# 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii © 1994 Dimitrios Antsos All Rights Reserved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ..................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) . . . . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. 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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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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., Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ppendix A Sample Matlab File Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 A ppendix A MathCAD File that Algebraically Verifies Equations (35) and (36) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 . EEsof D - L ib ra A N G t S i l] + FIL - Thu O c t 8 1 3 :0 3 :0 8 1 9 9 2 - 5dbm new 4 A N G tSll] YBCO 3. 5 0 0 \ t , \\ V\ R h \ V ra d w I 0 . 000 \\\ \ \\ \ \ Y X \ \\ Y \ \\V \ V\ vj -3 . 500 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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)) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 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 . 216 Appendix E Touchstone Circuit File that Verifies the Conjecture o f Section 7.5.4 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 222 iRANGE 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. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 . R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 . R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 R ep ro d u ced with p erm ission of th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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. R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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 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 . 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. ------------------------------------------------- ------------------............ ...................... ' ** ' ’ ............................................. . . . . ------------ ' * * _ _ ...... _ —- .............................................. .----- ------_ ______ ... . ' ........................... ............ ............................. ...................... ... , .................. ................. .......... ----------. . . . . -------- _ . , ...................... ..... ~ —~ .. - - .......................... __ .___ __ . . . _ ........................ . . . r , , . .. . -- - - - - - - - - . „ . _ ................ ................................ . .... .. .......................................... - ............... -----.---------- ---------- .---------------------------------------- 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 I t i r i \\\' \ ' i t I i t I I I / I I I I ! / II it I r if if I I I ! 11 I III •I / II II \ \\ \\ \ %\\ \\ \ \\\ \\\ \\\ II II S1» I I! I !I I I 11 1 I I I 11 1I I1 I I I ! 11 1 1\ 1 I\ 1 I\ I1\ I 11 I \1 I I 1 V M I \ \ \\ \ '' \ '1 \ 1I 1 11 1 I1 111 V1I 1 1I I 11 I m i M i l I M I I I 1 1 I 1 I 1 I I I I I I I i I I 1 I 1 I I 1 i I I II I lII I I I i I / /l I '/ ) } /// 1 1 1 ' I1 '' 0 .9 4 IC-0 I 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. .v.S N\ \ \ . S\ \ \ \ \ . \\ \ \ \ I t t \ III I // i i i i \ i i i i i i t v \ \ \ , t I I I ^s S / / I ^ ^ y s .♦/ 1 \ \ / \1 I/ / \ \ \ \ \ \ ' ! I \ I It 1 1\ \\ \ / / I / I / I i I I I I 1 i : i t / /i ! It * I II 1 / / / •' \ \ I ! III- I] i/ 1\1' IV\’ \ \ N ' \ \\ f / I J / \\ I // / / 1 \ .•/ / // M ! t / I ! I t \ \ \ \ 1 ; t / •s s / I i i ! \ I N\\\ \ 1 \ I I I ' t // fI I I\ 1i I / / \ \ \ N \ \ \ V \ \ \ '. \ I I I I I I I ■ / f t /I // - *- s»■.\ \ I \ M ••’ *\ \ t 1 ''till ‘ ‘ I I I I 1 1! I [ | •' ••' III / 1 0.0'-WE-0.1 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 Il I I I i 1 1 1 I 1 I1 1 J I I 11 1 I I: I 11I l 1M I I1 I I I M i l Mil 1 1 I I 1 I 1 I J I 1 I J i lli I I I ItI It I itt iii I i M l I i i i i i i M il iii !! it i I I I! I i i J M M I 1 ! t I I I 1 i I M I N I (lilt! 11 l it i i I I I 1 1 i i J J \ \ \ I I I I I I t I I M I I I I M I i i; i i i 1i i i Il i I1I 1I 1 M i M I I1 1 1 I 1 !1 I M I M IM I I I I 1I i I i 1 1 1 1 1I I I I i i 1 i I M 1 M i l l M M I i i i i i i o.:c *e-03 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 A a A A A A A A M t i t I 1 i t A A a A A A A A A I 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 cxi c-.i -10 -20 - u, r -t ■■ ■■*■W o:* -— C'l IB"' ?■J-rrWT+h-i-h-h-i-i-h-i-h-h --r t'O r^- O —r t-o —r co -.r 05 -t cxj m uO uo co — un co / -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 \ 1 I ' N \ \ \ >> \ \ \ ' \ \ \ ' V \ \ 'Ml 'Ml ‘' V I 'III ‘III \ II II 11 11 1I II \i ti \i ' ' 1 1 ‘ M l M M i t i i Mil t i ti M M M| j M J / M / / M ii ti M ‘ III 'III ‘ III 'III 'III ' III JI J J J ! 1I II I i t i j \ \ l ii ii o.t.7.:L-o? 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 V \l/ Y Y \J/ Y Y Y Y Y Y Y Y v Y Y Y \1/ Y Y Y Y Y Y Y \i \|/ \/ \ V Y V Y Y Y Y V v Y Y Y Y V \/ \/ V \/ V Y Y Y v ! \/ \i v Y \!/ V Y Y Y Y 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= () * / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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|: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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))-. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 | :*/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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*/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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.\): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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|; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) ,4493c-03,2.2970c+02) .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) logout Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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