
ANG
1.000
'
dB
0.100
rad
_pq----- — R - — p - ------- E
0 . 000
... -W- — — s
—
- e — 1—e — ^
s>—
* — ■- - X — j
--------
-1.000
\
■0 . 000
-0. 100
0 .0 0 0
7 .0 0 0
FREQ-GHZ
15. 00
Figure 5 S-paramctcr differences between LAD and L1NE2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
169
EEsof - Touchstone - Mon Sep 14 15:30:56 1992 - narrow
PI i 3
OUTEQN
P213
OUTEQN
R213
OUTEQN
ANG
RU3
OUTEQN
ANG
1.000
0.100
dB
rad
0 . 000
0 . 000
-1.000
-
0 . 000
7. 000
FREQ-GHZ
0.100
15. 00
Figure 6 S-paramctcr differences between LAD and LINES.
Figure 7 shows the three candidate impedances plotted versus frequency.
7.3 Application of the High Loss Model to a LPF of Chapter 5
When equations (3),(6),(7),(9) and (3.41) are incorporated into the loss model of the
CPW LPF described in chapter 5, a better fit between measured and modeled data is
achieved.
Specifically, the integrated squared error between the modeled and the
measured S-parameters decreases from 0.06710 to 0.06059, a 9.7 % decrease.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
170
EEsof - Touchstone - Mon Sep 14 15:30:56 1992 - narrow
□ RE[Z11]
TEST1
+ RE[Z11]
TEST2
o RE[ ZU]
TEST3
111.0
Ohms
102.0
93. 00
0 . 000
7. 000
FREQ-GHZ
15.00
Figure 7 The three candidate impedances plotted versus frequency.____________________________________
Figures 8-15 show the old and the new fit o f the model to the measured S-parameters of
the HTS CPW LPF o f chapter 5. As mentioned above, a device with high penetration
depth is chosen to accentuate the differences between the first-order and the high-loss
model. As a result, the pass-band insertion-loss o f the device o f this chapter is larger than
that of chapter 5 (compare figures 8 and 5.4).
Appendices A and B include the corresponding Touchstone circuit files.
The greatest
improvement is seen in the better modeling o f the phase of SI 1 from 9 to 12 GHz (figures
14 and 15).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
171
EEsof - T ouchstone - Tue Sep
□
DB[ S21]
FIL
+
8 10: 41: 53 1992 - -5dbm
DB[ S21]
YBCO
0 . 000
-
20 . 00
-40. 00
0 . 000
6 . 000
FREQ-GHZ
1 2 . 00
Figure 8 First-order model, magnitude o f S21, measured (YBCO) versus modeled (FIL).
EEsof - Touchstone - Tue Sep
□
DB[ S21]
FIL
+
B 10: 55: 52 1992 - -5dbmn
DB[ S21]
YBCO
0 . 000
-
20 . 00
-40. 00
0 . 000
6 . 000
FREQ-GHZ
12 . 00
Figure 9 High-loss model, magnitude o f S 2 1, measured (YBCO) versus modeled (FIL).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
172
EEsof - Touchstone - Tue Sep
□
ANG[ S21]
FIL
+
8 1 0 :4 2 :0 5 1992 - -5dbm
ANG[ S21]
YBCO
3. 500
rad
0 . 000
-3. 500
0 . 000
6 . 000
FREQ-GHZ
1 2 . 00
Figure 10 First-order model, phase o f S21. measured (YBCO) versus modeled (FIL).
EEsof - Touchstone - Tue Sep
□
ANG[ S21]
FIL
+
B 10: 55: 55 1992 — 5dbmn
ANG[ S21]
YBCQ
3. 500
rad
0 . 000
-3 . 500
0 . 000
6 . 000
FREQ-GHZ
12 . 00
Figure 11 High-loss model, phase o f S 2 1. measured (YBCO) versus modeled (FIL).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
173
EEsof - Touchstone - Tue Sep
□
DB[ S l l ]
FIL
+
B 10: 42: OB 1992 - -5dbm
D B tS ll]
VBCO
0 . 000
dB
-14. 00
-
28 . 00
0 . 000
B. 000
FREQ-GHZ
12 . 00
Figure 12 First-order model, magnitude o f SI 1, measured (YBCO) versus modeled (FIL).
EEsof - Touchstone - Tue Sep
□
D B tS ll]
FIL
+
B 10: 55: 56 1992 — 5dbmn
D B tSll]
YBCD
0 . 000
- 14 . 00
-28. 00
0 . 000
6 . 000
FREQ-GHZ
1 2 . 00
Figure 13 High-loss model, magnitude o f SI 1. measured (YBCO) versus modeled (FIL).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
174
EEsof - T ou ch stone - Tue Sep
ANG[ S l l ]
FIL
B 10: 4 2 :1 2 1992 - -5dbm
ANG[SU]
YBCQ
3 .5 0 0
rad
0. 000
-3 . 500
0 . 000
6 . 000
FREQ-GHZ
12 . 00
Figure 14 First-order model, phase o f SI 1, measured (YBCO) versus modeled (FIL).
EEsof - T ouchstone - Tue Sep
ANGtSll]
FIL
B 1 0 : 5 5 : 5B 1992 — 5dbmn
ANG[ S l l ]
YBCO
3. 500
rad
0 . 000
-3 . 500
0 . 000
B. 000
FREQ-GHZ
12 . 00
Figure 15 High-loss model, phase o f S 2 1, measured (YBCO) versus modeled (FIL).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
7.4 Application of the High Loss Model to High Power Modeling
7.4.1 Introduction
The superconducting state is maintained in a superconductor as long as it is energetically
favorable for electrons to be paired-up into Cooper pairs [2], When a high magnetic field
is applied to a superconductor (higher than a value called the critical field, Hc), the lower
energy state for the electrons is not the paired state anymore. Hence the electron pairs are
destroyed and with them the superconducting properties o f the material.
In Type I
superconductors this transition is an abrupt one with respect to the applied magnetic field.
In Type II superconductors and high temperature superconductors the transition is more
gradual.
It starts at the low critical field value Hc i, when the first electron pairs are
broken up, and is complete at the high critical field value H c 2, when all electron pairs have
been destroyed and the material is not superconducting anymore.
In the case o f high-temperature superconductors, which, due to their crystalline nature, are
insulators when non-superconducting, this is a very important effect, as it totally alters the
material properties. When the applied magnetic field is in the range from Hc ] to Hc 2, the
HTS appears lossy due to the deficiency in superconducting electron pairs. Associated
with the critical magnetic fields are critical currents, which produce magnetic fields that
can also drive the material non-superconducting. Hence the behavior of superconducting
devices is non-linear with respect to input power. Proper modeling o f this behavior may
provide the capability for new innovative circuit designs (e.g., a power-sensitive, switching
band-selection filter to protect the input of a sensitive receiver) and put bounds on the
power-handling capability of superconducting devices. An attempt at first-order modeling
o f this behavior is presented below, using the high-loss PEM model presented in 7.1.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
176
7.4.2 High Power M easurements
T o m odel th is p o w e r-d e p e n d en t beh av io r o f H T S s, m easu rem en ts o f devices a t different
input p o w ers are required. O n e o f th e H T S Y B C O C P W L P F s o f ch ap ter 5 is selected fo r
th e se m easurem ents.
T h e experim ental setup is sh o w n in fig u re 16.
A n H P 8 5 IOC
n e tw o rk analyzer is used in
HP 8510
N etw ork
A nalyzer
tan d em w ith an H P 8349B
SMA - N c o n n ecto r
so lid -state am plifier and a 10
dB
a tte n u a to r
(to
avoid
ov erlo ad in g p o rt 2
o f th e
— 3.5 nrt c a b le s
•3.5 - 3.5 nn c o n n e c to r
10 dB a t t e n u a t o r
8510).
The
8510
is
HP 8349B amplifier
LPF
calibrated fo r p o w er flatness,
D ew ar
using an H P
8 4 8 IB
p o w er
and
sen so r
an
LN2
highHP
F i g u r e 16 H ig h - p o w c r m e a s u r e m e n t s e tu p .
43 7 B p o w e r m eter (i.e., th e
o u tp u t p o w e r o f p o rt 1 is c o n sta n t th ro u g h o u t th e frequency sw eep from 1 to 12 G H z).
U n fortunately w ith this setu p , w hich is necessary fo r input p o w ers o f 5 dB m and higher,
only S21 can be m easured (sin ce a thru-calibration, th e only o n e possible, d o e s n o t c o rrect
for the retu rn loss o f th e am plifier). N ine m easurem ents are tak en , at input p o w e rs o f -20,
-15, -10, -5, 0, 5, 10, 15 and 2 0 dB m respectively. F o r th e first 6 m easurem ents all fo u r
S -p aram eters are m easured, w hile fo r the last 3 only S21 is m easured. F igures 17 and IS
sh o w th e m agn itu d e o f S21, in d B , fo r th e higher seven in p u t p o w e r m easurem ents (i.e., 10 to 20 dB m ). T h e cu rv e co rre sp o n d in g to 5 dB m has been p lo tted on b o th figures fo r
reference and th e scale is identical on both plots.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
177
EEsof DB[ S21]
MlODBM
T ou ch stone - Tue Sep
DB[ S21]
M5DBM
0 1 2 :1 7 :4 5 1992 - Ip f
o DBIS21]
ODBM
x
D B[S2i]
5DBM
0. 000
dB
- 2 5 . 00
-4 5 . 00
'1. 000
FREQ-GHZ
6. 500
12 . 00
F i g u r e 17 M e a s u r e d m a g n itu d e o f S 21 a t i n p u t p o w e r s - 1 0 , - 5 , 0 a n d 5 d B m .
EEsof - T ouchstone - Tue Sep
QB[ S21 ]
5DBM
0BES21]
10DBM
B 12:17: 45 1992 - lp f
DB[ S21]
15DBM
QB[ S21]
20DBM
0 . 000
■25. 00
-4 5 . 00
1 . 000
6. 500
FREQ-GHZ
1 2 . 00
F i g u r e 18 M e a s u r e d m a g n itu d e o f S21 a t in p u t p o w e rs 5, 10. 15 a n d 2 0 d B m .
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
178
T h e n o n -lin ear b e h a v io r o f th e circuit is evident.
T h e cu rv e s a t -10 and -5 dBm input
p o w e rs a re alm o st identical, b u t th e o n e at 0 d B m is significantly different from th ese tw o
w h ich in d icates th a t so m e c o m p o n en t o f th e filter (pro b ab ly th e n a rro w lines) reaches its
lo w critical field, H c j ,a t an in p u t p o w e r b etw een -5 and 0 dB m . A s th e p o w er is further
increased an d th e tran sm issio n lines o f th e filter b eco m e in su lato rs, th e lo w insertion loss
p ass-b an d o f th e filter d isap p ears and the device b eco m es v ery lossy (o v e r 15 dB insertion
lo ss at 20 dB m in p u t p o w er).
7.4.3 The Power-Dependent Model
T h e first e ffo rt is to asc e rta in w h e th er th e m odel o f c h a p te r 3, w ith th e enhancem ents o f
sectio n 7.1, w h ich enable m ore acc u ra te m odeling o f hig h -lo ss lines, is sufficient for
m odeling th e p o w e r-d e p e n d en t b eh av io r seen in figures 17 and 18. T h e param eters o f the
m odel a re o p tim ized fo r m inim um integrated sq u ared e rro r re la tiv e to th e m easurem ents o f
th e S -p a ra m e te rs o f th e filter at an input p o w er o f 5 dB m . T h e d e p en d en t variables o f the
o p tim izatio n a re th e z e ro -te m p e ra tu re penetration d epth, A0, and the norm al part o f the
co n d u ctiv ity a n. T his is equivalent to having optim ized th e d en sity o f superelectrons and
norm al electro n s, b ecau se th e real and im aginary p arts o f th e co m plex conductivity are
b o th o p tim ized in d ep en d en tly (see equation 3.9).
T he T o u c h sto n e file used to perform
this analysis is included as appendix C and the resulting fit is sh o w n in figures 19-22.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
179
EEsof - Touchstone - Tue Sep
DB[ S21]
FIL
8 1 4 :0 7 :2 9 1992 - 5dbmn
DB[ S21]
VBCO
0 . 000
-2 5 . 00
-45. 00
0 . 000
6 . 000
FREQ-GHZ
12 . 00
F i g u r e 19 M a g n i tu d e o f S 21 o f m o d e l v e r s u s m e a s u re m e n t a t 5 d B m in p u t p o w e r.
EEsof - Touchstone - Tue Sep
ANG[ S21]
FIL
8 14: 0 7 :33 1992 - 5dbmn
ANG[ S21]
VBCO
3. 500
rad
0 . 000
-3 . 000
0 . 000
6 . 000
FREQ-GHZ
1 2 . 00
F i g u r e 2 0 P h a s e o f S 2 I o f m o d e l v e rs u s m e a s u re m e n t a t 5 d B m in p u t p o w e r.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
180
EEsof - Touchstone - Tue Sep
DB[ S U ]
FIL
8 1 4 :0 7 :3 5 1992 - 5dbmn
DB[ S l i ]
YBCO
- 2 . 000
dB
- 16 . 00
-30. 00
0 . 000
6 . 000
FREQ-GHZ
12 . 00
F i g u r e 21 M a g n itu d e o f S I 1 o f m o d e l v e r s u s m e a s u r e m e n t a t 5 d B m in p u t p o w e r.
EEsof - T ouchstone - Tue Sep
ANG[ S U ]
FIL
8 1 4 :0 7 :3 7 1992 - 5dbmn
ANG[ S U ]
YBCO
3. 500
rad
0 . 000
-3. 500
0 . 000
6 . 000
FRFQ-GHZ
12 . 00
F i g u r e 22 P h a s e o f S 1 1 o f m o d e l v e r s u s m e a s u r e m e n t a t 5 d B m in p u t p o w e r.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
181
7.4.4 Discussion o f the Fit of the Model to the Measured Data
T h e in te g ra te d sq u a re d e rro r b etw ee n m odel and m easu rem en t is 0 .0 9 3 0 8 , 5 3 .6 % m ore
th a n fo r th e lo w p o w e r case, as re p o rte d above. T h e higher erro r is p artly d u e to th e fact
th a t th e m odel d o e sn 't pick u p th e lo catio n s o f th e poles o f th e filter (see fig u re 21 and
c o m p a re to fig u re 13) and u n d erestim ates th e insertion loss at th e low e n d o f th e passb an d (se e fig u re 19).
T h ese discrepancies m ay be due to inaccuracies in th e im pedance
calcu latio n s (se e d iscu ssio n in section 7.4.1. above).
T he fit o f th e p h ase s is q u ite g o od,
h o w ev er, w hich in dicates th a t eq uation (4) for the co rre cted p ro p a g a tio n co n stan t
c o rre c tly a c c o u n ts fo r increased d istributed internal inductance effects. T h e failure o f the
m odel to p in p o in t th e po les o f th e filter probably indicates th a t the m odel d o es not
calcu late im p ed an ce p roperly.
T his m ay b e b ecau se th e C P W L P F , w h ich com prises
n a rro w (h ig h -im p ed an ce) and w ide (lo w -im pedance) lines, is operatin g in its non-linear
p o w e r-d e p e n d e n t region.
In th e frequencies o f th e pass band o f th e filter, th e fo rw ard -
c u rre n t (from th e w a v e incident at th e input p o rt) g o e s th ro u g h every line o f th e filter
largely u n sc a tte re d (w ith o u t m any reflections). T h e to tal cu rre n t th ro u g h ev ery line m ust
be alm o st th e sam e.
T h e cu rren t density is, therefore, higher in the n a rro w lines than in
th e w ide lines. T h is m eans that the fields in the n a rro w lines are clo ser to th e high-critical
fields th an in th e w ide lines and hence th e num ber o f superco n d u ctin g elec tro n s is higher in
th e w id e than in th e n a rro w lines. T his points to th e need fo r tw o im p ro v e m e n ts on the
m odel:
1.
T h e n u m b er o f su p erelectro n s m ust be dependent on th e w idth o f th e m o deled line.
2.
T h e n u m b er o f su p erelectro n s and norm al electrons should not be d e p e n d en t only
on tem p eratu re.
T h ese im p ro v em en ts, to g e th e r w ith the fu rth er refinem ent o f considering th e d e p en d en ce
o f th e m odel on th e collision relaxation tim e r, are built into th e m odel using the follow ing
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
derivation.
7.5
First-Order Effects due to Collision Relaxation
7.5.1
The Analysis
L e t th e variable X (T ,H ;T C,H C) re p rese n t th e fractio n o f the to tal electrons th at are in
norm al state, i.e.,
^
(10)
= X { T , H - T c, H e)
and
i =l-X
II
( II)
w here n d e n o te s e lectro n v o lu m e density. F ro m eq u atio n (3.8) w e obtain
2
a = CT, - jcr2 = — <Jn ~ j — [n, + //„ ( cot) 2 ]
n
m co
(12)
w here ( o r ) 2« l has been assum ed. H o w ev er, this assum ption can n o t be used to simplify
th e im aginary p art o f (1 2 ) since, in h ig h -p o w er n a rro w lines, nn» n s. C om bining (10) and
(12) w e obtain
cr, =Xa„ .
(13)
T h e co rresp o n d in g exp ressio n for cr, isslightly m o re com plicated. By the definition o f the
p enetration d ep th w e have
=
J j—
p 0ns( I , H ) e
- ,
'
(14)
w here f.t0 is th e m ag n etic perm eability o f va cu u m and m and e are th e m ass and ch arg e o f
th e electron, respectively. W hen T and FI a re z e ro all th e electrons are superelectrons, i.e.,
us=n. U sing th e se v alues and eq u ation (1 1 ) to elem in ate ns from eq u atio n (14) w e obtain
a 2( 0, 0 )
——
,
„
= \ - X
.
(15)
C om bining (12), ( 14 ) and (1 5 ) gives
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
183
l-X
+ X c na>T
<y//0A2(0,0)
(16)
and thhee new
n ew ex p ressio n fo r th e conductivity becom es
l-X
<*=X<7n ~ j
- + X crncjr
(18)
(OjU0X-(0,0)
7.5.2 The Fit of the Model to Measurement
E q u a tio n (1 8 ) is in c o rp o ra te d in to a new circuit file w hich is included as appendix D. An
initial g u e ss is u sed fo r r and then it is perm itted to optim ize.
A different X and
co rresp o n d in g a is u sed fo r each o f the line w idths o f th e C P W L P F (hence th ree different
A’s are u sed , o n e fo r each o f th e narrow , th e w ide and th e 50 Q lines). T he fit betw een
m odel an d m easu rem en ts im p ro v es from 0 .09308 in teg rate d sq uared error (as reported
above) to 0 .0 6 6 7 5 , a 2 8 .3 % d ecrease. T h e im proved fit is show n in figures 23-26.
EEsof
□
DB[ S21 ]
FIL
- L ib ra
+
-
Mon O c t
5 12: 13: 51 1 9 9 2 - 5dbm new
DB1S21]
YBCO
0 . 000
-25. 00
0 . 000
6 . 000
FREQ-GHZ
12 . 00
F i g u r e 2 3 M a g n itu d e o f S 2 1 , m o d e l v e r s u s m e a s u re d .
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
184
E E s o f - L i b r a - Mon O c t
D
ANG[ S21]
FIL
+
5 1 2 : 1 3 : 5 5 1 9 9 2 - 5dbm new
ANG[ S21]
YBCO
3. 5 0 0
ra d
0 . 000
-3. 500
0. 000
6 . 000
FREQ-GHZ
12 . 00
F i g u r e 2 4 P h a s e o f S 2 1 . m o d e l v e rs u s m e a s u re d .
EEsof
□
-
D B fS ll]
FIL
- L ib ra
+
- Mon O c t
5 1 2 :1 3 :5 7
1 9 9 2 - 5dbmnew
0B[S11]
YBCO
2 . 000
dB
-
18 . 00
-34. 00
0 . 000
6 . 000
FREQ-GHZ
12 . 00
F i g u r e 2 5 M a g n itu d e o f S 11. m o d e l v e rs u s m e a s u re d .
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
185
EEsof
ANG[ S l l ]
FIL
-
L i b r a - Mon O c t
5 1 2 :1 3 :5 9 1992 -
5db(iinew
ANG[ S l l ]
YBCO
3. 5 0 0
rad
0 . 000
-3. 500
0 . 000
6 . 000
FREO-GHZ
12 . 00
F i g u r e 2 6 P h a s e o f S I 1, m o d e l v e r s u s m e a s u re d .
7.5.3 Discussion of the Results
C om p ariso n o f figures 2 3 -2 6 to figures 19-22 rev eals that th e new m odel m atches th e
m easured lo catio n o f th e poles o f the L P F better, a lth o u g h still n o t as accu rately as in th e
low p o w e r case. It is en co u rag in g that the optim ized values o f X fo r each o f th e n arrow ,
50 Q and w ide lines obey X n.ln.ow> X 5()Q > X wide (se e appendix D ).
Indeed, by th e
rationale p re se n te d above, w e w ould ex p ect th e n a rro w line to carry th e m ost norm al
electrons and th e w ide th e least.
A lthough th e fit o f figures 23-26 is prom ising, the
m ism atch o f th e locatio n o f th e poles b etw een m odel and m easurem ent indicates that th ere
is som ething m o re g o in g on in th e physics th a t the m odel is n o t capturing. T he reason fo r
the d iscrepancy is th o u g h t to be a se co n d -o rd er effect o f th e non-linear behavior w ith
resp ect to input p o w er.
As th e input p o w e r in creases th e n arro w lines "go norm al" first,
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
186
du e to th e h ig h er cu rre n t density th at they carry. T h e profile o f th e current density along
th e w id th o f th e line is ex p ected to b e a m inim um at th e m idpoint and sym m etric ab o u t it,
p eak in g at th e e d g e s (th e discontinuity). T h e sam e is tru e fo r th e m agnetic field. H en c e
th e e d g e s o f th e line m ust g o norm al at low input p o w e r and th e m iddle at higher input
po w er.
Since Y B C O is a crystalline substance, it b eco m es an in su lato r w hen it g o e s
norm al.
T h is m ean s th a t th e line becom es effectively n a rro w e r w hen a stro n g en o u g h
input field is applied (w ith its edges having b eco m e insulating).
T his affects n ot only th e
cro ss-se c tio n a l area A and th e increm ental ind u ctan ce fa c to r G (equation (3.32)), b u t also
th e im p ed an ce o f th e line even b efo re th e P E M is ap p lied fo r c o rrec tio n o f th e pen etratio n
d e p th effects, i.e., th e im ped an ce o f the line as if it w e re m ade o u t o f a norm al conductor.
7.5.4 Testing the Variable Effective Line-Width Hypothesis
T o te st th e ab o v e co n jectu re, a new circuit file is created , this tim e also optim izing A, G
and Z fo r each ty p e o f line (see appendix E o f this ch a p ter). T he in tegrated squared erro r
b etw een m odel and m easu rem en ts is reduced to 0 .0 2 2 4 0 , a 6 6 .4 % d ecrease relative to th e
p rev io u s m odel. T his d ram atic decrease in erro r confirm s th e co n jecture p ut forth above.
E x am in atio n o f app en d ix E sh o w s that, as expected, th e p aram eters o f the narro w lines are
th e o n es m o st affected. W hile ZWO and Z 500, th e im p ed an ces o f th e w ide and 50 Q lines
respectively, d o n o t ch an g e significantly as a result o f optim ization (from 22.64 to 23.93
Q fo r th e first and from 4 9 .5 6 to 50.15 D. for th e seco n d ) Z N 0, the im pedance o f the
n arro w line, increased from 83.38 to 114 E>, w hich confirm s th e "narrow ing" o f the line
d u e to high input po w er. T h e resulting fit is show n in figures 27-30.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
187
□
EEsof
- L ib ra
DB[ S21 ]
FIL
+
-
Thu O c t
8 1 3 :0 2 :5 9 1992 -
5dbm new 4
DB[ S21]
YBCO
0 . 000
•25. 00
0 . 000
6 . 000
FREQ-GHZ
1 2 . 00
Figure 27 M a g n i tu d e o f S l l , m o d e l v e r s u s m e a s u re d .
EEsof
□
- L ib ra
ANG[ S21]
FIL
+
-
Thu O c t
8 1 3 : 0 3 : 0 3 1 9 9 2 - 5dbm new 4
ANG[ S21]
YBCO
3. 5 0 0
ra d
0 . 000
-3. 500
0 . 000
6 . 000
FREQ-GHZ
12 . 00
Figure 2 8 P h a s e o f S 2 1. m o d e l v e r s u s m e a s u re 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.
188
EEsof
□
- L i b r a - Thu O c t
DB[ S l l ]
+
F IL
-
2 . 000
-
18 . 00
8 13: 03: 0 6 1 9 9 2 - 5dbm new 4
D B tS ll]
YBCO
- 3 2 . 00
0 . 000
12 . 00
FREQ-GHZ
6 . 000
F ig u r e 2 9 M a g n itu d e o f S 11, m o d e l v e r s u s m e a s u re d .
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A N G t S i l]
+
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-
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8 1 3 :0 3 :0 8
1 9 9 2 - 5dbm new 4
A N G tSll]
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3. 5 0 0
\
t
,
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X
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0. 000
6 . 000
FREQ-GHZ
1 2 . 00
F i g u r e 3 0 P h a s e o f S 1 1. m o d e l v e r s u s m e a s u re d .
R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
189
7.5.5 Conclusions
A lth o u g h th e a b o v e fit is en co u rag in g and su g g ests th a t th e re m ay b e a w ay to in co rp o rate
th e line "narrow ing" effect into th e m odel w hile retain in g its pred ictiv e value, th e problem
b ec o m e s n o n -lin ear and will only accept an iterativ e solution.
M o re im portantly, th e
m e asu rem en ts o f th e H T S C P W L P F are to o intrinsic and involve to o m any unknow n
p a ra m e te rs th a t h av e to be optim ized. W ith th e in crease o f th e n u m ber o f th ese unknow ns
(as th e m o d el p ro g re sse s form its initial sim pler fo rm to th e c u rren t m ore com plicated
o n e), th e h y p er-su rface describing th e in teg rated sq u ared e rro r b etw een m easurem ents and
m odel n o w has a m u ltitu d e o f local m inim a (ra th e r th a n o n e global tru e m inim um th a t can
b e fo u n d by a g ra d ie n t algorithm ). E ven if th e tru e m inim um am o n g th ese m inim a can be
fo u n d (w ith resp ect to all th e variables th at are to be o p tim ized ), th e re is little confidence
th a t this so lu tio n will, in fact, co rrespond to th e tru e physical values o f the unknow ns.
H en ce, alth o u g h th e tre n d s and results are en co u rag in g , this fo rm ulation o f th e m odel
should b e u sed w ith caution.
7.6 References
[1]
S. R a m o , J. R. W hinnery and T. V an D uzer, Fields and Waves in Communication
Electronics , W iley, N e w Y ork, 1965.
[2] T. V an D u z e r and C. W. T urner, Principles o f Superconductive Devices and Circuits ,
Elsevier, N e w Y ork, 1981.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Appendix A
First-Order HTS CPW LPF Touchstone Model (Low Power Response)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
! MODEL OF THE YBCO FILTER FIT TO DATA. USING LOSS AND LOSS TANGENT
! ALSO USING S2P FILE FOR 50 OHM TAPER K-CONNECTOR TO CPWG INTERFACE
! ALSO USING CONSTANT LOSS S-PAR FILE (ADJUST.S2P) TO CORRECT FOR
! CAL STANDARDS BEING AT ROOM TEMP (NOT DUNKED)
! USING LOW LOSS EQUATIONS FOR ALL ATTENUATIONS AS A FUNCTION OF LD AND
Sn
! BY DIMITRIOS ANTSOS (JULY 28, 1992)
1 FILENAME REFERS TO (TRUE) POWER INTO FILTER
i FILTER USED IS LAST PACKAGED CPW FILTER LEFT
DIM
LNG UM
ANG RAD
VAR
LD# 100E-9 7.77e-07 5E-6
T = 77
Tc = 83
Sn #1E4 1133285. 1E8
eO = 8.854E-12
AGn = 5e-12
Line )
Gn = 8.43479e4
Var (Narr
AGw = le-10
Gw = 1.7336e4
AG5 = 2.5e-ll
G5 = 2.3501e4
!PENETRATION DEPTH FOR YBCO
!TEMPERATURE OF MEASUREMENT
!Critical Temperture of Sample
!Normal Conductivity of Sample
!Permittivity of free space
IDimension Variable ( Narrow
!Incremental Inductance Rule
Z500 = 49.56
ZNO = 75.6614
ZWO = 22.64
KI #12 14.26254 18
K500 = 12.52
KNO = 12.468
KWO = 12.49
LI # 2000 2446.013 2800
L50
LI
L2
L3
L4
L5H
#
=
=
=
=
=
1800 2221.961 2800
720.8
997.0
1369.7
761.2
924.0
ACI0 #1E-10 1.00e-08 IE-6
(=5.02
ADI00 # 3E-4 0.009000 9e-3
! COPPER LOSS OF INPUT TAPER
! = 3.3E-4
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
192
TAND # 0 0.000104 .01
EQN
ADNO = 3.086E-4*TAND
AD500 = 3.093E-4*TAND
ADWO = 3.089E-4*TAND
ADIO = ADIOO*TAND
! DIELECTRIC LOSS OF INPUT CPW 50 OHM TAPER
LI1 = -LI
AI
= ACIO
* FREQ**2 + ADIO * FREQ
Computation of exact loss for narrow line
Constants
U0 = 4*PI*le-7
IMagnetic Permeability of vacuum
f = FREQ*le9
!Frequency in Hz
Sr = Sn*(T/Tc)**4
IReal Part of conductivity of YBCO
Si = (1-(T/Tc)**4)/(2*PI*f*U0*LD**2) llmaginary Part of conductivity
P = ATAN(Si/Sr)-2*PI
lAngle of conductivity
Th= PI/4-P/2
lAuxiliary angle definition
r = SQRT(SQR(Sr)+SQR(Si))
!Norm of conductivity
c = l/sqrt(e0*U0)
IVelocity of light
Bn=
Cn=
Dn=
En=
Un=
Wn=
Fn=
Mn=
Nn=
Rn=
Narrow Line Parameters
Gn*AGn*SQRT(2*PI*f*U0*r)
IFudge Factors
EXP(2*Bn*COS(Th))
C O S (2*Bn*SIN(Th))
SIN(2*Bn*SIN(Th))
SQRT(SQR(Cn*Dn-l)+SQR(Cn*En))
ATAN(Cn*En/(Cn*Dn-l))
Bn/(AGn*r*Un)
2*Bn*SIN(Th)
COS(PI/4+P/2-Wn)
COS(Mn+PI/4+P/2-Wn)
ReZn = Fn*(Nn+Cn*Rn)
IReal Part of Internal Impedance / Meter
NIn= SIN(PI/4+P/2-Wn)
RIn= SIN(Mn+PI/4+P/2-Wn)
ImZn = Fn*(NIn+Cn*RIn)
Zn = SQRT(SQR(ZN0)-(c*ZN0)/(2*PI*SQRT(KN0)*f)*ImZn)
ACN = -(8.686e-6) * ReZn/(2*Zn)
EFFN = ZN / ZN0
Bw=
Cw=
Dw=
Ew=
Uw=
Wide Line Parameters
Gw*AGw*SQRT(2*PI*f*U0*r)
IFudge Factors
E X P (2*Bw*COS(Th))
COS(2*Bw*SIN(Th))
S I N (2*Bw*SIN(Th))
SQRT(SQR(Cw*Dw-l)+SQR(Cw*Ew))
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
193
Ww=
Fw=
Mw=
Nw=
Rw=
ATAN{Cw*Ew/(Cw*Dw-l))
Bw/(AGw*r*Uw)
2*Bw*SIN(Th)
COS(PI/4+P/2-Ww)
COS(Mw+PI/4+P/2-Ww)
ReZw = Fw*(Nw+Cw*Rw)
IReal Part of Internal Impedance / Meter
NIw= SIN(PI/4+P/2-Ww)
RIw= SIN(Mw+PI/4+P/2-Ww)
ImZw = Fw*(NIw+Cw*RIw)
Zw = SQRT(SQR(’ZwO)-(c*ZwO)/(2 *PI*SQRT(KwO)*f)*ImZw)
ACw = -(8.686e-6) * ReZw/(2*Zw)
EFFw = Zw / ZwO
B5=
C5=
D5=
E5=
U5=
W5=
F5=
M5=
N5=
R5=
ReZ5
NI5=
RI5=
ImZ5
Z50=
AC50
50 Ohm Line Parameters
G5*AG5*SQRT(2*PI*f*U0*r)
IFudge Factors
EX P (2*B5*COS(Th ) )
CO S (2*B5*SIN(Th ) )
SIN(2*B5*SIN(Th))
SQRT(SQR(C5*D5-1)+SQR(C5*E5))
ATAN(C5*E5/(C5*D5-1))
B5/(AG5*r*U5)
2*B5*SIN(Th)
C O S (PI/4+P/2-W5)
COS(M5+PI/4+P/2-W5)
= F5*(N5+C5*R5)
IReal Part of Internal Impedance / Meter
S IN(PI/4+P/2-W5)
SIN(M5+PI/4+P/2-W5)
= F5*(NI5+C5*RI5)
SQRT(SQR(Z500)-(c*Z500)/(2*PI*SQRT(K500)* f )*ImZ5)
= -(8.686e-6) * ReZ5/(2*Z50)
EFF50 = Z50 / Z500
K50
KN
KW
= K500
= KNO
= KWO
* EFF50 * EFF50
* EFFN * EFFN
* EFFW * EFFW
A50
AN
AW
= AC50 + AD500 * FREQ
= ACN + ADNO * FREQ
= ACW + ADWO * FREQ
CKT
S2PA 1 2 0 costepl.s2p
DEF2P 1 2 BIG_STEP
S2PB 1 2 0 costep2.s2p
DEF2P 1 2 SML_STEP
S2PC 1 2 0 -5dbm.s2p
DEF2P 1 2 YBCO RAW
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission.
194
S2PD 1 2 0 adjust.s2p
DEF2P 1 2 ADJ
S2PE 1 2 0 hts_50.s2p
DEF2P 1 2 FIFTY
ADJ 1 2
YBCOJRAW 2 3
DEF2P 1 3 YBCO
TLINP 1 2 Z=50 L'Ll K'KI A'AI F=0
TLINP 2 3 Z=50 L'LIl K'KI A=0 F=0
FIFTY 3 4
TLINP__T1 4 5 Z~Z50 L'L50 K'K50 A'A50 F=0.0000000
SML_STEP
5 6
TLINP_T2 6 7 Z'ZW L'Ll K'KW A'AW F=0.0000000
BIG_STEP
7 8
TLINP_T3 8 9 Z'ZN L'L2 K'KN A'AN F=0
BIG_STEP 9 10
TLINP_T4 10 11 Z'ZW L'L3 K'KW A'AW F=0.0000000
BIG_STEP
11 12
TLINP_T5 12 13 Z'ZN L'L4 K'KN A'AN F=0.0000000
BIG_STEP 13 14
TLINP_T6 14 15 Z'ZW L'L5H K'KW A'AW F=0.0000000
DEF2P 1 15 HALF
HALF 1 2
HALF 3 2
DEF2P 1 3 FIL
!TLINP_T1
1 2 Z'Z50L'L50 K'K50 A'A50 F=0.0000000
!TLINP_T2
2 3 Z'ZW L'Ll K'KW A'AW F=0.0000000
!TLINP_T3 3 4 Z'ZN L"L2 K'KN A'AN F=0
!TLINP_T4
4 5 Z'ZW L'L3 K'KW A'AW F=0.0000000
!TLINP_T5
5 6 Z'ZN L'L4 K'KN A'AN F=0.0000000
!TLINP T6 6 7
Z'ZW L'L5H K'KW A'AW F=0.0000000
1DEF2P 1 7
HALF1
!HALF1 1 2
!HALF1 3 2
!DEF2P 1 3 NOSTEP
IRES 1 0 R~rez5
!DEF1P 1 TEST
FREQ
SWEEP 0.5 12 .115
OUT
FIL D B [S2 1 ] GR1
YBCO D B [S2 1 ] GR1
!SILVER D B [S2 1 ] GR1
!NOSTEP D B [S21] GR1
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
195
FIL D B [Sll] GR3
YBCO D B [Sl l ] GR3
!SILVER D B [Sll] GR3
FIL A N G [S21] GR2
YBCO ANG[S21] GR2
!SILVER A N G [S21] GR2
FIL ANG[Sll] GR4
YBCO ANG[Sll] GR4
1SILVER ANG[Sll] GR4
FIL S21 SC2
YBCO S21 SC2
! TEST R E [Zll] GR5
! FIL D B [S2 1 ] GR6
! YBCO D B [S2 1 ] GR6
GRID
RANGE 0 12 1
!RANGE 7 9 .2
! GR6 -2 0 .5
OPT
RANGE 1 11
YBCO MODEL FIL
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
Appendix B
High-Loss CPW LPF Touchstone Model (Low Power Response)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
197
! MODEL OF THE YBCO FILTER FIT TO DATA USING LOSS AND LOSS TANGENT
! ALSO USING S2P FILE FOR 50 OHM TAPER K-CONNECTOR TO CPWG INTERFACE
! ALSO USING CONSTANT LOSS S-PAR FILE (ADJUST.S2P) TO CORRECT FOR
! CAL STANDARDS BEING AT ROOM TEMP (NOT DUNKED)
! USING LOW LOSS EQUATIONS FOR ALL ATTENUATIONS AS A FUNCTION OF LD AND
Sn
! BY DIMITRIOS ANTSOS (JULY 28, 1992)
! FILENAME REFERS TO (TRUE) POWER INTO FILTER
! FILTER USED IS LAST PACKAGED CPW FILTER LEFT
DIM
LNG UM
ANG RAD
VAR
T = 77
Tc =83
of Sample
LD# 100E-9 5.42e-07 5E-6
Sn #1E4 3418408. 1E8
eO = 8.854E-12
AGn = 3e-12
Line )
Gn = 1.255e5
Var (Narr
AGw = le-10
Gw = 1.7336e4
AG5 = 2.5e-ll
G5 = 2.3501e4
!TEMPERATURE OF MEASUREMENT
!Critical Temperture
!PENETRATION DEPTH FOR YBCO
!Normal Conductivity of Sample
!Permittivity of free space
!Dimension Variable ( Narrow
!Incremental Inductance Rule
Z500 = 49.56
ZNO = 83.38
ZWO = 22.64
KI #12 17.17221
18
K500 = 12.52
KNO = 12.455
KWO = 12.49
LI # 2000 2000.038 2800
L50
LI
L2
L3
L4
L5H
#
=
=
=
=
=
1800 2443.082 2800
720.8
997.0
1369.7
761.2
924.0
ACI0 #1E-10 1.00e-10 IE-6
(=5.02
AD100 # 3E-4 0.000300 9e-3
!# 4000
4335.155 4400
! COPPER LOSS OF INPUT TAPER
! = 3.3E-4
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
198
TAND # 0 0.000248 .01
EQN
ADNO = 3.086E-4*TAND
AD 500 = 3.093E-4*TAND
ADWO = 3.089E-4*TAND
ADIO = ADIOO*TAND
! DIELECTRIC LOSS OF INPUT CPW 50 OHM TAPER
LI1 = -LI
AI
= ACIO
* FREQ**2 + ADIO * FREQ
! Computation of exact loss for narrow line
!
Constants
U0 = 4*PI*le-7
IMagnetic Permeability of vacuum
f = FREQ*le9
!Frequency in Hz
Sr = Sn*(T/Tc)**4
!Real Part of conductivity of YBCO
Si = (1-(T/Tc)**4)/(2*PI*f*U0*LD**2) llmaginary Part of conductivity
P = ATAN(Si/Sr)-2*PI
!Angle of conductivity
Th= PI/4-P/2
JAuxiliary angle definition
r = SQRT(SQR(Sr)+SQR(Si))
INorm of conductivity
c = 1/sqrt(e0*U0)
!Velocity of light
Vn=
Bn=
Cn=
Dn=
En=
Un=
Wn=
Fn=
Mn=
Nn=
Rn=
Narrow Line Parameters
c/sqrt(KN0)
IPhase velocity in line
Gn*AGn*SQRT(2*PI*f*U0*r)
IFudge Factors
EXP(2*Bn*COS(Th))
COS(2*Bn*SIN(Th))
SIN(2*Bn*SIN(Th))
SQRT(SQR(Cn*Dn-l)+SQR(Cn*En))
ATAN(Cn*En/(Cn*Dn-l))
Bn/(AGn*r*Un)
2*Bn*SIN(Th)
COS(PI/4+P/2-Wn)
CO S (Mn+PI/4+P/2-Wn)
ReZn = -Fn*(Nn+Cn*Rn)
Meter
NIn= SIN{PI/4+P/2-Wn)
RIn= SIN(Mn+PI/4+P/2-Wn)
ImZn = -Fn*(NIn+Cn*RIn)
LMn = ZN0/Vn+ImZn/(2*PI*f)
CMN = 1/(ZN0*Vn)
Zn = sqrt(LMn/CMn)
RATn= ReZn/(2*PI*f*LMn)
ANGn= 0.5 *ATAN(RATn)
IReal Part of Internal Impedance /
ACN = (8.686e-6) *
(2 *PI*f)*sqrt(LMn*CMn)*sqrt(s q r t (l+SQR(RATn)))*sin(ANGn)
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
199
KN = LMn*CMn/(eO*uO)*sgrt(l+SQR(RATn))*SQR(cos(ANGn))
1
Vw=
Bw=
Cw=
Dw=
Ew=
Uw=
Ww=
Fw=
Mw=
Nw=
Rw=
Wide Line Parameters
c/sqrt(KWO)
'Phase velocity in line
Gw*AGw*SQRT(2*PI*f*UO*r)
iFudge Factors
E X P (2 *Bw*COS(T h ))
COS(2*Bw*SIN(Th))
SIN(2*Bw*SIN(Th))
S Q R T (SQR(Cw*Dw-l)+SQR(Cw*Ew))
ATAN(Cw*Ew/(Cw*Dw-l))
Bw/(AGw*r*Uw)
2*Bw*SIN(Th)
COS{PI/4+P/2-Ww)
COS(Mw+PI/4+P/2-Ww)
ReZw =-Fw*(Nw+Cw*Rw)
NIw= S I N (PI/4+P/2-Ww)
RIw= SIN(Mw+PI/4+P/2-Ww)
ImZw =-Fw*(NIw+Cw*RIw)
LMw = ZWO/Vw+ImZw/(2*PI*f)
CMw= 1/(ZWO*Vw)
Zw = sqrt(LMw/CMw)
RATw= ReZw/(2*PI*f*LMw)
ANGw= 0.5 *ATAN(RATw)
IReal Part of Internal Impedance / Meter
ACW = (8.686e-6) *
(2*PI*f)*sqrt(LMw*CMw)*sqrt(sqrt(1+SQR(RATw)))*sin(ANGw)
KW = LMw*CMw/(eO*uO)*sqrt(l+SQR(RATw))*S Q R (cos(ANGw))
!
V5=
B5=
C5=
D5=
E5=
U5=
W5=
F5=
M5=
N5=
R5=
50 Ohm Line Parameters
c/sqrt(K500)
!Phase velocity in line
G5*AG5*SQRT(2*PI*f*U0*r)
IFudge Factors
E X P (2*B5*COS(Th))
C O S (2*B5*SIN(Th))
S I N (2*B5*SIN(Th))
S Q R T (SQR(C5*D5-1)+SQR(C5*E5))
ATAN(C5*E5/(C5*D5-1))
B5/(AG5*r*U5)
2*B5*SIN(Th)
C O S (PI/4+P/2-W5)
COS(M5+PI/4+P/2-W5)
ReZ5 =-F5*(N5+C5*R5)
NI5= S I N (PI/4+P/2-W5)
RI5= SIN(M5+PI/4+P/2-W5)
ImZ5 =-F5*(NI5+C5*RI5)
LM5 = Z500/V5+ImZ5/(2*PI*f)
CM5 = 1/(Z500*V5)
Z50 = sqrt(LM5/CM5)
RAT5= ReZ5/(2*PI*f*LM5)
ANG5= 0.5 *ATAN(RAT5)
IReal Part of Internal Impedance / Meter
AC50 = (8.686e-6) *
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission.
200
(2 *PI* f )* sqrt(LM5 *CM5)* sqrt(sqrt(1+SQR(RAT5)))*sin(ANG5)
K50 = LM5*CM5/(eO*uO)*sqrt(1+SQR(RAT5))*SQR(c os(AN G 5 ))
A50
AN
AW
= AC50 + AD500 * FREQ
= ACN + ADNO * FREQ
= ACW + ADWO * FREQ
CKT
S2PA 1 2 0 costepl.s2p
DEF2P 1 2 BIG_STEP
S2PB 1 2 0 costep2.s2p
DEF2P 1 2 SML_STEP
S2PC 1 2 0 -5dbm.s2p
DEF2P 1 2 YBCO_RAW
S2PD 1 2 0 adjust.s2p
DEF2P 1 2 ADJ
S2PE 1 2 0 hts_50.s2p
DEF2P 1 2 FIFTY
ADJ 1 2
YBCO_RAW 2 3
DEF2P 1 3 YBCO
TLINP 1 2 Z=50 L'LlK'KI A'AI F=0
TLINP 2 3 Z=50 L'LIl K'KI A=0 F=0
FIFTY 3 4
TLINP_T1 4 5 Z~Z50 L'L50 K'K50 A~A50 F=0.0000000
SML_STEP
5 6
TLINP_T2 6 7 Z'ZW L'Ll K'KW A'AW F=0.0000000
BIG_STEP
7 8
TLINP_T3 8 9 Z'ZN L'L2 K'KN A'AN F=0
BIG_STEP 9 10
TLINP_T4 10 11 Z'ZW L'L3 K'KW A'AW F=0.0000000
BIG_STEP
11 12
TLINP_T5 12 13 Z'ZN L'L4 K'KN A'AN F=0.0000000
BIG_STEP
13 14
TLINP_T6 14 15 Z'ZW L"L5H K'KW A'AW F=0.0000000
DEF2P 1 15 HALF
HALF 1 2
HALF 3 2
DEF2P 1. 3 FIL
!TLINP__T1
!TLINP__T2
!TLINP__T3
!TLINP__T4
!TLINP T5
1
2
3
4
5
2 Z'Z50 L'L50 K'K50 A'A50 F=0.0000000
3 Z'ZW L'Ll K'KW A'AW F=0.0000000
4 Z'ZN L'L2 K'KN A'AN F=0
5 Z'ZW L'L3 K'KW A'AW F = 0 .0000000
6 Z'ZN L “L4 K'KN A'AN F = 0 .0000000
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
201
!TLINP_T6 6 7
1DEF2P 1 7
Z'ZW L'L5H K'KW A'AW F=0.0000000
HALFI
IHALF1 1 2
1HALF1 3 2
IDEF2P 1 3 NOSTEP
RES 1 0 R'AC50
DEF1P 1 TEST
FREQ
SWEEP 0.5 12 .115
ISTEP 5
OUT
FIL D B [S2 1 1 GR1
YBCO D B [S 2 1 ] GR1
!SILVER D B [S 2 1 ] GR1
!NOSTEP D B [S2 1] GR1
FIL D B [S ll] GR3
YBCO DB[Sll] GR3
!SILVER D B [S l l ] GR3
FIL A N G [S 2 1 ] GR2
YBCO A N G [S21] GR2
!SILVER A N G [S2 1 ] GR2
FIL ANG[Sll] GR4
YBCO ANG[Sll] GR4
!SILVER ANG[Sll] GR4
FIL S21 SC2
YBCO S21 SC2
TEST R E [Z11] GR5
FIL D B [S 2 1 ] GR6
YBCO D B [S21] GR6
GRID
RANGE 0 12 1
!RANGE 7 9 .2
! GR6 -2 0 . 5
OPT
RANGE 1 11
YBCO MODEL FIL
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202
Appendix C
High-Loss Touchstone Model (5 dBm Input Power)
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
203
1 MODEL OF THE YBCO FILTER FIT TO DATA USING LOSS AND LOSS TANGENT
! ALSO USING S2P FILE FOR 50 OHM TAPER K-CONNECTOR TO CPWG INTERFACE
! ALSO USING CONSTANT LOSS S-PAR FILE (ADJUST.S2P) TO CORRECT FOR
! CAL STANDARDS BEING AT ROOM TEMP (NOT DUNKED)
! USING LOW LOSS EQUATIONS FOR ALL ATTENUATIONS AS A FUNCTION OF LD AND
Sn
! BY DIMITRIOS ANTSOS (JULY 28, 1992)
! FILENAME REFERS TO (TRUE) POWER INTO FILTER
! FILTER USED IS LAST PACKAGED CPW FILTER LEFT
DIM
LNG UM
ANG RAD
VAR
T = 77
Tc = 83
LD# 100E-9 5.72e-07
5E-6
Sn #1E4 9776496. 1E8
eO = 8.854E-12
AGn = 3e-12
Line )
Gn = 1.255e5
Var (Narr
AGw = le-10
Gw = 1.7336e4
A G 5 = 2.5e-ll
G5 = 2.3501e4
!TEMPERATURE OF MEASUREMENT
!Critical Temperture of Sample
!PENETRATION DEPTH FOR YBCO
!Normal Conductivity of Sample
!Permittivity of free space
!Dimension Variable ( Narrow
!Incremental Inductance Rule
Z500 = 49.56
ZNO = 83.38
ZWO = 22.64
KI #10 10.74451
18
K500 = 12.52
KNO = 12.455
KWO = 12.49
LI # 2000 2314.353 2800
L50
LI
L2
L3
L4
L5H
#
=
=
=
=
=
1800 2375.978 2800
720.8
997.0
1369.7
761.2
924.0
ACI0 #1E-10
1.03e-10 IE-6
(=5.02
ADI00 # 3E-4 0.000542 9e-3
!# 4000
4335.155 4400
! COPPER LOSS OF INPUT TAPER
! = 3.3E-4
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
204
TAND # 0 0.010000 .01
EQN
ADNO = 3.086E-4*TAND
AD500 = 3.093E-4*TAND
ADWO = 3.089E-4*TAND
ADIO = ADIOO*TAND
! DIELECTRIC LOSS OF INPUT CPW 50 OHM TAPER
L I 1 = -LI
AI
= ACIO
* FREQ**2 + ADIO * FREQ
! Computation of exact loss for narrow line
!
Constants
U0 = 4*PI*le-7
[Magnetic Permeability of vacuum
f = FREQ*le9
!Frequency in Hz
Sr = Sn*(T/Tc)**4
IReal Part of conductivity of YBCO
Si = (1-(T/Tc)**4)/(2*PI*f*U0*LD**2) !Imaginary Part of conductivity
P = ATAN(Si/Sr)-2*PI
[Angle of conductivity
Th= PI/4-P/2
[Auxiliary angle definition
r = S Q R T (SQR(S r )+SQR(Si))
[Norm of conductivity
c = 1/sqrt(e0*U0)
[Velocity of light
Vn=
Bn=
Cn=
Dn=
En=
Un=
Wn=
Fn=
Mn=
Nn=
Rn=
Narrow Line Parameters
c/sqrt(KN0)
!Phase velocity in line
Gn*AGn*SQRT(2*PI*f*U0*r)
[Fudge Factors
E X P (2*Bn*COS(Th))
COS(2*Bn*SIN(Th))
SIN(2*Bn*SIN(Th))
S Q R T (SQR(Cn*Dn-l)+SQR(Cn*En))
A T A N (Cn*En/(Cn*Dn-l))
Bn/(AGn*r*Un)
2*Bn*SIN(Th)
C O S (PI/4+P/2-Wn)
COS(Mn+PI/4+P/2-Wn)
ReZn = -Fn*(Nn+Cn*Rn)
[Real Part of Internal Impedance /
Meter
NIn= S I N (PI/4+P/2-Wn)
RIn= S I N (Mn+PI/4+P/2-Wn)
ImZn = -Fn*(NIn+Cn*RIn)
LMn = ZN0/Vn+ImZn/(2*PI*f)
CMN = 1 / (ZN0*Vn)
! Zn = sqrt(LMn/CMn)
RATn= ReZn/(2 *PI*f*LMn)
ANGn= 0.5 *ATAN(RATn)
Zn= sqrt(LMn/CMn)*sqrt(sqrt(1+SQR(RATn)))
! Zn2= sqrt(LMn/CMn)*sqrt(sqrt(1+SQR(RATn)))*cos(ANGn)
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205
ACN = {8.686e-6) *
(2 *PI* f )* sgrt(LMn *CMn)* sqrt(sqrt(1+SQR(R A T n )))* s i n (AN G n )
KN = LMn*CMn/(eO*uO)*sqrt(1+SQR(RATn))*SQR(cos(ANGn))
!
Vw=
Bw=
Cw=
Dw=
Ew=
Uw=
Ww=
Fw=
Mw=
Nw=
Rw=
Wide Line Parameters
c/sqrt(KW0)
IPhase velocity in line
Gw*AGw*SQRT(2*PI*f*U0*r)
IFudge Factors
E X P (2 *Bw*COS(T h ))
COS(2*Bw*SIN(Th))
S I N (2*Bw*SIN(T h ) )
SQRT(S Q R (Cw*Dw-l)+SQR(Cw*Ew))
A T A N (Cw*Ew/(Cw*Dw-l))
Bw/(AGw*r*Uw)
2*Bw*SIN(Th)
COS(PI/4+P/2-Ww)
COS(Mw+PI/4+P/2-Ww)
ReZw =-Fw*(Nw+Cw*Rw)
IReal Part of Internal Impedance / Meter
NIw= SIN(PI/4+P/2-Ww)
RIw= S I N (Mw+PI/4+P/2-Ww)
ImZw =-Fw*(NIw+Cw*RIw)
LMw = ZWO/Vw+ImZw/(2*PI*f)
CMw= 1/(ZW0*Vw)
! Zw = sqrt(LMw/CMw)
RATw= ReZw/(2*PI*f*LMw)
ANGw= 0.5*ATAN(RATw)
Zw= sqrt(LMw/CMw)*sqrt(sqrt(l+SQR(RATw)))
! Zw2= sqrt(LMw/CMw)*sqrt(sqrt(l+SQR(RATw)))*cos(ANGw)
ACW = (8.686e~6) *
(2*PI*f)*sqrt(LMw*CMw)*sqrt(sqrt(l+SQR(RATw)))*sin(ANGw)
KW = LMw*CMw/(eO*uO)*sqrt(1+SQR(RATw))*SQR(cos(ANGw))
!
V5=
B5=
C5=
D5=
E5=
U5=
W5=
F5=
M5=
N5=
R5=
50 Ohm Line Parameters
c/sqrt(K500)
IPhase velocity in line
G5*AG5*SQRT(2*PI*f*U0*r)
!Fudge Factors
E X P (2*B5*COS(T h ))
C O S (2*B5*SIN(T h ))
SIN(2*B5*SIN(Th))
SQRT(S Q R (C5*D5-1)+SQR(C5*E5))
ATAN(C5*E5/(C5*D5-1))
B5/(AG5*r*U5)
2 *B5*SIN(T h )
COS(PI/4+P/2-W5)
COS(M5+PI/4+P/2-W5)
ReZ5 =-F5*(N5+C5*R5)
NI5= S I N (PI/4+P/2-W5)
RI5= SIN(M5+PI/4+P/2-W5)
ImZ5 =-F5*(NI5+C5*RI5)
LM5 = Z500/V5+ImZ5/(2*PI*f)
CM5 = 1/(Z500*V5)
! Z50 = sqrt(LM5/CM 5 )
IReal Part of Internal Impedance / Meter
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206
RAT5= ReZ5/(2*PI*f*LM5)
ANG5= 0.5*ATAN(RAT5)
Z50= sqrt(LM5/CM5)*sqrt(sqrt(1+SQR(RAT5)))
• ! Z52= sqrt(LM5/CM5)*sqrt(sqrt(1+SQR(RAT5)))*cos(ANG5)
AC50 = (8.686e-6) *
(2 *PI*f )* sqrt(LM5 *CM5)* sqrt(sqrt(1+SQR(RAT5)))* s i n (ANG5)
K50 = LM5*CM5/(e0*u0)*sqrt(1+SQR(RAT5))*SQR(cos(A NG5))
A50
AN
AW
= AC50 + AD500 * FREQ
= ACN + ADNO * FREQ
= ACW + ADWO * FREQ
CKT
S2PA 1 2 0 costepl.s2p
DEF2P 1 2 BIG_STEP
S2PB 1 2 0 costep2.s2p
DEF2P 1 2 SML_STEP
S2PC 1 2 0 5dbm.s2p
DEF2P 1 2 YBCO_RAW
S2PD 1 2 0 adjust.s2p
DEF2P 1 2 ADJ
S2PE 1 2 0 hts_5Q.s2p
DEF2P 1 2 FIFTY
ADJ 1 2
YBCO_RAW 2 3
DEF2P 1 3 YBCO
TLINP 1 2 Z=50 L'LI K'KI A'AI F=0
TLINP 2 3 Z=50 LTLIl K'KI A=0 F=0
FIFTY 3 4
TLINP_T1 4 5 Z'Z50 L"L50 K'K50 A'A50 F=0.0000000
SML_STEP
5 6
TLINP_T2 6 7 Z'ZW L'LI K'KW A'AW F=0.0000000
BIG_STEP
7 8
TLINP_T3 8 9 Z'ZN L'L2 K'KN A'AN F=0
BIG_STEP 9 10
TLINP_T4 10 11 Z~ZW L~L3 K~KW A'AW F=0.0000000
BIG_STEP
11 12
TLINP_T5 12 13 Z'ZN L'L4 K'KN A'AN F=0.0000000
BIG_STEP
13 14
TLINP_T6 14 15 Z'ZW L'L5H K'KW A'AW F=0.0000000
DEF2P 1 15 HALF
HALF 1 2
HALF 3 2
DEF2P 1 3 FIL
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
207
!TLINP__T1
!TLINP^_T2
!TLINP__T3
!TLINP*_T4
!TLINP~~T5
!TLINP*_T6
1DEF2P 1 7
1
2
3
4
5
6
2 Z'Z50 L'L50 K'K50 A'A50 F=0.0000000
3 Z'ZW L'LI K'KW A'AW F=0.0000000
4 Z'ZN L~L2 K'KN A'AN F=0
5 Z'ZW L'L3 K'KW A'AW F=0.0000000
6 Z'ZN L'L4 K'KN A'AN F=0.0000000
7
Z'ZW L'L5H K'KW A'AW F=0.0000000
HALF1
1HALF1 1 2
1HALF1 3 2
1DEF2P 1 3 NOSTEP
RES 1 0 R'Zn
DEF1P 1 TEST1
IRES 1 0 R'Z51
1DEF1P 1 TEST2
!RES 1 0 R'Z52
!DEF1P 1 TEST3
FREQ
SWEEP 0.5 12 .115
!STEP 5
OUT
FIL DB[S21] GR1
YBCO D B [S 21] GR1
!SILVER D B [S 2 1] GR1
!NOSTEP DB[S21] GR1
FIL D B [Sll] GR3
YBCO D B [S l l ] GR3
!SILVER D B [S l l ] GR3
FIL AN G [S 2 1 ] GR2
YBCO A N G [S21] GR2
!SILVER A N G [S 2 1 ] GR2
FIL AN G [Sl l ] GR4
YBCO ANG[Sll] GR4
!SILVER ANG[Sll] GR4
FIL S21 SC2
YBCO S21 SC2
TEST1 R E [Z11] GR5
! TEST2 R E [Z 11] GR5
! TEST3 R E [Zl l ] GR5
FIL D B [S21] GR6
YBCO D B [S21] GR6
GRID
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208
RANGE 0 1 2 1
!RANGE 7 9 .2
1 GR6 -2 0 .5
OPT
RANGE 1 11
YBCO MODEL FIL
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
209
Appendix D
N ew Model with Improved Conductivity Equation
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
210
! MODEL OF THE YBCO FILTER FIT TO DATA USING LOSS AND LOSS TANGENT
! ALSO USING S2P FILE FOR 50 OHM TAPER K-CONNECTOR TO CPWG INTERFACE
! ALSO USING CONSTANT LOSS'S-PAR FILE (ADJUST.S2P) TO CORRECT FOR
! CAL STANDARDS BEING AT ROOM TEMP (NOT DUNKED)
! USING LOW LOSS EQUATIONS FOR ALL ATTENUATIONS AS A FUNCTION OF LD AND
Sn
! BY DIMITRIOS ANTSOS (JULY 28, 1992)
! FILENAME REFERS TO (TRUE) POWER INTO FILTER
! FILTER USED IS LAST PACKAGED CPW FILTER LEFT
DIM
LNG UM
ANG RAD
VAR
! T = 77
! Tc = 83
LD1 # .5
0.588172
.75
YBCO
Sn #1E4 9607145. 5E8
XN # 0.73 0.885900 .93
electron
XW # .050 0.072625 .251
X5 # 0.08 0.092468 .281
tau#le-18 1.27e-ll le-10
eO = 8.854E-12
AGn = 3e-12
Line )
Gn = 1.255e5
Var (Narr
AGw = le-10
Gw = 1.7336e4
AG5 = 2.5e-ll
G5 = 2.3501e4
!TEMPERATURE OF MEASUREMENT
!Critical Temperture of Sample
!PENETRATION DEPTH FOR
!Normal Conductivity of Sample
[Percentage of normal
[Function of H and T
[Collision Relaxation time
[Permittivity of free space
[Dimension Variable ( Narrow
[Incremental Inductance Rule
Z500 = 49.56
ZNO = 83.38
ZWO = 22.64
KI #5
14.44599
25
K500 = 12.52
KNO = 12.455
KWO = 12.49
LI # 1000 2391.998 3000
L50
LI
L2
L3
L4
L5H
#
=
=
=
=
=
1800 2561.258 2900
720.8
997.0
1369.7
761.2
924.0
!# 4000
4335.155 4400
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
211
ACIO #1E-10 4.10e-09 IE-6
ADIOO # 3E-4 0.000707 9e-3
TAND
! COPPER LOSS OF INPUT
! = 3.3E-4
# 0 0.010000 .01
EQN
LD = LDl*le-6
ADNO = 3.086E-4*TAND
AD500 = 3.093E-4*TAND
ADWO = 3.089E-4*TAND
ADIO = ADIOO*TAND
! DIELECTRIC LOSS OF INPUT CPW 50 OHM TAPER
LI1 = -LI
AI
= ACIO
* FREQ**2 + ADIO * FREQ
! Computation of exact loss for narrow line
!
Constants
UO = 4*PI*le-7
[Magnetic Permeability of vacuum
f = FREQ*le9
!Frequency in Hz
SrN = Sn*XN
IReal Part of conductivity of YBCO
SiNa = (1-XN)/ (2*PI*f*U0*LD** )+XN*Sn*2*PI*f*tau
!Imaginary Part of conductivity
SrW = Sn*XW
IReal Part of conductivity of YBCO
SiW = (1-XW)/(2*PI*f*U0*LD**2 +XW*Sn*2*PI*f*tau
[Imaginary Part of conductivity
Sr5 = Sn*X5
[Real Part of conductivity of YBCO
Si5 = (1-X5)/(2*PI*f*U0*LD**2 +X5*Sn*2*PI*f*tau
[Imaginary Part of conductivity
PN = ATAN(SiNa/SrN)-2*PI
[Angle of conductivity
ThN= PI/4-PN/2
[Auxiliary angle definition
rNa= SQRT(SQR(SrN)+SQR(SiNa))
[Norm of conductivity
PW = ATAN(SiW/SrW)-2*PI
[Angle of conductivity
ThW= PI/4-PW/2
[Auxiliary angle definition
rWi= SQR T (SQ R (SrW)+SQR(SiW) )
[Norm of conductivity
P5 = ATAN(Si5/Sr5)-2*PI
[Angle of conductivity
Th5= PI/4-P5/2
[Auxiliary angle definition
r50= SQR T (SQR(Sr5)+SQR(Si5))
[Norm of conductivity
c = 1/sqrt(eO*UO)
[Velocity of light
Vn=
Bn=
Cn=
Dn=
En=
Un=
Wn=
Narrow Line Parameters
c/sqrt(KN0)
[Phase velocity in line
Gn*AGn*SQRT(2*PI*f*U0*rNa)
[Fudge Factors
E X P (2*Bn*COS(ThN))
COS(2*Bn*SIN(ThN))
SIN(2*Bn*SIN(ThN))
S Q R T (S Q R (Cn*Dn-l)+SQR(Cn*En))
ATAN(Cn*En/(Cn*Dn-l))
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212
Fn=
Mn=
Nn=
Rn=
Bn/(AGn*rNa*Un)
2*Bn*SIN(ThN)
COS(PI/4+PN/2-Wn)
COS(Mn+PI/4+PN/2-Wn)
ReZn = -Fn*(Nn+Cn*Rn)
Meter
NIn= SIN(PI/4+PN/2-Wn)
RIn= SIN(Mn+PI/4+PN/2-Wn)
ImZn = -Fn*(NIn+Cn*RIn)
LMn = ZN0/Vn+ImZn/(2*PI*f)
CMN = 1/(ZN0*Vn)
Zn = sqrt(LMn/CMn)
RATn= ReZn/(2*PI*f*LMn)
ANGn= 0.5 *ATAN(RATn)
[Real Part of Internal Impedance /
ACN = (8.686e-6) *
(2*PI*f)*sqrt(LMn*CMn)*sqrt(sqrt(1+SQR(RATn)))*sin(ANGn)
KN = LMn*CMn/(eO*uO)*sqrt(1+SQR(RATn))*SQR(cos(ANGn))
!
Vw=
Bw=
Cw=
Dw=
Ew=
Uw=
Ww=
Fw=
Mw=
Nw=
Rw=
Wide Line Parameters
c/sqrt(KWO)
IPhase velocity in line
Gw*AGw*SQRT(2*PI*f*U0*rWi)
IFudge Factors
EXP(2*Bw*C0S(ThW))
C O S (2*Bw*SIN(ThW))
SIN(2*Bw*SIN(ThW))
SQR T (SQR(Cw*Dw-l)+SQR(Cw*Ew))
ATAN(Cw*Ew/(Cw*Dw-l))
Bw/(AGw*rWi*Uw)
2*Bw*SIN(ThW)
C O S (PI/4+PW/2-Ww)
C O S (Mw+PI/4+PW/2-Ww)
ReZw =-Fw*(Nw+Cw*Rw)
NIw= SIN(PI/4+PW/2-Ww)
RIw= SIN(Mw+PI/4+PW/2-Ww)
ImZw =-Fw*(NIw+Cw*RIw)
LMw = ZWO/Vw+ImZw/(2*PI*f)
CMw= 1/(ZWO*Vw)
Zw = sqrt (LMw/CMw)
RATw= ReZw/(2*PI*f*LMw)
ANGw= 0.5*ATAN(RATw)
IReal Part of Internal Impedance / Meter
ACW = (8.686e-6) *
(2 *PI*f)*sqrt(LMw*CMw)*sqrt(sqrt(1+SQR(RATw)))*sin(ANGw)
KW = LMw*CMw/(e0*u0)*sqrt(l+SQR(RATw))*SQR(cos(ANGw))
i
V5=
B5=
C5=
D5=
E5=
50 Ohm Line Parameters
c/sqrt(K500)
G5*AG5*SQRT(2*PI*f*U0*r50)
E X P (2*B5*COS(Th5))
C O S (2*B5*SIN(Th5))
SIN(2*B5*SIN(Th5))
IPhase velocity in line
IFudge Factors
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
213
U5=
W5=
F5=
M5=
N5=
R5=
S Q R T (SQR(C5*D5-1)+SQR(C5*E5))
ATAN(C5*E5/(C5*D5-1))
B5 / (AG5*r50*U5)
2*B5*SIN(Th5)
COS(PI/4+P5/2-W5)
C O S (M5+PI/4+P5/2-W5)
ReZ5 =-F5*(N5+C5*R5)
NI5= SIN(PI/4+P5/2-W5)
RI5= SIN(M5+PI/4+P5/2-W5)
ImZ5 =-F5*(NI5+C5*RI5)
LM5 = Z500/V5+ImZ5/(2*PI*f)
CM5 = 1/(Z500*V5)
Z50 = sqrt(LM5/CM5)
RAT5= ReZ5/(2*PI*f*LM5)
ANG5= 0.5 *ATA N (RAT5)
IReal Part of Internal Impedance / Meter
AC50 = (8.686e-6) *
(2*PI*f)*sqrt(LM5*CM5)*sqrt(sqrt(1+SQR(RAT5)))*sin(ANG5)
K50 = LM5*CM5/ (eO*uO)*sqrt(1+SQR(RAT5))*SQR(cos(ANG5))
A50
AN
AW
= AC50 + AD500 * FREQ
= ACN + ADNO * FREQ
= ACW + ADWO * FREQ
CKT
S2PA 1 2 0 costepl.s2p
DEF2P 1 2 BIG_STEP
S2PB 1 2 0 costep2.s2p
DEF2P 1 2 SML_STEP
S2PC 1 2 0 5dbm.s2p
DEF2P 1 2 YBCO_RAW
S2PD 1 2 0 adjust.s2p
DEF2P 1 2 ADJ
S2PE 1 2 0 hts_50.s2p
DEF2P 1 2 FIFTY
ADJ 1 2
YBCO_RAW 2 3
DEF2P 1
3 YBCO
TLINP 1
2 Z=50
L'LIK'KI A'AI F=0
TLINP 2
3 Z=50
L'LIl K'KI A=0 F=0
FIFTY 3 4
TLINP_T1 4 5 Z'Z50 L'L50 K'K50 A"A50 F=0.0000000
SML_STEP
5 6
TLINP_T2 6 7 Z'ZW L'LI K'KW A'AW F=0.0000000
BIG STEP
7 8
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214
TLINP_T3 8 9 Z'ZN L'L2 K'KN A'AN F=0
BIG_STEP 9 10
TLINP_T4 10 11 Z'ZW L “L3 K'KW A'AW F=0.0000000
BIG_STEP
11 12
TLXNP_T5 12 13 Z'ZN L'L4 K'KN A'AN F=0.0000000
BIG_STEP
13 14
TLXNP_T6 14 15 Z'ZW L~L5H K'KW A'AW F=0.0000000
DEF2P 1 15 HALF
HALF 1 2
HALF 3 2
DEF2P 1 3 FIL
!TLINP_T1
!TLINP_T2
!TLINP_T3
!TLINP_T4
!TLINP_T5
!TLINP_T6
1DEF2P 1 7
1
2
3
4
5
6
2
3
4
5
6
7
Z 'Z50 L'L50 K~K50 A~A50 F=0.0000000
Z'ZW L'LI K'KW A'AW F=0.0000000
Z'ZN L'L2 K'KN A'AN F=0
Z'ZW L'L3 K'KW A'AW F=0.0000000
Z'ZN L'L4 K'KN A'AN F=0.0000000
Z'ZW L'L5H K'KW A'AW F=0.0000000
HALF1
!HALF1 1 2
!HALF1 3 2
1DEF2P 1 3 NOSTEP
RES 1 0 R'Zn
DEF1P 1 TEST
FREQ
SWEEP 0.5 12 .115
!STEP 2
OUT
FIL DB[S21] GR1
YBCO D B [S21] GR1
!SILVER D B [S 2 1 ] GR1
!NOSTEP DB[S21] GR1
FIL D B [S l l ] GR3
YBCO D B [S l l ] GR3
!SILVER D B [S l l ] GR3
FIL AN G [S21 ] GR2
YBCO A N G [S2 1 ] GR2
!SILVER A N G [S21] GR2
FIL ANG[Sll] GR4
YBCO ANG[Sll] GR4
!SILVER ANG[Sll] GR4
FIL S21 SC2
YBCO S21 SC2
TEST R E [Zll] GR5
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215
! FIL DB[S21] GR6
! YBCO DB[S21] GR6
GRID
RANGE 0 12 1
!RANGE 7 9 .2
! GR6 -2 0 .5
OPT
RANGE 1 11
YBCO MODEL FIL
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216
Appendix E
Touchstone Circuit File that Verifies the Conjecture o f Section 7.5.4
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217
! MODEL OF THE YBCO FILTER FIT TO DATA USING LOSS AND LOSS TANGENT
! ALSO USING S2P FILE FOR 50 OHM TAPER K-CONNECTOR TO CPWG INTERFACE
! ALSO USING CONSTANT LOSS S-PAR FILE (ADJUST.S2P) TO CORRECT FOR
! CAL STANDARDS BEING AT ROOM TEMP (NOT DUNKED)
! USING LOW LOSS EQUATIONS FOR ALL ATTENUATIONS AS A FUNCTION OF LD AND
Sn
! BY DIMITRIOS ANTSOS (JULY 28, 1992)
! FILENAME REFERS TO (TRUE) POWER INTO FILTER
1 FILTER USED IS LAST PACKAGED CPW FILTER LEFT
! ALL Z'S AND AG, G OPTIMIZED
DIM
LNG UM
ANG RAD
VAR
LD1 # .5
0.749436
.75
YBCO
Sn #1E4 9.22e+07 5E8
XN # 0.73 0.929888 .93
electrons
XW # .05 0.077046 .25
X5 # .08 0.105440 .28
tau#le-18 2.28e-13 le-10
eO = 8.854E-12
AGn #le-14 1.03e-12 3e-9
Gn #le2 634085.0 le8
AGw #le-12 1.00e-08 le-8
Gw #lel 77356.57 le7
AG5 #2.5e-13 2.42e-08 2.5e-8
G5 #2.5el
37.64339 2.3e7
Z500 # 40 50.17075
ZN0 #75 114.7819
ZW0 #20 23.93472
KI #5
24.99001
!PENETRATION DEPTH FOR
INormal Conductivity of Sample
!Percentage of normal
!Function of H and T
ICollision Relaxation time
!Permittivity of free space
!= 3e-12
!Dimension
!= : .255e5
! Icremental Indue
!= le-10
!= 1.7336e4
!= 2.5e-ll
! 2.3501e4
190
250
150
!= 49.56
! = 83.38
!= 22.64
25
K500 = 12.52
KN0 = 12.455
KW0 = 12.49
LI # 1500 2178.428 2900
L50
LI
L2
L3
L4
L5H
#
=
=
=
=
=
1800 2664.809 2900
720.8
997.0
1369.7
761.2
924.0
ACIO #1E-10 1.47e-09 IE-6
ADI00 # 3E-4 0.005616 9e-3
!# 4000
4335.155 4400
! COPPER LOSS OF INPUT
! = 3.3E-4
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218
TAND
# 0 0.006364 .01
EQN
LD = LDl*le-6
ADNO = 3.086E-4*TAND
AD500 = 3.093E-4*TAND
ADWO = 3.089E-4*TAND
ADIO = ADIOO*TAND
! DIELECTRIC LOSS OF INPUT CPW 50 OHM TAPER
LI1 = -LI
AI
= ACIO
* FREQ**2 + ADIO * FREQ
! Computation of exact loss for narrow line
!
Constants
U0 = 4*PI*le-7
IMagnetic Permeability of vacuum
f = FREQ*le9
!Frequency in Hz
SrN = Sn*XN
IReal Part of conductivity of YBCO
SiNa = (1-XN)/(2*PI*f*U0*LD** )+XN*Sn*2*PI*f*tau
!Imaginary Part of conductivity
SrW = Sn*XW
IReal Part of conductivity of YBCO
SiW = (1-XW)/(2*PI*f*U0*LD**2 +XW*Sn*2*PI*f*tau
!Imaginary Part of conductivity
Sr5 = Sn*X5
IReal Part of conductivity of YBCO
Si5 = (l-X5)/(2*PI*f*U0*LD**2 +X5*Sn*2*PI*f*tau
!Imaginary Part of conductivity
PN = ATAN(SiNa/SrN)-2*PI
lAngle of conductivity
ThN= PI/4-PN/2
lAuxiliary angle definition
rNa= SQRT(SQR(SrN)+SQR(SiNa))
!Norm of conductivity
PW = ATAN(SiW/SrW)-2*PI
lAngle of conductivity
ThW= PI/4-PW/2
lAuxiliary angle definition
rWi= SQRT(SQ R (SrW)+SQR(S iW))
INorm of conductivity
P5 = ATAN(Si5/Sr5)-2*PI
1Angle of conductivity
Th5= PI/4-P5/2
lAuxiliary angle definition
r50= SQRT(SQ R (Sr5)+SQR(Si5))
INorm of conductivity
c = l/sqrt(eO*UO)
1Velocity of light
Vn=
Bn=
Cn=
Dn=
En=
Un=
Wn=
Fn=
Mn=
Nn=
Rn=
Narrow Line Parameters
c/sqrt(KN0)
IPhase velocity in line
Gn*AGn*SQRT(2*PI*f*U0*rNa)
IFudge Factors
EXP(2*Bn*COS(ThN))
COS(2*Bn*SIN(ThN))
SIN(2*Bn*SIN(ThN))
SQRT(SQ R (Cn*Dn-l)+SQ R (Cn*En))
ATA N (Cn*En/(Cn*Dn-l))
Bn/(AGn*rNa*Un)
2*Bn*SIN(ThN)
COS(PI/4+PN/2-Wn)
COS(Mn+PI/4+PN/2-Wn)
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219
ReZn = -Fn*(Nn+Cn*Rn)
Meter
NIn= SIN(PI/4+PN/2-Wn)
RIn= SIN(Mn+PI/4+PN/2-Wn)
ImZn = -Fn*(NIn+Cn*RIn)
LMn = ZNO/Vn+ImZn/(2*PI*f)
CMN = 1 / (ZNO*Vn)
Zn = sqrt(LMn/CMn)
RATn= ReZn/(2*PI*f*LMn)
ANGn= 0.5*ATAN(RATn)
IReal Part of Internal Impedance /
ACN = {8.686e-6) *
(2*PI*f)*sqrt(LMn*CMn)*sqrt(sqrt(1+SQR(RATn)))*sin(ANGn)
KN = LMn*CMn/(eO*uO)*sqrt(1+SQR(RATn))*SQR(cos(ANGn))
!
Vw=
Bw=
Cw=
Dw=
Ew=
Uw=
Ww=
Fw=
Mw=
Nw=
Rw=
Wide Line Parameters
c/sqrt(KW0)
JPhase velocity in line
Gw*AGw*SQRT(2*PI*f*U0*rWi)
!Fudge Factors
E X P (2*Bw*COS(ThW))
COS(2*Bw*SIN(T h W ))
SIN(2*Bw*SIN(ThW))
SQRT(SQR(Cw*Dw-l)+SQR(Cw*Ew))
ATAN(Cw*Ew/(Cw*Dw-l))
Bw/(AGw*rWi*Uw)
2 *Bw*SIN(ThW)
COS(PI/4+PW/2-Ww)
COS(Mw+PI/4+PW/2-Ww)
ReZw =-Fw*(Nw+Cw*Rw)
NIw= S I N (PI/4+PW/2-Ww)
RIw= SIN(Mw+PI/4+PW/2-Ww)
ImZw =-Fw*(NIw+Cw*RIw)
LMw = ZWO/Vw+ImZw/(2*PI*f)
CMw= 1/(ZW0*Vw)
Zw = sqrt(LMw/CMw)
RATw= ReZw/(2*PI*f*LMw)
ANGw= 0.5*ATAN(RATw)
IReal Part of Internal Impedance / Meter
ACW = (8.686e-6) *
(2*PI*f)*sqrt(LMw*CMw)*sqrt(sqrt(1+SQR(RATw)))*sin(ANGw)
‘ KW = LMw*CMw/(e0*u0)*sqrt(1+SQR(RATw))*SQR(cos(ANGw))
!
V5=
B5=
C5=
D5=
E5=
U5=
W5=
F5=
M5=
N5=
50 Ohm Line Parameters
c/sqrt(K500)
IPhase velocity in line
G5*AG5*SQRT(2*PI*f*U0*r50)
IFudge Factors
E X P (2*B5*COS(Th5))
C O S (2*B5*SIN(Th5))
S I N (2*B5*SIN(Th5))
SQRT(S Q R (C5*D5-1)+SQR(C5*E5))
A T A N (C5*E5/(C5*D5-1))
B5/(AG5*r50*U5)
2*B5*SIN(Th5)
COS(PI/4+P5/2-W5)
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220
R5= C O S (M 5+P I/ 4 + P 5 /2 - W 5 )
ReZ5 =-F5*(N5+C5*R5)
NI5= SIN(PI/4+P5/2-W5)
RI5= SIN(M5+PI/4+P5/2-W5)
ImZ5 =-F5*(NI5+C5*RI5)
LM5 = Z500/V5+ImZ5/(2*PI*f)
CM5 = 1/ (Z500*V5)
Z50 = sqrt(LM5/CM5)
RAT5= ReZ5/(2*PI*f*LM5)
ANG5= 0•5*ATAN(RAT5)
IReal Part of Internal Impedance / Meter
AC50 = (8.686e-6) *
(2 *PI*f )* sqrt(LM5*CM5)* sqrt(sqrt(1+SQR(RAT 5 )))* s in(A NG5)
K50 = LM5*CM5/(e0*u0)*sqrt(1+SQR(RAT5))*SQ R (c o s (ANG5))
A50
AN
AW
= AC50 + AD500 * FREQ
= ACN + ADNO * FREQ
= ACW + ADWO * FREQ
CKT
S2PA 1 2 0 costepl.s2p
DEF2P 1 2 BIG_STEP
S2PB 1 2 0 costep2.s2p
DEF2P 1 2 SML_STEP
S2PC 1 2 0 5dbm.s2p
DEF2P 1 2 YBCO_RAW
S2PD 1 2 0 adjust.s2p
DEF2P 1 2 ADJ
S2PE 1 2 0 hts_50.s2p
DEF2P 1 2 FIFTY
ADJ 1 2
YBCO_RAW 2 3
DEF2P 1 3 YBCO
TLINP 1
TLINP 2
FIFTY 3
TLINP_T1
SML_STEP
TLINP_T2
BIG_STEP
TLINP_T3
BIG_STEP
TLINP_T4
BIG_STEP
TLINP T5
2 Z=50 L'LIK'KI A'AI F=0
3 Z=50 L'LIl K'KI A=0 F=0
4
4 5 Z'Z50 L'L50 K'K50 A'A50 F=0.0000000
5 6
6 7 Z'ZW L'LI K'KW A'AW F=0.0000000
7 8
8 9 Z'ZN L~L2 K'KN A'AN F=0
9 10
10 11 Z'ZW L'L3 K'KW A'AW F=0.0000000
11 12
12 13 Z'ZN L'L4 K'KN A'AN F=0.0000000
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221
BIGJ3TEP 13 14
TLINPJT6 14 15 Z'ZW L'L5H K'KW A'AW F=0.0000000
DEF2P 1 15 HALF
HALF 1 2
HALF 3 2
DEF2P 1 3 FIL
!TLINP_T1
!TLINP_T2
!TLINP_T3
!TLINP_T4
!TLINP_T5
!TLINP T6
!DEF2P 1 7
1
2
3
4
5
6
2Z'Z50 L'L50 K"K50 A'A50 F=0.0000000
3 Z'ZW L'LI K'KW A'AW F=0.0000000
4 Z'ZN L'L2 K'KN A'AN F=0
5 Z'ZW L ‘L3 K'KW A'AW F=0.0000000
6 Z'ZN L'L4 K'KN A'AN F=0.0000000
7
Z'ZW L'L5H K'KW A'AW F=0.0000000
HALF1
IHALF1 1 2
1HALF1 3 2
1DEF2P 1 3 NOSTEP
RES 1 0 R'Zn
DEF1P 1 TEST
FREQ
SWEEP 0.5 12 .115
!STEP 2
OUT
FIL D B [S 2 1 ) GR1
YBCO DB[S21] GR1
!SILVER D B [S21] GR1
!NOSTEP D B [S21] GR1
FIL D B [S l l ] GR3
YBCO D B [Sll] GR3
!SILVER D B [Sll] GR3
FIL A N G [S21] GR2
YBCO A N G [S21] GR2
!SILVER A N G [S 2 1 ] GR2
FIL A N G [S l l ] GR4
YBCO ANG[Sll] GR4
!SILVER ANG[Sll] GR4
FIL S21 SC2
YBCO S21 SC2
TEST R E [Z11] GR5
! FIL D B [S21] GR6
! YBCO D B [S21] GR6
GRID
RANGE 0 12 1
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222
iRANGE 7 9 .2
! GR6 -2 0 .5
OPT
RANGE
1 11
YBCO MODEL FIL
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223
CHAPTER 8
CLOSED RECTANGULAR HTS W AVEGUIDES
8.1 Introduction
This c h a p te r is m o tiv ated by a q u estion posed to m e by D r. C harles Elachi o f JPL , during
my oral candidacy exam ination.
H e inquired a b o u t th e viability and usefulness o f H T S
w aveguides. In th e analysis th a t follow s, closed re c ta n g u la r H T S w aveguides a re assum ed
and th e analysis c o n c e n tra tes on th e T ran sv erse E lec tric p ro p ag atio n m ode o f o rd e r (1,0)
(T E 1 0 ).
T his is b e cau se T E 1 0 , th e lo w est o rd e r m o d e (th e o n e w ith the low est c u t-o ff
frequency), is th e m o st w idely u sed o n e for tra n sfer o f p o w e r in real-life applications.
A
so ftw a re p ro g ram , p resen ted below , using C lan g u ag e and incorporating m any F ortran
routin es, has been created as a to o l in accurately calculating th e fundam ental propagation
p aram eters, fields and cu rren ts o f arbitrary m odes in w av eguides o f different sizes and
m aterial p ro p erties. T h e cro ss-sectio n o f a typical w av eg u id e to be analyzed is show n in
figure 1.
It is b eyond th e sco p e o f this
M e t a l W a ll
c h ap ter to ad d ress h o w a stru c tu re like the
one show n in figure 1 could be fabricated,
alth o u g h fab rication should b e possible.
T h e c ro ss-sectio n show n in figure 1 is that
o f a re g u la r closed rectan g u lar m etallic
w aveguide, th e inside o f w hich has been
uniform ly co v ered w ith a layer o f H T S
H T S L aycf
F i g u r e 1 C r o s s - s c c lio n o f th e H T S w a v e g u id e ,
(possibly Y B C O ). T h e cro ss-h atch ed part
in figure 1 is th e m etal and th e single hatched part is th e H T S layer. This layer is assum ed
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to b e o f th ick n ess d, w h ere — » 1
(actually — > 5 should b e sufficient fo r th e desired
b o u n d ary co n d itio n s to hold; see discussion at th e end o f section 3.3).
S om e o f th e q u estio n s ad d ressed in this c h a p te r are, is th ere an advantage to using H T S
w av eg u id es
o v e r re g u la r w av eguides
and
if so w hen, w hat do th e
p ro p ag atio n
ch aracteristics o f H T S w aveguides look like and h o w high in p u t-p o w er can be applied
b efo re g ettin g in to tro u b le attrib u table to th e n atu re o f the H T S.
8.2 The Cross-Over Frequency
W hat I desig n ate as th e cro ss-o v e r frequency, f x , is defined as th e frequency at w hich th e
surface resistan ce o f a norm al c o n d u c to r is equal to th at o f an H T S .
c o n d u c to r and th e H T S can be at the sam e o r at different tem peratures.
T h e norm al
T h e cro ss-o v er
freq uen cy is independent o f th e g eom etry o f th e w aveguide, and is rath er a p ro p erty o f the
m aterial it is m ade of.
It is a sim ple yet useful m easure that can be used to determ ine
w h e th e r it is a d v a n tag eo u s or not, from a loss p o in t o f view , to u se an H T S w aveguide,
since th e losses in a w aveguide, assum ing a perfect dielectric filling, o c cu r in the surface
c u rre n ts flow ing in th e w alls, w ithin a few sk in -depths o f the surface. A s is d em o n strated
in eq u atio n (3.3 1 ), rep ro d u ced below for convenience, and in equation (1), th e surface
resistan ce o f an H T S is p ro p o rtio n al to th e sq u are o f frequency w hereas th at o f a norm al
m etal is p ro p o rtio n al to th e sq uare ro o t o f frequency [1],
/
Z,=j<»V<A
= -A o V /X
V
21/.
)
2
V\ /
fo r an H T S and
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225
\
<j
a
V
( 1)
cr
fo r a n o rm al c o n d u c to r, w h e re Zs is th e su rface im pedance, a> is th e an g u lar freq u en cy o f
th e ex citatio n , /.i0 is th e m ag n etic perm eability o f vacu u m , X0 is th e z e ro -te m p e ra tu re
p e n etratio n d e p th o f th e elec tro m a g n etic fields in th e H T S , n n is th e density o f norm al
electro n s in th e H T S , ns is th e density o f superco n d u ctin g electro n s in th e H T S , r is th e
electro n co llision re la x a tio n tim e, cr„ is th e norm al con d u ctiv ity o f th e H T S , T is th e
ab so lu te te m p e ra tu re o f th e H T S , Tc is th e critical te m p e ra tu re o f the H T S , / is the
freq u en cy o f th e ex c ita tio n an d a is th e conductivity o f the m etal.
H en ce, in sim plest
term s, it is b e tte r to u se a n o rm al m etal above th e cro ss-o v e r frequency and an H T S belo w
th e c ro ss-o v e r frequency. T h e eq u atio n fo r th e c ro ss-o v e r frequency is easily derived by
equ atin g th e real p a rts o f eq u a tio n s (3 .3 1 ) and (1 ) as
/,=
1
f jA 8
4 crcr
(2)
T able 1 sh o w s a "w o rst", "av erag e" and "best" case fo r th e cro ss-o v er freq u en cy w ith
p a ra m e te r values d ra w n from th e discussions in chapters 3-7.
In th e w o rst c a se a "bad"
H T S is co m p ared to g o ld at 77 K. In th e "best" case an "excellent" H T S is co m p a re d to
an im aginary "h o t", lo w co n d u ctiv ity m etal. In the "average" case an H T S like th e ones
m easured and d escrib ed in c h a p te rs 3 -7 is com pared to g o ld at room tem p eratu re.
In all
o f th e cases th e H T S s a re assum ed at the L N 2 tem p eratu re o f 77 K.
A.n (nm )
a (S/m )
a n (S/m )
T (K )
T r (K )
fv (G H z)
B est
140
107
1.1*106
77
90
5376
A verage
430
0 .4 4 * 1 0 8
3*106
77
87
163
W o rst
800
2* 1 0 8
10*106
77
85
12
C a se ...
T a b le 1 T h r e e c a s e s for th e c r o s s-o v e r freq u en cy
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226
It is o b v io u s fro m tab le 1 th a t th e c ro ss-o v e r freq u en cy is very sensitive to its param eters
and sp an s a h u g e ra n g e (o v e r 3 o rd e rs o f m agnitude) depending on their values.
In th e
w o rst case, an X -b an d w av eg u id e (W R 9 0 ) is th e h ig h est frequency H T S w aveguide th a t is
less lossy th a n a c o rresp o n d in g m etal one, w hereas, in th e b est c a se an H T S w av eg u id e is
less lo ssy th a n th e co rresp o n d in g m etal o n e in practically all frequencies.
H o w e v e r, th e b est and w o rst cases o f tab le 1 are u p p e r and lo w er limits th a t are rarely
en c o u n te re d in practice. F ig u res 2-4 are p lo ts o f th e c ro ss-o v e r frequency versus z e ro te m p e ra tu re p en etratio n d epth, critical tem p e ratu re and norm al conductivity, respectively,
w ith th e re st o f th e p aram eters k e p t at th e ir average valu es (see tab le 1).
400
400
500
600
X Q (nm )
F ig u r e 2 C ro ss-o v e r freq u en cy p lotted v ersu s zcro -lcm p cra tu re p c n e lr a lio n d ep th .
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227
190
180
170
150
140
on
130 Ojw---------55---------u
*H
Ok
D -------IJU
ij
T c (K)
F ig u r e 3 C ro ss-o v er freq u en cy p lo tted v ersu s c r itic a l tem p eratu re.
350
250
200
150
cJn ( M S /m )
F ig u r e 4 C r o ss-o v er freq u en cy p lotted versu s n orm al co n d u ctiv ity .
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228
F ig u res 2 -4 w e re calcu lated and p lo tted using M ath C A D .
file is inclu d ed as ap p en d ix A o f this chapter.
A p rin t-o u t o f th e M ath C A D
F ig u re s 2 -4 b ea r o u t th e fa ct th a t th e
averag e ra n g e o f valu es o f th e c ro ss-o v er frequency is n o t as extrem e as table 1 indicates.
In m o st cases th e c ro ss-o v e r frequency will b e fo u n d to lie b etw een 100 and 3 00 G H z.
H en ce th e re is usually a n u m b er o f w aveguide ty p e s (up to W R 10 and up to W R3
respectively) w hich w ill exhibit less loss if H T S w alls a re used.
8.3 A Contrast of the Exponential Attenuation of Normal and HTS Waveguides
In p h a so r n o tatio n , th e electric and m agnetic field so lu tio n s o f th e closed rectan g u lar
w av eg u id e problem have a z -d ep en d en ce and tim e d e p en d en ce pro p o rtio n al to
e U<^-TP) _ 0[j«*-(«+j/O=]
_
w h ere th e positive z-axis is th e longitudinal axis alo n g w hich th e energy p ro p a g a te s [1],
a, th e exponential a tte n u a tio n coefficient, is th e reciprocal o f th e distance one has to m ove
d o w n th e w av eg u id e b e fo re th e electric and m agnetic fields are atten u ated by a fa cto r o f
e~'.
F o r th e T E 1 0 m o d e, th e m ost com m only u se d m o d e o f closed rectan g u lar
w aveguides, th e exponential atten u atio n coefficient is given by [1]
(4 )
w h ere a and b are th e w idth and height o f the w aveguide cro ss-sectio n respectively (a>b is
needed; see figure 1), ?/ is th e im pedance o f the dielectric filling o f th e w aveguide, Rs is
th e su rface resistan ce o f w a v e g u id e w alls a n d / t . is th e c u to ff frequency o f th e T E 1 0 m ode.
In th e derivation o f eq u a tio n (4 ), a zero surface im p ed an ce (perfectly co n d u ctin g w alls) is
first assum ed, then th e lossless fields are calculated and from th ese th e exponential
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229
a tte n u a tio n co efficient is deriv ed , given th e su rface resistance. In this sense, eq uation (4)
is a first-o rd e r p e rtu rb a tio n to th e lossless field solution and is th erefo re expected to
p erfo rm optim ally fo r "small" surface im pedances. H ence, at least for frequencies b elo w
th e c ro s s -o v e r frequency, eq u atio n (4 ) should ad eq u ately m odel th e behavior o f an H T S
w av e g u id e w ith th e surface re sistan c e as given b y equation (3.31), fo r reasonable
p a ra m e te r values.
S u b stitu tin g th e real p art o f (3 .3 1 ) into (4) w e ob tain the expression fo r the exponential
atte n u a tio n o f an H T S w av eg u id e as
/
a-
./
\ 1cJ _______ a
J c
(5)
E q u a tio n (5 ) is p lo tte d v e rsu s frequency in fig u re 5 for four different ty p es o f H T S
w av egu id es (W R 9 0 , W R 2S , W R 1 0 and W R 5). F ig u re 5 also includes a p lo t o f eq u atio n
(4) fo r a go ld w aveg u id e, at ro o m tem p e ratu re, fo r com parison purposes.
T he vertical
axis is n orm alized to on e w ith re sp e ct to th e exponential loss coefficient o f a T E M w av e
p ro p a g a tin g b etw e e n tw o infinite parallel gold plates th at are spaced a distance b ap art at
th e c u to ff freq u en cy o f each ty p e o f w aveguide.
T h e horizontal axis is norm alized w ith
resp ect to th e c u to ff freq u en cy o f each type o f w aveguide.
H ence, w ith th e ab o v e
norm alization con d itio n s, th e cu rv e o f th e gold w av eg u id e is th e sam e for all o f W R 90,
W R 2 8 , W R 1 0 and W R 5 typ es (since
« 0.5 fo r all o f these types). The M athC A D file
used to plo t eq u a tio n s (4) and (5), as show n in figure 5, is included as appendix B o f this
chapter.
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230
TEM
HTS, W R d
HTS, W R
G o ld , A ll, @ 3 0 0 K
H T S .W R 2 8
H TS, W R 90
i~
3.5
T E 2 0 c u to lT
'
f
‘c
TE10
F i g u r e 5 N o r m a liz e d a t te n u a t i o n o f d if fe r e n t ty p e s o f H T S a n d g o ld w a v e g u id e .
T able 2 lists th e c u to ff frequencies o f th e T E 1 0 and T E 2 0 m odes o f the types o f
w av eg u id e co n sid e re d ab o v e and in addition
W R3 (co n sid ered la te r in th e chapter).
th ese w a v eg u id es fo r w hich % ~ 0 . 5
T E 1 0 c u to ff
T E 2 0 c u to ff
(GHz)
(GHz)
W R90
6.6
13.2
W R 28
21.1
42.2
W R 10
59.0
118.0
W R5
115.8
231.5
VVR3
173.3
346.6
All o f
have
( / c ) nr » = 2 ( / c ) n n o * ( / « ) n roi (the excePtion
b 4
is W R 9 0 fo r w hich — = —, but this small
a
T y p e ...
9
T a b i c 2 W a v e g u id e cu lo lT fre q u e n c ie s.
d eviation will be neg lected ).
As expected, for
th e a v erag e se t o f p a ra m e te rs , it is adv an tag eo u s, from th e stan dpoint o f loss, to use an
H T S w a v e g u id e up to W R 1 0 but not W R 5 (see table 1).
I f w e exam ine, as an exam ple,
th e freq u en cy at w hich th e go ld w aveguide and the W R 10 H T S w aveguide loss curves
cross, w e read th e c o rre sp o n d in g norm alized frequency as 2.75.
M ultiplying this by the
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231
T E 1 0 c u to ff freq u en cy o f 59 G H z found in tab le 2 w e obtain 162 G H z, which, is the
av erag e case c ro ss-o v e r frequency, as expected (see tab le 1).
A n ad v an tag e o f H T S o v er m etal w aveguides is th e location o f th e m inim um loss
frequency.
In o rd e r to m inim ize distortion o f th e input signal as it travels in the
w av eg u id e, w av eg u id es a re alm ost alw ays used in th eir single m ode frequencies. H ence,
th e hig h est freq u en cy th a t is recom m ended fo r use w ith a particular w aveguide is ju st
b elo w th e T E 2 0 m o d e c u to ff frequency. In fig u re 5 it can b e seen that th e m inim um loss
o f m etal w a v eg u id es is a t a frequency higher th a n th e T E 2 0 c u to ff frequency and hence
can n o t be used. In co n trast, in H T S w aveguides, the m inim um loss lies below th e TE 20
c u to ff frequ en cy , an ad v an tag e n o t to be o v erlooked.
A low er limit on th e usable
freq u en cies ex ists also, b u t is th e sam e fo r H T S and m etal w aveguides, dictated by the
T E 1 0 c u to ff freq u en cy (it is not recom m ended to o p e ra te to o close to th e latter since the
g ro u p v elo city has a steep slope w ith respect to frequency and even a n arrow band signal
ex p erien ces high disp ersio n at th ese frequencies).
T h e frequency o f the minim um
atte n u a tio n can be found by differentiating (5) w ith re sp ec t to frequency and setting the
deriv ativ e equal to zero: It is given by
min
In particular, fo r ^ / = 0 .5 , w e obtain / mm = ^ 3 + J \ 7 f c ~ 1.33 f c < 2 f c, as claim ed
above. An eq u atio n fo r th e m inim um attenuation is o b tained by substituting (6) into (5 ) as
E q u atio n (7) is p lo tted in figure 6 versus the w idth o f th e w aveguide cro ss-section, a ,
assum ing a co n stan t ratio y = 0 .5 .
3
/a
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232
WR3
W R5
0.01
^X [W R 28
0.001
0.03
0.1
0.2
0.3
0.-1
0.5
a (in)
Figure 6 Minimum attenuation o f different types o f HTS waveguides.
8.4 A Finite-Difference Numerical Solution for the Modes o f HTS Waveguides
8.4.1 The Problem
As mentioned in section 8.3, the exponential attenuation coefficient given by equations (4)
and (5) is only a first-order perturbation approximation to the "true" value o f the
coefficient.
In the derivation o f equation (4), a zero surface impedance (perfectly
conducting walls) is first assumed, then the lossless fields are calculated, and from these
the exponential attenuation coefficient is derived, given the surface resistance [1],
In
reality, of course, the walls have a finite surface impedance which affects the solution of
the fields in the waveguide.
Roughly speaking, one would expect the real part o f the
surface impedance, the surface resistance, to affect the exponential attenuation coefficient,
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233
a, and the imaginary part of the surface impedance, the surface reactance, to affect the
propagation constant, fi, and the cutoff properties o f the waveguide. However, this is not
completely true, because the existence o f a surface reactance, which affects the lossless
solution o f the fields, also indirectly affects the exponential attenuation coefficient, since,
the fields and currents that dissipate energy at the walls are different in the presence o f the
surface reactance. Hence, a is expected to be a strong function o f the surface resistance
and a weak function o f the surface reactance, whereas /? is expected to be a strong
function o f the surface reactance and a weak function o f the surface resistance.
The
question then becomes, how well does equation (5) represent the "true" exponential loss
coefficient o f an HTS waveguide?
To compound matters, in the case o f HTSs, the complex surface impedance spans a wide
range o f arguments (angles) over the frequency range under consideration (1-300 GHz),
whereas in the normal metal case, the argument o f the complex surface impedance is
S fJ
Y
arctan(-^L- )
X .
(Degrees)
70
00
50
200
f (GHz)
Figure 7 The argument (angle) of the surface impedance o f an HTS.
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constant and equal to — or 45° [ 1], Figure 7 is a plot o f the argument o f the surface
impedance o f an HTS versus frequency, using the average set o f parameter values (table
1). Over the usable frequency range the angle o f the surface impedance spans a range of
25 degrees for the average set o f parameter values (figure 7) and a range o f about 50
degrees from the worst-case to the best-case sets o f parameter values. In this frequency
range the surface impedance goes from being almost purely reactive to being almost
equally resistive and reactive. Figure 8 is a log-log plot o f the magnitude o f the surface
impedance o f an HTS versus frequency, using the average set o f parameters.
10
0.1
0.001
f (G H z)
Figure 8 The magnitude o f the surface impedance o f an HTS.
Over the 1 to 300 GHz frequency domain, the magnitude o f the surface impedance o f the
HTS spans a range o f 2.5 decades, from 0.003 to 1 in magnitude.
This large range is
spanned because the surface resistance is proportional to the square o f the frequency and
the surface reactance is proportional to the frequency (as compared to both the surface
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235
resistance and reactance being proportional to the square root o f frequency in normal
metals). Figures 7 and 8 bear out the fact that the equation for the boundary condition at
the HTS walls is modified as E,anscnlial = \Zs\eJ^ ( n x
from the corresponding
equation E tangcntial = 0 for the perfectly conducting walls, where Zs spans a range from
0.003 to 1 in magnitude and 90 to 65 degrees in phase in the frequency domain from 1 to
300 GHz. With such a great range o f boundary conditions there is the need for a way to
validate and define the limits of usage o f equation (5), and also accurately predict the
magnetic fields and currents at the HTS walls.
These are important since when they
approach or exceed their critical values the behavior o f the HTS changes drastically.
8.4.2
The Solution
The above problem is solved by a numerical solution o f Maxwell's equations, with the
appropriate boundary conditions for the HTS. Specifically, a finite-difference formulation
o f the Helmholtz wave equations, derived from Maxwell's equations, is employed.
We
start from the Helmholtz wave equations [1], [2]
(8)
V;.E + ( r +A2) e = 0
and
V2rH + (y 2 + A2)H = 0 ,
(9)
where k 2 = <y2//0£\ V~. is the transverse Laplacian operator, and sinusoidal phasor waves
propagating in the positive z-direction, proportional to
have been assumed.
Actually, only the z-components of ( 8) and (9) need to be solved for. The other four field
components (Ex, E v, Hx and H v) may be obtained from Ez and Hz using the four auxiliary
equations listed below [ 1]:
f
y +k
dE.
ox
.
dH }
8y
( 10)
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236
1
dE ,
.
dH.
F =
-Y
-Z
T
+
W
lh
y r2+k2
dy
dx
H= —
dE ,
J03S-
y +k
dy
( 11)
3H,
- r -
( 12)
dx
1
Hy
. dE,
dH.
j c o e - ^ + y' y 2+ k 2 V
dx
dy y
(13)
The waveguide is first sub-sectioned in a uniform rectangular grid, as shown in figure 9.
/\
x= 0, i=0
Symmetry
Plane
x=a/2, i=M
Symmetry
Plane
• ©
• - • - © -©■ • e 4 ----------------------- /
>••■©■ ■ © ■ • © " • © " • ■ • © - © • © . . . © . . . © . •
• ►
|
! •"© ■ • ♦ - © - © • - © - © - © © ■■ ■♦ • • ■© • © ' • >
/
/
1
© ...© ...© .
/
* • " © • •©■-■©■•■©■•■©■.-©■•.©
© •
f
y=b/2,j=N
« • ©
©
©
©
»-© •
|
«
•
©
©
i
>■•©■ • © ■ • © ■ ■ • © ■ • • • ■ • © - ©
©
f t ©
/ v yj
y= 0,j= 0 '
x ,i
Figure 9 The cross-seclion o f the HTS waveguide sub-sectioned by a uniform rectangular grid.________
Because of the symmetry planes, shown in figure 9, only one quarter o f the waveguide
cross-section needs to be sub-sectioned.
The uniform rectangular grid comprises M + 1
points in the x-direction (/ = 0, 1. . . M ) and N+\ points in the y-direction ( j = 0,1... JV).
Hence, the grid spacing is
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237
4* = ^
(H )
in the x-direction and
b
Ay = ----2N
(15)
V ’
in the y-direction.
The z-components of equations ( 8) and (9) are discretized, by approximating derivatives
with ratios o f finite (but small) differences. The discretized equations are [3]
E . (/ +1, j ) + E z (i - 1 J ) + R % (/, j +1) + R 2E : ( i j - 1) + [(/- + k 2)Ax2 - 2(l + R 2)]£. (/, j ) = 0
(16)
and
H : (i
+1 , j )
+ H .( i -
1, j )
+ R 2H : ( i , j
+1) +
R 2H . ( i , j -
1)
+ [(r +
k 2
)Ax2 - 2(1 +
R 2) ] H . ( i j )
- 0
(17)
where
( i j ) e { l,2 ...M - l} x { l,2 ...iV - l} ,
7? = —
and
a
center-differences
at
discretization is employed to maintain second-order accuracy.
It will be noted that
equations (16) and (17) can only be used at the "interior" points, away from the walls, and
they constitute a total o f 2(M-1)(AM) equations. The total number o f unknowns (Ez and
Hz at every grid point) is 2(M +\)(N+\). Hence, a further 4(M+N) equations are required
to give the problem a unique solution. There are 2{M+N) grid points on the walls. Hence
2 equations are needed from each grid point. The equations used are shown below for
each boundary.
a.
B O TTO M W A LL ( i = 0. i = 1. 2 ... M - H
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238
This is an HTS
wall.
Equation (3.31) is used for the surface impedance.
E .{ i ,j) = - Z sH x ( i ,j ) combined with a discretized version o f equation (12) give
2Ax ( k 2 + cc - 0 1 + j 2 a $ ) E ,( i , j ) +
+ {j2R R sc o s -2 R X ,m ) [ E I (i j +1) - E , (/,./)] +
~[aRs - P X S + j (J3Rs + a X s)][H: (/ +1J )
-
H : (/ -1,./)]
=0 ■
(18)
E x (J ,j) = ZsH .( i ,j ) combined with a discretized version o f equation ( 10) give
2 A x { l t i e + a 1 - / ) - 2 a / W , + ./[ * ,( t ! + a 2 - 0 ‘) + 2 a l3 R ^ H ,(iJ ) +
+j2(o/.i0R[H : (/,./ +1 ) - H ; (i,j)] +
(19)
+ ( « + jP)[E-. (i +1 J ) ~ E. ( / - 1J ) ] = 0 .
b. L E F T W A L L ( / = 0. ; = 1 .2 ...W - n
This is an HTS wall.
E .( i,j ) = ZsH v( i ,j ) combined with a discretized version o f
equation (13) give
2Ax ( lr + a ~ - p 1 + j 2 a $ E : { i,j) +
+ ( j2 R /o e - 2 X,<os)[E. (/' + 1,7') - E . (/,./')] +
+ [a /? /r,- fi R X , + j(p R R s + a R X ,) \H : { i ,j + \ ) - H : { i J - 1) ] - 0
.
(20)
E v( R j) = - Z sH . ( i ,j ) combined with a discretized version o f equation ( 11) give
2Ax{t?,(a 3 + a 2 - p 2) - 2 a p X s + j[ X s( k 2 + a 2 - p 2) + 2 a p R s^ H : ( i ,j ) +
+./2
[ H. (/’ + 1 , 7 ) - IE (/',./)]+
(21)
- { R a + jR p )[ E : (/,./ + 1) - E : (/, 7 - 1)] = 0.
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239
c. TOP SYMMETRY PLANE ( i = N . i = 1.2... M - n
This is a perfect electric wall, if a TEmn or TMmn mode is sought, where n is even, and a
perfect magnetic wall otherwise.
i. Perfect Electric Symmetry Plane
E,(i,j)
=
0
(22)
and E x( i J ) = 0 combined with a discretized version o f equation (10) give
j2Rcoit0[H. (/, j ) - H . ( /',./'- 1)] + ( a + j 0 )[ E z (/ +1, j ) - E . (/ - 1 , j ) ] = 0 .
(23)
ii. Perfect M agnetic Symmetry Plane
^
=0
(24)
and Hx( i , j ) - 0 combined with a discretized version o f equation ( 12) give
j2Rco8{E: ( i , j ) - E : ( i , j - } ) ] - { a + j P)[H: (i + \ J ) - H :( i - \ J ) ] = 0 .
(25)
d. RIGHT SYMMETRY PLANE (i = M . / = 1.2.. . N -11
This is a perfect electric wall, if a TEmn or TMmn mode is sought, where m is even, and a
perfect magnetic wall otherwise.
i. Perfect Electric Symmetry Plane
E :( i,j ) = 0
(26)
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240
and E y { i , j ) = 0 combined with a discretized version of equation ( 11) give
j 2 o ) ^ [ H 2( i j ) - H .( i -1 J ) ] - { R a + jR .p ) [ E t ( i j + 1) - E , ( i , j - 1)] = 0 .
(27)
ii. Perfect M agnetic Sym m etry Plane
=0
(28)
and Hy { i,j) = 0 combined with a discretized version o f equation (13) give
j2codjL. (/,./) - E : (/ - \ J ) ] + {R a+ jR (S)[H : (i j + 1) - H : (i , j -1 )] = 0 .
(29)
e. C O R N ER S
i. B ottom L eft C orn erf / = 0, j = 0)
This is an HTS corner on both sides. Equations (18) and (19) are modified as follows, so
as not to contain terms like
which do not make sense.
A x(/r + c r - j } + j 2 a fi ) E .( i ,j) +
+{jRRsa e - R X sm ) [ E : ( i , j + 1) - £ .(/,./)] +
(30)
and
Ax{
r s (k 2 + a 2 -J32)
- 2 afiX , + j [ x j j r + a 2 - ( ? ) + 2 apRs]}/■/.(/, j ) +
+ja>f40R[H. (/, j + 1) - H : (/, j)] +
(31)
+ { a + j/3)[E; (i + ] J ) - E :0 J ) ] = 0.
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241
ii. Bottom Right Corner (i - M , j = 0)
This corner is HTS on one side and perfect electric or magnetic on the other side.
However, by contrasting theoretical and computer results in a low-loss case, it is
determined that the best accuracy solutions are obtained when both boundary conditions
are drawn from the perfect boundary side.
1. Perfect Electric Symmetry Plane
E:VJ)=
and E
(32)
0
= 0 combined with a discretized version o f equation ( 11) give
jcoM0[H: ( i J ) - H :( i - l J ) ] - { R a + jR/3)[E: ( i , j + l ) - E : ( iJ ) ] = 0 .
(33)
2. Perfect Magnetic Symmetry Plane
=
o
and
(34)
= 0 combined with a discretized version o f equation (13) give
j(o t[E : ( i , j ) - E : { i - \ J ) ] + {R a + jR(3)[H: ( i , j + \ ) - H : {i,j)] = 0 ■
(35)
iii. Top Left Corner (/ = 0, / = N \
The boundary conditions from the perfect boundary are used, as in case c.ii above.
1. Perfect Electric Symmetry Plane
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242
E:( i j ) = 0
(36)
and E x ( i , j ) = 0 combined with a discretized version o f equation (10) give
jRcojii0[Hz( i , j ) - - \ ) ] + {a + jfi)[ E : (i + ] , j ) ~ E z( i,j)] = 0 .
(37)
2. Perfect M agnetic Symmetry Plane
n :( i j ) = 0
(38)
and H x ( / , j ) = 0 combined with a discretized version o f equation (12) give
JR cot{E: (/,./) - E : (/,./' - 1 )] - ( a + jp )[H : (/ +1, j ) - H . ( / J ) ] = 0 .
(39)
iv. Top Right Corner (i = M , j = N \
The applicable boundary conditions are as follows.
1. Top Perfect Electric Symmetry Plane
E .( i ,j ) = 0
(40)
and Ex (i, j) = 0 combined with a discretized version o f equation (10) give
j R &>//(,[//_(/, j ) - H .( / ,./ - ! ) ] + (« + jP )[E : (/,./) - E . (/' -1 ,./)] = 0 .
(41)
2. Top Perfect M agnetic Symmetry Plane
H : (.i>j) = 0
(42)
and Hx (J ,j) = 0 combined with a discretized version o f equation (12) give
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243
jR a f{ E : { i , j ) - E z( i , j - 1) ] ~ { a + jp )[ H s( / ',./ ) - H : ( / - 1, 7 )] = 0 .
(43)
3. Right Perfect Electric Symmetry Plane
E s( i,j ) = 0
(44)
and Ey ( i,j) = 0 combined with a discretized version o f equation (11) give
(/,7 ) - / ^ (/* - 1, 7 ) ] - ( ^ a +
(
/
,
y ) - ^ (/,y - 1)] = 0 .
(45)
4. Right Perfect Magnetic Symmetry Plane
=Q
(46)
and H v( i,j) - 0 combined with a discretized version o f equation (13) give
j m [ E z (/', j ) - E . (i - 1 j)] + { R a + jR p )[H . (/, j ) - H . (/, 7 - 1)] = 0 .
If the corner is electric-electric, (40) and (41) are used.
(47)
If the corner is magnetic-
magnetic, (42) and (43) are used. If the corner is electric-magnetic (40) and (42) are used.
The 2 (M -\)(N -\) interior equations and the 4{M+N) boundary equations, two from each
boundary grid point, constitute a total of 2(M+1)(A^+1) homogeneous equations with
2(A7+l)(yV+l) unknowns {IE and Hz at every grid point). They may be written in matrix
notation as
A ( /) X = 0 ,
(48)
where X is the vector o f the unknowns,
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244
'
£-(o,o)
"
£-(0,0)
£ -0 ,0 )
£ -0 ,0 )
E .( M , 0)
H .( M , 0)
x = £-(o,i)
(49)
>■2 { M +1 ) ( N + 1) rows,
£-(o,i)
£ / M , I)
£ -( M,1)
E.(M,N)
A is a 2( M +1 )(N + 1) x 2 {M +1 ) (N +1) matrix o f coefficients and the right-hand side of
(48) is the 2(M +\){N+\) element zero column-vector. A is very sparse (there are at most
5 non-zero elements in any row) and a function o f y, with the frequency/as a parameter.
Equation (48) has a non-trivial solution if and only if it is singular, i.e., if and only if
det[A (y)] = 0. This condition resembles an eigenvalue problem. Unfortunately, because
A is a function of both y and y-, it is a non-linear eigenvalue problem that cannot be
reduced to a linear one. This increases the computational complexity o f finding the y that
make the determinant o f the matrix zero, by one to two orders o f magnitude.
These y
correspond to the modes that are launched in the waveguide (both propagating and
evanescent). They are found by locating the roots o f the characteristic polynomial. In the
P{y) = det[A (y)]
is a complex-valued
polynomial o f a complex variable, with complex coefficients.
To demonstrate the
general case, the characteristic polynomial
principle and study an example o f a polynomial, A is entered into Malhemctfica (TM) by
Wolfram Inc. for the next to smallest possible grid: four interior points and twelve
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245
boundary points, or M=N=2.
M athematica is used here because o f its symbolic
manipulation capability. Appendix C contains the matrix and its factored determinant as
returned by Mathematica. In this example symmetry planes are not used (i.e., the whole
waveguide cross-section is sub-sectioned) and perfectly conducting walls are assumed.
The characteristic polynomial for this case is
p ( r ) =
8(
Ax Ay
y-+ o)-/.t0£~
r 2 + o ) \ t 0 s ) \ y 2 + ( D 2n 0 s
-
^
y-+a>-fi0e-
2
Ax~
\
A_y2
A x2
Av2
Ay2
->
2
2
y
r +<» M oA y2 v
y 2 + a 2j u 0 e -
1
Ax2
2 + c o 2M 0 £
2
A r2
A_y
Ay1
(50)
1
Ay
The first factor in the right-hand side o f equation (50) represents the TEM solution.
Unfortunately this pops up because the characteristic polynomial does not "recognize" that
the TEM solution has the X vector identically equal to zero.
The second factor is the
factor o f interest. Remembering that in this case A x ~ ^ , the zero o f this factor may be
3
expressed as
18
r = j p = j
m-/j0s
a "u r.£
/
a
\
co~
(51)
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246
Combining this with the equation for the propagation constant o f a waveguide [1]
(52)
we find that the effective cutoff frequency o f the TE10 mode, as predicted by the
characteristic polynomial, is
(53)
only 35 % off from its true value for such a coarse grid.
Also we notice that the
characteristic polynomial root has the correct dimensions (o f angular frequency).
Here we briefly digress to explain why equation (52), for the propagation constant, still
applies, despite the presence o f the surface reactance, Xs, which might affect the
propagation constant.
The proof extends the first-order perturbational solution for the
attenuation coefficient, which assumes the lossless modes in the waveguide and from these
calculates the extra loss, due to the surface resistance.
The extension uses a complex
surface impedance in place o f the surface resistance and interprets the imaginary part o f
the attenuation coefficient as a perturbation on the propagation constant. First, however,
we have to motivate this approach from physical laws. We start with Poynting's equation;
equation (3.22). The last term, e j , represents the instantaneous power per unit volume
converted to heat.
If instead we switch to phasors and use a surface current density
instead o f a volume current density, we obtain the expression
the real part o f
which represents the instantaneous power per unit surface area lost to heat on the walls.
The imaginary part o f this expression represents a power density that "sloshes" back and
forth between different forms o f energy, but is not dissipated because the current is 90
degrees out o f phase with the electric field.
If we allow this component to enter our
calculation o f the exponential attenuation, we obtain an imaginary component o f the latter
which may be interpreted as a correction to the propagation constant, as a result o f the
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247
presence o f the surface reactance.
The formula for calculating a , the exponential
attenuation, is [ 1]:
Power Lost /
/U n it Length
— .
a = -------------2 • Power Transmitted
(54)
The now complex numerator o f equation (54) is given by
(55)
using the bottom wall as an example. We continue the derivation as follows:
Reactive
The last part o f (56) is almost identical to the standard equation used to derive the power
loss per unit length in a waveguide [1], only Rs is substituted by Zs. What this means is
that we do not have to perform the long series o f calculations that equation (54) calls for,
but instead we can use equation (4) with Xs in place o f Rs. Hence
A fi=
(57)
Combining equations (57) and (3.31) we obtain the final expression for the expected
perturbation in the propagation constant due to the existence o f surface reactance:
(58)
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248
Table 3 shows the fractional perturbation,
calculated using equations (58) and (52)
for WR90, WR28, WR10, WR5 and WR3
waveguides, of average HTS parameters, at a
frequency that is at the upper limit of the
Type...
f (GHz)
W R90
W R 28
W R10
W R5
W R3
12.4
40.0
110.0
220.0
325.0
W
7fi
7.33-10_s
2.14 •10-4
6. 12 -10'4
1.17 -10~3
4.28-10‘3
T able 3 Fractional Perturbation o f the
Propagation Constant.
recommended operating range for the TE10
mode, according to the Handbook for Normal Waveguides. Table 3 indicates that for all
intents and purposes equation (52) is accurate enough to use, and also agrees with the
output o f the finite-difference program reported below (see figures 31, 33, 35, 37 and 39).
After legitimizing the use o f equation (52), we return to the characteristic polynomial,
P( y ) = d e t[A (y )]. In general, the roots of the characteristic polynomial can be located by
locating a local minimum o f \P{y)\. An initial guess is required, which, in this case, can be
provided by equation (5) for a and equation (52) for /?, given the right cutoff frequency of
the mode sought. Once the value y0 has been found, that makes the matrix A(y) singular,
for a given frequency, the matrix is then evaluated at y0 and the resulting matrix, say A 0 is
singular. Therefore the rank of A 0 is less than full. The unknown vector X is then a
member o f the nullspace o f A0. If the nullity o f A 0 is 1, X is unique modulo a scale factor.
If, however, the nullity o f A 0 is /?>1, then there are n orthonormal vectors X that satisfy
AoX=0. This corresponds to the case in which there is no loss (a= 0) and higher order TE
and a TM solutions exist for the same y0. Hence, n is 1, when loss is included or the
TE10 mode is considered, and 2 when loss is neglected and modes higher than TE10 are
considered. Therefore, in the "interesting" cases n is always 1 and a unique solution X 0
(modulo a scale factor) exists. The null vector X 0 can be found by performing a singular
value decomposition (SVD) on the matrix A0. This is to express A 0 in the form
A 0 = UZV" ,
(59)
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249
where H is the hermitian conjugate operator, £ is a square diagonal matrix with the
singular values in the main diagonal, and U and V are square unitary matrices.
The
diagonal entries o f E, cr,., are always non-negative and can be made to decrease in value
with respect to i. The nullity of A 0 is equal to the number o f cr that are equal to zero (or,
in practice, very small compared to cr,).
The columns o f V whose same-numbered
elements cr,. are zero are an orthonormal basis for the nullspace. In practice this means
that, in all interesting cases, the last column o f V is the unknown null vector X0. Using
the SVD method has the advantage that if, for some reason, the y0 located by the
minimization process does not truly render the matrix A singular, then the last singular
value is not small compared to the other singular values, and the error is thus uncovered.
Conversely, when A 0 is singular with nullity one it is
<<; j an^ ^
resuiting
^(M + lX W + O -l
X 0 is known to be a good solution.
8.4.3 T he P rogram
A program is written in C language, utilizing many Fortran library routines, and compiled
on the JPL CRAY Y-MP2E computer system, voyager, and the Goddard Space Flight
Center CRAY C98 computer system, chcirmy. The former uses the Unicos 7.0 operating
system and the latter Unicos 7.C.3.
A listing o f the program C-code is included as
appendix D o f this chapter.
The name o f the program is \vg_plot.c and it implements the-steps described in section
8.4.2 above to calculate the propagation and attenuation constants, electric and magnetic
fields, total power and maximum tangential fields on the walls o f a closed rectangular FITS
waveguide. SI units are used consistently throughout the program for all variables. In the
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250
description o f the code that follows, it may be helpful to the reader to follow along in
appendix D.
The first variables defined in the code are M , N and ROWLENGTH. M and N correspond
to the homonymous variables defined above. ROW LENGTH is equal to 2 (M + 1)(/V+1),
the number o f rows and columns o f the matrix A.
The first function called by the function main, which is the function the program launches
into and calls all other functions from, is veclor. This function, which is adopted from the
Numerical Recipes in C software package [4], reserves a memory chunk o f a given size
for use by the program and returns a pointer to it. In this case 2- ROW LENGTH 2 words
(units o f memory equal to 8 bytes long that are used by the Cray for one single precision
number) are reserved for the matrix A and a pointer called matrix is assigned to that
memory location.
Henceforth in the program matrix refers to what has hitherto been
called A.
The next set of commands asks the user to input the parameters o f the waveguide to be
analyzed. These parameters are assigned to an array o f eight real numbers called params,
with params[ 0] the width o f the waveguide cross-section, params[ 1] the height o f the
waveguide cross-section, params[2] the relative dielectric constant o f the waveguide
interior, params[3] is frequency to be analyzed, params[4] the effective zero temperature
penetration depth o f the HTS walls, params[5] the normal conductivity o f the HTS walls.
params[6] the temperature o f the HTS walls and p a ram s[l] the critical temperature o f the
HTS walls. The programs asks for each o f these parameters to be input from the standard
input sequentially.
Then the program asks for the m and the n o f the mode you are
seeking (assuming TEmn or TM m n is being sought, depending on the symmetries used in
the function matr\ see below).
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251
Then params,. m and n are passed to the next function called by main, which is called
guess_gamma. As its name implies, the purpose o f this function is to provide a starting
point for locating the zero o f the characteristic polynomial. It calculates the theoretical
attenuation and propagation constant, using equations (5) for the attenuation o f TE/??0
modes and the corresponding one for other modes (see [1]) and equation (52) for the
propagation constant. It returns these values, via a pointer, to a real array o f two numbers
called gamma, with gam m a[\] the guess for a and gamma[2] the guess for /?.
The next function called is determinant. This function takes gamma, params and matrix
as its input and returns the magnitude of the determinant o f matrix divided by the
magnitude of the factor ( y 2 + colp 0s f 1
'.
This is done to remove the corresponding
TEM factor from the characteristic polynomial, because it acts as a strong "attractor" o f
the root-finding process.
Frequently the minimization inadvertently ends at the TEM
solution, when this factor is not divided out. The function itself calls two Fortran library
routines: CGEFA and CGED. These are modifications o f the corresponding well known
U N PAC K routines [5], which are included in the JPL custom Fortran mathematical library
called M ATH77 [ 6], CGEFA performs the LU decomposition [7] of a general complex
matrix (in this case matrix). Then CGED takes the result and computes the determinant of
the original matrix by multiplying the diagonal elements o f the upper triangular matrix [7],
The function determinant also calls matr.
The function matr initializes matrix to the
correct values for all its elements. The function matr calls two other functions: putreal
and putimag. These two functions merely assign a value, real or imaginary respectively, to
a certain element o f matrix. The value to be assigned and the row and column coordinates
o f the element are passed to putreal and putim ag as arguments. The function matr also
calls Ez and Hz, as arguments to putreal and putimag.
These functions translate grid
coordinates to matrix row coordinates. For example, for A7=2, N=2 (3 by 3 grid, 18 by 18
matrix) Ez( l ,l ) is in the 9 ^ column o f the matrix (see equation (49)).
The functions
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252
putreal and putim ag place all the appropriate coefficients, seen in equations (16)-(45), of
every equation in the proper memory locations reserved as matrix. The function matr also
calls another function called initmatrix which zeros out every element o f matrix.
The function determinant is called thrice by main, via a for-loop, and initializes the 3element real a rra y s to the determinant values o f three points in the complex gamma plane,
one at the value returned by guessjgam m a and the other two lalpha and Ibeta away. The
minimization routine (called next) works optimally (minimum number o f iterations) when
lalpha and Ibeta are chosen such that the three initial points in the complex-gamma plane
are approximately equidistant from the true minimum.
The optimum lalpha and Ibeta
depend on the grid size (.M and N), but a good rule o f thumb is found to be
lalpha = 0.03 a and Ibeta = 0.02/?.
The function amoeba is called by main next. The function amoeba (and its sister function
amotry, which it calls) has been adopted from the Numerical Recipes in C software
package [4] and modified,
amoeba implements Nelder and Mead's downhill simplex
minimization method in multi-dimensions (two dimensions in our case) [8], This method
is found to be faster than successive one-dimensional minimizations in normal directions.
The function takes as its arguments the three complex gamma-plane points guessed, as
described above, and their respective determinants (pre-calculated in p), and iterates until
it locates a minimum to within a given fractional tolerance o f the determinant value. It is
modified so that it also stops iterating when the minimum has been located to within a
fractional tolerance o f the determinant's arguments (i.e., gamma). Hence the calculation
o f gamm a does not go beyond the specified number o f significant figures and CPU time is
not wasted. The magnitude o f the characteristic polynomial equals zero only at the zeros
o f the polynomial. Hence, amoeba searches and locates the closest to the initial guess
zero o f the characteristic polynomial. As amoeba iterates searching for the minimum, the
values o f gamma it guesses are printed to the standard output for traceability purposes.
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253
When amoeba exits, this zero is stored in the first row o f a two-dimensional real array
named p. Hence,/?[1][1] is the attenuation and p[ 1][2] is the propagation constant.
After these variables are printed to the standard output for the information of the user, the
solution vector X 0 is calculated, via the SVD process described in section 8.4.2. This is
done by the next function called by main, called svd. The function svd takes the array p
(which now contains the value o f y that makes matrix singular), matrix and params as its
arguments. First it calls matr to re-initialize matrix for the singular value o f gamma (i.e.,
matrix is now what has been called A () above).
Then it uses three Fortran routines
adopted from the Fortran mathematical routine package LAPACK [9]. The names o f the
three routines are CGEBRD, CUNGBIl, and CBDSOR.
CGEBRD reduces a general
complex m-by-n matrix A to upper or lower bi-diagonal form B by a unitary
transformation Q /'A P , = B . CLINGBR generates one o f the unitary matrices Q, or P "
determined by CGEBRD when reducing a complex matrix to bidiagonal form. CBDSOR
computes the SVD o f a real n-by-n bidiagonal matrix B: B = Q 2SP ,r , where S is a
diagonal matrix with non-negative diagonal elements (the singular values o f B) and Q 2 and
P, are orthogonal matrices. Combining CGEBRD and CBDSOR we obtain
A = Q ,Q 2SP 27P," = (Q ,Q 2)S(P,P2)// ,
' ~ u ~ y ‘T
(60)
’"
which is equivalent to equation (59). The diagonal elements of S, the singular values, are
returned to main in a variable called diag. The last 10 are printed to the standard output
to assure the user o f a converged solution (see last paragraph of section 8.4.2). V (the last
column of which is X0, the solution) is calculated by feeding V ” , calculated by CUNGBR,
into CGEBRD. The resulting matrix V is stored in the same memory location as matrix
(which has by now been overwritten as it is no longer needed) to conserve memory, main
then stores X(), the last column o f V, into a new complex vector called res.
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254
The function main then calls fre e v e c to r , another function from Numerical Recipes in C
which reverses the function performed in the beginning by vector and frees the memory
locations reserved for storing matrix.
Next, E H Power calc is called by main. This function calculates all the complex phasor
electric and magnetic field components
(E x( x ,y ) ,E y( x ,y ) ,E .{ x ,y ) ,H x{ x ,y ) ,H v{ x ,y ) ,H .{ x ,y ) ) and all the real electric and
magnetic field components
(ex{ x ,y ,z ,t ) ,e y { x ,y ,z ,t ) ,e !{ x ,y ,z ,t) ,h x{ x ,y ,z ,t) ,h y { x ,y ,z ,t) ,h ;{ x ,y ,z ,t) ) , given res, z,
and /. It also calculates the power flowing down the waveguide at the z=0 plane and the
maximum tangential field at the bottom and left walls. The latter is important for HTS
waveguides, because if the tangential fields and surface currents in the walls exceed their
respective critical values, the HTS ceases to be superconducting and becomes an insulator
and power propagation becomes impossible. EH_Power_caIc calculates the transverse
field phasor quantities from res (i.e., the longitudinal field components) using
appropriately discretized versions o f equations (10)-(13). Then it uses
(61)
to calculate the real field quantities in the lower left quadrant o f the waveguide crosssection ((/',/') = { o , l , . . . , A z / } x { o , l , . . . , A f } , see figure 9). Then, depending on the types o f
symmetry used in calculating the original matrix, the following equations are used to
calculate the fields in the whole waveguide cross-section.
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255
i. Top M agnetic Symmetry Plane. Right M agnetic Symmetry Plane
e M 2 N - j ] = ex[i][J]
eym N - j ] = - e y [i}[j]
ex[ 2 M - i J 2 N - j ] = - e x\ i l j ]
ey [ 2 M - i ] [ 2 N - j ] = - e y[i][j]
e:m N - j ] = e:[ i i n
ez[ 2 M - i ] [ 2 N - j ] = eXi}{j]
K m N - j ] = - h x[i}[j ]
K m N - j } = hy[i}[j]
hx[ 2 M - i ] [ 2 N - j ] = - h x[i][j]
hy [ 2 M - i l 2 N - j ] = - hy[ i l J]
m [ 2 N - j ] = - h :[ a j ]
h: [ 2 M - i ] [ 2 N - j ] = h:[i][j]
FIELDS
KNOWN
e ,[2 M - i ] [ j } = - e x[i][j}
ey [ 2 M - i ] [ j ] = ey [i][j]
e: [ 2 M - i ] [ j ] = ez[i][j]
hx[ 2 M - i ] [ j ] = hx[i)[j)
hy [2 M - i ] [ j ] = - hy[i][j]
k [ 2 M - i ] [ j ] = - h:[i][J]
ii. Ton M agnetic Symmetry Plane. Right Electric Symmetry Plane
e M M - j ] = ex[ i l j ]
ey [ i I 2 N - j ] = - e y{i}[j]
ex[ 2 M - i][2N - j ] - <?v[/'][ /]
e M 2 N - j } = e: [i][.n
e: [ 2 M - i ] [ 2 N - j ] = - e :[i][j]
hxm * t - j ] = - h M n
hym N - j ] = hy [ i l j )
hx[ 2 M - i ] [ 2 N - j ] = hx[iJj ]
hy [ 2 M - i ] [ 2 N - j ] = hy[i][j]
h .[ i\2 N - j ] = -/?_-[/][/]
h: [ 2 M - i ] [ 2 N - j ] = - h :[i)[j]
FIELDS
KNOWN
ex[2 M - i l j ] = ex[ i l j ]
ey [ 2 M - i I j ] = - e yV I j ]
a . [ 2 M - i ] [ j ] = - e . [/][./]
K V - M -/][./] = - h x[ i i j ]
M 2 M - i l j ] = hy [iJj ]
k [ 2 M - i ] [ j ] = h:[i][J}
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256
iii. Top Electric Symmetry Plane, Right M agnetic Symmetry Plane
exm N - n = - e x [ i i n
ey m N - j } = ey [i}[j]
ex[ 2 M - i l 2 N - j ] =e M n
ey[ 2 M - i ] [ 2 N - j ] = ey[i][j]
e M 2 N -j] =-e M j]
K m N - j ] = h x[ i i j }
e:[ 2 M - i ] [ 2 N - j ] = - e z[i][j]
hx[ 2 M - i ] [ 2 N - j ] = hx[ i \ j ]
K m N - n = - h y[iij]
hy[ 2 M - i ] [ 2 N - j ) = hym ]
K m N -j)= k[i][ji
K [ 2 M - i ] [ 2 N - j ] = - h :[ i l j ]
FIELDS
KNOWN
ex[ 2 M - i l j } = - e x[i][j}
ey [2 M - i ] [ j ] = ey [i][j]
m
M - i ] [ j ] = e:[i][J]
K [ 2 M - i ] [ j ) = hx[ i I j 1
K [ 2 M - i ] [ j ] = - h y[i][j]
k [ 2 M -/][./] = - h :[i][j)
iv. Ton Electric Symmetry Plane. Right Electric Symmetry Plane
e * m N - j ] = - e x[ i I j ]
ey [ i ] [ 2 N - j ] = ey [i][j]
ex[ 2 M - i ] [ 2 N - j ] = - e x[i][j]
ey [ 2 M - i ] [ 2 N - j ] = - e y[i][j]
e:[ 2 M - i ] [ 2 N - j ] = e:[i][j]
K m N - j ] =h M j ]
K m N - m - h M n
hx[ 2 M - i ] [ 2 N - j ] = - h x[ i l f \
hy[2 M - /][2 N - j ] = -/?,.[/][./]
K m N - j ] = h :[ i i j ]
h _ [ 2 M - i ] [ 2 N - j ] = h:[i][j]
FIELDS
KNOWN
ex[ 2 M - / ] [ . / > <?,[/][./]
e v[ 2 M - i l j ] = - e y [ i l j ]
e:[ 2 M - i I j ] = - e ;[ i l j ]
K V M - i ] [ j ] = - h x[ ilJ ]
W 2 M - i I j ] = hy [i][j]
h. [ 2 M - i ][_/'] = h. [/' ][ /]
The total power at z=0 is calculated using [1]
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257
p 4 f j > ( * xH- ) ' ^
■
(«2)
where h is a unit vector normal to the surface o f integration.
In our case n = z, and
therefore (62) becomes
■
(63)
When discretized equation (63) becomes
( i=0 J=0
+ 2 2 [ je jc (/• , N ) H ; (i, N ) - E y ( i , N ) H ; (/, a o ] +
;=0
+ 2 Z [ E x ( M J ) H ; { M J ) - E v( M J ) H ; ( M J ) ] +
j =0
(64)
+ E x ( M , N ) H ; { M , N ) - E y ( M , N ) H X* ( M , N )}A x A y
which is used for the power calculation.
The maximum tangential magnetic fields at the bottom and left walls (and therefore, by
symmetry, also to the top and right wall) are calculated next.
It can be shown, by
differentiating and setting the derivative equal to zero, that the maximum real field value,
with respect to phase, for two vectorially added phasors normal to each other is (using H x
and H z at the bottom wall as an example)
hmax =
-4 -9
J{H ;r - H l + H ; r - H : X + 4{HxrHxi + H :rH :i)2
(65)
where H xr = R e(H v), l i xi = Iin (llv), H.r = Re(H ; ) and I~l:j = Im (ll; ).
The function
EH_Power_ca/c returns the calculated power and maximum tangential magnetic fields to
wain, which prints these values to the standard output.
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258
Next main calls openplot.
This function calls four Fortran library routines (OPNGKS,
GOCNTN, G SELNT and WTSTR) that initialize the NCAR Graphics software package on
the CRAY [10], This Graphics package uses a number o f Fortran routines that create
graphs and write the output in Computer Graphics Metafiles (CGM) which can then be
viewed on an X I 1 host via the Internet, or plotted in Hewlett-Packard Graphics Language
(HPGL) file format.
The function main then calls plotm air which takes and plots a vector velocity field, given
two matrices o f vector components.
Th function ploim atr uses the NCAR Graphics
routine EZVEC. plotm air is called twice, to plot the cross-sectional view of the electric
and magnetic fields in the z= 0 plane, respectively. The plot goes to an output CGM called
gmela, by default.
Then calc J o n g is called.
The function calc J o n g calls E H J 3ower_calc NOFDZ times
(NOFDZ is defined before main and presently equals 65) and calculates all the real field
quantities on NOOFDZ uniformly spaced cross-sectional planes. The spacing between the
planes is necessarily equal to A r, because the plotting routine EZVEC assumes the plotted
grid points to be equi-spaced in both directions.
The field components along three
different planes are stored in new variables for plotting purposes. The planes chosen are
the j - N (x = ~ ) plane for both e and h, the / = M (x - ^-) plane for e and the / = 2 M
(x = a ) plane for h.
Next plotm atr is called four times successively to plot the four field snapshots described
above in gmela.
Finally, closeplot, whose purpose is to close gm ela, is called.
The
function closeplot simply calls the NCAR Graphics Fortran library routine CLSGKS and
returns. Then the program returns and execution terminates. On exit, three views o f the
electric and magnetic fields are saved in gmeta\ a cross-sectional view, a surface view and
a longitudinal view.
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259
8.4.4 Running the Program
The program is compiled on the JPL CRAY Y-MP2E computer voyager and on the
Goddard Space Flight Center CRAY C98 computer charney.
On August 30, 1992
charney was upgraded from an 8 processor CRAY Y-MP2E to a 6 processor CRAY C98.
The difference in CPU time from voyager is dramatic.
The program runs 4 to 5 times
faster on voyager than on charney. Also the maximum memory available in the queues of
voyager is 16 MW (MegaWords) whereas on charney it is 60 MW.
Because o f these
advantages the program is mainly run on charney and these statistics will be reported here.
The SVD and plotting parts o f the program are only required for plotting the fields out
and are CPU intensive, time-costly and memory-hungry.
If one only wants to know
accurate propagation characteristics o f an HTS waveguide for a range o f frequencies, it is
better to implement another program without these features (i.e., with the code
terminating right after amoeba and iterating for a range o f frequencies). This strippeddown version of the program is called wg_sweep.c.
The purpose o f wg sweep is to
accurately calculate Y for a range o f frequencies in the minimum possible time, whereas
that o f w g j)lo l is to calculate and plot the fields for a single frequency and mode.
In
w g sw eep a grid as dense as possible is required (i.e., maximum M and N given memory
and time constraints). In wg plot, too dense a grid would not only make execution very
slow, but also would make the final plots o f the fields crowded and almost unreadable.
Following are some o f the statistics o f the program.
An example o f the output of
wg sweep is included as appendix E o f this chapter.
8.4.4.1 CPU Tim e and M emory Usage
The main memory storage requirement is due to the need of storing matrix.
Its size is
2 ■ROWIJCNCjTH " Words, or 8( M + 1): (Y + 1)2 Words. It has been empirically found
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260
th a t th e p ro g ram is m o st accu rate w hen Ax = Ay.
H ence, fo r th e usual w av eguides
( ^ = 0 .5 ), M = 2 N is a g o o d rule o f thum b to observe. G iven an u p p er m em ory limit o f
60 M W (o r 6 2 9 1 4 5 6 0 W o rd s) th e m axim um M usab le is 72, noting th a t som e ex tra
m em ory has to b e reserv ed fo r th e rest o f th e variables.
A s only one q u a rte r o f th e
w aveg u id e is su b -sectio n ed , th e effective grid size, in th is case, is 144-by-72. W ith this
grid size ch arney can analyze o n e frequency in a b o u t 3.5 C P U hours. T he m axim um C P U
tim e lim it on any jo b is 4 hours.
well m atched.
T h erefo re th e m em ory ceiling and th e tim e ceiling are
H o w e v e r th e 144-by-72 grid is fo u n d to b e unnecessarily fine and th e
values fo r M and N u sed in wg_sweep are A /= 58, N = 29 , w hich are equivalent to a 1 16-by58 effective grid size. W ith this grid size \vg_sweep needs 25 M W o f m em ory and charney
can analyze ab o u t 9 freq u en cies in 4 C P U h o urs.
It has been found th at th e C P U tim e
required to ru n a g iv en jo b roughly obeys th e law CPU Time oc ROWLENGTH 22.
T he standard M and N values used in w g jjlo t are M= 30, A M 5. T hese have been fo u n d to
provide fairly a c c u ra te results w ith g o o d arro w density in th e o u tp u t field plots. W ith this
grid size \vg p lo t n e e d s 3 M W o f m em ory and charney ex ecu tes it in 5 C P U m inutes.
8.4.5 The Results
F igure 10 defines th e cu t-planes,
across
the
bo d y
of
Figs. 13, 18,2(5
the
w aveguide, on w h ich th e fields
are
p lo tted
in
th e
follow ing
figures.
Fics. U. U», 27
Figures 11-15 sh o w exam ples o f
the o u tp u t o f w g jjlo t fo r th e
^
Fit-s. 11, 12, 16. 1 7 . 2 1 , 2 5
Fi a s . 15. 2 0 . 2K '
T E 1 0 m ode o f a W R 9 0 H T S
F ig u r e 10 D e f in itio n s o f th e c u l-p la n c s o f the field p lots.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
261
w av eg u id e, o f a v e ra g e H T S param eters, a t 12 G H z.
In this lo w e st frequency exam ple,
th ere is n o visible differen ce b etw een th e calcu lated fields and th e ideal fields o f a perfectc o n d u c to r W R 9 0 w av eg u de.
TT
IT
Q . ‘1 B 7 E « o t
MAXIMUM
VECTOR
F ig u r e 11 C r o s s-sc c tio n a l v ie w o f th e electric fie ld o f th e T E 10 m o d e in a W R 9 0 H T S w a v e g u id e o f
a v era g e H T S p a ra m ete rs at 12 G H z.
------------------------------------------------- ------------------............ ......................
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1 0 3 E -01
MAXIMUM
VECTOR
Fi gurc 12 C r o s s-sc c tio n a l v ie w o f th e m ig n e lie fiel d 0! ’ the T E K ) m od e in a W R 9 0 H T S w a v e g u id e o f
av cra g c H T S p a ra m ete rs a t 12 G H z.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
262
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MAXIMUM VECTOR
F ig u r e 13 S u r fa c e v ie w o f th e m a g n etic fie ld o f th e T E 1 0 m o d e in a W R 90 H T S w a v eg u id e o f a v era g e
H T S p a ra m eters at 12 G H z._____________________________________________ __ _________________________________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
263
0.62^-01
MAXIMUM
VECTOR
F ig u r e 1 4 L o n g itu d in a l v ie w o f th e m a g n e tic fie ld o f th e T E 1 0 m o d e in a W R 9 0 H T S w a v e g u id e o f
a v era g e H T S p aram eters at 12 G H z.
0..1£7E^02
MAXIMUM
VECTOR
F ig u r e 15 L o n g itu d in a l v ie w o f the electric fie ld o f th e T E 1 0 m od e in a W R 9 0 H T S w a v e g u id e o f
a v era g e H T S p aram eters at 12 G H z.
F igures 16-20 sh o w exam ples o f the o u tp u t o f wg pJol fo r the T M 3 2 m ode o f a W R90
H T S w aveguide w ith th e av erag e set o f p aram eters at 12 G H z. T h e T M 32 m ode is m ore
"interesting" and has m o re salient featu res th an the T E 10 m ore, and is, therefore, a b etter
exam ple to help illu strate th e validity o f the results o f w g j)lo t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
264
0.1-13E-O1
MAXIMUM*VECTOR
F ig u r e 1 6 C ro ss-sec tio n a l v ie w o f th e e le c tr ic fie ld o f th e T M 3 2 m o d e in a W R 90 H T S w a v e g u id e o f
a v e r a g e H T S p aram eters at 4 0 G H z.
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MAXIMUM
VECTOR
F ig u r e 17 C ro ss-scc tio n a l v ie w o f the m a g n e tic fie ld o f th e T M 3 2 m o d e in a W R 90 H T S w a v e g u id e o f
a v era g e H T S p aram eters at 4 0 G H z.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
265
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VECTOR
F ig u r e 1 8 S u r fa c e v ie w o f th e m a g n e tic field o f th e T M 3 2 m o d e in a WR9Q H T S w a v e g u id e o f av era g e
H T S p a ra m eters at 4 0 G H z.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
266
0 . 1 53E-03
MAXIMUM
VECTOR
F ig u r e 1 9 L o n g itu d in a l v ie w o f the m a g n e tic fie ld o f th e T M 3 2 m o d e in a W R 9 0 H T S w a v eg u id e o f
a v e r a g e H T S p a ra m eters at 4 0 G H z.
. 'V U ? C - 0 :
MAXIMUM
F ig u r e 2 0
VEClM
L o n g itu d in a l v ie w o f the e lectric fie ld o f th e T M 3 2 m o d e in a VVR90 H T S w a v e g u id e o f
a v e r a g e H T S p a ra m eters at 4 0 G H z.
In b o th th e ex am p les above th e average H T S p a ram eters are u sed and therefore th e
surface resistan ce o f th e walls is small (less than 0.2, see figure 8).
H ence, as expected,
th e fields in fig u res 11-20 look like the fields in a w av eg u id e w ith perfectly conducting
w alls. T o c o n sid er an exam ple w here th ere is a visible difference betw een the fields o f a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
267
w aveguide w ith p erfectly cond u cting w alls and a w aveguide w ith H T S w alls, the H T S
param eters m u st b e cho sen so th at th e m agnitude o f th e su rface im pedance is large.
H en ce th e b est exam ple is at a high frequency and w ith the w o rst case H T S param eters
(see table 1). F ig u re 21 is an arro w plot o f the cross-sectional view o f the electric field o f
th e T E 10 m o d e in a W R 3 H T S w aveguide o f th e w o rst case H T S p aram eters at 380 G H z.
OR
F ig u r e 21 C r o ss-sc c tio n a l v ie w o f the electric fie ld o f the T E 1 0 m o d e in a W R 3 H T S w a v e g u id e o f w orstc a s e H TS p aram eters at 3 8 0 G H z._____________________________________________________________________________
F igure 22 is a b lo w -u p o f th e region o f figure 21 below and to the right o f th e cen ter o f
th e to p wall, as sh o w n in figure 21. F igures 23 and 24 are plots o f th e angle o f deviation
o f th e electric field v e c to rs from their (ideal) vertical orientation, along th e vertical line
and horizontal lines sh o w n in figure 21, respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
268
A
A
A
A
A
A
A
A
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a
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1
F ig u r e 2 2 B lo w -u p o f th e r e g io n o f fig u r e 19 b e lo w and to the righ t o f th e m id d le o f the top w a ll. T h e
v ertica l lin e in th e m id d le is d raw n for cye-referen ce.
C learly, th e ele c tric field is n o t purely y-directed as in th e case o f a w av eg u id e w ith
perfectly c o n d u c tin g w alls.
H ere, th e electric field visibly bow s in aw ay from the
p erp en d icu lar b is e c to r o f th e to p and b o tto m walls. T h e x -com ponent o f th e electric field
is required to p ro d u c e th e surface cu rren t that su p p o rts th e necessary m agnetic field at the
wall.
F ig u re 25 sh o w s a cross-sectional view o f the m agnetic field fo r th e sam e
w aveguide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N u m b e r o f P o in t
F ig u r e 23 D e v ia tio n a n g le o f th e e le c tr ic field v ecto rs fro m the v e r tic a l, a lo n g a lin e p a ra llel to th e y a x i s
50
40
30
x = a /2
x=0
20
10
*
0 H-l-H-H-t-l-l-I^HHH-md-i-4-ii
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-10
-20
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-30
x=a
-40
-50
N u m b e r o f P o in t
F ig u r e 24 D e v ia tio n a n g le o f th e e le c tr ic field v ecto rs from the v e r tic a l, a lo n g a lin e p a ra llel to the x -a x is.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
270
0 .2 H IE -0 2
MAXIMUM
VECTOR
F ig u r e 2 5 C r o s s-se c tio n a l v ie w o f th e m a g n e tic fie ld o f th e T E 1 0 m o d e in a W R 3 H T S w a v e g u id e o f
w o r s t-c a s e H T S p a ra m eters a t 3 8 0 G H z.
A lth o u g h th e re are slight differences b etw e en figures 11 and 22, th e re is no noticeable
system atic d ifference as in th e case o f figures 10 and 20.
T h e m agnetic field is not
affected as stro n g ly by th e ex isten ce o f th e high surface im pedance.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
271
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MAXIMUM
VECTOR
F ig u r e 26 S u rfa c e v ie w o f the m a g n e tic fie ld o f the T E 1 0 m o d e in a W R 3 H T S w a v e g u id e o f w o rsl-ca se
H T S p a ram eters at 3 8 0 G H z.
F ig u re 26 and figure 13 are very simitar, and th e re a re no salient differences betw een them .
T his confirm s th at th e m agnetic field solution is n ot very affected by th e high surface
im pedance o f th is exam ple. F igure 27 show s th e longitudinal view o f the m agnetic field,
w hich su p p o rts th e sam e conclusion (com pare w ith figure 14).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
272
0 .3 J 9 E - 0 2
MAXIMUM
VECTOR
F ig u r e 2 7 L o n g itu d in a l v ie w o f th e m a g n e tic fie ld o f th e T E 1 0 m o d e in a W R 3 H T S w a v e g u id e o f w orstc a s e H T S p aram eters at 3 8 0 G H z._______________________
F igure 28 show s a longitudinal view o f th e electric field in th e w aveguide.
TTTi
II''
0.283C
MAXIMUM
01
CTOR
F ig u r e 2 8 L o n g itu d in a l v ie w o f th e e le c tr ic fie ld o f th e T E 1 0 m o d e in a W R3 H T S w a v e g u id e o f w o rstc a s c H TS p aram eters at 3 8 0 G H z.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
273
H ere again w e see th e electric field solution d e p art from th e p u re T E ch aracter,
longitudinal c o m p o n e n t appears.
and a
F igure 29 is a b low -up o f a reg io n o f figure 28, right
u n d er th e to p w all, sh o w in g this effect, as sh o w n in figure 28.
Y
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F ig u r e 2 9 B lo w -u p o f th e r e g io n o f fig u r e 25 b e lo w a n d to th e right o f th e m id d le o f th e top w a ll. T h e
v ertica l lin e in th e m id d le is d raw n for ev e-rcferen ce.
T h e fo llo w in g 10 fig u res are p lo ts o f a tten u a tio n and p ro p a g atio n co n sta n t v ersu s
freq u en cy co m p arin g th e th eoretical values (equations (5 ) and (52 )) to the o u tp u t o f
\vg_sweep.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
274
4 .0 0 E - 0 3
3 .5 0 E - 0 3
3 .0 0 E -0 3
2 .5 0 E - 0 3
2 .0 0 E -0 3
1 .5 0 E -0 3
\v g _ sw e e p
1 .0 0 E -0 3
T heory
5 .0 0 E - 0 4
0 .0 0 E + 0 0
6 .5 6
1 3 .1 2
2 6 .2 4
19.68
f (G H z )
F ig u r e 3 0 A tten u a tio n v ersu s freq u en cy in a W R 9 0 H T S w a v eg u id e.
500
450
400
350
300
o
250
200
w g sw eep
150
X
C l.
T heory
100
6 .5 6
1 3 .1 2
19.68
2 6 .2 4
f (G H z )
F ig u r e 31 P rop agation c o n sta n t versu s freq u en cy in a W R 9 0 H T S w a v eg u id e.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
275
0.12
0.10
0 .0 8
0 .0 6
w g _ sw e e p
0 .0 4
T heory
0.02
0.00
21.1
4 2 .2
6 3 .3
8 4 .4
f (G H z )
F ig u r e 3 2 A tten u a tio n v e r su s freq u en cy in a W R 2 8 H T S w a v eg u id e.
1600
1400
1200
•S 1000
800
o
600
w g_sw cep
400
T heory
200
21.1
4 2 .2
8 4 .4
F ig u r e 3 3 P rop agation c o n sta n t v ersu s freq u en cy in a W R 28 H T S w a v eg u id e.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
276
3 .0 0
2 .5 0
2.00
1 .5 0
w g_sw cep
1.00
T h eory
0 .5 0
0.00
5 9.1
1 1 8 .2
2 3 6 .4
1 7 7 .3
f (G H z )
F ig u r e 3 4 A tte n u a tio n v e r su s freq u en cy in a W R 1 0 H T S w a v e g u id e .
5000
4500
E
4000
•-§ 3 5 0 0
3000
°
2500
w g_sw eep
'•S 2000
c_
T h eory
15 0 0
1000
500
5 9 .1
118.2
1 7 7 .3
2 3 6 .4
f (G H z )
F ig u r e 3 5 P ro p a g a tio n c o n sta n t versu s freq u en cy in a W R 1 0 H T S w a v e g u id e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
277
1 8 .0 0
1 6 .0 0
1 4 .0 0
12.00
o
10.00
\v g _ s\v e e p
8.00
T heory
6.00
4 .0 0
116
232
34S
464
f (G H z )
F ig u r e 3 6 A tte n u a tio n v e r su s freq u en cy in a W R 5 H T S w a v e g u id e .
9000
8000
7000
£
6000
5000
4000
w g _ s \v c e p
3000
T h eory
2000
------
1000
116
232
348
464
I'(G H z)
F ig u r e 3 7 P ro p a g a tio n co n sta n t v ersu s freq u en cy in a W R 5 H T S w a v e g u id e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
278
£
25
\v g _ sw e e p
T h eory
174
34S
52 2
f (G H z)
F ig u r e 3 8 A tten u a tio n v e r s u s freq u en cy in a WR.3 H T S w a v eg u id e.
10000
9000
8000
OOH
7000
6000
5000
\v g _ s w c c p
4000
T heory
3000
2000
174
348
52 2
f (G H z)
F ig u r e 3 9 P rop agation c o n sta n t v ersu s freq u en cy in a W R3 H T S w a v e g u id e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
279
8.4.6 Conclusions
A p ro g ram h as been p resented that accurately p re d icts th e electric and m agnetic fields in
an H T S w a v eg u id e, as w ell as the atten u atio n and p ro p a g a tio n fo r arbitrary choice o f H T S
p aram eters and m ode.
Figures 30-39 sh o w a very g o o d ag reem en t betw een th e sim ple
p e rtu rb a tio n th e o ry based on ideal fields and the p rogram .
F igures 21 , 33, 35 , 37 and 39 show co m p ariso n s o f th e calculated and predicted
p ro p ag atio n co n stan ts.
T he agreem ent is to w ithin 0.1 % e rro r in the w o rst case,
co n sisten t w ith a sim ple perturbational refinem ent o f the pro p ag atio n constant, as
exp ressed by e q u a tio n (58). The p ro p agation co n sta n t pro v es virtually unaffected by the
increase o f th e su rface reactan ce o f the w alls w ith frequency. T he latte r is tru e b o th fo r a
single w a v e g u id e acro ss th e p lo tted frequency ra n g e and in com parison o f different
w aveguides. T h e co nclusion is, therefore, th at e q u atio n (52) is very accurate and may be
used w ith a w id e ran g e o f FITS param eters, resulting in surface im pedances o f m agnitude
as high a s 2 O h m s p e r square.
F igures 30, 32, 34, 3 6 and 38 show com parisons o f th e predicted exponential attenuation
constan ts.
T h e re is g o o d agreem ent betw een eq u atio n (5) and the predictions o f the
program . T h e typical e rro r betw een the tw o is 2.5 % , fo r th e 1 18-by-60 grid. T his error,
how ever, is d u e to th e discretization o f the equ atio n s (q u an tizatio n erro r) and clearly does
n o t rep resen t a "real" difference since increasing the g rid to its m axim um (from a m em ory
limit sta n d p o in t) size o f 132-by-72 reduces th e e rro r to 1.9 % . B y observing the tendency
o f the e rro r w ith resp ect to grid size, it becom es a plausible con jectu re that th e e rro r does
not in fact a sy m p to te to zero w ith respect to grid size.
Also, in th e case o f th e tw o highest frequency w av eg u id es considered (W R5 and W R 3)
the e rro r in creases from ab o u t 2.2 % at the frequency o f m inim um attenuation to 2.9 % at
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
280
th e highest freq u en cy co n sid ered (ab o u t th ree tim es th e c u to ff frequency), although in the
tw o low est frequency w a v eg u id es considered (W R 90 and W R 2 8 ) th e erro r is alm ost
co n sta n t w ith re sp e c t to freq u en cy (2 n d decim al p o in t v ariatio n s in percen tag e).
This
tren d is believed to b e a "true" e v en t and verifies th e c o n jec tu re p o sed in section 8.4.1.
T h at is, at th e high end o f th e frequency spectrum considered th e real loss d o es start to
increase a little faster th an th e eq u atio n (5) predicts. N ev erth eless, th e above results show
th at equations (5) and (5 2 ) a re "accurate enough" even fo r th e w o rst-c ase H T S
w aveguides and th eir u se is recom m ended.
A lso th ese re su lts v alid ate th e analysis
presented in section 8.3.
8.5 Power Handling Capability
H T S s have a m axim um cu rren t carrying capability, beyond w hich they turn into their
norm al crystalline and th e re fo re insulating state.
T his m axim um cu rren t density, J c, is
called th e critical cu rren t d ensity and is an intrinsic p ro p erty o f th e H T S . H ence, th ere is a
limit on th e input p o w e r th at any H T S w aveguide can carry.
T his section investigates
w hat this limit is for th e H T S w aveguides o f av erage H T S p aram ete rs used above as
exam ples.
W e start w ith eq u atio n s (3 .2 8 ) and (3.29).
T h e surface c u rren t p e r unit w idth flow ing
u n d er th e w alls is equal to th e m agnetic field tangential to th e walls. H ence, w e have
w h ere C, is defined in eq u atio n (3.25).
S ubstituting eq u atio n (3 .2 9 ) into (66) and
rem em bering th at P cc H 2, w e obtain
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A
A
cm'
nr
A typical v alu e o f th e critical c u rre n t density o f Y B C O is ([1 1 ], [12]) 107 — - = 10" — .
U sin g th is v alue fo r th e critical c u rren t density and eq u atio n (67), table 4 is obtained for
H T S w av eg u id es o f a v e ra g e H T S p ara m e ters at th eir resp ectiv e frequencies o f m inim um
atten u atio n .
Type of W aveguide
Maximum Input Power (dBm)
W R 90
2 1 .2
W R 28
1.56
W R 10
-1 6 .2
W R5
-2 7 .9
W R3
-3 5 .0
T a h le 4 M a x im u m P o w ers o f H T S W a v eg u id es.
8.6
References
[1]
S. R am o , J. R. W hinnery and T. V an D u zer, F ields a n d Waves in Com m unication
Electronics, W iley, N e w Y o rk , 1965.
[2]
D o u g las G. C o rr and J. B rian D avies, "C o m p u ter A nalysis o f th e F undam ental and
H ig h er O rd e r M o d es
in
Single
and
C o u p led
M icrostrip,"
IE E E
T ran sactio n s on
M icro w av e T h e o ry and T ech n iq u es, V ol. M T T -2 0 , N o. 10, pp. 6 69-678, O cto b er 1972.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
282
[3]
J. S. H o rn sb y and A. G opinath, "N um erical A nalysis o f a D ielectric-L oaded
W av eg u id e w ith a M icro strip L ine - Finite D ifference M eth o d s," IE E E T ransactions on
M ic ro w a v e T h e o ry a n d T echniques, Vol. M T T -1 7 , N o . 9, pp. 6 8 4 -6 9 0 , S eptem ber 1969.
[4]
W illiam H . P re ss et al., N um erical R ecipes in C, C am bridge U niversity Press,
C am bridge, 1988.
[5]
J. J. D o n g a ra , et al., U N P A C K U ser’s G uide , S o ciety fo r Industrial and A pplied
M ath em atics, Philadelphia, 1979.
[6] M A T H 7 7 R elease 4.0, JP L D o cum ent JP L -D -1 3 4 1 , R ev. C, M ay 1992.
[7] G en e H . G olub and C harles F. V an L oan, M atrix C om putations, T he Johns H opkins
U niversity P re ss, B altim ore, 1983.
[8] J. A. N e ld e r and R . M ead, C omputer Journal, V ol. 7, p. 308, 1965.
[9]
E. A n d e rso n et al., L A P A C K User's Guide, S o ciety fo r Industrial and A pplied
M ath em atics, P hiladelphia, 1992.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
283
[10]
F re d C la re and D av e K ennison, N C A R G raphics Guide to Utitlities, Version 3.0,
N atio n al C e n te r fo r A tm ospheric R esearch, Scientific C om puting D ivision, B o u ld er,
C o lo ra d o , 1989.
[11]
W . C h e w et al., "D esign and P e rfo rm an ce o f a H ig h -T c S u p erco n d u cto r C o p la n ar
W av eg u id e F ilter," IE E E T ransactions on M ic ro w av e T heory and T echniques, V ol. 39,
N o . 9, pp. 1 4 5 5 -1 4 6 1 , S ep tem b er 1991.
[12] D . E. O a te s, A. C. A nderson and P. M . M ankiew ich, "M easurem ent o f th e S urface
R esistan ce
o f Y B a 2C u 307 _x Thin
Film s
U sin g
Stripline
R esonators,"
Jo urnal
S u p erco n d u ctiv ity , V ol. 3, N o. 3, 1990.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of
284
A p p en d ix A
MathCAD File Used to Calculate and Plot the Cross-Over Frequency
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
285
Cross-over Frequency
jx0 := 4 *tu"10 7
4-o-av2 •
f x [A .,<y,ov . T , T
T
W orst Case:
fj
800-10 9 , 2 - 1 0 8 , 1 0- 10 6 , 7 7, 85
= 11.956109
Best C ase:
f x L 140-10 " , 10' , 1.1 -10 , 7 7 , 9 0 J = 5375.768 -109
Average Case:
f x [ 43 0* 1 0 9 , 0. 4 4 * 1 0 8 , 3 ' 1 0 6 , 7 7 , 8 7 ] = 162.753 -109
1 := 140-10 9 , 150-10 9 .. 700-10 9
500
500
400
300
~9
.77.87
200
100
0
200
100
400
600
A*10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
700
286
T c := 82,82.5.. 92
200
180
4 3 0 * 1 0
9,0 .4 4 -1 0 8
160
, 3 *106 . 77. T ,
140
120
80
85
90
T ,
Cv := 1 *106 , 1.1 *106 .. 6 -1 0 6
4 0 0
300
-9
f x
4 3 0 - 1 0
8
.0 .4 4 -1 0
.a.,, 7 7 . 87
10
-9
200
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
287
A ppendix B
MathCAD File Used to Calculate and Plot the Exponential Attenuation versus
Frequency
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Attenuation d u e conductor and HTS los ses
in c l o s e d r e c t a n g u l a r w a v e g u i d e s
8-6-93
CONSTANTS:
\i
■■=
4 -7C-10 7
0 := 0 . 4 4 * 1 0
X
:= 8 . 8 5 4 * 1 0 ~ 12
e
8
o v := 3 * 1 0 6
1
c .=
:= 4 3 0 * 1 0
T := 77
H*e]
T c := 87
[ 2 * a * 2 . 5 4 * 10
2 . 5 4 * 1 0 2 *al 2 +[ 2 . 5 4 • 10 2 *b 1 2
f c . T E l l ( a ’ b)
TE10:
R s c ( 0 : =
R sc[x * fc (a)]
alpha
(a , b , x ) :=
L
-
2
1-
R s c [ f c (a)]L
L
v
(z- b ) .
a
2
X
1
X
HT S
R ss ( f ) := 2
T
■li 2 - n 2 - f 2
-X3 - o v
-
R ss x *fc (a)
alpha
(a , b , x ) :=
i 4 2- l b ) .
a
T
2’
X
R sc f c (a)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
289
x := 1 . 0 0 1 . 1 . 0 1 . . 4
alpha c [0.1 ,0.05 ,x]
alphas [0.051 ,0.0255 ,x]4
alpha s [ o . 1,0.05.x]
alpha s [0.28,0.14,x]
alpha s [0.9.0.4,x]
0
1
1.5
2
2.5
3
3.5
x
—
’ '
—
Gold WG
WR5
W R 10
WR28
WR90
T,
rr
,2
It'll -A,
4 - a v 2 -Oj1
f'x
= 1.406
f c (0.051)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
290
Appendix C
Mathematica Results on Characteristic Equation of 32-by-32 Lossless AMatrix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
291
Xn[ 5 ] : =
!cc e h 3 3 _ s t d o u t . c
I n [5]:= In [l]
Out [5] = ( { 0 ,
2
u y , 0,
0,
0,
0,
2
0, ux
,
>
(0, 0,
0,
>
2
ux , 0 ,
>
{0, 0 , 0 ,
0, 0 ,
0,
0,
2
uy , 0 ,
0,
0,
0, u y
2
0, uy , 0,
0,
2
0, u y ,
g
2
- 2 uy
0, 0, 0,
0,
0, 0, 0,
0,
0, 0,
0,
2
g
2
2 ux
-
0,
0, 0,
2
0, 0, 0, 0, 0, u x , 0 , g
,
0, 0,
0,
0,
0, 0,
2
uy,
2
>
2
- 2 ux
2
•r e p s mu w ,
0,
2
0, 0, 0),
0 , 0,
2
2
- 2 u y + e p s mu w , 0,
0,
0,
2
- 2 ux
0,
0),
2
!
:
- 2 uy
+ e p s mu w ,
2
0, ux
(0, 0,
,
2
0, 0 , 0 , 0, ux , 0,
2
>
2
g
2
>
>
2
0, ux , 0,
0,
0,
2
- 2
,
0, 0, 0,
0, 0 , 0 ,
2
ux
0, uy
- 2 uy
0,
2
+ e p s mu w
0,
0, 0, 0,
0,
0, 0,
0),
2
0, ux , 0,
2
, 0, ux ,
;>
0, 0 ,
0, 0,
, 0, 0 , 0 , 0 ,
0 , uy
2
>
0, 0,
0,
>
2
ux , 0 ,
>
0, 0,
0, 0,
0),
2
g
{0,
0, 0,
- 2
2
uy +
2
- 2 ux
0,
0, 0,
0,
0, 0, 0, u y ,
2
eps mu w , 0,
0,
2
ux , 0 , 0,
0,
0,
0,
0,
2
0,
C I, 0,
uy , 0,
2
0, 0 ) ,
2
g - 2
>
0,
>
0),
>
g
>
{0, 0,
>
g
>
(0, 0,
>
0, 0,
>
0,
>
0,
{0,
0,
2
ux - 2
0, 0, 0,
2
uy + e p s
0,
0, 0,
2
muw,
0,
2
0, 0, u y
2
0, u x ,
,
0, 0,
0,
0,
0,
2
>
0, 0,
0, 0, 0,
- 2 uy
2
+ e p s mu w
2
, 0, ux ,
0, 0,
0, 0, 0, 0, 0,
2
- 2
ux
0,
0,
2
>
0, 0, 0, 0, 0,
2
2
- 2
ux
1,
0,
0,
0, 0,
0,
0,
I
- 2 uy
2
2
+ e p s mu w
0, 0,
0, 0, 0, 0, 0,
0,
0, 0,
0),
0,
0, 0,
0,
uy
0, 0,
uy ,
0, 0,
- (g ux)
{-------------- , 0, 0,
2
0, 0,
0,
2
, 0,
0, 0,
0, 0,
0,
0,
2
0, uy ,
0, 0,
ux , 0,
0, 0 , 0 ) ,
2
2
0, uy , 0,
2
, 0, ux ,
ux ,
2
0,
2
{0,
0,
0,
0,
0, 0,
ux , 0,
2
0, 0,
0 , 0,
0, 0,
0, 0,
0, uy
0,
0, 0 ),
,
0, 0, 0, 0,
0,
- I mu u y w,
g ux
------ ,
2
0,
0,
0,
0,
mu uy w, 0, 0, 0,
0, 0,
0, 0, 0,
0, 0, 0,
0,
0,
0,
0,
0,
0,
0,
0),
{0,
0, 0,
0, 0, 0,
0, 0, 0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
I mu uy w,
0,
0, 0, 0, 1, 0,
0, 0,
- ( g ux)
{0, 0,--- --------------,
2
0,
0, 0, 0, 0, 0 ) ,
g ux
0, - I mu uy w , --------- , 0,
2
0, 0,
0, 0,
0,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
292
>
0 , 0, 0,
0, 0,
0 , 0, 0 ,
0, 0, 0,
0, 0, 0,
0, 0 , 0 ) ,
>
{ 0 , 0, 0,
0 , 0,
0 , 0, 0 ,
0, 0, 0,
0, 0, 0 ,
0, 0 , 0, 0,
0, 0, 0,
0,
0,
0,
>
0 , 0, 1, 0 , 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
>
0 , 0, 0,
0 , - I mu u y w,
- (g ux)
0,--------------- ,
0, 0, 0,
g ux
0 , 0, I mu u y w,
, 0,
2
>
0,
o,
0),
{0,
0,
0,
0, 0 ,
0, 0,
0,
1,
0,
0, 0 } ,
{0,
0,
0, 0 ,
0, 0,
0,
0,
0,
0, 0,
0,
0,
0,
0,
0,
0,
- (g ux)
------------ ,
0,
0,
I mu uy w,
- I mu u y w,
0,
0,
0,
0,
0,
0,
0,
0,
0,
2
0,
0,
0,
0,
0,
0,
0,
0,
0,
{0 , 0, 0,
0, 0, 0, 0, 0,
0,
0,
0,
0,
0,
0,
0,
0,
g ux
-------- , 0 ) ,
2
>
0,
2
1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
- (g uy)
>
0 , 0,
0,
0,
0,
0,
0,
0),
(
,
0, 0,
0,
0,
0, 0,
0,
0,
I mu ux w,
2
>
0 , - I mu
g uy
ux w, 0, 0 , 0 , 0 , -------- , 0, 0 ,
0,
0, 0, 0,
0, 0 ,
0,
0,
0,
0,
0,
2
>
0 , 0 ), {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 1, 0, 0, 0, 0,
>
0, 0, 0,
0, 0, 0, 0, 0,
0, 0,
>
{0, 0, 0,
0, 0, 0, 0, 0 ,
- (g uy)
---------, 0, 0,
0 ),
0,
0, 0, 0,
0,
0,
0, 0 ,
X mu ux w,
0,
2
>
- I mu ux w,
0,
0,
0,
0,
g uy
------- , 0,
0,
0,
0,
0),
2
>
{0, 0, 0,
>
0, 0, 0,
0, 0, 0, 0, 0,
0, 0,
0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0 ), {0,0, 0, 0, 0, 0 ,
0, 0, 0, 0, 0,
- < g uy)
, 0, 0, 0, 0, 0, 0,
2
I mu ux w,
0,
- I mu ux w,
0, 0,
0,
0,
g uy
0 , ------- ,
0,
0, 0,
0,
0, 0,
0, 0,
2
0,
0 ),
{0,
0,
0, 0,
0,
0,
0, 0,
0,
0,
0,
0, 0,
0, 0 , 0,
0, 0,
0,
0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0),
- ( g uy)
{0,
0, 0,
0,
0,
0, 0,
0,
0,
0,
0,
0, 0,
0,
,
0,
0, 0,
0,
0,
0,
2
>
I mu ux w,
0,
- I mu ux w,
0,
0,
0,
0,
0,
0,
g uy
----------, 0 ) ,
2
>
( 1, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0, 0,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
293
0,
0,
0,
0,
0,
0,
O
o
o
o
>
0,
0)
o
o
o
o,
0,
1,
O
O
o
o
>
0,
0,
o
o
>
{ 0,
0,
0 ( o,
-
f {-(g
o
O
>
0,
0,
o
o
o
o
>
to,
0,
o
o
o
o
w
,
0,
0,
o
o
0,
o,
0,
0 , 0,
0,
0,
-
1,
0
0,
0,
0,
0,
0,
0,
o,
0,
0
0,
0,
0,
o,
0,
0,
0,
0,
0,
0,
0,
0,
o,
0,
0,
0,
0,
0,
o,
o,• 0,
o,
0,
0,
0
g ux,
0,
0,
0
0,
0,
o,
0
o,
- (g u x ) , i mu uy w,
o,
0,
0),
o,
0,
0,
o,
0),
o
o
o
0,
o
-I mu uy
0,
O
>
0,
0,
o
o
0,
0 , o,
0,
o
o
>
o
o
-I mu u y w, 0,
o
o
0,
o
o,
o,
(g ux) , o,
o
>
0,
o
0) / (0 ,
0,
o
0,
0,
o
0,
0
0,
o
0,
o
0,
o
0,
o
0,
o
1,
o
>
0 , o,
o
0,
o
0,
o
(0,
o
>
0,
0,
o
0,
o
o,
o
0,
O
0,
0,
o,
-I mu u y w, 0,
O
0,
0,
0),
(g u x ) , 0, g u x ,
0,
0,
o
0 , o,
0,
o
o
o
0,
u x ) , -I mu uy W, g u x ,
o
0)
o
o
I mu u y w,
>
O
I mu u y w,
0,
o
>
o
o
0,
o
o
>
0,
0,
g ux, I mu uy
0,
0,
0,
o,
0,
0,
w))
I n [ 6 ] : = I n [2]
O ut[6]=
(32,
32)
I n [ 7 ] := d =De t [ A]
Factor[d]
16
12
mu
Out [ 7 ] = g
12
>
58 g
>
193 g
>
36
>
124 g
>
1016 g
>
4 92 g
>
58 g
>
1662 g
>
1820 g
>
144 g
4
12
mu
8
8
12
12
12
6
12
-
10
12
mu
10
10
ux
uy
12
12
ux
8
12
14
ux uy
w
10
w
4 14
12
u x uy
w
8
14
12
10
6
8
ux
12
mu
10
uy
12
mu
6
16
ux uy
2
45 6g
12
muux
10
w
10
uy
12
mu
4
12
12
mu ux
8
+ 1016 g
12
+
+
12
w
+
12
w
+
12 12
w
14
uy
10
+
12
12
14
uy
6
ux
12
w
12 10
12
uy
w
ux
10
12
6
mu u x uy
- 2568 g
12
+
w
12
2
12
w -
8
12
uy w
+
4
uy
mu
104 4g
4
12 12
uy
w
- 504 g
12
12
ux
uy
12
12
muux
- 72 g
14
ux
10
12
12
mu
12
14
12 g
12
w -
8
12
12
u x uy
w
mu
12
mu
- 504 g
mu
10
6
- 132 g
12
u x uy
mu
6
16
8
uy
w
12 6
8 12
ux u y w
+
10
12
10
8
12
144 g mu
ux
uy w
+
8 12
uy w
mu
4
4
12
ux
4
12
g mu
ux
12
mu
8 12
uy w
ux
mu
8
8
12
14
uy w
- 12 g
mu
ux
12
w
14
12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
294
25 6 8 g
mu
2
12
muu x
10 0 8 g
6
104 4 g
2
1 00 8 g
w
12
uy
+ 272 8 g
14
w
12
12
muux
12
6
18
uy
ux
12
4
mu
20
uy
ux
14
13
12
4
ux u y
8 eps g
mu
348 e p s
10 13
g muux
8
uy
77 2 e p s
6
13
g mu
12
ux
8
10
6
4064 e p s g
13
mu
ux
13
mu
ux
2
13
984 e p s g
mu
10
14
w
- 84 e p s g
12
8
8
14
4
13
uy w
- 39 6 e p s g
mu
6
10
w
8
- 2520 eps g
10
14
4
w - 3132 ep s g
uy
12
14
4
ux
14 14
6
uy w
+ 4064 e p s g mu
8
12
8
4
14
14
uy w
13
13
mu
6
ux
4
ux
18 14
uy w
20
uy
8
ux
10 14
w +
uy
10
10 14
uy
w
+
14
w
6
ux
13
+
12 14
w +
uy
6
13
g mu
eps g
13
13
8
ux
4
ux
8
6
ux
16 14
w
uy
18
14
uy w
+
18
uy
20
uy
14
w
16
14
uy w
ux
13
- 72 e p s mu
14
4
ux
mu
4
13
g mu
- 456 e p s mu
14
14
uy w
13
10
mu ux u y
2
+3640
6
ux
2
14
w
13
13
10
12 14
mu ux u y
w
+
+ 5456 e ps g
13
10
16
14
mu ux
uy
w
- 396 e p s
g mu
10
14
uy w
+
14
13
14
12
14
w - 45 6 e p s mu
ux uy
w
13
6
16
14
mu
ux uy
w
2
4
ux
mu
- 7704 e p s g
13
12
14
14
mu ux
uy
w
+ 772 e p s
4
8 14
uy w
+
13
- 2520 eps g
12
14
uy w
12
13
8
mu u x
w
14
ux
16
uy
uy
uy
8 14
uy w
+
13
12
mu ux
14
10
14
13
ux
uy
w
- 72 e p s muux
4
14
w +
10
ux
mu
ux
4
g
2
12
13
g mu
14
uy
10
eps
8
uy
mu
13
mu
13
6
13
- 720 e p s g
2
7704 e p s
984 e p s
13
ux
mu
14
w
20 12
w
+
uy
ux
g mu
1008 e p s
6
13
mu
8
31 3 2 e p s g
18 12
uy w
+
6
g
72 0 e p s
1008 e p s
12
8
mu ux
ux
3640 e p s g
72 e p s g
12
4
18 12
mu ux
uy w
+
13
mu
348 e p s g
6648 e p s
6
2
12
72 g mu ux
2
13
16
8 14
72 e p s g mu
ux
uy
w
- 84
744 e p s g
12
8
16 12
mu ux
uy w
-
8
w
4
2
- 45 6 g
w
uy
12
4
16 12
mu u x
uy w
- 132 g
12
w
ux
8
+ 1820 g
10 16
uy
w
12
mu
+ 193 g
12
mu
4
uy
12
6 16
mu ux u y
w
4
492 g
36 g
ux
14
w
14
w +
+
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
295
2
>
28 e p s
>
870 e p s
>
11 5 8 e p s
>
36 e p s
12
g
2
8
g
2
14
2
>
6096 e ps
>
492 e p s
>
5040
>
770 4 e p s
>
144 0
>
77 0 4 e p s
>
1158
>
18 2 0 e p s
14
2
14
mu
2
14
2
2
4
2
ux
14
10
g mu
772 e p s
>
420 e p s
>
504 0 e p s
>
1044 e p s
>
5040 eps
>
2568 e p s
14
14
2
mu
3
3
3
4
15
ux
8
16
uy
4
uy
8
g
15
mu
4
ux uy
15
mu
3
g
16
w
8
12
ux u y
mu
15
ux
12
ux
4
15
mu ux
15
mu
4
10
ux
8
14
mu
16
w
+
10
16
w
16
8
12
ux
396 e p s
16
uy
2
eps
14
mu
w
14
3
20
uy
16
18
uy
10
12
uy
12
14
8
18
6
ux
uy
w
3 2
15
10
eps g
mu
ux uy
10
3
3
- 1440 e p s
10
6
15
4
12
g mu
ux
uy
3 2
6648 e p s g
18
w
10
8 18
ux u y
w
+
vj
+ 1160 e p s
18
w +
18
w +
uy
18
w
+
16
w +
3 4
g
3 6
15
g
mu
+
16
- 14 4 0 e p s
18
w +4064
w
w
8 18
w
10
uy
16
uy
8
15
6
8
g mu
ux uy
18
3
15
w - 132 e p s mu ux
16
w +
6
420 e p s
15
mu
16
w
14
uy
ux
4
ux
-
14
uy
10
ux
mu
4
ux
6
ux
14
mu
2 2 14
g mu
10 18
w + 2 4 80 e p s
6
14
mu
+
w
18
-
8
uy
uy
12
uy
ux
2 2
- 3132 e p s g
w
w
15
mu
3
2
+ 2728 e p s
16
uy
15
8
mu ux uy
4
2 8 14
g mu
10
uy
14 12
ux uy
mu
2 4
+ 6096 e p s g
16
w
16
ux
2
g
g
16
w
6
18
16
u x uy
w+ 3 6
6
g
3
3
14
uy
8
14
ux uy
mu
mu
eps
8
ux
12
16
2
14
12
12 16
uy w
+ 18 2 0 e p s
mu ux
uy
w
4
ux
14
mu
g
16
w +870
10
mu
g
eps
>
10
16
2
2
uy w
- 3132 e p s
g
ux
6
g
14
mu
6
12 16
2 4
ux u y
w
+ 9972 e p s g
mu
g
2
11 6 0 e p s
10
ux
6
g
16
2 2
14
14
8 16
w
- 39 6 e p s
g mu
ux
uy w
+
6
10
16
2 6
ux uy
w
- 5040 eps g
14
14
10
mu
ux uy
eps
>
8
8 16
w
+
8 16
2 10
14
4
10
16
uy w
- 252 e p s
g
mu
ux uy
w
+
14
mu
2
3
12
uy
14
4
2
56 e p s
16
2 6
14
10
w - 1440 e p s
g mu
ux u y
mu
g
eps
>
8
8
ux uy
16
ux
g
2
492 e p s
14
8
eps
2
>
16
2 10
14
6
8 16
w - 252 e p s
g
mu
ux u y w
+
14
muux
mu
186 0
4
8
ux u y
mu
2 4
g
>
14
mu
18
18
w
18
w
15
8
12 18
mu
ux u y
w
4
15
4
14
g mu
ux
uy
18
w
+
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
296
3
>
4 064 ..eps
>
77 2
eps
>
132
eps
>
42 0
eps
>
720
eps
>
420
eps
>
2520
>
87 0
>
1662
eps
>
1016
eps
2
3
2
g
3
15
mu
2
4
16
10
mu u x
6
g
2
4
>
504
eps
>
5 04
eps
17
4
>
8 eps
>
12 e p s
20
w
-
4
ux
8
uy
22
4 4
g
17
4
mu ux
10
uy
22
w+ 3 4 8
17
mu
6
ux
12
uy
22
w
4
4 2
g
6
eps
7
18
ux
8
8
uy
18
mu
6
ux
10
uy
7
19
mu
4
ux
+ 58 e p s
26
-
12
26
w
2
g
8
+ eps
10
20
uy w
10 20
w
+
12
uy
16
20
w +
6
8 22
ux uy
w
+
8
22
w
4
ux
4
14
uy
18
muux
6
uy
4
uy
4
ux
10 22
w
12
22
uy w
ux
18
mu ux
20
w +
20
w
+
10
uy
17
6 18
mu
8 20
w
2 17
6
mu
ux uy
mu
6
18
4
14
ux uy
4
uy
17
mu
6 2
eps g
- 84 e p s
24
vj
10
uy
2
g
5
24
w
2 19
4 8
mu
ux uy w
g
5
12
uy
16
mu
5
eps g
eps
- 84
+
w
16
6
mu ux
16
mu ux
eps
- 14 4 e p s
4 18
4
8 24
mu
ux uy
w
g
16
4
16
10
mu
ux u y
eps
2520 ep s
10 22
uy w
+744
8
ux
18
w
16
6
mu ux
5 4
17
- 25 2 e p s g
mu
w
17
mu
6
124
186 0 e p s
2
17
8
8 22
5
17
g mu
ux uy
w
- 144 e p s
muux
5
>
4 12
uy
ux
mu
5
6
58 e p s mu
+
14
uy
16
8
8 20
mu ux u y
w
4
16
193 e p s mu
ux
16
6
14 20
mu ux
uy
w+ 1 9 3
g
>
6
uy
4
5
28 e p s
15
muux
16
8
12 20
4 2
mu ux
uy
w
- 720 ep s
g
6
>
3
4
5
eps
16
mu
g
252
20
w
8
ux
8
16
4
8 20
g mu
ux u y w
eps
8 20
uy w
+
4 10
uy
ux
4
g
5
>
4
15
mu
1044 e p s
16
8
10 20
mu ux u y
w
+ 1016
g
eps
eps
16
mu
4
eps
348
18
w -
4
18
18
uyw
+70
ux
g
>
16
uy
3
eps
6
16
6
8 20
4 4
g mu
ux u y
w
+ 870 e p s
g
4
5 6 eps
6
14 18
ux u y
w
- 2568
4
ux
15
mu
4
>
15
mu
g
22
w
+
8
w
10
24
+
24
w
12
uy
+
24
w +
7
eps
19
6 8
26
mu ux
uy w
20
mu
4
8 28
ux uy
w
I n [ 8 ] : = N [ Pi ]
12
4 8
mu ux uy
Out [ 8 ] =
2
>
(g
2
- 3 ux
12
2
v; (g + e p s
2
- 3 uy
mu
2
+ e ps mu w )
2 2
w ) (g - 2 ux
2
(g
2
- ux
2
+
2
- 3 uy
2
e p s mu w )
2
+ e ps mu w )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
297
2
2
- 2 uy
>
(g
2
2
>
(g
- 3 ux
2
2
2
+ eps mu w ) ( g
2
- uy
- 2 ux
2
2
- 2 uy
2
+ eps mu w ) (g
2
- ux
2
+ eps mu w )
2
- uy
2
+ eps mu w )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
298
Appendix D
C-code Listing of the wgjplot.c Program
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
299
# in c lu d e < std lib .h >
A in clu d e < s td io .h >
//in c lu d e < m a th .h >
A in clu d e < m a llo c .h >
//in c lu d e < c o m p lc x .h >
A in clu d e "nrutil.h"
A d efin e M 30
//d e fin e N
15
//d e fin e N O O F D Z 6 5
A d elin e R O W L E N G T H (2 * (M + 1 )* (N + 1 ))
//d e fin e N M A X 5 0 0 0
A d elin e G E T _ P S U M \
for (j= l;j< = n d im ;j+ + ) { \
for ( s u m = 0 .0 .i= l; i< = m p ls ; i+ + ) su m + = p [ij[j];\
p su m fj]= su in ;}
A d efin e S W A P (a .b ) {s\v a p = (a );(a )= (b );(b )= sw a p ;}
A d elin e J C M P L X F (0 .0 .1.0)
v o id m a in ()
flo a t * sv d (). * v ccto r(). c l3 ( ). d c lc r m in a n t(),* g u c ss_ g a m m a (),* E H _ P o \v c r _ c a lc ();
v o id a m o e b a ().o p c n p lo t().p lo tm a ir (),c lo s c p lo t(),c a lc jo n g ();
v o id free _ v c c lo r ();
flo a t p a r a m s |8 |, * g a m m a . p [41[3 ].y [4 ].* m a tri.x frcs[R O W L E N G T H ][2 ],* d ia g ;
fio a l e x [ 2 * N + l |[ 2 * M + 1 |. c y [ 2 * N + l ] [ 2 * M + l ] , e z |2 * N + l ] [ 2 * M + l ] ? h x [ 2 * N + lJ [ 2 * M + l] ,
h y [ 2 * N + 11 [2 * M + 1 ]. h z (2 * N + 1 ] [2 * M + 1], liz _ I o n g [2 * M + 1J [N O O F D Z + 1],
h .\ J o n g [ 2 * M + I |[ N O O F D Z + l |. c z J o n g |2 * M + l|[ N O O F D Z + l] , c.xJ o n g [2 * M + 1 1| N O O F D Z + 1].
h z J o n g 2 [ 2 * N + l ] [N O O F D Z + l |. h y _ lo n g |2 * N + 1][ N O O F D Z + 1], e z J o n g 2 [ 2 * N + 1|[N O O F D Z + 1 1.
c v _ I o n g [2 * N + I ||N O O F D Z + l |:
llo a l la lp lia , Ibcta. * P o w cr_ H m a x ;
in i i, iter, m .n:
n ia lr i.\= v c c to r (().2 * R O W L E N G T H * R O W L E N G T H -l);
/*
p a r a m s |0 |= () .0 2 2 8 6 ;
W id th o f B ox
p a ram sl 11=().0 1 0 1 6 ;
H eig h t o f B ox
p a ra m sl 2 1= 1.0;
R ela tiv e D ielectric C on stan t o f Interior
p a ra m sl 3 J= 10 c9 :
F req u en cy
p a r a m s [4 |= 4 5 0 e -9 :
Z ero T em perature P en etration D ep th
p a rn m s[5 |= .3 c6 ;
N orm til C on d u ctivity in S /m
p a r a m sl6 [= 7 7 ;
T em p eratu re o f W G
p a r a m s |7 |= 9 0 :
C ritical T em p eratu re o f H T S
*/
p rin tf(" \n E n tcr w id th o f w a v e g u id e cro ss-sec tio n :");
sca n f(" % f" .& p a ra m s|0 |);
p rin tf(" E n tcr h e ig h t o f w a v e g u id e cro ss-sec tio n :" );
sca n f(" % f" .& p aram s| 1 1);
p rin lf(" E n tcr rela tiv e d ie le c tr ic con stan t o f w a v eg u id e in t e r io r :" );
sc a n f(" % f" ,& p a ra m s|2 |);
p rin tf(" E n tcr e ffe c tiv e zcro -tcin p era lu rc p en etration d ep th o f H T S w a lls :" );
scanf("% f".< fcparains|4|):
p rin tf(" E n lcr n orm al c o n d u c tiv ity o f H TS w a l l s :");
sea n f( "%f". & pa ra i n s 15 ]):
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
300
p rin tf("E n ter tem p eratu re o f H T S w a lls :" );
sc a n f(" % f',& p a r a m s[6 ]);
p rin tf("E n ter critica l tem p era tu re o f H T S w a lls :" );
sca n f(" % f',& p a r a m s[7 ]);
p rin tf(" E n ter'm ' o f th e m o d e y o u are in te rested in : ");
scanf("% d " ,& m );
prin tf("E n ter 'n' o f th e m o d e y o u are in te rested in :" );
scanf("% d",& n);
p rin tf(" E n ler freq u en cy :" );
sc a n f(" % f',& p a r a m s[3 ]);
g a m m a = g u e s s_ g a m m a (m ,n ,p a r a m s);
la lp h a = 0 .0 3 * g a m m a [l] ;
lb e la = 0 .0 2 * g a m m a [2 ];
p [l][l]= g a m m a [l]:
p [ l] [ 2 |= g a m m a [ 2 ] ;
p [ 2 ] [ l |= p [ l ] [ l ] + l a l p h a ;
P [ 2 ][ 2 |= p | 1 ][2 J-lbeta;
P[3][lJ=p[l]Ll]-IaIpha;
p [3 J [2 J = p [l]|2 |-lb e ta ;
fo r ( i= l; i< = 3 ; i+ + )
y | i]= d clcrm in a n t(& p [ i | [ 0 ] ,m a trix,p aram s);
a m o c b a (p ,y . 2. l e - 6 ,l c - 5 , d eterm in a n t,& itcr,m a tr ix ,p a ra m s);
prin lf(" \n % d Itc r a tio n s .W .ite r );
p rin tf(" \n A t freq u en cy % c H z, ( a lp h a .b c la ) = (% .5 c .% .5 e ) \n " ,p a r a m s [3 ] .p [l] [ l] ,p [ l] [ 2 ]) ;
fflu sh (N U L L );
p rin tf(" \n C a lcu la tin g so lu tio n vector..A n " );
d ia g = sv d (& p [ 1 1[0 |.m a lr i.\,p a rains);
p rin tf(" M a x im u m S in g u la r V alu e: % A n " ,d iag[0]);
p rin lf(" L ast 5 S in g u la r V a lu e s: \n"):
fo r (i= R O \V L E N G T H -5 ;i< = R O W L E N G T H -l;i+ + )
prinlf("% d
% .3 c \n " ,i+ l.d ia g |i]) ;
fo r ( i= 0 ;i< = R O W L E N G T H -l;i+ + )
f
r c s li||0 ]= c !3 (m a tr ix .i,R O W L E N G T H -1 .0 .R O W L E N G T H .R O W L E N G T H ,2 );
r c s |i|| ll= -l.(> * c l3 (m a lr i.\.i.R O W L E N G T H -l,l,R O W L E N G T H .R O W L E N G T H .2 );
)
i
fr c c _ v c c lo r (m a tr ix .().2 * R O W L E N G T H * R O W L E N G T H -l);
P o w c r _ H m a x = E H _ P o w c r _ c a lc (C M P L X F (p f l ] f 1 1, p [ l] [ 2 ] ) ,r e s , p a ra m s, 0 .0 .0 .0 , e x , e y ,c z ,h x .h y .h z );
p rin tf(’’\n P o w e r at z = ( ) : % c \V \n B o tlo m W all M a x im u m T a n g e n tia l M a g n e tic F ield : % e A /m \n L c ft W all
M a x im u m T a n g e n tia l M a g n e tic F ield : %c A /m \n " .P o w e r _ H m a x [2 |.P o w c r _ H m a x |0 |.P o w e r _ H m a x | 11):
fflu sh (N U L L ):
o p cn p lo lO ;
p lo lin a t r ( c x .c y .2 * M + 1 ,2 * N + 1):
p lo t m a t r ( h x .h y .2 * M + l,2 * N + l) ;
c a lc _ lo n g ( h x _ lo n g .h z _ lo n g .e x _ lo n g .c z _ lo n g .h v _ lo n g .h z _ lo n g 2 ,e y _ lo n g .c z _ lo n g 2 .c x .c y .c z .h x .h y .h z .p a r a
m s.C M P L X F (p | 11|1 |.p | l ] |2 |) ,r c s ) ;
p lo tm a t r (h z _ lo n g .h x _ lo n g .N O O F D Z + l,2 * M + l) :
p lo tm a lr ( c /.J o n g .c x J o n g .N O O F D Z + 1 .2 * M + l) ;
p lo tm a t r (h /.J o n g 2 .h y J o n g .N O O F D Z + 1 .2 * N + l) :
p lo t m a t r ( c /.J o n g 2 .c y _ lo n g ,N O O F D Z + l,2 * N + l) ;
c lo scp lo lO ;
return:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
flo a t * g u e ss _ g a m m a (m ,n ,p a r a m s)
flo a t *p a ra m s;
in t m ,n ;
{
flo a t fc ,R s,m u ,e p s,w ;
sta tic flo a t g a m m a [3 ];
m u = 4 * M _ P I * le - 7 ;
e p s = p a r a m s [2 ]* 8 .8 5 4 e -1 2 ;
w = 2 * M _ P I * p a r a m s [3 ];
R s = 0 .5 * S Q R (m u )* S Q R (w )* p o \v (p a r a m s [4 ],3 .0 )* p a r a m s [5 ]* p o w ((p a r a m s[6 ]/p a r a m s[7 ]),4 .0 );
fc = l/(2 * s q r t(m u * c p s ))* s q r t(S Q R (n i/p a r a m s [0 ])+ S Q R (n /p a r a m s [l]));
i f (p a r a m s[3 ]> fc )
1f
i f (n = = 0 )
ii
g a m m a | l]= R s/(s q r t(m u /e p s)* p a r a m s [ 1] * sq rt( 1 S Q R (fc /p a r a m s [3 ])))* (l+ 2 * p a r a m s [l]/p a r a m s [0 ]* S Q R (fc /p a r a m s [3 ]));
}
e ls e
(
\
g a m m a [ l]= 2 * R s /(s q r t( m u /e p s )* p a r a m s [l] * s q r t (lS Q R (f c /p a r a m s [ 3 ]) ) )* ( ( l+ p a r a m s [ l] /p a r a m s [0 ] )* S Q R ( fc /p a r a m s [3 ] )+ ( lS Q R (fc /p a r a m s[3 J ))* ((p a r a m s[lJ /p a r a m s [0 ]* (p a r a m s [l]/p a r a m s[0 ]* S Q R (m )+ S Q R (n )))/(S Q R (p a r a m sll]*
m /pa ra m s| 0 1)+ S Q R (n ))));
>
}
g a m m a [2 )= 2 * M _ P I* p a r a m s [3 ]* sq r t(m u * e p s)* s q r t(l-S Q R (fc /p a r a m s |3 |)):
)
e ls e
{
g a m m a [lJ = 2 * M _ P I * fc * s q r t(m u * e p s )* s q r t(l-S Q R (p a r a in s |3 |/fc )):
g a m m a |2 j = 0 .0 ;
>
i
p riiitf("\n F c: % e H xA iiThcorelical TE % d% d a lp h a : % c N e p c r s /m \n L o sslc ss bcla: % c
r a d /in \n \n " .fc .m .n .g a m m a |lj.g a m m a |2 ]);
rctu rn (g a m m a );
i<
float c l3 (m n tr ix d .m ,n .o .p .q )
float * m a lrix ;
int l.m .n .o .p .q ;
\(
r c tu r n (m a lr ix [I* p * q + m * q + n ]):
)
I
v o id m a lr fg a m m a . m atrix.p aram s)
float * g a m m a . *m a lrix . *param s;
(
v o id im tm a lr ix (). p u lic a l(). piitim agO'.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
302
in t E z (), H z();
in t i,j,ro w in d e .\;
flo a t a lp h a ,b e ta ,D x ,D y ;
flo a t a ,b ,c p s,m u ,w ,R ,R s ,X s,k s q ;
a = p a ra m s[0 ];
b = p a r a m s [l];
c p s= p a r a m s[2 ]* 8 .8 5 4 e -1 2 ;
\v = 2 * M _ P I* p a r a m s[3 ];
m u = 4 * M _ P [ * le -7 ;
R s= 0 .5 * S Q R (m u )* S Q R (\v )* p o \v (p a r a m s[4 ],3 .0 )* p a r a m s[5 ]* p o \v ((p a r a m s [6 ]/p a r a m s[7 ]),4 .0 );
X s = m u * \v * p a r a m s[4 ];
D x = a /(2 * M );
D y = b /(2 * N );
R = D x /D y ;
a lp h a = g a m m a [l];
b cta = g a in m a [2 ];
p rin tf(" \n (% .4c,% .4e)" , a lp h a , beta);
fflu sh (N U L L );
k sq = \v * \v * m u * e p s+ a lp h a * a lp h a -b c ta * b e ta ;
ro \v in d cx = 0 :
in itn ia trix (m a trix );
/* In sid e P o in ts. H e lm h o ltz E q u a tio n . * /
for ( j = l: j < = N - l; j + + )
for ( i= l ; i < = M - l : i + + )
{
p u tr c a l(r o w in d e x ,E z (i+ 1.j), 1 .0 ,m atrix);
p u tr c a l(r o w in d e x ,E z (i-l j ) , 1.0, m atrix);
p u tr c a I(r o w in d e .x ,E z (i.j+ l).R * R , m atrix);
p u lr e a l(r o w in d c .x ,E z (i.j-l),R * R , m atrix);
p u tr c a l(r o w in d e x .E z (i,j).k s q * D x * D x -2 ‘i:(l+ R * R ).m a tr i.\);
p u tim a g (r o \v in d e x ,E z (i.j).2 * a Ip h a * b e ta * D x * D x ,m a tr ix );
ro \v in d cx + + :
p u tr e a l(r o w in d e x .H z (i+ l,j), 1 .0 ,m atrix);
p iitrc a l(ro w in d c.x ,H z(i-l ,j), 1 .0 ,m atrix);
p u tr c a l(r o w i\\d c x .H z (i.j+ l),R * R .m a tr ix );
p u lr c a !(r o w in d e x ,H z (i.j-l),R * R , m atrix);
p u tr c a l(r o w in d c x ,H z (i.j).k sq :|<D x * D x - 2 :i:(l+ R * R ),m a tr ix );
p u tim a g (r o \v in d c x ,H z (i.j).2 * a lp h a * b c la * D x * D .x . m atrix);
ro \v in d cx + + ;
J
/ * B O T T O M W A L L */
j= 0 ;
for ( i = l ; i < = M - l ; i+ + )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
303
{
/* E z = -Z s * H x * /
p u tr e a l(r o v v in d e x ,E z (i,j),2 .0 * D x * k sq , m atrix);
p u tim a g (r o w in d e x ,E z (i,j),4 * D x * a lp h a * b e la , m a trix );
p u tr e a l(r o w in d e x ,E z (i,j+ l),-2 * R * X s * w * e p s ,m a tr ix );
p u tim a g (r o v v in d e x ,E z (i,j+ l),2 * R * R s* w * e p s , m a trix);
p u tr e a l(ro w in d ex ,E z(i.j),2 * R * X s* -w * ep s,m a trix );
p u tim a g (r o w in d e x ,E z (i,j),-2 * R * R s* \v * e p s, m a trix);
p u tr e a l(r o \v in d e x ,H z (i+ l,j),-(a lp h a * R s -b e ta * X s ),m a tr ix );
p u tim a g (r o w in d e x ,H z (i+ l,j),-(b e ta * R s+ a I p h a * X s ), m atrix);
p u tr e a l(r o \v in d e x ,H z (i-l,j),a lp h a * R s-b e ta * X s, m a trix );
p u tim a g (r o w in d c x ,H z (i-l,j),b e ta * R s+ a Ip h a * X s , m atrix);
ro w in d ex + + ;
/* E .x= Z s*H z * /
p u tr e a l(r o \v in d e x ,H z (i,j),2 * D x * (R s* k sq -2 * a lp h a * b e ta * X s ), m atrix);
p u tim a g (r o w in d e x ,H z (i,j),2 * D x * (X s * k s q + 2 * a lp h a * b e ta * R s), m atrix);
p u tim a g (r o \v in d e x ,H z (i,j+ l),2 * \v * m u * R ,m a tr ix ):
p u tim a g (r o w in d e x ,H z (i.j),-2 * w * m u * R .m a lr ix ):
p u lr e a l(r o \v in d e x ,E z (i+ l j ),o lp iia , m atrix);
p u tim a g (r o w in d e x ,E z (i+ l,j),b e ta ,m a tr ix );
p u tr e a l(r o \v in d c x ,E z (i-l,j),-a lp h a ,m a tr ix );
p u lin ia g (r o w in d c .x ,E z (i-l,j),-b e ta ,m a tr ix );
ro w in d cx + + ;
}
/* T O P W A L L * /
j= N :
for ( i= l ; i < = M - l ; i+ + )
(
\
/* P erfect E lectric B o u n d ary * /
/* E z = 0 * /
p u trc a l(ro \v in d e.\.E z(i.j), 1.0. m atrix);
ro \v in d ex + + ;
/* E x = 0 * /
p iU im a g (r o \v in d c x ,H z (i.j),2 * \v * m u * R , m atrix);
p u tim a g (r o \v in d e x ,H z (i.j-l).-2 * \Y * m u * R .m a lr ix );
p u tr c a l(r o \v in d c x ,E z (i+ l.j),a lp h a ,m a tr ix ):
p u tim a g (r o \v in d c x ,E z (i+ l.j),b e ta ,m a tr ix ):
p u tr c a l(r o \v in d c x ,E z (i-l.j),-a lp h a , m atrix):
p u tim a g (r o \v in d c x ,E z (i-l,j).-b c ta ,m a tr ix ):
ro\vm d cx+ + :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
304
/* P erfect M a g n e tic B o u n d a r y
Hz=0
p u tr e a l(r o \v in d e x ,H z (i,j), 1 .0 ,m atrix);
r o w in d e x + + ;
H x=0
p u tim a g (r o \v in d e x ,E z (i,j),2 * v v * e p s* R , m atrix);
p u tim a g (r o w in d e x ,E z (i,j-l),-2 * \v * e p s * R ,m a tr ix );
p u tr e a l(r o \v in d e x ,H z (i+ l,j),-a lp h a ,m a tr ix );
p u tim a g (r o \v in d e x ,H z (i+ I,j),-b e ta ,m a tr ix );
p u tr e a l(r o \v in d e x ,H z (i-1.j),a lp h a ,m a trix );
p u tim a g (r o w in d e x ,H z (i-1,j), b eta, m atrix);
ro \v in d ex + + :
*/
>
/* L E F T W A L L * /
fo r ( j = 1 ; j< = N - 1 ;j+ + )
/* E z = Z s* H v * /
p u tr c a l(r o \v in d c x ,E z (i.j).2 * D x * k s q , m atrix):
p u tim a g (r o \v in d e x .E z (i,j).4 * D x * a lp h a * b c ta , m atrix);
p u tr e a l(r o \v in d c x ,E z (i+ l.j),-2 * X s * w * c p s ,m a tr ix );
p u tim a g (r o \v in d c x ,E z (i+ l.j).2 * R s * \v * c p s , m atrix);
p u tr c a I(r o \v in d c x ,E z (i.j).2 * X s* \v * c p s , m atrix);
p u tim a g (r o \v in d c x ,E z (i.j),-2 * R s * \v * c p s,m a tr ix );
p u tr e a l(r o \v in d c x ,H z (i.j+ l),a lp h a * R * R s -b c la * R * X s .m a tr ix );
p u lim a g (r o w in d c x .H z (i.j+ I).b c la * R * R s+ a lp h a * R * X s,m a tr ix );
p u lr e a l(r o \v in d e x ,H z (i.j-l).-(a Ip h a * R * R s -b e la * R * X s ),m a tr ix );
p u tim a g (r o \\a n d c x .H z (i.j-l),-(b e ta * R * R s+ a lp h a * R * X s),m a lr ix );
r o w in d c x + + ;
/* E y = -Z s * H z */
p u tr e a I(r o \v in d c x .H z (i.j).2 * D x * (R s * k s q -2 * a lp lia * b c la * X s).m a tr ix );
p u tim a g (r o \v in d c x ,H z (i.j),2 * D .x * (X s* k s q + 2 * a lp h a * b c ta * R s),m a tr ix );
p iitim a g (r o \v in d c x ,H z (i+ l.j).2 * \v * m u , m atrix);
p u lim a g (r o w in d e x .H z (i.j).-2 * w * m u ,m a tr ix );
p u tr e a l(r o w in d c x ,E z (i,j+ l).-R * a lp h a .m a tr ix );
p u tim a g (r o \v in d c x .E z (i.j+ l).-R * b c ta .m a tr ix );
p u tr c a I(r o \v in d c x ,E z (i.j-l).R * a lp h n .m a lr ix );
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
p u tim a g (r o \v in d e x ,E z (ij - l) ,R * b e la , m a trix);
r o \v in d e x + + ;
}
/* R IG H T W A L L * /
i= M ;
fo r ( j = I : j < = N - l: j + + )
{
/* P e r fe c t M a g n e tic B o u n d a ry * /
/* H z = 0 * /
p u tr e a I(r o \v in d c x ,H z (i.j), 1.0, m atrix);
r o \v in d e x + + ;
/* H y = ()* /
p u tim a g (r o \v in d c x ,E z (i.j),2 * \v * e p s .m a tr ix );
p u tim a g (r o \v in d e x ,E z (i-1 .j),-2 * \v * c p s7m a trix ):
p u tr e a l(r o w in d c x ,H z (i.j+ l),R * a lp h a , m atrix);
p u tim a g (r o w in d c x ,H z (ij+ l),R * b c ta ,m a tr ix )'.
p u tr e a l(r o w in d c x ,H z (i.j-l).-R * a I p h a ,m a tr ix ):
p u tim a g (r o w in d e x ,H z (i.j-l),-R * b c ta . m atrix);
r o \v in d c x + + :
/* P erfect E lec tric B ou n d ary
E z=()
p u trc a l(ro rv in d ex ,E z(i.j), 1.0, m atrix):
r o w in d c x + + ;
Ey=0
p v \lim a g (r o \v in d c x .H z (i.j),2 * w * m u ,m a tr ix );
p iilim a g (r o \v in d e x .H z (i-l.j),-2 * w * n n i, m atrix)
p u tr c a l(r o \v in d e x .E z ( i.j + l) ,- R :|<a lp h a .m a trix );
p u lim a g (r o \v in d c x .E z (i.j+ I),-R * b e la .m a tr ix ):
p u tr c a l(r o \v in d c x .E z (i.j-1).R * a lp h a .m a trix );
p u lim a g (r o \v in d c x .E z (i.j-l).R * b c ta .m a tr ix );
ro w in d c x + + :
*/
>
/* C O R N E R S * /
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
306
i=0;
j=0;
/* E z = -Z s * H x * /
p u tr e a l(r o w in d e x ,E z (i,j),D x * k sq , m atrix);
p u tim a g (r o w in d e x ,E z (i,j),2 * D x * a lp lia * b e la ,m a tr ix );
p u tr e a l(r o \v in d e x ,E z (i,j+ l),-R * X s* \v * e p s ,m a tr ix );
p u lim a g (r o w in d e x ,E z (i,j+ l),R * R s* w * e p s,m a tr ix );
p u tr e a l(r o w in d e x ,E z (i,j),R * X s* \v * e p s,m a tr ix );
p u tim a g (r o \v in d e x ,E z (i,j),-R * R s* \v * c p s,m a tr ix );
p u tr e a I(r o w in d e x ,H z (i+ l,j),-(a lp h a * R s -b e ta * X s),m a tr ix );
p u tim a g (r o w in d e x ,H z (i+ l,j),-(b e ta * R s+ a lp h a * X s),m a tr ix );
p u tre a I(ro w in d ex ,H z(i,j),a lp h a * R s-b eta * X s, m atrix);
p u tim a g (r o w in d e x ,H z (i,j),b e ta * R s+ a lp h a * X s,m a tr ix );
ro\vin d ex+ + ;
/*
E x = Z s * H z
* /
p u tr e a l(r o \v m d e x ,H z (i,j),D x * (R s* k sq -2 * a lp h a * b c la * X s),m a tr ix );
p u tim a g (r o w in d e x ,H z (i,j),D x * (X s * k s q + 2 * a lp h a * b c ta * R s), m atrix);
p u tim a g (r o w in d c x ,H z (i.j+ l),w * m u * R .m a tr ix ):
p u tim a g (r o \v in d e x ,H z (i,j),-\v * m u * R , m atrix);
p u lr e a l(r o w in d e x ,E z (i+ l.j),a lp h a ,m a tr ix );
p u tim a g (r o \v in d e x ,E z (i+ l,j), b eta, m atrix);
p u trc a l(ro rv in d cx ,E z(i.j).-a lp h a .m a trix );
p u lim a g (r o w in d e x ,E z (i.j),-b e ta ,m a tr ix );
ro w in d cx + + ;
i= M ;
j= 0 ;
/ * E z = -Z s* H x
p tilr e a l(r o \v in d c x ,E z (i.j).D x * k sq ,m a tr ix );
p u tim a g (r o w in d c x ,E z (i,j).2 * D x * a lp h a * b c ta ,m a tr ix ):
p u lr c a l(r o \v in d e x ,E z (i.j+ l).-R * X s* \v * e p s .m a lr ix );
p u tim a g (r o \v in d c x ,E z (i,j+ l),R * R s * \v * c p s.n ia lr ix ):
p u trc a I(ro \v in d c x ,E z(i.j).R * X s* \v * e p s,m a trix );
p tilim a g (r o \v in d e x ,E z (i.j),-R * R s* \v * e p s.m a tr ix );
p u tre a l(r o \v in d c x ,H z (i,j).-(a lp h a * R s-b c ta * X s).m a lr ix );
p u tim a g (r o \v in d c x ,H z (i,j).-(b c la * R s+ a Ip h a * X s). m atrix);
p u tr c a l(r o w in d c x .H z (i-l.j).a lp h a * R s -b c la * X s ,m a tr ix );
p iitim a g (r o \v in d c x ,H z (i-l.j),b c ta * R s + a lp h a * X s . m atrix):
row in d cx+ + ;
E x = Z s* H z
p u lr c a l(r o \v in d c x .H z (i.j).D x * (R s* k sq -2 * a Ip h a * b c (a * X s).n ia lr i.\):
p u tim a g (r o \v in d c x ,H z (i.j).D x * (X s* k sq + 2 * a lp Iia * b c ta * R s).m a tr ix ):
p iilim a g (r o \v m d c x .H z (i.j+ l).\v * m u * R .m a lr ix ):
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
307
p u tim a g (r o \v in d e x ,H z (i,j),-\v * m u * R , m atrix);
p u tr e a l(r o w in d e x ,E z (i,j), a lp h a, m atrix);
p u tim a g (r o w in d e x ,E z (i.j),b e ta ,m a tr ix );
p u tr e a l(r o w in d e x ,E z (i-l.j),-a lp h a ,m a tr ix );
p u tim a g (r o \v in d e x ,E z (i-l,j),-b e ta , m atrix);
r o \v in d ex + + ;
*/
/* P erfect M a g n e tic B ou n d ary * /
/* H z = 0 * /
p u tr e a l(r o \v in d e .\,H z (i,j), 1 .0 ,m atrix);
ro \v in d cx + + ;
/* H y = 0 * /
p u tim a g (r o w in d e x ,E z .(i.j),w * c p s, m atrix);
p u tim a g (r o \v in d e x ,E z (i-1 .j),-\v * c p s, m atrix);
p u tr e a l(r o \v in d e x ,H z (i.j+ l).R * a lp h a ,m a tr ix );
p u tim a g (r o w in d c x ,H z (i j + 1),R * b cta , m atrix);
p u tre a l(ro w in d ex ,H z(i,j),-R * a Ip h a ,m a trix );
p u tim a g (r o \v in d c x ,H z (i.j).-R * b e ta , m atrix):
ro \v in d cx + + ;
/* P erfect E le c tr ic B ou n d ary
Ez=()
p u trc a I(ro \v in d c x ,E z(i.j), 1.0 .m a trix );
ro w in d c x + + :
Ey=()
p u tim a g (r o \\in d c x ,H z (i.j),w * m u .m a tr ix ):
p u tim a g (r o \v in d c x ,H z (i-l.j),-\v * m u ,m a tr ix ):
p u tr c a l(r o \v in d e x ,E z (i.j+ l).-R * a Ip h a ,m a tr ix ):
p u lim a g ( r o \v in d c x .E z ( i,j+ l) .-R !|,bcta, m atrix):
p iitr c a l(r o w in d c x ,E z (i.j).R * a Ip h a , m atrix);
p u tim a g (r o \v in d c x .E z (i.j).R * b c ta . m atrix);
ro w in d cx + + :
*/
i= 0;
j= N ;
/* E z = + Z s* H x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
308
p u tr e a l(r o \v in d e x ,E z (i,j).D x * k s q , m atrix);
p u tim a g (r o \v in d e x ,E z (i,j),2 * D x * a Ip h a * b e ta , m atrix);
p u tr e a l(r o \v in d e x ,E z (i,j),R * X s * \v * e p s, m atrix);
p u tim a g (r o \v in d e x ,E z (i.j),-R * R s * w * e p s,m a tr ix );
p u tr e a l(r o w in d e x ,E z (i,j-l).-R * X s* \v * e p s ,m a tr ix );
p u tim a g (r o w in d e x ,E z (i.j-l),R * R s * \v * e p s , m atrix);
p u tr e a l(r o \v in d e x ,H z (i+ l,j).a lp h a * R s -b e ta * X s, m atrix);
p u tim a g (r o w in d e x ,H z (i+ l,j),b e ta * R s+ a lp h a * X s,m a tr ix );
p u lre a I(r o \v in d e x ,H z (i,j),-(a lp h a * R s-b e ta * X s),m a tr ix );
p u tim a g (r o w in d e x ,H z (i.j),-(b e la * R s+ a lp h a * X s),m a tr ix );
ro w in d ex + + ;
E x = -Z s* H z
p u tr e a l(r o \v in d e x ,H z (i,j),D .\* (-R s * k s q + 2 * a lp h a * b e ta * X s).m a lr ix );
p u tim a g (r o w in d e x ,H z (i,j).-D x * (X s* k s q + 2 * a lp h a * b e ta * R s),m a tr ix );
p u tim a g (r o \v in d e x ,H z (i.j).\v * m u * R ,m a lr ix ):
p u tim a g (r o \v in d e x ,H z (i.j-l),-\v * m u * R , m atrix);
p u lr e a l(r o w in d e x ,E z (i+ 1.j), a lp h a , m atrix);
p u tim a g (r o \v in d e x .E z (i+ 1 ,j),b eta.m atrix);
p u tr c a l(r o \v in d e x .E z (i,j).-a lp h a ,m a tr ix ):
p u tim a g (r o \v in d e .\,E z (i.j).-b e ta ,m a tr ix );
ro\vin d ex+ + ;
/* P erfect E le c tr ic B o u n d ary * /
/* E z = 0 * /
p u tr c a l(ro \v in d cx ,E z(i.j). 1 .0 ,m atrix):
ro \v in d cx + + ;
/* E x = 0 * /
p u tim a g (r o \v in d c x ,H z (i.j).w * n u i* R ,m a tr ix ):
p u tim a g (r o \v in d e x .H z (i.j-l).-\v * m u * R ,m a tr ix ):
p u tr c a l(r o \v in d c x .E z (i+ 1 .j),a lp h a , m atrix);
p u tim a g (r o \v in d e x ,E z (i+ 1 .j).b c tn ,m a tr ix ):
p u tr c a l(ro \v in d cx ,E z(i.j).-a lp h a ,m a trix ):
p u lim a g (r o \v in d c x .E z (i.j).-b c ln . m atrix):
ro\vin d cx+ + :
/* P crfccl M a g n e tic B o u n d a ry
H z=0
p u tre a l(ro \v in d cx .H /.(i.j), 1. (I. m atrix):
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
309
r o \v in d e x + + ;
H x=0
p u tim a g (r o \v in d e x ,E z (i,j),w * e p s* R , m atrix);
p u lim a g (r o \v in d e x ,E z (i,j-l),-\v * e p s * R , m a trix );
p u tr c a l(r o \v in d c x ,H z (i+ l,j),-a lp h a ,m a tr ix );
p u tim a g (r o w in d c x ,H z (i+ 1 ,j),-b eta ,m a trix );
p u tr c a l(r o \v in d e x ,H z (i,j).a lp h a ,m a tr ix );
p u tim a g (r o \v in d e x ,H z (i.j),b c ta ,m a tr ix );
r o \v in d e x + + :
*/
i= M :
j= N ;
/* E z= + Z s* H x
p u tr e a I(r o \v in d e x ,E z (i.j).D x * k sq ,m a tr ix );
p u tim a g (r o \v in d e x ,E z (i.j),2 * D x * a lp h a * b e ta ,m a tr ix );
p u tr c a l(r o \v in d c x ,E z (i.j),R * X s * \v * e p s, m a trix );
p u tim a g (r o \v in d e x ,E z (i.j),-R * R s * \v * c p s.m a tr ix );
p ittr c a l(r o \v in d c x .E z (i,j-1 ),-R * X s * w * c p s,m a tr ix );
p u tim a g (r o \v in d c x ,E z (i.j-I),R * R s* \v * e p s , m a trix );
p u tr c a l(r o \v in d c x ,H z (i.j).a lp lia * R s-b c ta * X s,m a tr ix );
p u tim a g (r o \v in d c x .H z (i.j),b e la * R s + a lp h a * X s ,m a tr ix );
p u tr c a l(r o \v iiid c x ,H z (i-l.j).-(a lp h a * R s -b e ta * X s ), m a trix);
p iU im a g (r o \v m d c x .H z (i-l.j).-(b c la * R s + a lp h a * X s ),m a tr ix );
r o \v in d c x + + ;
E x = -Z s * H z
p u tr c a l(r o \v iiid c x ,H z (i.j).D .x * (-R s * k s q + 2 * a lp h a * b c ta * X s), m atrix);
p u tim a g (r o \v in d c x .H /.(i.j).-D x * (X s * k sq + 2 * a lp h a * b c ia * R s ), m atrix);
p u tim a g (r o \v in d c x ,H z (i.j),\v * m u * R ,m a tr ix );
p u tim a g (r o \v in d c x ,H z (i,j-l),-\v * m u * R ,m a tr ix );
p iU rca l(ro \v in d ex ,E z(i,j).a lp h a .m a trix ):
p iitim a g (r o \v in d c x .E z (i.j), beta, m atrix);
p u tr e a l(r o \v in d c x .E z (i-1.j),-a lp h a ,m a trix );
p u tim a g (r o \v in d c x .E /.(i-1 . j).-b cla ,m a trix ):
r o \v in d c x + + ;
*/
/* H /.= 0 * /
p u tr c a l(r o \v in d c x .H z (i.j). 1.0. m atrix);
r o \v in d c x + + :
/* E z= () * /
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310
p u tr e a l(r o w in d e x ,E z (i,j), 1 .0 ,m atrix);
ro w in d ex + + ;
/* E x = 0
p u tim a g (r o w in d e x ,H z (i,j),\v * m u * R ,m a tr ix );
p u tim a g (r o \v in d e x ,H z (i,j-l),-w * m u * R ,m a tr ix );
p u tr e a l(r o \v in d e x ,E z (ij), a lp h a , m atrix);
p u tim a g (r o w in d e x ,E z (ij),b e ta ,m a tr ix );
p u tr e a I(r o w in d e x ,E z (i-l,j),-a lp h a ,m a tr ix );
p u tim a g (r o w in d e x ,E z (i-l.j),-b e ta ,m a tr ix );
ro \v in d c.\+ + ;
*/
return;
>
v o id in itm a tr ix (A )
flo a t A fR O W L E N G T H ) [R O W L E N GTHJ ) 2 ] ;
f
X
int i.j.k;
for ( i= 0 :i< = R O W L E N G T H -l;i+ + )
for (j= ():j< = R O W L E N G T H -l:j+ + )
for ( k = 0 ; k < = l; k + + )
A [ illj ] |k l= 0 .( );
return:
i
v o id p u lr c a l(r o \v in d c x ,c o lin d c x , v a lu e , m a trix )
int r o w in d c x .c o lin d e x ;
n o a t v a lu e. m a tr ix |R O W L E N G T H ]lR O W L E N G T H ]|2 ];
(
X
/ * For F O R T R A N c a lc u la tio n s rev erse c o lin d e x an d ro w in d ex
i.e ., T r a n sp o se the m a trix
*/
m a lr ix |c o lin d c x |[r o \v in d e x |f()|+ = v a h ie :
return;
f
v o id p u lim a g (r o \v in d e x .c o lin d c x , v a lu e , m a trix )
int ro w in d e x .c o lin d e x ;
flo a t v a lu e . m a tr ix |R O \V L E N G T H ||R O W L E N G T H |[2 ];
(
X
/* For F O R T R A N c a lc u la tio n s rev erse c o lin d e x an d ro w in d ex
i.e .. T r a n sp o se th e m atrix
*/
m a lr ix [c o ! in d c x |[r o u in d c x || I )+ = v a lu e;
return:
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
311
in t E z (i.j)
in tij;
{
r e tu r n (2 * i+ 2 * j* (M + l));
}
in t H z(i.j)
in t i j ;
{
r e t u r n (2 * i+ 2 * j * (M + l) + l) ;
flo a t d eterm in a n t(g a n im a ,m a trix ,p a ra m s)
flo a t * g a m m a , * m a trix , *param s;
{
v o id m atr();
flo a t m a g ;
fo r tr a n v o id C G E F A ();
fortran v o id C G E D ();
sta tic flo a t d e l[3 ];
int in fo , ip v t[R O W L E N G T H ];
in a tr(g am m a, m a trix , param s);
C G E F A (m a trix ,R O W L E N G T H ,R O W L E N G T H .ip v t.& in fo );
C G E D (m a trix ,R O W L E N G T H ,R O W L E N G T H ,ip v t,d ct);
m a g = 0 .5 * I o g l0 (S Q R ( d e t [0 ] )+ S Q R ( d e t[ l]) ) + d e l[ 2 ] -( fio a t ) (M + N + l)* lo g l( ) (S Q R (g a m in a [2 ])+ S Q R (2 * M _ P I * p a r a m s [3 ])* 4 * M _ P I * le -7 * 8 .8 5 4 c -1 2 * p a r a m s |2 |):
rctu rn (m ag);
i
flo a t * sv d (g a m m a , m atrix, p aram s)
flo a t * g a m n ia , * m atrix. *param s;
{
v o id m atr();
fortran v o id C G E B R D ();
fortran v o id C 'B D SQ R ();
fortran v o id C U N G B R Q ;
int in fo;
ch ar upper, vect;
flo a t E [R O V V L E N G T H -l |, T A U P fR O W L E N G T H ][2 ], T A U Q [R O W L E N G T H |[2 ].
w ork [64*R O V V L E N G T H ] [2];
sta tic flo a t D [R O W L E N G T H l;
uppcr='U':
vcct='P';
111
a t r(ga m m a , 111 a t ri x , pa ra m s ) ;
C G E B R D (R O \V L E N G T H ,R O \V L E N G T H .m a tr ix .R O W L E N G T H .D .E .T A U Q .T A U P .\v o r k .6 4 * R O W L E N
G T H .in fo);
p rin lf(" \n F irst Info: ‘X>d\n",info);
C U N G B R (& v ect,R O W L E N G T H ,R O V V L E N G T H ,R O V V L E N G T H .n ia lrix .R O W L E N G T H ,T A U P .\v o rk .6 4 *
R O W L E N G T H .in fo );
p rin lf(" S cco n d Info: ‘K>d\ii".info);
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
312
C B D S Q R (& u p p e r ,R O W L E N G T H ,R O W L E N G T H ,0 ,0 ,D ,E ,m a tr ix ,R O W L E N G T H .& in fo , 1,& in fo , 1.w o rk ,
& in fo );
p rin tf(" T h ird In fo: % d \n ",in fo);
retu rn (& D [0 ]);
v o id a m o e b a (p ,y ,n d im ,fto l,a r g to l,fu n k ,n fim k ,m a tr ix ,p a r a m s)
flo a t ( * fu n k )(),p [4 ] [3 ],fto l,a rg to l,y [],* m a trix ,* p a ra m s;
in t * n fu n k ,n d im ;
{
f lo a t am otry();
in t i,ih i,ilo ,in h i,p ilii[ 3 ] ,p ilo [ 3 ],p in h i[3 ] j ,m p t s = n d im + l;
f lo a t rtol, su m , sw a p , y sa v e ,y try ,* p su m ,p to l[3 ];
p su in = v e c to r ( 1 ,n d im );
* n fu n k = 0 ;
G E T _PSU M
for (;;) {
i l o = l;
ih i = y [ l ] > y [ 2 ] ? ( i n h i = 2 , l ) : ( in h i= l,2 ) ;
for ( i= l; i< = m p ls ; i+ + ) {
i f (y [i] < = y filo ]) ilo = i;
if(y [i]> y [ih i]){
in h i= ih i;
ih i= i;
} e ls e i f (y [i] > y [in h i] & & i != ih i) in h i= i:
}
fo r ( j= l:j< = n d im :j+ + )
'{
pilofj]=l;
p iliifj] = p |l]L j]> p |2 J [il ? ( p i n h i | j ] = 2 , l ) : (p in lii|j|= 1 .2 ) :
fo r ( i= l; i< = m p ts ;i+ + ) {
i f (P [i]|j] < = P fp iIo |j]]fj]) p ilo |j]= i;
i f ( p |i|[ j l > P fp il'i|j]J ljl) {
p in h ifj]= p ih ifj];
pihiLj]=i;
} e ls e i f ( p lf llj] > p [p in U i[j]||jl & & i != p ih ifj]) p in h i|j]= i;
}
p t o l[ i] = 2 .0 * r a b s (p ip ih i[j ] ]|j ]- p [p ilo |j] ] |jl) /(f a b s ( p |p ilii|jlllj l)+ f a b s ( p [ p ilo |il||j ]) ):
}
r to l= 2 .0 * f a b s ( y |ih i]- y [ilo |) /(f a b s ( v lih i] ) + f a b s ( v |ilo l) ) :
i f (rtol < fto l) {
S W A P (y [lJ ,y [ilo ])
for (i= 1 ;i< = n d i m ;i+ + ) S \V A P (p [ 1] I i | ,p | ilo 11 i | )
break:
}
i f ( p t o l [ l ] < = argtol & & p to l|2 ] < = a rg to l) break:
i f (* n fu n k > = N M A X ) n rcrror(" N M A X e x cee d e d " ):
* n fu n k + = 2;
y lr y = a m o tr y (p ,y ,p su m .n d im .fu n k ,ih i.-I.O .m a lr ix .p a r n m s):
i f (v tiy < = y |ilo ] )
ytry= am otry(p ,y,p siiin .iK lim , funk. ih i. 2 .0 . m atrix, para ms);
e ls e i f (vtry > = y |in h i] ) {
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
313
y sa v e = y [ih i];
y tr y = a m o lr y (p ,y ,p su m ,n d im ,fu n k , ih i, 0 .5 , m a tr ix , p aram s);
i f (ytry > = y sa v e ) {
for ( i= l; i< = m p t s ; i+ + ) {
i f (i != ilo ) {
for (j= l; j< = n d im ; j+ + )
p [i]0 ]= p su m L j)= 0 .5 * (p [i][j]+ p [ilo ]L i]);
y [i]= (* fu n k )(p su m ,n ia tr ix .p a r a m s):
}
}
*n fu n k + = n d im :
G ET _PSU M
}
} e ls e --(* n fu n k ):
}
fr e e _ v e c to r (p su n i, l.n d im ):
}
#u n d ef SW A P
#u n d ef G E T _PSU M
#undef N M A X
flo a t am otiy'(p , y .p s u m .n d im . fu n k .ih i.fa c , m atrix, p aram s)
flo a t (* fu n k )(),p [4 ][3 ],fa c ,p s iu n |],y ||.* m n tr ix ,* p a r a m s ;
in t ih i.n d iin :
{
in t j;
flo a t fa c l.fa c 2 ,y tr y ,* p tr v ;
p tty = v e c to r ( 1 .n d im ):
f a c l= ( 1 .0 - fa c ) /n d im ;
f a c 2 = fa c l- fa c ;
for ( j= l ;j< = n d im ;j+ + ) p try| jl= p su m [j |* fa c 1-p |ih i|fjl" T a c 2 ;
y try = (* fu n k )(p try , m atrix,p aram s):
i f (ytry- < y f ih i] ) {
y |ih i] = y lr y :
for ( j= l;j< = n d im :j+ + ) [
p siu n fjl + = p try |.jj-p (ih i|fj|:
*/
p |ilii] |j |= p lr y |j |:
/
fr c e _ v c c to r (p tr y . 1 .n d im ):
return vtrv:
}
flo a t * E H J 3o \v c r _ c a Ic (g a n u n a ,r e s u lt,p a r a m s.z .l.c x ,c y .c z ,h x .Iiy .h z )
flo a t c o m p le x g a m m a , resu ll[R O W L E N G T H ):
flo a t
* p a r a m s , z , t , e x [ 2 * N + r i |2 * M + l l ,c y |2 * N + l l |2 * M + l ] .l i x |2 * N + l l [ 2 * M + l ] .h v |2 * N + l ] |2 * M + l ] .c / .[ 2 * N + l ||
2 * M + l ] ,h z |2 * N + l ||2 * M + 1 1 :
(
int E z () , H z();
int i.j:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
314
flo a t D x .D y ;
flo a t a ,b ,e p s,m u ,w ;
flo a t c o m p le x p refac;
flo a t c o m p le x E x ,E y ,H x ,H y ;
flo a t c o m p le x P o \v e r = C M P L X F (0 .0 ,0 .0 );
flo a t c o m p le x P te m p = C M P L X F (0 .0 ,0 .0 );
static flo a t P o w e r _ H in a x [3 ] = { 0 .0 ,0 .0 ,0 .0 } ;
flo a t H _ b o t_ tem p , H _ lef_ tem p ;
a = p a ram s[0];
b = p a r a m s[l];
e p s= p a ra m s[2 ] * 8 .8 5 4 e - 12;
w = 2 * M _ P I* p a r a m s[3 ];
m u = 4 * M _ P I * le -7 ;
D .\= a /(2 * M );
D y = b /(2 * N );
p r e fa c = -I /(\v * \v * m u * e p s+ g a in n ia * g a n n n a );
H _ b o t_ tem p = 0 .0 ;
H _ le f_ te n ip = 0 .0 ;
/♦ I N S ID E * /
for ( j= l; j < = N - l; j + + )
for ( i = l; i< = M - l : i+ + )
<
I
E x = p r c fa c * (g a m n ia * (r e s u I t|E z (i+ l.j)]-r e s u lt[E z (il.j )] )/( 2 * D x ) + J * \v * m u * (r c s u lt [H z ( i.j+ l) ]- r e s u lt [ H z (i.j -l) ] )/( 2 * D y )) ;
H y = p r c fa c * (J * w * c p s * (r c s u lt[E z (i+ l,j)]-r c s u lt|E z (il.j ) ] ) /( 2 * D x ) + g a m n ia * (r c s u lt |H z ( i.j + l) |-r c s u lt|H z ( i.j- l) J )/( 2 * D y )) :
E y = p r c fa c * (g a m m a * (r c s u I t[E z (i.j+ I )]-r c s u lt[E z (i.j-l)])/(2 * D y )J * w * m u * (r c s iiI l[H z (i+ I .j)]-r e s u lt[H z (i-l.j)])/(2 * D x ));
H x = p r c fa c * (-J * w * c p s* (r c s u It[E z (i.j+ l)'|-r c su ll[E z (i.jl)]) /(2 * D y ) + g a in n ia * ( r c s u I t |H z (i+ l.j) J - r e s u lt |H z (i- l.j) |)/( 2 * D x ) );
P tc m p + = E x * c o n j(H y )-E y * c o n j(H x );
c x [j||i]= c r c a l(E .\* c e .\p (J * w * t-g a n im a * z )):
l'y [jJ |i]= crc a l(H y * c c x p (J * w * t-g a n iin a * z ));
c y [j]|i|= c r c a l(E y * c c x p (J * w * t-g a n im a * z )):
lix |j]|i|= c r c a l(H x * c c N p (J * \v * t-g a n im n * z ))'.
c z |j|[i|= c r c a I (r c s u h |E z (i.j)]* c e x p (J * w * t-g a n im a * z ));
h z |j]|i]= c r e a l(r c s u U |H z (i,j)]* c e x p (J * w * l-g a n in in * z ))'.
/♦ D e p e n d in g oil type o f R igh t and T o p w a lls p ick the righ t sy m m etr y eq u a tio n * /
/* c x | j ][ 2 * M - i]= e x |j |[ i j ; * /
c x |j |i2 * M - i|= - c x [ j |'[ i|:
/* c x [ 2 * N - j ] |i] = c x |j |f i |; * /
cn [ 2 * N -j
] | i |= -c x |j | [ i ];
c x [ 2 * N - j ||2 * M - i] =
cn
|j jf i
|;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
315
/* e x [2 * N -j ] [2 * M - i] = - e x [ j] [ i] ; * /
h x [j ] [2 * M - i] = h x [ j ][ i];
/* h x fj] [2 * M -i]= -h x [j] [i] ;* /
lix [2 * N -j ][ i] = h xfj][i];
/* h x [2 * N - j ][ i]= - h x [j ] [i] ;* /
h x [2 * N -j ][ 2 * M - i]= h x [j][i]:
/* lix [2 * N - j ][ 2 * M - i]= - h x f j ][ i]; * /
e y [ j] [ 2 * M -i] = ey [j][i];
/* c y [j] [2 * M -i]= -e y [j] [i];* /
c y [2 * N -j ] [i] = e y [jjfij;
/* c y [ 2 * N -j][i]= -e v [j][i];* /
c y [ 2 * N -j ] [2 * M - i] = e y [j][i];
/* e y [2 * N -j ] [2 * M - i] = - e y |j ][ i); * /
/* h v [j ] [2 * M - i] = h y [j][i]:* /
l'y [il[2 * M -i]= -h y [j][i];
/* h y [2 * N - j ][ i]= h y [j][i]:* /
liy |2 * N -j][i]= -h y [j][i];
h y f2 * N -j ]|2 * M - i] = hy|JJ[i];
/* h y [2 * N -j][2 * M d ]= " -h y [j][i]:* /
c z |j ||2 * M - i ] = c z |j][ij:
/* c z [ j ] |2 * M - i|= - e z [ j ] [ i] : * /
/ * c z |2 * N - jJ [ i]= c z [j][i];* /
c z [ 2 * N -j ] i i |= - c z [ j] j i] ;
/ * c z |2 * N - j ] [ 2 * M - i] = c z [ j ] |i] : * /
c z [2 * N -j]f2 * M -iJ = -e z [j]fi]:
/ * h z |j ||2 * M - i |= h z ( j] |ij: * /
lizL i]|2 * M -iJ = -liz |j|r il;
li7 .[2 * N -jj|i]= l.zfjH i];
/ * h z |2 * N - j J |i] = - h z [ j l|i] ; * /
/* li z |2 * N - j ] |2 * M - i l= h z |j j [ i|; * /
h z [ 2 * N - j j i2 * M - i|= - h z |j ||i |;
i
P o \v er= 4 *P tcm p ;
P lcm p = C M P L X F (0.0.O .O );
/* S I D E S * /
/* B o llo m S id e * /
j=0:
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316
for (i= l;i< = M -l;i+ + )
{
Ex=prefac*(gam m a*(result[Ez(i+l.j)]-result[Ez(i-l.j)])/(2*D x)+J*w *m u*(result[H z(ij+l)]result[H z(ij)])/Dy);
H y=prefac*(J*w *eps*(resuIt[Ez(i+l.j)]-rcsult|Ez(i-l,j)])/(2*D x)+gam nia*(result[H z(ij+l)]result[Hz(i.j)] )/Dy);
Ey=prefac*(gamnia*(rcsuItlEz(i.j+l)]-rcsult[Ez(ij)])/Dy-J*\v*mu*(result[Hz(i+l.j)]-result[Hz(il,j)))/(2*D x));
Hx=prefac*(-J*w*eps*(rcsuIt[Ez(i.j+I)]-resuIt[Ez(i.j)])/Dy+gamnia*(result[Hz(i+l.j)]result[Hz(i'l,j)])/(2*Dx));.
H_bot_temp=sqrt(0.5*(SQR(cabs(Hx))+SQR(cabs(result[Hz(i.j)]))+sqrt(SQR(SQR(creal(Hx))SQR(cimag(Hx))+SQR(crcal(rcsult[Hz(i.j)]))SQR(cimag(rcsull[Hz(i.j)])))+4*SQR(crcal(Hx)*ciniag(Hx)+creal(resuU[Hz(i.j)])*cimag(rcsull[Hz(i.j)]))))
);
i f (H_bot_teinp>Powcr_Hmax[0|) H_bot_temp=Power_Hmax[0]',
Ptemp+=Ex*conj(Hy)-Ey*conj(Hx);
ex[j]li]=creal(Ex*cexp(J*w*t-gamma*z));
hy[j][i]=crcal(Hy*ccxp(J*\v*t-gamma*z));
ey[j]|i]=crcal(Ey*cexp(J*w*t-gainnia*z)):
hx[j][i]=creal(Hx*cexp(J*w*l-gamnia*z));
cz|j][i]=creal(rcsull[Ez(i.j)]*ccxp(J*w*l-ganuna*z));
hzfj][i]=crcaI(rcsull|Hz(i.j)]*ccxp(.J*w*l-gamina*z));
/♦Depending on type o f Right and Top walls pick the right symmetry equation */
/*cx(j][2*M -i|= cx[jl(i|;*/
cxfj ] 12*M-i ]=-cx|j] [i]:
/*cx[2*N -j][i|= ex|j][i|:*/
cx[2*N-j]iiJ=-cx[j]'|i]'.
exl2*N -j|12*M -i]= ex |j)|i|;
/*cx|2*N-j][2*M d]=-cxfjJ[iJ;*/
h x |j]|2 * M -i|= h x |j||i|;
/* h x [j|[ 2 *M -i|= -h x|j||i];*/
h x |2 * N -j|[i]= h x[j||i|;
/*h x|2*N -j][i]=-h x|j|[i];*/
h x[2*N -j||2*M -i|= hx|j|li.|;
/* h x [2 * N -j||2 * M -i|= -h x |j||i|:* /
cy[jJ|2*M-iJ= cy|j] | i]:
/*cy|j]|2*M -i'J=-cyfjl|i|;*/
c y |2 * N -j]|i|= c y |j||i|;
/*cy [2 *N-j |1 i]=-ey [j 11i |;*/
c v [2 * N -j||2 * M -i|= c y |j||i|:
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317
/*ey[2*N-j][2*M -i]=-cyfj][i];*/
/*hyLj][2 5l'M -i]= hy[j][i];*/
hy[j][2*M -i]=-hy[j][i];
/*hy[2*N -j][i]= hy[j][i];*/
Iiy [2*N-j] [i] =-hy [j] [i];
hy[2*N -j][2*M -i]= hyfj][i];
/*hy[2*N -j][2*M -i]=-hy|j][i];*/
ez[j][2*M -i]= ez[j][i];
/*cz[j] [2 *M -i]=-ez[j] [i];*/
/*ez[2*N -j][i]= ez[j][ij :*/
ez[2*N -j][i]=-ezfj][i];
/*ez[2*N -j][2*M -i]= ez[j][i]:*/
ez[2*N -j][2*M -i]=-ez[j||j];
/*lrz|j][2*M -i|= hz[j][i];*/
liz[j][2*M -i]=-hz[j][i|;
hz[2*N -j][i]= hzfj] [i];
/*liz[2*N -j]|i|= -liz[j][i];*/
/*hz[2*N -j][2*M -i]= h z|jj|i];*/
hz[2*N -j][2*M -i]=-hz[j][i];
}
Po\vcr+=4*Ptemp;
Plemp=CM PLXF(0.0.0.0);
/♦Left Side*/
for (i=l'.j<=N -i:j++)
\i
Ex=prefac*(ganim a*(rcsuIl|Ez(i+I.j)|-rcsull|Ez(i.j)|)/D x+J*\v*nui*(result[H z(i.j+l)]result [H z(i.j-l)])/(2*D y»;
Hy=prefac*(J*\v*cps*(rcsult[Ez(i+I.j)|-rcsiilt|Ez(i.j)])/D.N+gamiii;i*(resiiIt[Hz(i.j+l)]rcsult|H z(i.j-l)])/(2*D v));
Ey=prcfac*(gam m a*(result[Ez(i.j+l)|-rcsult|Ez(i.j-l)])/(2*D y)-J*\v*inu*(result|H z(i+l,j)lrcsu 11 fHz(i j )_| )/D x):
Hx=prefac*(-J*\v*cps*(rcsult[Ez(i.j+l)|-rcsull|Ez(i.j-l)])/(2*D y)+gam m a*(rcsull[H z(i+l.j)]rcsult[Hz(i.j)])/Dx);
Plcinp+=E.\*conj(Hy)-Ey*conj(H.\):
ex[j | li |=crcai(Ex*ccxp(J*\v*t-gnnuua*z));
l'y|jlI'l=crci>KHy*ccxp(J*w*t-gamnia*z));
e y I j||i|=creal(Ey*ccxp(J*\v*t-gamnia*z));
hx|j]|i|=crcal(Hx*ccxp(J*\v*l-ganim a*z));
cz|j]|i|=creal(rcsult|Ez(i.j)]*ccxp(J*\v*l-ganuna*z));
hz[j||i|=crcal(rcsult|Hz(i.j)|*ccxp(J*\v*t-gainma*z))-.
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318
/♦Depending on type o f Right and Top walls pick the right symmetry equation */
/*e.\'[j][2*M-i]= e x [j||i|;* /
c.x[jl[2*M -i]=-ex|j][i];
/*cx[2*N -j][i]= cx[j][i];*/
exf2*N-j][i]=-exUl[i];
ex[2*N -j][2*M -i|= ex[j][i];
/*ex[2*N -j][2*M -i|=-ex[j][i];*/
hx[j][2*M -i}= hx[j][i];
/*hx[j][2*M -i]=-hx[j][i];*/
h x |2 * N -j||i]= hxfj|[i|;
/*h xl2*N -j]|i]=-h x|j][i];*/
hx(2*N -j|f2*M -i]= hxfj][i];
/* h .\|2*N -j|[2*M -i|= -h x[j||i];*/
cy [i]| 2 *M -i]= ey |j]|i|;
/*cy |j ] [2 *M-i |=-cy |j 11i | ;*/
cy|2*N -j]|ij= eyfjjlij;
/*ey [2 * N-j | [i J=-cy [j 11i1;* /
cy[2*N-j] |2*M -i |= cyfj] |i |;
/*cyf2*N-j][2*M -iJ=-cy[j|fi|;*/
/*hv|iH 2*M -i]= h v |j]|i|:* /
hy UJ12 *M - iJ=-hv|j | [i | :
/* h v |2 * N -j]|i]= h V|iin i:* /
hy 12 * N- j] | iJ = -h | j 1[i I;
hvf2*N -j||2*M -i|= h y |j||i|:
/*hy|2*N -j]|2*M -iJ=-hy[j|[i|;*/
cz|j|(2*M -i}= e /|j||ij:
/*cz|j][2*M -i|=-c/.|j||iJ:*/
/* cz[2 * N -jj|i|= czfjlfij;*/
czf 2*N-j | f i]=-cz|j |"| il;
/* cz|2 * N -jl|2 * M -i|= e z [j||i|;* /
ez|2*N -i|j2*M -i]= -cz|j|[il;
/*h z|i|[2 * M -i]= hz|j]|i]:*/
hz|jJ|2*M -i]=-Iiz|jlii|;
hz|2*N -j][i]= h z |j||i|:
/* hz[ 2* N-jl | i |=-hz| j 11i | :*/
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319
/*hz[2*N-j] [2*M -i]= hz[j] [i] ;*/
hz[2*N-j][2*M -i]=-hz[j][i];
}
Power+=4*Ptemp;
Ptemp=CMPLXF(0.0,0.0);
/♦Top Side*/
j=N;
for (i= l;i< = M -l;i+ + )
{
Ex=prefac*(gam ma*(result[Ez(i+l,j)]-result[Ez(i-l.j)])/(2*Dx)+J*\v*mu*(result[Hz(i.j)]result[H z(ij-l)])/D y);
Hy=prefac*(J*w*eps*(result[Ez(i+l.j)]-result[Ez(i-lj)])/(2*Dx)+gamm a*(result[Hz(i.j)]result[H z(ij-l)])/D y);
Ev=prefac*(gam m a*(result[Ez(i.j)]-resuIt[Ez(ij-l)])/Dy-J*w*m u*(result[Hz(i+l,j)]-result|Hz(i1-j)])/(2*Dx));
Hx=prefac*(-J*w*eps*(rcsult[Ez(i.j)]-result[Ez(i.j-l)])/Dy+gamma*(result[Hz(i+l.j)]result[Hz(i-l j)])/(2*D x));
Ptemp+=Ex*conj(Hy)-Ey*conj(H.\);
ex[j][i]=creal(Ex*ccxp(J*w*t-gainma*z));
hv|j | [i]=crcal(Hy*cexp(J*w*t-gainma*z));
cy[j|[i]=creal(Ey*cexp(J*w*t-ganima*z));
hx|j][i]=crcal(Hx*ccxp(J*w*t-gamma*z));
ez.|j|[i]=crcal(resuIt[Ez(i,j)]*cexp(J*w*t-gamma*z));
hz| j|[i|=crcal(rcsiilt|Hz(i.j)]*ccxp(J*w*t-gainma*z));
/♦Depending on type o f Riglu and Top walls pick the right symmetry equation */
/*cx[j]|2*M -i|= ex|j|[i]:*/
cx| j 112*M-i i=-ex[jJl i |;
h x|j|[2*M -ij=hx[j]|i];
/*hx|j][2*M -i]=-hx|j|[i|;*/
cv |j|l2 * M -i|= ey [j]|i|;
/*cy|j][2*M -i]=-cy|j||i];*/
/*h v|j]|2*M -iJ=h v|j||i]:*/
h y |j|f 2 *M -i]=-hy[j|iil;
cz[j][2*M -i|=cz[j][i|;
/* cz[j||2*M -i|= -cz|j|[i]:*/
/*hz|j][2*M -i]= h z|j|[i|:*/
l'z Ijl|2*M -i|= -h z|j||i|:
i
Powcr+=2*Ptemp:
Ptcnip=CMPLXF(0.().().0);
/♦Right Sidc*/
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320
i=M;
for (j= l:j< = N -l J++)
{
Ex=prefac*(ganima*(result[Ez(i,j)]-resull[Ez(i-l.j)])/Dx+.I*w*mu*(result[Hz(i.j+l)]rcsul t [Hz(i ,j -1)] )/(2 *D y));
Hy=prefac*(J*\v*eps*(result[Ez(i,j)]-result[Ez(i-1j)])/Dx+gamnva*(result[Hz(i,j+1)] rcsult[H z(i.j-l)])/(2*D y));
Ey=prefac+(gam ma*(result[Ez(i.j+l)]-result[Ez(i.j-l)j)/(2*Dy)-J*\v*mu*(resuH[Hz(i,j)]result|H z(i-l.j)])/D x);
Hx=prefac*(-J*w*eps*(result[Ez(i,j+l)]-result[Ez(i.j-l)])/(2*Dy)+gam m a*(result[Hz(i,j)]rcsult[H z(i-l.j)])/Dx);
Ptemp+H2x*conj(Hy)-Ey*conj(Hx);
cx[j][i]=creal(Ex*cexp(J*w*t-gamma*z));
hy[.iHil=crcal(Hy*cexp(J*w*t-gamma*z));
ey[j][i]=crcal(Ey*cexp(J*\v*t-gamina*z));
hx[j][i]=creal(Hx*cexp(J*w*t-gamma*z));
cz[j][i]=creal(result[Ez(i,j)]*cexp(J*w*t-gamma*z));
hz|j][i]=creal(result[Hz(i.j)]*cexp(J*w*t-gamma*z));
/♦Depending on type o f Right and Top walls pick the right symmetry equation */
/*cx|2*N -j][i]= ex[j][i]:*/
cx[2*N-j][i]=-cx[jJ[i];
hx|2*N -j)[i]= hx[j][ij;
/*hx[2*N -j][i]=-hx|j][i];*/
cy|2*N -j][i]=eyfjJ[i];
/*cy|2*N -j]|i]= -ey|j][i];*/
/*hv|2*N -j][iJ= hy|fl[i]:*/
hy[2*N-j]|"i]=-hy[j]'[i];
/*czl2*N -j][i]= c z|j]|i]:* /
cz(2 *N-j] f i J=-ez[j] [i];
lrz|2*N-j 1[i]= hz[j] |i |:
/* h z |2 * N -j|[i|= -h z |j|[i|:* /
)
Powcr+=2*Ptcmp;
Ptemp=CM PLXF(0.0.0.0);
/♦CORNERS*/
i= 0 ;
j= 0 ;
Ex=prcfnc*(gam m a*(result[Ez(i+l.j)]-rcsull[Ez(i.j)])/Dx+J*w*nm *(result|H z(i.j+l)]-rcsult|Hz(i.j)|)/Dy)
Hy=prefac*(.)*w*cps*(rcsultlEz(i+l ,j)]-rcsult[Ez(i.j)])/Dx+gamma*(resiiItfHz(i.j+I)]-rcsult[Hz(i.j)J)/Dy)
Ey=prcfac*(gom m a*(rcsult[Ez(i.j+l)]-rcsult[Ez(i.j)])/Dy-J*w*m u*(result|H z(i+l,j)|-rcsuIt|H z(i.j)|)/Dx):
Hx=prcfac*(-J*w*cps*(rcsiilt|Ez(i.j4d)]-rcsiilt[Ez(i.j)])/Dy+gam ma*(rcsull|Hz(i+I.j)]rcsult|Hz(i.j)])/D.\):
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321
H_bot_temp=sqrt(0.5*(SQR(cabs(Hx))+SQR(cabs(result[Hz(i.j)]))+sqrt(SQR(SQR(creal(Hx))SQR(cimag(Hx))+SQR(creal(result[Hz(ij)]))SQR(cim ag(rcsult[Hz(ij)])))+4*SQR(creal(Hx)*cimag(Hx)+creal(result[Hz(i.j)])*cim ag(result[Hz(i.j)]))))
);
if (H_bot_temp>Power_Hmax[0]) Po\ver_Hmax[0]=H_bot_temp;
H_lef_temp=sqrt(0.5*(SQR(cabs(Hy))+SQR(cabs(result[Hz(ij)]))+sqrt(SQR(SQR(creal(Hy))SQR(cimag(Hy))+SQR(creal(result[Hz(i,j)]))SQR(cimag(resultlHz(ij)])))+4*SQR(creaI(Hy)*cimag(Hy)+creal(result[Hz(i.j)])*cimag(result[Hz(i.j)]))))
);
if(H_Ief_tem p>Power_H m ax[l]) Power_Hniaxfl]=H_lef_temp;
Ptemp+=Ex*conj(Hy)-Ey*conj(Hx);
ex[j][i]=creal(Ex*cexp(J*w*t-ganima*z));
by[.i][i]=crcal(Hy*cexp(J*w*t-gamma't!z));
eyfj][i]=crcal(Ey*cexp(J*w*t-gamma*z));
hx|j][i]=creal(Hx*cexp(J*w*t-gamma*z));
ez[j][i]=creal(rcsultIEz(i.j)]*cexp(J*\v*t-gamma*z));
hz[j][iJ=creal(result[Hz(i.j)]*cexp(J*w*t-gamma*z));
^'Depending on type o f Right and Top w alls pick the right symmetry equation */
/*ex[j][2*M -i]= ex[j][i];*/
ex |j]i 2 *M -i]=-ex[j]fi];
/*cxf2*N -j][i]= ex[j][i];*/
cx[2*N -jlii]=-cx[j][i];
cxl2*N -j][2*M -i]=ex[j][i]:
/*ex[2*N -j][2*M -i]=-ex[j][i];*/
bx|j][2*M -i]= hx[j|[i];
/*hx[j|[2*M -i]=-hx[j][i];*/
h x|2*N -j][i|= lix[j][i];
/*hx|2*N -jl[i]=-h x|j][i];*/
hx|2*N -j)[2*M -i]= hx|j]fi];
/*Iix[2*N-j]|2*M -iJ=-hx[j][i];*/
cy|j][2*M -i]= cy[jl[i],
/*cv [j] [2*M -i]=-ey| j] | i] ;*/
ey|2*N -j]|i]= cvfj][i];
/*ey [2 *N-j ] [i J=-ey | j] [i ]; */
cy|2*N -j][2*M -i]= cy[j][i];
/* cy 12 *N-j] 12 * M -i ] =-cy [j ] [i ]; */
/*hy[j]| 2 *M -i|= h y [j||i|;* /
by[i I|2*M -i |=-hy[j 11i]:
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322
/*hy[2*N -j][i]= hy[j][i];*/
hy[2*N-j] [i]=-hy [j] [i];
hy[2*N -j][2*M -i]= hy[j][i];
/*hy[2*N-j][2*M -i]=-hy[j][i];*/
ez[j][2*M -i]= ez[j][i];
/*ez[j] [2 *M -i]=-ez[j] [i] ;*/
/*ez[2*N -jl[i]= ezfjiri];*/
ez[2*N -j] [i]=-ez[jj [i];
/*ez[2*N -j][2*M -i]= ez[j][i]:*/
ez[2*N -j] [2*M -i]=-ez[j] [i];
/*hz[j][2*M -i]= liz[j][i];*/
hz[j][2*M -i]=-hz[j][i];
hz[2*N -j][i]=hz[j][i];
/*hz[2*N -j] [i]=-hz[j] [i] ;*/
/*hz[2*N -j][2*M -i]= hz[j][i];*/
hz[2*N-j] [2*M -i]=-hz[j] [i];
Po\ver+=4*Ptemp;
Ptemp=CMPLXF(0.0,0.0);
i=M;
j= 0 ;
Ex=prefac*(gam ma*(rcsult|Ez(i.j)]-rcsiilt[Ez(i-l.j)|)/Dx+J*\v*m u*(rcsuU[Hz(i.j+l)]-rcsull|Hz(i.j)])/Dy);
Hy=prefac*(J*w*eps*(resiilt[Ez(ij)]-rcsultlEz(i-I.j)])/Dx+gamma*(rcsuIt|Hz(i,j+l)]-resuIt[Hz(i..j)])/Dy):
Ey=prefac*(ganima*(resull[Ez(i,j+l)]-i'csull|Ez(i.j)])/Dy-J*w*imi*(rcsiiIl[Hz(i.j)]-resuIt|Hz(i-l,j)])/Dx);
Hx=prcfac*(-J*w*eps*(rcsuU|Ez(i.j+l)|-rcsullfEz(i.j)|)/Dy+gamnia*(resuItfHz(i.j)]-rcsull[Hz(i-l.j)J)/Dx):
H_bot_temp=sqrt(0.5*(SQR(cabs(Hx))+SQR(cabs(rcsult|Hz(i,j)|))+sqrl(SQR(SQR(crcal(Hx))SQR(cimag(Hx))+SQR(crcaI(rcsiill[Hz(i.j)]))SQR(cimag( result[Hz(i. jy|)))+4*SQR(crcal(Hx)*cimag(Hx)+crcal(rcsull[ Hz(i.j)|)*cimag( result) Hz(i.j)]))))
):
if (H_bot_temp>Power_Hmax|()J) Po\vcr_Hmax|0|=H_bot_tcmp:
Ptemp+=Ex*conj(Hy)-Ey*conj(Hx);
cx[j][i]=creal(E.\*cexp(J*\v*t-gamma*z));
hyfj] [i]=creal(Hy*ccxp(J*w*t-gamma*z)):
ey|j]|i]=creal(Ey*cexp(J*\v*t-gamma*z)):
hx[j]|i]=creaI(Hx*ccxp(J*w*t-ganima*z));
czfjJ[i]=creal(rcsult[Ez(i.j)]*ccxp(J*\v*l-gammn*z)):
hz[j][i]=creal(result[Hz(i,j)|*ccxp(J*\v*t-ganima*z))..
/*Dcpcnding on type o f Right and Top walls pick the right symmetry equation */
/*ex[2*N -j|[i]= ex[j|[i|;* /
cx[2*N -j|[i]=-cx[j]ji|;
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323
hx[2*N-j][i]= hxUlti];
/*hx[2*N-j][i]=-hx[j][i];*/
ey[2*N -j][i]= ey[j][i];
/*cy[2*N-j][i]=-ey[j][i];*/
/*hy[2*N-j][i]= hy[j][i];*/
hy[2*N-j][i]=-hy[j][iJ;
/*ez[2*N -j][i]= ez[j][i];*/
ez[2*N-j][i]=-ez[j][i];
hz[2*N -jJ[i]=hz|j][i|;
/*hz[2*N-j][i]=-hz[j][i];*/
Power+=2*Ptemp;
Ptemp=CMPLXF(0.0,0.0);
i= 0 ;
j=N;
Ex=prefac*(ganinia*(rcsult|Ez(i+l.j)]-rcsuIt[Ez(i.j)|)/D,\+J:fw*niu*(rcsult[Hz(i.j)]-rcsult[Hz(i.j-l)l)/Dy):
Hy=prefac*(J*w*eps*(result[Ez(i+l ,j)]-resull|Ez(i,j)])/Dx+ganima*(result[Hz(i,j)]-resull[Hz(i.j-l)])/D)0;
Ey=prerac*(gamnia*(rcsulttEz(i.j)]-result[Ez(i.j-l)|)/D)'-J*\v*mu*(resiiltfHz(i+l,j)]-result[Hz(i,j)|)/Dx);
Hx-prefac*(-Jhv*eps*(rcsult[Ez(i.j)]-resultlEz(i,j-l)])/Dy+gainnia*(resultfH z(i+I.j)]-rcsull|Hz(i,j)])/Dx);
HJef_tenip=sqrt(0.5*(SQR(cabs(Hy))+SQR(cabs(rcsult[Hz(i.j)]))+sqrl(SQR(SQR(crcal(Hy))SQR(cimag(Hy))+SQR(creal(resull(Hz(i.j)]))SQR(cimag(result[Hz(i.j)l)))+4*SQR(crcal(Hy)*cimag(Hy)+crcaI(rcsultfHz(i.j)|)*ciniag(resuIt|Hz(i.j)J))))
);
if (H_lef_tcinp>Po\vcr_Hinax[ 11) Po\vcr_Hmax[l]=H_lef_tcmp;
Ptcnip+=Ex*conj(H\j-Ey*conj(Hx):
ex[j]fij=creal(Ex*cc.\p(J*w*t-gamma*z)):
hy|j][il=crcaI(Hy*cc.\p(J*w*t-gamma*z))'.
cyfj][i|=creaI(Ey*cexp(J*w*t-gamma*z));
hx[j|[iJ=creal(Hx*cc.\p(J*w*l-gamnia*z));
ezLil|i]= crcal(result|Ez(i,j)J*ce.\p(J*w*l-gamma*z));
hz[j||i]=creal(result|Hz(i.j)]*ccxp(J*w*l-gainnia*z));
/*Depcnding on type o f Right and Top walls pick the right symmetry equation */
/*ex|j]|2*M -i]= ex|j][i];*/
ex fj ] j2 * M-i I=-ex [j] j i);
hx[jJ|2*M -i]= hx|j][ij;
/*hxfj][2*M -i]=-hx|j]|i];*/
cy|j|[2*M -i]= cyl j][ i |;
/*cy|j | [2 *M-i |=-ey |j 11i |;*/
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
324
/*hy[j][2*M -i]=hy[j][i];*/
hy[j][2*M -i]=-hy|j][i];
ez[j][2*M -i]= ez[j][i];
/*ez[fl[2*M -i]=-ez[j][i];*/
/*hz[j][2*M -i]= hzfj][i];*/
hz[j]i2*M -i]=-hz[j][i];
Po\ver+=2*Ptemp;
Ptemp=CMPLXF(0.0,0.0);
i=M;
j=N;
Ex=prefac*(gam ma*(result[Ez(i.j)]-rcsull[Ez(i-l.j)])/Dx+J*w*mu*(result[Hz(i,j)]-resull[Hz(i,j-l)])/Dy);
Hy=prefac*(J*\v*eps*(result[Ez(i.j)]-resull[Ez(i-l j)l)/Dx+gam m a*(result[Hz(i,j)l-result[H z(i,j-l)])/Dy)',
Ey=prefac*(gam m a*(resull[Ez(i,j)]-rcsult[Ez(i,j-l)l)/Dy-J*\v*mu*(rcsull[Hz(i.j)]-resultfHz(i-l.j)])/Dx);
Hx=prefac*(-J*\v*cps*(rcsuIt[Ez(i.j)]-rcsuIl[Ez(i.j-l)l)/Dy+ganinia*(resuIt|Hz(i.j)l-result[Hz(i-l.j)])/Dx);
Ptemp+=Ex*conj(Hy)-Ey*conj(Hx);
exfjl|i]=creal(Ex*ccxp(J*\v*t-gamma*z));
hy[ j|[i|=crcal(Hy*ccxp(J*\v*t-gamma*z));
ey|j||iJ=creal(Ey*cexp(J*\v*t-gamma*z)):
Iixf|j|i]=crcal(Hx*cexp(J*\v*t-gamma*z));
cz[j][i]=crcal(rcsull[Ez(i.j)|*ccxp(J*\v*t-gamma*z));
hz|j]|i]=creaI(rcsuIt[Hz(ij)]*ccxp(J*\v*t-gamma*z));
Po\ver+=Pteinp;
Po\ver_Hmax[2|=0.5*creal(Po\vcr)*Dx*Dy:
rcturn(Powcr_Hmax):
)
f
void calcJong(lix_long, lv /J o n g . c x jo n g , ez_long. liy jo n g . hz_long2. c y jo n g . cz_long2. ex. cy, cz. h.x,
liy. hz. params, gamma, result)
noat lix_long|2*M +l ]|N O O F D Z +l], hz_long|2*M +lJ[NO OFDZ+IJ. c zJ o n g |2 * M + l||N O O F D Z + l |.
ex J o n g [2 * M + 1]|NOOFDZ+1 J, lr/Jon g2 [2* N + l|[N O O F D Z + l]. liy J o n g |2 * N + l|[N O O F D Z + l|.
c zJ o n g 2 |2 * N + l][N O O F D Z + l|, cy_long| 2 *N+1] [NOOFDZ+11, *params:
Hoat cx [2 *N + l][2*M + l |, cy |2 * N + r||2 * M + l |. lix[2*N+I ||2 * M + I[. hv|2*N +l |[2*M +I J.
ez|2 * N + l[[2 * M + l |. liz[2*N +l ]|2*M +1]:
float complex gamma,resull|ROWLENGTH|:
<
float *EH_Po\vcr_caIc():
float Dz;
int ij:
Dz=pa ra ms [0 ]/(2* M);
for (i=0;i<=NOOFDZ;i++)
<
\
EH_Po\ver_calc(gamma.result,params.i*Dz,().().cx.cy.cz.lix.hy.liz):
for ( j = ( ) ; j < = 2 * M ;j + + )
h x J o n g |j|[i|= h x |N ||j|:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
325
h z jo n g f j] [i]=hz[NJ [jj;
cx_long[j] [i]=ex[N] [j];
ez_Iongfj][i]=ez[N]|j];
}
for (j=0;j<=2*N:j++)
{
hy_Iong[j][i]=hy[j][2*M];
hz_long2 [j] [i]=hz[j] [2*M ];
eyJongLj] [i]=eyfj] [M];
ez_long2|j][i]=ez[j][M ];
}
}
return;
void pIotniatr(xniatr,yniatr,ni,n)
float *xmatr, *ymatr;
int 111, 11;
{
fortran void EZVEC();
EZVEC(xniatr,yniatr,&ni,&n);
return;
void openpIot()
<
I
fortran void OPNGKS();
fortran void GQCNTN():
fortran void GSELNT():
fortran void WTSTR();
int icrr.icn;
static char string|]="Titlc":
OPNGKSO;
GQCNTN(&ierr,&icn):
GSELNT(O);
WTSTR(. l,.96.string.2.0.-1):
GSELNT(&icn);
return;
J
\o id closeplot()
\i
fortran void CLSGKS():
CLSGKS():
re lurn;
j
float *vcctor(nl.nh)
long 11l1.nl;
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
326
/* allocate a float vector with subscript range v[nl..nh] */
{
float *v;
v=(float *)malloc((unsigned int) ((nh-nl+2)*sizeof(float)));
if (!v) nrcrror("allocation failure in vcctor()");
relurn(v-nl+l);
void frec_vector(v,nl.nh)
float *v;
long nh.nl;
/* free a float vector allocated with vector() */
{
free ((char*) (v+nl-l));
}
void nrcrror(error_te.\t)
char error_text|];
/* Error print routine */
fprintf(stderr,"Run-time error... \n");
fprintf(stderr."%s\n".crror_tc\l):
fprintf(stdcrr."... now exiling to system ... \n"):
cxit(l):
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
327
Appendix E
Sample Output (for a WR90 HTS Waveguide) of m'g sweep
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
328
Warning: no access to tty; thus no job control in this shell...
» » » » NCCS Cray Y-M P C98/6256 UNICOS 7.C.3 Node: charney « « « «
Technical Assistance Group Bldg 28 Rni S201 tag@ nccs.gsfc.nasa.gov
MVS support 0900-1100, (301)286-9120; other hours, leave message
Cray/Convex support 0830-1700, Tue & Thu 0830-1600, (301)286-CR AY /2729
Current system status: enter 'slatinfo' or call (301)286-1392.
======== URGENT INFORMATION! as o f 17:37 Fri, Sep 3 ,1 9 9 3
09/02 Account renewal forms were due to the NCCS on September 1, 1993.
See "consult news 1295" for more information.
======== SCHEDULED SYSTEM UNAVAILABILITY as o f 17:37 Fri, Sep 3, 1993
F rom -T o System(s) - Reason
09/06 Mon 0 8 0 0 -1 6 0 0 UniTree - UltraNet and silo testing
09/07 Tue 0 6 0 0 -0 9 0 0 UniTree/Convex — Backups and silo testing
09/07 Tue 0 6 0 0 -0 9 0 0 Cray C98 — scheduled preventive maintenance
09/08 Wed 0 6 0 0 -0 9 0 0 UniTree — Convex 3820 UltraNet and silo testing
09/09 Thu 0 6 0 0 -0 9 0 0 UniTree — Convex 3820 UltraNet and silo testing
09/10 Fri 1200 to
09/14 Tue 0900
UniTree/Convex - Upgrade to Convex 3820
= ======= Recent CONSULT ARTICLES as of 17:35 Thu, Sep 2, 1993
09/02 NEW S
1297 Planned Schedule for UniTree M ove From Convex
C 3240 to C 3820 (same as UNITREE 1010)
08/30 CRAY
1216 Cray C98 NQS Job Queue Scheduling (updated)
08/30 CRAY
1210 Transition to the Cray C98 (updated)
08/27 CRAY
1214 How to be Certified for Multitasking on the C98
news: statinfo cray.consult cray.,1210 cray.l 120 cray.1216 cray.1021 cray.1206
SCC30.news C F7750.ncws cray.1214 cray.1211 CF7760.news Pascal4.2 cray.1213
cray.2160 crav. 1212 cray.l 118 cray.l009 cray.2070 cray. 1050 cray.1028
cray. 1130 crav. 1048 cray. 1010 cray. 1174 cray. 1072 cray. 1015 cray. 1089
cray. 1088 cray. 1069 cray.l 109 cray. 1139 cray.l 155 cray. 1044 cray. 1022
cray. 1004 cray.J023 cray. 1208 cray. 1063 cray.l 144 cray. 1064 cray. 1158
cray.l 105 cray. 1027 cray. 1053 cray.l 157 cray. 1046 cray. 1202 cray. 1127
cray. 1207 cray. 1209 cray. 1205 unicos70.ro cray. 1203 cray. 1037 cray. 1201
cray.1200 cray.l 187 cray.2132 cray.2142 cray.l 196 cray.l 195 cray.l 194
cray. 1193 cray. 1078 crav. 1101 cray. 1094 cray. 1036 cray. 1192 cray. 1188
cray. 1024 cray. 1019 cray. 1045 cray. 1186 cray. 1128 cray. 1185 cray. 1183
cray. 1182 cray. 1181 cray. 1180 cray. 1171 cray. 1005 cray. 1179 cray.l 178
cray. 1177 cray.l 125 cray.l 176 cray.l 175 cray.l 173 cray.l 172 cray. 1111
cray. 1170 cray. 1169 cray. 1166 cray. 1168 cray. 1167 cray. 1165 cray. 1164
cray.l 163 cray. 1161 cray. 1160 cray.2150 cray.2140 cray.2130 cray.2050
cray.2040 cray.2030 cray.2020 cray.2010 cray.2000 cray. 1159 cray.l 153
cray.l 151 cray.l 150 cray.l 149 cray.l 147 cray. 1146 cray.l 145 cray.l 141
cray.l 140 cray.l 138 cray. 1137 cray.l 136 cray.l 135 cray.l 134 cray.l 132
cray. 1131 cray. 1126 cray. 1124 cray. 1123 cray. 1119 cray. 1115 cray. 1108
cray.l 107 cray. 1098 cray. 1097 cray. 1096 cray. 1095 cray. 1090 cray. 1085
cray. 1084 cray. 1076 cray. 1075 cray. 1074 cray. 1062 cray. 1055 cray. 1052
cray.1051 cray. 1043 crav.1039 cray.1033 cray. 1030 cray. 1026 cray. 1012
cray. 1000 cray. 1154 C F77_5.0 cray. 1148 mvs_disk_fail ncws.consult cray. 1034
cray. 1018
Sal Sep 4 13:26:25 PDT 1993
Enter width o f waveguide cross-section : Enter height o f waveguide cross-section : Enter relative dielectric
constant o f waveguide interior: Enter effective zero-temperature penetration depth o f HTS walls : Enter
normal conductivity o f HTS w a lls : Enter temperature o f HTS w a lls : Enter critical temperature o f HTS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
329
w alls : Enter 'nV o f the mode you are interested in : Enter 'n' o f the mode you are interested in : Enter
starting frequency : Enter stopping frequency : Enter number o f points in frequency range :
Starting on frequency 8.000000e+09 Hz
Fc: 6 .5 5 7 2 10e+09 Hz
Theoretical TE10 alpha : 1.056854c-03 Nepers/m
Lossless beta: 9.604952e+01 rad/m
(1 ,0569e-03f9 .6050c+ 01)
(1 ,0780e-03,9.5089e+01)
(1,0357e-03,9.5089e+01)
(1.0569e-03.9.412 9 c+ 0 1)
(1.0569e-03,9.3168c+01)
(1.0991e-03,9.4129e+01)
(1.0780e-03,9.3168e+01)
(1,0780e-03,9.4609e+01)
(1.0780e-03,9.3648c+01)
(1 ,0780e-03,9.4369c+01)
(1.0357e-03.9.4369e+01)
(1 ,0569e-03.9 ,4609e+01)
(1.0569e-03.9.4249c+01)
(1 .0569e-03.9.4489e+01)
(1.0569e-03.9.4309c+01)
(1 .0569e-03,9.4429e+01)
(1.0569e-03,9.4339e+01)
(1.0991e-03,9.4339e+01)
(1.1308e-D3,9.4324c+01)
(1.0780e-t)3,9.4309c+01)
(1.0780c-03.9.4354c+01)
( 1 .1 2 0 3 e -0 3 .9 .4 3 5 4 c + 0 1)
(1.1520c-03.9.436 le+ ()l)
(1.0991 c-03,9.4369e+01)
(1.099 lc-03.9.4346c+ 01)
(1.0991c-03.9.4361c+01)
(1.0991 e-03.9.4350c+ 01)
(1.0991c-03.9.4357c+ 01)
(1.0991c-03.9.4352e+01)
(1.1414c-03.9.4352c+01)
(1.1203c-03,9.4350e+01)
( 1 .1203c-03,9.4353e+01)
(1.1203 c-0 3.9.43 5 1c + 0 1)
( 1 .1203c-03,9.4352c+01)
(1.12 03c-03.9.4351e+() 1)
(1 .12()3c-03.9.4352e+01)
(1.0780c-03.9.4352e+01)
(1.0991e-03.9.4352c+01)
(1.0991 c-03,9.43 52c+01)
(1.1414c-03,9.43 5 2c+ 01)
(1.1203e-03.9.4352c+01)
(1.1203c-03.9.4352c+01)
(1 ,0780c-03.9.43 5 2c+ 01)
(1.1255c-03.9.4352c+ 01)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
330
(1.1044e-03,9.4352e+01)
(1 .1 163e-03,9.4352e+01)
(1 .1427e-03,9.4352e+01)
(1 .1 100e-03,9.4352e+01)
(1.1008e-03,9.4352e+01)
(1,0945e-03,9.4352e+01)
(1.1109e-03,9.4352e+01)
(1.1201e-03,9.4352e+01)
(1.1056e-03,9.4352e+01)
(1.1064e-03,9.4352e+01)
(1.1091e-03,9.4352e+01)
(1 .1039e-03,9.4352e+01)
(1.1091e-03,9.4352e+01)
(1.1056e-03,9.4352e+01)
(1.1082e-03,9.4352e+ 01)
(1.1117e-03,9.4352e+01)
(1.1071e-03,9.4352e+01)
(1.1063e-03,9.4352e+ 01)
(1.1084e-03,9.4352e+01)
(1.1095e-03,9.4352e+01)
(1.1089c-03,9.4352c+01)
(1.1088e-03,9.4352e+01)
(1.1085e-03,9.4352e+01)
(1.1092c-03,9.4352e+ 01)
(1.1085e-03,9.4352e+01)
(1.1081e-03,9.4352e+01)
(1.1081 e-03,9.4352e+01)
(1 .1084e-03,9.4352e+01)
(1 .1088c-03,9.4352c+01)
(1 .1082e-03,9.4352e+01)
( 1 .108Ic-03,9.4352e+01)
(1.1082e-03,9.4352e+01)
( 1 .1083e-03,9.4352c+01)
( 1 .1083e-03.9.4352e+01)
( 1 .1081c-03,9.4352e+01)
(1.1082e-03.9.4352e+01)
(1 .1082c-03.9.4352e+01)
(1 .1084c-03.9.4352e+01)
(1 .1082c-03.9.4352c+01)
At frequency 8.000000c+09 Hz, (alpha,bcta)=(1.10820c-03,9.43519c+01)
80 Iterations.
Starting on frequency 8.800000c+09 Hz
Fc: 6 .5 5 7 2 10e+09 Hz
Theoretical TE10 alpha : 1.027177e-03 Ncpcrs/m
Lossless beta: 1.230000c+02 rad/m
(1.0272c-03.1.2300c+02)
(1,0477e-03.1.2177c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
331
(1,0066e-03,1.2177e+02)
(1.0272e-03,1.2054e+02)
(1.0272e-03,1.2115e+02)
(1.0272e-03,1.2238e+02)
(1.0272e-03,1.2146e+02)
(1.0272e-03,1.2208e+02)
(1.0272e-03,1.2162e+02)
(1.0683e-03,1.2162e+02)
(1.0991e-03,1.2154e+02)
(1,0477e-03,1.2146e+02)
(1.0477e-03,1.2169e+02)
(1 ,0888e-03,1.2169e-t 02)
(1,0683e-03,1.2177e+02)
(1.0683e-03,1.2165e+02)
(1.0683e-03,1.2173e+02)
(1 .0683e-03,1.2167e+02)
(1 ,0272e-03,1 .2 167e+02)
(1 .0477e-03,1.2165e+02)
(1.0477e-03,1.2168e+02)
(1 ,0888e-03,1.2168e+02)
(1 .1 196e-03,1.2169e+02)
(1.0683e-03,1.2169e+02)
(1,0683e-03.1.2168e+02)
(1.0683c-03,1.2169e+02)
(1,0683e-03,1.2168e+02)
(1.10 9 4 c-0 3 ,1.2168e+02)
(1 ,0888e-03.1.2168e+02)
(1,0SS8c-03,1.2168e+02)
(1,0477c-03,1.2168c+02)
(1.0683e-03,1.2168e+02)
(1,0683e-03,1.2168e+02)
(1.10 9 4 c-0 3 ,1.2168e 1-02)
(1,0888c-03,1.2168e+02)
(1 .088Se-03,1.2168e+02)
(1,0477c-03,1.2168e+02)
(1,0683c-03,1.2168c+02)
(1,0683e-03,1.2168e+02)
(1.1094c-03,1.2168e+02)
(1,0888c-03.1.2168c+02)
(1,0888c-03,1.2168e+02)
(1,0477c-03.1.2168e+02)
(1,0683c-()3,1.2 l68c+02)
(1.0683c-03,1.2168c+02)
(1.1094e-()3,1.2168c+02)
(1.063 lc -0 3 ,1.2168e+02)
(1,0426c-03.1.2168e+02)
(I.0773e-03.1.216Sc+02)
(1,0824c-03.1 .2 168c+02)
(1 ,()679c-03,1.2168e+02)
(1.0590c-03.1.2168c+02)
(1.0727c-03.1.2168c+02)
(1,0724c-03.1 .2 16Se+02)
(1,0693c-03,1.2168e+02)
(1,0740c-03.1.2168c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
332
(1.0725e-03,1.2 168e+02)
(1.0691e-03,1.2168e+02)
(1.0718e-03,1.2168e+02)
(1.0686e-03,1.2168c+02)
(1,0666e-03,1.2168e+02)
(1.0661e-03,1.2168e+02)
(1 ,0704e-03,1.2168e+02)
(1 ,0696e-03,1.2168e+02)
(1 ,0694e-03,1.2168e+02)
( 1.0676e-03,1.2 168e+02)
(1 ,0697e-03,1.2 168e+02)
(1.0705e-03,1.2168e+02)
(1,0690e-03,1.2168e+02)
(1.0693e-03,1.2168e+02)
(1.0694e-03,1.2168e+02)
(1,0687e-03,1.2 168e+02)
(1 ,0690e-03,1 .2 168e+02)
(1.0687e-03,1.2168c+02)
(1.0692e-03.1.2168c+02)
(1.0691e-03,1.2168c+02)
(1 .0691e-03,1 .2 168c+02)
(1.0693e-03,1.2 168e+02)
(1.0692e-03,1.2168c+02)
(1 ,0692e-03,1 .2 168c+02)
(1.0690e-03,1.2168e+02)
(1 .0692e-03,1.216Se+02)
(1.0693e-03,1.2168c+02)
(1.0691e-03,I.2168c+02)
At frequency 8.80()000e+()9 Hz, (alpha.beta)=( 1.06 9 2 0 e-0 3 ,1.21682e+02)
81 Iterations.
Starting on frequency 9.600000c+09 Hz.
Fc: 6.5572 lOc+09 Hz
Theoretical T E 10 alpha : 1.057287c-03 Nepcrs/in
Lossless beta: 1.469514c+02 rad/m
(1.0573e-03,1.4695c+02)
(1 ,0784c-03,1.4548c+02)
(1.036 lc -0 3 ,1,4548c+02)
( 1.0573e-03.1.4401c+02)
(1 .0573e-03.1,4622e+02)
(1.0573e-03.1.4475c+02)
(1 ,0573c-03,1,4585c+02)
(1.0996c-03,1.4585c+02)
(1.1313e-03,1.4603c+02)
(1.0784c-03.1.4622c+02)
(1 .0784e-03,1 ,4567c+02)
(1 .0784e-03.1,4603c+02)
(1.0784c-03.1.4594c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(1.078 4 e-0 3 ,1,4576e+02)
(1.0784e-03,1.4590e+02)
(1.0784e-03,1.4580e+02)
(1.0784e-03,1.4587e+02)
(1.0784e-03,1.4583e+02)
(1.0784e-03,1.4586e+02)
(1.0784e-03,1.4584e+02)
(1.0784e-03,1.4586e+02)
( 1 .1207e-03,1.4586e+02)
(1 ,099 6 e-0 3 ,1,4586e+02)
(1.0996e-03,1.4585e+02)
(1.0573e-03,1.4585e+02)
(1.0784e-03,1.4585e+02)
(1.0784e-03,1.4585e+02)
(1.0784e-03,1.4585e+02)
(1.078 4 e-0 3 ,1,4585e+02)
(1.1207e-03,1.4585e+02)
(1.0996e-03.1.4585e+02)
(1.0996e-03,1.4585e+02)
(1.0573e-03,1.4585e+02)
(1.0784e-03,1.4585c+02)
(1 ,0 78 4 e-0 3 .1.4585e+02)
(1.12 0 7 e -0 3 ,1,4585e+02)
(1,0 9 9 6 e-0 3 ,1.4585e+02)
(1.0 9 9 6 c-0 3 .1.4585e+02)
(1.0573e-03,1.4585e+02)
(1,0 78 4 c-0 3 ,1.4585e+02)
(1.0784e-03.1.4585c+02)
(1.1207e-03.1.4585e+02)
(1.0996c-03.1.4585e+02)
(1.0996e-03.1.4585e+02)
(1.0573e-03J.4585e+ 02)
(1.1049c-03.1.4585e+02)
(1.12 6 0 c-0 3 ,1,4585e+02)
(1.0903c-03.1.4585c+02)
(1,0956e-03.1 ,4585c+02)
(1.0986e-03.1.4585c+02)
(1.084 le - 0 3 .1,4585c+02)
(1.0997e-03.1.4585c+02)
(1.1079c-03.1.4585c+02)
(1,0 9 4 7 c-0 3 .1,4585c+02)
(1 .0 95 8 e-0 3 ,1,4585c+02)
(1.0979e-03.1.4585e+02)
(1,0 93 0 c-0 3 .1,4585e+02)
(1.0 9 8 0 c-0 3 .1.4585c+02)
(1.101 le-03.1.45S5c+02)
(1.096.3c-03.1,4585c+02)
(1,0 9 6 4 c-0 3 .1,4585e+02)
(1,0975c-03.1.4585c+02)
(1.0959c-03.1.4585c+02)
(1.0964c-03.1.4585c+02)
(1,0 9 5 2 c-0 3 .1,4585c+02)
( 1,0969c-03.1.4585c+02)
(1,0 9 6 9 c-0 3 .1,4585c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
334
( 1.0965e-03,1.4585e+02)
(1.0959e-03,1.4585c+02)
(1.0967c-03,1 ,4585e+02)
(1,096 5 c-0 3 ,1,4585e+02)
(1.0965c-03,1,4585e+02)
(1,0969e-03,1,4585e+02)
(1.0965c-03,1.4585e+02)
(1.0963c-03,1.4585e+02)
(1.0964e-03,1.4585e+02)
(1,0966c-03,1 ,4585e+02)
(1.0965c-03,1.4585e+02)
At frequency 9.600000e+09 Hz, (alpha,beta)=(l.0965 le-03,1.45853e+02)
75 Iterations.
Starting on frequency 1.040000e+10 Hz
Fc: 6 .5 5 7 2 10e+09 Hz
Theoretical TE10 alpha : 1.1 16978e-03 Ncpers/m
Lossless beta: 1.691824e+02 rad/m
(1 .1 170e-0.3,1.6918e+02)
(1. 1393c-03,1.6749c+02)
(1.0946e-03.1.6749c+02)
(1.117 0 e-0 3 .1,6580c+02)
(1 .1 170e-03.1.6834c+02)
(1.1617 c -0 3 ,1,6834e+02)
(1.19 5 2 c-0 3 ,1.6876c+02)
(1.13 9 3 e -0 3 .1.691 Sc+02)
( 1.13 9 3 e-0 3 .1.679 lc+02)
(1 .1393c-03.1,6876e+02)
(1.13 93e-03.1.6812c+02)
At frequency 1.0400()0e+10 Hz, (alpha.beta)=(1.13932e-03.1.68I25c+02)
8 Iterations.
Starting on frequency 1.120000e+10 Hz
Fc: 6 .5 5 7 2 10e+09 Hz
Theoretical TE10 alpha : 1.195677c-03 Nepers/ni
Lossless beta: 1.902968e+02 rad/m
(1 .1957e-03,1.9030c+02)
(1.2196 c-0 3 .1.8839e+02)
(1.1718 c-0 3 ,1.8839c+02)
( 1.1479c-03.1.9()30c+()2)
(1 .1 120e-03.1.9125c+t)2)
(1 .1 7 18c-03.1.9220c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
335
(1.1718e-0 3 ,1.8935c+02
(1.1239e-03,1.8935e+02
(1.1479e-03,1.8839e+02
(1.1479e-03,1.89S2e+02
(1.1479e-03,1.8887e+02
(1 .1479e-03,1,8958e+02
(1 .1479e-03,1.891 le+02
(1.1479e-03,1 .8946e+02
(1 .1957e-03,1,8946c+02
(1.2315e-03,1.8952c+02
(1.1718e-03,1.8958e+02
(1.1718e-03,1.8940e+02
(1.1718e-03,1.8952e+02
(1.1718e-03,1.8943e+02
(1.1718e-03,1 .8949e+02
(1.1718e-03,1.8945e+02
(1.2196e-03,1.8945e+02
(1,2555e-03,1,8944e+02
(1.1957e-03,1.8943e+02
(1.1957e-03,1 ,8946e+02
(1.2435e-03,1.8946e+02
(1.2794e-03,1.8946e+02
(1.2196c-03,1 ,8946c+02
(1 .2 196e-03,1,8945e+02
(1.2674e-03,1.8945e+02
(1.2435e-03,1.8945c+02
(1.2435e-03,1.8945e+02
(1.1957 e-0 3 ,1,8945c+02
(1.2196e-0 3 ,1,8946c+02
(1.2196 e-0 3 ,1,8945c+02
(1.2674c-03.1.8945e+02
(1.2435e-03,1.8945e+02
(1.2435c-03.1.8945c+02
(l.2435c-03,1.8945c+02
(1.2435c-03.1.8945c+02
(1.1957c-03,1,8945c+02
(1.2196e-03,1,8945c+02
(1.2196e-03,1,8945c+02
(1.2196c-03,1.8945c+02
(1.2196c-03,1,8945e+02
(1.2196c-03,1,8945c+02
(1.2196e-()3.1,8945c+02
(1.2674c-03,1.8945c+02
(1.2136 c-0 3 .1,8945c+02
(1,2375e-03,1,8945c+02
(1,2330c-03,1.8945c+02
(1.2629c-03,1.8945e+02
(1 ,2259c-03.1.8945c+02
(1,2364c-03.1.8945c+02
(1.2381 e -0 3 .1.8945c+02
(1,2540c-03,1,8945c+02
(1,2329c-03,1.8945c+02
(1.2258c-03.1.8945c+02
(1.2391c-()3.1.8945c+02
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(1.2356e-03,1.8945e+02)
(1.2358e-03,1.8945e+02)
(1.2297e-03,1.8945e+02)
(1.2367e-03,1.8945e+02)
(1.2339e-03,1.8945e+02)
(1.2353e-03,1.8945e+02)
(1.239 le -0 3 ,1,8945e+02)
(1.2345e-03,1.8945e+02)
(1 ,2359e-03,1.8945e+02)
(1.2362e-03,1.8945e+02)
(1.2337e-03,1.8945e+02)
(1.2360e-03,1.8945e+02)
(1.2374e-03,l.S945e+02)
(1.2352e-03,1.8945e+02)
(1.2353c-03,1,8945e+02)
(1.2354e-03,1.8945c+02)
(1.2347e-03,1.8945e+02)
(1.2356c-03,1.8945e+02)
(1.2359e-03,l.S945e+02)
(1,2354e-03,1,8945e+02)
(1.2356e-03,1.8945e+02)
(1.2355e-03,1.8945e+02)
(1.2358e-03,1.8945e+02)
(1.2355e-03,1.8945e+02)
( 1.2356e-03,1,8945e+02)
(1.2356e-03,l.S945c+02)
(1.2357c-03,1.8945e+02)
(1.2355e-03.1.8945c+02)
At frequency 1 .120000e+10 Hz, (alpha,bcta)=(1.23554c-03.1.89454c+02)
85 Iterations.
Starting on frequency 1.200000e+I0 Hz
Fc: 6.55 7 2 10e+09 Hz
Theoretical T E I0 alpha : I.288667c-03 Nepcrs/ni
Lossless beta: 2 . 106307e+02 rad/m
(1.2887c-03,2. l()63e+02)
(1 .3 144e-03.2.0852c+02)
(1.2629c-03.2.0852c+02)
(1.3402c-03.2.1063c+02)
(1.3789c-03.2.1168c+02)
(1 .3 144e-03,2.1274c+02)
(1 .3 144c-03,2.0958c+02)
(1 .3660c-03,2.0958c+02)
(1.4046c-03.2.0905c+02)
(1.3402c-03.2.0852c+02)
(1.3402c-03,2. IOlOc+02)
(1.3918 c-03.2.1010c+02)
(1 .3660c-03.2.1063c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
337
(I.3660e-03,2.0984e+02)
(1.3144e-03,2.0984e+02)
(1.2758e-03,2.0971e+02)
(1.3402e-03,2.0958e+02)
(1.3402e-03,2.0997e+02)
(1.3402e-03,2.0971e+02)
(1.3402c-03,2.099 le+ 02)
(1,3402e-03,2.0978e+02)
(1.3402e-03,2.0987e+02)
(1,2887e-03,2.0987e+02)
(1 .3 144e-03,2.099 le+02)
(1.3144e-03,2.0986e+02)
(1 .3 144e-03,2.0989e+02)
(1 .3 144c-03,2.0987e+02)
(1 .3 144e-03,2.0988e+02)
(1.3144e-03,2.0987e+02)
(1.3144c-03,2.0988e+02)
(1 .3 144e-03,2.0987c+02)
(1.3660c-03,2.0987c+02)
(1.3402c-03,2.0987c+02)
(1,3402e-03,2.0987e+02)
(1.3402c-03,2.0987c+02)
(1.3402e-03,2.09S7e+02)
(1.28S7e-03,2.0987c+02)
(1 .3 144e-03,2.0987c+02)
(1 .3 144c-03,2.0987e+02)
(1 .3 144e-03,2.0987e+02)
(1 .3 144e-03,2.0987e+02)
(1.3144c-03,2.0987e+02)
(1 .3 144e-03,2.0987c+02)
(1.3660e-03,2.0987e+02)
(1.34670-03,2.09870+02)
(1.3209c-03.2.0987c+02)
(1 ,3354c-03,2.0987c+02)
(1.3676e-03,2.0987c+02)
(1 ,3277c-03,2.0987c+02)
(1 .3 165c-()3,2.0987c+02)
(1.3240c-03,2.0987c+02)
(I.3164c-03.2.0987c+02)
(1.3306c-03,2.0987c+02)
(1.3269e-()3,2.0987c+02)
(l.3275c-03,2.0987e+02)
(1.334 lc-03,2.0987c+02)
(1 .3 3 16e-03.2.0987c+02)
(1 .3285e-03,2.0987c+02)
(1.330 lc-03.2.0987c+02)
(1.3342c-03,2.0987c+02)
(1.3292e-()3,2.0987c+02)
(1.3325c-03,2.0987e+02)
(1.3300c-03.2.09S7c+02)
(1,3285c-03,2.0987c+02)
(1.3308c-03,2.0987c+02)
(1.3309c-03,2.0987c+02)
( 1,3302c-03,2.0987c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
338
(1.3295e-03,2.0987e+02)
(1,3305e-03,2.0987e+02)
(1.3306e-03,2.0987e+02)
(1.3302e-03,2.0987e+02) •
(1.3305e-03,2.0987e+02)
(1.3303e-03,2.0987e+02)
(1.3300e-03,2.0987e+02)
(1.3304c-03,2.0987e+02)
(1.3303e-03,2.0987e+02)
(1.3303e-03,2.0987c+02)
(1.3305e-03,2.0987e+02)
(1.3303e-03,2.0987e+02)
At frequency 1.200000e+10 Hz, (alpha,bela)=(1.33038e-03,2.09872e+02)
76 Iterations.
Starting on frequency 1.280000e+10 Hz
Fc: 6 .5 5 7 2 10e+09 Hz
Theoretical TE10 alpha : 1.393510e-03 Ncpers/m
Lossless beta: 2.303909e+02 rad/m
(1.3935c-03.2.3039c+02)
(1.4214e-03,2.2809c+02)
(1.3656c-03.2.2809c+()2)
(1.3378c-03.2.3()39c+02)
(1.2960c-03.2.3154e+02)
(1.3656c-03.2.3269c+02)
(1.3656c-03,2.2924e+02)
(1.3099e-03,2.2924c+02)
(1.3378c-03.2.2809e+02)
(1.337Sc-03.2.2981e+02)
(1.3935c-03.2.2981c+02)
(1.4353c-03.2.3010c+02)
(1.3 656c-03.2.3 03 9e+02)
(1.3656c-03.2.2953c+02)
(1.3656c-03.2.3010e+02)
(1.3656c-03.2.2967c-K)2)
(1.4214c-03,2.2967e+02)
(1.46.32c-03,2'.2960e+02)
(1.3935c-03.2.2953c+02)
(1.3935c-03.2.2974e+()2)
(1.3935c-03.2.2960c+02)
(I.3935e-03.2.2971c+02)
(1.4493e-03.2.2971c+02)
(1.491 le-03,2.2972c+02)
(1 .4 2 14c-03.2.2974c+02)
(1 .4 2 14c-03.2.2969c+02)
(1.4214c-03.2.2972e+02)
(1.4214e-03.2 .2970c+02)
(1.477 le-03.2.2970c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
.4493e-03,2.2969e+02)
.4493e-03,2.2970e+02)
,4493e-03,2.2969e+02)
,4493e-03,2.2970e+02)
,3935c-03,2.2970e+02)
.4214e-03,2.2970e+02)
,4214e-03,2.2970e+02)
.4 2 14e-03,2.2970e+02)
.4214e-03,2,2970e+02)
.4214e-03.2.2970e+02)
.4 2 14e-03,2.2970e+02)
.4771e-03,2.2970e+02)
,4493e-03,2.2970e+02)
,4493e-03,2.2970e+02)
.4493e-03,2.2970e+02)
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.3935e-03,2.2970c+02)
,4562c-03.2.2970e+02)
.4283e-03.2.2970c+02)
,4440c-03,2.2970e+02)
,4789c-03,2.2970e+02)
.4358e-03,2.2970c+02)
,4236c-03.2.2970e+02)
.4 3 18c-03.2.2970c+02)
.434Sc-03.2.2970e+02)
.4552c-03.2.2970c+02)
.4 3 15c-03,2.2970e+02)
,4222c-03.2.2970e+02)
.4386c-03.2.2970e+02)
.4419e-03,2.2970e+02)
,4393c-03.2.2970c+02)
.443 le-03,2.2970c+02)
.4368c-03.2.2970e+02)
,4375c-03.2.2970c+02)
,4383c-03,2.2970e+02)
,4359c-03,2.2970e+02)
,4374c-03.2.2970c+02)
,4370c-03.2.2970e+02)
,4394c-03,2.2970c+02)
,4368c-03.2.2970c+02)
,4354e-03.2.2970e+02)
.4370c-03,2.2970e+02)
,4378e-03.2.2970e+02)
.4370c-03.2.2970e+02)
.4376c-03.2.2970c+02)
.437 le-03.2.2970e+02)
.4366c-03.2.2970e+02)
.4373c-03.2.2970c+02)
.4375c-03.2.2970c+02)
.4377c-03.2.2970c+02)
,4377c-03,2.2970e+02)
.4373c-03.2.2970c+02)
.4374c-03.2.2970c+02)
,4374c-()3.2.2970c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
(1.4372e-03,2.2970e+02)
(1.4374e-03,2.2970e+02)
At frequency 1.280000e+10 Hz, (alpha,beta)=(1.43738e-03,2.29700e+02)
82 Iterations.
Starting on frequency 1.360000e+10 Hz
Fc: 6.557210e+09 Hz
Theoretical TE10 alpha : 1.508819e-03 Nepers/m
Lossless beta: 2.497136e+02 rad/m
(1 .5088e-03,2.497 le+ 02)
(1.5390e-03,2.4722e+02)
(1.4786c-03,2.4722e+02)
(1.4485e-03,2.4971e+02)
(1.4032e-03,2.5096e+02)
(1.4786e-03,2.5221e+02)
(1.4786e-03,2.4847e+02)
(1.4183e-03,2.4847e+02)
(1 ,4485e-03,2.4722e+02)
(1 ,4485e-03,2.4909e+02)
(1.5088e-03,2.4909e+02)
(1.554 le-03,2.4940e+02)
(1.4786e-03,2.497 le+ 02)
(1.4786e-03,2.4878e+02)
(1.4786e-03,2.4940e+02)
(1.4786e-03,2.4893e+02)
(1.4786e-03,2.4925e+02)
(1.4786e-03,2.490 le+02)
(1 ,4786e-03,2.4917e+02)
(1.4786e-03,2.4905e+02)
(1.4786c-03,2.4913c+02)
(1.4786c-03,2.4907e+02)
(1.5390c-03,2.4907e+02)
(1 .5843e-03,2.4906e+02)
(1.5088e-03,2.4905e+02)
(1.5088c-03,2.4908e+02)
(1.5692c-03,2.4908e+02)
(1.6144e-03,2.4908e+02)
(1.5390c-03.2.4909e+02)
(1.5390c-03,2.4907c+02)
(1.5390e-03,2.4908e+02)
(1.5390c-03,2.4908e+02)
(1.53 90c-03,2.4908e+02)
(1.5390c-03.2.4908e+02)
(1.5993e-03,2.4908e+02)
(1.5692c-03,2.4908c+02)
(1.5692c-03,2.4908c+02)
(1.5692e-03.2.4908e+02)
(1.5692e-03.2.4908e+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
341
(1.5088e-03,2.4908c+02)
(1.5767e-03,2.4908e+02)
(1,6069e-03,2.4908e+02)
(1.5560e-03,2.4908e+02)
(1.5635e-03,2.4908e+02)
(1.5678e-03,2.4908e+02)
(1.5470e-03,2 ,4908e+02)
(1.5544e-03,2.4908e+02)
(1.5662e-03,2.4908e+02)
(1.5585e-03,2.4908e+02)
(1.5452e-03,2.4908e+02)
(1.5508e-03,2.4908e+02)
(1.5549e-03,2.4908e+02)
(1.5546e-03,2.4908e+02)
(1.5469e-03,2.4908e+02)
(1.5556e-03,2.4908e+02)
(1.5593e-03,2.4908e+02)
(1.5530e-03,2.4908e+02)
(1.5540e-03,2.4908e+02)
(1.5544e-03,2.4908e+02)
(1.5518e-03,2.4908e+02)
(1.5527e-03,2.4908e+02)
(1.5542e-03,2.4908e+02)
(1.5533e-03,2.4908e+02)
(1.5550e-03,2.4908e+02)
(1.556 lc-03,2.4908e+02)
(1.556 le-03,2.4908e+02)
(1.5540e-03,2.4908e+02)
(1.5545e-03,2.4908e+02)
(1.5545e-03.2.4908e+02)
(1.5554e-03,2.4908e+02)
(1.5562e-0.3,2.4908e+02)
(1 ,5559e-03,2.4908c+02)
(1.5556c-03.2.4908e+02)
(1.5560e-03,2.4908e+02)
(1.5552e-03.2.4908c+02)
(1.555 lc-03.2.4908e+02)
(1.5555c-03,2.4908c+02)
(1.5557e-03.2.4908e+02)
(1.5553c-03.2.4908e+02)
At frequency 1.360000e+10 Hz, (alpha.bela)=(1.55545c-03,2.49078c+02)
76 Iterations.
Starting on frequency 1.440000c+10 Hz
Fc: 6 .5 5 7 2 10c+09 Hz
Theoretical TE10 alpha : 1.633745e-03 Nepers/in
Lossless beta: 2.68693 le+02 rad/m
(1.6337c-03.2.6869c+t)2)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
342
(1,6664e-03,2.660 le+ 02)
(1.601 le-03,2.6601e+02)
(1,5684e-03,2.6869e+02)
(1.5194e-03,2.7004e+02)
(1.601 le-03,2.7138e+02)
(1.601 le-03,2.6735e+02)
(1.601 le-03,2.7004e+02)
(1.601 le-03,2.6802e+02)
(1.5357e-03,2.6802e+02)
(1.5684e-03,2.6735e+02)
(1.5684e-03,2.6836e+02)
(1.5684e-03,2.6769e+02)
(1.5684e-03,2.6819e+02)
(1.5684e-03,2.6785e+02)
(1.5684e-03,2.681 le+ 02)
(1.6337e-03,2.681 le+ 02)
(1,6828e-03,2.6815e+02)
(1.601 lc -0 3 ,2 .6 8 19c+02)
(1.601 le-03,2.6806e+02)
(1.601 le-03,2.6815e+02)
(1.601 lc-03,2.6808e+02)
(1.601 le-03,2.6813e+02)
(1.601 le-03,2.6809e+02)
(1.601 le-03,2.6812e+02)
(1.601 le-03,2.6810e+02)
(1.601 le-03,2.681 le+ 02)
(1.601 lc-03,2.6810c+02)
(1.601 le -0 3 ,2.681 le+ 02)
(1.601 le-03,2.6810c+02)
(1.601 le-03,2.681 le+ 02)
(1.6011 c-0 3 .2.6810e+02)
(1.601 le-03.2.681 le+02)
(1.601 le-03,2.681 le+ 02)
(1.601 lc-03.2.6811c+02)
(1.6011c-03.2.681 le+ 02)
(1.6664c-03.2.681 le+02)
(1.7154e-03.2.6Sl le+ 02)
(1.6337C-03,2.681 le+ 02)
(1.6337e-03.2.681 le+ 02)
(1.6337e-03,2.681 le+ 02)
(1.6337c-03,2.681 le+ 02)
(1.6991e-03,2.681 le+ 02)
(1.7318c-03,2.681 le+ 02)
(1.7073e-03,2.681 le+ 02)
(1.6746e-03,2.681 le+ 02)
(1.6930e-03.2.6Sl le+ 02)
(1.6521e-03.2.681 le+ 02)
(1.6935c-03.2.681 le+ 02)
(1.7200C-03,2.681 le+ 02)
(1.6798c-03,2.681 le+ 02)
(1.6803c-03,2.681 Ic+02)
(1.6898e-03,2.681 le+02)
(1.6762C-03.2.6S1 le+02)
(I.6805C-03.2.68I le+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(1.6905e-03,2.681 le+ 02)
(1.6825e-03,2.681 le+ 0 2 )
(1.6732e-03,2.681 le+ 02)
(1.6856e-03,2.681 le+ 0 2 )
(1.6876e-03,2.681 le+ 0 2 )
(1.6823e-03,2.681 le+ 0 2 )
(1.6854e-03,2.681 le+ 02)
(1.6847e-03,2.681 le+ 02)
(1.6813e-03,2.6811e+02)
(1.6824e-03,2.681 le+ 02)
(1.6848e-03,2.681 le+ 02)
(1.6829e-03,2.681 le+ 02)
(1.6806e-03,2.6811e+02)
(1.6837e-03,2.681 le+ 02)
(1.6842e-03,2.681 le+ 02)
(1.6837e-03,2.681 le+ 02)
(1.6845e-03,2.681 le+ 02)
(1.6833e-03,2.681 le+ 02)
(1.6832e-03,2.681 le+ 02)
(1.6830e-03,2.681 le+ 02)
(1.6836e-03,2.681 le+ 02)
(1.6832e-03,2.681 le+ 02)
(1.6829e-03,2.6811 e+02)
(1.6828e-03,2.681 le+ 02)
(1.6834e-03,2.681 le+ 02)
(1.6834e-03,2.681 le+ 02)
(1,6836e-03.2.681 le+02)
( 1.6833e-03,2.681 le+02)
At frequency 1.440000c+10 Hz, (alpha,bela)=(1.68342e-03.2.68I05c+02)
80 Iterations.
Starting on frequency 1.520000c+10 Hz
Fc: 6.557210e+09 Hz
Theoretical TE10 alpha : 1.767737c-03 Ncpcrs/m
Lossless beta: 2.873976e+02 rad/m
(l.7677c-03.2.8740c+ 02)
(1.803 le-03,2.8452C+02)
(1.7324c-03.2.8452c+02)
(1.6970e-03,2.8740c+02)
(1.6440e-03,2.8883e+02)
(1.7324c-03,2.9027e+02)
(1.7324c-03,2.8596e+02)
(1.7324c-03,2.8883c+02)
(1.7324c-03.2.8668c+02)
(1 .6 6 17c-03,2.866Se+02)
(1.6970c-03,2.8596e+02)
(I.6970e-03.2.8704c+02)
(1.697()c-03.2.8632c+02)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(1.6970e-03,2.8686e+02)
(1 .7677e-03,2.8686e+02)
( 1.8208e-03,2.8695e+02)
(1.7324e-03,2.8704e+02)
(1.7324e-03,2.8677e+02)
(1.7324e-03,2.8695e+02)
(1 ,7324e-03,2.8681e+02)
(I.7324e-03,2.8690e+02)
(1.7324e-03,2.8684e+02)
(1.7324e-03,2.8688e+02)
(1.7324e-03,2.8685e+02)
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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